YSVA Booklet 2012

CONTENTS
Young Stress and Vibration Analyst Competition Finalists
A validated point-wise approach to the analysis of stresses and strains
around complex composite geometries using Digital Image Correlation
and Thermoelastic Stress Analysis
George Crammond
University of Southampton (UK) ............................................................................. 1
Assessment of corneal deformation using optical coherence tomography and
digital volume correlation
Jaiwei Fu
Arts et Métiers ParisTech, Châlons-en-Champagne (France) and Loughborough
University, UK ........................................................................................................ 6
Experimental verification of finite element analysis of locally thinned areas in
pressure equipment
Danielle Lowe, University of Liverpool (UK) ............................................................. 11
Experimental study of mechanical disturbances caused by a cylindrical inclusion
in a soft material subjected to contact loading
Xiaohua Tan, University of Liverpool (UK)................................................................ 15
A validated point-wise approach to the analysis of stresses and strains around complex composite
geometries using Digital Image Correlation and Thermoelastic Stress Analysis
G Crammond, S W Boyd and J M Dulieu-Barton
University of Southampton, Engineering Sciences, Southampton, SO17 1BJ, UK
1
Introduction
Composite materials are weak normal to the plane of the laminate due to a lack of throughthickness reinforcement, with strength dictated by the brittle epoxy matrix. Therefore it is important
to evaluate the through-thickness load transfer in composite bonded joints because of their
discontinuous nature to improve confidence in the join and inform more efficient joint designs.
Joint structure
A double butt strap joint (DBSJ), Figure 1, was constructed with 800g/m2 unidirectional and 450g/m2
chopped strand matt glass fibre in a [CSM8 904]s sequence using Gurit Prime 20lv epoxy resin using
the resin infusion process. Araldite 2015 epoxy adhesive was used to bond the adherends.
Area of
analysis
Strap
length
Root of
discontinuity
Overlap
length
Outer
adherend
(strap)
Adhesive
Inner
adherend
Figure 1: Schematic of Double butt strap joint
Meso scale analysis
Component strains within the DBSJ were evaluated using 2D DIC. A Canon mp-e65 macro lens
connected to a 5Mp LaVsion 5Mp E-Lite camera was used to image an area of 3.1 mm x 2.6 mm
around the discontinuity between adherends. The specimen was mounted in an Instron 5569 test
1
machine and loaded at 1mm/min up to failure. Complex localised strain distributions were revealed,
shown in Figures 2 and 3. Detailed analysis of the strain fields identifies small, yet critical, throughthickness and shear strains evolving within the joint. To validate the DIC strain values, Thermoelastic
Stress Analysis (TSA) was conducted. TSA provides a fast and accurate experimental technique, to
capture the complex behaviour around the join at similar length scales as the DIC. The two
independent experimental data sets provide sufficient detail to fully inform numerical models.
5
5
4
10
10
3
Inner
adherend Adhesive
Outer
adherend
Inner
adherend
0.04
Outer
adherend
Adhesive
0.035
0.03
15
2
Peel strain
15
Edge of joint
x 10
5
20
1
0.025
25
25
0
0.02
30
30
-1
0.015
35
35
-2
40
40
-3
50
5
CSM
10
15
20
0.01
Adhesive
45
Adhesive
20
45
-4
25
CSM
30
35
50
-5
Shear strain
-3
Edge of joint
5
Figure 2: Peel strain distribution at 18kN
10
15
CSM
20
0.005
25
30
CSM
35
0
Figure 3: Shear strain distribution at 18kN
Validation of DIC strains using Thermoelastic Stress Analysis
When a material experiences a stress change it is accompanied by a small temperature change [1],
ΔT. The relationship between ΔT and the change in the sum of principal stresses, σ1, σ2 is as follows
for an orthotropic material
(
)
(1)
where α1 and α2 are the coefficients of thermal expansion in the principal stress directions, ρ is the
material density, Cp is the specific heat at constant pressure, and T is the specimen temperature.
These constants form the ‘thermoelastic constants’ in the principal stress directions K1, K2, providing
the relationship between the sum of the principal stresses,
2
(
)
(2)
T and ΔT are scalar quantities. This equation can be rearranged into a form that provides a calibrated
stress metric, but this changes equation (2) into the following tensorial quantity:
(3)
In [2] and [3] methodologies were established to obtain the thermoelastic constant in the principal
material directions parallel and transverse to the fibre direction, KP and KT.. These were used in
subsequent joint analysis making the assumption that the material directions and the principal stress
directions were coincident. This assumption worked well away from the region of the discontinuity.
In the current work detailed analysis takes place close to the root of the join between adherends.
Due to the geometric discontinuity there is significant variation in the principal stress directions,
hence the thermoelastic constants in the principal stress directions are required.
The DBSJ was loaded at 20 Hz with a mean load of 6 kN and loading amplitude of 3 kN in an Instron
8800 test machine. TSA was conducted by imaging the root of the join using a Flir Silver SC5000
infra-red detector fitted with a G1 high resolution macro lens.
The DIC strain and TSA stress data were manipulated into the form of the calibrated stress sum in
equation (3) for direct comparison. The DIC data showed that the principal strain (stress) directions
are not coincident with the principal material directions, and rotate significantly in the vicinity of the
discontinuity. This means that the KP and KT values obtained using the calibration specimens in [2]
and [3] are not valid. Therefore they must be transformed into values associated with the principal
stress directions using data from the DIC point by point around the geometric discontinuity for each
TSA data point. Likewise, it is necessary to transform the DIC component strains to principal strains
and use transformed material stiffness matrices to obtain the principal stresses, which are
substituted into equation (3), along with the transformed thermoelastic constants. The TSA/DIC
transformation methodology is shown in Figure 4a and 4b.
3
IR Detector
Principal material
direction thermoelastic
constant coupon tests
Principal
direction
transformation
P
1
P,
2
12
,
θ
1
P
S
σ
σ
Figure 4a: Methodology for transformation of TSA data
Component
strains from
DIC
Materials
properties from
coupon tests
Q
Principal
direction
transformation
Q
12
L
ε
εL
γL
θ
θ
1
σ1
σ2
τ12
Q
ε1
ε2
γ12
L
Q
12
P,
T
θ
1
ε
εL
γL
ε1
ε2
γ12
σ
Principal material
direction thermoelastic
constant coupon tests
1
2
P
1
12
σ
ε (Q
Q )
P
ε (Q
Q )
Figure 4b: Methodology for transformation of DIC data into principal stresses
Figures 5 and 6 show the two comparable DIC and TSA data sets. There is very good agreement
showing both techniques provide accurate measurements of the complex joint behaviour as well as
validating the two experimental techniques. The position of the stress concentration corresponds
with the failure initiation sites observed from experiments. The combination of high peel strain and
stress concentrations at the root of the discontinuity in the inner adhered lead to the initiation of
cracks, before propagating along the preferential high strain, low stress, region at the
adhesive/adherend interface in the outer adhered.
4
150
150
44
44
66
ADHESIVE
88
100
100
10
10
12
12
14
14
16
16
50
50
66
88
100
100
ADHESIVE
Stress metric / MPa
22
Stress metric / MPa
150
150
22
10
10
12
12
14
14
16
16
5050
18
18
18
18
20
20
22
20
20
55
10
10
15
15
20
20
25
25
30
30
Figure 5: DIC calibrated principal stress metric
00
22
22
55
10
10
15
15
20
20
25
25
30
30
00
Figure 6: TSA calibrated principal stress metric
Conclusions
The principal strain directions obtained from DIC studies have been used to enable TSA of a complex
composite structural component using a point by point calibration. For these complex geometry
situations the assumption that the principal material and principal stress axes are coincident cannot
be made. Analysis using the two experimental techniques has shown how the heterogeneous stress
and strain fields are intricately linked to the initiation and propagation of damage, in addition to final
failure behaviour of the joint. Validation of these two experimental techniques adds a greater depth
to the analysis of the joint under load, allowing evaluation of both stresses and strains within the
structure.
[1]
G. Pitarresi and E. Patterson, “A review of the general theory of thermoelastic stress
analysis” The Journal of Strain Analysis for Engineering Design, vol. 38, no. 5, pp. 405417, Jan. 2003.
[2]
S. W. Boyd, J. M. Dulieu-Barton, O. T. Thomsen, and A. Gherardi, “Development of a
finite element model for analysis of pultruded structures using thermoelastic data”
Composites Part A: Applied Science and Manufacturing, vol. 39, no. 8, pp. 1311-1321,
Aug. 2008.
[3]
S. W. Boyd, J. M. Dulieu-Barton, O. T. Thomsen, and S. El-Gazzani, “Through thickness
stress distributions in pultruded GRP materials” Composite Structures, vol. 92, no. 3,
pp. 662-668, Feb. 2010.
5
Assessment of Corneal Deformation Using Optical Coherence
Tomography and Digital Volume Correlation
J. Fu1, 2, F. Pierron1, P. D. Ruiz2
1
Arts et Métiers ParisTech, Rue St Dominique, 51006 Châlons-en-Champagne Cedex, France
2
Loughborough University, Leicestershire, LE11 3TU, UK
Introduction
The study of the mechanical behavior of the cornea under intraocular pressure is important for the
assessment of corneal biomechanics for instance for pathology assessment. Recent advances in optical
coherence tomography (OCT) [1, 2] enable the non-invasive and non-destructive reconstruction of the
volume microstructure of semitransparent inhomogeneous samples such as corneas. Swept Source
Optical Coherence Tomography (SS-OCT) systems are able to provide sample microstructure volumes
with high sensitivity and at video rates. Digital image correlation can be applied to measure surface
full-field deformations of the cornea [3]. However, 3D full-field deformation measurements have rarely
been obtained yet. In the present work, digital volume correlation is applied to measure the volume
displacement and strain fields from the volume images generated through SS-OCT of phantom and
porcine cornea samples under posterior inflation, which is more representative of the physiological
state than the more standard uniaxial tests [4]. An important objective of this work is to study in detail
the metrological performances that can be expected from the digital volume correlation procedure
when using SS-OCT volume images.
Methods
A schematic and photo of the experimental set-up is shown in Fig 1. The specimen was mounted and
fixed on the artificial anterior chamber (AAC) which has inlet and outlet for the fluid and a pressure
transducer. The simulated intraocular pressure was achieved by adjusting a 1 ml micro-syringe. At each
pressure increment, a 3D volume image sequence of the specimen was acquired using a Swept Source
Optical Coherence Tomography system (Thorlabs OCS1300SS). Both silicon resin phantom (seeded
with titanium oxide particles) and porcine cornea were used as the specimen in the inflation test. The
outer edge diameter and thickness of the phantom are 15.6 mm and 0.58 mm, respectively. Regarding
the porcine cornea, the diameters of the maximum and minimum meridians are 13 mm and 10 mm
respectively, while the average cornea thickness for the central part of the cornea is 1.2 mm. The cornea
and phantom samples were inflated from 2 to 2.25 kPa as illustrated in Fig. 1. The reconstructed
volumes at both load steps were recorded for digital volume correlation and displacement and strain
fields computed. Fig 2 shows a central transverse slice of the phantom and porcine cornea, respectively.
6
Due to the less adequate speckle contrast in the bottom regions, geometrical mask was added to mask
out the regions with low quality, as illustrated in Fig 2 (b). Then three dimensional digital volume
correlation was performed on the volume images using the Davis (LaVision) commercial package
based on a fast Fourier transform algorithm.
Fig 1. Schematic diagram and photo of the experimental set-up
(a) Phantom
(b) Porcine cornea
Fig 2. Central transverse slice of the phantom and porcine cornea generated through the SS-OCT
system, scales show dimensions in the phantom and porcine cornea specimens
Noise study
The effect of noise and reconstruction uncertainties are evaluated through correlation of two subsequent
reconstructed volumes of the stationary phantom. The displacement and strain resolutions of the DVC
are evaluated after calculating their mean and standard deviation values. Then several reconstructed
volumes are recorded one after the other, introducing a rigid body translation of 10 microns between
successive images to examine the performance of the DVC algorithm in producing displacement and
strain fields. Their mean and standard deviation values are then calculated and related to the quality of
image recording. From Fig 3 (a) it can be observed that for the stationary test the mean values of all
strain components are very small and close to zero, while their standard deviations are between 0.05%
and 0.1%. Similar result is found for the rigid body translation test in Fig 3 (b) of which the strain
standard deviation is slightly larger.
7
(a) Stationary test
(b) Rigid body translation
Fig 3. Strain resolution for stationary test and rigid body translation with 24×24×24-voxel sub-volume,
50% overlap and without smoothing
The influence of sub-volume size on the correlation precision is analyzed quantitatively then. 4
different sub-volume sizes were chosen for the stationary test. The strain resolutions are compared in
Fig 4 for εxx and εyy only, for the sake of legibility. Among different sub-volume sizes, 24×24×24-voxel
sub-volume is found to be a good compromise between the strain resolution and spatial resolution.
Further increasing it gives slight improvement for the resolution, which, however, degrades the spatial
resolution. This is particularly critical here because of the small thickness of the phantom and cornea
compared to the spatial resolution of the SS-OCT system. As a consequence, a sub-volume size of
24×24×24 will be kept for the rest of this study.
Fig 4. Strain resolutions under different sub-volume sizes
Results and Discussion
Typical displacement fields along the X, Y and Z axes for a central z-slice for both phantom and
porcine cornea specimens under inflation conditions are shown in Fig 5 and compared with an elastic
finite element (FE) model (E=0.9 MPa, ν=0.48) of the porcine cornea. The displacement fields of the
phantom are close to that of the porcine cornea and both of them match the results of the FE model,
which is consistent with the inflation condition. The difference observed between the z-displacement
maps for the phantom and porcine cornea is due to the z-slice chosen in each case. For the porcine
8
cornea, z-slice is located in the front section, while that of the phantom is in the back section.
(a) Phantom
(b) Porcine cornea
(c) FE model
Fig 5. Displacement fields for: (a) phantom, (b) porcine cornea and (c) FE model inflated from 2 to
2.25 kPa
The strain fields for the same transverse central slices of phantom and porcine cornea were calculated
from the displacement data and are shown in Fig 6. As expected from the displacement fields, the strain
maps of the phantom and porcine cornea are also consistent with those from the FE model. The results
show that the major deformation occurs in the inner central part of the cornea, while the outer part is
less deformed. Comparing the strain values from the inflation tests to the strain resolution, it can be
observed that the strain values are much greater, which verifies that these distributions are the results of
material behavior rather than noise artifact. Although there are some apparent irregularities such as the
negative εzz in the central z-slice, generally the strain maps are consistent with expectations for both
specimens.
Conclusions
In the present study, digital volume correlation was carried out on the volume images generated
through swept source optical coherence tomography. A 24×24×24-voxel sub-volume was found to be a
good compromise between resolution and spatial resolution. Based on this sub-volume size, reasonable
displacement and strain results on the inflation test were obtained. The strain standard deviation is
significantly lower than the strain values. Future work will focus on the use of these data to identify the
elastic behavior of the cornea using the Virtual Fields Method [5].
9
(a) Phantom
(b) Porcine cornea
(c) FE model
Fig 6. Strain fields for: (a) phantom, (b) porcine cornea and (c) FE model inflated from 2 to 2.25 kPa
References
[1] Kaluzy BJ, Kaluzny JJ, Szkulmowska A, Gorczynska I, Szkulmowski M, Bajraszewski T, et al.
Spectral optical coherence tomography a novel technique for cornea imaging. Cornea, 25: 960-965,
2006.
[2] Fujimoto JG. Optical coherence tomography for ultrahigh resolution in vivo imaging. Nat
Biotechnol, 23: 1361-1367, 2003.
[3] Boyce BL, Grazier JM, Jones RE, Nguyen TD. Full-field deformation of bovine cornea under
constrained inflation conditions. Biomaterials, 29: 3896-3904, 2008.
[4] Elsheikh A, Alhasso D, Rama P. Assessment of the epithelium’s contribution to corneal
biomechanics. Exp Eye Res, 86: 445-451, 2008.
[5] Pierron F, Grédiac M. The Virtual Fields Method, Springer New-York, ISBN 978-1-4614-1823-8,
536 p., 2012.
10
Experimental Verification of Finite Element Analysis of Locally
Thinned Areas in Pressure Equipment
Danielle Lowe
University of Liverpool
Year 3 M Eng Mechanical Engineering
Project in conjunction with INEOS ChlorVinyls and Eann Patterson (University of Liverpool)
In a commercial environment, equipment life and availability is of critical commercial importance, but
safe operation is vital. In industry, corrosion and erosion damage to pressure vessels and pipework is
common, leading to defects such as localised thinning. The thickness in these defect areas are often
beyond the allowable limits specified for general thickness in the applicable design codes, but general
thickness requirements may be over conservative for local thinning. If adequate justification could be
provided to allow continued safe operation in the damaged condition, equipment availability could be
improved. One method of justification is to model the defects using finite element analysis (FEA) to
determine the resulting stress state and hence, margin before failure. Validation of the results to ensure
they are reliable and accurate must be carried out before it can be used with confidence. The aim of this
project was therefore to determine a conservative methodology of creating and analyzing simplified
models of locally thinned pressure equipment using a specified finite element package, Solidworks that
could be employed to confidently prolong equipment life at INEOS ChlorVinyls and to validate it
through experimental testing. INEOS ChlorVinyls wish to use Solidworks for routine assessment of
defective pipework.
Defects were artificially introduced into a new 6” carbon steel, seamless pipe and aligned along the
pipe at appropriate intervals following the guidelines of Saints Venant’s Principle [1]. A 40bar
analogue pressure gauge was attached to the pipe and water pressurisation used to exert an internal
pressure. The flange rating limited the experimental pressure to 21bar.
An internally pressurised steel pipe was modelled and analysed using the FEA package Solidworks
following the recommendations given in BS7910 [2]. An element size of 13mm was used for the no
defect and defect flaw areas following mesh convergence analysis. If INEOS ChlorVinyls adopted the
FEA method to assess defective pipes, the equipment being investigated would be in-service, thus any
material properties not given on the material test certificate could not be determined through
experimental testing. Therefore to follow the procedures that would be adopted by INEOS
ChlorVinyls, Poisson’s ratio and Young’s modulus were investigated parametrically to obtain
appropriate uncertainties for the simulation results.
The experimental data was obtained using a commercially available digital image correlation (DIC)
system (Dantec Dynamics Q-400, Ulm, Germany) and used to validate the displacement and strain
results obtained from the FEA model. A 3D DIC technique was used due to the non-planar nature of
the samples being tested [3]. The cameras were set-up parallel to the axis of the pipe as shown in figure
1. Prior to the experimental testing a full system calibration was carried out following the procedures
given by the SPOTS consortium [4]. A system uncertainty of 3.8% for values of strain in the region of
149µ strain was found; combining this with the experimental uncertainties gave a total DIC uncertainty
of 4.6%. In the analysis of the DIC results a facet size of 17 pixels and a grid spacing of 14 pixels were
used based on a convergence analysis.
Comparisons of the FEA and DIC displacement maps for a defect are shown in figure 2.The simulation
predicts similar behaviour to that obtained using the DIC, with the maximum values of strain in the
same location, but the poor quality mesh on the FEA does not allow for detailed analysis of the defect
area.
Comparison of the results for the maximum out of plane displacement predicted by the FE model with
those found using DIC and thin walled pressure vessel theory were within 8% and 15% respectively for
an internal pipe pressured up to 20bar. The corresponding difference for a pipe with a defect was 31%
when comparing the FE and DIC (figure 3). This large difference suggests that the FE analysis
11
provides a conservative result, i.e. it predicts higher deformation and stresses than are experienced inpractice. An attempt was made to calculate values of strain from the DIC results for displacement using
the algorithms supplied with the DIC system. However the results were obviously incorrect and
unreliable following a comparison with a calculated gross value circumferential strain from the DIC
displacements. A difference of 57% was found, with the actual DIC algorithm results being the larger,
it was therefore logical to disregard the DIC strain results as the time-scale of the project did not allow
this issue to be resolved.
The validation of the results by a finite element analysis software package is an important step in
ensuring they are accurate and reliable. Many sources of errors and uncertainties in the DIC
experiments and FEA make it difficult to validate the results precisely. Despite this, when the FEA is
used more widely in an industrial setting these uncertainties will always be present and therefore it is
important to be aware of them and choose appropriate values for the material properties. In addition,
the experimental DIC results were obtained in an industrial setting which in itself will introduce more
uncertainty than if they had be obtained in a laboratory setting where conditions can be more easily
controlled. Also, Solidworks does not allow refinement of the mesh, it only allows the size and not the
triangular shape to be altered and hence the results tend to be conservative. Due to the unreliable strain
results calculated from the DIC system algorithms it was logical not to use these results in the
validation comparison. Despite this, all strains are calculated from displacement and therefore the
correlation of the displacement values was deemed to be appropriate to validate the FEA software and
demonstrate its accuracy. The FEA results agreed with those from the analytical theory sufficiently for
both the displacements and strains for a pipe with no defects. It also produced displacements, which
either correlated closely to the appropriate DIC results, or were larger for the defective pipe tests,
which means the FEA is producing results which are conservative.
These findings confirm the reliability and give engineers confidence in the results produced from the
Solidworks package which could lead to more sustained use of FEA in industry to aid decision making
on the prolonging of locally thinned equipment life and be used in routine assessment at INEOS
ChlorVinyls.
Figure 1: DIC Experimental Set-up
12
Figure 2: Comparison of maximum value locations on the displacement maps for both the DIC
and FEA for a defective pipe
0.0280
0.0270
0.0260
0.0250
0.0240
0.0230
0.0220
0.0210
0.0200
0.0190
Displacement (mm)
0.0180
DEFECT - Solidworks
Maximum
NO DEFECT Solidworks Maximum
NO DEFECT - DIC
Maximum
NO DFECT - Analytical
Theory
DEFECT DIC Maximum
0.0170
0.0160
0.0150
0.0140
0.0130
0.0120
0.0110
0.0100
0.0090
0.0080
0.0070
0.0060
0.0050
0.0040
0.0030
0.0020
0.0010
0.0000
0
2
4
6
8
10
12
14
16
18
20
22
24
Applied Internal Pressure (bar)
Figure 3: Plot of maximum displacement results at different load pressures for a no defect and
defect pipe, comparing FEA (Solidoworks), DIC and analytical theory.
13
References
[1] C.O.Horgan. (1989). Recent developments concerning Saint Venant's principle: an update. Appl.
Mech. Rev , Vol 42, Pgs 295-303.
[2] British Standards Institution. (2005). BS7910: Guide to methods assessing the acceptability of flaws
in meatallic structures. BSI.
[3] Y-K Zhu, G-Y.Tian, R-S.Lu, H.Zhang (2011). A Review of Optical NDT Technologies. Sensors ,
11, 7773-7798.
[4] SPOTS (2010). Guidelines for the Calibration and Evaluation of Optical Systems for Strain
Measurements, SPOTS. www.opticalstrain.org.
[5] Dantec Dynamics. (2011). Q-400 Operation Manual.
14
EXPERIMENTAL STUDY OF MECHANICAL DISTURBANCES
CAUSED BY A CYLINDRICAL INCLUSION IN A SOFT MATERIAL
SUBJECTED TO CONTACT LOADING
X.H. Tan1, Y.L. Kang2(*), Eann Patterson3
1, 2
Department of Mechanics, Tianjin University, Tianjin, 300072, PR China
3
School of Engineering, University of Liverpool, Liverpool, L69 3GH, United Kingdom
(*)
Email: [email protected]
Introduction
Soft materials, such as rubber, biological tissue, and foam, play an important role in a
wide range of technological applications (Hamley, 2007). They are often loaded through
transferring forces from other components which often results in large deformation and stress
concentrations in vicinity of the contacting areas. It is known that the contact properties of
materials can be significantly affected by the presence of heterogeneities (Leroux, 2011).
Analysis of the changes induced by material heterogeneities in an elastic medium is a
fundamental physical and engineering problem. The first solution to this problem was given
by Eshelby (1957), who proposed the equivalent inclusion method to deal with an elliptical
heterogeneity in an infinite medium subjected to uniform loading. After that, many
researchers have devoted their efforts to this field. Recently, in a numerical study (ChangHung, 2007) demonstrated that the contact stress distribution is locally increased by a hard
inhomogeneity near the contact region. However, most of the previous works are carried out
within the limitations of small strain theory of elasticity, which is not applicable to soft
materials as additional inequalities and nonlinearities are introduced. The singular
characteristics such as stress concentration within these materials have not been well
identified due to their multi-disciplinary nature, complicated physical properties, and variable
boundary conditions.
Objective: this work is specifically aimed at developing an experimental technique for
measuring and comparing the mechanical differences in a soft material induced by a rigid
cylindrical inclusion. Considering the fact that soft materials could be easily deformed by
small stresses or thermal fluctuations, a technique has been developed by integrating the
digital moiré method with embedded gratings to investigate the mechanical behaviour of a
vulcanized silicon rubber under contact loading.
Experiment & specimen
An experiment consisting of a homogeneous bulk material indented by a rigid wedge
was performed in the first place; then the same procedure was implemented for a material
containing a rigid cylindrical inclusion. Both the specimens, with and without inclusion, were
cast from the same room temperature vulcanized silicone rubber and indented by the same
rigid wedge (as shown in figure 1). Orthogonal moiré gratings composed of black toner were
prefabricated on the plane of symmetry of the specimen, which ensured that the embeddedgrating layer would be deformed in plane. In this case, the error caused by out-of-plane
deformation could be neglected and which enabled the analysis of the in-plane mechanical
behaviour of the soft material under various loads. The specimens were loaded at a constant
15
rate of 1mm/min. Meanwhile, deformation of the gratings was recorded by a CCD image
acquisition system. Afterwards, the images were processed by a digital image processing
method as shown in figure 2.
Steel cylinder
Silicone rubber
30mm
Embedded-gratings
25mm
35mm
ϕ
30mm
30mm
Fig.1 Profile of a soft material indented by a rigid wedge, with the coordinate system located at the apex of
indenter (left); a soft material containing a cylindrical inclusion indented by a rigid wedge (middle);
Diagrammatic sketch of the specimen with inclusion (right).
(a)
(b)
(c)
Separation &
logical operation
Indenter
Inner-gratings
(d)
CCD Camera
four-step phase
shifting
60mm
Specimen
Unwrapping
30mm
60mm
3D Movable
Platform
Computer &
Image Card
Deformed Gratings
under Different Load
Displacement
Fig.2 Schematic diagram of experimental procedure. (a) Embedded moiré grating and specimen under loading;
(b) image acquisition system; (c) examples of experimental images; (d) flowchart of digital image processing.
Discussion & conclusion
From the displacement data obtained from digital moiré analysis, the corresponding
strain fields were calculated within the framework of large strain theory. As can be seen from
the strain maps shown in figure 3, two deformation sectors could be observed from the results
as predicted theoretically by Gao and Mai (2002). For example, consider the horizontal strain
ε x , there is an Expansion Sector (EX) below the indenter, which is very narrow before
deformation and then becomes very wide after deformation. On the contrary, there is a
Shrinking Sector (SS) occupying the majority domain on both sides of the rigid wedge before
deformation, which is squeezed into a narrow area after deformation. In the presence of the
rigid inclusion, the EX was expanded and a symmetric cross area can be observed from figure
3(a2) & (b2), which indicates that the strain outside this area was reduced by the inclusion. As
for the shear strain maps, an asymmetrical deformation area can be found below the rigid
inclusion.
16
SS
SS
200
0.06
0.06
200
ε x =0
ε x =0
0.04
400
400
0.02
0.02
600
600
0
EX
800
ε x =0
ε x =0
1000
εx
εx
200 400 600
800 1000
0
-0.02
800
-0.02
1000
200 400 600 800 1000
(a1)
(a2)
ε y =0
0.02
200
400
0.04
εx
εx
ε y =0
ε y =0
εx
εx
600
200
0
-0.02
400
-0.04
600
-0.06
800
ε y =0
εx
εx
0
-0.02
-0.04
-0.06
800
-0.08
-0.08
1000
200 400 600 800 1000
1000
200 400 600 800 1000
(b1)
-0.1
(b2)
γ xy =0
200
0.02
γ xy =0
εx
εx
0.1
200
0.1
γ xy =0
0.05
γ xy =0
εx
400
400
εx
0
0.05
0
600
600
-0.05
-0.05
800
800
-0.1
-0.1
1000
200 400 600
800 1000
1000
200 400 600 800 1000
(c1)
(c2)
Fig.3 Strain maps obtained from experimental images under load of 38N. (a1) & (a2):
without inclusion; (b1) & (b2):
εy
maps with and without inclusion; (c1) & (c2):
γ xy
εx
maps with and
maps with and without
inclusion;
After the comparison of strain fields, the change in stress near the boundary of
cylindrical inclusion cause by the heterogeneity was derived by means of the Mooney-Rivlin
model. The plots of Cauchy stresses are illustrated in figure 4. It is obvious that the
distributions of contact stresses are locally increased by the hard heterogeneity, and the most
significant changes can be found right below the rigid cylinder.
It seems that the proposed experimental approach makes it easy and effective to measure
and analyze large deformation and large strains. The obtained strain and stress distributions
qualitatively reveal the deformation behaviour and features of the soft material.
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Horizontal stress without inclusion
Horizontal stress with inclusion
Vertical stress without inclusion
Vertical stress with inclusion
ϕ
Shear stress without inclusion
Shear stress with inclusion
ϕ
ϕ
Fig.4 Normalized stress distributions along the boundary of cylindrical inclusion under load of 38N.
Comparison of Cauchy stress component σ x (left), σ y (middle), τ (right).
References
Chang-Hung, K. Stress disturbances caused by the inhomogeneity in an elastic half-space
subjected to contact loading. International Journal of Solids and Structures, 2007, 44(3-4), p.
860-873.
Eshelby, J.D.. The determination of the elastic field of an elliptical inclusion, and related
problems. Proc. R. Soc. A, 1957, 241, p. 376–396.
Gao, Y.C., Mai, Y.W.. The contact problem of a rubber half-space dented by a rigid cone
apex. Archive of Applied Mechanics, 2002, 72, p. 213-228.
Hamley, I.W.. Introduction to Soft Matter-Revised Edition. John Wiley & Sons, Ltd, West
Sussex, 2007
Leroux, J. and D. Nélias. Stick-slip analysis of a circular point contact between a rigid sphere
and a flat unidirectional composite with cylindrical fibers. International Journal of Solids and
Structures, 2011, 48(25-26), p. 3510-3520.
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