Thesis - Open Research Exeter (ORE)

An Analytical and Numerical Investigation of
Auxeticity in Cubic Crystals and Frameworks
T. P. Hughes
Submitted by Thomas Peter Hughes, to the University of Exeter as a thesis for the
degree of Doctor of Philosophy in engineering, August 2012.
This thesis is available for Library use on the understanding that it is copyright material
and that no quotation from the thesis may be published without proper acknowledgement.
I certify that all material in this thesis which is not my own work has been identified
and that no material has previously been submitted and approved for the award of a
degree by this or any other University.
(Signature)..............................
1
Abstract
Negative Poisson’s ratio, or auxetic, materials present the possibility of designing structures and components with tailored or enhanced mechanical properties.
This thesis explores the phenomenon of auxetic behaviour in cubic crystals using
classical and quantum modelling techniques and assesses the validity of these techniques
when predicting auxetic behaviour in cubic elemental metals. These techniques are then
used to explore the mechanism of this behaviour.
The findings of the atomistic modelling are then used as a template to create networks
of bending beams with tailored Poisson’s ratio behaviour.
2
Contents
1
Background: Properties of materials
22
1.1
Negative Poisson’s ratio materials . . . . . . . . . . . . . . . . . . . . . . .
22
1.1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.1.2
Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.1.2.1
Simple Crystals . . . . . . . . . . . . . . . . . . . . . . . .
24
1.1.2.2
Silicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.1.2.3
Paratellurite . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.1.2.4
Metal organic frameworks . . . . . . . . . . . . . . . . . .
27
Fabricated structures . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.1.3.1
Honeycombs . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.1.3.2
Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.1.3.3
Molecular Auxetics . . . . . . . . . . . . . . . . . . . . . .
29
1.1.3.4
Composites . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.1.3.5
Granular materials . . . . . . . . . . . . . . . . . . . . . . .
31
1.1.3.6
Applications of mono-crystalline structures . . . . . . . .
31
Investigative Methodologies . . . . . . . . . . . . . . . . . . . . . .
31
1.1.4.1
Crystallography . . . . . . . . . . . . . . . . . . . . . . . .
32
1.1.4.2
Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . .
32
1.1.4.3
Molecular Modelling Techniques . . . . . . . . . . . . . .
33
1.1.4.4
Analytical Modelling . . . . . . . . . . . . . . . . . . . . .
34
1.1.3
1.1.4
1.1.4.4.1
1.2
Rotating Rigid Units . . . . . . . . . . . . . . . .
35
Other unusual material behaviour . . . . . . . . . . . . . . . . . . . . . . .
36
3
1.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
1.2.2
Negative Thermal Expansion . . . . . . . . . . . . . . . . . . . . . .
36
1.2.2.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
1.2.2.2
Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
1.2.2.3
1.2.2.4
1.2.3
2
1.2.2.2.1
Cubic crystals . . . . . . . . . . . . . . . . . . . .
36
1.2.2.2.2
Metal Organic Frameworks . . . . . . . . . . . .
37
1.2.2.2.3
Other framework structures . . . . . . . . . . . .
37
Mechanisms of NTE . . . . . . . . . . . . . . . . . . . . . .
39
1.2.2.3.1
Increasing symmetry . . . . . . . . . . . . . . . .
39
1.2.2.3.2
Positive expansion of bonds . . . . . . . . . . . .
39
1.2.2.3.3
Electron valence transition/cation movement . .
40
1.2.2.3.4
Rotation of rigid units . . . . . . . . . . . . . . .
40
Methodologies . . . . . . . . . . . . . . . . . . . . . . . . .
40
1.2.2.4.1
Experimental . . . . . . . . . . . . . . . . . . . . .
40
1.2.2.4.2
Analytical Modelling - Rotating unit modes . . .
41
1.2.2.4.3
Molecular modelling . . . . . . . . . . . . . . . .
41
Negative Linear Compressibility . . . . . . . . . . . . . . . . . . . .
42
Background: Theoretical framework and computational modelling
43
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.2
General elasticity equations for stress and strain . . . . . . . . . . . . . . .
43
2.2.1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.2.1.1
Mandel notation . . . . . . . . . . . . . . . . . . . . . . . .
44
2.2.1.2
Voigt Notation . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.2.2
The strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.2.3
The stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2.2.4
Generalised Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . .
48
2.2.5
Conversion between stiffness and compliance . . . . . . . . . . . .
49
2.2.6
Rotations of constants . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.2.7
Relationship between elastic constants and engineering constants .
53
2.2.7.1
54
Young’s Modulus . . . . . . . . . . . . . . . . . . . . . . .
4
2.2.7.2
Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.2.7.3
Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.2.7.4
Bulk Modulus . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.2.7.5
Zener ratio . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Application to auxetic media . . . . . . . . . . . . . . . . . . . . . .
56
Classical atomistic modelling techniques . . . . . . . . . . . . . . . . . . .
58
2.3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2.3.2
Two Body Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2.3.2.1
The Morse Potential . . . . . . . . . . . . . . . . . . . . . .
59
2.3.2.2
The Lennard-Jones Potential . . . . . . . . . . . . . . . . .
60
2.3.2.3
The Cauchy relation . . . . . . . . . . . . . . . . . . . . . .
60
Many Body Potentials . . . . . . . . . . . . . . . . . . . . . . . . . .
61
2.3.3.1
’Basic’ many-body potentials . . . . . . . . . . . . . . . . .
61
2.3.3.2
Embedded atom model for metals . . . . . . . . . . . . . .
62
2.3.3.3
The Finnis-Sinclair potential . . . . . . . . . . . . . . . . .
63
2.3.3.4
Sutton-Chen potential . . . . . . . . . . . . . . . . . . . . .
64
2.3.3.5
The Cleri-Rosato potential . . . . . . . . . . . . . . . . . .
64
Quantum Atomistic Modelling Techniques . . . . . . . . . . . . . . . . . .
66
2.4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
2.4.2
Application to auxetic media . . . . . . . . . . . . . . . . . . . . . .
66
2.4.2.1
Selected applications of DFT to find elastic properties . .
66
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
The finite-element method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
2.2.8
2.3
2.3.3
2.4
2.4.3
2.5
3
Fundamental atomistic modelling of the elastic properties of cubic crystals
70
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.2
Modelling methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.2.2
Derivation of elastic energy . . . . . . . . . . . . . . . . . . . . . . .
71
3.2.3
Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.2.3.1
73
Calculation of bulk modulus . . . . . . . . . . . . . . . . .
5
3.2.3.2
Calculation of tetragonal shear modulus . . . . . . . . . .
74
3.2.3.3
Calculation of rhombohedral shear modulus . . . . . . . .
74
3.2.3.4
Summary: Finding elastic properties from strains . . . . .
75
3.2.3.5
Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . .
75
Geometry considerations for cubic crystal systems . . . . . . . . . .
76
3.2.4.1
Face-centred cubic unit cell
. . . . . . . . . . . . . . . . .
76
3.2.4.2
Body-centred cubic unit cell . . . . . . . . . . . . . . . . .
78
Elastic properties calculated from analytical modelling . . . . . . . . . . .
81
3.3.1
A simple model: volume dependence . . . . . . . . . . . . . . . . .
81
3.3.1.1
Bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.3.1.2
Tetragonal shear . . . . . . . . . . . . . . . . . . . . . . . .
82
3.3.1.3
Rhombohedral shear . . . . . . . . . . . . . . . . . . . . .
82
3.3.1.4
Poisson’s ratios . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.3.1.5
Material properties from volume potential . . . . . . . . .
83
A simple pair potential: The spring model . . . . . . . . . . . . . .
84
3.3.2.1
84
3.2.4
3.3
3.3.2
3.3.2.2
3.3.3
Application to a face-centred cubic unit cell . . . . . . . .
3.3.2.1.1
Bulk modulus: . . . . . . . . . . . . . . . . . . . .
84
3.3.2.1.2
Tetragonal shear: . . . . . . . . . . . . . . . . . .
84
3.3.2.1.3
Rhombohedral shear: . . . . . . . . . . . . . . . .
85
3.3.2.1.4
Poisson’s ratio: . . . . . . . . . . . . . . . . . . . .
85
Application to the body-centred cubic unit cell . . . . . .
86
3.3.2.2.1
Bulk modulus: . . . . . . . . . . . . . . . . . . . .
86
3.3.2.2.2
Tetragonal Shear modulus:
. . . . . . . . . . . .
86
3.3.2.2.3
Rhombohedral shear modulus: . . . . . . . . . .
86
3.3.2.2.4
Poisson’s ratio: . . . . . . . . . . . . . . . . . . . .
87
3.3.2.3
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.3.2.4
Combination of the spring and volumetric potentials . . .
87
The Lennard-Jones Potential . . . . . . . . . . . . . . . . . . . . . .
89
3.3.3.1
89
Face Centred Cubic cells . . . . . . . . . . . . . . . . . . .
3.3.3.1.1
First nearest neighbours . . . . . . . . . . . . . .
6
89
3.3.3.1.2
3.3.3.2
3.3.3.3
3.4
92
Body centred cubic cells . . . . . . . . . . . . . . . . . . . .
96
3.3.3.2.1
First nearest neighbours . . . . . . . . . . . . . .
96
3.3.3.2.2
Second nearest neighbours . . . . . . . . . . . . .
98
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.3.3.1
First nearest neighbours . . . . . . . . . . . . . . 101
3.3.3.3.2
Second nearest neighbours . . . . . . . . . . . . . 102
3.3.3.3.3
Influence of potential range . . . . . . . . . . . . 102
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.1
4
Second nearest neighbours . . . . . . . . . . . . .
Summary of calculated properties . . . . . . . . . . . . . . . . . . . 105
Atomistic modelling of the elastic properties of cubic crystals
4.1
4.2
106
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.1
Elemental cubic metals . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.2
Covalently bonded zincblende compounds . . . . . . . . . . . . . . 107
Classical Potentials: Methodology . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.2
Numerical modelling methodology . . . . . . . . . . . . . . . . . . 109
4.2.2.1
Calculation of Poisson’s ratio for elemental cubic metals . 110
4.2.2.1.1
Optimisation method . . . . . . . . . . . . . . . . 111
4.2.2.1.2
Direct measurement method . . . . . . . . . . . . 111
4.2.2.2
Parameter space exploration . . . . . . . . . . . . . . . . . 112
4.2.2.3
Modelling of a coupled three-body/harmonic system . . . 113
4.2.2.3.1
Fitting of potentials to covalently bonded cubic
crystals . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3
Elastic properties calculated from classical potentials . . . . . . . . . . . . 115
4.3.1
Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.2
Elastic and engineering constant data from potential models . . . . 117
4.3.2.1
Accuracy of calculated properties . . . . . . . . . . . . . . 117
4.3.2.2
Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.2.2.1
Analysis: Morse Potential . . . . . . . . . . . . . 119
7
4.3.2.3
Finnis-Sinclair potential . . . . . . . . . . . . . . . . . . . . 121
4.3.2.3.1
4.3.2.4
Sutton-Chen potential . . . . . . . . . . . . . . . . . . . . . 123
4.3.2.4.1
4.3.2.5
4.3.3
4.3.4
Analysis: Sutton-Chen Potential . . . . . . . . . . 124
Cleri-Rosato potential . . . . . . . . . . . . . . . . . . . . . 125
4.3.2.5.1
4.3.2.6
Analysis: Finnis-Sinclair . . . . . . . . . . . . . . 122
Analysis: Cleri-Rosato Potential . . . . . . . . . . 126
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Deformed structures calculated from classical potentials . . . . . . 127
4.3.3.1
Body-centred cubic crystals
. . . . . . . . . . . . . . . . . 128
4.3.3.2
Face-centred cubic crystals . . . . . . . . . . . . . . . . . . 128
Variation of predicted properties with phase space exploration . . 131
4.3.4.1
Morse potential . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3.4.2
Finnis-Sinclair potential . . . . . . . . . . . . . . . . . . . . 133
4.3.4.3
Sutton-Chen Potential . . . . . . . . . . . . . . . . . . . . . 134
4.3.4.4
Analysis: Variation of predicted Poisson’s ratio with phasespace exploration . . . . . . . . . . . . . . . . . . . . . . . 135
4.3.4.5
4.3.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Elastic constants calculated from coupled three-body/harmonic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.3.5.1
Influence of the relationship of three-body and axial forces
on predicted properties . . . . . . . . . . . . . . . . . . . . 139
4.4
5
Can classical potentials predict auxetic behaviour?
. . . . . . . . . . . . . 140
Density Functional Theory modelling of elastic properties of cubic elemental
metals
141
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.1
5.3
Convergence example . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.1
LDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.3.2
GGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8
5.4
6
5.3.3
PAW LDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.3.4
PAW PBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.5
Magnetic moment consideration . . . . . . . . . . . . . . . . . . . . 163
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Finite Element Modelling of cubic frameworks
168
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.2
From crystalline bonds to beam networks . . . . . . . . . . . . . . . . . . . 169
6.3
Analytical derivation of mechanical properties of cubic beam structures . 171
6.4
6.3.1
Assumptions used in analytical modelling . . . . . . . . . . . . . . 171
6.3.2
Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.3.3
Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.3.4
Type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.3.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Finite element modelling methodology . . . . . . . . . . . . . . . . . . . . 178
6.4.1
Element type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4.2
Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.4.3
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4.3.1
Representative volume element . . . . . . . . . . . . . . . 181
6.4.3.2
Panel model . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4.3.2.1
6.4.4
Comparison of RVE and Panel Model . . . . . . 182
6.4.3.3
Material Properties . . . . . . . . . . . . . . . . . . . . . . 182
6.4.3.4
Loading conditions . . . . . . . . . . . . . . . . . . . . . . 182
High throughput techniques . . . . . . . . . . . . . . . . . . . . . . 183
6.4.4.1
Varying material properties . . . . . . . . . . . . . . . . . 183
6.4.4.2
Automatic generation of beam networks from crystallographic structure data . . . . . . . . . . . . . . . . . . . . . 183
6.4.4.3
6.4.5
6.5
Calculation of properties . . . . . . . . . . . . . . . . . . . 184
Validation of finite element modelling with analytical model . . . . 185
Results of finite element modelling . . . . . . . . . . . . . . . . . . . . . . . 186
6.5.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9
6.5.2
Comparison of panel model and RVE . . . . . . . . . . . . . . . . . 187
6.5.3
Elastic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.5.3.1
Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.5.3.2
Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . 193
6.5.3.3
Shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.5.3.4
Comparison with [100] axis . . . . . . . . . . . . . . . . . 196
6.5.3.5
Summary of results . . . . . . . . . . . . . . . . . . . . . . 197
6.5.3.6
Mechanical performance of beam networks relative to conventional honeycomb . . . . . . . . . . . . . . . . . . . . . 198
6.5.3.7
7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Discussion
7.1
7.2
7.3
201
Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.1.1
Classical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.1.2
Ab-initio modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.1.3
Mechanical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.2.1
Exploration of auxetic behaviour in crystals . . . . . . . . . . . . . . 206
7.2.2
Beam networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Key Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
A Methodology of DFT
210
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
A.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
A.2.1
The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . 210
A.2.2 Born Oppenheimer Approximation . . . . . . . . . . . . . . . . . . 211
A.2.3 Hohenberg-Kohn Theorem and Kohn-Sham Theorem . . . . . . . . 211
A.2.4 Self consistent loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.2.5 Hellman-Feynman Theorem . . . . . . . . . . . . . . . . . . . . . . 214
A.2.6 Choice of Pseudpotentials . . . . . . . . . . . . . . . . . . . . . . . . 215
A.2.7 Periodicity, reciprocal space and k-point mesh . . . . . . . . . . . . 215
10
A.2.8 Basis, plane waves and energy cutoff . . . . . . . . . . . . . . . . . 216
A.2.8.1
Magnetic moment consideration . . . . . . . . . . . . . . . 216
B The Finite Element Method
219
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
B.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
B.2.1
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
B.2.2
Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
B.2.2.1
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
B.2.2.2
Material properties . . . . . . . . . . . . . . . . . . . . . . 221
B.2.2.3
Analysis type, boundary conditions and loading . . . . . 221
B.2.3
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B.2.4
Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
C Derivation of off-axis elastic constants from on-axis deformation using the finite
element method
223
11
List of Figures
1.1
Schematic orthographic projection of non-auxetic, auxetic and completely
auxetic material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.2
The auxetic mechanism of a re-entrant honeycomb . . . . . . . . . . . . . .
28
1.3
The auxetic mechanism of one cell of a tetrachiral honeycomb . . . . . . .
28
2.1
Schematic showing normal and shear stresses . . . . . . . . . . . . . . . . .
47
2.2
Angles used to describe rotated properties . . . . . . . . . . . . . . . . . .
51
2.3
Schematic representation of a classical potential energy function . . . . . .
59
2.4
Schematic showing a two-body (axial), and three-body (angular spring)
potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.1
A face centred cubic unit cell . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.2
Strains applied for c’ shear in an FCC structure.
. . . . . . . . . . . . . . .
77
3.3
Strains applied for c44 shear in an FCC structure.
. . . . . . . . . . . . . .
78
3.4
A body centred cubic unit cell . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.5
Strain for C‘ shear for body-centred cubic, (110) plane . . . . . . . . . . . .
79
3.6
Shear displacement for c44 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.7
Variation of the (11̄0, 110) Poisson’s ratio with change in cutoff, for an
arbitrary FCC system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.8
Variation of the lattice constant (normalised relative to the lattice constant
for the first nearest neighbour value) with change in cutoff, for an arbitrary
FCC system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.9
Variation of the (11̄0, 110) Poisson’s ratio with change in cutoff, for an
arbitrary BCC system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
12
3.10 Variation of the lattice constant (normalised relative to the lattice constant
for the first nearest neighbour value) with change in cutoff, for an arbitrary
BCC system.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.1
Relative orientation of a body-centred cubic unit cell . . . . . . . . . . . . . 111
4.2
Schematic representation of the Keating model . . . . . . . . . . . . . . . . 113
4.3
Structure of covalently bonded zincblende type crystal structure . . . . . . 114
4.4
Deformation mechanism of body-centred cubic crystal . . . . . . . . . . . 128
4.5
A ’side’ view and a ’top’ view of the deformed unit cell . . . . . . . . . . . 129
4.6
The ’external’ deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.7
The hinging of the inter-layer bonds . . . . . . . . . . . . . . . . . . . . . . 130
4.8
The ’internal’ deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.9
ν with variation of Morse potential α . . . . . . . . . . . . . . . . . . . . . . 131
4.10 ν with variation of Morse potential r0 . . . . . . . . . . . . . . . . . . . . . . 132
4.11 ν with variation of Finnis-Sinclair potential density . . . . . . . . . . . . . 133
4.12 ν with variation of Finnis-Sinclair potential density . . . . . . . . . . . . . 133
4.13 ν with variation of Sutton-Chen potential density . . . . . . . . . . . . . . 134
4.14 ν with variation of Sutton-Chen potential repulsive component . . . . . . 134
4.15 Poisson’s ratio (ν110,11̄0 ) and relative spring stiffnesses calculated from the
three-body/harmonic model for selected body-centred cubic elemental metals136
4.16 Poisson’s ratio (ν110,11̄0 ) and relative spring stiffnesses calculated from the
three-body/harmonic model for selected covalently bonded cubic metal
compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1
Convergence of elastic constants with change in energy cutoff . . . . . . . 143
5.2
Poisson’s ratio in the off-axis direction for cubic elemental metals, calculated from LDA pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3
Poisson’s ratio in the off-axis direction for cubic elemental metals, calculated from GGA pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . 151
5.4
Poisson’s ratio in the off-axis direction for BCC elemental metals, calculated
from PAW-LDA pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . 154
13
5.5
Poisson’s ratio in the off-axis direction for FCC elemental metals, calculated
from PAW-LDA pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . 154
5.6
Poisson’s ratio in the off-axis direction for BCC elemental metals, calculated
from PAW-PBE pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.7
Poisson’s ratio in the off-axis direction for FCC elemental metals, calculated
from PAW-PBE pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.1
Representative unit cell for the simple cubic, body-centred cubic and facecentred cubic cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2
Structures under consideration . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.3
Type I structure, [110] strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.4
Free body diagram of simplified structure . . . . . . . . . . . . . . . . . . . 173
6.5
Type II structure, [100] strain . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.6
Load configuration for type II structure ([110] direction), side view . . . . 174
6.7
Load configuration for type II structure ([110] direction), top view
6.8
Type III cubic structure, [110] strain. . . . . . . . . . . . . . . . . . . . . . . 175
6.9
Type III cubic structure, [110] strain
. . . . 175
. . . . . . . . . . . . . . . . . . . . . . 176
6.10 Schematic showing the deformed shape of a cantilevered beam . . . . . . 179
6.11 Evolution of Poisson’s ratio with mesh density for an arbitrary type II beam
network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.12 Schematic showing the boundary conditions applied for axial and shear
loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.13 Representative volume element . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.14 Poisson’s ratio of structure for all considered cases, for strains in the [110]
direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.15 In-plane deformed configuration of panel model . . . . . . . . . . . . . . . 188
6.16 Deformed shape of the RVE model . . . . . . . . . . . . . . . . . . . . . . . 189
6.17 Representative unit cell for the structure with two dominant types of beams 190
6.18 Poisson’s ratio of structure for cases where two types of beams are dominant191
6.19 Tensile modulus of structure for cases where two types of beams are dominant193
6.20 Shear modulus of structure for cases where two types of beams are dominant195
14
7.1
Possible network structures of MOF5 and α-cristobalite.
. . . . . . . . . . 207
A.1 Flow diagram of DFT process . . . . . . . . . . . . . . . . . . . . . . . . . . 213
C.1 Poisson’s ratio in the (110, 11̄0) directions . . . . . . . . . . . . . . . . . . . 225
15
List of Tables
3.1
Elastic constants calculated for each of the potential models . . . . . . . . 105
3.2
Poisson’s ratio calculated for each of the potential models using the elastic
constants shown in table 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1
Poisson’s ratio maxima and minima for elemental cubic metals . . . . . . . 108
4.2
Varied potential parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3
Elements used as basis for parameter space exploration.
4.4
Experimentally observed elastic properties of elemental metals . . . . . . 116
4.5
Elastic constants for elemental metals, calculated using Morse potential . 118
4.6
Engineering constants for elemental metals, calculated using Morse potential119
4.7
Elastic constants for elemental metals, calculated using Finnis-Sinclair po-
. . . . . . . . . . 113
tential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.8
Engineering constants for elemental metals, calculated using Finnis-Sinclair
potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.9
Elastic constants for elemental metals, calculated using Sutton-Chen potential123
4.10 Engineering constants for elemental metals, calculated using Sutton-Chen
potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.11 Elastic properties for elemental metals, calculated using Cleri-Rosato potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.12 Engineering constants for elemental metals, calculated using Cleri-Rosato
potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.13 Elastic properties for body-centred cubic metals, calculated using Threebody/harmonic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
16
4.14 Elastic constant data from Three-body/harmonic model for covalently bonded
cubic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1
Experimentally observed elastic constant data for elemental cubic metals.
Data taken from Landolt-Bornstein (1985). Poisson’s ratio in the [110]
direction calculated from these data. . . . . . . . . . . . . . . . . . . . . . . 144
5.2
Elastic properties of elemental metals calculated using LDA pseudopotentials.
5.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Engineering constants of elemental metals calculated using LDA pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.4
Elastic properties of elemental metals calculated using GGA pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.5
Engineering constants of elemental metals calculated using GGA pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.6
Elastic properties of elemental metals calculated using PAW-LDA pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.7
Engineering constants of elemental metals calculated using LDA-PAW
pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.8
Elastic properties of elemental metals calculated using PAW-PBE pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.9
Engineering constants of elemental metals calculated using PAW-PBE pseudopotentials.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.10 Elastic properties of elemental metals calculated using four pseudpotentials 164
5.11 Engineering constants of elemental metals calculated using using four
pseudpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.1
Displacements of nodal points for BCC structure loaded in [110] direction
177
6.2
Comparison of panel and RVE model . . . . . . . . . . . . . . . . . . . . . 182
6.3
Displacements of nodal points for Type I structure . . . . . . . . . . . . . . 185
6.4
Comparative (dimensionless) values for tensile and shear modulus, and
Poisson’s ratio for maximum component beam stiffness. . . . . . . . . . . 197
17
6.5
Parameters used in the finite element analysis, and subsequently in determining normalised material properties. . . . . . . . . . . . . . . . . . . . . 199
6.6
Apparent and relative mechanical properties for structures where one, two
and three beams are present within the structure, and the component beams
have equal material properties (see table 6.5).
6.7
. . . . . . . . . . . . . . . . 199
Apparent and relative mechanical properties for conventional honeycomb
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.1
Poisson’s ratio calculated for each of the potential models . . . . . . . . . . 202
7.2
Ability of each potential to predict Poisson’s ratio . . . . . . . . . . . . . . 204
C.1 Comparison of Poisson’s ratio calculated directly from strains in the (110)
direction, and from rotating the elastic constants derived from strains in
the (100) direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
18
Acknowledgements
Foremost, I would like to express my sincere gratitude to Dr Arnaud Marmier for his
continuous support and guidance, for his patience, motivation and enthusiasm. I could
not have imagined having a better advisor and mentor. I would like to thank Professor
Ken Evans for his encouragement, insightful comments, and challenging questions.
I thank my fellow researchers: Chris Taylor, Nunzio Palumbo, Marc Boucher, Rob
Allen, Matt Johns, Ste Mellor and David Barnes. A special thanks goes to Dr. Wayne
Miller who has supported me both academically and pastorally throughout my time at
Exeter.
Last but not the least, I would like to thank my family: my parents Paul and Wendy,
for all the support and help that they have given me throughout the years.
19
Introduction
This thesis details a thorough investigation of negative Poisson’s ratio, or auxetic, behaviour in cubic systems, from metallic and covalent crystals to macroscopic beam
networks. Negative Poisson’s ratio materials are those which contract laterally when
compressed and expand laterally when stretched. The main strength of the study is that
many techniques, analytical and numerical, have been employed to understand the main
drivers of auxeticity in cubic structures. Despite the range of the techniques used, and
the variety of subject materials, the conclusions are strikingly similar, showing that the
structure of the bonding networks dominates the details of the "interaction".
While auxetic behaviour has been shown to exist in face centred cubic and body
centred cubic crystals, the underlying causes and mechanisms are still little known. Understanding causes and mechanisms of auxeticity in crystals is important in itself for
fundamental reasons, but the main attraction of a multi-scale, multi-techniques approach
is that it opens the way for the design of macroscopic, very lightweight, three dimensional
beam structures with very specific elastic properties. The idea is to take advantage of large
databases of measured elastic moduli in crystals (or of cheap computational techniques)
to provide already very good candidate structures for beam networks, where the beams
are analogous to the bonds.
The first two chapters review the current literature in the field, at first exploring
materials known to exhibit negative Poisson’s ratio behaviour, then presenting the tools
used to describe and calculate elastic properties. Purely analytical techniques are used in
the third chapter to isolate the role of various effects of bonding; structure and range of
interaction within cubic crystals. The fourth chapter demonstrates that empirical classical
potential models can be used to access the elastic properties of cubic elemental metals,
and highlights the mechanism of this behaviour, in the case of face-centred cubic crystals,
for the first time. The limitations of such model are analysed at length. In the following
chapter, quantum modelling provides a more robust methodology for finding the elastic
properties of cubic crystals, mitigating the shortcomings of empirical models. The sixth
chapter abandons crystals and details of interatomic bonding, as the insights obtained
from the crystallographic modelling are applied to scale-independent beam networks.
20
In this section of the work, a combination of analytical and finite element modelling
techniques results in findings which are very similar to those obtained at the atomistic
scale.
In the final chapter, the conclusions from diverse techniques at various scales are
summarised and generic, scale independent, rules for Poison’s ratios in cubic structures
are postulated. As the generic method of using bond structures from promising crystals
to generate real three dimensional structures has been validated, I finally suggest possible
ways to extend this work towards realistic applications.
21
Chapter 1
Background: Properties of materials
1.1
1.1.1
Negative Poisson’s ratio materials
Introduction
Poisson’s ratio (ν) is the ratio of lateral to axial strain,
ν=−
εlateral
,
εaxial
(1.1)
and for many common materials ranges from 0.25 to 0.3 (Callister, 2003).
Negative Poisson’s ratio, or "auxetic" (Evans et al., 1991) materials exhibit the counterintuitive property that they expand laterally when stretched, or conversely, contract
laterally when compressed, in one or more directions. Auxetic materials are worthy of
investigation; as a by-product of this behaviour they often exhibit high shear modulus
and indentation resistance (Evans and Alderson, 2000). Another potential application
of auxetic materials is where sinclastic (doubly curved, or bowl shaped) curvature is
required. The nature of auxetic materials means that in bending, the tensile/compressive
behaviour occurs both parallel and perpendicular to the profile of the curvature, unlike in
conventional materials which tend to form anticlastic shapes when bent in one direction.
Applications of auxetic materials are wide ranging, including arterial prosthesis (Caddock
and Evans, 1995), blast curtains for protection of building occupants from debris (Wright
et al., 2010) and seat cushions (Lowe and Lakes, 2000).
22
In this context, materials may be classified as non-auxetic, where Poisson’s ratio is
positive in all directions for all strains; auxetic, where they exhibit negative Poisson’s
ratio behaviour in one direction when strained in a specific direction (accompanied by
a corresponding positive orthogonal strain); averagely auxetic, where for most strains,
negative Poisson’s ratio behaviour is observed in one or more directions or completely
auxetic, where for a specific directional strain, the corresponding negative Poisson’s ratios
are all negative (see figure 1.1).
Figure 1.1: Schematic orthographic projection showing, left to right, deformed shape of
non-auxetic, auxetic and completely auxetic material (unstrained green, strained cream)
One of the earliest recorded instances of a material with a negative Poisson’s ratio is
iron pyrite, calculated to be - 17 (Love, 1892). More recently, Gibson et al. (1982) showed
both experimentally and analytically that it is possible to create a honeycomb structure
with a Poisson’s ratio approaching -1. Work by Lakes (1987) showed that it was possible
to modify a conventional open-celled foam structure to form a re-entrant structure with
a Poisson’s ratio of -0.7.
As well as the aforementioned Iron Pyrite, auxetic behaviour can also be seen in a
variety of crystal structures. It has been shown that 69% of cubic metals exhibit a negative
Poisson’s ratio in the (110) plane (Baughman et al., 1998). More complex crystal structures
such as Zeolites and Cristobalites also exhibit a negative Poisson’s ratio due to their
23
framework structure; Lethbridge et al. (2010) conducted a systematic investigation into
the Poisson’s ratio of 472 materials, both elements and compounds of many symmetries,
and found 36% of them to be auxetic in one or more directions.
1.1.2
1.1.2.1
Crystals
Simple Crystals
Auxetic behaviour has been observed in simple crystal structures: face-centred and bodycentred cubic, and hexagonal close packed structures (FCC, BCC and HCP respectively).
Milstein and Huang (1979) used experimental values of elastic moduli to determine the
Poisson’s ratio for face-centred cubic crystals. Of the twelve elements investigated, only
aluminium did not show negative Poisson’s ratio behaviour, the remaining 11 showed
some degree of auxetic behaviour in the [11̄0] direction.
It has been shown analytically (Milstein and Huang, 1979; Baughman et al., 1998),
that BCC metals can exhibit negative Poisson’s ratio in the (110) plane. Experimentally
derived elastic constants are used to compute the Poisson’s ratio for 32 metals with cubic
phases. 69% are found to exhibit non-axial negative elastic behaviour. 20 hexagonal
structures are studied and only Zn and Be found to be auxetic.
Baughman suggests a mechanism for negative elastic behaviour due to hinging of the
bonds in the crystal. By representing the inter-atomic forces in two adjacent body-centred
cubic cells as rigid beams able to hinge at the atoms, it is shown that for a stretch in the
[110] direction the corresponding atoms within the cell are displaced in a scissor-jack like
mechanism and there is a resulting contraction in the vertical plane.
Tokmakova (2005) used stereographic projection to investigate Poisson’s ratio in auxetic crystals. Tokmakova defines the Poisson’s ratio in the direction m for a stress in the
direction q as a function of two Euler angles (to identify the direction of q) and the lateral
strain in the cross section. The Poisson’s ratio for any possible stretch for cubic, monoclinic and hexagonal crystals is investigated using this reference system, and plotted. The
minimum and maximum Poisson’s ratios are found for Copper, Silver and Iron alloys. All
the alloys considered in this study were calculated to have a negative Poisson’s ratio in
one or more directions. Tokmakova found labradorite to have a Poisson’s ratio of -0.102
24
for the stretch direction (0°, 65.4 °, 90°) and augite to have a Poisson’s ratio of -0.008 for
the stretch direction (0°, 81.4°, 90°).
Investigation into hexagonal structures gave a negative Poisson’s ratio for Zn, as reported by Lubarda and Meyers (1999), molybdenum sulphide, graphite polypropylene,
and carbon. Common monoclinic crystals were not found to exhibit negative elastic
behaviour. Auxetic behaviour is reported by Li (1976) in Cadmium, however no experimental data or elastic constants are provided, and this result is not found by other
studies.
1.1.2.2
Silicates
Silicates are a form of silicon dioxide based minerals. Common examples of silicates
include sand and quartz. Generally the structure of silicates consists of one central Silicon
(Si) atom surrounded by four Oxygen (O) atoms in a tetrahedral configuration. These
tetrahedra are linked at the vertices, which then gives an open framework structure. It
has been thought that this open framework structure could lead to negative Poisson’s
ratio as there is the possibility for rotational displacements within the structure.
Quartz is made up of a continuous framework of SiO4 silicon-oxygen tetrahedra,
with each oxygen being shared between two tetrahedra. Negative Poisson’s ratio has
been found in quartz, and the analytical model attributes this to both the rotation and
dilation of tetrahedra (Alderson and Evans, 2009) .
Cristobalite is a polymorph of silicon dioxide that has a high temperature (β) and a low
temperature (α) modification (Yeganeh-Haeri et al., 1992). Using Brillouin spectroscopy,
Yeganeh-Haeri showed that a single crystal of α-cristobalite has a Poisson’s ratio (ν) of
between 0.08 to -0.5, but predominantly ν is negative. The elastic constants of α-cristobalite
showed it is highly anisotropic, and the shear modulus was found to be around 2.4 times
the bulk modulus, indicating that it is far more rigid than compressible. The transition
from α to β phase was modelled using molecular dynamics by Kimizuka et al. (2000). The
β phase was calculated to also be auxetic (νmin = −0.28 at 1800K).
The mechanism for the auxetic behaviour of α-cristobalite is due to its tetrahedral
framework structure. The unit cell can be described as a tetragon, comprised of four
25
smaller tetrahedra (Alderson and Evans, 2001). Rotation of the rigid tetrahedra gives rise
to the negative Poisson’s ratio mechanism.
An alternative rotating squares or rectangles mechanism is suggested (Grima et al.,
2006, 2005). This theory is suggested after conducting molecular modelling investigations. Rectangular units can be observed in the structure but this model is only a two
dimensional, projected, representation of the tetrahedral model.
Other SiO2 framework structures that exhibit a negative Poisson’s ratio include the
zeolites Natrolite, Thompsonite and Analcime (Grima et al., 2000).
Natrolite and Analcime were investigated using Brillouin scattering (Sanchez-Valle
et al., 2005). The Poisson’s ratios found by experiment were found to be all positive, in
contrast with the computational study by Grima (2000).
Experimental measurements of the bending, compression and indentation behaviour
of Natrolite were conducted and compared to force-field calculations (Lethbridge et al.,
2006). The experimental ultrasonic data were subject to large uncertainties due to the
shape of the crystals examined. However despite this, all the simulated and experimental
data were found to be in reasonable agreement. It was concluded that there was no
evidence for on-axis auxetic behaviour. The predicted off-axis negative Poisson’s ratio
behaviour is attributed to rotation of the tetrahedral units in an idealised structure. In
the actual Natrolite structure, cations not present in the idealised structure prohibit this
rotation, and the positive Poisson’s ratio originates from the deformation of the tetrahedral
units. Contrary to this, negative Poisson’s behaviour was found in the (001) plane by
Grima et al. (2007) from analysis of the elastic constants, and the mechanism is attributed
to rotating squares.
1.1.2.3
Paratellurite
Paratellurite (α−TeO2 ) is a crystal that is widely used in a variety of applications due to its
piezoelectric properties. It also has remarkable elastic anisotropy and negative Poisson’s
ratio behaviour (Ogi et al., 2004). This behaviour is attributed to a star shaped truss
structure that when subjected to a compression in the [110] direction, creates contraction
in the perpendicular direction. Further investigation revealed that ν is negative for
26
displacements around the b and c axis (Ogi et al., 2006). Ab-initio calculations support
the experimental findings (Ceriotti et al., 2006).
1.1.2.4
Metal organic frameworks
Metal organic framework structures (MOFs) are single-phase crystalline hybrid organicinorganic materials (Tan and Cheetham, 2011). These materials can demonstrate the
functionality of organic materials whilst exhibiting the thermal stability of inorganic materials. MOFs are an open framework structure, and it is this that makes them both useful
for molecular separation, gas storage and catalysis applications, as well as presenting the
possibility of negative Poisson’s ratio behaviour. Synthesis of MOFs is still in its infancy
and very much a developing field, there are to date, around 2000 MOF structures (Yaghi,
2012), and the nomenclature is yet to be fully developed. Most mechanical characterisation of MOFs has thus far employed nano indentation techniques to find the elastic
constants however the accuracy of this technique, particularly given the dependence of
elastic properties on material orientation in such highly anisotropic structures, combined
with the size of the crystals being investigated means that these results may benefit for
further validation and often a range of values is to be found in the literature for a specific
crystal.
1.1.3
1.1.3.1
Fabricated structures
Honeycombs
Several auxetic honeycomb structures have been devised, these can mainly be classified
as either re-entrant, chiral, or anti-chiral. The first auxetic honeycomb structures to be
developed were re-entrant honeycomb structures, a two dimensional structure with a
Poisson’s ratio of -1 (Gibson et al., 1982; Almgren, 1985) and variations thereof (Evans,
1989). The deformation mechanism is such that inplane, on axis compression results in
hinging of the ribs of the structure which is forced to occur in the orthogonal axis and
thus the structure has a Poisson’s ratio of -1 (see fig. 1.2). A unit cell for this mechanism
in three dimensions is suggested by Lakes (1987). The effect on the Poisson’s ratio of
varying the geometry of re-entrant honeycombs was considered by Wan et al. (2004) who
27
showed that varying the aspect ratio of the cells could give a Poisson’s ratio as low as
-4.211.
Figure 1.2: The auxetic mechanism of a re-entrant honeycomb. (Taken from
http://silver.neep.wisc.edu/ lakes/Poisson.html)
Chiral honeycombs first suggested by Lakes (1991) are composed of nodes of equal
radius joined by ribs of equal length (Prall and Lakes, 1997), and may have three, four or
six ribs at each node. The deformation mechanism is such that an on-axis compression
forces the ribs of the structure to bend and the nodes to rotate (see fig. 1.3), drawing in the
ribs that are off-axis and thus giving the structure auxetic behaviour with a Poisson’s ratio
of -1. Anti-chiral honeycombs utilise the same mechanism but have the ribs arranged in
such a fashion that they exhibit symmetry.
Figure 1.3: The auxetic mechanism of one cell of a tetrachiral honeycomb (left undeformed, right - deformed)
1.1.3.2
Foams
Conventional thermoplastic and polymeric foams can be modified to create an auxetic
structure by triaxial compression followed by heat treatment; the resulting structure has
curved ribs that create a re-entrant structure under compression. Foams generally have
a degree of anisotropy owing to the manufacturing process (Weaire and Hutzler, 2005).
28
A negative Poisson’s ratio of up to -0.25 in one plane, and -0.15 orthogonal to this was
achieved in preliminary investigations into this technique using a polyester urethane
foam (Friis et al., 1988). Polyethylene foams can be processed to give a Poisson’s ratio
of around -0.7 for small strains (Brandel and Lakes, 2001). More recently, foams with a
Poisson’s ratio as low as -1.4 in the ν yx direction have been manufactured Bianchi et al.
(2011).
1.1.3.3
Molecular Auxetics
Polymers synthesised to create materials with a negative Poisson’s ratio at the molecular
scale have been developed. These employ a variety of mechanisms. At a molecular
level the structure ’(n,m)-reflexyne’ was proposed (Evans et al., 1991). This gives a two
dimensional re-entrant honeycomb structure with a theoretical Poison’s ratio, calculated
using molecular dynamics, of -0.96.
Polytetraflouroethylene (PTFE) was found to be highly anisotropic with a Poisson’s
ratio as extreme as -11 due to the complex microstructures from which it is formed
(Caddock and Evans, 1989). This can be approximated as a particle-string model to
give an analytical model in close agreement to experimental data (Evans and Caddock,
1989). Ultra high molecular weight polyethylene (UHMWPE) can be processed to give it
a similar structure to PTFE (Neale et al., 1993) and thus a Poisson’s ratio of approximately
-7 (Alderson and Evans, 1993). A similar network of hinging interconnecting rods was
proposed by He et al. (2005) who used para-quaterphenyl rods laterally attached to a
polymer backbone to give a possible mechanism for auxetic behaviour at the molecular
scale.
Polyphenylacetylene networks comprise triangular units of benzene rings ’nodes’
joined by acetylene chains to make a structure of triangular repeating units. Molecular
dynamics calculations show that by a ’rotating triangles’ mechanism (similar to the rotating squares mechanism of zeolites) this configuration can produce a Poisson’s ratio of
-0.97 (Grima and Evans, 2000).
29
1.1.3.4
Composites
Composite materials can be designed such that their properties are optimised for their intended use and thus are generally high strength, low weight materials. The key parameter
in making a negative Poisson’s ratio composite from auxetic inclusions embedded in a
conventional matrix is the ratio of auxetic to non-auxetic components (Wei and Edwards,
1998). By alternating plates of auxetic and conventional materials, homogenization theory
can be used to estimate the global modulus of the structure, and it is found that inclusion
of auxetic layers results in a stiffer structure (Donescu et al., 2008; Lim, 2007). Composite
inclusions of various shapes can be used to vary the properties of the composite, and the
exact response of the material is dependent not only on the ratio of auxetic to conventional
materials but also the shape of the auxetic inclusions. A composite containing star shaped
rigid inclusions set in a compressible matrix can be found to give auxetic behaviour by
either dilation of the triangles within the structure (Milton, 1992) and by a re-entrant like
structure (Theocaris et al., 1997).
Auxetic fibre reinforced composites may be made entirely from conventional materials by choosing an optimal stacking configuration (Alderson et al., 2005). A material
manufactured in this way from standard carbon epoxy prepreg was shown to have a
Poisson’s ratio of -0.156, and demonstrated greater resistance to indentation than a similar conventional composite material. A high/negative Poisson’s ratio elastomer-matrix
composite has the potential to produce a structure with extraordinary Poisson’s ratio:
from 100 to -60 dependent on composition and angle of lay-up (Peel, 2007).
Double-helix yarn fibres comprises a low-modulus central strand, with a thinner high
modulus strand wound around it. Under tension, the higher modulus strand forces the
lower modulus strand to adopt a helical shape, and drives a perpendicular expansion.
The magnitude of this effect is dependent on the parameters of the system; comparative
radius of the yarns and angle of the wrap, but Poisson’s ratios of -2.1 have been shown
Miller et al. (2012a). Using this system, it has been possible to construct a negative
Poisson’s ratio Carbon fibre composite (Miller et al., 2012b).
30
1.1.3.5
Granular materials
It has been shown analytically that, in both two and three dimensional systems the
Poisson’s ratio of a granular solid is dependent on the ratio of the axial to tangential
stiffness, λ (Bathurst and Rothenburg, 1988). For λ greater than 1, Poisson’s ratio is
negative. Numerical simulations of bonded disc assemblies show that in two dimensions,
an assembly of 1000 discs with a λ value of 3 gives a Poisson’s ratio of -0.301.
The link between granular materials and a network of bending beams was made by
Koenders (2008), modelling the interactions between particles as beams and the particles
themselves as nodes. It was shown than auxetic behaviour could result from anisotropy
in the packing arrangement of the beams and that this problem was analogous to that of
a granular system.
1.1.3.6
Applications of mono-crystalline structures
Micro electro-mechanical systems (MEMS) are being widely used for a variety of both
mechanical and electronic applications. A recent application for a micro-mechanical
system is in the control of lightweight mirrors for space telescopes (see, for example
Hishinuma et al. (2005)). Current developments in this field are based around silicon based
structures, however, by incorporating a highly anisotropic, monocrystalline structural
element into the actuator at a specific orientation to maximise the Poisson effect, there
is the possibility that the small strain of the silicon component could be magnified by
the monocrystalline element, and thus a far greater range of movement made available.
Accurate modelling of these off-axis properties is critical to implementation of these
material in this, and other developing technologies.
1.1.4
Investigative Methodologies
Investigation of auxetic properties has taken several routes: experimental, mathematical
and more recently computational modelling have been used to investigate auxetic properties. For a more comprehensive review of the techniques employed for this thesis refer
to chapter 2.
31
1.1.4.1
Crystallography
Various crystallographic techniques have been applied to the study of elasticity, however
generally all involve interaction of a wave with lattice phonons and extracting the elastic
constants from dispersion curves.
Neutron scattering methods are capable of probing lattice dynamics (Dove, 2005). A
neutron beam is directed at the sample being considered and is diffracted as it passes
through the sample. The phonon spectrum of the material can be observed from the
resulting change in wave energy and wave vector. Neutron diffraction can be used for
both single and poly-crystalline samples. Neutron diffraction allows the phonons within
the crystal to be observed, and from these the elastic properties can be calculated.
Crystalline materials can be investigated using x-ray scattering techniques (for example (Mendham et al., 2000)) in a process similar to neutron scattering however at a far
greater energy. X-ray scattering is said to be in its infancy compared to neutron scattering techniques, however, with the development of synchrotron x-ray sources, dispersion
curves comparable to that produced by neutron scattering can be produced (Dove, 2005).
The theory of both X-ray and neutron scattering is discussed in depth by Ashcroft and
Mermin (1976) and by Burkel (2001) amongst others.
1.1.4.2
Spectroscopy
Raman scattering uses light to probe the lattice phonons of the sample. The technique has
a high resolution and is suitable for both single and polycrystalline samples, however is
particularly useful for investigating the spectra of single crystals at differing alignments
to investigate the symmetries of the vibrations (Palmer et al., 1994).
Brillouin spectroscopy has been widely used as a method for determining crystal
structures, for example Yeganeh-Haeri et al. (1992). Brillouin scattering uses acoustic
modes to determine the elastic constants and Poisson’s ratio of a single crystal (SanchezValle et al., 2005). The energy of the scattered light may be increased or decreased;
usually referred to as Brillouin shift. This is equal to the energy of the interacting phonon
and magnon and thus Brillouin scattering can be used to measure phonon and magnon
energies, and to determine elastic properties.
32
1.1.4.3
Molecular Modelling Techniques
Molecular mechanics provides a ’virtual experiment’ to model the interactions of a system
of atoms. The potential of the system can be calculated using either classical, or quantum
mechanics.
The use of molecular modelling techniques to model crystalline structures and their
elastic properties is widespread. For the modelling of Silicates, a specific force field model
has been developed from experimental data, the Burchart force field. It assumes that the
frameworks are largely covalent however, one shortcoming of this method is that it does
not accurately reproduce the on-axis Poisson’s ratio found experimentally (Alderson et al.,
2005). Grima and Evans (2000) use Burchart, BKS, Universal and Constant Valence forcefield modelling (using Cerius2 ) to predict the negative Poisson’s ratio in idealised zeolite
cage structures. Similarly, the CVFF force-field model was applied by Grima et al. (2006)
to model the deformation mechanism in alpha-cristobalite. Cerius2 was also used by
Grima et al. (2005) to model calix-[4]-arene network structures using four force-fields.
The ’egg-rack’ microstructure was found to exhibit negative Poisson’s ratio in two planes
and through variation of the structure, the Poisson’s ratio could be altered from -0.51 to
-0.95.
The use of ’dummy-atoms’ to investigate the properties of a re-entrant honeycomb at
nano-scale showed that this scale structure can be created with auxetic behaviour (Grima
et al., 2005). The EMUDA (Empirical modelling using dummy atoms) approach can
be used as a tool for modelling new structures where analytical methods are yet to be
developed, and is a cheaper alternative to experiment.
Density functional theory was used to investigate the lattice dynamics of solid benzene
(Kearley et al., 2006). It was found that elongation of the crystal along the b axis leads to
an increase in the a and c directions.
Monte-Carlo simulations of theoretical materials (Tretiakov and Wojciechowski, 2005)
show that, as found in the analytical model (Wojciechowski, 2004), auxetic behaviour can
occur. Simulations of an auxetic ferrogel shows that magnetic effects can be combined
with auxetic behaviour (Dudek et al., 2007).
33
1.1.4.4
Analytical Modelling
Modelling of auxetic microporous polymers was investigated using a ’node/fibril’ model
to try to reproduce the mechanisms seen with PTFE and UHMWPE (Alderson and Evans,
1995). The fibrils are modelled as free to rotate, and connected by hinged inextensible
rods. The Poisson’s ratio and Young’s modulus can be found in this way. Comparing
the models to experimental data it is found that the model is valid for up to around 10%
strain, at which point the Poisson’s ratio begins to become less negative as the deformation
mechanism becomes extension of the fibrils.
A model for negative Poisson’s ratio behaviour in foams was developed by Masters
and Evans (1996). It was noticed that samples of both auxetic and conventional foams
had broken ribs within their structure as a result of the manufacturing process, this was
used as the basis of the ’missing rib’ model (Smith et al., 2000).
When modelling a structure such as a polymeric foam, the behaviour of the foam
at the macro scale is far different from the behaviour of the foam at the micro scale. A
large sample of foam would behave as though a solid homogeneous medium whereas at
the microscale, it is composed of a complex network of ribs. It is not possible to model
this microscale using continuous elasticity and thus homogenization must be employed
(Gaspar et al., 2008). Using homogenisation, Gaspar showed that for an isotropic small
scale material where the modulus fluctuates it is possible to produce an isotropic auxetic
material. For anisotropic small-scale media, to produce auxetic behaviour at the macro
scale the fluctuations in modulus allowed are reduced.
Modelling of hard spheres as a framework for auxetic behaviour has been conducted
using molecular dynamics (MD) calculations - the MD framework being applied to a
purely theoretical structure. Modelling hard cyclic multimers can be shown to exhibit
negative Poisson’s ratio under certain conditions (Wojciechowski, 2004) where the arrangement is such that the structure forms a triangular lattice at close packing. Variation
of the ratio of diameters of this model and introduction of disorder into the system
gives a model that shows the effect of molecular shape asymmetry on elastic properties
(Narojczyk and Wojciechowski, 2008).
Classical elasticity provides an ideal framework for the modelling of negative Pois34
son’s ratio materials, and is discussed in detail in section 2.2.
1.1.4.4.1
Rotating Rigid Units
Rotating polygons (and in 3 dimensions) polyhedra provide an accurate and convenient
general model for negative behaviour in a variety of structures, and are comparable with
the Rigid Unit Modes models suggested by Dove (1997). The rotating rectangles method
(see, for example (Grima et al., 2005b) and similar), provides a 2D model for planar
structures exhibiting auxetic behaviour.
The rotating rectangles model assumes that the structure being considered, generally a
framework structure such as a zeolite, can be discretised into a system of rigid rectangular
elements linked by bonds that allowed the rigid units to hinge. Deformation of the
structure results in hinging of the rigid units allowing a Poisson’s ratio of -1. Using this
model a compliance matrix for the structure can be derived. This model can be applied
to zeolites and silicates and is also proposed as an alternative mechanism for auxeticity
in foam structures (Grima et al., 2005a). A similar model was proposed by Ishibashi and
Iwata (2000) and reviewed in detail by Dmitriev (2007), whereby a system of squares
linked by springs is employed to model the structure.
A variation on this analytical model was developed by Williams et al. (2007) for
auxetic behaviour in Zeolites. Rotating linked parallelogram structures are used to give
a 2D model. This model gives a theoretical Poisson’s ratio of -1. Including a term for the
distortion of the parallelograms gives the possibility of a lower ν.
Stretching, rather than rotation of squares is a further mechanism for auxetic behaviour
(Grima et al., 2008). This mechanism has potential to be adjustable to give the required
properties in the structure being designed. As a model it is valid but seemingly does
not describe any pre-existing behaviour, and is conditional on the unit cells having the
described ’piston like’ structure thus seems unlikely to be applicable to any molecular or
honeycomb like structure as well as presenting various manufacturing difficulties.
35
1.2
1.2.1
Other unusual material behaviour
Introduction
Whilst this work is concerned with the discovery and exploitation of negative Poisson’s
ratio (NPR) behaviour, considering other ’negative’ elastic properties (negative thermal
expansion (NTE) and negative linear compressibility (NLC)) could provide an insight
into the mechanism of NPR behaviour. The discovery, prevalence and mechanism of
NTE and NLC materials are briefly reviewed in the next section.
1.2.2
Negative Thermal Expansion
1.2.2.1
Background
Negative thermal expansion (NTE), the phenomenon whereby a material shrinks on
heating or expands on cooling can be observed in many natural structures, the most
commonly known example is the expansion of water during freezing (although this is
brought about by a phase change) but NTE is also seen in more complex crystals and
in fabricated structures (Oruganti et al., 2004). There are several mechanisms by which
materials exhibit negative thermal behaviour, dependent on the structure of the material.
Investigation of negative thermal expansion materials provides a useful background from
which novel composites, with potential for zero thermal expansion, can be developed.
1.2.2.2
1.2.2.2.1
Crystals
Cubic crystals
Whilst NTE behaviour is generally observed in complex framework structures such as
those discussed below, it is also found in Caesium. There are six known phases of
caesium at room temperature. Ab-initio calculations by Christensen et al. (2000) revealed
the coefficient of thermal expansion for Caesium in its II (face-centred cubic) phase to be
negative at pressures above 3 GPa for all temperatures. Whilst there is no experimental
evidence offered to support this for FCC, the results for BCC caesium are compared to
experimental data and found to be in agreement.
36
1.2.2.2.2
Metal Organic Frameworks
Metal organic framework (MOF) structures comprise atoms or clusters of metals with organic linkers (Zhou et al., 2008). MOF-5 (or Zn4 O13 − (C8 H4 )) has been shown in molecular
dynamics simulations to have a thermal expansion coefficent as low as -20E-6K−1 (Dubbeldam et al., 2007). MOF-5 comprises ZnO4 clusters linked by a 1,4-benzenedicardoxylate
FCC crystal (Li et al., 1999). The mechanism for NTE suggested by Zhou et. al. is that
the ZnO4 tetragonal units are bridged to the benzene rings in the structure by carboxyl
groups. Large amplitude vibrations of these carboxyl groups leads to negative thermal
expansion. Experiment found that the linear thermal expansion coefficient of MOF-5
was -16E-6K−1 . The lattice phonon modes were investigated and it was found that low
frequency lattice modes were responsible for the NTE in the structure. It is suggested that
MOFs that do not have a flexible bridging group such as the carboxyl group in MOF-5
would not exhibit NTE. Wu et al. (2008) investigated Cu3 (1, 3, 5 − benzenetricarboxylate)2
experimentally, it was found to have a negative thermal coefficient owing to translation
of the aromatic ring in a similar fashion to that of MOF-5.
1.2.2.2.3
Other framework structures
As with auxetic behaviour, framework structures such as zeolites have potential to behave
as negative thermal expansion materials. One such structure is silver hexacyanocobalate.
The structure is that of a ’Kagome’ lattice, hexagonal units formed by Ag atoms with
cyanocobalate ions positioned above the Kagome holes (Goodwin et al., 2008). Experimental investigation using x-ray diffraction by Goodwin et al. revealed the coefficient of
thermal expansion to be in the order of -120E-6 to -130E-6 K−1 , an order of magnitude
larger than is generally found with framework structures. DFT calculations reveal that
this behaviour was due to the structure behaving like a garden trellis; the bond lengths
undergo very small changes under heating but the bond angles change considerably
(Calleja et al., 2008).
Zirconium tungstate exhibits negative thermal expansion over a very wide temperature range (Evans et al., 1997). Despite that there is a structural phase transformation
within this temperature range, it exhibits negative thermal expansion in both phases and
37
thus this leads to isotropic thermal expansion (Pryde et al., 1996). A similar effect is seen
in the compound ZrV2 O7 .
The negative thermal expansion of Zirconium Tungstate is caused by transverse vibrations of the Oxygen molecule in Zr-O-W bonds which lead to shortening of the Zr-W
distance and thus a negative thermal expansion (Mary et al., 1996). The structure of
zirconium tungstate is such that it comprises WO4 tetrahedra linking ZrO6 octahedra,
the combination of linked octahedra and tetrahedra forming a framework structure. Decreasing the distance between the adjacent Zirconium and Tungsten leads to opposing
rotations of the polyhedra. The mechanism of NTE by rotation of these polyhedra without
distorting is known as a rotating unit mode (RUM) (Giddy et al., 1993).
The shape of a typical metal-oxide potential implies that longitudinal oxygen vibrations creates an increase in M-O and M-M distances, however a transverse vibration in
which M-O bond distances are unchanged can result in a reduction in the M-M distance
(Evans et al., 1997). Tao and Sleight (2003) investigated NTE in five oxide structures with
varying ratios of octahedral and tetrahedral cations.
Negative thermal expansion is found in Sc2 (WO4 )3 from 10 to 1073K (Evans et al.,
1998). The structure of Sc2 (WO4 )3 comprises ScO6 octahedra joined at the corners to six
WO4 tetrahedra, each tetrahedron being joined to four octahedra. It has been observed
that an increase in temperature results in minimal change in bond angles in the polyhedra
and thus the rigid unit modes model is applicable to this structure.
Where the hinging of the rigid units is enabled due to M-O-M oxide based frameworks,
this can also be seen in cyanide bridged frameworks where the bridging unit becomes
M-CN-M (Goodwin and Kepert, 2005). Unlike in oxide frameworks where the linking
element is formed of just one molecule, in CN bonded structures, the diatomic linking
element gives rise to a larger range of RUMs. For the perovskite structure, investigated
by Goodwin, it was found that whilst with the M-O-M linkages the was a single RUM at
one wave vector, using a diatomic linkage results in RUMs at all wave vectors across the
Brillouin zone.
The framework structure of zeolites make them a likely class of solids that exhibits
negative thermal expansion. Computational investigations into the effect of tempera-
38
ture on siliceous faujasite conducted by Couves et al. (1993) predicted a reduction in
lattice parameters under heating. These data were verified by high-resolution powder
diffraction studies. Lattice dynamics calculations for 18 zeolite structures found NTE
behaviour which was attributed to the structure expanding into the pores within the
network (Tschaufeser and Parker, 1995). Strong NTE was observed by Woodcock et al.
(1999). It was thought that water trapped within the pores of the structure may be evacuated at high temperatures, and this may be the cause of this behaviour however further
investigation did not confirm this.
Reisner et al. (2000) investigated the zeolite RHO experimentally and found there to
be two stages of NTE, rapid contraction from 25-75°Cand slower contraction up to 500°C.
Cation relocation caused by dehydration of the structure under heating means that the
structure changes shape. In dehydrated zeolite RHO this was not observed.
1.2.2.3
1.2.2.3.1
Mechanisms of NTE
Increasing symmetry
Increasing symmetry thermal expansion is seen in PbTiO3 and similar AMO3 compounds
that have a perovskite structure (Sleight, 1998a). Such compounds contain MO6 and AO12
polyhedra that become more regular with increasing temperature. As the polyhedra
become more regular, the M-O distances decrease. This leads to a thermal expansion in
two planes, and thermal contraction in the third plane.
1.2.2.3.2
Positive expansion of bonds
Positive thermal expansion of some M-O bonds in hexagonal structures can contribute to
negative thermal expansion (Sleight, 1998b). The mechanism for this is described in detail
for Cordierite, Mg2 Al2 Si5 O18 by Sleight (1998a). Assuming that the thermal expansion
of the AL-O, Si-O, P-O and Zr-O bonds is very small, then the mechanism is dependent
on the thermal expansion of the Mg-O, Li-O and Na-O bonds. Because in this class of
oxides the structure is comprised of face/edge sharing polyhedra, it is possible for the
polyhedron to expand in only one or two directions (expansion is not possible in all three
planes due to cation-cation repulsion across shared edges) and some negative expansion
39
is seen.
1.2.2.3.3
Electron valence transition/cation movement
The movement of cations is used by Sleight (1998a) to explain the negative thermal
behaviour in β-eucryptite. The Li+ cations migrate from tetrahedral to octahedral sites
to contribute to negative volume expansion. Electronic valence transition is proposed as
a mechanism for zero volumetric thermal expansion in YbGaGe by Salavdor et al. (2003).
As the material is cooled, electrons depopulate the conduction band and increase the
fraction of Yb2+ ions. As Yb2+ ions are larger than Yb3+ ions, there is an expansion in the
a-b plane. This mechanism, and the existence of zero thermal expansion in YbGaGe is
disputed by Bobev et al. (2004) and Janssen et al. (2005) as in similar studies this effect was
not observed, and it seems that the stoichiometry of the sample may define its behaviour.
Janssen found the thermal expansion to be comparable to that of copper. Delta plutonium
is suggested as a possible material that does exhibit electronic valence transition to give
rise to NTE behaviour.
1.2.2.3.4
Rotation of rigid units
In structures composed of polyhedra, such as tetragon or hexagons, linked by shared
atoms the corners of the structure, rotation of these units can lead to a negative thermal
expansion. Co-operative rotation of the polyhedra leads to a reduction of the spaces
between the polyhedra.
1.2.2.4
1.2.2.4.1
Methodologies
Experimental
Experimental methodolgies used in the literature are generally the same as those employed for observation of negative Poisson’s behaviour; neutron diffraction (Evans et al.,
1998, 1996), x-ray diffraction (Lightfoot et al., 2001; Woodcock et al., 1999; Wu et al., 2008)
and raman spectroscopy. Infra-red spectroscopy is used by Evans et al. (1996). Different
vibration modes give differing absorption spectra, thus the behaviour of the sample over
a range of temperatures can be analysed.
40
1.2.2.4.2
Analytical Modelling - Rotating unit modes
Rigid unit modes (RUMs) has been used as a model for negative thermal expansion
in framework structures (for example: (Pryde et al., 1996; Heine et al., 1999; Tao and
Sleight, 2003)). In a RUMs model the framework structure, such as zirconium tungstate,
is modelled as rigid polyhedral units joined at corners. These units are assigned classical
properties such as mass, moment of inertia and three vibrational and translational degrees
of freedom (Tao and Sleight, 2003). The only connectivity in the model is that between
the units. The units share oxygen atoms, which serve to link them together.
Investigations of compounds with these structures has been conducted using the
program CRUSH (Giddy et al., 1993), where a split atom model has been implemented.
The split atom model treats the oxygen molecules linking the units as two atoms with a
separation of zero, and the constraint that the two split atoms are bound by an invented
potential. From this model it is possible to predict if RUMs occur in a given structure,
and thus explain the mechanism of NPR in the compound under consideration.
The degree to which an expansion is seen is dependent on the degree of rotation,
which is a function of the Boltzmann constant, temperature and the frequency of the
rotational phonon (Pryde et al., 1996). Summing these over all the units in the lattice gives
a net volume change in the structure. Where the modes result in a small distortion of the
polyhedra, the RUMs become known as quasi-RUMS.
1.2.2.4.3
Molecular modelling
As well as CRUSH, developed for determining the Rigid Unit Modes in structures, several
other computational methods have been employed to investigate NTE behaviour. A dynamic version of the split atom method was developed to investigate RUMs in cristobalite
and quartz (Gambhir et al., 1999).
Free energy minimisation conducted with the program PARAPOC was used to in some
of the earliest work on zeolites to model the cell dimension at a range of temperatures
(Tschaufeser and Parker, 1995). Interatomic potentials were used by Pryde et al. (1996) to
determine the behaviour of zirconium tungstate.
41
1.2.3
Negative Linear Compressibility
Negative linear compressibility is a phenomenon whereby a material expands in one
direction when subjected to hydrostatic loading. Negative linear compressibility has
been found in several compounds by Skelton et al. (1976) in paratellurite (TeO2 ), and by
Mariathasan et al. (1985) in lathanum niobate.
Baughman et al. (1998) investigated around 500 noncubic crystal compounds, using
experimental elastic constants. He found 13 compounds that exhibit this behaviour.
Baughman suggests a ’wine-rack like’ deformation structure. The possible mechanisms
for this behaviour are suggested to be dependent on the structure, but are all based
around interconnected helices. Two helices, opposingly wound would result in a hinged
structure that decreases in volume when subjected to an axial tensile load, or if subjected
to a pressure from within the helices, the structure can be seen to expanded axially.
Negative compressibility was also considered by Grima et al. (2008) who proposes
a truss structure formed of triangular hinged units. The model suggested is a triangle
whereby one side is comprised of a material with lower compressibility than the other
two sides. This gives structure that exhibits negative linear compressibility in the plane
perpendicular to the side with lowest compressibility.
A further structure that exhibits negative compressibility is suggested by Gatt and
Grima (2008), comprising of a structure similar to the chiral structures of Prall and Lakes
(1997) however the round central nodes are replaced by square rectangular blocks and the
interconnecting ribs are comprised a bi-metallic strips. Applications for negative linear
compressibility structures are yet to be realised.
42
Chapter 2
Background: Theoretical framework
and computational modelling
2.1
Introduction
Negative Poisson’s ratio, and other unusual material properties have been observed in
variety of crystal and fabricated structures. This chapter describes techniques used to
investigate these properties both analytically and numerically.
2.2
General elasticity equations for stress and strain
The linear elastic behaviour of an anisotropic material or structure can be fully described
by a set of 21 elastic constants, relating the stress and strain in any given direction (Nye,
1985). This section details the relationship between stress and strain, the conversion from
elastic constants to so called "engineering" constants, the relationship between stiffnesses
and compliances, and the derivation of elastic properties in any direction from the elastic
constants for the standard orientation.
2.2.1
Notation
When discussing properties of elastic media, it is usual to talk in terms of stresses and
strains within a reference framework. Axial directions are generally referred to as 1, 2, 3,
43
and thus stresses and strains can be expressed as σii , or σi j dependent on whether they are
in axial or shear directions. The full components of the stiffness (and compliance) tensor
that relate stress and strain and described the elastic properties of the material can be
expressed as si jkl . The most commonly employed notation schemes are the Mandel and
Voigt notation.
2.2.1.1
Mandel notation
The Mandel notation allows a symmetric second rank tensor to be expressed as a vector
(albeit one of higher dimension). This is useful as it also allows a rank four symmetric
tensor (such as that described in section 2.2.4) to relate stresses and strains in an elastic
solid) to be reduced from 81 components to a six by six matrix. The symmetry dictates
that Di jkl = D jikl and Di jkl = Di jlk and thus the tensor is reduced to:
DM

 D
 1111


 D2211



 D3311
=  √
 2D

1211
 √

 2D2311

 √

2D1311
√
√
D1122
D1133
2D1112
√
D2222
D2233
2D2212
D3333
√
2D1233
√
2D2333
√
2D1333
2D2223
√
√
D3322
√
2D1222
√
2D2322
√
2D1322
2D1123
√
2D3312
2D3323
2D1212
2D1223
2D2312
2D2323
2D1312
2D1323

√
2D1113 

√

2D2213 


√
2D3313 


2D1213 


2D2313 


2D1313 
(2.1)
This transformation is generic and symmetric, but rarely used.
2.2.1.2
Voigt Notation
For historical reasons, the notation of Voigt is more commonly used (Nye, 1985). This
also transforms the three-dimensional fourth-order tensor to a 6 x 6 matrix. The notation
follows the rules that a double index e.g. xx becomesx, and for indices of the type xy, the
Voigt index becomes 9 − (x + y).
The Voigt notation is similar to the Mandel notation, but does describe stress and
stiffness, for which the transformation is more convenient. As a consequence, the strain
and compliance transformation must account for double or quadruple counting. For
44
convenience when performing such operations as rotations, it is possible to modify the
matrix accordingly; when transforming from the true tensor notation to Voigt (matrix)
notation, numerical factors must be introduced to identify x11 with x1 and similarly x13
with 12 x5 etc. (Hearmon, 1957). These factors are:
sxxyy = sqr (When q and r = 1, 2 or 3)
1
sxxyz = sqr (When q = 1, 2 or 3, and r = 4, 5, or 6)
2
1
sxyxy = sqr (When q and r = 4, 5 or 6)
4
(2.2a)
(2.2b)
(2.2c)
An example of this is given in equation 2.15.
2.2.2
The strain tensor
The strain tensor can be used to describe linear elastic strains; for a solid cuboid, of side
lengths l1 , l2 and l3 , with a uniform normal pressure applied to one face in the 1 direction
(where 1,2 and 3 are equivalent to x, y and z), the face is displaced by du1 . The axial
pressure (for almost all materials) results in a corresponding contraction or expansion of
the material in the lateral directions. The axial strain in the load direction can be described
as:
ε11 =
du1
l1
(2.3)
Similarly, the corresponding orthogonal strains (in the 2 and 3 direction) can be written
as:
ε22 =
du2
l2
(2.4a)
ε33 =
du3
l3
(2.4b)
Applying a displacement parallel to the faces of cuboid results in corresponding shear
deformation. In pure shear the deformation is characterised by deformations in the x and
y directions. The deformation can be expressed in terms of the angle.
45
θ1 =
du1
l2
θ2 =
(2.5a)
du2
l1
(2.5b)
The total shear strain can be written as
ε12
1 du1 du2
+
=
2 l2
l1
!
(2.6)
This can be generalised to the form:
1 dui du j
εi j =
+
2 lj
li
!
(2.7)
Combining the equations for the axial and shear strains, the total strain tensor can be
written as:

 ε
 11 ε12 ε13


 ε21 ε22 ε23



ε31 ε32 ε33
 
 
 
 
 
 = 
 
 
 
du1
l1
1
2
du
1
2
du
2
l1
l1
3
+
+
1
2
du1
l2
du1
l3
du
1
l2
+
du2
l1
du2
l3
du2
l2
1
2
du
l2
3
+



l3

du

du3 
1
2
+

2 l3
l2


du3

1
2
du
1
+
du3
l1
(2.8)
l3
The symmetry of the tensor (ε12 = ε21 etc.) can be used to replace the strain tensor to a
six dimensional vector:

 ε
 11


 ε22



 ε33
ε = 
 1 ε
 2 23

 1
 ε31
 2

 1
2 ε12
46




















(2.9)
2.2.3
The stress tensor
Figure 2.1: Schematic showing normal (S1,2,3 ) and shear stresses (S12,13,23 ) for am
arbitrary cuboidal solid.
The stress tensor can also be constructed in a similar way to the strain tensor. Considering
the cuboidal example above, a displacement has a force associated with it. A force over
a given area results in a stress in the specified direction. Components of the stress tensor
take the form of σi j where i represents the normal to the surface on which the stress acts,
and j is the direction of the stress component.
The stress tensor can be represented as:

 σ
 11 σ12 σ13


σ =  σ21 σ22 σ23


 σ31 σ32 σ33









(2.10)
Where components of the form σii represent the axial stresses, and components of the
from σi j represent the shear stresses. As with the strain tensor, the stress tensor can be
represented as a six dimensional vector.
47










σ = 








2.2.4
σ11
σ22
σ33
σ23
σ31
σ12




















(2.11)
Generalised Hooke’s Law
Hooke’s law states that:
F = kx
(2.12)
where F is the applied force, k the stiffness of the material and x the displacement.
Extending this to stress and strain for isotropic materials, the stress is related to the strain
by the Young’s modulus:
ε = Eσ
(2.13)
For anisotropic materials the strain can be related to the stress by the elastic compliance
matrix, thus:
εi j = si jkl σkl
(2.14)
In the matrix form:




















ε11
ε22
ε33
ε23
ε31
ε12
 
  s
  1111 s1122 s1133
 
 
  s2211 s2222 s2233
 
 
 
  s3311 s3322 s3333
 = 
 
 2s
  2311 2s2322 2s2333
 
 
 2s3111 2s3122 2s3133
 
 
 2s1211 2s1222 2s1233
2s1123 2s1131
2s2223 2s2231
2s3323 2s3331
4s2323 4s2331
4s3123 4s3131
4s1223 4s1231
48

2s1112  
 
 
2s2212  
 
 
2s3312  
 

4s2312  
 
 
4s3112  
 
 
4s1212  
σ11
σ22
σ33
σ23
σ31
σ12




















(2.15)
Similarly, the elastic stiffness matrix relates the stress to the strain:
σi j = ci jkl εi jkl
(2.16)
Or, in full:




















σ11
σ22
σ33
σ23
σ31
σ12
 
 
 
 
 
 
 
 
 
 
 = 
 
 
 
 
 
 
 
 
 
c1111 c1122 c1133 c1144 c1155 c1166
c2211 c2222 c2233 c2244 c2255 c2266
c3311 c3322 c3333 c3344 c3355 c3366
c4411 c4422 c4433 c4444 c4455 c4466
c5511 c5522 c5533 c5544 c5555 c5566
c6611 c6622 c6633 c6644 c6655 c6666

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ε11
ε22
ε33
ε23
ε31
ε12




















(2.17)
Each index (i, j, k, l) may be either 1, 2 or 3, signifying its direction in three dimensional
space, hence the 81 constants. Due to the symmetry of the tensor, this can be reduced to
21 unique elastic constants.
2.2.5
Conversion between stiffness and compliance
Compliances are the inverse of the stiffnesses, thus by inverting the stiffness matrix the
compliance matrix can be found and vice versa. For cubic media where symmetry dictates
the elastic constants as c11 , c12 and c44 , the inversion is straightforward:
−c11 − c12
+ 2c212 − c11 c12
−c21
= 2
−c11 + 2c212 − c11 c12
1
s44 = s55 = s66 =
c44
s11 = s22 = s33 =
s12 = s13 = s23
−c211
(2.18)
(2.19)
(2.20)
At lower symmetries, the number of unique elastic constants increases the number
of terms in the tensor increases. For tetragonal media, the symmetry dictates that seven
49
constants are required, thus the compliance matrix is:


s
0
s16 
 11 s12 s13 0




s12 s11 s13 0

0
−s
16 





s13 s13 s33 0
0
0 


s = 

 0

0
0
s
0
0

44





 0
0
0
0
s
0

44





s16 −s16 0
0
0
s66 
(2.21)
The inverse of this matrix describes the stiffnesses:
c11 =
s11 s33 − s213
s33 s211 − 2s213 s11 + 2s213 s12
(2.22a)
c33 =
s11 + s12
s11 s33 − 2s213 + s12 s33
(2.22b)
c13 =
s13
s11 s33 − 2s213 + s12 s33
(2.22c)
−s16
s11 − s12 s66
1
=
s44
1
=
s66
c16 =
(2.22d)
c44
(2.22e)
c66
(2.22f)
(2.22g)
2.2.6
Rotations of constants
A full set of elastic constants for a given direction can be used to determine the elastic
constants for any arbitrary direction. Consider the original reference axes x, y, z , these
can be rotated through angles θ and φ into the position x0 , y0 and z0 (see figure 2.2).
Methodolgies for this are discussed at length in the literature (see (Nye, 1985),(Hearmon,
1957), (Lieberman and Zirinsky, 1956)).
50
Figure 2.2: Angles θ and φ used to describe rotated properties. χ shows the plane
orthogonal to the vector described by these angles (a), b is the vector orthogonal to this.
As a simple example, consider the rotation of the elastic constants of a cubic crystal
0
from the [100] to [110] direction. The constants Ti jkl is rotated to the new orientation Ti jkl
according to:
Ti0jkl = aim a jn ako alp Ti jkl
(2.23)
The rotation can be defined by the matrix, A, where

 a
 11 a12 a13


A =  a21 a22 a23


 a31 a32 a33









(2.24)
The tensor can be rotated about the z axis in a revised configuration using the formula:


 cosθ sinθ 0 







A =  −sinθ cosθ 0 




 0
0
1 
51
(2.25)
Thus:
a11 = a22 = cosθ
(2.26a)
a33 = 1
(2.26b)
a12 = sinθ
(2.26c)
a21 = −sinθ
(2.26d)
For a cubic crystal, the symmetry of the crystal means that there are only three unique
elastic constants (in Voigt notation), for the principal axes (s11 , s22 , s33 ), the shear component (s44 , s55 , s66 ) and the perpendicular components (s12 , s32 , s23 ). This leads to any terms
where the index is not one of these, or an equivalent index, being disregarded. Furthermore, as the rotation is about the z axis, any index (in tensor notation) where the 3 term
is present can be ignored as this does not change.
Thus:
s01111








= 






s1111
−
−
−
−
−
s1112 s1121 s1211 s2111
s1122 s1212 s1221 s2121 s2211 s2112
s2221 s1222 s2122 s2112
−
s2222
−
−
−
−
−
















(2.27)
This can be simplified using the symmetry of the crystal to become:
s01111







= 





s1111
−
−
−
−
−
s1122 s2211 s1212 s1221 s2121 s2112
s2112
−
−
−
−
s2222
−
−
−
−
−













(2.28)
Thus, converting the tensor notation to matrix notation, and including the above
52
0
0
factors, and applying the same methodology for s12 and s44 :
0
s11 = R · 2s11 + 2s12 + s44
0
s12 = R · 2s11 + 2s12 − s44
(2.29a)
(2.29b)
0
s44 = R · 8(s11 − s12 )
(2.29c)
For the example of mapping to the constants from [100] to [110]:

 cos π

4


A =  −sin π4



0
sin π4
cos π4
0

0 


0 


1 
(2.30)
Thus:
0
1
s11 = (2s11 + 2s12 + s44 )
4
0
1
s12 = (2s11 + 2s12 − s44 )
4
0
1
s44 = (8(s11 − s12 ))
4
2.2.7
(2.31a)
(2.31b)
(2.31c)
Relationship between elastic constants and engineering constants
Traditionally, materials are characterised by the engineering constants, as these can be
easily measured experimentally, and provide a useful indicator of a materials suitability
for a specific application. These constants can be summarised as the Young’s (elastic),
shear and bulk moduli and the Poisson’s ratios.
The elastic constants can be used to derive the engineering constants for isotropic
media (Ashby and Gibson, 1997):
53




















 
εxx  
 
 
ε yy  
 
 
εzz  
 = 
 
γ yz  
 
 
γzx  
 
 
γxy  
1
E
− Eν
− Eν
0
0
− Eν
1
E
− Eν
0
0
− Eν
− Eν
1
E
0
0
0
0
0
0
1
G
0
0
0
0
1
G
0
0
0
0
0

0  
 
 
0  
 
 
0  
 

0  
 
 
0  
 
1 
 
G

σxx 


σ yy 


σzz 


σ yz 


σzx 


σxy 
(2.32)
Where ε signifies extensional strain, γ represents shear strain, ν is Poisson’s ratio, E
is Young’s modulus, G is shear modulus and σ stress in a given direction. Engineering
constants in terms of the elastic constants are as follows:
2.2.7.1
Young’s Modulus
Young’s modulus relates the extensional or compressive strain to the stress in the direction of loading. For a given sample there are up to three Young’s (or elastic) moduli,
corresponding to the principal axes. For n = 1,2,3:
En =
2.2.7.2
1
snn
(2.33)
Poisson’s ratio
An isotropic material has a Poisson’s ratio bounded by thermodynamics; the Young’s
modulus, bulk modulus and Poisson’s ratios must have positive values, thus Poisson’s
ratio must fall between -1 and 0.5 (see eqn. 2.2.7.4). Anisotropic materials are not restricted
by these bounds (Ting and Chen, 2005). For an anisotropic material, an axial strain results
in a major and a minor Poisson’s ratio corresponding to the ratios of the axial and lateral
strains in each perpendicular direction.
s21
s31
s32
, ν13 = − , ν23 = −
s11
s11
s22
s12
s13
s23
= − , ν31 = − , ν32 = − .
s22
s33
s33
ν12 = −
ν21
54
(2.34)
2.2.7.3
Shear Modulus
Shear modulus relates the shear stress to the shear strain. Shear modulus can be be
known as modulus of rigidity as it describes the materials resistance to lateral rather than
compressive loads. Again, there are three shear moduli corresponding to the faces normal
to each principal axis.
G23 =
2.2.7.4
1
,
s44
G31 =
1
,
s55
G12 =
1
.
s66
(2.35)
Bulk Modulus
Bulk modulus is a measure of a materials resistance to uniform volumetric compression.
For anisotropic media, this is
k=
1
s11 + s22 + s33 + 2s12 + 2s13 + 2s23
(2.36)
For cubic media, this factorises to:
K=
1
,
3(s11 + 2s12 )
or for isotropic media:
K=
E
.
3(1 − 2ν)
(2.37)
This form shows how the restricton that E, ν and K must be greater than zero which
limits Poisson’s ratio for isotropic media, as the inequality (1 − 2ν) > 0 must be satisfied.
2.2.7.5
Zener ratio
The Zener ratio is a measure of a materials anisotropy, based on the stiffnesses, for cubic
materials:
Z=
2C44
.
c11 − c12
55
(2.38)
A Zener ratio of 1 indicates isotropic behaviour. It is possible to characterise materials
of lower symmetry by comparing the ratio of the maximum and minimum shear wave
velocities in the material (Ledbetter and Migliori, 2006).
2.2.8
Application to auxetic media
Investigation of Poisson’s ratio using classical elasticity theory for auxetic media has been
developed for well over a hundred years and has provided an accurate and robust method
for determining the elastic behaviour of materials but seldom reveals the mechanisms of
this behaviour.
In most materials, the Young’s modulus, shear modulus and Poisson’s ratio are directional. In an isotropic elastic medium, the stresses and strains are linked by Hooke’s law,
thus = Eσ . For isotropic cubic media, Poisson’s ratio must be between -1 and 0.5 for the
material to be stable. In an anisotropic material this is not the case. Ting and Chen (2005)
show that Poisson’s ratio for anisotropic media can be unbounded. This is then developed
by Ting (2004) [n.b.(Ting and Chen, 2005) was written and submitted prior to (Ting, 2004) but
publication appears to have been delayed] who shows that the large transverse strain in this
model does not comply with linear elasticity theory, and that transverse strain must be
bounded to comply with these laws.
Some symmetries have been investigated in detail; the stress-strain relationships
for cubic crystals are shown by Thomas (1966), Hooke’s law can be generalised for an
anisotropic cubic solid to take into account the direction of stresses and resulting strains
and in a cubic system, the resulting compliance tensor can then be transformed depending
on the orientation of the stress. Symmetry can be used to reduce the number of terms from
21 to 3 for a cubic anisotropic homogeneous material (Turley and Sines, 1971b) (Turley
and Sines, 1971a). The resulting equations allow the elastic properties of the crystal under
consideration to be plotted for different planes, rotating about a specified axis.
Negative Poisson’s ratio in anisotropic media is considered by Ting and Barnett (2005).
Using the equation of Hayes and Shuvalov (1998), it is shown that for a cubic crystal to be
completely auxetic the compliance S12 must be greater than 0, however searches have yet
to reveal such a crystal. The conditions for a non cubic/hexagonal crystal to be completely
56
auxetic are derived in terms of the compliances of the crystal, giving parameters from
which auxetic behaviour can be found from elastic constants published in the literature.
The relationship between Poisson’s ratio and elastic properties of an anisotropic crystal
is further investigated by Guo and Wheeler (2006) who define the criteria for ν to be
negative in terms of directional compliances and compared the analytical finding to the
experimental work of Yeganeh-Haeri et al. (1992).
The directions of auxeticity for monoclinic crystals, which have one plane of elastic
symmetry, are considered by Rovati (2004). The number of elastic constants can be
reduced to 13. Using these constants to find an expression for ν, it is shown that when
the material is stretched in a plane orthogonal to the symmetry plane, the Poisson’s ratio
in the same plane is always negative.
Brańka et al. (2009) show analytically that it is possible for some crystals to exhibit a
Poisson’s ratio as low as -5 in what they term a V3 direction, that is a direction very close
to the [111] direction but not defined by Miller indices. Branka et al. discuss the condition
by which this may be seen, based on the ratios of bulk, tetragonal shear and simple shear
moduli.
57
2.3
2.3.1
Classical atomistic modelling techniques
Introduction
Atomic forces and structure can be modelled as mathematical functions which describe
the potential energy of the system as a function of the interatomic separation (Finnis,
2003).
These functions, known as classical potential models, can be used to model the elastic
constants and lattice parameters of materials in order to determine their behaviour under
pressure, temperature, strain etc. at an atomic level. Classical potentials are based
on empirical data; the calculations are generally not computationally expensive but are
reliant on the accuracy and availability of the experimental (or sometimes quantum) data
on which they are based, for fitting parameters to be determined.
Use of classical potential models can provide the elastic properties of the crystals
under consideration, as well as insight into the mechanism of negative Poisson’s ratio
behaviour.
2.3.2
Two Body Potentials
There are many ways to classify chemical potentials, but for this study it is convenient
to group them into two-body and many-body potentials. The two-body approximation
assumes the interaction of a set of n atoms can be replaced by the sum of n(n−1) pointwise
interactions. This is often reduced to those points where the separation is less than a
specified distance, known as a cutoff; the presence of surrounding atoms does not affect
the interaction of a pair of atoms. Two-body potentials accurately capture the directional
nature of ionic or Van der Waals bonding, but are less well suited for metallic or covalent
materials. Despite this, they can be a good first approximation, and reproduce both the
structure and energy of many materials.
Several such potential models have been developed over the last century to describe
the inter-atomic potential energy of atoms. The first of these was by Morse (1929) who
derived a two-body potential, based on the forces between two neighbours, many-body
potentials followed to describe include near-neighbour effects.
58
2.3.2.1
The Morse Potential
Figure 2.3: Schematic representation of a classical potential energy function, showing
the variation of energy with interatomic spacing (arbitrary units).
Morse proposed a solution to Schrödingers’ equation based on a simplified form of the
energy function. Previous work has been based on series forms however these only work
over a small range, and contain further approximations.
Morse determined the energy function must fulfil several criteria; it must tend to a
finite value as the inter-atomic separation tends to infinity, it must tend to infinity when
r = 0 and it must have a minimum at r = r0 . The proposed potential takes the form:
U(r) = De (1 − e−2α(rij −r0 ) − 2e−α(rij −r0 ) )
(2.39)
D describes the energy, or the depth of the potential well, r0 the separation at the
minimum energy and α the width of the potential well. The Morse potential was applied
to cubic metals by Girifalco and Weizer (1959) to find the elastic constants, who concluded
that the Morse potential could be used to study lattice deformations in cubic metals where
the crystals were perfect however if defects were present, the electron distribution of the
crystal would become distorted and the derived constants would no longer be applicable.
Figure 2.3 shows an example schematic of the change in energy relative to interatomic
spacing.
59
Figure 2.4: Schematic showing (left) a two-body, and (right) three-body angular spring
potential
2.3.2.2
The Lennard-Jones Potential
Lennard-Jones (1931) derived the two-body potential based on the Van der Waals forces
in the system. A repulsive and a cohesive term are included in the model. The shape of
the potential is such that for separations of r < r0 , the potential is very steep, indicating
the strength of the repulsive interaction. The Leonard-Jones potential has been widely
used to study the properties of liquids (Broughton et al., 1982) and is a good model for
the solid states of the noble gases (Ashcroft and Mermin, 1976).
" 12 6 #
σ
σ
−
U(r) = 4
r
r
(2.40)
The ( 1r )12 term describes the repulsive element of the potential, the ( 1r )6 the cohesive
term. and σ are the depth of the potential well and distance at which the energy becomes
zero. The Leonard-Jones potential was used by Barron and Domb (1955) to determine the
stability of hexagonally close packed structures.
2.3.2.3
The Cauchy relation
Elastic constants calculated from a two-body potential are governed by the Cauchy relation. A two-body potential is only able to describe the change in energy resulting from
a change in the interatomic spacing of two atoms, there is no rotational component of
this energy. The potential is unable to account for the difference in energy change for a
strain in the shear (ε1212 ) or lateral (ε1122 ) directions and subsequently, elastic constants
predicted using a pair potential will always show s44 = s12 .
60
2.3.3
Many Body Potentials
Two-body potentials have the advantage of being comparatively easy to solve, however
they are obviously limited by only describing the forces on two adjacent atoms. To
consider the effect of clusters of atoms, many-body potentials based on near-neighbour
interactions have been developed. A comparison of pairwise and many-body forces
is conducted by Holian et al. (1991). Two dimensional simulations are also conducted
for high velocity impacts where the projectile penetrates the wall. It is found that the
formation of defects in the structure is better represented in the many-body model.
2.3.3.1
’Basic’ many-body potentials
The most simple three-body potentials depend on the change of angle between three atoms
(see figure 2.4). These can be extended to include a fourth atoms, and to include non linear
effects. Examples of these types of potential include those based solely on change of angle
such as the Linear-threebody and Three-body harmonic; those dependent solely on the
separation, such as the Urey-Bradley potentials; and those dependent on both angle and
separation such as the Axilrod-Teller potential which is particularly suited to modelling
Van der Walls forces.
1
1
1
U = k2 (θ − θ0 )2 + k3 (θ − θ0 )3 + k4 (θ − θ0 )4
2
6
24
U=
1
k4 (θ − θ0 )4
24
1
U = − (r23 − r023 )2
2


 1 + 3 cos γ cos γ cos γ 
i
j

k

U = E0 

3


ri j r jk rik
(2.41a)
(2.41b)
(2.41c)
(2.41d)
Equation 2.41a shows the three-body linear potential, 2.41b the three-body harmonic,
2.41c the Urey-Bradley and 2.41d the Axilrod-Teller. θ represents a change in angle, r the
interatomic separation, k the spring constant, γ the angle between the vectors ri j and E0
is a function of the energies of the p orbitals.
61
2.3.3.2
Embedded atom model for metals
Daw and Baskes (1983) devised a semi-empirical model based on a generalisation of
quasiatom theory, which they called the embedded atom method; each atom is viewed
as being an impurity embedded in a host lattice comprising all other atoms in the model.
Stott and Zaremba (1980) showed that the energy of an impurity in a host is a functional
of the electron density of the host, without the impurity. It can be assumed that the
embedding energy depends only on the environment around the impurity or the impurity
has a uniform local electron density.
This equation can be used to calculate the ground state properties of solids, however
Daw and Baskes used experimental data to fit their model to enable it to calculate the
elastic properties correctly. The equation is tested using a model of a slab of Nickel, a
strain is applied to the model and the result compared to that of existing elastic theory
and the data that were used to fit the model. The model is then modified to simulate
vacancies in the slab, and then include a Hydrogen atom in the slab to model Hydrogen
embrittlement.
Daw and Baskes (1984) then develop the model to include a core-core repulsion term
(modelled as a short pair-wise repulsion between cores). The assumptions that the host
density is close to the sum of the atomic densities of the constituents means that the energy
becomes a function of the position of the atoms. The lattice vectors can be calculated as a
function of the position vectors to the near neighbours.
Foiles et al. (1986) derive a set of embedding energies, pair interactions and atomic
densities to be used with the embedded atom method for 6 transition metals. These were
fitted using the most simple parameterised forms possible with the least parameters in
order to maximise the number of alloys that could be modelled using these data. The
functions were then validated by applying them to both the pure metals and alloys for
bulk and surface calculations . Comparison with experimental data shows that generally
agreement is found However in some cases, for instance calculation of surface energies,
the experimental value is around 50% higher than the calculated value.
Avinc and Dimitrov (1999) derive a generalised long-ranged potential using the
Leonard-Jones potential, with an EAM element. The advantage of this potential over
62
the Sutton-Chen potential is that it provides a better estimation of the elastic constants,
and an exact fit for the lattice parameters. This modified Leonard-Jones potential is most
applicable to FCC transition metals with computed values being within around 10% of
experiment data. When applied to BCC metals (Fe, Mo, Nb, W and V), the calculated
values around generally far lower than the experimental data, around 50% in some cases.
2.3.3.3
The Finnis-Sinclair potential
Finnis and Sinclair (1984) highlight the shortcomings of the Morse, and similar two-body
potentials. The standard way of dealing with this is to include a term dependent on
the macroscopic volume of the crystal. This leads to the problem whereby the bulk
modulus calculated from constant volume differs from the bulk modulus calculated from
a homogeneous deformation, a problem that is more apparent when internal cavities or
cracks are present in the simulation. To combat this, Finnis and Sinclair propose a model
that includes the band character of metallic cohesion. The cohesive energy is modelled
√
as z where z is atomic co-ordination depending on the structure being considered, 1 for
a diatomic molecule, 8 for a BCC crystal.
The total energy is the sum of the many-body (UN ) and pair potential (UP ) terms:
UFS = UN + UP
X
UN = −A
f (ρi )
(2.42)
(2.43)
i
UP =
1X
V(Ri j )
2
(2.44)
ij
ρi =
X
φ(Ri j )
(2.45)
j
Ri j = |Ri j | = |R j − Ri |
(2.46)
where R represents the interatomic separation, U the total energy, ρi the local charge
density, σ the atomic charge densities and V the volume of the system. Unlike the twobody potentials, it is not possible to plot the change in energy relative to interatomic
separation because of the potentials dependence on the local density.
63
The Finnis-Sinclair potential was extended by Dai et al. (2006) by extending the repulsive component. The result of this modification is that for BCC metals the pressure-volume
relationship can be more accurately modelled, and also, some of the properties of FCC
metals can be modelled, something not possible with the original formulation.
2.3.3.4
Sutton-Chen potential
The potential suggested by Sutton and Chen (1990) is a long range version of the FinnisSinclair potential, adding a long-range Van der Waals term. The potential always favours
the FCC structure over the BCC structure, and the cohesive energy of the BCC structure
is always predicted to be less than the the energy of the FCC structure. For this reason,
fitting parameters are only given for a range of FCC metals.
The potential comprises an EAM functional term, and EAM density term and a pair
potential.
Fi (ρ̄i ) = −
X
p
Ai ρ̄i
(2.47)
i
ρ̄i =
X
Cr−6
ij
(2.48)
A
r7
(2.49)
i
φi j (ri j ) =
For the EAM functional term (equation 2.47), A is a fitting parameter, expressed in eV.
For the EAM density term, C is a fitting parameter expressed in Å6 . For the pair potential
component, A is expressed in eV Å7 .
2.3.3.5
The Cleri-Rosato potential
Like the free-electron-gas model, the tight-binding model assumes electrons are independent of the atomic structure but contrary to the free-electron picture, the tight-binding
model describes the electronic states starting from the limit of isolated-atom orbitals. This
simple model gives good quantitative results for bands derived from strongly localised
atomic orbitals (Cannini, 2008).
64
The Cleri-Rosato potential uses the second-moment approximation of the tight-binding
model (Cleri and Rosato, 1993) to model the metallic nature of cohesive bonds up to the
fifth neighbour. Applied to FCC metals it is found that the elastic constants, atomic
volume and cohesive energies calculated are comparable with experimental data.
The Cleri-Rosato potential calculates the energy of the system (Ec )as the sum of the
band energy (EiB ) and the ion-ion repulsions (EiR ).
Ec =
X
EiB + EiR
(2.50)
i

 21
X




EiB = − 
ξ−2

αβ
−2qαβ rij /r −1 


0
e
j
EiR =
X
j
(2.51)
αβ
A
αβ
−Pαβ(rij /r −1)
0
αβe
(2.52)
αβ
r0 is the first nearest neighbour bonding distance, ξ is an effective hopping integral,
and q describes the dependence on the interatomic spacing. P describes the compressibility of the bulk metal.
65
2.4
2.4.1
Quantum Atomistic Modelling Techniques
Introduction
The empirical, potential based methods described in the previous section have been very
successful in calculating elastic properties of crystals, but in some cases, the explicit,
electronic nature of the interaction must be considered. This is especially true in the
case of metals, where electrons are delocalised. The electrons must be treated quantum
mechanically, and many theoretical methods have been developed to this effect. These
are often referred to as ab initio, first principles, or quantum techniques. One of the
most successful ab initio methods is Density Functional Theory (DFT). DFT has been
chosen for this study as it is well adapted to periodic crystals and is implemented in
mature, convenient to use software which allows complex analyses such as elastic constant
calculations to be carried out quite easily. As is the case for most quantum based methods,
DFT is complex and very subtle; its formalism has been described in many publications
(see for example: (Parr, 1983; Koch et al., 2001; Sholl and Steckel, 2009)). A brief description
of the methodology employed by DFT is given in Appendix A, this section provides an
overview of applications of DFT to determine the elastic properties of materials.
2.4.2
Application to auxetic media
Density functional theory has been used to investigate the structural and elastic properties
of a range of crystals. Studies employing ab intio methods to investigate auxetic behaviour
have primarily been confined to silicates and other similar framework molecules, as
discussed in section 1.1.2. The use of density functional theory to investigate the elastic
properties of cubic metals has been conducted, however these studies have generally
focussed on the structural, and on-axis properties, rather than auxetic behaviour.
2.4.2.1
Selected applications of DFT to find elastic properties
Pour et al. (2009) used DFT to investigate the properties of biprismanes. These are
polycyclic hydrocarbons, made up of an internal and external ring structure, that exhibit
negative Poisson’s ratio. It was initially suggested that this behaviour was driven by the
66
bow-tie shape of the external rings of the molecule. This would mirror the mechanism of
auxetic re-entrant honeycombs (see section 1) however due to the way that the external
rings shrink under stretching, this was not found to be the case. It was found from DFT
calculations that whilst the external rings of the structure undergo a positive Poisson’s
ratio deformation, it is the internal rings of the structure that drive the auxetic behaviour
through a change in internal angles.
Hammerschmidt et al. (2007) used both analytical modelling and DFT to determine
the elastic behaviour of InAs and GaAs crystals under a biaxial strain. Negative Poisson’s
ratio as low as -3.21 is found in the InAs for very high, non-linear strains.
Cubic metals are explored in this thesis, using both classical and quantum methods.
The partially filled d shells of the cubic transition metals complicates prediction of their
elastic properties. Wills et al. (1992) investigate the properties of the transition metals
using DFT to try to find trends in the total energies and the values for c’
c0 =
c11 − c12
.
2
(2.53)
It is found that for crystals where there is a very large difference between the energies
of the BCC and FCC configurations, the value of c’ is very high.
Iridium and its Ir3 X compounds have a face-centred cubic structure. Chen et al.
(2003) use a Generalised Gradient Approximation (see Appendix A) approach to find the
elastic constants of these structures by distorting the lattice and computing the resulting
energies. The Poisson’s ratio of the crystals is computed, and this is correlated to the ratio
of shear modulus to bulk modulus to find a measure of a material’s brittleness in terms
of Poisson’s ratio.
Sin’ko and Smirnov (2002) used ab-initio methods to model Aluminium in its BCC,
HCP and FCC phase to determine its structural and elastic properties under a range of
pressures and temperatures. Keskar and Chelikowsky (1992) explicitly explore auxetic
behaviour in α-cristobalite, comparing the results gleaned from DFT to those from a
pair potential. The Poisson’s ratios found are -0.2 and -0.17, and from these calculations
the mechanism for NPR in α-cristobalite is derived. It is shown that the mechanism is
dependent on the rigidity of the tetrahedra. This was determined by varying the rigidity
67
of the tetrahedra, straining the structure and finding that Poisson’s ratio becomes more
negative when the rigidity of the tetrahedra is increased.
2.4.3
Conclusions
Whilst no investigations have explicitly considered negative Poisson’s ratio in cubic elemental metals using DFT, studies have been conducted where auxetic behaviour of
crystals has been modelled, and where the elastic properties of cubic metals have been
calculated when modelling on-axis properties.
68
2.5
The finite-element method
The finite element method has been used to explore the properties of a variety of structures
that exhibit negative Poisson’s ratio, such as those described in section 1.1.3. A brief
explanation of the methodology of the finite-element method is given in Appendix B.
Most of the studies where the finite element method has been used to model auxetic
media have characterised the auxetic behaviour in honeycombs. The re-entrant honeycomb, first reported by Lakes (1987) has been investigated extensively. Evans et al. (1994)
modelled re-entrant cells using both analytical and finite element methods and showed
that the Poisson’s ratio of the structure could be varied from 0.6 to -1.2 dependent on the
geometry of the structure being considered. Whitty et al. (2003) used the finite element
method to investigate the effect of geometry variation on the failure method for both
conventional and auxetic honeycombs. Bezazi et al. (2008) also used the finite element
method to investigate the properties of honeycombs with differing geometries. Scarpa
et al. (2008) extended the use of this technique to auxetic tubes.
In addition to re-entrant structures, studies have investigated the mechanical properties of chiral honeycombs. The structure of chiral honeycombs, being made up of a central
cylinder and tangential ribs gives the possibility of designing a structure with a Poisson’s
ratio of -1 whilst tailoring the elastic properties. The original structure was proposed by
Prall and Lakes (1997), but more recently these structures have been investigated using
both experimental testing and the finite-element method. The finite element has been
used to investigate the buckling behaviour (Spadoni et al., 2005), the wave propagation
(Tee et al., 2010), the out of plane properties (Miller et al., 2010; Lorato et al., 2010) and
in-plane properties (Alderson et al., 2010).
In summary, from the literature surveyed it is apparent that the use of the finite
element method provides a method for investigating the structural properties of auxetic
properties in an efficient and robust manner, particularly where a range of geometries are
to be considered.
69
Chapter 3
Fundamental atomistic modelling of
the elastic properties of cubic crystals
3.1
Introduction
In this chapter, analytical expressions of the elastic constants and Poisson’s ratio of BCC
and FCC crystals are derived from simple models of interatomic interactions. In principle,
this is very similar to the discussions of Young’s modulus for simple cubic crystals found in
basic materials science textbooks (for example Ashby (2011)), but these analyses determine
the influence of the form of the potential model and the number of nearest neighbour
interactions on the bounds of Poisson’s ratio. For more realistic systems, albeit still rather
simple, the mathematical effort is already considerable and limits the analysis to second
nearest neighbours and simple potential forms.
The analytical nature of the models allow the unambiguous discovery of trends and
relationships. This section aims to explore the hypotheses that the BCC structure is inherently more auxetic than the FCC one, the inclusion of second neighbour interactions also
increases the auxeticity, while a volume based many-body contribution decreases it. This
information is very useful when interpreting the results obtained from the sophisticated
models of interaction used in the following chapters.
70
3.2
3.2.1
Modelling methodology
Introduction
Whilst numerical modelling of crystal structures allows prediction and visualisation of
the elastic behaviour, analytical modelling allows a deeper understanding of how the
potential form can affect the predicted properties and explain the results that are observed.
This section details the foundation of the classical potentials, and the strains imposed in
order to calculate the elastic properties.
3.2.2
Derivation of elastic energy
The elastic energy per unit volume is the integral of stress with respect to strain:
Eelas =
1X
1X
σi · ε j =
ci j · εi · ε j .
2
2
i,j
(3.1)
i, j
For a cubic crystal, the elastic stiffness matrix is greatly simplified, with only three
independent components, owing to the symmetry of the crystal.










ci j = 








c11 c12 c13
0
0
0
c21 c22 c23
0
0
0
c31 c32 c33
0
0
0
0
0
0
c44
0
0
0
0
0
0
c55
0
0
0
0
0
0
c66










 .









(3.2)
Therefore, the elastic energy (equation 3.1) simplifies to:
Eelas =
1 2
c11 ε1 + ε22 + ε23 + c12 (ε1 ε2 + ε2 ε3 + ε3 ε1 ) + c44 ε24 + ε25 + ε26 .
2
(3.3)
It can be seen that application of specific strains makes it possible to vary the dependence of the elastic energy, and thus to determine the elastic constants. In the cubic case,
three strains are sufficient (axial compression, tetragonal shear and rhombohedral shear),
but note that c12 can never be obtained directly by itself (it requires two strain components
71
to be non-zero, and this in turn requires c11 ).
3.2.3
Strain
To apply strain to lattice cells, it is more convenient to use the tensor notation and to
separate the direction of strain from its amplitude by using a scalar, γ, and a symmetric
matrix (order 2 tensor), T:
εαβ = γ · Tαβ .
(3.4)
The new lattice vectors coordinates, a0iα are then obtained from the original lattice
vector coordinates, aiα , by:
a0iα = aiα + γ
X
Tαβ aiβ .
(3.5)
β
The strained volume is the determinant of the matrix a0iα . The elastic energy per
volume can be rewritten in terms of the strain (transformation) matrix (Finnis, 2003),
which in turn can be shown to be the sum of the energy contributions from the axial (K),
tetragonal shear (c0 ) and rhombohedral shear (c44 ) deformations:
Eelas
K =
elas
elas
Eelas = Eelas
K + Ec0 + Ec44
(3.6)
1
(c11 + 2c12 )(Txx + T yy + Tzz )2 γ2
6
(3.7)
1
2
2
2
2
Eelas
c0 = (c11 − c12 )(Txx + T yy + Tzz − Txx T yy − T yy Tzz − Tzz Txx )γ
3
(3.8)
2
2
2
2
Eelas
c44 = 2c44 (Txy + T yx + Tzx )γ .
(3.9)
The energy can be rearranged to determine the bulk modulus,
1
K = (c11 + 2c12 ),
3
and a cubic shear modulus,
72
(3.10)
c‘ = (c11 − c12 ).
(3.11)
To determine the elastic constants for a cubic medium the bulk, tetragonal and rhombohedral shear strains can be applied, the change in energy calculated and the elastic
constants determined.
3.2.3.1
Calculation of bulk modulus
The obvious strain to obtain the bulk modulus (K) is a triaxial compression (equal compression in x, y and z). This gives the change in energy, from which the axial elastic
constants can be determined:

 1 0 0



T =  0 1 0


 0 0 1









(3.12)
thus the scaled lattice parameters become:

 a(1 + γ)
0
0



a0 = 
0
a(1 + γ)
0



0
0
a(1 + γ)









(3.13)
The determinant of this matrix is a3 (γ + 1)3 and so for this simple isotropic strain, the
energy can then be expressed in terms of the applied strain (γ) and the elastic constants
c11 and c12 :
Eelas
=
K
3
(c11 + 2c12 )γ2
2
(3.14)
or, the change in energy with respect to the change in strain:
2 elas
1 d (EK )
(c11 + 2c12 ) =
3 dγ2
73
(3.15)
3.2.3.2
Calculation of tetragonal shear modulus
In order to obtain the first, or tetragonal, shear modulus c’ , the strain

 1 0 0



T =  0 −1 0


 0 0 0









(3.16)
is applied (for example, in the case of an FCC system, figure 3.2). From this, the
strained lattice parameters are:


 a(1 + γ)
0
0 





a0 = 
0
a(1 − γ) 0  .





0
0
a 
(3.17)
The energy can then be calculated as:
Eelas
= (c11 − c12 )γ2 ,
c0
(3.18)
and the lattice parameters in terms of the energy expressed as:
(c11 − c12 ) =
3.2.3.3
1 d2 Ec‘
2 dγ2
(3.19)
Calculation of rhombohedral shear modulus
To find the three elastic constants that describe the elastic properties of cubic media, it is
necessary to find the rhombohedral shear modulus, C44 . As with the first two strains, the
strain corresponding to the rhombohedral shear deformation is applied:


 a aγ 0 






a0 =  aγ a 0 




 0 0 a 
This results in an energy of
74
(3.20)
2
Eelas
c44 = 2c44 γ
(3.21)
and as with the previous strains, the corresponding elastic constant calculated:
c44 =
3.2.3.4
1 d2 Ec44
.
4 dγ2
(3.22)
Summary: Finding elastic properties from strains
The methodology to obtain the elastic constants from a model of the interaction energy
can be summarised as four stages that are applicable to all the potentials considered:
• Apply the relevant strains described in section 3.2.3
• Express the geometric changes (distances, volume) as functions of γ
• Find the energy. The second derivatives of the energy with respect to γ then lead to
combinations of elastic constants.
• Solve these linear equations to find the elastic constants
This methodology can then be applied to a range of simple potential models to determine the predicted elastic properties of the cubic crystals.
3.2.3.5
Poisson’s ratio
Poisson’s ratio, in terms of elastic compliances, is equal to
tions,
−s012
s011
−s12
s11 ,
or for rotated configura-
. As the compliance matrix is the inverse of the stiffness matrix, the compliance
ratios can also be expressed as functions of the stiffnesses. Most elemental cubic metals
exhibit negative Poisson’s ratios, when they occur, in the [110] (and perpendicular) directions. As an aside, it should be noted that whilst experimental data usually shows auxetic
behaviour in the [110] direction, theoretically Poisson’s ratios as low as -5 can be seen for
strains in specific directions close to the [111] direction (Ting and Chen, 2005; Ting, 2004;
Brańka et al., 2009).
75
The following formulae (found using the methodology described in section 2.2.6)
enables the determination of in-plane Poisson’s ratio for an axial strain and corresponding
directions of interest.
−s12
c12
=
s11
c11 + c12
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
=
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
4c12 c44
=
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
ν[010][100] =
ν[11̄0][110]
ν[001][110]
3.2.4
(3.23)
(3.24)
(3.25)
Geometry considerations for cubic crystal systems
Potential energy models usually depend on interatomic distances or angular change and
it is therefore useful to compute these discretely in order that they can be applied in the
potential functions under consideration. This section calculates the geometry change for
face-centred and body-centred cubic crystals, and these will later be applied to section
3.3.
3.2.4.1
Face-centred cubic unit cell
Figure 3.1: A face centred cubic unit cell, showing the 4 atoms (large, numbered), and
some periodic images (small). The 12 nearest neighbour bonds of atom 4 are shown in
red, using the standard chemical convention for perspective.
76
There are 4 atoms in a FCC unit cell, and each atom has 12 first nearest neighbours (4
in each basal plane (see Fig. 3.1)). The first nearest neighbour distance is r, and for the
original cell:
√
r = r0 =
Tri-axial compression:
2
a
2
(3.26)
Tri-axial compression, for calculation of the bulk modulus is
simply a scaling of the lattice parameters:
√
r = (1 + γ)r0 =
2
(1 + γ)a
2
(3.27)
Tetragonal shear: For the first (tetragonal) shear, c0 , each basal plane must be treated
independently, thus resulting in equations 3.28-3.30 to describe the xy, xz and yz planes,
as shown in figure 3.2:
q
q
a
2
2
rxy =
(1 + γ) + (1 − γ) = r0 1 + γ2
2
r
q
a
1
rxz =
(1 + γ)2 + 1 = r0 1 + γ + γ2
2
2
r
q
a
1
r yz =
(1 − γ)2 + 1 = r0 1 − γ + γ2
2
2
Figure 3.2: Strains applied for c’ shear in an FCC structure.
77
(3.28)
(3.29)
(3.30)
Rhombohedral shear: Calculation of the second shear, c44 for the face-centred cubic
configuration has three components, for strains in the xy, xz and yz planes (see figure 3.3).
q
a
(1 + γ)2 + (1 + γ)2 = r0 (1 + γ2 )
=
2
q
a
r−xz =
(1 + γ)2 + (1 − γ)2 = r0 (1 − γ)
2
r
q
γ2
a
1 + (1 − γ)2 = r0 1 −
r yz =
2
2
r+xy
(3.31)
(3.32)
(3.33)
Figure 3.3: Strains applied for c44 shear in an FCC structure.
3.2.4.2
Body-centred cubic unit cell
Figure 3.4: A body centred cubic (BCC) unit cell, showing the 2 atoms (labelled 1 and 2),
and some periodic images (small). The 8 nearest neighbour bonds of atom 2 are shown
in red, using the standard chemical convention for perspective.
Using the methodology outlined above for the face centred unit cell, the body-centred
cubic configuration can be modelled in the same way, accounting for the change in the
primitive unit cell.
78
The body centred cubic crystal has a primitive unit cell of two atoms, which can be
visualised as being located at a vertex and centre point of a cube (labelled atoms 1 & 2 in
figure 3.4). Each atom within the structure has eight nearest neighbours, and each plane
adjacent to the central atom comprises four atoms. The first nearest neighbour distance
is therefore
√
3
r = r0 =
a.
2
Bulk modulus:
(3.34)
Applying a tri-axial compression to the body-centred cubic unit cell
gives a simple scaling of the lattice vector, to derive the bulk modulus:
√
3
a
r = (1 + γ)r0 = (1 + γ)
2
(3.35)
Figure 3.5: Strain for C‘ shear for body-centred cubic, (110) plane
Tetragonal shear: The first shear modulus, c‘, is described in terms of the (110) and
(101) planes, as shown in figure 3.5:
r110 = r101
a
=
2
r
q
2
2 + 2γ2 + 1 = r0 1 + γ2
3
(3.36)
Figure 3.6: Shear displacement for c44 , showing a) (110), b) (11̄0), c) (101) planes
79
Rhombohedral shear modulus:
To find the second shear modulus, c44 , the bonds
on the sheared (110), (11̄0)& (101) planes (see figure 3.6) must be considered separately:
r
q
2
4
r11̄0
2(1 + γ)2 + 1 = r0 1 + γ + γ2
3
3
r
q
a
2
4
r110 =
2(1 − γ)2 + 1 = r0 1 − γ + γ2 ,
2
3
3
r
q
a
2
r101 =
3 + 2γ2 = r0 1 + γ2 .
2
3
a
=
2
(3.37)
(3.38)
(3.39)
Having described the strains required to calculate the elastic properties from the
fundamental potentials, these can be applied to a volumetric, spring and Lennard-Jones
potentials and the results are considered in section 3.3.
80
3.3
Elastic properties calculated from analytical modelling
This section describes the elastic properties calculated from the simple potential models,
starting with a very simple volume dependent model, increasing in complexity to a simple
spring model and then a Lennard-Jones potential.
3.3.1
A simple model: volume dependence
To model the relationship between the energy and elastic properties of solids, a fundamental volumetric model is used:
Eelas
V =
1 kv
· (V − V0 )2
2 V0
(3.40)
where kv is an equivalent spring constant (the parameter used to describe the linear
bond stiffness) and V and, V0 , are the strained and original cell volume. For a cubic cell
of length a, the original volume V 0 is a3 . This simple model neglects the Bravais lattice
structure of the crystal (whether BCC or FCC) and as there is only one fitting parameter,
it is simple to examine the characteristics of the model.
3.3.1.1
Bulk modulus
Under tri-axial compression, the new volume is simply obtained from equation 3.13, and
thus the energy can be found by substituting this into equation 3.40:
Eelas
K =
V1 = a3 (1 + γ)3
(3.41)
kv
· (a3 (1 + γ)3 − a3 )2
2a3
(3.42)
A Taylor expansion yields:
9 3 2
a kv γ + . . .
2
Equating this with equation 3.14 gives
81
(3.43)
(c11 + 2c12 ) = 3a3 kv .
3.3.1.2
(3.44)
Tetragonal shear
The same process is repeated for the first shear modulus with the associated tetragonal
strain:
V0 = a3 (1 + γ)(1 − γ).
(3.45)
The energy can be found from equation 3.18
Eelas
c0 =
kv
· (a3 (1 + γ)(1 − γ) − a3 )2
2a3
−1 3 4
Eelas
a kv γ ,
c0 =
2
(3.46)
(3.47)
Recalling equation 3.21, it is apparent that there is no term in γ2 , and thus no contribution from this potential energy form to the c‘ shear:
3.3.1.3
(c11 − c12 )γ2 = 0.
(3.48)
V0 = a3 (1 − γ2 )
(3.49)
Rhombohedral shear
Eelas
c44 =
kv
· (a3 (1 − γ2 ) − a3 )2
2a3
1 3 4
Eelas
c44 = a kv γ .
2
(3.50)
(3.51)
As with the first shear derivation, expanding equation 3.50 it is apparent that there
is no term in γ2 , so again there is no contribution from this potential energy form to the
shear, and c44 = 0.
82
3.3.1.4
Poisson’s ratios
Equations 3.14, 3.47 and 3.50 can be solved to obtain:
c11 = c12 = a3 kv
(3.52)
c44 = 0
(3.53)
Using these values the Poisson’s ratios for specific strains can be found :
s12
c12
1
=
=
s11
c11 + c12 2
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
= −1
ν[11̄0][110] =
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
4c12 c44
ν[001][110] =
=0
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
ν[010][100] =
3.3.1.5
(3.54)
(3.55)
(3.56)
Material properties from volume potential
The volume potential gives only a primitive representation of material properties but
is still effectively a many-body potential, being a function of the homogenised material
rather than describing specific inter-atomic interactions.
The elastic constants generated from this model are not governed by the Cauchy
relation, however there is no shear rigidity in this model as it only describes the energy
under a volume change. A pure shear conserves volume, hence c44 = 0.
The axial (c11 ) and lateral (c12 ) components are equal, as the model is unable to account
for the mode of the volume change. The negative Poisson’s ratio for the ν(11̄0,110) direction
shows that the ’default’ behaviour for the simple potential is to predict auxetic behaviour
even when the relationships between the axial and tangential stiffnesses are not fully
defined.
83
3.3.2
A simple pair potential: The spring model
This potential assumes that the bonds can be represented by linear springs. It is the
simplest way to represent pairwise bonded interactions, and the energy (per volume) of
a bond is given by Hooke’s Law:
Eelas
=
r
3.3.2.1
1 kr
· (r − r0 )2
2 V0
(3.57)
Application to a face-centred cubic unit cell
As described in section 3.2.4.1, there are 4 atoms in a one primitive face-centred cubic unit
cell, and each has 12 first neighbours. Each bond is shared between two atoms, thus there
are 24 bonds, 8 in each basal plane.
3.3.2.1.1
Bulk modulus:
By replacing r with a triaxial strain (see section 3.2.4.1), the energy for this strain, and thus
the bulk modulus are derived:
Eelas
r
3.3.2.1.2
√ !2
1 kr
kr
2
kr
2
= 24 ·
· (r − r0 ) = 12 3
a (1 + γ − 1)2 = 6 γ2
2 V0
2
a
a
kr
3
(c11 + c12 )γ2 = 6 γ2
2
a
kr
(c11 + c12 ) = 4
a
(3.58)
(3.59)
(3.60)
Tetragonal shear:
The shear strain, as described in section 3.2.4.1, has components to describe the strains in
the xy, xz and yz directions.
Eelas
r

 r
2
 r
2 
q
2





 
1
1
1 kr 



=
8 r0 1 + γ2 − r0 + 8 r0 1 + γ + γ2 − r0  + 8 r0 1 − γ + γ2 − r0  

2 V0 
2
2
(3.61)
84
Expanding this Taylor series gives:
(c11 − c12 ) =
3.3.2.1.3
kr
a
(3.62)
Rhombohedral shear:
For the strains described in section 3.2.4.1, there are three strains that are applied. These
are then summed to give the total energy:
Eelas
r
1 kr
=
2 V0

2 
 r

 

1


2
2
4 r0 (1 + γ) − r0 + 4 r0 (1 − γ) − r0 + 16 r0 1 + γ2 − r0  


2
Eelas
=
r
3.3.2.1.4
1
(4γ2 + 4γ2 )
4a
kr
c44 =
a
(3.63)
(3.64)
(3.65)
Poisson’s ratio:
Having determined the relationship between the bulk modulus and shear modulus, the
elastic constants can be found from equations 3.60, 3.62 & 3.65:
c11 = 2
kr
a
c12 = c44 =
(3.66)
kr
a
(3.67)
The simple spring model is a two-body potential and thus has no rotational component, resulting in the Cauchy relation being obeyed. From the elastic constants, the
analytical values of Poisson’s ratio for the directions shown in section 3.3.1.4 are
c12
1
=
c11 + c12 3
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
=0
ν[11̄0][110] = −
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
4c12 c44
1
ν[001][110] =
= .
2c11 c44 + (c11 − c12 )(c11 + 2c12 ) 2
ν[010][100] =
85
(3.68)
(3.69)
(3.70)
3.3.2.2
Application to the body-centred cubic unit cell
The same paradigm can be applied to the body-centred cubic unit cell, to give the elastic
properties in terms of the pair potential:
3.3.2.2.1
Bulk modulus:
Each atom within the body-centred cubic unit cell has eight nearest neighbours, at a
√
spacing of
3
2 a,
so for a triaxial compression
√
Eelas
r
3.3.2.2.2
!2
3
a (1 + γ − 1)2
2
kr
Eelas
= 3 γ2
r
a
3
kr
(c11 + 2c12 )γ2 = 3 γ2
2
a
kr
(c11 + 2c12 ) = 2
a
kr
=8 2
2a
(3.71)
(3.72)
(3.73)
(3.74)
Tetragonal Shear modulus:
Eelas
r
kr
=4 3
a
r
2


2
 1 + γ2 − 1


3
(3.75)
Expanding this as a Taylor series, there is no term in γ2 ,
E=
4 k γ4
+ ...
9 a3
(3.76)
and thus c11 − c12 = 0.
3.3.2.2.3
Eelas
r
Rhombohedral shear modulus:
  r
2
 r
2
 r








1 
4
2
4
2 
2 



= 2 r0 1 + γ + γ2 − r0  + 2 r0 1 − γ + γ2  + 4 r0 1 + γ2  (3.77)

2
3
3
3
3
3
2c44 γ2 =
c44 = c12
86
4 kr 2
γ
3a
2 kr
=
3a
(3.78)
(3.79)
3.3.2.2.4
Poisson’s ratio:
From equations 3.74, 3.76 & 3.79, we find the elastic constants are equal for this model,
and thus the Poisson’s ratios can be calculated:
c11 = c12 = c44
c12
1
=
c11 + c12 2
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
= −1
ν[11̄0][110] = −
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
4c12 c44
ν[001][110] =
= 2.
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
ν[010][100] =
3.3.2.3
(3.80)
(3.81)
(3.82)
(3.83)
Analysis
The spring model is a more sophisticated model than the volume model, as it includes
terms to describe the deformations of specific interatomic attractions in the crystal, hence
having a different form for the body-centred and face-centred cubic lattices.
The face-centred cubic crystal model predicts elastic constants which obey the Cauchy
relation; the body-centred cubic form predicts that c11 = c44 = c12 . The predicted Poisson’s
ratios calculated from equations 3.81, 3.82 and 3.83 show that for the face-centred cubic
unit cell, the predicted Poisson’s ratio in the (110,11̄0) direction is zero, not negative,
despite the Cauchy relation being obeyed. This is consistent with the theory that the
negative Poisson’s ratio behaviour in the face centred cubic crystal is not directly governed
by first-nearest neighbour axial interactions, but by the rotational rigidity of the bonds,
something which is not accounted for in this model (This effect is explored in depth
using numerical modelling techniques, and described in chapter 4). The magnitude of
the stiffness of the interactions is also greater for the face-centred cubic cell than the
body-centred cubic cell, presumably owing to the greater number of interactions.
3.3.2.4
Combination of the spring and volumetric potentials
The volumetric potential alone is not able to successfully predict Poisson’s ratios, however,
by combining the spring and volume potentials the many-body character of the volume
87
potential can offset the Cauchy relation characteristics of the spring potential.
Using the calculated values for the volume and spring potentials for each lattice, the
ratios of the elastic constants can be simplified. If the potential parameters are described
by the term ε, the elastic constants for the face-centred cubic become
c11 = a3 kv + 2
kr
= 1 + 2ε
a
kr
=ε
a
kr
= a3 kv + = 1 + ε
a
1
ν(11̄0,110) =
4
c44 =
c12
(3.84)
(3.85)
(3.86)
(3.87)
This indicates that by adding a many-body component to the potential form, the predicted
Poisson’s ratio changes from negative to positive.
88
3.3.3
The Lennard-Jones Potential
The Lennard-Jones (1931) potential is a non-linear, two-body potential which describes
the interatomic potential energy as:
Er =
B
−A
+ 12
r6
r
(3.88)
Where A and B are fitting parameters, and r is the interatomic separation. The
Lennard-Jones potential is an anharmonic potential that accounts for the fact that there is
both an attractive and repulsive component in interatomic bonding. Given that only two
parameters govern the potential, elastic constants can be derived analytically.
3.3.3.1
Face Centred Cubic cells
3.3.3.1.1
First nearest neighbours
Initially, the potential was truncated such that only first nearest neighbours interact. In
this case, the total energy (for a cell containing 4 atoms) is



 −A
B
Er = 24  6 + 12 
r0
r0






B 
 −A

Er = 24  √ 6 + √ 12 
 2

2


2 a
2 a
(3.89)
(3.90)
where a is the lattice parameter and r0 is the first nearest neighbour bond distance (for
√
FCC, r0 =
2
2 a).
To find the equilibrium lattice parameter a1nn , the derivative of the energy is set to
zero. The first derivative of the interatomic energy with respect to strain is the inter
atomic force. At equilibrium the force is zero thus:


dE  6 · A 12 · B 
=
− 13 
dr0  r70
r0
and
89
(3.91)
a1nn
√
= 2
r
6
2B
A
(3.92)
The strains applied for the previous potentials under consideration can be applied in
the same manner, using the lattice parameter calculated in equation 3.92.
Bulk Modulus strains:
r
B
=4
A
r 6 2B
r1 = γ + 1
A


2


A2
A

E1 = 24 
6 
12 −
4 γ+1 B 2 γ+1 B
V1nn =
E1 = −
a31nn
6 A2 216 A2 γ2 1512 A2 γ3
+
−
B
B
B
216A2
3
(c11 + 2c12 ) =
2
B
144A2
(c11 + 2c12 ) =
B
(3.93)
(3.94)
(3.95)
(3.96)
(3.97)
(3.98)
Tetragonal shear:
r q
q
6 2B
1 + γ2
r1xy = r0 1 + γ2 =
A
r
r r
γ2
γ2
6 2B
r1xz = r0 1 + γ +
=
+1
γ+
2
A
2
r
r r
γ2
γ2
6 2B
r1yz = r0 1 − γ +
=
−γ +
+1
2
A
2
90
(3.99)
(3.100)
(3.101)

 

 
 −A
B   −A
B   −A
B 

E1 = 8  6 + 12  +  6 + 12  +  6 + 12 
r1xy r1xy
r1xz r1xz
r1yz r1yz






A2
A2
A2



+
− E1 = 8 −
3 . . . 
2
 2 γ2 + 13 B 4 γ2 + 16 B
γ


+γ+1 B
2
(3.102)
(3.103)
2
E1 = −
6 A2 36 A2 γ2 195 A2 γ4
+
+
+ ...
B
B
B
A2
c11 − c12 = 36 .
B
(3.104)
(3.105)
Rhombohedral shear:
r1xyp = r0 (1 + γ)
r1xym = r0 (1 − γ)
r
γ2
r1x = r0 1 +
2

 
 


 −A

  −A

−A
B
B
B


E1 = 4  6
+ 12  +  6
+ 12  +  6 + 12 
r1xyp r1xyp
r1xym r1xym
r1x r1x







2
2



A2
A
A2
A


 + 4 

E1 = 16  − −



6
3
6
12
 γ2

γ2
2
γ
+
1
B
4
γ
+
1
B
4

2
+
1
B
+
1
B
2
2


2
2


A
A
+4 
6 
12 −
4 1−γ B 2 1−γ B
E1 = −
6 A2 72 A2 γ2
+
+ ...
B
B
A2
c44 = 36
B
91
(3.106)
(3.107)
(3.108)
(3.109)
(3.110)
(3.111)
(3.112)
Poisson’s ratio:
c11 =
2A2
q
B AB
A2
q
c12 = c44 =
B
(3.113)
(3.114)
B
A
1
c12
=
c11 + c12 3
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
ν[11̄0][110] = −
=0
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
1
4c12 c44
= .
ν[001][110] =
2c11 c44 + (c11 − c12 )(c11 + 2c12 ) 2
ν[010][100] =
3.3.3.1.2
(3.115)
(3.116)
(3.117)
Second nearest neighbours
The interatomic distance varies if the Lennard-Jones is also allowed to act between the
second nearest neighbours. Therefore, the equilibrium lattice parameters must first be
calculated. Recalling the structure of the face-centred cubic unit cell, there are 24 first
√
and 12 second nearest neighbour bonds to be considered at a separation of
2
2 a
and a
respectively. The strains applied to calculate the properties for the first nearest neighbour
interactions can be applied to find the second nearest neighbour interactions, and the
geometry change of these bonds summed with those of the first-nearest neighbour bonds
to find the total energy.
E2nn








 A
A
B
B 



= 24 − √
6 + √
12  + 12 − 6 + 12 

a2nn a2nn

2
2

2 a2nn
2 a2nn


 (2 · 8 + 1)A (2 · 64 + 1)B 

+
E2nn = 12 −

a62nn
a12
2nn
(3.118)
(3.119)
From the first derivative of this, the interatomic spacing is:
r
a2nn =
6
92
258B
17A
(3.120)
Comparing this to equation 3.92 it is evident adding nearest neighbours contracts the
cell. This revised lattice parameter can be used to calculate the energies of the strained
crystal so that the effect off adding a greater number of nearest neighbour bonds in the
model can be observed.
Bulk modulus:
Applying the relevant strains, the energy component for the bulk modulus, from both
the first and second nearest neighbour can be found. The distances for the first, r1 , and
second nearest neighbour, r2 , under a strain γ can be found.
q
6
r1 =
q
6
r2 =
258 AB (γ + 1)
√ √
6
2 17
(3.121)
258 AB (γ + 1)
√
6
17
(3.122)
The total energy is is the sum of the energy of the first and second nearest neighbour
energies:
Etot


= 12 
Enn1 =
−A
B
4624 A2
68 A2
+
=
−
6
r61
r12
16641 γ + 1 12 B 129 γ + 1 B
1
(3.123)
Enn2 =
−A
B
289 A2
17 A2
+
=
−
6
r62
r12
66564 γ + 1 12 B 258 γ + 1 B
2
(3.124)
Etot = Enn1 + Enn2
(3.125)



2
2



289 A2
17 A2
4624
A
68
A
6  + 24 
6 
12 −
12 −
66564 γ + 1 B 258 γ + 1 B
16641 γ + 1 B 129 γ + 1 B
(3.126)
Expanding this as a Taylor series yields
93
Etot
289 A2 10404 A2 γ2 72828 A2 γ3
=−
+
−
+ ...
43 B
43 B
43 B
3
10404 A2
(c11 + 2c12 ) =
2
43B
(3.127)
(3.128)
Tetragonal shear modulus:
The shear strains (rnm ) can then be applied to each of the shear planes, to find the
resulting energies (Enm ), which can then be summed to find the total energy.
q
γ2 + γ + 1
rxy =
√ √
6
2 17
q
q
γ2
6 258B
A
2 +γ+1
rxz =
√ √
6
2 17
q
q
γ2
6 258B
A
2 −γ+1
r yz =
√ √
6
2 17
q
q
q
6 258B
6 258B
6 258B
(1
+
γ)
(1
−
γ)
A
A
A
rx =
ry =
rz = √
√
√
6
6
6
17
17
17


2
2

4624 A
68 A


Exy = 8 
−

6
3
16641 γ2 + 1 B 129 γ2 + 1 B






4624 A2
68 A2


Exz = 8 
−
2
6
2
3 

 16641 γ + γ + 1 B 129 γ + γ + 1 B 
2
2
6
258B
A
p
94
(3.129)
(3.130)
(3.131)
(3.132)
(3.133)
(3.134)
E yz






68 A2
4624 A2


−
= 8 
2
6
2
3 

 16641 γ − γ + 1 B 129 γ − γ + 1 B 
2
2




17 A2
289 A2

Ex = 4 
6 
12 −
66564 γ + 1 B 258 γ + 1 B




17 A2
289 A2

E y = 4 
6 
12 −
66564 1 − γ B 258 1 − γ B
Ez = −
4097A2
16641B
Etot = Ex + E y + Ez + Exy + Exz + E yz
(3.135)
(3.136)
(3.137)
(3.138)
(3.139)
A Taylor expansions reduces this to
Etot = −
289 A2 61064 A2 γ2 369410 A2 γ4
+
+
+ ...
43 B
1849 B
1849 B
(3.140)
From this,
(c11 − c12 ) =
61064 A2
1849 B
(3.141)
Rhombohedral shear modulus
q
6
258 AB (1 + γ)
2
rxyp =
ann2 (1 + γ) =
√ √
6
2
2 17
q
6
√
258 AB (1 − γ)
2
rxym =
ann2 (1 − γ) =
√ √
6
2
2 17
q
q
r
γ2
6
√
B
258 A 2 + 1
2
1 2
r yz =
ann2 1 + γ =
√ √
6
2
2
2 17
q
6
258 AB
rz = ann2 = √
6
17
q
p
6
q
258 AB γ2 + 1
rx = ann2 1 + γ2 =
√
6
17
√
95
(3.142)
(3.143)
(3.144)
(3.145)
(3.146)
The energy for each strain can then be found, these summed and expanded as a Taylor
series:
289 A2 474368 A2 γ2 13929766 A2 γ4
+
+
+ ...
−
43 B
5547 B
5547 B
237184A2
c44 =
5547 B
(3.147)
(3.148)
Poisson’s ratio:
420310A2
5547B
237184A2
= c44 =
5547B
c11 =
c12
(3.149)
(3.150)
c11 ' 1.7 · c44
(3.151)
c12
= 0.36
c11 + c12
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
ν[11̄0][110] = −
= −0.09
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
4c12 c44
= 0.6.
ν[001][110] =
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
ν[010][100] =
3.3.3.2
3.3.3.2.1
(3.152)
(3.153)
(3.154)
Body centred cubic cells
First nearest neighbours
√
For the body-centred cubic cell the first nearest neighbour bond distance is
3
2 a.
Each
atom has eight nearest neighbour bonds, and 6 second nearest neighbour bonds. There are
two atoms in the primitive unit cell. The energy for the first nearest neighbour interaction
is:
Enn1 = 8
−A
B
+ 12
6
a
a
From the first derivative of the energy:
96
(3.155)
r
r0 =
6
2B
A
(3.156)
The strains for bulk, tetragonal and rhombohedral shear can then be applied, as with
the face centred unit cell.
Bulk modulus strains:
r
6 2B
r1 = r0 (1 + γ) = γ + 1
A




2
 −A


B 
A2
A


E1 = 8  6 + 12  = 8 
6 
12 −
r1
r1
4 γ+1 B 2 γ+1 B
E1 = −
2 A2 72 A2 γ2 504 A2 γ3
+
−
+ ...
B
B
B
72A2 γ2
3
(c11 + 2c12 ) =
2
B
48A2 γ2
(c11 − c12 ) =
B
(3.157)
(3.158)
(3.159)
(3.160)
(3.161)
Tetragonal shear modulus:
r1p =
a1nn
p


E1 = 8 
q
2 γ2 + 3
= 2 γ2 + 3
2
A
B
E1 = 8 6 + 12
r1p r1p
729 A2
r B √
6
2
3
A
27 A2
6 −
3
4 2 γ2 + 3 B 2 2 γ2 + 3 B
E1 = −
(3.162)
(3.163)




2 A2 8 A2 γ4
+
+ ...
B
B
(3.164)
(3.165)
There is no γ2 term in the energy for the tetragonal shear
c11 − c12 = 0.
Rhombohedral shear modulus:
97
(3.166)
q
2 + 4 γ + 3 6 2B
2
γ
A
a1nn
r1p =
√
2
3
q
p
6
2
q
2 γ − 4 γ + 3 2B
A
a1nn
r1m =
3 − 4γ + 2γ2 =
√
2
3




 −A −B 
 −A −B 


E1 = 4  6 + 12  + 4  6 + 12 
r1p
r1m r1m
r1p


2
2


27
A
729
A
E1 = 4 
6 −
3 
2
2
4 2γ + 4γ + 3 B 2 2γ + 4γ + 3 B


2
2


27
A
729
A

−
+4 
6
3 
4 2 γ2 − 4 γ + 3 B 2 2 γ2 − 4 γ + 3 B
p
q
3 + 4γ + 2γ2 =
E1 = −
2 A2 32 A2 γ2
+
+ ...
B
B
A2
c44 = 16
B
(3.167)
(3.168)
(3.169)
(3.170)
(3.171)
(3.172)
Poisson’s ratio:
From equations 3.161, 3.166 & 3.172, the elastic constants can be found, and Poisson’s
ratios derived.
c11 = c12 = c44
1
c12
=
c11 + c12 2
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
= −1
ν[11̄0][110] = −
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
4c12 c44
= 2.
ν[001][110] =
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
ν[010][100] =
3.3.3.2.2
(3.173)
(3.174)
(3.175)
(3.176)
Second nearest neighbours
As with the first nearest neighbours, to find the lattice parameter for the the crystal
including the second nearest neighbour interactions it is necessary to find both the first
and second nearest neighbour separation:
98

 −A

= 6  2 6 +
 √ r
0



 −A

B 
B


Enn2
+
8
+




2 12 
√
r60
r12
r0
0
( 3
( 3
!
!
6 A 12 B
81 A
2187 B
dE
+8
=6
−
− 13
da
32 r0 7 1024 r0 13
r0 7
r0
r
6 (55713 · 16 B)
r0 =
(1011 · 512 A)
q
√
3
6
379B
2
7
A
2
a2nn = √ r0 = √ √
6
3
3 10784
(3.177)
(3.178)
(3.179)
(3.180)
Bulk modulus:
The strain for bulk modulus can then be applied using the same methodology as for
the face-centred cubic unit cell, with the revised separation from equation 3.179:
√
3
q
7(γ + 1) 379B
A
r1 = r0 (1 + γ) =
√
6
10784
q
√
3
6
27(γ + 1) 379B
A
r2 = a2nn (1 + γ) =
√ √
6
3 10784







 −A
−A
B
B


E2 = 8  6 + 12  + 6  6 + 12 
r1
r2
r1
r2


2
2


82791801
A
9099
A
E2 = 6 
6 
12 −
1379528164 γ + 1 B 37142 γ + 1 B


2
2


116294656
A
10784
A
+8 
6 
12 −
344882041 γ + 1 B 18571 γ + 1 B
E2 = −
6
113569 A2 2044242 A2 γ2 2044242 A2 γ3
+
−
+ ...
37142 B
18571 B
2653 B
3
2044242A2
(c11 + 2 c12 ) =
2
18571B
99
(3.181)
(3.182)
(3.183)
(3.184)
(3.185)
(3.186)
Tetragonal shear modulus:
q
p
√
3
2+3 6 B
7
2γ
A
a2nn
r1p =
√ √
6
2
3 10784
q
√
3
6
7(γ + 1) AB
rx = a2nn (1 + γ) = √ √
6
3 10784
q
√
3
6
7(1 − γ) AB
r y = a2nn (1 − γ) = √ √
6
3 10784
q
√
3
6
7 AB
rz = a2nn = √ √
6
3 10784




!
!
 −A
 −A B 
−A
−A
B 
B
B



E2 = 8  6 + 12  + 2 6 + 12 + 2  6 12  + 2 6 + 12
r1p r1p
rx
ry ry
rz
rx
rz
q
3 + 2 + γ2 =
E2 = −
113569 A2 165201444 A2 γ2
−
+ ...
37142 B
49268863 B
−165201444A2
c11 − c12 =
49268863B
(3.187)
(3.188)
(3.189)
(3.190)
(3.191)
(3.192)
(3.193)
Rhombohedral shear modulus:
q
p
√
3
6
2
7 2γ + 4γ + 3 379B
A
a2nn
r1p =
3 + 4γ + 2γ2 =
√ √
6
2
3 10784
q
p
√
3
2 − 4γ + 3 6 379B
q
7
2γ
A
a2nn
r1m =
3 − 4γ + 2γ2 =
√ √
6
2
3 10784
q
p
√
3
6
2
q
7 2γ + 1 379B
A
r2xy = a2nn 1 + γ2 =
√ √
6
3 10784
q
100
(3.194)
(3.195)
(3.196)
q
√
3
6
7 379B
A
rz = a2nn = √ √
6
3 10784









 −A
 −A


 −A
 −A
B 
B
B
B



E2 = 8  6 + 12  + 4  6 + 12  + 4  6 + 12  + 2  6 + 12 
r1p r1p
r1m r1m
r1xy r1xy
r1z
r1z
E2 = −
(3.197)
(3.198)
2780889963 A2 517114368 A2 γ 26206931712 A2 γ2 14544769024 A2 γ3
−
+
−
+ ...
689764082 B
344882041 B
344882041 B
147806589 B
(3.199)
2c44 =
26206931712 A2 γ2
344882041 B a32nn
(3.200)
Poisson’s ratio:
c12
≈ 0.53
c11 + c12
2c11 c44 − (c11 − c12 )(c11 + 2c12 )
ν[11̄0][110] = −
≈ −1.55
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
4c12 c44
ν[001][110] =
≈ 2.93
2c11 c44 + (c11 − c12 )(c11 + 2c12 )
ν[010][100] =
3.3.3.3
3.3.3.3.1
(3.201)
(3.202)
(3.203)
Discussion
First nearest neighbours
The Lennard-Jones potential includes terms for both the attractive and repulsive atomic
interactions. Considering only the first nearest neighbour bonds, the properties calculated
show that for face-centred cubic cells, negative Poisson’s ratio is not predicted in the
(11̄0, 110) direction, but the predicted value is zero. The predicted elastic constants for the
body-centred cubic cell are all equal. This mimics the result of the simple spring potential,
and it can be seen that when considering only the first nearest neighbour interactions the
predicted Poisson’s ratio in the off-axis direction is
−1
2 .
These results again show that the
first nearest neighbour bonding in the face-centred cubic cell does not give rise to auxetic
behaviour. For the body-centred cubic cell, negative Poisson’s ratio is predicted when
considering the only the first nearest neighbour bonding.
101
3.3.3.3.2
Second nearest neighbours
Considering the second nearest neighbour interactions, the calculated properties differ
from those found when only the first nearest neighbours are considered. The predicted
values for the face centred cubic unit cell show c11 > c44 and c44 = c12 . The Cauchy
relation is respected, and the crystal shows greater axial stiffness than shear stiffness. The
predicted Poisson’s ratio for the (100,010) directions is positive, however, the Poisson’s
ratio for the (11̄0, 110) direction is negative (-0.09). This supports the theory that the
negative Poisson’s ratio behaviour in the face-centred cubic metals is contingent not on
the first nearest neighbour bonding, but on the second and third nearest neighbour bonds.
The calculated values for the body-centred cubic unit cell show c11 < c12 . This leads
to a Poisson’s ratio of -1.55 for the (11̄0, 110) direction, with an associated very high (2.9)
corresponding orthogonal positive strain. This result is perhaps counter intuitive; it is
expected that negative Poisson’s ratio behaviour in the body-centred cubic cell can be
attributed to the first nearest neighbour bonding, however this result shows how adding
additional nearest neighbours accentuates this.
3.3.3.3.3
Influence of potential range
It is implied from the analysis of the Lennard-Jones potential that the range of the potential
has a greater effect on the calculated properties than any other parameter. To explore this
numerical analysis was conducted computationally (further explanation of this technique
is presented in chapter 4). It can be seen from section 3.3.3.2.2 that using just two
nearest neighbours results in a lengthy calculation, adding further nearest neighbours
becomes increasingly so, even with the aid of a symbolic mathematics program. An
arbitrary face-centred cubic unit cell is used as a starting point for this analysis. The
cutoff of the potential is varied so that the interactions up to the ninth nearest neighbour
are modelled. The calculated (11̄0, 110) Poisson’s ratio, calculated from the numerical
methods, is shown in figure 3.7. The change in interatomic separation is shown in figure
3.8. It can be seen that adding more neighbours contracts the cell, as has been shown
with the analytical modelling. The variation in Poisson’s ratio mimics the change when
neighbours are added, and the Poisson’s ratio converges to less than 1% variation with 5
102
nearest neighbours, and less than 10% variation with two nearest neighbours. Repeating
this analysis (figures 3.9 and 3.9) shows the BCC crystal system mirrors this.
Figure 3.7: Variation of the (11̄0, 110) Poisson’s ratio with change in cutoff, for an
arbitrary FCC system.
Figure 3.8: Variation of the lattice constant (normalised relative to the lattice constant for
the first nearest neighbour value) with change in cutoff, for an arbitrary FCC system.
103
Figure 3.9: Variation of the (11̄0, 110) Poisson’s ratio with change in cutoff, for an
arbitrary BCC system.
Figure 3.10: Variation of the lattice constant (normalised relative to the lattice constant
for the first nearest neighbour value) with change in cutoff, for an arbitrary BCC system.
104
3.4
3.4.1
Conclusions
Summary of calculated properties
Potential
c11
c44
c12
Volume
a3 kv
0
a3 kv
Spring - FCC
2 kar
kr
a
kr
a
Spring - BCC
2 kr
3a
2 kr
3a
2 kr
3a
Lennard-Jones - FCCnn1
2
2A
√
B AB
2
A
√
Lennard-Jones - BCCnn1
16 AB
Lennard-Jones - FCCnn2
2
75 AB
Lennard-Jones - BCCnn2
22.2 AB
B
2
16
2
2
A
√
B
A
A2
B
16
B
B
A
A2
B
2
42.7 AB
42.7 AB
2
25.7 AB
25.7 AB
2
2
Table 3.1: Elastic constants calculated for each of the potential models
ν(010,100)
ν(11̄0),(110)
ν(001,110)
Volume
1
2
-1
0
Spring - FCC
1
3
0
1
2
Spring - BCC
1
2
-1
2
Lennard-Jones - FCCnn1
1
3
0
1
2
Lennard-Jones - BCCnn1
1
2
-1
2
Lennard-Jones - FCCnn2
0.36
-0.09
0.62
Lennard-Jones - BCCnn2
0.53
-1.55
2.93
Potential
Table 3.2: Poisson’s ratio calculated for each of the potential models using the elastic
constants shown in table 3.1
It has been demonstrated that simple two-body potentials exhibit negative Poisson’s
ratio for strains in the (11̄0, 110) direction. The predicted values are sensitive to number
of nearest neighbours included in the analysis. The Bravais lattice of the crystal has an
effect on both the interatomic separation and the number of nearest bonds in the cell, and
this in turn changes the way in which the potential predicts the elastic properties.
105
Chapter 4
Atomistic modelling of the elastic
properties of cubic crystals
4.1
Introduction
This chapter describes the numerical investigation of the elastic properties of crystalline
materials. The primary aim of this study is to evaluate whether classical atomistic modelling techniques are able to predict negative Poisson’s ratio behaviour in cubic elemental
metals. Both classical potentials and quantum mechanical modelling are considered in
this thesis. In this chapter the numerical results of the classical studies are used to investigate the underlying cause of auxetic behaviour, and to provide a template for the finite
element modelling investigation of scale independent beam networks (see chapter 6).
4.1.1
Elemental cubic metals
Elemental cubic metals are the focus of this chapter as the elastic constant data for these
materials is readily available and the existence of negative Poisson’s ratio behaviour is
well documented, allowing the techniques used to be validated. What is not explicitly
known is why some crystals exhibit this behaviour whilst those that are extremely similar
in structure do not.
The elastic constant data from Landolt-Bornstein (1985), a compilation of experimentally derived elastic constants, is used to calculate the experimentally derived Poisson’s
106
ratios for elemental cubic metals. The experimental data is published for only the principal
crystallographic orientations, however as discussed in section 1.1.2.1, negative Poisson’s
ratio is usually observed in off-axis directions. It is possible to use the on-axis data to
compute the Poisson’s ratio for all directions as shown in section 2.2.6, however to do this
manually is very laborious, as all angles must be considered, and even with spreadsheets,
this is very time consuming. For convenience, the program ElAM (Marmier et al., 2010)
is used to perform these calculations . ElAM provides an automated methodology to
calculate elastic properties in all directions (for a specified resolution) and report the maximum and minimum values, and there associated directions. The calculated maximum
and minimum Poisson’s ratios are shown in table 4.1.
4.1.2
Covalently bonded zincblende compounds
In addition to the elemental cubic metals that are the main focus of this work, the properties of covalently bonded compounds are investigated. The focus of the analysis is to
determine whether there is a link between the axial and rotational bonding within the
crystal and the resulting Poisson’s ratio. Covalent bonding is far more directional in character than the metallic bonding of the elemental metals considered. Whilst it would have
been logical to use covalently bonded BCC and FCC structures, the analysis depends on
having experimentally measured elastic properties with which to validate the modelling
and thus the zincblende structures of these compounds were used in the analysis as this
data.
107
Element
Symmetry
Group
νmin
νmax
Cs
bcc
1
-0.46
1.22
Na
bcc
1
-0.44
1.20
K
bcc
1
-0.43
1.21
Rb
bcc
1
-0.40
1.15
Ba
bcc
2
-0.29
0.82
Ca
fcc
2
-0.26
0.89
Sr
fcc
2
-0.26
0.85
Ta
bcc
5
0.17
0.50
Nb
bcc
5
0.21
0.61
V
bcc
5
0.29
0.45
Cr
bcc
6
0.13
0.32
Mo
bcc
6
0.21
0.40
W
bcc
6
0.28
0.28
Fe
bcc
8
-0.06
0.62
Rh
fcc
9
0.07
0.44
Ir
fcc
9
0.10
0.38
Ni
fcc
10
-0.07
0.67
Pd
fcc
10
-0.05
0.81
Pt
fcc
10
0.21
0.57
Cu
fcc
11
-0.13
0.82
Ag
fcc
11
-0.09
0.83
Au
fcc
11
-0.04
0.88
Al
fcc
13
0.27
0.42
Pb
fcc
14
-0.19
1.01
Table 4.1: Experimentally determined Poisson’s ratio maxima and minima for elemental
cubic metals (listed according to the 1990 IUPAC group system (Fluck, 1988)). Data
taken from (Landolt-Bornstein, 1985).
108
4.2
4.2.1
Classical Potentials: Methodology
Introduction
Classical potentials are used to find the energy of the crystals under consideration and
from this calculate the elastic properties of a range of elemental cubic metals. Simple
potentials have been used in chapter 3 to investigate how the Bravais lattice and the
number of nearest neighbours in the model affect the predicted properties of the crystal.
By using more sophisticated potentials, the properties of the elemental cubic metals can be
calculated, and the understanding gained in the previous chapter can be used to analyse
how well the potentials predict negative Poisson’s ratio behaviour.
The total internal energy of a crystal structure is the sum of the energies resulting from
interactions of atoms within the crystal (nearest neighbour, second nearest neighbour etc.).
The accuracy of the description of the energy is to some extent conditional on the number
of interactions considered but it is also true that, as the order of the interaction increases,
its contribution to the total energy diminishes; second order interactions have less effect
on calculated properties than first. The elastic properties of the crystals are calculated
from the energy, and thus has a fundamental effect on the accuracy of the predicted
properties.
This section first details how the elastic properties are calculated using potential
forms and parameters found in the literature. The parameters of these potentials are then
varied to determine how these affect Poisson’s ratio. A fundamental potential model is
then applied to selected covalent compounds to investigate how Poisson’s ratio varies
with comparative bond energies.
4.2.2
Numerical modelling methodology
The numerical modelling is conducted using well established potentials (outlined in
section 2.3), the two-body Morse potential, and the many-body Finnis-Sinclair, SuttonChen and Cleri-Rosato potentials. The General Utility Lattice Program (Gulp) (Gale, 2007)
is used to perform the analysis of the crystal structures. Fitting parameters for the Morse
potential are taken from Girifalco and Weizer (1959), and for the many-body potentials
109
from the library within Gulp (these are from respective original publications; (Finnis and
Sinclair, 1984; Sutton and Chen, 1990; Cleri and Rosato, 1993)). Structural information
is obtained from the literature (Ashcroft and Mermin, 1976; Landolt-Bornstein, 1985) to
determine the lattice constants, elastic constants and Poisson’s ratios in both the [100] and
[110] directions for a range of cubic metals. These input parameters are used to construct
the Gulp input files.
The potential modelling methodology can be described in three stages: an optimisation; followed by a relaxation; followed by calculation of the bulk properties. The
structural optimisation is conducted by finding the local energy minimum, closest to the
starting configuration. The energy can then be expanded as a Taylor series:
U(x + δx) = U(x) +
1 ∂2 U
∂U
(δx)2
δx +
2! ∂x2
∂x
(4.1)
The first derivative of this Taylor series is the gradient vector, and the second derivative
is the Hessian matrix. Gulp uses a second order truncated form of this, and a NewtonRaphson iterative scheme to find the local minimum (Gale, 2007). Once the local energy
minimum has been found and the resulting structure calculated, it is possible to derive
the physical properties of the structure from the gradient of the energy surface. The elastic
constants are the second derivative of the energy density (with respect to strain):
1 ∂2 U
ci j =
V ∂εi ∂ε j
!
(4.2)
Having determined the elastic constant matrix it is then possible to further derive the
elastic properties using the procedures outlined in section 2.2.6.
4.2.2.1
Calculation of Poisson’s ratio for elemental cubic metals
To investigate the Poisson’s ratio (measured in the [110], [11̄0] direction) for the cubic
crystals, two methodologies are employed; calculation of the elastic constants in the
standard orientation and computation of the off-axis properties from these data, and
direct measurement from a rotated structure.
110
4.2.2.1.1
Optimisation method
For each of the potential functions, the experimentally calculated structure was used as a
starting point. The structure is optimised, and allowed to fully relax. The relaxed structure
and elastic stiffnesses are calculated, and Poisson’s ratio in the [110, 11̄0] directions (the
direction for which a minimum Poisson’s ratio has been observed from experimental data)
calculated from the stiffness tensor. The accuracy of the potential was then appraised
relative to the published experimental data.
4.2.2.1.2
Direct measurement method
The direct measurement method is employed to verify that the off-axis strains calculated
by the optimisation method are replicated when measured directly. The unit cell is rotated
through 45◦ about the [001] axis. The atomic positions in the 1’ (revised [100]) direction
are constrained at a value of 1.005 times the rotated lattice constant, to generate a 0.5%
strain in the original [110] direction, and the position of the (unconstrained) orthogonal
atoms calculated after optimisation (see figure 4.1). The resulting strains are calculated
from the lattice constants resulting from the relaxation and the Poisson’s ratio verified
against that produced from the optimised elastic constants. The main advantage of the
direct measurement method is that the deformed structure can be visualised in order to
determine the mechanism of the deformations.
Figure 4.1: Relative orientation of a body-centred cubic unit cell for optimisation (left)
and direct measurement methods (right) (n.b. axes, bottom left)
111
4.2.2.2
Parameter space exploration
Each of the potentials used is controlled by fitting parameters that govern the shape of
the potential function, for example the width and depth of the potential well and the
interatomic separation at equilibrium. Given that the elastic constants are calculated
from the second derivative of the energy, changing these parameters and thus the energy
function affects the calculated constants and resulting Poisson’s ratio.
The parameters in the model are fitted to the experimental data, usually the lattice
parameters and c11 , c12 . For many-body potentials, which are not bound by the Cauchy
relation, c44 is also included. Each parameter models a different physical property in the
crystal (electron density etc.) which has a different effect on the calculated energy and
Poisson’s ratio.
To understand which potential parameters regulate the negative Poisson’s ratio behaviour crystals and align this with physical properties of the crystals, a parameter space
exploration is conducted. Fitting parameters for the potentials are varied to investigate
the controlling factors of the calculation of the elastic constants and subsequent Poisson’s
ratio. The parameters varied for each potential are shown in table 4.2. Values from the
literature were used as a starting point and high throughput techniques help to conduct
the analyses.
Potential
Varied parameters
Morse
Separation at minimum energy (r0 )
Well width (α)
Finnis-Sinclair
EAM functional (A)
EAM Density (φ)
Sutton-Chen
Lennard-Jones (repulsive) component (ρ)
EAM Density (φ)
Table 4.2: Varied potential parameters. (n.b. Potentials detailed in section 2.3)
The geometry of an elemental cubic metal, and associated fitting parameters are used
as the basis for the analysis. High throughput computing techniques are then employed
112
to vary each of the parameters of the potential. The respective element used in each
analysis is shown in table 4.3.
Potential
Element
Morse
Al
Finnis-Sinclair
Fe
Sutton-Chen
Ni
Table 4.3: Elements used as basis for parameter space exploration.
4.2.2.3
Modelling of a coupled three-body/harmonic system
Figure 4.2: Schematic representation of the Keating model
A simple model of the bonding in crystals is the so called "Keating model" where the bonding is modelled by the combination of a harmonic and a three-body potential. The original
model, published by Keating (1966a) and subsequently extended (Keating, 1966b), models the interaction of diamond like crystals to find the elastic properties. This fundamental
model allows the relationship between the ratio of axial to rotational forces and Poisson’s
ratio to be analysed. The potentials used for this analysis are the harmonic and simple
three-body potentials;
1
Eharm = Kharmonic (r − r0 )2
2
1
Ethree = Kthree (θ − θ0 )2 ,
2
(4.3)
(4.4)
where En is the respective energy, Kn the associated spring constant, (r − r0 ) the axial
deformation, and (θ − θ0 ) the angular deformation.
113
The contribution to the overall energy from each potential can be used to determine
the influence each has on the overall properties and the ratio of lateral to axial stiffness
can be found.
4.2.2.3.1
Fitting of potentials to covalently bonded cubic crystals
The bonding within the elemental cubic metals is, implicitly, metallic bonding. The nature
of metallic bonding is such that it is non-directional but can be approximated as such. In
contrast, covalent bonding is highly directional and thus is more suited to being described
by directional potentials.
The covalently bonded crystals considered all have a zincblende structure (see figure
4.3). The fitting parameters for covalent cubic crystals are not readily available and
thus it is necessary to use the fitting capabilities of Gulp to conduct this analysis. As
the focus of this thesis is the elastic properties of BCC and FCC crystals, analysis of
BCC and FCC covalent compounds would have been more directly relevant, however
the analysis depends on accurate experimentally measured elastic data and thus the
zincblende structures were modelled as these data were available in the literature.
The elastic constants of the crystals are obtained from the experimental data (Vukcevich, 1972). The structure is constrained to that observed experimentally, the elastic
constants used as observables, and the spring constants of the potential allowed to relax.
Having found the fitting parameters, we can then perform an optimisation to find the
energy of the system.
Figure 4.3: Structure of covalently bonded zincblende type crystal structure - metal grey,
non-metal yellow. Unconnected atoms are periodic images.
114
4.3
Elastic properties calculated from classical potentials
This section presents the results of the potential modelling. It is divided into three
sections. Initially, the elastic properties calculated from the classical potentials is shown.
These models are then used as a basis for the exploration of how the parameters of
the model vary the calculated Poisson’s ratio. Finally, the results of the fundamental
three-body modelling is presented.
4.3.1
Experimental data
N.b. For convenience, the experimental data from the literature is provided below in
table 4.4 to enable comparison with the predicted elastic properties.
115
Element
Symmetry
Group
Period
Lattice const.
c11
c44
c12
ν110
ν12
K
G
E
Na
b
1
3
4.2
7.6
4.3
6.3
-0.4
0.5
6.8
4.3
1.8
K
b
1
4
5.2
3.7
1.9
3.1
-0.4
0.5
3.3
1.9
0.8
Rb
b
1
5
5.6
3.0
1.6
2.4
-0.4
0.5
2.6
1.6
0.8
Cs
b
1
6
6.0
2.5
1.5
2.1
-0.5
0.5
2.2
1.5
0.6
Ca
f
2
4
5.6
22.8
14.0
16.0
-0.3
0.4
18.3
14.0
9.6
Sr
f
2
5
6.1
15.3
9.9
10.3
-0.3
0.4
12.0
9.9
7.0
Ba
b
2
6
5.0
12.6
9.5
8.0
-0.3
0.4
9.5
9.5
6.4
V
b
5
4
3.0
230.0
43.1
120.0
0.5
0.3
156.7
43.1
147.7
Nb
b
5
5
3.3
245.0
28.4
132.0
0.6
0.3
169.7
28.4
152.6
Ta
b
5
6
3.3
264.0
82.6
158.0
0.2
0.4
193.3
82.6
145.7
Cr
b
6
4
2.9
348.0
100.0
67.0
0.3
0.2
160.7
100.0
326.4
Mo
b
6
5
3.1
465.0
109.0
163.0
0.4
0.3
263.7
109.0
380.4
W
b
6
6
3.2
523.0
160.0
203.0
0.3
0.3
309.7
160.0
409.5
Fe
b
8
4
2.9
230.0
117.0
135.0
-0.1
0.4
166.7
117.0
130.1
Rh
f
9
5
3.8
413.0
184.0
194.0
0.1
0.3
267.0
184.0
289.0
Ir
f
9
6
3.8
580.0
256.0
242.0
0.1
0.3
354.7
256.0
437.5
Ni
f
10
4
3.5
247.0
122.0
153.0
-0.1
0.4
184.3
122.0
130.0
Pd
f
10
5
3.9
221.0
70.8
171.0
-0.1
0.4
187.7
70.8
71.8
Pt
f
10
6
3.9
347.0
76.5
251.0
0.2
0.4
283.0
76.5
136.3
Cu
f
11
4
3.6
169.0
75.3
122.0
-0.1
0.4
137.7
75.3
66.7
Ag
f
11
5
4.1
122.0
45.5
92.0
-0.1
0.4
102.0
45.5
42.9
Au
f
11
6
4.1
191.0
42.2
162.0
0.0
0.5
171.7
42.2
42.3
Al
f
13
3
4.0
108.0
28.3
62.0
0.3
0.4
77.3
28.3
62.8
Pb
f
14
6
5.0
48.8
14.8
41.4
-0.2
0.5
43.9
14.8
10.8
Table 4.4: Experimentally observed elastic properties of elemental metals (Ashcroft and
Mermin, 1976; Landolt-Bornstein, 1985). Elastic stiffnesses in GPa.
116
4.3.2
4.3.2.1
Elastic and engineering constant data from potential models
Accuracy of calculated properties
The numerical modelling is conducted on both body-centred and face-centred cubic metals. The elastic constants for each metal are computed, and the engineering constants
calculated from these. The accuracy of the properties calculated using the classical potentials could then be compared to experimental data. This section discusses the accuracy of
the values calculated from the numerical modelling, and considers the shortcomings of
these techniques.
117
c11 Expt
c44 Expt
c12 Expt
c11 Potential
c44 Potential
c12 Potential
Error c11 %
Error c44 %
Error c12 %
Morse Potential
Element
4.3.2.2
Cs
2.5
1.5
2.1
2.4
1.7
1.7
-4.6
13.7
-18.3
Rb
3.0
1.6
2.4
3.2
2.3
2.3
7.2
44.9
-5.0
K
3.7
1.9
3.1
3.7
2.8
2.8
-1.2
49.2
-10.9
Na
7.6
4.3
6.3
7.3
5.8
5.8
-4.3
35.1
-8.2
Ba
12.6
9.5
8.0
10.6
9.2
9.2
-15.7
-3.5
14.6
Sr
15.3
9.9
10.3
14.9
10.8
10.8
-2.4
9.3
5.1
Ca
22.8
14.0
16.0
20.8
15.1
15.1
-8.7
7.9
-5.6
Cr
348.0
100.0
67.0
202.1
197.1
197.1
-41.9
97.1
194.1
Mo
465.0
109.0
163.0
290.5
289.3
289.3
-37.5
165.4
77.5
W
523.0
160.0
203.0
340.7
330.0
330.0
-34.9
106.2
62.5
Fe
230.0
117.0
135.0
180.6
167.0
167.0
-21.5
42.8
23.7
Ni
247.0
122.0
153.0
233.0
163.4
163.4
-5.7
34.0
6.8
Ag
122.0
45.5
92.0
133.2
90.1
90.1
9.2
98.1
-2.0
Cu
169.0
75.3
122.0
173.6
122.6
122.6
2.8
62.8
0.5
Al
108.0
28.3
62.0
94.4
67.5
67.5
-12.6
138.7
8.9
Pb
221.0
70.8
171.0
55.8
37.2
37.2
-74.8
-47.5
-78.3
Table 4.5: Elastic constants for elemental metals, calculated using Morse potential.
Elastic stiffnesses in GPa.
118
ν(11̄0,110)
ν(11̄0,110)
ν(010,100)
K
G
E
Error ν(11̄0,110) %
Error ν(010,100) %
Error K %
Error G %
Error E %
-0.28
-0.46
0.45
2.20
1.48
0.60
24.69
8.35
13.16
-13.74
-59.72
Na
-0.35
-0.40
0.45
6.75
4.30
1.83
-16.30
2.27
6.76
-35.10
-14.48
K
-0.65
-0.43
0.46
3.34
1.88
0.82
-3.42
5.57
7.32
-49.25
-50.79
Rb
-0.37
-0.44
0.45
2.61
1.60
0.75
5.68
6.58
0.39
-44.88
-61.08
Ba
-0.93
-0.29
0.39
9.53
9.50
6.39
-121.98
-19.32
-1.24
3.50
66.78
Ca
-0.34
-0.26
0.41
18.27
14.00
9.60
-39.41
-1.96
6.87
-7.90
15.47
Sr
-0.33
-0.26
0.40
11.97
9.90
7.01
-42.79
-4.45
-1.89
-9.33
16.80
W
-0.79
0.32
0.28
309.67
160.00
409.48
425.18
-75.97
-7.71
-106.24
96.11
Cr
-0.44
0.40
0.16
160.67
100.00
326.37
388.90
-205.81
-23.70
-97.07
97.71
Mo
-0.99
0.28
0.26
263.67
109.00
380.39
344.51
-92.26
-9.88
-165.44
99.55
Fe
-0.51
-0.06
0.37
166.67
117.00
130.14
-1170.70
-29.91
-2.92
-42.75
84.57
Ni
-0.32
-0.07
0.38
184.33
122.00
129.96
-338.41
-7.80
-1.24
-33.98
24.46
Cu
-0.26
-0.09
0.42
137.67
75.30
66.70
-146.34
1.28
-1.42
-62.84
-8.14
Ag
-0.38
-0.13
0.43
102.00
45.50
42.90
-195.71
6.12
-2.43
-98.07
-40.89
Al
-0.36
0.27
0.36
77.33
28.3
62.78
228.33
-14.37
1.09
-138.67
39.41
Pb
-0.91
-0.05
0.44
187.67
70.8
71.81
-394.87
8.37
76.89
47.52
63.65
Expt
Element
Cs
Table 4.6: Engineering constants for elemental metals, calculated using Morse potential.
Experimental off-axis PR shown for comparison. K,G & E in GPa.
4.3.2.2.1
Analysis: Morse Potential
The Morse potential is used to investigate 16 metals. The calculated elastic constants are
shown in table 4.5, the derived elastic properties in table 4.6. The most apparent feature of
these data is that the Cauchy relation, c44 = c12 , is obeyed for all of the metals considered.
The computed values of c11 show good agreement for group one and two elements, with
less than 10% error. The computed values for c44 and c12 are generally an average of the
true values of c44 and c12 . For the group one and two elements, these are still less than
119
50% error. For the transition metals and poor metals, the magnitude of error for c11 is
greater than 50%.
Engineering constants are often used as a measure of accuracy when analysing potential data. The computed values for ν(010,100) show an error of less than 20% for all except
four of the transition metals. Similarly, the predicted values for bulk modulus show good
agreement with the experimentally calculated values. It is only when considering the
values that have off-axis components (ν(11̄0,110) and shear modulus, G) that the erroneous
prediction of the elastic constants is evident. The values for Young’s modulus are calculated from the compliances, which are in turn calculated from the inverted stiffness
matrix. Because of this, the Young’s modulus value, whilst only being a function of axial
compliances, is not accurately predicted due to the errors in the stiffness values.
Experimental elastic constants for the body-centred cubic metals show a negative
Poisson’s ratio for the group one and two metals, and a positive Poisson’s ratio for the
transition metals, with the exception of body-centred cubic Iron. Comparison of the
computed Poisson’s ratio of the group one elements to experimental results shows that
for these metals the Morse potential is able to accurately predict the Poisson’s ratio, both
in magnitude and sign. The result for Barium, group two, predicts auxetic behaviour
however, the calculated value is double that of the experimental value.
The face-centred cubic transition metals have a negative Poisson’s ratio when calculated from experimental data. The Morse potential correctly predicts this auxetic
behaviour however generally over predicts the value of ν by a factor of around 3.
The shear modulus calculated from the Morse potential shows good agreement for
the group two metals but generally has greater than 50% error for the remaining metals.
120
c11 Expt
c44 Expt
c12 Expt
c11 Potential
c44 Potential
c12 Potential
Error c11 %
Error c44 %
Error c12 %
Finnis-Sinclair potential
Element
4.3.2.3
V
230.0
43.1
120.0
227.9
42.6
118.7
-0.9
-1.2
-1.1
Nb
245.0
28.4
132.0
246.6
28.1
133.2
0.7
-1.1
0.9
Ta
264.0
82.6
158.0
266.0
82.4
161.2
0.8
-0.2
2.0
Cr
348.0
100.0
67.0
387.1
100.8
103.5
11.2
0.8
54.5
Mo
465.0
109.0
163.0
464.7
108.9
161.5
-0.1
-0.1
-0.9
W
523.0
160.0
203.0
522.4
160.6
204.4
-0.1
0.4
0.7
Fe
230.0
117.0
135.0
243.1
121.1
138.1
5.7
3.5
2.3
ν(11̄0,110)
ν(11̄0,110)
ν(010,100)
K
G
E
Error ν(11̄0,110) %
Error ν(010,100) %
Error K %
Error G %
Error E %
V
0.45
0.45
0.34
155.10
42.60
146.60
-0.31
0.11
1.00
1.16
0.76
Nb
0.62
0.61
0.35
171.01
28.10
153.18
-0.79
-0.17
-0.79
1.05
-0.41
Ta
0.17
0.17
0.38
196.15
82.41
144.36
0.61
-0.78
-1.46
0.23
0.91
Cr
0.37
0.32
0.21
198.05
100.81
343.44
-14.25
-30.68
-23.27
-0.81
-5.23
Mo
0.40
0.40
0.26
262.57
108.90
381.40
-0.14
0.64
0.42
0.09
-0.27
W
0.28
0.28
0.28
310.43
160.62
407.46
1.08
-0.58
-0.25
-0.38
0.49
Fe
-0.04
-0.06
0.36
173.11
121.09
143.04
38.55
2.05
-3.87
-3.50
-9.91
Expt
Element
Table 4.7: Elastic constants for elemental metals, calculated using Finnis-Sinclair
potential. Elastic stiffnesses in GPa.
Table 4.8: Engineering constants for elemental metals, calculated using Finnis-Sinclair
potential. Experimental off-axis PR shown for comparison. K,G & E in GPa.
121
4.3.2.3.1
Analysis: Finnis-Sinclair
The Finnis-Sinclair potential is used to predict the behaviour of seven body-centred cubic
transition metals, shown in tables 4.7 and 4.8. The calculated elastic constants are in good
agreement for all of the metals considered. The engineering constants calculated from
these values are thus in good agreement with the experimental results.
For V, Nb, Ta, Cr, W and Mo the Poisson’s ratio is found to be positive experimentally
and this is replicated in the simulations. For Fe, the experimental Poisson’s ratio is
negative, and which is also found with the computed value.
Not only does this model correctly predict whether the Poisson’s ratio is positive or
negative but also the value of the result is essentially correct. The main conclusions that
can be drawn are that the Finnis-Sinclair is not only able to accurately predict whether
Poisson’s ratio will be positive or negative, but also can accurately determine the magnitude of ν. The Morse potential, being a two-body potential, is not able to predict when an
element has a positive Poisson’s ratio, due to the Cauchy relation as discussed in section
2.3.
122
c11 Expt
c44 Expt
c12 Expt
c11 Potential
c44 Potential
c12 Potential
Error c11 %
Error c44 %
Error c12 %
Sutton-Chen potential
Element
4.3.2.4
Rh
413
184
194
339.0
142.7
231.9
-17.9
-22.4
19.5
Ir
580
256
242
475.2
210.0
309.5
-18.1
-17.9
27.9
Pd
221
70.8
171
248.5
93.4
175.8
12.5
32.0
2.8
Ni
247
122
153
230.8
79.6
177.1
-6.5
-34.7
15.8
Pt
347
76.5
251
314.5
73.7
258.4
-9.4
-3.7
3.0
Ag
122
45.5
92
139.9
58.8
95.6
14.6
29.3
4.0
Cu
169
75.3
122
168.7
58.2
129.4
-0.2
-22.8
6.1
Au
191
42.2
162
179.9
42.2
147.8
-5.8
-0.1
-8.7
Al
108
28.3
62
81.2
15.5
71.6
-24.8
-45.3
15.4
Pb
48.8
14.8
41.4
49.4
15.5
38.0
1.3
4.7
-8.2
Table 4.9: Elastic constants for elemental metals, calculated using Sutton-Chen potential.
Elastic stiffnesses in GPa.
123
ν(11̄0,110)
ν(11̄0,110)
ν(010,100)
K
G
E
Error ν(11̄0,110) %
Error ν(010,100) %
Error K %
Error G %
Error E %
-0.06
0.07
0.41
267.60
142.69
150.63
182.20
-27.09
-0.23
22.45
47.88
Ir
-0.05
0.10
0.39
364.75
210.04
231.06
150.34
-33.97
-2.84
17.95
47.19
Pd
-0.03
-0.05
0.41
200.05
93.43
102.87
41.38
5.03
-6.60
-31.97
-43.24
Ni
-0.08
-0.07
0.43
195.03
79.62
77.01
-6.08
-13.51
-5.81
34.74
40.74
Pt
0.00
0.21
0.45
277.12
73.69
81.41
98.54
-7.46
2.08
3.67
40.27
Ag
-0.06
-0.09
0.41
110.38
58.85
62.18
38.30
5.53
-8.22
-29.34
-44.94
Cu
-0.08
-0.13
0.43
142.52
58.17
56.32
42.12
-3.55
-3.53
22.75
15.58
Au
0.00
-0.04
0.45
158.54
42.16
46.57
108.09
1.72
7.65
0.10
-10.08
Al
-0.08
0.27
0.47
74.78
15.49
14.12
128.28
-28.47
3.30
45.26
77.51
Pb
-0.03
-0.19
0.43
41.81
15.50
16.36
82.39
5.27
4.68
-4.71
-51.51
Expt
Element
Rh
Table 4.10: Engineering constants for elemental metals, calculated using Sutton-Chen
potential. Experimental off-axis PR shown for comparison. K, G & E in GPa
4.3.2.4.1
Analysis: Sutton-Chen Potential
The data from the Sutton-Chen potential is shown in tables 4.9 and 4.10. Where Poisson’s
ratio is experimentally predicted to be negative, the potential predicts a negative Poisson’s
ratio to a reasonable degree of accuracy. Where Poisson’s ratio is predicted to be positive,
the potential predicts a Poisson’s ratio of zero. Predictions of bulk modulus are generally
within 10% accuracy, however the potential over-estimates the shear and Young’s moduli
in the majority of cases.
124
c11 Expt
c44 Expt
c12 Expt
c11 Potential
c44 Potential
c12 Potential
Error c11 %
Error c44 %
Error c12 %
Cleri-Rosato potential
Element
4.3.2.5
Rh
413.0
184.0
194.0
399.0
201.9
242.3
-3.4
9.8
24.9
Ir
580.0
256.0
242.0
554.5
261.4
345.8
-4.4
2.1
42.9
Pd
221.0
70.8
171.0
231.9
72.6
178.5
5.0
2.5
4.4
Ni
247.0
122.0
153.0
298.2
157.2
184.1
20.7
28.8
20.3
Pt
347.0
76.5
251.0
341.1
90.6
273.4
-1.7
18.4
8.9
Ag
122.0
45.5
92.0
131.7
50.6
96.7
7.9
11.2
5.0
Cu
169.0
75.3
122.0
176.7
82.2
125.2
4.6
9.2
2.6
Au
191.0
42.2
162.0
187.4
44.7
154.4
-1.9
6.0
-4.7
Al
108.0
28.3
62.0
95.0
37.0
74.5
-12.1
30.8
20.2
Pb
221.0
70.8
171.0
48.4
12.8
39.8
-78.1
-82.0
-76.8
Table 4.11: Elastic properties for elemental metals, calculated using Cleri-Rosato
potential. Elastic stiffnesses in GPa.
125
ν(11̄0,110)
ν(11̄0,110)
ν(010,100)
K
G
E
Error ν(11̄0,110) %
Error ν(010,100) %
Error K %
Error G %
Error E %
-0.08
0.07
0.38
294.53
201.95
215.86
-3.40
-18.22
-10.31
-9.76
25.31
Ir
-0.05
0.10
0.38
415.36
261.39
288.93
-4.39
-30.45
-17.11
-2.11
33.96
Pd
-0.03
-0.05
0.43
196.32
72.61
76.68
4.95
0.30
-4.61
-2.55
-6.79
Ni
-0.10
-0.07
0.38
222.14
157.17
157.68
20.74
0.21
-20.51
-28.83
-21.34
Pt
-0.01
0.21
0.44
295.99
90.61
97.83
-1.69
-6.00
-4.59
-18.44
28.22
Ag
-0.08
-0.09
0.42
108.32
50.62
49.84
7.92
1.53
-6.20
-11.25
-16.19
Cu
-0.14
-0.13
0.41
142.40
82.23
72.84
4.57
1.07
-3.44
-9.20
-9.19
Au
-0.01
-0.04
0.45
165.40
44.72
47.89
-1.89
1.56
3.65
-5.96
-13.19
Al
-0.17
0.27
0.44
81.34
37.00
29.45
-12.06
-20.55
-5.19
-30.76
53.09
Pb
-0.05
-0.05
0.45
42.64
12.75
12.61
-78.08
-3.32
77.28
81.99
82.44
Expt
Element
Rh
Table 4.12: Engineering constants for elemental metals, calculated using Cleri-Rosato
potential. Experimental off-axis PR shown for comparison. K, G & E in GPa.
4.3.2.5.1
Analysis: Cleri-Rosato Potential
The results of modelling using the Cleri-Rosato potential are shown in tables 4.11 and
4.12. A range of transition metals were analysed. The potential is not able to accurately
recreate the elastic constants for Lead (group 14). For the remaining metals (groups 9 13) the potential underestimates the value of c11 and under estimates the values of c12 and
c44 . The most accurate results are for the group 10 and 11 metals.
Comparing the experimental and predicted properties for the off-axis Poisson’s ratio,
the predicted value is always negative, even when the experimental value is positive. The
computed values for the bulk modulus are generally within 10% error, and the values for
shear modulus and Young’s modulus generally within a 30% error.
126
4.3.2.6
Summary
In general, predictions for group one and group two metals are more accurate than those
for transition metals, with the exception of those calculated using the Finnis-Sinclair
potential.
4.3.3
Deformed structures calculated from classical potentials
The classical potential method allows the deformed structure of the crystal to be easily
calculated from the direct measurement method: the deformed lattice parameters are
automatically generated by Gulp. There are four possible deformed structures that can
be observed:
• Body-centred cubic with positive Poisson’s ratio
• Body-centred cubic with negative Poisson’s ratio
• Face-centred cubic with positive Poisson’s ratio
• Face-centred cubic with negative Poisson’s ratio
The crystals are subjected to a 0.05% strain in the [110] direction, and the magnitude of
the atomic displacements is recorded. These displacements are replicated and translated
so that the deformation of the unit cell can be observed.
127
4.3.3.1
Body-centred cubic crystals
(a) Positive ν
(b) Negative ν
Figure 4.4: Schematic of the deformation mechanism of body-centred cubic crystals.
Left: Negative Poisson’s ratio, with corresponding perpendicular positive Poisson’s
ratio deformation. right: positive Poisson’s ratio in both planes.
The mechanism for auxetic behaviour in body-centred cubic crystals is suggested by
Baughman et al. (1998). Baughman suggested a scissor type, hinging arrangement where
the perpendicular out of plane behaviour is a high positive Poisson’s ratio behaviour and
the in-plane behaviour is negative. This gives a theoretical minimum in-plane Poisson’s
ratio of -1, and the bond lengths remain unchanged. By computing the deformed structure
of the body-centred cubic crystals that are shown to have a negative Poisson’s ratio,
it can be seen that the mechanism suggested by Baughman is replicated in the Gulp
models (see figure 4.4). The positive Poisson’s ratio mechanism is perhaps less intuitive
when considering the atomistic picture, it corresponds to a contraction of the planes
perpendicular to the strained direction. The varying interatomic distances make this a
more complex deformation that the auxetic behaviour.
4.3.3.2
Face-centred cubic crystals
The mechanism for auxetic behaviour in face-centred cubic crystals is far less obvious
than that for body-centred cubic. The unit cell of the face-centred cubic cell consists
of four atoms, however, in the rotated configuration the unit cell assumes tetragonal
symmetry (the x and y axes are rotated, z is not). This results in an eight atom unit cell.
128
Computing the displacements of this cell alone does not provide a clear understanding of
the deformation mechanism, and thus it is necessary to further manipulate the resulting
configuration, taking care to preserve the symmetry of the crystal. Initially, a macro-cell
is generated, and from this it is possible to observe the auxetic behaviour. As with the
body-centred cubic unit cell there is a large positive orthogonal strain associated with the
negative in-plane Poisson’s deformation.
Visualising the deformed unit as the cubic cell most readily associated with a facecentred cubic crystal, the strains can be more readily observed. A possible mechanism
is derived from the Cleri-Rosato potential modelling of Iridium: The deformed unit cell
appears to be subject to two deformation mechanisms. The first is analogous to the
deformation mechanism of the body-centred unit cell described above, the second is a
combination of stretching and hinging within the cell.
Figure 4.5: A ’side’ view (left) and a ’top’ view (right) of the deformed unit cell.
Deformed atomic positions shown as dashed circles, undeformed as solid. Not to scale.
Figure 4.5 shows a view down the (11̄0) direction, a side view, and a perpendicular
view down the (001) axis, a top view. Considering the cubic cell as being constructed of 3
layers, the upper and lower layers deform in a manner very similar to the body-centred
cubic unit cell, the ’in-plane’ bonds undergoing a stretching, the ’out of plane’ bonds
being shortened, as shown as dashed lines in figure 4.6. It can be seen that the bonds
connecting the central layer to the upper and lower layers undergo hinging, as shown in
figure 4.7. The bonds within the central layer undergo stretching, as indicated by dashed
lines in figure 4.8. The magnitude of the displacements of the ’mid-layer’ atoms is less
than that of the top layer atoms.
129
Figure 4.6: The ’external’ deformation. Deformed atomic positions shown as dashed
circles, undeformed as solid. Not to scale.
Figure 4.7: The hinging of the inter-layer bonds. Initial bonds shown as solid lines,
deformed bonds shown as dashed. Deformed atomic positions shown as dashed circles,
undeformed as solid. Only half of the unit is shown for clarity.
Figure 4.8: The ’internal’ deformation. Deformed atomic positions shown as dashed
circles, undeformed as solid. Not to scale.
This mechanism suggests that the auxetic behaviour of the face-centred cubic cells
is conditional on a combination of both rotational and axial forces, the stretching of the
upper and lower layers and the rotation of the inter-layer bonding.
This mechanism gives a lower magnitude of Poisson’s ratio than the body-centred
cubic deformation, in agreement with the result shown by the simple spring models in
section 3.3.
130
4.3.4
Variation of predicted properties with phase space exploration
Figures 4.9 to 4.14 show the change in calculated Poisson’s ratio as the respective potential
parameter is varied. The calculated ’virtual’ value is shown by a blue rhombus. The value
calculated from the fitted value is shown as a pink square. An explanation of the potential
forms, and their component terms is given in section 2.3.3.2.
4.3.4.1
Morse potential
Figure 4.9: Poisson’s ratio, calculated from the Morse potential, showing the change in
ν11̄0,110 with variation of the well width (α) component of the Morse potential. Pink
square shows the fitted values for Aluminium, for reference.
131
Figure 4.10: Poisson’s ratio, calculated from the Morse potential, showing the change in
ν110 with variation of the interatomic separation (r0 ) component component of the Morse
potential. Pink square shows the fitted values for Aluminium, for reference.
132
4.3.4.2
Finnis-Sinclair potential
Figure 4.11: Poisson’s ratio, calculated from the Finnis-Sinclair potential, showing the
change in ν110 with variation of the density component fitting parameter. Pink square
shows the fitted values for Iron, for reference.
Figure 4.12: Poisson’s ratio, calculated from the Finnis-Sinclair potential, showing the
change in ν110 with variation of the EAM functional component fitting parameter. Pink
square shows the fitted values for Iron, for reference.
133
4.3.4.3
Sutton-Chen Potential
Figure 4.13: Poisson’s ratio, calculated from the Sutton-Chen potential, showing the
change in ν110 with variation of the density component fitting parameter. Pink square
shows the fitted values for Copper, for reference.
Figure 4.14: Poisson’s ratio, calculated from the Sutton-Chen potential, showing the
change in ν110 with variation of the pairwise repulsive (Lennard-Jones) component. Pink
square shows the fitted values for Copper, for reference.
134
4.3.4.4
Analysis: Variation of predicted Poisson’s ratio with phase-space exploration
The results of the analysis are shown in figures 4.9 to 4.14. The analysis is conducted to
investigate the sensitivity of the calculated Poisson’s ratio to changes in the potential parameters. The components of each model were varied and the resulting ν(11̄0,110) Poisson’s
ratio computed. Where there appears to be little, or significant variation the sensitivity of
the analysis was adjusted accordingly to capture this effect.
The Morse potential uses three terms, to describe the interatomic separation and both
the depth, and width of the potential well. Variation of the depth of the potential well
has no impact on the calculated Poisson’s ratio once a minimum value has been found.
Variation with the width of the potential well, α, is shown in figure 4.9. The Poisson’s ratio
is very sensitive to variation in α, until a maximum value of around zero is found, but as
the Morse is a two-body potential, the Poisson’s ratio never becomes positive. Varying
the atomic separation at minimum energy results in a variation of the Poisson’s ratio from
-0.6 to almost zero.
The Finnis-Sinclair potential has two main parameters that affect the calculated properties. As discussed in section 2, the Finnis-Sinclair is a form of embedded-atom model
(EAM). Within the potential are terms for the density and the EAM functional. Varying
the density component (shown in figure 4.11) has the effect of varying the Poisson’s ratio
from -0.3 to -0.2. Varying the EAM functional term gives the possibility of varying the
Poisson’s ratio from -1 to around 2.
The Sutton-Chen and Cleri-Rosato component are very closely related in form, and
thus only the Sutton-Chen potential is considered. The Sutton-Chen potential is controlled
by the an embedded atom model and by a Lennard-Jones repulsive component. The
Sutton-Chen potential is far less sensitive to variation in potential parameters than the
Finnis-Sinclair and Morse potentials, however it can be seen that the computed results
deviate from the trend for some cases. Investigation of these shows that this is an
artefact of the modelling; as the density or repulsive component changes, the separation
at minimum energy also changes. The cutoff in the simulation is fixed, and thus this small
change in cutoff determines the number of nearest neighbours in the analysis which in
turn leads to a change in the overall energy, and thus the elastic constants derived from
135
this.
4.3.4.5
Conclusions
The sensitivity of the potentials to input parameters depends greatly on the form used.
The Poisson’s ratio of the Finnis-Sinclair and Morse potential is very sensitive to small
changes in input parameters, but the Sutton-Chen potential less so. From this analysis, it
seems impossible to generalise. Fortunately it is computationally inexpensive to conduct
such sensitivity analyses on a case-by-case basis.
4.3.5
Elastic constants calculated from coupled three-body/harmonic system
This model is described in section 4.2.2.3. The values presented here are those derived
from the fitted data.
Figure 4.15: Poisson’s ratio (ν110,11̄0 ) and relative spring stiffnesses calculated from the
three-body/harmonic model for selected body-centred cubic elemental metals. Violet Calculated Poisson’s ratio. Burgundy - Poisson’s ratio as reported in the literature.
Cream - ratio of three-body and harmonic spring constants.
136
Crystal
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
Φharmonic
Φthree
kharm
kthree
Normalised kthree
Figure 4.16: Poisson’s ratio (ν110,11̄0 ) and relative spring stiffnesses calculated from the
three-body/harmonic model for selected covalently bonded cubic metal compounds.
Green - Calculated Poisson’s ratio, red - Poisson’s ratio as reported in the literature, blue
- ratio of three-body and harmonic spring constants.
Ba
12.44
9.87
7.80
-1.30
3.85
-2.54
3.50E-07
2.34E-04
0.44
0.11
0.58
Cr
320.64
161.30
33.09
-7.86
61.30
-50.62
6.20E-07
2.75E-03
3.48
1.34
3.87
Cs
2.24
1.99
1.78
-9.21
34.31
-13.56
9.10E-07
4.63E-04
0.11
0.02
0.14
Fe
212.17
157.16
112.88
-7.75
34.32
-16.38
8.20E-07
9.35E-04
3.92
0.46
1.31
K
3.28
2.89
2.58
-11.58
53.96
-18.08
4.77E-05
3.81E-05
0.14
0.02
0.10
Li
12.62
11.35
10.33
-5.83
18.24
-8.59
2.70E-07
3.88E-05
0.36
0.02
0.07
Mo
409.61
234.55
93.34
-11.91
115.19
-42.74
1.49E-03
3.91E-03
5.84
1.91
6.03
Na
6.84
6.03
5.37
-9.87
40.14
-15.21
1.00E-05
4.37E-05
0.23
0.02
0.09
Nb
199.40
130.66
75.34
-18.61
360.07
-42.93
1.00E-07
1.78E-03
3.61
0.87
2.88
Rb
2.64
2.31
2.05
-10.72
44.49
-16.16
0.00E+00
4.17E-05
0.12
0.02
0.11
Ta
227.75
164.08
112.83
-13.73
98.64
-28.59
5.54E-06
1.66E-03
4.68
0.81
2.70
V
193.15
126.89
73.43
-16.02
194.41
-38.81
3.73E-04
1.30E-03
3.20
0.64
1.93
W
468.78
283.63
134.29
-10.37
77.27
-33.85
1.28E-03
4.18E-03
7.25
2.04
6.46
Table 4.13: Elastic properties for body-centred cubic metals, calculated using
Three-body/harmonic model. Elastic stiffnesses in GPa
137
Element
c11
c44
c12
ν(11̄0,110)
Error c11 %
Error c44 %
Error c12 %
Φharmonic
Φthree
kharm
kthree
kharm
k3
kthree
a
Φthree
Φharm
kthree norm
kharm
138
ZnS
99.1
63.0
44.3
-0.10
15.4
24.6
9.1
0.015
18.96
6.29
2.06
3.05
11.68
2.56
1.86
ZnSe
87.2
51.2
39.9
-0.06
-11.1
-18.3
-10.8
0.028
11.73
5.85
1.90
3.08
10.29
0.85
1.76
ZnTe
72.1
41.1
30.8
-0.04
1.2
1.0
-1.4
0.021
8.86
5.05
2.04
2.48
12.45
0.85
2.47
InP
103.4
58.2
45.4
-0.03
1.2
1.1
-1.3
0.026
10.68
7.05
2.54
2.78
14.88
0.83
2.11
InAs
84.2
45.8
39.1
-0.03
1.1
1.1
-1.1
0.026
13.10
7.16
3.49
2.05
24.85
0.99
3.47
InSb
66.7
36.5
30.2
-0.02
0.0
0.0
0.0
0.026
7.58
5.10
2.19
2.33
14.19
0.59
2.78
AlSb
90.3
45.0
41.1
0.02
1.0
1.5
-1.1
0.018
3.77
6.53
2.28
2.86
13.93
0.43
2.13
GaSb
89.4
41.1
42.7
0.05
1.1
2.0
-1.1
0.018
1.81
6.61
2.10
3.15
12.81
0.20
1.94
GaAs
120.2
55.0
58.1
0.05
1.8
3.3
-1.8
0.046
1.89
8.25
2.15
3.84
12.16
0.08
1.47
GaP
142.4
63.5
69.9
0.06
0.9
1.7
-0.9
0.011
1.61
9.55
2.34
4.08
12.74
0.28
1.33
Table 4.14: Elastic constant data from Three-body/harmonic model for covalently bonded cubic crystals
4.3.5.1
Influence of the relationship of three-body and axial forces on predicted properties
The original three-body/harmonic or Keating model was devised to model the interatomic
interactions of Diamond (Keating, 1966a). In the previous section this technique was
adapted to investigate the bonding in both elemental body-centred cubic metals and
some covalently bonded compounds which have a zinc-blende structure. By analysing
whether the Poisson’s ratio (in the (110) direction) predicted by the potential is negative,
and how the two components of the potential relate, a link between the bond order and
auxetic behaviour may be seen.
Body-centred cubic metals, were the first to be examined followed by covalent crystals
with a zinc-blende structure. Results are shown in table 4.13. The Keating model is not
able to perfectly reproduce the elastic properties of any of the elemental metals, however,
it is able to reproduce c11 to within a 10% error for most of the crystals and c12 to within a
30% error. c44 is calculated to within a 50% error for 5 of the crystals, but for the remaining
crystals, a good fit could not be found. Results where the experimental and calculated
Poisson’s ratio are in close agreement are shown in figure 4.15. The analysis of these data
encompasses comparison of the energies and spring constants for the three-body and
harmonic components of the potential form.
No clear trends can be observed, but comparing the ratio of the three-body and
harmonic spring constants for the metals where a satisfactory fit can be found, there is
some correlation between the spring stiffness ratio and the Poisson’s ratio. Comparing
the energy values shows a several orders of magnitude of variation, as the fitted values
are not able to accurately reflect the bonding in the crystal.
The covalent cubic crystals are modelled far more accurately, with less than 30% error
for all the elastic constants, which given the simplicity of the model seems a reasonable
level of accuracy. Both the energy and the stiffness value, for each component of the
potential, can be compared. The ratio of the harmonic and three-body contributions to
the energy are shown in figure 4.16, and the numerical values for both the spring constants
and energy values can be seen in table 4.14. There is a clear trend that shows that as the
harmonic/three-body energy ratio increases Poisson’s ratio becomes more positive, again
139
showing how the Poisson’s ratio is sensitive to on the ratio of axial to lateral stiffness in
the bonding. In the metallic crystals, the bonding is non-directional, and thus the lateral
stiffness of the bonds is provided by the higher order bonding, however, for the covalent
bonding, the bonds are highly directional, and thus the lateral contribution from the first
neighbour bonding dominates.
4.4
Can classical potentials predict auxetic behaviour?
Classical potentials are used to investigate the elastic properties of elemental cubic metals.
Two-body potentials, whilst able to accurately predict the structure of the crystals, are
unable to predict when positive Poisson’s ratio behaviour occurs, as they are governed
by the Cauchy relation. Many-body potentials are more successful, the Finnis-Sinclair
potential in particular is able to predict elastic properties for a range of transition metals.
Of course, this can be attributed to how well the model is fitted to experimental data.
Investigation of the deformation mechanism based on the potential modelling shows
that the scissor type mechanism is responsible for the negative Poisson’s ratio deformation
in body-centred cubic metals. For the face-centred cubic metals, the mechanism is more
complex, but appears to be a combined stretching and rotation mechanism, similar to that
of the body-centred cubic crystals.
Analysis of the relationship of the three-body and harmonic spring stiffnesses on
Poisson’s ratio shows that there is no obvious link between Poisson’s ratio and the ratio
of the stiffnesses for metals, but by considering the data as a whole, particularly the
covalently bonded crystals, a weak trend emerges. In other words, the degree of auxeticity
of cubic metals can be accounted for by some specific potential models but only if the
experimental elastic constant data was intrinsic to the derivation of the model. This is
clearly not the case for such potential as the Cleri-Rosato.
The conclusion should be heeded by those wishing to use potential energy methods
for modelling nano-scale structures.
140
Chapter 5
Density Functional Theory
modelling of elastic properties of
cubic elemental metals
5.1
Introduction
The classical potentials described and used in chapter 4 show that in many cases it is
possible to accurately model the Poisson’s ratio, both negative and positive, for elemental
cubic metals and alloys. This study also shows that predicting the elastic properties
of these materials using this technique, whilst being computationally inexpensive, is
conditional on the validity of the potential chosen, and the accuracy of the parameters.
This dependence on empirical data makes this analysis impossible when using these
techniques to model systems where experimental data is not available.
In order to provide an additional source of numerical data from which the Poisson’s
ratios of elemental metals can be calculated, a first principal quantum based technique is
required: Density Functional Theory (DFT). DFT techniques are not dependent on fitting
to experimental data to derive model parameters and provide more generic solutions.
They allow calculation of Poisson’s ratio for a range of crystals without dependence on
the accuracy of the fitting parameters.
141
5.2
Methodology
The purpose of this investigation is to appraise whether negative Poisson’s ratio can be
accurately predicted by DFT techniques, using a set of crystals for which the behaviour
is well characterised experimentally. An outline summary of the principles behind DFT
is given in Appendix A.
The elastic properties for the elemental cubic metals are calculated using Vienna Abinitio Software Project (VASP) (Kresse and Hafner, 1993, 1994a; Kresse and Furthmüller,
1996a,b). The input files are generated from established lattice constant data (Ashcroft
and Mermin, 1976). The elastic constants are calculated during the analysis using the
built-in functionality of the software package, negating the need for any manual post
processing.
The pseudopotentials chosen are ultrasoft Vanderbilt LDA and GGA pseudopotentials. Strictly speaking, LDA and GGA are approximations of the exchange correlation
functional (Exc ), but because the pseudopotentials are generated from all-electron solutions, it is normal to refer to them as pseudopotentials. Additionally, LDA-PAW and PBE
projector augmented wave methodologies were also used (Kresse and Hafner, 1994b;
Blöchl, 1994; Perdew et al., 1996, 1997; Kresse and Joubert, 1999). Where a range of
methodologies were available, all were used for comparison as differing pseudopotentials formulations are better suited to specific crystal types. These are noted in the results
as follows:
• h - a harder potential than standard
• s - a softer potential than standard
• pv, sv - the p or s semi-core states are treated as valence states.
• new - a revised version of the original potential
5.2.1
Convergence example
A convergence for the k-point mesh and cut off was conducted, and suitable values for
mesh density and energy cut-off decided upon based on these simulations. To aid this
process, scripting was used to generate the input files, conduct the analysis and extract
142
the elastic constants. Periodic (Bloch) boundary conditions are used in the analysis (see
section A.2.7).
For each crystal/pseudopotential combination, an optimisation was conducted to find
the calculated elastic constants. Once this had been determined, a final simulation could
be conducted to find the elastic properties of the crystals.
Figure 5.1: Example convergence of elastic constants with change in energy cutoff, using
GGA pseudpotential . c11 - blue, c12 - green, c44 - red.
Figure 5.1 shows an example convergence test for the GGA pseudopotential methodology. These tests were conducted to determine the optimum energy cutoff and k-point
mesh density for the analysis. In this case, it is sufficient to use a plane wave cutoff of 350
eV.
5.3
Results
The results of the analyses are shown in the following tables. Each pseudopotential yields
a set of elastic constants. These elastic constants are used to calculate the engineering
constants for each metal, which are presented in a separate table.
First, the results of the LDA pseudopotential modelling is presented. This is followed
by the results of the GGA, PAW-LDA and PAW-PBE models. The accuracy of the elastic
143
properties calculated from the DFT modelling is appraised relative to that found in the
literature. For convenience, this is shown in table 5.1.
Element
Group
Period
c11
c44
c12
ν110
Na
1
3
7.59
4.3
6.33
-0.44
K
1
4
3.71
1.88
3.15
-0.43
Rb
1
5
2.96
1.6
2.44
-0.4
Cs
1
6
2.47
1.48
2.06
-0.46
Ca
2
4
22.8
14
16
-0.26
Sr
2
5
15.3
9.9
10.3
-0.26
Ba
2
6
12.6
9.5
8
-0.29
V
5
4
230
43.1
120
0.45
Nb
5
5
245
28.4
132
0.61
Ta
5
6
264
82.6
158
0.17
Cr
6
4
348
100
67
0.32
Mo
6
5
465
109
163
0.4
W
6
6
523
160
203
0.28
Fe
8
4
230
117
135
-0.06
Rh
9
5
413
184
194
0.07
Ir
9
6
580
256
242
0.1
Ni
10
4
247
122
153
-0.07
Pd
10
5
221
70.8
171
-0.05
Pt
10
6
347
76.5
251
0.21
Cu
11
4
169
75.3
122
-0.13
Ag
11
5
122
45.5
92
-0.09
Au
11
6
191
42.2
162
-0.04
Al
13
3
108
28.3
62
0.27
Pb
14
6
48.8
14.8
41.4
-0.19
Table 5.1: Experimentally observed elastic constant data for elemental cubic metals.
Data taken from Landolt-Bornstein (1985). Poisson’s ratio in the [110] direction
calculated from these data.
144
Symmetry
Group
Lattice const.
Lattice error %.
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
LDA
Element
5.3.1
Na
b
1
1.71
4.26
8.26
6.22
7.19
-8.83
-44.65
-13.59
K
b
1
1.84
4.02
3.98
2.65
3.33
-7.28
-40.96
-5.71
Rb
b
1
1.59
3.76
3.03
1.93
2.62
-2.36
-20.62
-7.38
Cs
b
1
1.96
-1.49
2.73
1.69
2.18
-10.53
-14.19
-5.83
Ca
f
2
2.75
1.47
22.30
14.30
15.10
2.19
-2.14
5.63
Sr
f
2
3.02
0.76
12.20
8.40
7.30
20.26
15.15
29.13
Ba
b
2
2.08
4.24
14.20
11.90
8.87
-12.70
-25.26
-10.87
V
b
5
1.46
3.05
312.90
58.15
146.30
-36.04
-34.92
-21.92
Nb
b
5
1.65
-0.01
280.50
-86.17
145.10
-14.49
403.41
-9.92
Cr
b
6
2.02
-1.04
561.80
89.10
150.00
-61.44
10.90
-123.88
Mo
b
6
1.77
1.27
523.70
96.00
179.10
-12.62
11.93
-9.88
W
b
6
1.40
1.27
578.00
143.00
210.00
-10.52
10.62
-3.45
Fe
b
8
1.90
2.70
110.00
218.30
472.60
52.17
-86.58
-250.07
Rh
f
9
1.88
0.82
482.80
208.50
214.42
-16.90
-13.31
-10.53
Ir
f
9
1.94
-1.20
563.10
234.00
216.90
2.91
8.59
10.37
Ni
f
10
1.76
-0.02
256.70
112.10
177.50
-3.93
8.11
-16.01
Pd
f
10
1.98
-1.87
192.70
60.00
148.90
12.81
15.25
12.92
Pt
f
10
2.00
-1.85
289.30
54.98
209.80
16.63
28.13
16.41
Cu
f
11
1.82
-0.94
171.00
62.70
126.50
-1.18
16.73
-3.69
Ag
f
11
2.08
-1.93
98.04
24.90
83.50
19.64
45.27
9.24
Au
f
11
2.09
-2.54
131.00
6.24
132.50
31.41
85.21
18.21
Al
f
13
2.02
0.13
113.60
45.20
53.60
-5.19
-59.72
13.55
Pb
f
14
2.52
-1.64
56.50
20.40
33.50
-15.78
-37.84
19.08
Table 5.2: Elastic properties of elemental metals calculated using LDA pseudopotentials.
145
ν11̄0,110
ν11̄0,110
ν010,100
K
G
E
Error ν11̄0,110 %
Error ν010,100 %
Error K %
Error G %
Error E %
-0.62
-0.44
0.47
7.55
6.22
1.57
29.19
-2.34
-11.8
-44.65
14.46
K
-0.51
-0.43
0.46
3.55
2.65
0.95
15.7
0.79
-6.29
-40.96
-15.78
Rb
-0.55
-0.4
0.46
2.76
1.93
0.6
27.66
-2.63
-5.48
-20.62
20.51
Cs
-0.41
-0.46
0.44
2.36
1.69
0.79
-13.42
2.36
-7.59
-14.19
-33.15
Ca
-0.26
-0.26
0.4
17.5
14.3
10.11
-2.78
2.09
4.2
-2.14
-5.24
Sr
-0.22
-0.26
0.37
8.93
8.4
6.73
-16.84
6.96
25.35
15.15
3.96
Ba
-0.33
-0.29
0.38
10.65
11.9
7.38
11.91
1
-11.68
-25.26
-15.55
V
0.47
0.45
0.32
201.83
58.15
219.68
5.14
7.08
-28.83
-34.92
-48.72
Nb
4.34
0.61
0.34
190.23
-86.17
181.56
85.94
2.63
-12.12
403.41
-19.01
Cr
0.56
0.32
0.21
287.27
89.1
498.58
42.65
-30.53
-78.8
10.9
-52.77
Mo
0.5
0.4
0.25
293.97
96
432.42
19.62
1.82
-11.49
11.93
-13.68
W
0.38
0.28
0.27
332.67
143
466.07
26.26
4.69
-7.43
10.62
-13.82
Fe
1.29
-0.06
0.81
351.73
218.3
-656.74
104.85
-119.32
-111.04
-86.58
604.65
Rh
0.1
0.07
0.31
303.88
208.5
350.91
26.34
3.78
-13.81
-13.31
-21.43
Ir
0.13
0.1
0.28
332.3
234
442.47
28.8
5.55
6.31
8.59
-1.13
Ni
-0.09
-0.07
0.41
203.9
112.1
111.58
14.07
-6.88
-10.61
8.11
14.14
Pd
-0.04
-0.05
0.44
163.5
60
62.89
-43.83
0.08
12.88
15.25
12.42
Pt
0.28
0.21
0.42
236.3
54.98
112.92
24.17
-0.15
16.5
28.13
17.15
Cu
-0.06
-0.13
0.43
141.33
62.7
63.42
-110.75
-1.42
-2.66
16.73
4.92
Ag
-0.12
-0.09
0.46
88.35
24.9
21.23
19.52
-6.99
13.39
45.27
50.51
Au
-2.14
-0.04
0.5
132
6.24
-2.25
98.22
-9.57
23.11
85.21
105.33
Al
0.13
0.27
0.32
73.6
45.2
79.23
-114.46
12.1
4.83
-59.72
-26.22
Pb
0.1
-0.19
0.37
41.17
20.4
31.56
287.08
18.9
6.16
-37.84
-192.33
Expt
Element
Na
Table 5.3: Engineering constants of elemental metals calculated using LDA
pseudopotentials
146
Figure 5.2: Poisson’s ratio in the off-axis direction from cubic elemental metals,
calculated from LDA pseudopotential. (ν12 DFT - Filled squares, ν12 Expt. - Hollow
squares, ν110 DFT - Filled triangles, ν110 Expt. - Hollow triangles)
The elastic properties calculated using LDA pseudopotentials are shown in tables 5.2 and
5.3, Poisson’s ratio is shown in figure 5.2.
The predicted lattice constants are all within 5% of the experimentally measured value
with the exception of Niobium. The predicted elastic constants vary in accuracy. General
agreement with c11 is good, and similar to those calculated by the GGA pseudopotential.
Group one and two elements show a good agreement with the exception of Strontium.
The results for the transition metals show less agreement, particularly for Cr and Fe. The
predicted values for c11 show better agreement than for c12 and c44 .
The Poisson’s ratios for the elemental metals, calculated using the LDA pseudopotentials generally shows agreement, where the Poisson’s ratio is observed to be negative
experimentally, this is shown in the pseudopotential calculations, similarly this is seen
where Poisson’s ratio is positive. Iron and Lead are exceptions to this, where the DFT
calculations are unable to predict the Poisson’s ratio. For the engineering constants, bulk
modulus is calculated to within a 25% error, aside from for Iron and Chromium. Shear
modulus shows a lesser agreement, even for the group one and two metals, but is still
within a 50% error for most of the elements. Finally, the Young’s modulus shows the
greatest agreement with only Iron, Lead, Silver and Gold not being able to be predicted
to within a reasonable degree of accuracy.
147
Symmetry
Group
Lattice const.
Lattice error %.
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
GGA
Element
5.3.2
Na
b
1
4.19
0.96
8.26
6.22
7.19
8.11
30.87
11.96
K
b
1
5.29
-1.09
3.98
2.66
3.33
6.78
29.24
5.41
Rb
b
1
5.67
-1.45
3.03
1.93
2.62
2.31
17.10
6.87
Cs
b
1
6.16
-1.74
2.11
1.35
1.82
-17.06
-9.63
-13.19
Ca
f
2
2.75
1.47
22.30
14.30
15.10
-2.24
2.10
-5.96
Sr
f
2
3.02
0.76
9.39
5.40
5.20
-62.94
-83.33
-98.09
Ba
b
2
5.01
0.12
12.20
10.50
7.73
-3.28
9.52
-3.49
V
b
5
2.99
1.04
265.80
4.21
132.60
13.47
-923.75
9.50
Nb
b
5
4.19
-26.95
249.20
-27.50
131.40
1.69
203.27
-0.45
Ta
b
5
3.30
0.43
275.60
63.00
157.30
4.21
-31.11
-0.44
Cr
b
6
2.79
3.27
482.80
83.79
130.50
27.92
-19.35
48.6
Mo
b
6
3.15
-0.10
462.00
86.70
156.00
-0.65
-25.72
-4.48
W
b
6
3.17
-0.38
516.80
130.10
189.90
-1.20
-22.98
-6.89
Fe
b
8
2.77
3.56
66.10
166.50
35.83
-247.96
29.73
-276.78
Rh
f
9
1.88
0.82
393.60
170.10
174.00
-4.93
-8.17
-11.49
Ir
f
9
1.94
-1.20
563.10
234.00
216.90
-3.00
-9.40
-11.57
Ni
f
10
1.76
-0.02
256.30
112.10
177.30
3.63
-8.83
13.705
Pd
f
10
1.98
-1.87
192.70
60.00
148.90
-14.69
-18.00
-14.84
Pt
f
10
2.00
-1.85
289.30
54.90
209.80
-19.94
-39.34
-19.63
Cu
f
11
1.82
-0.94
171.04
62.84
126.50
1.19
-19.83
3.55
Ag
f
11
2.08
-1.93
131.70
6.24
132.60
7.37
-629.17
30.61
Au
f
11
2.09
-2.54
131.60
5.72
132.50
-45.14
-637.76
-22.26
Al
f
13
2.02
0.13
113.60
45.20
53.60
4.93
37.39
-15.67
Pb
f
14
2.52
-1.64
56.50
20.40
33.50
13.63
27.45
-23.58
Table 5.4: Elastic properties of elemental metals calculated using GGA pseudopotentials
148
ν11̄0,110
ν11̄0,110
ν010,100
K
G
E
Error ν11̄0,110 %
Error ν010,100 %
Error K %
Error G %
Error E %
-0.62
-0.44
0.47
7.55
6.22
1.57
29.19
-2.34
-11.8
-44.65
14.46
K
-0.51
-0.43
0.46
3.55
2.66
0.95
15.87
0.79
-6.29
-41.33
-15.78
Rb
-0.55
-0.4
0.46
2.76
1.93
0.6
27.66
-2.63
-5.48
-20.62
20.51
Cs
-0.55
-0.46
0.46
1.92
1.35
0.42
15.87
-1.84
12.75
8.78
28.86
Ca
-0.26
-0.26
0.4
17.5
14.3
10.11
-2.78
2.09
4.2
-2.14
-5.24
Sr
-0.1
-0.26
0.36
6.6
5.4
5.68
-155.05
11.42
44.87
45.45
18.94
Ba
-0.35
-0.29
0.39
9.22
10.5
6.2
16.7
0.13
3.29
-10.53
2.86
V
0.94
0.45
0.33
177
4.21
177.53
52.52
2.92
-12.98
90.23
-20.19
Nb
1.59
0.61
0.35
170.67
-27.5
158.47
61.57
1.4
-0.59
196.83
-3.87
Ta
0.34
0.17
0.36
196.73
63
161.29
49.35
2.95
-1.76
23.73
-10.71
Cr
1.03
0.32
0.98
87.93
83.79
-252.76
68.77
-506.46
45.27
16.21
177.45
Mo
0.49
0.4
0.25
258
86.7
383.24
18.27
2.75
2.15
20.46
-0.75
W
0.37
0.28
0.27
298.87
130.1
414.74
24.63
3.9
3.49
18.69
-1.29
Fe
-0.68
-0.06
0.35
45.92
166.5
40.91
90.85
4.96
72.45
-42.31
68.56
Rh
0.1
0.07
0.31
247.2
170.1
286.92
26.63
4.08
7.42
7.55
0.72
Ir
0.13
0.1
0.28
332.3
234
442.47
28.8
5.55
6.31
8.59
-1.13
Ni
-0.09
-0.07
0.41
203.63
112.1
111.3
15.18
-6.9
-10.47
8.11
14.35
Pd
-0.04
-0.05
0.44
163.5
60
62.89
-43.83
0.08
12.88
15.25
12.42
Pt
0.28
0.21
0.42
236.3
54.9
112.92
24.36
-0.15
16.5
28.24
17.15
Cu
-0.06
-0.13
0.43
141.35
62.84
63.48
-108.36
-1.41
-2.67
16.55
4.84
Ag
-1.56
-0.09
0.5
132.3
6.24
-1.35
93.91
-16.7
-29.71
86.29
103.15
Au
-1.62
-0.04
0.5
132.2
5.72
-1.35
97.65
-9.32
22.99
86.45
103.19
Al
0.13
0.27
0.32
73.6
45.2
79.23
-114.46
12.1
4.83
-59.72
-26.22
Pb
0.1
-0.19
0.37
41.17
20.4
31.56
287.08
18.9
6.16
-37.84
-192.33
Expt
Element
Na
Table 5.5: Engineering constants of elemental metals calculated using GGA
pseudopotentials
149
Table 5.4 displays elastic properties calculated using the GGA pseudopotentials, table 5.5
shows the engineering constants derived from these data. In addition the Poisson’s ratio,
in the (11̄0,110) direction, is shown in figure 5.3.
The calculated lattice parameters for the cubic metals are all within 5% of the experimentally predicted value with the exception of Niobium. The failure of DFT to
successfully predict the elastic properties of Niobium was encountered by Louail et al.
(2004) who investigated the elastic constants of transition metals and found the calculated
values for Niobium to be erroneous. No explanation of this is offered by Louail, however
it is thought that the electronic configuration of transition metals being such that the 3d
and 4s orbitals are both partially occupied (Hayward, 2002) results in the DFT approximation not being able to fully model the behaviour of these elements without a specific
non-general approach.
Both the bulk modulus and Young’s modulus are accurately predicted by the GGA
methodology to within 10% error in most cases. Poisson’s ratio in the off-axis direction
shows good agreement with the experimental result. The potential correctly predicts
whether Poisson’s ratio is negative or positive for all of the elements except Lead. For most
elements, the error is less than 30%, and generally it can be seen that the smaller group
one and two body-centred cubic elements show greater agreement with experimental
data than the face-centred cubic group 11-14 elements.
150
Figure 5.3: Poisson’s ratio calculated from GGA pseudopotential. (ν12 DFT - Filled
squares, ν12 Expt. - Hollow squares, ν110 DFT - Filled triangles, ν110 Expt. - Hollow
triangles)
151
Symmetry
Group
Lattice const.
Lattice error %.
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
PAW LDA
Element
5.3.3
Na
b
1
4.06
4.13
10.01
7.13
8.89
-31.9
-65.87
-40.38
Rb
b
1
5.37
3.86
3.91
2.29
3.5
-32.08
-43.41
-43.43
K
b
1
5.73
3.73
0.52
4.41
2.08
85.9
-134.73
34.06
Cs
b
1
19.53
5.22
0.57
2.17
2.58
76.78
-46.35
-25.3
Ba
b
2
4.77
4.93
14.08
11.89
8.84
-11.74
-25.21
-10.48
Ca
f
2
5.58
0
19.82
11.92
15.1
-15.04
-17.45
-5.96
Sr
f
2
5.8
4.68
16.62
14.01
5.2
7.92
29.33
-98.08
V
b
5
2.91
3.6
325.68
-5.1
159.71
-41.6
111.84
-33.09
Ta
b
5
3.25
1.88
313.87
64.9
177.67
-18.89
21.42
-12.45
Nb
b
5
3.26
1.11
277.5
10.96
154.58
-13.27
61.4
-17.11
Cr
b
6
2.78
3.52
586.43
93.99
166.9
-68.51
6.01
-149.1
W
b
6
2.91
7.87
1003.43
250.33
460.54
-91.86
-56.46
-126.87
Mo
b
6
3.11
1.42
530.48
98.47
190.53
-14.08
9.66
-16.89
Fe
b
8
2.69
6.11
97.23
207.14
443.57
57.72
-77.05
-228.57
Rh
f
9
3.77
0.9
500.55
219.4
174
17.49
16.13
-11.49
Ir
f
9
3.82
0.56
675.19
282.18
216.9
14.1
9.28
-11.57
Ni
f
10
3.42
2.79
319.11
141.08
177.3
22.6
13.53
13.71
Pd
f
10
3.85
0.91
266.38
83.4
148.9
17.04
15.11
-14.84
Pt
f
10
3.91
0.36
378.78
80.52
209.8
8.39
5
-19.64
Cu
f
11
3.52
2.4
221.26
84.54
126.5
23.62
10.93
3.56
Ag
f
11
4.01
1.84
155.14
39.28
128.27
21.36
-15.83
28.28
Au
f
11
4.06
0.45
199.42
26.63
132.5
4.22
-58.48
-22.26
Al
f
13
3.99
1.58
127.58
51.8
53.6
15.35
45.36
-15.67
Pb
f
14
4.91
0.8
67.16
23.68
33.5
27.33
37.51
-23.58
Table 5.6: Elastic properties of elemental metals calculated using PAW-LDA
pseudopotentials
152
ν11̄0,110
ν11̄0,110
ν010,100
K
G
E
Error ν11̄0,110 %
Error ν010,100 %
Error K %
Error G %
Error E %
-0.64
-0.44
0.47
9.26
7.13
1.65
31.66
-3.41
-37.2
-65.87
9.76
K
4.48
-0.43
0.8
1.56
4.41
-2.8
109.52
-73.97
53.28
-134.73
442.1
Rb
-0.6
-0.4
0.47
3.64
2.29
0.6
33.73
-4.53
-39.14
-43.41
20.04
Cs
1.55
-0.46
0.82
1.91
2.17
-3.65
129.7
-79.93
12.97
-46.35
712.02
Ca
-0.33
-0.26
0.43
16.67
11.92
6.76
21.21
-4.86
8.72
14.86
29.6
Sr
-0.2
-0.26
0.24
9.01
14.01
14.14
-26.04
40.76
24.74
-41.51
-101.63
Ba
-0.34
-0.29
0.39
10.59
11.89
7.26
13.5
0.69
-11.04
-25.21
-13.72
V
1.06
0.45
0.33
215.03
-5.1
220.59
58.12
4.03
-37.25
111.84
-49.33
Nb
0.84
0.61
0.36
195.56
10.96
166.89
27.71
-2.18
-15.26
61.4
-9.39
Ta
0.38
0.17
0.36
223.07
64.9
185.42
55.51
3.46
-15.38
21.42
-27.28
Cr
1.02
0.32
0.98
112.19
93.99
-325.55
68.49
-509.26
30.17
6.01
199.75
Mo
0.5
0.4
0.26
303.85
98.47
429.78
18.48
-1.81
-15.24
9.66
-12.99
W
0.35
0.28
0.31
641.5
250.33
713.67
20.24
-12.51
-107.16
-56.46
-74.29
Fe
1.27
-0.06
0.82
328.13
207.14
-630.41
104.92
-121.76
-96.88
-77.05
584.42
Rh
0.12
0.07
0.26
282.85
219.4
410.78
38.12
19.29
-5.94
-19.24
-42.14
Ir
0.14
0.1
0.24
369.66
282.18
569.71
33.26
17.41
-4.23
-10.23
-30.22
Ni
0.03
-0.07
0.36
224.57
141.08
192.45
349.36
6.62
-21.83
-15.64
-48.09
Pd
0.2
-0.05
0.36
188.06
83.4
159.6
126.8
17.81
-0.21
-17.8
-122.26
Pt
0.38
0.21
0.36
266.13
80.52
229.22
44.04
15.08
5.96
-5.26
-68.18
Cu
0.09
-0.13
0.36
158.09
84.54
129.23
247.26
13.24
-14.83
-12.28
-93.74
Ag
-0.05
-0.09
0.45
137.23
39.28
39.02
-95.22
-5.28
-34.54
13.67
9.03
Au
0.49
-0.04
0.4
154.81
26.63
93.63
107.78
13.02
9.82
36.9
-121.31
Al
0.14
0.27
0.3
78.26
51.8
95.87
-100.09
18.88
-1.2
-83.03
-52.71
Pb
0.17
-0.19
0.33
44.72
23.68
44.86
212.25
27.49
-1.94
-60.02
-315.5
Expt
Element
Na
Table 5.7: Engineering constants of elemental metals calculated using LDA-PAW
pseudopotentials
153
Figure 5.4: Poisson’s ratio: Calculated using PAW-LDA, for BCC metals (ν12 DFT - Filled
squares, ν12 Expt. - Hollow squares, ν110 DFT - Filled triangles, ν110 Expt. - Hollow
triangles)
Figure 5.5: Poisson’s ratio: Calculated using PAW-LDA, for FCC metals (ν12 DFT - Filled
squares, ν12 Expt. - Hollow squares, ν110 DFT - Filled triangles, ν110 Expt. - Hollow
triangles)
The elastic properties calculated using the LDA-PAW methodology are shown in tables
5.6 and 5.7. Poisson’s ratio is shown in figure 5.4.
With the exception of Tungsten and Iron, the lattice constant is calculated to less than
a 5% error. The elastic constants for these metals are also in poor agreement with the
experimental data. For group one metals, the elastic constant data shows c11 and c12 are
154
in closer agreement with the experimental data than c44 but the general agreement is poor.
The results for the transition metals generally shows agreement with less than 50% error
for the majority of the elements, in contrast to the GGA pseudopotentials.
The Poisson’s ratio does not reflect the sign for eight of the 24 elements. Poisson’s
ratio is however predicted within 30% error for the group one and group two elements
indicating that the methodology can determine the relative, if not absolute values of the
constants. Bulk modulus shows less than 30% error for all except Chromium and Iron,
shear modulus and Young’s modulus are also in good agreement for the group one and
two metals, but less so for elements in group 10 and above.
155
Symmetry
Group
Lattice const.
Lattice error %.
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
PAW PBE
Element
5.3.4
Na
b
1
4.19
0.95
8.59
6.27
7.55
11.61
31.4
16.12
Napv
b
1
4.19
0.95
0.57
10.41
11.67
-1223.6
58.71
45.75
Nasv
b
1
4.19
0.95
9.42
6.45
7.94
19.42
33.37
20.25
Nbpv
b
1
3.29
0.30
258
10.04
142.1
98.85
84.07
98.28
Nbsv
b
1
3.29
0.30
258
10.04
142.1
98.85
84.07
98.28
Nbsv new
b
1
3.29
0.30
258
10.04
142.1
98.85
84.07
98.28
Cs
b
1
5.73
5.29
-2.1
-2.02
-2.19
217.54
173.21
194.1
Cssv
b
1
5.73
5.29
2.08
2.54
3.22
-18.53
41.69
35.97
Ca
f
2
5.56
0.36
22.78
14.74
15.57
-0.1
5.04
-2.78
Capv
f
2
5.58
0.00
21.17
14.02
13.83
-7.72
0.16
-15.7
Casv
f
2
5.58
0.00
20.83
13.84
13.64
-9.46
-1.16
-17.26
Srsv
f
2
6.08
0.00
12.9
8.82
7.86
-18.63
-12.23
-30.98
Ba
b
2
4.77
4.98
16.02
12.45
9.63
21.36
23.72
16.96
V
b
5
2.91
3.64
337.6
8.87
171.4
31.87
-386.11
30
Vpv
b
5
2.91
3.64
345.7
24.81
177.3
33.46
-73.74
32.32
Vsv
b
5
2.99
0.99
359.5
52.63
172.2
36.02
18.11
30.33
Vsvh
b
5
2.99
0.99
259.3
-1.37
127.3
11.31
3257.3
5.72
Vsv new
b
5
2.99
0.99
279.6
26.77
143.7
17.74
-60.99
16.52
Nb
b
5
3.29
0.30
258
10.04
142.1
5.02
-182.77
7.13
Ta
b
5
3.3
0.30
276.7
58.59
158.7
4.59
-40.98
0.46
Tapv
b
5
3.3
0.30
288
81.9
170.4
8.34
-0.85
7.28
Cr
b
6
2.78
3.47
600.2
109.4
176.2
42.02
8.63
61.98
Crpv
b
6
2.78
3.47
615.9
128.1
187.5
43.49
21.93
64.26
Crpv new
b
6
2.78
3.47
611.3
127.2
186.2
43.07
21.39
64.02
156
Element
Symmetry
Group
Lattice const.
Lattice error %.
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
Mo
b
6
3.11
1.27
538.8
107.8
196.6
13.7
-1.11
17.08
Mopv
b
6
3.15
0.00
478.6
106.3
175.5
2.83
-2.5
7.13
Mopv new
b
6
3.15
0.00
59.73
25
59.93
-678.45
-336.08
-172
Mosv
b
6
3.15
0.00
477.4
107.2
168.5
2.59
-1.67
3.26
W
b
6
3.13
0.95
603
157.7
235.6
13.27
-1.46
13.84
Wpv
b
6
3.17
-0.32
542.3
149.1
211.4
3.55
-7.32
3.97
Wpv new
b
6
2.91
7.91
1030
310.6
500.5
49.24
48.49
59.44
Fe
b
8
2.69
6.27
123.9
224.9
454.3
-85.61
47.98
70.28
Feh
b
8
2.69
6.27
1535
647
605.3
85.02
81.92
77.7
Fepv
b
8
2.69
6.27
122.4
234.9
461
-87.86
50.19
70.72
Fepv new
b
8
2.69
6.27
116.5
230
452
-97.41
49.14
70.13
Fesv
b
8
2.69
6.27
391.4
324.9
552.4
41.24
63.99
75.56
Rh
f
9
3.8
0.00
452.2
196.9
195.7
8.67
6.57
0.86
Rhnew
f
9
3.8
0.00
452.2
196.9
195.7
8.67
6.57
0.86
Rhpv
f
9
3.8
0.00
452.2
196.9
195.7
8.67
6.57
0.86
Rhpv new
f
9
3.8
0.00
452.2
196.9
195.7
8.67
6.57
0.86
Ir
f
9
3.83
0.26
663.5
285.1
270.5
12.59
10.19
10.54
Ni
f
10
3.52
0.00
234
101.7
157.7
-5.57
-20.01
2.98
Ninew
f
10
3.52
0.00
234
101.7
157.7
-5.57
-20.01
2.98
Nipv
f
10
3.52
0.00
234
101.7
157.7
-5.57
-20.01
2.98
Pd
f
10
3.89
0.00
249.5
83.64
188.3
11.42
15.35
9.17
Pdnew
f
10
3.89
0.00
238.4
78.99
180
7.28
10.36
5.02
Pdpv
f
10
3.89
0.00
250.3
84.17
190.1
11.69
15.89
10.03
Pdpv new
f
10
3.89
0.00
237.6
72.9
176.6
6.99
2.88
3.17
Pdvnew
f
10
3.89
0.00
238.4
78.99
180
7.28
10.36
5.02
157
Element
Symmetry
Group
Lattice const.
Lattice error %.
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
Pt
f
10
3.92
0.00
372.23
84
265.13
6.78
8.93
5.33
Ptnew
f
10
3.92
0.00
362.31
80.67
256.83
4.23
5.17
2.27
Ptpv
f
10
3.92
0.00
365.4
80.37
258.15
5.03
4.81
2.77
Ptpv ZORA
f
10
3.92
0.00
365.77
80.49
258.58
5.13
4.96
2.93
PtZORA
f
10
3.92
0.00
361.73
74.59
256.3
4.07
-2.56
2.07
Cu
f
11
3.61
0.00
167.24
57.87
123.41
-1.05
-30.12
1.14
Cu f
f
11
3.61
0.00
167.24
57.87
123.41
-1.05
-30.12
1.14
Cunew
f
11
3.61
0.00
167.24
57.87
123.41
-1.05
-30.12
1.14
Cupv
f
11
3.61
0.00
167.24
57.87
123.41
-1.05
-30.12
1.14
Cupv f
f
11
3.61
0.00
188.19
74.36
136.38
10.2
-1.26
10.54
Ag
f
11
4.08
0.24
132.67
37.02
109.45
8.05
-22.9
15.94
Agnew
f
11
4.08
0.24
126.36
34.29
104.62
3.45
-32.7
12.07
Agpv
f
11
4.08
0.24
129.21
34.85
106.75
5.58
-30.55
13.82
Au
f
11
4.07
0.25
205.08
35.19
190.5
6.86
-19.92
14.96
Aunew
f
11
4.07
0.25
196.27
29.82
184.43
2.68
-41.52
12.16
Al
f
13
4.04
0.25
118.56
48.89
57.39
8.91
42.11
-8.03
Pb
f
14
4.95
0.00
61.96
21.47
37.16
21.24
31.07
-11.41
Table 5.8: Elastic properties of elemental metals calculated using PAW-PBE
pseudopotentials
158
ν11̄0,110
ν11̄0,110
ν010,100
K
G
E
Error ν11̄0,110 %
Error ν010,100 %
Error K %
Error G %
Error E %
-1.14
-0.46
0.51
-2.16
-2.02
0.13
59.7
-12.19
198.32
236.60
77.9
Cssv
-21.87
-0.46
0.61
2.84
2.54
-1.8
97.89
-33.46
-29.26
-71.50
405.3
Na
-0.63
-0.44
0.47
7.89
6.27
1.53
30.2
-2.86
-16.93
-45.77
16.7
Napv
1.09
-0.44
0.95
7.97
10.41
-21.6
140.0
-109.61
-18.08
-142.19
1282.3
Nasv
-0.53
-0.44
0.46
8.43
6.45
2.1
17.2
-0.57
-24.91
-50.09
-17.8
Ba
-0.28
-0.29
0.38
11.76
12.45
8.79
-4.57
3.3
-23.39
-31.09
-37.6
Ta
0.37
0.17
0.36
198.05
58.59
160.97
53.73
2.64
-2.44
29.07
-10.5
Tapv
0.22
0.17
0.37
209.61
81.90
161.33
23.09
0.72
-8.42
0.84
-10.7
V
0.90
0.45
0.34
226.81
8.87
222.14
50.45
1.78
-44.77
79.43
-50.4
Vpv
0.75
0.45
0.34
233.42
24.81
225.47
40.27
1.1
-48.99
42.44
-52.6
Vsv
0.55
0.45
0.32
234.65
52.63
247.89
19.54
5.52
-49.77
-22.12
-67.8
Vsvh
1.02
0.45
0.33
171.29
-1.37
175.51
56.36
3.98
-9.33
103.17
-18.8
Vsvn
0.67
0.45
0.34
189.02
26.77
181.97
33.94
0.96
-20.65
37.88
-23.2
Nbpv
0.85
-0.40
0.36
180.74
10.04
156.98
146.9
21.4
-6816.07
-527.72
-20692.4
Nbsv
0.85
-0.40
0.36
180.74
10.04
156.98
146.9
21.4
-6816.07
-527.72
-20692.4
Nbsvn
0.85
-0.40
0.36
180.74
10.04
156.98
146.9
21.4
-6816.07
-527.72
-20692.4
Nb
0.85
0.61
0.36
180.74
10.04
156.98
27.99
-1.46
-6.53
64.64
-2.9
Cr
0.51
0.32
0.23
317.54
109.44
520.19
36.93
-40.6
-97.64
-9.44
-59.4
Crpv
0.46
0.32
0.23
330.28
128.09
528.35
29.89
-44.6
-105.57
-28.09
-61.9
Crpvn
0.46
0.32
0.23
327.90
127.21
524.31
29.85
-44.6
-104.09
-27.21
-60.65
Mo
0.47
0.40
0.27
310.65
107.81
433.73
13.28
-2.99
-17.82
1.09
-14.0
Mopv
0.42
0.40
0.27
276.52
106.34
384.35
4.60
-3.4
-4.88
2.44
-1.0
Mopvn
-1.02
0.40
0.50
59.86
25.00
-0.29
139.5
-92.95
77.30
77.07
100.0
Mosv
0.42
0.40
0.26
271.45
107.21
389.45
4.12
-0.5
-2.95
1.64
-2.4
Expt
Element
Cs
159
Element
ν11̄0,110
ν11̄0,110
Expt
ν010,100
K
G
E
Error ν11̄0,110 %
Error ν010,100 %
Error K %
Error G %
Error E %
W
0.35
0.28
0.28
358.07
157.70
470.62
20.02
-0.48
-15.63
1.44
-14.9
Wpv
0.33
0.28
0.28
321.68
149.09
423.68
14.67
-0.31
-3.88
6.82
-3.5
Wpvn
0.25
0.28
0.33
677.10
310.60
703.02
-10.02
-16.93
-118.65
-94.13
-71.7
Fe
1.39
-0.06
0.79
344.17
224.91
-589.96
104.49
-112.43
-106.50
-92.23
553.3
Feh
0.12
-0.06
0.28
915.32
647.01
1193.04
149.94
23.55
-449.19
-453.0
-816.8
Fepv
1.39
-0.06
0.79
348.15
234.90
-606.10
104.49
-113.64
-108.89
-100.77
565.7
Fepvn
1.37
-0.06
0.80
340.15
230.02
-602.19
104.55
-114.96
-104.09
-96.60
562.7
Fesv
-36.63
-0.06
0.59
498.71
324.91
-255.17
99.83
-58.24
-199.22
-177.70
296.1
Ca
-0.27
-0.26
0.41
17.97
14.74
10.14
1.46
-1.58
-1.65
5.04
5.3
Capv
-0.25
-0.26
0.40
16.27
14.02
10.24
-6.30
-4.35
-12.24
0.16
6.2
Casv
-0.25
-0.26
0.40
16.04
13.84
10.03
-5.02
-4.19
-13.89
-1.16
4.2
Srsv
-0.22
-0.26
0.38
9.54
8.82
6.94
-13.91
-6.22
-25.42
-12.23
-1.0
Rh
0.10
0.07
0.30
281.19
196.93
334.01
26.26
-5.82
5.05
6.57
13.5
Rhnew
0.10
0.07
0.30
281.19
196.93
334.01
26.26
-5.82
5.05
6.57
13.5
Rhpv
0.10
0.07
0.30
281.19
196.93
334.01
26.26
-5.82
5.05
6.57
13.5
Rhpvn
0.10
0.07
0.30
281.19
196.93
334.01
26.26
-5.82
5.05
6.57
13.5
Ir
0.11
0.10
0.29
401.52
285.06
506.85
14.54
-1.66
11.67
10.19
13.7
Ni
-0.06
-0.07
0.40
183.12
101.66
106.96
-16.34
5.00
-0.66
-20.01
-21.5
Ninew
-0.06
-0.07
0.40
183.12
101.66
106.96
-16.34
5.00
-0.66
-20.01
-21.5
Nipv
-0.06
-0.07
0.40
183.12
101.66
106.96
-16.34
5.00
-0.66
-20.01
-21.5
Pd
-0.04
-0.05
0.43
208.67
83.64
87.57
-24.46
-1.43
10.06
15.35
17.9
Pdnew
-0.04
-0.05
0.43
199.48
78.99
83.42
-39.56
-1.38
5.92
10.36
13.9
Pdpv
-0.05
-0.05
0.43
210.13
84.17
86.18
-1.24
-1.06
10.69
15.89
16.7
Pdpvn
0.02
-0.05
0.43
196.94
72.90
87.04
365.56
-2.32
4.71
2.88
17.459
Pdvnew
-0.04
-0.05
0.43
199.48
78.99
83.42
-39.56
-1.38
5.92
10.36
13.9
160
Element
ν11̄0,110
ν11̄0,110
Expt
ν010,100
K
G
E
Error ν11̄0,110 %
Error ν010,100 %
Error K %
Error G %
Error E %
Pt
0.21
0.21
0.42
300.83
84.00
151.66
1.52
-0.90
5.93
8.93
10.13
Ptnew
0.23
0.21
0.41
291.99
80.67
149.23
6.18
-1.18
3.08
5.17
8.67
Ptpv
0.23
0.21
0.41
293.90
80.37
151.64
9.68
-1.38
3.71
4.81
10.12
Ptpvzora
0.23
0.21
0.41
294.31
80.49
151.59
9.38
-1.35
3.84
4.96
10.09
Ptzora
0.26
0.21
0.41
291.44
74.59
149.16
19.28
-1.21
2.90
-2.56
8.63
Cu
-0.03
-0.13
0.42
138.02
57.87
62.44
-318.17
1.26
0.26
-30.12
-6.83
Cu f
-0.03
-0.13
0.42
138.02
57.87
62.44
-318.17
1.26
0.26
-30.12
-6.83
Cunew
-0.03
-0.13
0.42
138.02
57.87
62.44
-318.17
1.26
0.26
-30.12
-6.83
Cupv
-0.03
-0.13
0.42
138.02
57.87
62.44
-318.17
1.26
0.26
-30.12
-6.83
Cupv f
-0.08
-0.13
0.42
153.65
74.36
73.59
-70.16
0.22
10.40
-1.26
9.35
Ag
-0.09
-0.09
0.45
117.19
37.02
33.73
-2.82
4.90
12.96
-22.90
-27.19
Agnew
-0.09
-0.09
0.45
111.87
34.29
31.58
-10.38
5.09
8.82
-32.70
-35.82
Agpv
-0.08
-0.09
0.45
114.24
34.85
32.63
-20.98
4.97
10.71
-30.55
-31.48
Au
-0.26
-0.04
0.48
195.36
35.19
21.59
85.12
4.70
12.13
-19.92
-95.95
Aunew
-0.27
-0.04
0.48
188.38
29.82
17.56
86.01
5.27
8.87
-41.52
-140.89
Al
0.10
0.27
0.33
77.78
48.89
81.12
-162.10
-11.81
0.58
42.11
22.61
Pb
0.12
-0.19
0.37
45.43
21.47
34.09
263.49
-22.42
3.43
31.07
68.33
Table 5.9: Engineering constants of elemental metals calculated using PAW-PBE
pseudopotentials.
161
Figure 5.6: Poisson’s ratio: Calculated from PAW-PBE, for BCC metals (ν12 DFT - Filled
squares, ν12 Expt. - Hollow squares, ν110 DFT - Filled triangles, ν110 Expt. - Hollow
triangles)
Figure 5.7: Poisson’s ratio: Calculated from PAW-PBE, for FCC metals (ν12 DFT - Filled
squares, ν12 Expt. - Hollow squares, ν110 DFT - Filled triangles, ν110 Expt. - Hollow
triangles)
The PAW-PBE parameterisation offers a range of pseudopotentials, each with a differing
form to reflect either a differing model for the valence electrons of the element, or a
differing assumption. The results for each available pseudopotential form are shown in
tables 5.8, 5.9 and figures 5.6 and 5.7.
The PAW-PBE data show that the from of the pseudopotential can have a large effect
on the calculated properties of the metals. In general, the calculated lattice constant is the
same for each of the pseudopotentials, only the elastic constants change. It is evident from
the data that for many of the pseudopotential variations that there is a trade-off between
c11 , c44 and c12 . An example of this is Vanadium, where the sv form predicts values with
the lowest average errors, whereas the updated form gives a greater accuracy for c11 and
c12 at the expense of the c44 prediction.
Using the form of the pseudopotential which gives the most accurate constants relative
to the experimental, it is possible to appraise the potential in the same fashion as the
162
alternate methodologies. The lattice constant is within 5% for each of the crystals. For
the group one and two metals, agreement within 30% error for all constants is achieved,
with the exception of Niobium. The variation in potential form also allows this level of
accuracy to be determined for the majority of the transition metals, and group 10 and 11
elements. As with the other methodologies, the value for c44 shows a far lower degree of
accuracy that that for the other two constants.
The sign (whether negative or positive) for Poisson’s ratio is correctly predicted.
Notable exceptions to this are Iron and Lead, where Poisson’s ratio is not successfully
predicted by any of the potential forms. The magnitude of Poisson’s ratio is also generally
predicted to within 30% error for the majority of the metals. The engineering constants
are not successfully predicted for Niobium, Iron and Caesium.
5.3.5
Magnetic moment consideration
For Chromium, Iron and Nickel, the DFT analysis is unable to correctly model the elastic
properties of the material to an acceptable degree of accuracy. These are ferromagnetic
materials. The analysis conducted has not considered the magnetic moment (spin) of
these materials, which could explain the anomalous results. These elements do not obey
the Madelung rule. The analyses for these elements is repeated, considering the magnetic
moment of the atoms and the results shown in tables 5.10 and 5.11. These analyses
were conducted using the built in functionality of VASP, however for completeness, an
overview of the techniques employed is given in appendix A.
163
Element
PSP
Symmetry
Lattice const.
c11
c44
c12
Error c11 %
Error c44 %
Error c12 %
Ni
LDA
FCC
3.52
359.5
215.8
160.3
31.3
43.5
4.6
Ni
GGA
FCC
3.52
380.7
231.8
174.4
35.1
47.4
12.3
Ni
PAW-LDA
FCC
3.52
275.8
159.6
129.6
10.4
23.6
-18.1
Ni
PAW-PBE
FCC
3.52
217.0
118.5
54.0
5.29
16.72
-36.5
Fe
LDA
BCC
2.87
215.4
114.2
59.3
-6.8
-2.5
-127.8
Fe
GGA
BCC
2.87
156
105.3
70.1
-47.5
-11.1
-92.7
Fe
PAW-LDA
BCC
2.87
218.2
108.9
68.7
-5.4
-7.4
-96.6
Fe
PAW-PBE
BCC
2.87
225.6
121.1
63.5
-1.9
3.4
-112.7
Cr
LDA
BCC
2.85
467.6
120
74.9
25.6
16.7
10.5
Cr
GGA
BCC
2.85
488.7
129.5
85.5
28.8
22.8
21.6
Cr
PAW-LDA
BCC
2.85
487.2
132
88.2
28.6
24.2
24
Cr
PAW-PBE
BCC
2.85
467.6
125.1
74.7
25.6
20.1
10.3
Table 5.10: Elastic properties of elemental metals calculated using four pseudpotentials
164
PSP
ν11̄0,110
ν11̄0,110
ν010,100
K
G
E
Error ν11̄0,110 %
Error ν010,100 %
Error K %
Error G %
Error E %
GGA
-0.08
-0.07
0.31
243.2
231.8
271.1
7.2
-21.7
24.2
47.4
52.1
Ni
LDA
-0.07
-0.07
0.31
226.7
215.8
260.7
-9.2
-24
18.7
43.5
50.1
Ni
PAWLDA
-0.06
-0.07
0.32
178.3
159.6
192.9
-25
-19.6
-3.4
23.6
32.6
Ni
PAWPBE
-0.03
0.07
0.30
161.5
146.5
193.61
-157.5
-27.4
-14.11
16.72
32.88
Fe
GGA
-0.13
-0.06
0.31
98.7
105.3
112.5
51
-19.3
-68.9
-11.1
-15.7
Fe
LDA
0.62
-0.06
2.04
80.5
84.1
-743.6
110.1
81.9
-107
-39.1
117.5
Fe
PAWLDA
0.06
-0.06
0.24
185.3
211.8
-54.5
-40.6
-7.4
29.8
-7.4
29.8
Fe
PAWPBE
0.02
-0.06
0.22
117.5
121.1
197.8
375.9
-68.5
-41.8
3.4
34.2
Cr
GGA
0.36
0.32
0.15
219.9
129.5
463.3
9.6
-8.4
26.9
22.8
29.6
Cr
LDA
0.37
0.32
0.14
205.8
120
447
12.5
-17
21.9
16.7
27
Cr
PAWLDA
0.35
0.32
0.15
98.7
105.3
112.5
7.2
-5.3
27.4
24.2
29.1
Cr
PAWPBE
0.35
0.32
0.14
205.7
125.1
447
8
-17.2
21.9
20.1
27
Expt
Element
Ni
Table 5.11: Engineering constants of elemental metals calculated using using four
pseudpotentials
For these methodologies, Poisson’s ratio is calculated for both Nickel and Chromium;
negative Poisson’s ratios are correctly modelled. For Iron, this is not the case, but the off
axis Poisson’s ratio value is of a very low magnitude, and this is captured by the analysis.
The inclusion of the magnetic moment in the analysis yields an improvement of the
calculated properties of the ferromagnetic materials for each of the potential type. The
results for Chromium give elastic constants within 30% for c11 and c12 . For LDA, all three
constants show improved values, for GGA, the value of c12 is much improved especially.
For the PAW analysis results values for c11 and c12 are improved with no detrimental
effect on the accuracy of the value of c44 . This gives far greater accuracy, with less than
30% error for both the elastic constant and derived engineering constants.
165
For Iron, the results show less agreement with the experimental properties, but still
an improvement over then analysis with no magnetic moment consideration. For LDA,
error in the analysis is reduced by a half when compared to the previous calculations.
Similarly, the results for GGA show an error of less than 50% for c11 and c44 , and less than
100% error for c12 , from the previous 276%. The PAW results show a similar trend with
c11 and c44 being predicted to less than 10% error, and c12 not being accurately modelled.
The result of this is that the derived engineering properties do not show good agreement.
The results for Nickel show a better agreement for the PAWs pseudopotentials, with
less than 30% error for c11 and c44 , and 35% error for c12 . The LDA and GGA pseudopotentials give a result with a greater degree or error. These erroneous results mirror those of
Louail et al. (2004), who found the elastic constants of Nickel could not be well predicted
by the DFT calculations employed.
5.4
Conclusion
Density functional theory techniques have been employed to predict the elastic properties
of elemental cubic metals. Unlike most classical potential modelling, density functional
theory is able to predict whether Poisson’s ratio is negative or positive. Differing pseudopotential methods are better suited to differing elements and in particular, transition
metals cannot be easily modelled. The usage of projector augmented waves provides a
methodology where the electronic structure can be accounted for enables a more accurate
prediction of the properties.
LDA and GGA pseudopotentials are able to predict the behaviour, however the PAW
methodologies provide far more accurate elastic constant data when compared to the
experimentally calculated properties.
The metals considered are a range of group one, group two and transition metals.
Generally, the small group one and two metals show a greater level of accuracy than
the transition metals. Additionally, for the ferromagnetic elements it is necessary to
consider the electron spin polarization. It should be noted that, as would be expected,
the calculated Poisson’s ratio and engineering constants are entirely dependent on the
accuracy of the elastic constant data. As with the classical potentials, C11 is generally well
166
replicated, but C12 and c44 show worse agreement.
The computational cost of the DFT simulations is considerably greater than that of
the simple potential models (hours rather than seconds), but the accuracy of the data
combined with the lack of dependence on empirical data gives a far better modelling
methodology.
167
Chapter 6
Finite Element Modelling of cubic
frameworks
6.1
Introduction
It has been shown in chapter 1 that crystal structures can exhibit negative Poisson’s ratio
behaviour. This chapter investigates whether by keeping the crystalline structures and
replacing the atomic bonding by linear-elastic beams it is possible to recreate the auxetic
behaviour at a scale independent level. Such beam networks could potentially find a variety of applications in situations where truss or beam networks are currently employed,
but where negative Poisson’s ratio, or tailored directional mechanical properties could
be advantageous. This section outlines the methodology used in this analysis before
presenting the structures considered and the corresponding results of these studies. It is
worth noting that the purpose of this study is not to expressly model or exactly recreate
the auxetic behaviour observed in the beam networks using the finite element method,
but to simply use these structures as a starting point to inspire scale independent network
structures. If successful, this approach could be applied to other complex atomic networks with unusual mechanical properties in order to create similar scale independent
structures.
The basis for this approach is to replace the nearest neighbour bonding in crystalline
structures with linear elastic beams in order to replicate the auxetic behaviour found in
168
crystal structures. A model that would allow bending of ribs rather that hinging at nodes
is used; the beams can both bend and axially stretch. This is justified because this type
of structure can be manufactured using both conventional and additive manufacturing
techniques, whereas a hinging design would be more costly and complicated to produce.
The following section describes how the beam networks are derived from the bonding
in body-centred, face- centred and simple cubic crystals. This is followed by an analytical
derivation of the deformation of the structures. Finally, the finite element method is used
to analyse the elastic properties of combinations of the different beam structures.
6.2
From crystalline bonds to beam networks
The crystallographic structures are obtained from experimental data published in the
literature. For convenience, the Crystallographic Database Service (CDS) (Fletcher et al.,
1996) is used to find the structural information for the crystals under consideration. As
well as providing information on the space group and lattice parameters of the crystals, it
is also possible to obtain structural information files that can be used both to graphically
display the structure and formulate the finite element model.
Having obtained the basic structure of the crystal, this data is used to construct
the network of beams in a finite element model. The atomic co-ordinates are used as
nodal points and the vectors corresponding to the nearest neighbour bonds provide the
geometry and location of the connecting beams.
There are several crystal space groups that exhibit negative Poisson’s ratio behaviour
however it is not possible to study all of these. This section will explore the possibility of
creating scale independent beam networks from cubic crystal inspired frameworks.
Nomenclature
The elastic properties of simple cubic crystals are discussed at length in previous chapters.
It has been shown both in the literature and also by classical and quantum modelling
techniques that negative Poisson’s ratio behaviour has been observed in elemental cubic
metal crystals. Cubic crystals have the highest symmetry and the least number of elastic
169
constants and thus form the logical starting point when trying to reproduce the negative
Poisson’s ratio behaviour observed in crystal as a scale independent beam network. The
cubic symmetry class contains the simple cubic, body-centred cubic and face-centred
cubic Bravais lattices.
(a)
(b)
(c)
Figure 6.1: Representative unit cell for the simple cubic (a), body-centred cubic (b) and
face-centred cubic (c) cells.
Figure 6.1 shows that the nearest neighbour bonding in simple cubic crystal is analogous to the second nearest neighbour bonding in body-centred and face-centred cubic
crystals. Auxetic behaviour is not seen in simple cubic crystals, but is in face-centred and
body-centred cubic crystals. Using the mechanism for body-centred cubic as a starting
point, the nearest first, second and third nearest neighbour bonds are modelled as beams
(see figure 6.2). For simplicity, simple cubic analogous beams are labelled as type I, facecentred cubic analogous as type II and body-centred cubic analogous as type III. In order
to tailor the Poisson’s ratio of the structure, the relative stiffness of each type of beam is
varied and the mechanical response of the structure calculated.
170
Figure 6.2: Structures under consideration, that form the macro scale beam network (left
to right) Type I - simple cubic analogous, Type II - face-centred cubic analogous, Type III
- body-centred cubic analogous.
6.3
Analytical derivation of mechanical properties of cubic beam
structures
In order to examine the deformation of the structures, each of the three beam configurations considered is modelled using analytical beam equations. In all cases, the in-plane
Poisson’s ratio of the structure is calculated for a strain in the [110] direction.
6.3.1
Assumptions used in analytical modelling
It is assumed that the structure is made up of linear elastic beams of solid, homogeneous,
isotropic circular cross section. These simple analytical models echo the modelling conducted in chapter 3, however in this case the linear springs are replaced with bending
beams.
In modelling the beams, assumptions must be made. Elastic deformations are assumed to be quasi-static; any inertial effects in the loading can be ignored, and only the
steady state solution is considered. Analytical beam modelling can follow two main
formalisms: either Euler-Bernoulli or Timoshenko. The Timoshenko approach improves
on Euler-Bernoulli by including terms for the shear deformation of the beams. However
as the modelling conducted in this work is limited to the static case for slender beams
where the deformation arises from bending and only for for small strains, this level of
sophistication is not necessary (Ashby and Gibson, 1997). Euler-Bernoulli beam theory
171
assumes that the cross-sectional area remains unchanged, and that the maximum stress
is less than the yield stress. This approximation provides a convenient but sufficiently
robust methodology for calculating the deformed structure and the resulting Poisson’s
ratio.
6.3.2
Type I
Figure 6.3: Type I structure, [110] strain
The strain in the [110] direction leads to an elongation of the top and bottom squares
of the frame, with an in-plane perpendicular contraction in each. For an applied strain,
rather than a uniformly distributed load, the axial displacement will be driven by this
deformation and thus for a displacement, δx , the Poisson’s ratio can be easily calculated.
From the symmetry of the structure, it can be seen that the displacement applied to each
beam is equal, and thus the total deformation can be found by considering the deformation
of one beam and using the symmetry to find the total deformation. Under symmetrical
loading, the beams can be considered as guided end cantilever beams, and beam theory
used to compute the displacement.
For a square configuration, the load is at an angle of 45◦ to the load direction (for [110]
loading). The load at each nodal point can be considered by summing the deflection of
each beam in turn.
Considering the top (or bottom) of the type I structure independently of the vertical
beams, we can compute the in-plane Poisson’s ratio for the structure. Considering the top
or bottom of the structure as a square framework (figure 6.4), made up of beams A,B,C,D,
connected at points 1,2,3,4, subjected to a load, P (the resultant of the force applied to the
172
vertical beams in the [110] direction). The symmetry dictates that this can be simplified
to a half and a quarter of the unit, see figure 6.4.
(a) Top/bottom frame
(b) Symmetric con-
(c) Single beam
ditions
Figure 6.4: Free body diagram of simplified structure (a): Top frame, (b):Symmetric
condition and (c): Single beam
Considering beam B, it is subjected to axial (Pax ) and transverse load (Plat ) (the components of force P, acting at an angle to the beam). Given that the of the load angle, θ for
a square configuration is 45◦ , Plat = Pax . The axial force is then exerted as the transverse
load on beam A, and an equivalent lateral displacement is seen. This loading is symmetrical about both sides, and thus the displacement at point 3 is equal to the sum of the y
displacements caused by deflection of beams A and B, and their symmetric equivalents.
The transverse load causes a bending deflection on the beam, the deflection of a slender
beam is given by
Plat =
p
2 sin
θ L3
12EI
,
(6.1)
where P is the load, L is the length of the beam, E is the Young’s modulus, I the second
moment of inertia and θ the angle of the load relative to the beam. This is a standard
solution as found in the literature (Roark and Young, 1982).
The Poisson’s ratio of the structure can be found from:
173
2δx
εlateral
=
εaxial
√point2 2
2lx
δy
point 3
√
.
(6.2)
2l2x
The Poisson’s ratio for the type I structure for a [110] displacement is +1.
6.3.3
Type II
Figure 6.5: Type II structure, [100] strain
The deflection of the type II structure in the [110] direction gives rise to purely axial
loading on beams A & D (see figure 6.6). This axial stretching dominates and gives an
in-plane Poisson’s ratio equivalent to that of the beams, thus almost zero. This loading
causes deflection of points in the z direction owing to bending of beams E & F, and K & L,
the reaction moment exerted by the junction of beams A, E & F, causing inwards (z plane)
curvature of beam A.
Figure 6.6: Load configuration for type II structure ([110] direction), side view
174
Figure 6.7: Load configuration for type II structure ([110] direction), top view
For a load, P, applied to beam A in the x direction the resulting strain ∆Ax can be
found:
∆Ax =
PAx
πr2
· LAx
E
(6.3)
The stretching of beam A causes no perpendicular in-plane deformation of beam B,
resulting in an in-plane deformation of zero, and thus an in-plane Poisson’s ratio of zero.
6.3.4
Type III
Figure 6.8: Type III cubic structure, [110] strain.
By straining the body-centred cubic analogous structure (figure 6.8), a stretch in the [110]
direction (points 2 and 4), results in a contraction in one perpendicular direction (points 1
and 3) which will in turn force an expansion in the corresponding perpendicular direction
175
(points 5 and 6) (the unit cell has been translated so that points 1 (or 3) correspond to the
centre of the cubic unit shown in figure 6.8). The plane on which the strain/contraction
occurs can be seen to be the same as that in figure 6.4(a).
Figure 6.9: Type III cubic structure, [110] strain, showing unit cell - points 1 and 3 are
central points of unit cell shown in figure 6.8.
The perpendicular plane undergoes the same deformation, rotated through 90 degrees,
the only variation in this case is that the deformation occurs in perpendicular planes and
thus the resulting load on beams A,B,C,D forces compression of beams EFGH at points 1
and 3, causing expansion at points 5 and 6. The symmetry of the structure dictates that
the deflection of all the component beams must be equal, and that thus the deflection
of the orthogonal points must also be equal. Using the methodology outlined in section
6.3.2, the deflection, γ at each point, and the resulting Poisson’s ratio can be found
γ=
P
4 sinθ
12 E I
176
l
(6.4)
Point
δx
δy
δz
1
0
0
γ
2
−γ
−γ
0
3
0
0
−γ
4
γ
−γ
0
5
0
γ
0
6
0
−γ
0
Table 6.1: Displacements of nodal points for BCC structure loaded in [110] direction
The strains are derived, and Poisson’s ratio computed:
2γ
−2γ
2γ
εy =
εz =
2l sinθ
2l θ
2l sinθ
εy
εz
νxy = − = 1 νxz = − = −1
εx
εx
εx =
6.3.5
(6.5)
(6.6)
Conclusion
The analytical modelling has shown the in-plane Poisson’s ratio behaviour for each of the
three beam types. Type I beams give a Poisson’s ratio of 1, type II zero and type III -1.
The analytical modelling gives an exact solution, however equations must be derived for
each case. Additional beams add to the complexity of the analysis. In order to investigate
the elastic properties of combinations of beam structures, the next section shows how the
finite element method can provide an automated, robust analysis tool.
177
6.4
Finite element modelling methodology
This section presents the finite element modelling of combinations of the beam networks
modelled in section 6.3. The parameters of the analysis are discussed, and the validity of
the results appraised. High throughput techniques are used to conduct analysis of beam
networks and the resulting elastic properties calculated. Finally the conclusions of this
study are presented.
6.4.1
Element type
When using the finite element method to perform calculations of beam networks, several
assumptions are made about the type of beams being used. The purpose of the study is
to investigate the feasibility of generating scale independent networks with negative, or
tailored Poisson’s ratio. The manufacturing techniques and constituent materials used to
assemble the networks are dependent on the scale of the structure being generated and
thus in this study, relative material properties are used. There are two possible methods
for varying the relative stiffness of the beams (EI); varying the Young’s modulus of the
constituent material (E), or varying the second moment of area, I by varying the thickness
(for circular beams, radius) of the beams. One dimensional or three dimensional elements
can be employed. Using one dimensional elements enables the linear-elastic deformation
of the beams to be captured at a low computational cost.
178
Figure 6.10: Schematic showing the deformed shape of a cantilevered beam. Top to
bottom - Blue - 1D beam elements, 3 nodes; Green - 1D beams elements, 15 nodes;
Orange - 3D tetrahedral elements, 1200 nodes; Red - 3D tetrahedral elements, 96000
nodes; Black - Analytical (Euler-Bernoulli) model.
6.4.2
Mesh convergence
To determine the validity of the finite element simulations, it is standard practice to conduct a mesh convergence. It is possible to capture the initial geometry of the beams with
one element however this does not fully replicate the deformed shape of the beams. The
mesh convergence determines the required mesh density in order to accurately simulate
the structure. A fixed geometry and loading condition (applied displacement in the (110)
direction) is issued the for analysis, and the Poisson’s ratio of the structure calculated
for increasing mesh densities, shown in figure 6.11. As the mesh density increases, the
Poisson’s ratio of the structure increases, as the finer mesh is able to replicate the lateral
and axial displacement of the component beams with greater accuracy (see figure 6.10)..
179
Figure 6.11: Evolution of Poisson’s ratio with mesh density for an arbitrary type II beam
network
6.4.3
Boundary Conditions
Boundary conditions within the finite element model determine the translational and
rotational constraints placed on the nodes within the model. When modelling crystalline
structures, it is normal to use periodic or Bloch boundary conditions as the number of
atoms in a crystal may run to several billion. The structures that are modelled in the
finite element study have geometries based on the crystalline structures, but all are of a
comparatively small number of unit cells. For this reason, true periodic boundary conditions are not imposed. A schematic representation of the applied boundary conditions is
shown in figure 6.12.
Figure 6.12: Schematic showing the boundary conditions applied for axial (left) and
shear (right) loading.
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6.4.3.1
Representative volume element
For this study a Representative Volume Element (RVE) is chosen as the unit cell, as shown
in figure 6.13. Whilst not the primitive cell of the reciprocal lattice, this structure was
chosen as it is the smallest unit cell that encompasses the unit cell shown by Baughman
et al. (1998). This RVE cell is subsequently verified against a larger lattice structure.
The RVE model enables rapid generation and analysis of a large number of models at
a comparatively low computational expense, as well as a simple way of visualising the
deformation mechanism of the structures.
Figure 6.13: Representative volume element, showing the Type I (red, dotted), Type II
(green, solid) and Type III (blue, dashed) component parts.
6.4.3.2
Panel model
An array of 10x10x2 RVE unit cells was constructed to represent a sandwich panel core
style structure. Not only does this serve as a verification of the RVE model, but also
shows how the RVE could be used in a sandwich panel or space-frame structure. Having
a large array also simplified the boundary conditions required for shear deformation, as
periodicity and edge effects can be disregarded.
181
6.4.3.2.1
Comparison of RVE and Panel Model
The panel model and RVE were modelled for the structures where the Type I, II and III
structures are dominant. The results are shown in table 6.2.
Type I
Type II
Type III
RVE
Panel
RVE
Panel
RVE
Panel
ν110
0.9
0.9
0
0
0.95
0.95
ν100
0
0
0.26
0.26
0.33
0.33
Table 6.2: Comparison of panel and RVE model
6.4.3.3
Material Properties
The combined structure was modelled in the finite element package. Each type of beam
was assigned discrete material properties, but the radius kept constant. The stiffnesses
were varied from 1 to 9000 arbitrary units. The relative stiffnesses of the beams control
the deformation of the structure, and govern the resulting Poisson’s ratio, shear modulus
and compressive modulus. The structure is scale independent and thus the units of force,
modulus and length are specified as relative, rather than absolute values. The dimensions
of the type one beams were specified as a length of 10, and a circular cross-section of 0.3,
kept constant for all three beam types.
Abaqus 6.8 (Dassault, 2008) was used to perform the finite-element analysis. Beams
were modelled as B31 type linear beam elements. Linear elastic stiffnesses were assigned
ranging across several orders of magnitude. Two geometries were used in the analysis in
order to determine the accuracy of the boundary conditions employed, a representative
volume element and a ’panel’ model.
6.4.3.4
Loading conditions
The structures under consideration were subjected to a known displacement in the [110]
direction and the deformations in plane were measured such that Poisson’s ratio of the
structure could be found. For the panel models, the structures are displaced at the nodes
along the flat faces, and the Poisson’s ratio calculated in plane. For comparison the
182
structures were also loaded in the [100] direction. Shear deformations were applied to the
flat faces of the panel model in both cases.For the RVE, load conditions were those used
in the analytic modelling, section 6.3.
6.4.4
6.4.4.1
High throughput techniques
Varying material properties
To conduct the high throughput analysis, the model is defined such that the each type of
beam can be assigned separate material properties. The input file from the initial model
is used as a template for generation of the large number of input files required to conduct
the study, and these can then be batch processed for analysis and post-processing. This
technique is computationally very efficient; the high level scripting languages Python or
Perl are used to edit the relevant text files.
6.4.4.2
Automatic generation of beam networks from crystallographic structure data
As outlined above, the commercial FEA code employed in this study, Abaqus, has the
facility for use of the Python scripting interface to generate and analyse models using
the finite element method. Having demonstrated the possibility of crystal structures to
provide a geometric framework for a negative Poisson’s ratio structure, an automated
methodology is used to construct these networks.
The crystalline structure of materials is well known from experimental data, and
repositories of such data are readily available from such sources as the Crystallographic
Database Service (Fletcher et al., 1996). This data is often available in the format of
rectangular coordinates which can then be used as a starting point for the geometry of
the finite element model.
The script is designed to be run from within Abaqus with minimal input, or knowledge
of the scripting interface from the user, and is designed to directly report stresses and
strains from compressive and shear tests in order to fully characterise the material.
The function of the code can be summarised as:
• The coordinate points are used to calculate the vectors that connect these points,
183
duplicate points are removed.
• The cutoff value is defined by the user, either as a nearest neighbour or a longest
length. This value is used to determine the lengths of the beams to be constructed.
Vectors longer than the cutoff value are discarded. Each vector is only constructed
once, and replicated in the part as necessary.
• The beams are drawn within the software.
• Each beam is translated and rotated to the correct position.
• The material properties for each length of beam can be individually assigned, or
assigned as a uniform value. In order to maintain slender beam assumptions, the
default diameter of the beams used in the calculations is
1
15
of the length value of
the shortest beam within the structure.
• The material properties are assigned to each beam, and the beams merged into
one uniform part such that the deformation modes are stretching and bending, not
hinging. Beam, not truss, elements are used so shear effects are included in the
analysis.
• Displacements are assigned for compressive strains in the 1,2,3 and shear in the 12,
23, 13 directions. Boundary conditions are such that the part is restrained in space,
but is free to expand/contract.
• After analysis, the stresses, strains and displacements (for Poisson’s ratio) are reported, from which elastic constants can be calculated.
6.4.4.3
Calculation of properties
Poisson’s ratios were calculated for deformations in both the [100] and [110] direction
to give ν(010,100) and ν(11̄0,110) . The behaviour was assumed to be linear elastic, and engineering strains were calculated. Data were taken from both the RVE and panel model for
comparison. When calculating the Poisson’s ratio in the (100) plane, the computed strains
were those in the [100] (load) direction and the [010] perpendicular in plane direction.
184
For loading in the [110] orientation, strains used were those in the [110] direction and the
in-plane perpendicular direction.
Loads were applied to edge nodes as a fixed displacement. Nodes were free to translate
in the corresponding orthogonal directions to avoid over constraining the structures. The
perpendicular displacements were recorded at the edge nodes of the structure.
The tensile and shear moduli were calculated from the panel model (rather than the
RVE) to avoid having to include periodic boundary conditions in the analysis periodicity
into the analysis. When calculating the moduli for cases where one, two, or three types
of beam are dominant the stiffness value of the dominant beams is the maximum value
of 9000, and the resulting moduli of the structure are in the same arbitrary units. Whilst
it may seem more sensible to normalise the values with respect to this maximum, this
would not allow comparison for values where the dominant class of beam does not have
the maximum stiffness value.
6.4.5
Validation of finite element modelling with analytical model
To validate the computational model, a direct comparison is made with the analytically
derived solution of the Poisson’s ratio. The result of this, compared to the finite element
analysis of the same structure is tabulated in table 6.3. Nodal displacements are shown;
for identical geometries, the displacements are equal (to three significant figures). This is
replicated in the other two structures.
Point
Analytical δx
Analytical δy
FE δx
FE δy
1
0
0
0
0
2
2.65
2.65
2.65
2.65
3
0
5.3
0
5.3
4
-2.65
2.65
-2.65
2.65
Table 6.3: Displacements of nodal points for Type I structure loaded in [110] direction.
Example geometry = length 100mm, diameter 1mm, Young’s modulus 1600MPa
185
6.5
6.5.1
Results of finite element modelling
Overview
Figure 6.14: Poisson’s ratio of structure for all considered cases, for strains in the [110]
direction. Horizontal axes show normalised comparative beam stiffness. Red region
shows structures where Type I beams dominate, green where Type II dominate, and blue
where III dominate.
This section provides a general overview of the Poisson’s ratio trends exhibited by the
structure when loaded in the [110] direction. The effects of varying two specific beam
types within the structure is analysed in order to characterise the specific role of each
beam type on the Poisson’s ratio, tensile and shear modulus. Comparisons are then
made between the behaviour of the structure in the auxetic and non-auxetic [100] load
directions. Data for the Poisson’s ratio was taken from the RVE model, and subsequently
verified against data from the panel model. Data for the elastic moduli were taken from
the panel model.
Figure 6.14 shows values for Poisson’s ratio for all combinations of beam stiffnesses
where the stiffness of the type I, II and III are varied by a factor of 20 (presented in
dimensionless units of beam stiffnesses normalised with respect to the stiffness of the
type three component).
As shown in section 6.3, each type of beam structure can be seen to have a differing
deformation mechanism resulting in a differing Poisson’s ratio. The influence of type I
beams on the deformation of the structure is to give a Poisson’s ratio of +1, the upper
186
bound for in-plane strains for a type I structure. Deformation of type II structures gives
a Poisson’s ratio of around zero when strained in the [110] direction. The deformation
mechanism of the structures where type III beams are dominant gives a Poisson’s ratio of
-1, as shown to be the lower limit by Baughman et al. (1998) (for an in-depth discussion
of Poisson’s ratio bounds in the cubic crystal system see Ting and Chen (2005)). The
in-plane negative Poisson’s ratio has an associated out of plane positive Poisson’s ratio
deformation. Increasing the stiffness of type I and II beams will reduce this effect as the
structure becomes stiffer.
6.5.2
Comparison of panel model and RVE
Panel model
The resulting structures for a positive, zero and negative Poisson’s ratio deformation is
shown in figure 6.15(a), 6.15(b) and 6.15(c). The panel model shows how the representative
volume element can be positioned, as would be employed in a sandwich panel or generic
truss structure. The Poisson’s ratio, relative shear modulus and relative Young’s modulus
of the structure are shown in figures 6.18, 6.19, 6.20. Absolute values are not given as
these are conditional on the parent material of the structure and manufacturing techniques
employed.
187
(a) ν = +1
(b) ν = 0
(c) ν = -1
Figure 6.15: In-plane deformed configuration of panel model showing (a) Positive
Poisson’s ratio, (b) Zero Poisson’s ratio and (c) Negative Poisson’s ratio (Deformed
green, Undeformed black)
RVE
Analysis of the RVE model allows the deformation mechanism to be observed in detail.
The deformation behaviour in these structures results from flexure of the beams in the
structure (see Fig. 2). The in-plane Poisson’s ratio for structures with all considered
relative component beam stiffnesses is shown in figure 6.16.
188
Figure 6.16: Deformed shape of the RVE model showing (left to right) Poisson’s ratio of
1, 0 and -1 for strains in the [110] direction (where load direction is perpendicular to
page. Deformed black, undeformed grey).
6.5.3
Elastic properties
Each substructure has a differing deformation mechanism under tensile load, thus by
varying the relationship of the component beam stiffnesses it is possible to vary the
Poisson’s ratio of the whole structure. If it is assumed that once the stiffness of one type
of beams is considerably lower than that of the other components its influence on the
deformation mechanism can be ignored then seven discrete types of substructure can be
identified:
• Structures comprising only type I, type II, or type III beams.
• Structures comprising type I and II, I and III, or II and III beams (see figure 6.17).
• Structures where all types of beams have an influence on the deformation mechanism.
189
(a)
(b)
(c)
Figure 6.17: Representative unit cell for (a) Type I and Type II; (b) Type I and Type III;
and (c) Type II and Type III Combined structures.
By considering each of the substructures independently it is then possible to analyse
the effect that each type of beams has not only the deformation mechanism and resulting
Poisson’s ratio of the structure but also on the tensile strength and shear moduli of the
combined structure. It should be possible to calculate the required material properties
to manufacture a structure with a desired Poisson’s ratio, shear and tensile modulus.
Considering structures where two types of beam are present enables the interaction of
the different sub structures to be analysed.
190
6.5.3.1
Poisson’s ratio
Figure 6.18: Poisson’s ratio of structure for cases where two types of beams are
dominant (strains are in the [110] direction). (a) Ratio: Type I/Type II structure. (b) Ratio:
Type I/Type III structure. (c) Ratio: Type II/Type III structure.
191
Considering the structures where two types of beam are dominant, trends can be observed
corresponding to the relative (thus dimensionless) stiffnesses of the component beams,
as one beam type increases in stiffness relative to the other. The Poisson’s ratio of these
substructures is shown in figure 6.18. The trend echoes that of the three component beam
structure; a type III dominated structure exhibits a negative Poisson’s ratio, type II a
Poisson’s ratio of around zero and type I a positive Poisson’s ratio.
192
6.5.3.2
Young’s modulus
Figure 6.19: Tensile modulus of structure for cases where two types of beams are
dominant. (a) Type I/Type II structure. (b) Type I/Type III structure. (c) Type II/Type III
structure.
193
Comparison of the maximum tensile moduli of the structures where two types of beams
govern the deformation shows that the maximum value for tensile modulus is found in
the type I/type II structure with a comparative modulus of 610, the type I/type III has a
comparative modulus of 395 and the type II/type III structure a comparative modulus of
561. This shows that the inclusion of type II beams in the structure leads to an increase in
the relative stiffness of the structure at the expense of auxeticity, as these beams link the
vertices displaced by the auxetic deformation.
Considering each substructure in detail, the influence of each individual type of beam
can be observed. Figure 6.19 shows the affect on tensile modulus of varying the component
beam stiffness for a type I and II; type I and III; and type II and III structure, it can be seen
that for a given ratio, the absolute magnitude of the stiffness increases as the magnitude
of the component beams increases, but each configuration shows the same trend.
For a fixed value of type II beams, increasing the stiffness of the type I beams by a
factor of nine has the effect of approximately doubling the modulus of the structure. For
a fixed value of type I beams, the effect of varying the type II beams by a factor of two is
to approximately double the stiffness of the structure, as the deformation mode is axial
stretching of these beams. As the initial value of the type I beam increases, this effect is
reduced slightly. A type I/type III structure shows a similar trend. The lower modulus
for values where the only one type of beam is present should be noted, as the solely type
III beam structure is far less stiff than one comprising type II beams alone. Also, for the
type I/type III structure, overall stiffness of the structure is far lower, highlighting the
compromise that must be reached when a negative Poisson’s ratio structure is sought.
The type II/type III structure shows that for a low value of type II beams, increasing the
stiffness of the type III beams has little effect on the overall modulus of the structure.
194
6.5.3.3
Shear modulus
Figure 6.20: Shear modulus of structure for cases where two types of beams are
dominant. (a) Type I/Type II structure. (b) Type I/Type III structure. (c) Type II/Type III
structure.
As with the tensile modulus, the maximum shear modulus of 383 is found in the type
I/type II structure, for a type I/type III structure Gmax =276 and for a type II/type III structure
Gmax =175. Considering each substructure independently, we see that for a type I/type
II structure the trend for shear is very similar to that seen in tension, at low values of
195
type II beams, increasing the stiffness of type II beams by a factor of nine increases the
global stiffness by a factor of around 2. For a type I/type III structure, and type II/type III
structures it should be again noted that the influence of the type II beams on the global
modulus is much higher than that of the type I beams.
6.5.3.4
Comparison with [100] axis
All previous data have been reported for the structure strained in the [110] direction.
For comparison, the structure is also loaded in the [100] direction. Comparisons can be
drawn between the structure when loaded in the extremal Poisson’s ratio, and the ’flat’
directions.
The Poisson’s ratio for seven asymptotic structures is shown in Table 1. It can be seen
that the Poisson’s ratio varies from around zero to around 0.33, but is always positive in
all cases.
The tensile moduli of the structures in the [100] orientation shows the same general
trend as in the [110] direction, however it should be noted that by rotating the unit cell,
the orientation of type I beams in the [100] direction means they are now loaded in the
same axis as type II beams were in the [110] direction (and conversely the orientation
of type II beams means that they now are loaded in the manner that type I were in the
previous orientation). It should also be noted that the flexural behaviour differs slightly
due to the change in relative beam lengths.
196
6.5.3.5
Summary of results
Structure
Tensile modulus
Shear modulus
Poisson’s ratio
Type I
583
1.8
0
Type II
397
192
0.26
Type III
3.25
5
0.33
Type I/II
1078
310
0.14
Type I/III
711
224
0.12
Type II/III
413
263
0.29
Type I/II/III
1135
673
0.17
Type I
2.5
42
0.9
Type II
415
152
0.01
Type III
177
5
-0.95
Type I/II
610
383
0.4
Type I/III
395
276
0.5
Type II/III
561
175
-0.2
Type I/II/III
905
453
0.12
[100] direction
[110] direction
Table 6.4: Comparative (dimensionless) values for tensile and shear modulus, and
Poisson’s ratio for maximum component beam stiffness.
Table 6.4 summarises the results. The first thing to notice in Table 1 is the increase in
global stiffness of the substructure by combination of two types of beams.
The second aspect is that while strains for [100] direction never result in a negative
Poisson’s ratio behaviour, deformations in the [110] direction give negative Poisson’s ratio
in cases where type III beams dominate.
The third point is that compromises can be reached where the structure is able to
demonstrate negative Poisson’s ratio by a combination of type II and type III beams
whilst maintaining a comparatively high tensile and shear modulus.
It is generally accepted that auxetic materials have a high shear modulus (see 1). For
197
these structures, the negative Poisson’s ratio behaviour is at the expense of the shear
rigidity as the mechanism depends on there being no cross linking within the structure.
6.5.3.6
Mechanical performance of beam networks relative to conventional honeycomb
Conventional (hexagonal) honeycombs are widely employed in engineering applications
due to their high stiffness and comparative low weight. Whilst their properties are well
known, and can be easily modelled (see, for example, Ashby and Gibson (1997)), the
Poisson’s ratio of such structures is fixed at 1 for regular hexagons, and always positive
for variations thereof. Re-entrant honeycombs (as discussed in section 1.1.3) can be
constructed with an in-Plane Poisson’s ratio of -1.
As with the structures presented in this work, the mechanical properties of honeycombs is dependent on the geometry and parent material of the structure. Engineering
design usually requires a low weight, high stiffness structure and thus to enable the comparison of the properties of differing honeycomb (or other substructures), performance is
usually expressed in terms of the properties of the structure relative to that of its parent
material. To enable comparison of the structures presented in this work with those of
commercially available honeycombs, relative density (ρR ), relative shear modulus (GR )
and relative Young’s modulus (ER ) will be used to describe the relationship between the
apparent properties (P∗ ) and the property of the parent material (Ps ). In order to normalise each of the structures, the parameters shown in table 6.5 are used. Whilst the finite
element analysis uses consistent units, units are shown for clarity. The relative material
properties can be calculated:
E∗
Es
G∗
GR =
Gs
ρ∗
ρR = .
ρs
ER =
198
(6.7)
(6.8)
(6.9)
Property
Value
Units
1000
mm3
0.1
mm
Es
1000
MPa
Gs
384
MPa
ρs
1
g mm−3
Apparent volume of unit cell
Radius of circular beams
Table 6.5: Parameters used in the finite element analysis, and subsequently in
determining normalised material properties.
ν110,11̄0
M (g)
ρ∗ (gmm−3 )
ρR
E∗ (MPa)
ER
G∗ (MPa)
GR
ER
ρR
GR
ρR
Type I
0.9
3.77
3.77E-03
3.77E-03
2.5
0
42
0.11
0.66
29.01
Type II
0.01
6.53
6.53E-03
6.53E-03
415
0.41
152
0.4
63.56
60.62
Type III
-0.95
1.78
1.78E-03
1.78E-03
177
0.18
5
0.01
99.6
7.33
Type I/II
0.4
10.3
10.3E-03
10.3E-03
610
0.61
383
1
59.23
96.84
Type II/III
0.5
8.31
8.31E-03
8.31E-03
395
0.4
276
0.72
47.55
86.53
Type I/III
-0.2
5.55
5.55E-03
5.55E-03
561
0.56
175
0.46
101.13
82.16
Type I/II/III
0.12
12.08
12.1E-03
12.1E-03
905
0.91
453
1.18
74.94
97.68
Structure
Table 6.6: Apparent and relative mechanical properties for structures where one, two
and three beams are present within the structure, and the component beams have equal
material properties (see table 6.5).
rho∗ (g/cm3)
ρR
E∗ (Mpa)
ER
G∗ (Mpa)
GR
ER
ρR
GR
ρR
0.18 mm
0.104
3.85E-02
1896.06
2.75E-02
275.79
1.08E-02
0.713
0.281
0.03 mm
0.147
5.46E-02
2895.80
4.20E-02
365.42
1.43E-02
0.769
0.263
0.04 mm
0.199
7.35E-02
4481.59
6.50E-02
448.16
1.76E-02
0.883
0.239
Gauge
Table 6.7: Apparent and relative mechanical properties for three examples of
conventional honeycomb structures. Data taken from Hexcel (2012).
Table 6.6 shows the normalised material properties of the structures. Table 6.7 shows
comparable data, taken from experimental measurements (Hexcel, 2012). The compressive and shear stiffness, as a function of the relative density of the structure is shown for
199
both the beam networks and the classical honeycombs. It can be seen that the relative
in-plane performance of the beam networks, when density is considered, is greater than
that for the conventional honeycombs, however the absolute mechanical properties of the
honeycombs are far superior, owing to the greater cross sectional area of the structures.
It should also be noted that many applications of cellular solids utilise the out of plane
properties of the structure (see for example Zhang and Ashby (1992)).
6.5.3.7
Conclusion
A structure consisting of a network of bending beams can exhibit a negative Poisson’s
ratio. The negative Poisson’s ratio behaviour is driven by the (bcc analogous) type III
beams, the type II (fcc like) beams result in a structure with a Poisson’s ratio of around
zero and type I (simple cubic configuration) beams result in a Poisson’s ratio of around +1.
The tensile and shear moduli of the type III beams can be augmented by addition of type
II and type III beams. By tailoring the relative stiffness of the component beams within
the structure it is possible to design an auxetic truss structure with a specific Poisson’s
ratio, shear modulus and tensile modulus.
200
Chapter 7
Discussion
7.1
Synopsis
This work has explored negative Poisson’s ratio in cubic structures using a variety of
analytic and modelling techniques and at different scales, from the atomic to the macroscopic.
The evidence for negative Poisson’s ratio in a variety of materials, both naturally occurring crystals and man made structures, is reviewed through a thorough analysis of the
literature. From this, it is apparent that while negative Poisson’s ratio in elemental cubic
metals has been considered by previous studies, no work has systematically explored this
phenomenon to the same extent, or used the same breadth of techniques as this thesis.
Previous studies have focussed on either experimental data, simple modelling of isolated
cases, or abstract mathematical consideration of the elastic tensors.
A mechanism for auxeticity in body-centred cubic materials was proposed as early
as 1998, but no mechanism for auxeticity in face-centred cubic metals has since been
determined, despite this being the more common behaviour. Away from the cubic system,
crystal structures of lower symmetries exhibit negative Poisson’s ratios; this has in rare
cases been experimentally observed directly, but in the majority of cases, the presence
of auxeticity in off-axis directions is calculated from either the experimentally observed
axial data, or data obtained from nano-indentation, as a result little is known about the
underlying mechanism of such behaviour.
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7.1.1
Classical Modelling
Models of differing levels of complexity can be used to explore elastic properties in general,
and auxeticity in particular, using the formalism of classical elasticity. The methodology
is conceptually simple as elastic stiffnesses can be obtained from the energy changes of
a few specific strains (three in the case of materials with cubic symmetry). The explicit
derivation of this approach, and its application to the study of auxetic media is shown
analytically for the simpler models of interaction. Table 7.1 displays the results from this
analytic work. The first model has a very simple harmonic volumetric dependence and
it shows that for an arbitrary cubic system, the Poisson’s ratio is positive in the axial
directions ((ν(100,010) ), and negative in the off axis direction ((ν(110,11̄0) ).
Modelling the first nearest neighbour bonding with linear springs results a more
sophisticated model which includes a description of the Bravais lattice of the structure.
For the body-centred cubic unit cell, Poisson’s ratio in the (110, 11̄0) direction is predicted
to be negative while in the face-centred cubic crystal, this is not the case and the Poisson’s
ratio is found to be positive, albeit possibly zero. This supports the hypothesis that the
negative Poisson’s ratio behaviour in BCC crystals is driven by the first nearest neighbour
bonds, but in FCC crystals additional bonding must be taken into account to allow
auxeticity.
ν(010,100)
ν(11̄0),(110)
ν(001,110)
Volume
1
2
-1
0
Spring - FCC
1
3
0
1
2
Spring - BCC
1
2
-1
2
Lennard-Jones - FCCnn1
1
3
0
1
2
Lennard-Jones - BCCnn1
1
2
-1
2
Lennard-Jones - FCCnn2
0.36
-0.09
0.62
Lennard-Jones - BCCnn2
0.53
-1.55
2.93
Potential
Table 7.1: Poisson’s ratio calculated for each of the potential models
Replicating the methodology applied to the simple linear springs, first nearest neighbour bonding is represented by the Lennard-Jones potential; a two-body potential that
202
accounts for the repulsive component within the atomic bonding. For the face-centred
cubic cells, negative Poisson’s ratio is not allowed, and for the body centred cubic unit
cells, the predicted value was -1. Adding second nearest neighbour bonding to the model
in this analytical model yields more realistic results with auxeticity being more prevalent.
For the body-centred cubic crystal, the anisotropic character of the deformation is captured, and the Poisson’s ratio is calculated to be -1.5 with a corresponding perpendicular
positive Poisson’s ratio at 2.9. For the face-centred cubic cell the in plane Poisson’s ratio
is finally negative, at -0.09.
The Lennard-Jones model not only gives a more accurate representation of the bonding
than the simple spring models as it includes a term for the modelling repulsive component
but by permitting second nearest neighbour bonding, it is possible to further improve the
accuracy of the model. Adding further neighbours increases the complexity of the model
and thus analytical derivations are abandoned in favour of computational methods to find
numerical solutions to the analyses. The potentials considered are the two-body Morse
potential (fitted to both BCC and FCC crystals) and the many-body Finnis-Sinclair (fitted
to BCC metals), Cleri-Rosato and Sutton Chen potentials (both fitted to FCC metals). The
fitted values for the elemental cubic metals are obtained from the literature and these are
used to investigate the off-axis Poisson’s ratio behaviour of the elements. Preliminary
cutoff convergence studies are carefully conducted to determine the optimal range for the
potentials.
The Morse data shows that the two-body potential gives values where the c12 and
c44 are equal, and thus always predicts negative Poisson’s ratio behaviour. The manybody potentials yield far more interesting results. It is possible to successfully model
the experimentally observed negative Poisson’s ratio with the many-body potentials, but
unlike the two-body potentials, these are also capable of modelling positive Poisson’s
ratio behaviour where it exists.
From this classical potential modelling, possible mechanisms for the negative Poisson’s ratio behaviour are derived. As previously mentioned, the mechanism for the
auxetic behaviour in BCC metals has been discussed in the literature but no mechanism
for the auxetic behaviour in face-centred cubic crystals has been suggested to date. By
203
using the deformation computed for negative Poisson’s ratio in the FCC crystals, a possible mechanism is found and detailed. The ability of the potentials to model Poisson’s
ratio is summarised in table 7.2.
BCC
FCC
ν+
ν-
ν+
ν-
Morse
7
3
7
3
Finnis-Sinclair
3
7
-
-
Sutton-Chen
-
-
3
3
Cleri-Rosato
-
-
3
3
Potential
Table 7.2: Ability of each potential to predict Poisson’s ratio
To understand the underlying cause of the auxetic behaviour, the numerical approach
was applied to the simple spring methodology used earlier. Cubic, covalently bonded
compounds are selected for the analysis as the bonding in these crystals is more directional
in character and thus better suited to being modelled in this way. A simple model, based
on the methodology of Keating, and originally used to investigate the properties of
Diamond is employed. The parameters of the potentials are fitted to the elastic constant
data available in the literature, and the calculated spring constants normalised to give a
comparative stiffness value for the axial and tangential bonding stiffnesses.
This analysis shows that, for the systems considered, the fitted constants are within
an acceptable margin for error and able to reproduce the elastic properties. As the ratio
of the axial to lateral stiffness energies increased, the Poisson’s ratio also increased.
The classical potential modelling has been conducted to determine whether these
relatively simple and computationally inexpensive models are able to accurately predict
negative Poisson’s ratio in cubic elemental metals, and whether any insight could be
gleaned as to what governs whether or not a crystal exhibiting auxetic behaviour. It
is apparent that for most elements a correctly fitted many-body potential can predict
Poisson’s ratio, but the accuracy is wholly dependent on the accuracy of the fitting
parameters and the suitability of the potential form.
204
7.1.2
Ab-initio modelling
To model the Poisson’s ratio of the elemental cubic metals without such a dependence on
the experimental data, density functional theory techniques have been employed. These
first principal techniques have been used to determine the elastic constants for a range
of materials. As with the classical potentials the exact form of the modelling, specifically
the pseudopotential used, has an effect on the accuracy of the solution. For ferromagnetic
elements it is also necessary to include the spin polarisation to better replicate the elastic
properties. Inclusion of the Projector augmented wave forms also yields an improved
solution, and in the majority of cases, can not only accurately predict the sign of Poisson’s
ratio but also its magnitude. The density functional theory approach, whilst incurring a
greater computational expense gives a more robust modelling methodology when predicting the elastic constants of the elemental metals. These techniques are able to predict
both positive and negative Poisson’s ratio behaviour, but the success of the analysis is
dependent on the accuracy of the c44 term.
7.1.3
Mechanical modelling
Finally, the information derived from the modelling of the interatomic interactions in the
crystals has been used to inspire the scale independent beam networks that are explored
using the finite element method. The beam networks are directly based on the first,
second and third nearest neighbour interactions in cubic crystals.
By replicating each of the component bonds as linear elastic bending beams of varying
elastic (Young’s modulus), structures with tailored elastic properties have been generated.
The same principles of how the bonding in the physical crystals can drive the auxetic
behaviour seem to hold. Simple cubic analogous (type I) beams give a positive Poisson’s
ratio, face-centred cubic analogous (type II) beams give Poisson’s ratio of zero and bodycentred cubic analogous (type III) beams lead to a negative Poisson’s ratio. Elastic and
shear stiffnesses of the structure are also governed by the relative component beams. By
generating structures where each component beam type has a differing elastic stiffness it is
possible to construct an array of beam with tailored mechanical properties: Poisson’sratio,
shear stiffness and axial stiffness.
205
7.2
Further Work
Extensions of the work conducted within this thesis should follow two streams.
• Further analysis of the beam networks modelled in this work
• Extension of the modelling techniques employed to investigate auxetic behaviour
in crystals of lower symmetry
7.2.1
Exploration of auxetic behaviour in crystals
This work has provided an analysis of the suitability of modelling techniques to the
observation of negative Poisson’s ratio. It shows that techniques that are able to model the
structure of a material to a high degree of accuracy are in some cases unable to successfully
predict one or more of the elastic constants, in particular c44 . As a result, the calculation
of Poisson’s ratio is not possible with any degree of certainty for some combinations of
material and methodology. As the symmetry of the crystal under consideration increases,
the number of elastic constants and thus the potential for inaccuracies in the analysis
increases.
In extending this methodology to other symmetries of crystal, it would be prudent to
start with structures where the symmetry of the structure provides a unit cell that can be
replicated and give a panel geometry that has planar edges - some of the lower symmetry
crystals will not provide such unit cells. In doing so, it will subsequently be possible to
characterise the structures using both finite element and experimental methods.
Consideration should be given to both the magnitude and orientation of the Poisson’s
ratio that has been observed experimentally, examples of such materials are discussed at
length in chapter 1. Structures such as MOF-5 and alpha cristobalite lend themselves well
to being the starting framework. Modelling of these materials could in the future yield
interesting structures with novel mechanical properties.
206
(b) α-cristobalite
(a) MOF 5
Figure 7.1: Possible network structures of MOF5 and α-cristobalite.
Classical potential modelling, whilst simple and not infallible has proven useful in
understanding the underlying mechanism of these behaviours. Density functional theory techniques have proved more reliable in determining the magnitude and direction of
negative Poisson’s ratio behaviour and would provide a good starting point when considering materials where the existence of auxetic behaviour is based on nano-indentation
or other techniques subject to a high degree of experimental uncertainty.
7.2.2
Beam networks
The beam networks investigated using the finite element method can be produced from
a variety of manufacturing methods, dependent on the scale required. For manufacture
where cell sizes are at the centimetre scale, additive manufacture would be most appropriate, even if materials anisotropy is present. Comparative beam stiffnesses could be
achieved either by varying the radius of the beams or by using a system that has the ability
to use different component materials to give a parent material with variable mechanical
properties. Experimental validation of the finite-element modelling at a variety of scale
could provide a starting point for the application of these structures.
Using the methodology developed in this work to explore beam networks based on
crystals of lower symmetries could yield beam networks with further enhanced mechanical properties; lower Poisson’s ratio structures could be found, based on crystals with a
high degree of anisotropy, or networks that are completely auxetic may be possible, based
on some silicate style frameworks. Further analysis could be conducted to investigate the
out-of-plane properties of the structures.
207
7.3
Key Findings
This work has explored negative Poisson’s ratio behaviour in cubic elemental metals using
a full range of numerical methods not employed in previous studies. The success of these
methods in predicting this behaviour has been appraised, and inferences drawn from
this analysis. Each of the methodologies employed have shown negative Poisson’s ratio
behaviour exists in cubic elemental metals. Additionally, it was shown using classical
potential modelling that auxetic behaviour in covalent cubic crystals can be successfully
modelled.
Fundamental analysis, using linear springs, showed the ’default’ behaviour of the
structure within the crystals using first nearest neighbour bonds; body-centred cubic
crystals exhibit negative Poisson’s ratio behaviour, face-centred cubic have a Poisson’s
ratio of around zero.
Classical potential modelling methodologies have demonstrated that two-body potentials, whilst perfectly adequate at producing structural and on-axis properties, are
highly unsuitable for calculating off-axis properties due to them being governed by the
Cauchy relation. Additionally, it was shown that increasing the ’many-body’ character of
the potential, increases the accuracy.
The experimental data found in the literature shows that a greater proportion of facecentred cubic crystals exhibit negative Poisson’s ratio than body-centred cubic, further
investigation into this reveals that as a general trend, the magnitude of the negative
Poisson’s ratio in the face-centred cubic is far lower than that of the body-centred cubic.
The negative Poisson’s ratio behaviour has been shown to derive from the body-centred
cubic bonding in the crystals. The interaction of the differing nearest neighbour bonding
types governs the magnitude of the Poisson’s ratio. Comparison of many-body and
two-body classical potential techniques supports this hypothesis. Even for a face-centred
cubic system, increasing the potential range gives a lower value of Poisson’s ratio. For
the body-centred cubic system, this is also true.
Investigation of the deformation mechanism of the face-centred cubic structures using
simple analytical springs, classical potential modelling and beam networks all shows
that the Poisson’s ratio of these bonds alone is around zero. Similarly, analysis of the
208
body-centred cubic frameworks shows a Poisson’s ratio of -1. The contribution of the
additional bonding within the structure is what appears to govern the negative Poisson’s
ratio behaviour in the face centred cubic crystals, but the long range of these bonds results
in a lower magnitude.
The finite element modelling showed that is possible to transfer the crystallographic
mechanism into a beam network, despite the difference in nature between the atomic
bonds in an elemental metal (or springs in simple models) and the linear elastic beams.
Not only does this give confidence in the findings of the atomistic modelling, but additionally, shows that for unusual elastic properties atomic scale principles can be applied
at the macro scale.
209
Appendix A
Methodology of DFT
A.1
Introduction
The following section of this thesis is limited to a pragmatic explanation of the background, methodology and factors affecting the efficacy of the technique.
A.2
A.2.1
Background
The Schrödinger Equation
DFT is a technique for calculating the electron density and subsequent material properties by solving an approximation to the Schrödinger equation. Solving the Schrödinger
equation for a realistic, many ion, many electron system is a great challenge due to the
large number of degrees of freedom and can only be achieved using approximations and
numerical methods.
The time-independent Schrödinger equation takes the form:
−
h2 2
∇ Ψ + VΨ = EΨ
2m
(A.1)
∂2
∂2
∂2
+
+
.
∂x2 ∂v2 ∂z2
(A.2)
where:
∇=
The potential operator, V, and the wavefunction, Ψ, of the system are functions of the
210
particles positions, thus equation A.1 can be re-arranged to give:
!
h2 2
− ∇ + V Ψ = EΨ
2m
(A.3)
2
h
where − 2m
+ V is the Hamiltonian operator, Ĥ. The Hamiltonian operator represents
the total energy of the system expressed as a function of the coordinates of the particles
and their conjugate moments and can be described as the sum of the kinetic and potential
energies.
A.2.2
Born Oppenheimer Approximation
The Born-Oppenheimer approximation states that as the electrons are much lighter than
nuclei and move at a greater velocity, their motion can be de-coupled from that of the
nuclei. This allows the wavefunctions to be split into the wavefunction of the nuclei and
the wavefunction of the electrons. The wavefunctions for the nuclei can then be treated
classically, and be a part of the hamiltonian for the electronic wave function.
A.2.3
Hohenberg-Kohn Theorem and Kohn-Sham Theorem
The electronic wavefunction is still a complex many-dimensional mathematical object.
Further steps must be taken to simplify the problem. The next step is to note that
the ground state of a quantum system can be calculated from electronic density alone
(Hohenberg et al., 1964). This dependence on the electronic density is referred to as the
Hohenberg-Kohn theorem.
This is powerful, as the complex diagonalisation procedure to obtain the eigenvalues
of the the Hamiltonian associated with the direct solution of Schrödinger’s equation is
replaced by a one to one mapping from electronic density to ground-state energy. In
other words, the ground state energy is a functional of the density, (a functional being a
function which takes a function as its input argument). Still, the exact form of this energy
functional is not known and is very difficult to approximate for systems of interacting
electrons.
The last, crucial approximation consists in replacing the system of interacting electrons
211
by a system of non-interacting electrons of the same density. As they are independent,
each electron has its own Shrödinger equation, where the effective potential operator Ve f f
must take into account the interaction with the other electrons (Kohn et al., 1965). Using
these assumptions, the energy functional of the DFT system reduces to:
Z
E[ρ] = TNI [ρ] +
ρ(r)ν(r)dr +
1
2
Z Z
ρ(r’)
drdr’ + Exc [ρ]
|r − r’|
(A.4)
This then gives the energy for a system as a sum of the non-interacting particle (Kohn
Sham) kinetic energy (TNI ), the external potential, the classically treated coulombic interacting energies and the exchange correlation functional. The exchange correlation functional captures everything that has been neglected, chiefly the quantum only exchange
energy. However this is still a functional at this stage, and therefore still a complex
mathematical object. A lot of the success of DFT can be attributed to the fact that this
exchange-correlation functional can be approximated quite easily. The two most common
schemes are the Local Density Approximation (LDA), and General Gradient Approximation (GGA), but more complex ’hybrid’ or schemes exist. The local density approximation
assumes that the density can be treated as that of a homogeneous electron gas, and the
exchange-correlation potential becomes the integral of the product of the electronic density by an exchange correlation function (derived from accurate homogeneous electron
gas calculations).
Z
ELDA
xc
= [ρ] =
ρ(r)xc (ρ)dr
(A.5)
There are many GGA approaches, but in general, they include a contribution due to
the gradient of the electron density, and can be more successful in situations where the
electron density varies significantly through the system.
A.2.4
Self consistent loop
There is an apparent shortcoming in the algorithm outlined previously. It possible to
access the electronic density from summing the squares of the the wave functions obtained
212
Figure A.1: Flow diagram of DFT process
from the Kohn-Sham equations, but the exchange correlation terms in these equations
depends on the electronic density. A well known solution to such correlated problems
is to use a so called ’self consistent loop’. The usual idea consists of starting from a
reasonable electronic distribution, based, for instance, on independent atoms, derive
the Kohn-sham equations, solve them and compare the resulting density with the input
density. If the difference is lower than a user chosen threshold, the loop is terminated, and
the observables (usually energy) can be calculated. If not, a new density is formed from
mixing the input and output densities (different mixing schemes exist), and the process
repeated, as shown in figure A.1.
213
A.2.5
Hellman-Feynman Theorem
The Hellman-Feynman (Feynman, 1939; Hellmann, 1941) theorem relates the derivative
of the total energy to the derivative of the Hamiltonian and allows the forces in the system
to be obtained. Once the electronic distribution has been found, classical electrostatics
can be used to find the forces in the system. In classical mechanics, force at a position R
can be expressed in terms of the energy:
F = −∇R U(R)
(A.6)
In quantum mechanics, this can be expressed as:
F = −∇R hEi
(A.7)
where:
hEi =
hΨHΨi
hΨΨi
(A.8)
To find hEi for the wave function Ψ, it can be expanded in terms of a set of fixed basis
functions (equation A.9), and then the energy can be minimised with respect to ci .
|Ψi =
X
ci |ϕi i
(A.9)
i
Using plane waves as the basis functions the general expression for the forces becomes:
*
+ X
*
+
X
∂hEi
∂ X ∗ ∗
∂H
∗
∗
=
ci ϕi (r)|H|
c j ϕ j (r) =
ci c j ϕi (r)|
|ϕ j (r)
∂R
∂R
∂R
i
j
(A.10)
i,j
The forces are then calculated from the from the expansion coefficients used to minimise the energy, reducing the computational cost.
The choice of functional can determine the accuracy of the result; the LDA has a
tendency to underestimate the ground state energy, the GGA to overestimate it. As a
consequence, care must be taken when selecting the methodology to be employed.
214
A.2.6
Choice of Pseudpotentials
Although it is by no means necessary, most modern implementations of DFT use an
additional approximation, which consists of incorporating the non valence electrons into
a pseudo-core, which replaces the original atomic nucleus. The benefits are obvious;
the number of electrons is reduced to just the valence electrons. The cost is that the
simple external, coulombic electron-nucleus potential is replaced by a quite complex
electron-core pseudopotential. These pseudopotential are generated by conducting full
electrons DFT calculations on atomic systems. Pseudopotentials are intimately linked
to the functional that they have been generated from: It does not make sense to use a
pseudopotential generated by full electron atomic LDA-DFT to simulate a crystal using
GGA-DFT for instance. Many schemes and implementations exist, but it goes beyond the
scope of this thesis to discuss them. In this work, psedupotentials of the LDA and GGA
form have been used.
The choice of pseudopotential can determine the accuracy of the result; the LDA
has a tendency to underestimate the ground state energy, the GGA to overestimate. An
alternative recent refinement to the pseudopotential methodology is to use so called
Projector Augmented Waves (PAW). PAW methodology uses the total energy functional,
rather than a potential, and the forces derived are from the total energy. The auxiliary
wave functions are transformed by a transformation operator such that the auxiliary
wave functions can be mapped onto the true wave functions and the total energy can
be expressed in terms of the auxiliary functions. Use of the PAW methodology can give
more accurate results with no increase in computational expense.
A.2.7
Periodicity, reciprocal space and k-point mesh
As customary in solid state physics (Ashcroft and Mermin, 1976), the crystal is assumed
to be an infinite periodic structure, as even a small crystal may span several billion atoms.
The periodic nature of crystals means that it is possible to simulate the large scale by
considering the reciprocal (spatial frequency) space where small wavevectors correspond
to large periods. In fact only one such reciprocal unit cell has to be considered, the first
Brillouin zone to be exact. The integrals over the infinite system are replaced by the
215
integrals over the Brillouin zone in reciprocal space (Bloch Theorem) (Bloch, 1929). At a
finite number of points within the Brillouin zone, the function can be evaluated. These
points are known as the k-point mesh. The mesh density and configuration of the k-point
mesh will determine the validity of the result and thus a convergence test is required in
order to determine the density at which results can be considered to be valid.
A.2.8
Basis, plane waves and energy cutoff
The actual calculations on wave functions (solutions of Kohn-Sham differential equations)
are carried out using linear algebra, by projecting the wave function onto a basis set, and
dealing with the resulting coordinates. Depending on the nature of the system, different
methods exist. For instance, for molecules where the electrons are well constrained, it is
convenient to use localised functions, usually centred on the atoms. Gaussian functions
are very common. The more such functions are included, the better the wave functions
are described.
In the case of periodic systems, a natural basis set is provided by simple plane waves
(spatial sine). Here again, the quality of the basis set is controlled by the number of functions: how small their period is allowed to be, or how high their frequency. Traditionally,
this maximum frequency is converted to energy, and this maximum energy is referred to
as the cutoff. As in the case of the k-point mesh, the energy cutoff must be optimised by
a preliminary convergence study.
A.2.8.1
Magnetic moment consideration
The magnetic moment is modelled using the inbuilt functionality of VASP. Ferromagnetism can be described by a simple Stoner model (Stoner, 1938) to relate the energies of
the spin up and spin down electron bands to the strength of the exchange correlation.
Is n↑
N
Is n↓
E↓ (k) = E(k) −
N
E↑ (k) = E(k) −
216
(A.11)
(A.12)
where Is is the Stoner parameter that describes the energy reduction due to electronic
spin correlation and n↑ and n↓ are the density of up and down spins.
The spin excess, R, can be found from
R=
n↑ − n↓
N
(A.13)
The energy can then be expressed as
E↑,↓ (k) = Ẽ(k) − Is
R
2
(A.14)
where
Ẽ = E(k) −
Is (n↑ + n↓ )
.
2N
(A.15)
The spin excess, R, can then be found from Fermi statistics
R=
1 X
f↑ − f↓ (k)
N
(A.16)
k
f↑,↓ = [exp(Ẽ(k) ± Is
R EF −1
−
)] .
2 kT
(A.17)
From a Taylor expansion:
1 X δ3 f (k)
1 X δ f (k)
(Is R) −
(I R)3 + ...
3 s
N
24N
δ
Ẽ(k)
δ
Ẽ(k)
k
k
Z
X δf !
V
−V
=
δk(−δ(Ẽ − EF )) =
DEF
3
δE
2
(2π) N
R=
(A.18)
(A.19)
k
where D is the density of states at the Fermi level. The Stoner condition for ferromagentism is dependent on the densty of states per atom per spin (D(EF )) and the Stoner
parameter.
If
217
D̃(EF ) =
V
D(EF )
2N
(A.20)
then
R = D̃(EF )Is R − O(3)
(A.21)
where O describes the third order terms in the analysis.
−O(3) = R(1 − D̃(EF )Is )
(A.22)
for the spin excess, R, to be positve 1 − D̃(EF )Is must be negative and thus for a material
to be ferromagnetic:
D(EF )Is > 1.
(A.23)
This is valid for Fe, Co and Ni. This band theory approach enables the magnetisation
owing to the mobile d band electrons to be described.
218
Appendix B
The Finite Element Method
B.1
Introduction
In this study the finite element method is employed to evaluate the mechanical response of
the structures that are derived from the crystalline structures. The finite element method
solves large geometric problems, often involving non-linear partial differential equations
which can be reduced to a linear algebraic problem. The finite element method can be
used to solve a variety of field problems (structural, thermal, magnetic) but this work
concentrates solely on structural analysis problems.
B.2
Methodology
B.2.1
Outline
The methodology of the finite element method may be described as three discrete stages;
pre-processing, analysis and post-processing. A general outline of each of these given
below, and a more specific description of the techniques employed in this thesis is given
in chapter 6.
Many commercial codes are available for conducting finite element analysis, and it
is not uncommon for users to write their own code tailored to specific applications. The
work conducted in this thesis uses the commercial finite element code Abaqus and thus
this introduction is tailored to the nuances of the particular package.
219
B.2.2
Pre-processing
The pre-processing stage compiles an input file based on information specified by the user.
The requires specification of the geometry, materials, loading and boundary conditions
and analysis type.
B.2.2.1
Geometry
Accurate description of the geometry under consideration is vital for a correction solution
to the finite element analysis problem. Geometry may be specified using either the
graphical user interface (GUI), or by calculating the vertices and connecting units within
the structure and using the Python scripting interface to assign the geometry to the model.
The geometry is then used to specify the mesh, the array of nodal points and connecting
elements that will be analysed. The geometric problem is discretized into a number of
finite elements (the mesh), each bounded by nodes (meshless techniques are also available
for specific analyses). For models containing multiple interacting parts, contact elements
can be used to describe the interface between surfaces.
The elements that make up the mesh may be: one dimensional where they describe a
beam, truss or wire; two dimensional where they describe a plate; or three dimensional
where they describe a continuum. Each element is bounded by nodes at the vertices (or
ends for 1d elements) and in higher order elements, may have central or mid side nodes.
Formulation errors are caused by incorrectly modelling the problem, and thus care
should be taken to to ensure that appropriate elements and constraints are used (e.g.
slender beam assumptions taken into account, and correct elements selected accordingly). Discretization errors are the main source of error, where the number of elements is
insufficient to describe the geometry under consideration. To minimise this error a mesh
convergence is performed where solution is computed for a range of mesh densities to
verify that the solution converges. Further validation can be achieved by comparing the
solution to experimental or analytical data.
220
B.2.2.2
Material properties
The material property specification required is dependent on the analysis type, for a simple linear elastic analysis of isotropic homogeneous materials only the Young’s modulus
and Poisson’s ratio need be specified. For orthotropic or anisotropic materials a frame
of reference must be assigned to define material orientation relative to the model, and
materials properties may be specified either as engineering constants (Exx , Gxy , νxy ), or as
an elastic stiffness matrix. Non-linear, for example elasto-plastic material properties can
be entered, and if necessary, user subroutines can be written to model fracture, or other
time dependent plastic effects.
B.2.2.3
Analysis type, boundary conditions and loading
The type of analysis used is entirely dependent on the physics of the situation being
considered. For structural simulations where no time, thermal or other external effects
are taken into account either a linear elastic, for small strains, or geometrically non-linear
analysis can be used.
Each node within the model has both rotational and translational degrees of freedom.
Constraints or boundary conditions can be placed on the model to replicate the physical
loading conditions by prescribing specific displacements or rotations to specified nodes.
Application of boundary conditions also has the effect of reducing the degrees of freedom,
thus reducing the problem size.
B.2.3
Solution
Once the geometry, material properties, boundary condition and analysis type has been
defined, this information is used by— the solver to compute the solution.
The solution of the finite element problem is found by relating the stress σab , strain
εab and constitutive (material properties) matrices dab . As an example, a simple two
dimensional example of a body in plain stress is thus described by σ = Eε :
221




σx






σy






 σxy
 

 d


 11 d12 0





 
=

 d21 d22 0


 



  0
0 d33

 
 εx
 

 

 
 
 εy
 

 

 

 2εxy

















(B.1)
For the structure to be in equilibrium, the body forces, fa , must equal the stresses
(σab , τab ) in the structure:
∂σx ∂σ y
+
+ fx = 0
∂x
∂y
∂τxy ∂σ y
+
+ fy = 0
∂x
∂y
(B.2)
(B.3)
The matrix of nodal strain displacements (B), and nodal displacement vectors (d) is
related to the global strains by σ = Bd. For each element, the strain energy is:
Z
1
(σx εx + σ y ε y + τxy γxy dV
U=
σ εdV =
2 v
v
Z
Z
1
1
(Eε)T εdV =
=
εT EεdV
2 v
2 v
Z
1
1 T
= d
Bt BEdVd = dT kd
2
2
v
Z
T
(B.4)
(B.5)
(B.6)
By minimising the energy function it is then possible to compute the nodal displacements and potential energy for the whole model.
B.2.4
Post-processing
From the displacement/energy solution, the forces and stresses within the model can
be derived. Stresses are computed at integration points within each element. It is also
possible to output the reaction forces at each node and from these compute the global
elastic properties of the structure. Data maybe visualized as either contour plots showing
loads, stresses and the deformed structure, or as numerical data for further post processing
and interpretation.
222
Appendix C
Derivation of off-axis elastic
constants from on-axis deformation
using the finite element method
When calculating the elastic properties of a continuum, given the full elastic constant
tensor for a given structure in a specific direction it is possible to calculate the elastic tensor
for the structure in any orientation. This methodology (described in section 2.2) is well
proven for continuum structures and thus it was assumed that this could be extended
to the beam networks modelled in this study. The cubic based beam networks were
constructed in two orientations; one such that it could be loaded in the [110] orientation,
and one such that it could be loaded in the [100] orientation. Both structures were analysed
using the finite element method and the directly measured Poisson’s ratio, and elastic
constants, for an axial stretch computed.
From strains in the (110) direction, the elastic constants are derived. These constants
are then rotated to derive the elastic constants in the (110) (or any other) direction.
223
ν110
ν110 rotated
Type I
Type II
Type III
1
1
1
0.07
0.11
10000
1
1
0.01
0.05
1
10000
1
0.98
0.99
10000
10000
1
0.29
0.35
1
1
10000
-0.92
-0.92
10000
1
10000
-0.24
-0.21
1
10000
10000
0.00
0.17
10000
10000
10000
0.07
0.11
f ea
Table C.1: Comparison of Poisson’s ratio calculated directly from strains in the (110)
direction, and from rotating the elastic constants derived from strains in the (100)
direction
Conducting these analyses for all possible configurations yields a large volume of
data. Table C.1 shows the Poisson’s ratio computed using both the direct measurement
and rotated constants methodologies for comparison. These data show that for structures
where type III beams are dominant the agreement between the rotated Poisson’s ratio and
measured result is good. For structures where type I is dominant, there is less agreement.
To determine the underlying reason for this poor agreement, it necessary to consider
the deformation of type I structures in the (100) direction as shown in section 6.3. It can be
seen that an axial applied strain results in a lateral strain of zero due to the beam stretching
mechanism and the lack of cross-linking in the structure. This gives a Poisson’s ratio of
zero, and a very high value of s12 . When these constant values are used to predict rotated
elastic properties, these values lead to skewed results. In contrast to this, the deformation
of the type III structure is primarily bending and there is a high degree of cross linking
in the structure. For structures where there are type III beams present there is sufficient
cross linking in the structure such that deformations in both the (100) and (110) gives high
axial deformations and thus the constants can be rotated.
224
Figure C.1: Poisson’s ratio in the (110, 11̄0) directions - calculated using (top) ElAM and
(bottom) direct measurement from FE
This analysis shows that when predicting elastic properties, applying continuum
methods to anisotropic beam networks is dependent on the nature of the beam networks
deformation mode. Networks where the deformation is accompanied by very small
orthogonal strains give erroneous values of rotated constants thus extreme caution should
be applied when using this methodology to predict off-axis material properties.
225
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