Numerically Efficient Finite Element Formulation for Modeling Active

Mechanics of Advanced Materials and Structures, 13:379–392, 2006
c Taylor & Francis Group, LLC
Copyright ISSN: 1537-6494 print / 1537-6532 online
DOI: 10.1080/15376490600777624
Numerically Efficient Finite Element Formulation
for Modeling Active Composite Laminates
Dragan Marinković, Heinz Köppe, and Ulrich Gabbert
Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany
Active systems have attracted a great deal of attention in the
last few decades due to the potential benefits they offer over the
conventional passive systems in various applications. Dealing with
active systems requires the possibility of modeling and simulation
of their behavior. The paper considers thin-walled active structures
with laminate architecture featuring fiber reinforced composite as
a passive material and utilizing piezoelectric patches as both sensor
and actuator components. The objective is the development of numerically effective finite element tool for their modeling. A 9-node
degenerate shell element is described in the paper and the main
aspects of the application of the element are discussed through a
set of numerical examples.
1. INTRODUCTION
The science and technology have made amazing developments in the design of structures using advanced materials, such
as multifunctional fiber reinforced composites including smart
material components, which offer the possibility of redefining
the concept of structures from a conventional passive elastic system to an active controllable system with inherent self-sensing,
diagnosis, actuation and control capabilities [1, 2]. Such new
materials and structural concepts also require advanced computational methods for modeling and design purposes. The paper considers modeling thin-walled smart laminate structures,
the smartness of which is provided by the inherent property
of embedded piezoelectric material to couple mechanical and
electrical fields. Dealing with active systems requires adequate
modeling tools in order to simulate their behavior. Nowadays
the numerically efficient formulations are practically by default
addressed to the finite element method. Since the pioneer work
of Allik and Hughes [3], various types of finite elements for
the coupled electro-mechanical field have been developed, including beam, plate, solid and shell elements [4]. Application
of solid elements for the analysis of thin-walled structures is
Received 15 March 2006; accepted 5 April 2006.
Address correspondence to Dragan Marinković, Institut für
Mechanik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, Magdeburg, D-39106, Germany. E-mail: Dragan.
[email protected]
numerically very time-consuming, especially in large-scale applications of laminated structures and, therefore, attention has
turned to shell type finite elements, which can provide a satisfying accuracy with acceptable numerical effort. Piefort [5] has
extended a layerwise facet element available in the commercial
finite element package SAMCEF (Samtech s.a.) by adding an
arbitrary number of piezoelectric layers. The 4-node element
is based on the Mindlin-Reissner kinematics, has a plane form
and takes the full piezoelectric coupling into account. Tzou and
Ye [6] have developed a triangular solid-shell element with biquadratic shape functions for the in-plane mechanical degrees
of freedom and linear shape functions for both mechanical and
electrical degrees of freedom in the thickness direction. Gabbert
et al. [7] have extended a Semi-Loof shell element to an electromechanical coupled element for modeling active composite laminates. The original mechanical Semi-Loof element was first published by Irons [8]. The formulation is based on the Kirchhoff
kinematics and does not take the transverse shear strains into
account. Hence, it is free of shear locking effect, but the membrane locking acts with this element. This element has proven to
give good results in modeling thin piezoelectric composite laminates and will be used in the present work for the comparison
purpose. Lammering [9] has described a bilinear shell element
with the electrical degrees of freedom for the upper and lower
piezoelectric layer at each node. The selectively reduced integration is included in the formulation. A similar approach has
been presented by Koegl and Bucalem [10] but the locking effects were treated by means of Mixed Interpolation of Tensorial
Components method. Mesecke-Rischmann [11] has formulated
a shallow shell element with a quadratic distribution of electric
potential over the thickness of piezoelectric layers.
A variety of higher order lamination theories has been proposed during the last decade in order to improve the transverse
shear stress calculation. Nevertheless, an analytical trade-off of
higher order theories carried out by Rohwer [12] showed, that for
finite element analysis the First-order Shear Deformation Theory (FSDT) is the best compromise between the accuracy and
effort. More recently developed are the so-called layerwise theories based on the polynomial distributions (zig-zag functions)
of the membrane displacements. If no constraints for the zigzag functions are introduced, the number of functional degrees
379
380
D. MARINKOVIĆ ET AL.
of freedom depends on the number of layers leading to a finite
element formulation that requires a computational effort in the
range of a full 3D-analysis. The number of functional degrees
of freedom can be significantly reduced by a priori ensuring the
continuity of the transverse shear stresses. Corresponding finite
element formulations either need C1 -continuity, based on mixed
formulations [13] or a weak form of the Hook’s law permitting
to eliminate the stress variables at the layer level [14].
This paper takes advantage of a relatively simple equivalent
single-layer formulation. As a new feature it offers the extension of the degenerate shell approach based on the kinematics of
the FSDT to modeling multilayered composite structures (directionally dependent properties) of arbitrary shape with embedded
piezoelectric active layers polarized in the thickness direction.
A similar approach was recently used by Zemčı́k et al. [15],
who developed a bilinear (4-noded) degenerate shell element
with the discrete shear gap (DSG) method and the enhanced
assumed strain (EAS) method used to resolve the shear locking and membrane locking problems, respectively. The present
work deals with a full biquadratic (9-noded) formulation of the
degenerate shell element, which is more suitable for generally
curved structures and less prone to the mentioned locking problems. The paper also investigates the possibility of alleviating
locking problems by means of a simpler and numerically more
efficient technique—uniformly reduced integration.
2. FORMULATION OF THE ELEMENT
2.1. Coordinate Systems and Geometry of the Element
The degenerate shell element was first developed by Ahmad
[16] from a 3D solid element by a degeneration process which
directly reduced the 3D field approach to a 2D one in terms of
mid-surface nodal variables. The most significant advantages
achieved in this manner are that the element is not based on the
classical shell theories and is applicable over a wide range of
thickness and curvatures.
The inherent complexity of the degenerate shell element requires the usage of several coordinate systems in order to describe the element geometry, displacement field and to develop
the strain field (Figure 1b). Besides the global (x, y, z) and the
natural (r, s, t) coordinate systems it is necessary to introduce
the local-running (co-rotational) coordinate system (x , y , z ).
FIG. 1.
The local coordinate system is defined so as to have one of its
axes (say z -axis) perpendicular to the mid-surface, while the
other two axes form the tangential plane. In the case of in-plane
isotropic material properties the orientation of in-plane axes may
be arbitrary. However, the kind of non-isotropy exhibited by the
fiber reinforced composite laminates requires the introduction of
a structure reference direction (defined by the user), with respect
to which the fiber orientation in the layers is given. In this case it
is reasonable to fix the orientation of the local in-plane axes with
respect to the structure reference direction. The simplest way is
overlapping one of the axes (say x -axis) with it (Figure 1a).
Using the full biquadratic Lagrange shape functions Ni [17],
the coordinates of a mid-surface point are given by:
 
 
x
x


9
 
  i

y =
Ni yi

 z
 i=1 
z 

i
(1)
with xi , yi and zi denoting global coordinates of the nine nodes.
The thickness of the shell is assumed to be in the direction normal to the mid-surface. Denoting the unity vectors of the local
coordinate system with respect to the global coordinate system
by eai , where a stands for x , y or z depending on the axis and
i denotes the node, the 3D shell geometry may be regenerated
from its mid-surface in the following way:
 
 
x
x


9
9
 
  i
 hi
y =
Ni y i +
Ni t ez i

 z
 i=1 
z 
 i=1 2
(2)
i
where hi denotes the shell thickness at node i and −1 < t < +1.
2.2. Element Displacement and Strain Field
The degeneration process performed by Ahmad is based on
the assumption that the thickness direction line of the shell remains straight after deformation but not necessarily perpendicular to the mid-surface (the Mindlin kinematical assumption).
Therefore, the displacement of any point within the volume of
the shell is given as a superposition of the corresponding midsurface point displacement and a linear function of the rotations
Element geometry and coordinate systems.
381
NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION
about the local x - and y -axis through the mid-surface point, θx
and θy :
 
 
u
u


9
9
 
  i
 θx
hi
v =
Ni t [ −ey i ex i ]
N i vi +
θy

 i=1 
 w
w 
 i=1 2
i
where, for example, u,x = ∂u/∂x. The global derivatives are
afterwards transformed to the local derivatives by means of the
transformation matrix [T] = ex ey ez :

u ,x
(3)
It is to be noted that the rotations are in the local coordinate
system and upon transformation to the global coordinate system
one gets:
 
 
 
9
9
 u  ui   θx 
hi
v =
Ni v i +
Ni t[ −ey i ex i ] [T ]T θy
  i=1   i=1 2
 
w
wi
θz
v ,x
v ,y
v ,z
u ,  y
u , z
∂u
∂x
T
(4)

x,r

[J] = x,s
x,t
y,r
y,s
y,t

−1
= [J]
u,r
u,
 s
u,t
z,r


u,x

z,s 
 ⇒ u,y
z,t
u,z

v,r w,r
v,s w,s 

v,t w,t
FIG. 2.
v,x
w,x

v,y
w,y 

v,z
w,z
(5)
v,x
v,y
v,z

w,x
w,y 
 [T]
w,z
(6)
The derivatives in Eq. (6) can be divided into a part related to
global displacements and a part related to global rotations only.
For example, observing the term u ,x one obtains:
where [T ] is the modified form of the transformation matrix
relating the local and the global coordinate systems, [T], and is
given in the form [T ] = ex ey .
Now, due to the directionally dependent material properties,
it is of crucial importance to develop the strain field in the local
coordinate system (Figure 2). This allows direct application of
the composite laminates constitutive matrix, the so-called ABD
matrix. The advantage of having the strain field with respect to
the local coordinate system is also obvious when the piezoelectric coupling within the thickness polarized piezopatch using the
e31 effect is considered.
The interpolations are performed in the natural coordinate
system. Hence, the displacement derivatives with respect to the
natural coordinates are directly obtained from Eq. (4). The transformation of derivatives from the natural to the global coordinate
system is achieved by means of Jacobian inverse:


u,x
w ,x
T u,
w ,y 
 = [T]  y
w ,z
u,z
9 ∂Ni
∂Nr
∂Ni
l1
=
l1 ui +
m 1 vi +
n 1 wi
∂x
∂x
∂x
i=1
∂Ni
∂Ni
∂Ni
+ m1
m1 vi +
l 1 ui +
n1 w i
∂y
∂y
∂y
∂Ni
∂Ni
∂Ni
+ n1
(7)
l1 ui +
m1 vi +
n 1 wi
∂z
∂z
∂z
and
∂u
∂x
=
R
9
∂Ni
∂Ni
hi
l1 t
+ J∗13 Ni + m1 t
+ J∗23 Ni
2
∂x
∂y
i=1
∂Ni
+ n1 t
+ J∗33 Ni · (n3i θyi − m3i θzi )l1
∂z
+ (l3i θzi − n3i θxi )m1 + (m3i θxi − l3i θyi )n1 (8)
where l1 , m1 , n1 and l3 , m3 , n3 are the cosine directions of the
unit vectors ex and ez with respect to the global coordinates,
respectively, i.e., ex = {l1 m1 n1 }T and ez = {l3 m3 n3 }T , those
with subscripting i are to be calculated at the ith node, and
those without i at integration points, J∗ab (a, b = 1, 2, 3) are the
constants of the Jacobian inverse, and:
∂Ni
∂Ni
∂Ni
= J∗11
+ J∗12
;
∂x
∂r
∂s
∂Ni
∂Ni
∂Ni
= J∗21
+ J∗22
;
∂y
∂r
∂s
∂Ni
∂Ni
∂Ni
= J∗31
+ J∗32
;
∂z
∂r
∂s
Mappings and transformations for the geometry and strain description.
(9)
382
D. MARINKOVIĆ ET AL.
It is quite common within a 2D formulation to give the strain
field in the form that makes a distinction between the in-plane
components and the out-of-plane components.

 


ε
xx 













ε y y 





 
 ∂u

γ
y
∂y
x
{ε } =
=



−−


− −






∂v






γ
∂z



y
z
 


 ∂u
γx z ∂z
∂u
∂x
∂v
∂y
+
∂v
∂x











− − −




+ ∂w

∂y



∂w
+ 
(10)
∂x
The formulation adopts the plane-stress state assumption, i.e.,
σz z = 0, and consequently, the normal transverse strain component εz z is not included in Eq. (10), since the product of the
mentioned strain and stress components does not contribute to
the strain energy. After determining all partial derivatives in the
local coordinate system (Eq. (6)), the discretized strain field can
be suitably given in the following form:
  

[Bmf ]
 {εmf } 
{ε } = −− =  − −  {d}
 
{εs }
[Bs ]


[BTm ]
t[BRlf ]


=  − − − + − − − − − − − − −  {d}
[BTs ]
[BR0s ] + t[BRls ]
= [Bu ]{d}
E=−
(11)
where {εmf } = {εx x εy y γx y }T is the membrane-flexural (inplane) strain field, {εs } = {γy z γx z }T comprises transverse
shear strains, [Bmf ] and [Bs ] are the corresponding straindisplacement matrices further suitably represented in terms of
the B-matrices having “m”, “f” and “s” in the subscript depending whether they contribute to the definition of the membrane,
flexural or shear strains, respectively, those with subscripting
“T” are related to the nodal translations and with “R” are related
to the nodal rotations, and finally, “0” denotes constant terms
while “1” denotes linear terms with respect to the natural thickness coordinate t. The vector {d} comprises nodal displacements
(translations and rotations).
2.3. Piezoelectric Layer
The constitutive equations of the piezoelectric material depend on the choice of the independent variables. Since the aim
is vibration suppression of the considered structures it is suitable to choose the mechanical strain and the electric field [18],
yielding:
{σ} = [CE ]{ε} − [e]T {E}
{D} = [e]{ε} + [dε ]{E}
where {σ} is the mechanical stress in vector (Voigt) notation,
{D} is the electric displacement vector, [CE ] is the piezoelectric material Hook’s matrix at constant electric field E, [dε ] is
the dielectric permittivity matrix at ε constant, and [e] is the
piezoelectric coupling matrix.
This paper considers piezoceramic elements with electrodes
on the top and bottom surfaces and poled in the thickness direction, where the in-plane strains are coupled with the perpendicularly applied electric field through the piezoelectric e31 effect.
The authors of the paper have made an investigation about the
accurate description of the electric potential distribution across
the thickness of the piezolayer [19]. The Gauss’ law for dielectrics (no free electric charge density) given as div{D} = 0
together with the here presented kinematical assumptions lead
to a quadratic distribution of the electric potential. Nevertheless, the investigation has shown a negligible difference in the
obtained results when the linear and the quadratic distribution
of the electric potential is accounted for. This is valid for most
of the typical piezoelectric laminate structures with relatively
small thickness of the piezolayers in comparison to the overall
laminate thickness. Therefore, the numerically more efficient
formulation with linear approximation is adopted here, thus:
∂ϕ
k
⇒ Ek = −
∂z
hk
where ϕ is the electric potential, k is the difference of the
electric potentials between the electrodes of the kth layer and
hk is the thickness of the piezolayer. The approximation in the
Eq. (13) defines a diagonal electric field—electric potential matrix [Bφ ] with typical term 1/hk on the main diagonal. The diagonal form of the matrix [Bφ ] results from the fact that the
difference of the electric potentials of a layer affects only the
electric field within the very same layer.
3. FINITE ELEMENT EQUATIONS
The governing equation of the structure’s dynamical behavior is given by the Hamilton’s principle, which says that the
system takes the path of the least action. Since it is dealt with
the piezoelectric continuum, the Lagrangian is properly adapted
in order to include the contribution from the electrical field besides the contribution from the mechanical field. The governing
thermodynamic equation has to correspond to the chosen independent variables (the mechanical strain and the electric field,
see Eq. (12)) and it is the electric enthalpy. Upon integration
by parts of the kinetic energy, the Hamilton’s principle for the
piezoelectric continuum can be given in the following developed
form:
[ρ{δu}T {ü} + {δε}T [CE ]{ε} − {δε}T [e]T {E}
(12)
(13)
v
−{δE}T [e]{ε} − {δE}T [dε ]{E}] dV
383
NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION
=
{δu}T {δFv } dV + {δu}T δFSi dS1
v
S1
+{δu}T {δFP } −
δφqdS2 − δφQ
-piezolayer electric charges:
{Qext } = −
(14)
where Fv , FS1 and FP are the external volume, surface (acting
on surface S1 ) and point loads, respectively, q and Q are the surface electric charge (acting on surface S2 ) and the point electric
charges, respectively. Due to the assumption of a constant difference of the electric potential over the surface of each piezoelectric layer, only the corresponding electrical loads are considered
in the sequel and those are the uniformly distributed surface
electric charges.
Performing the discretization of the structure one comes up
with the well-known semi-discrete form of the finite element
equations:
[M]{d̈} + [C]{ḋ} + [Kuu ]{d} + [Kuφ {φ} = {Fext }
Kφu {d} + Kφφ {φ} = {Qext }
(15)
(16)
with vector {φ} comprising piezolayers electrical degrees of
freedom (layer-wise differences of electric potentials, assumed
to be constant per single piezoelectric layer within an element)
and the following matrices and vectors are introduced:
-element mass matrix:
[Mu ] =
[Nu ]T ρ[Nu ] dVe
-element mechanical stiffness matrix:
[Kuu ] = [Kmf ] + [Ks ] + [Kt ]
-element piezoelectric coupling matrix:
([Bu ]T [e]T [Bφ ]) dV = [Kφu ]T
(18)
Ve
-element dielectric stiffness matrix:
[Kφφ ] = −
[Bφ ]T [dε ][Bφ ] dVe
(19)
Ve
-element damping matrix (Rayleigh):
[C] = α[M] + β[Kuu ]
(20)
-nodal mechanical loads:
T
{Fext } =
[Nu ] {Fv } dVe + [Nu ]T FS1 dS1 + [Nu ]T {Fp }
Ve
(22)
where the matrix [Nu ] is the element shape-functions matrix, Ve
is the volume of the element and {q} is the vector comprising
surface electric charges for all piezoelectric layers within the
element (each component of the vector corresponding to one
piezolayer). It should be noted that Eq. (22) takes advantage of
the assumption of uniform electric charge distribution over the
surface of the element, for each piezolayer (i.e., Nq ≡ 1).
The element stiffness matrix comprises the contribution from
the stiffness matrix related to the membrane-flexural (in-plane)
strains [Kmf ], the stiffness matrix related to the shear strains
[Ks ] and the so-called torsional stiffness [Kt ]. The former two
are determined by taking advantage of the developed strain field,
hence:


[BTm ]T


[Kmf ] =
(23)
− − −−  [Cm ][[BTm ] t[BR1f ]] dVe
Ve
T
t[BR1f ]


[BTs ]T


T
[Ks ] =
− − − − − − −−  [Cs ][[BTs ] [BR0s ]
Ve
[BR0s ]T + t[BR1s ]T
+ t[BR1s ]] dVe
(24)
(17)
Ve
[Kuφ ] =
{q} dS2
S2
S2
S1
(21)
where [Cm ] is the part of the Hook’s matrix relating the inplane stresses and strains, while [Cs ] relates the transverse shear
stresses and strains.
The element has five degrees of freedom in the local-running
c.s. (only two rotations, Eq. (3)), but due to their transformation
there are in general six degrees of freedom in the global c.s. (all
three rotations, Eq. (4)). Consequently, for a special position of
the element, when the z -axis of the local-running c.s. coincides
with one of the global coordinate axes, the stiffness constant
corresponding to the rotational degree of freedom around that
axis is equal to zero. The element will not be constraint enough,
and any slight disturbance in the load corresponding to this degree of freedom could cause an erratic behavior of the element.
The solution of the problem is achieved following Zienkiewicz
and Taylor [20] through the introduction of additional torsional
stiffness, by defining the governing torsional strain energy, Et ,
that behaves as a penalty function forcing the local rotation θz
be approximately equal to 0.5·(∂v /∂x −∂u /∂y ) at integration
points:
1
Et = αn Yhn
2
A
1
θ z −
2
∂v
∂u
−
∂x
∂y
2
dA,
(25)
(r,s,o)
where Y is the Young’s elasticity module and αn is a fictitious
elastic parameter (torsional coefficient) that can be determined
384
D. MARINKOVIĆ ET AL.
following the suggestions of Krishnamoorthy [21]. More details
regarding the calculation of the torsional stiffness matrix can be
found in [22].
As pointed out earlier, the formulation covers the piezoelectric coupling achieved between the in-plane strains and the electric field in the thickness direction. Thus, only the matrix [Bmf ]
contributes in [Kuφ ], which is therefore calculated by the following integral:

[BTm ]T
− − −−  [e][Bφ ] dVae
[Kuφ ] =
Vae
t[BR1f ]T

(26)
where Vae denotes that the integration in Eq. (26) involves only
the active, i.e., piezoelectric layers of the element and the dimension of the matrix [Kuφ ] is 54 × npe , where npe denotes the
number of piezoelectric layers.
Finally, the matrix [Kφφ ] has the dimension npe × npe and is
of a diagonal form, because the piezolayers are electrically not
coupled and there is only one electrical degree of freedom per
layer.
Altogether the element has 54 mechanical and npe electrical
degrees of freedom.
4. NUMERICAL INTEGRATION AND ITS ORDER
The calculation of the previously given matrices and vectors
requires an integration of the respective terms. The integration in
the thickness direction is performed analytically in a layerwise
manner since the mechanical, piezoelectric and dielectric properties are generally different for different layers. The advantage
of differentiating between the B matrices related to the constant
and linear strain terms with respect to the thickness coordinate
(Eq. (11)) becomes obvious when the mentioned analytical integration in the thickness direction is considered. The integration
in the in-plane directions, r- and s-direction, is to be performed
by means of numerical integration procedures. Among different
procedures for the numerical integration the Gauss quadrature
formulas are very attractive from the finite element method point
of view due to their efficiency (n sampling points required for
the exact integration of the polynomial of the order (2n − 1)).
The exact integration of the matrices and vectors for the full
biquadratic degenerate shell element requires 3 × 3 integration
rule (3 Gauss integration points for both in-plane directions).
Nevertheless, the susceptibility of the degenerate shell element
to both shear and membrane locking is a well-known fact. As
Prathap [23] has pointed out, referring to the purely mechanical
single-layer formulation of the 9-node degenerate shell element,
it “does not lock severely,” which is due to the full biquadratic
shape functions. This means that the refinement of the mesh
always leads to a converged solution (which might not be always the case with a bilinear element), but the element suffers a
suboptimal convergence. A number of researchers have made investigations regarding the use of uniformly reduced integration,
selectively reduced integration, assumed and enhanced strain
methods, addition of incompatible (bubble) modes etc. All of the
offered methods are based on the experience and are not without
certain drawbacks. It is the inherent complexity of the element
that inhibits a solution for the problem that would be variationally consistent and this represents one of the most important
drawbacks of the mentioned methods. Prathap has argued that
a quite simple, satisfactory and numerically very efficient solution is given by the reduced integration techniques, since it
does not require reconstitution of either displacement nor strain
fields. For a rectangular plane form of the element a selective
integration technique (the 2 × 3 rule for εx term and the 3 × 2
rule for εy term) would lead to a locking free solution. However,
for a general quadrilateral double-curved geometry of the element the uniformly reduced integration technique introduced by
Zienkiewicz et al. [24] (the 2 × 2 rule) is the optimal one. An
important drawback of the technique is that it might cause the
unwanted zero-energy (“hour-glass”) mechanisms. Except for a
few cases where they act up, the uniformly reduced 9-node degenerate shell element represents a general shell element acceptably free of shear and membrane locking that can be gainfully
used for the analysis of doubly curved “passive” as well as piezoelectric active laminate structures. This will be demonstrated on
a few examples covering both purely mechanical cases and the
cases involving the piezoelectric coupling.
5. NUMERICAL EXAMPLES
Literature provides a great number of examples successfully
demonstrating the application of the reduced integration to relieve the locking effects for purely mechanical field. The aim
here is to show that the same technique may be just as well
used for the coupled electro-mechanical formulation given previously. A special emphasis is put on the simplicity and high
numerical efficiency as an advantage of this method.
The examples are calculated by means of COSAR (www.
femcos.de), a general purpose finite element package developed in cooperation between the Institute of Mechanics of the
TABLE 1
Material properties of the layers in the principal
material directions
Material properties
PZT G1195
piezoceramic layer
Elastic properties
63
63
0.3
24.2
Piezoelectric properties
C/mm2 )
2.286
C/mm2 )
2.286
T300/976
graphite/epoxy
E11 (GPa)
E22 (GPa)
ν12
G12 (GPa)
150
9
0.3
7.1
e31 (10−5
e32 (10−5
0
0
NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION
FIG. 3.
Shape control of a simply supported composite plate.
Otto-von-Guericke University of Magdeburg and the engineering company FEMCOS mbH. Two different shell type elements
are compared—the here presented piezoelectric Shell9 element
and the previously developed and in praxis already proven to
be accurate [7, 25–27] piezoelectric 8-node Semi-Loof element.
The obtained results are compared to each other and to solutions
from other authors, for the cases where they are available.
All the below given examples consider composite structures
with two types of layers—the graphite/epoxy layer and the PZT
layer. The properties of the layers are given in Table 1, which is
in full (except for thermo-elastic properties) taken from the reference [28]. Not specified piezoelectric constants are considered
to be equal to zero. The paper is focused on the actuator function
of the piezoelectric layers, hence the dielectric constants are not
needed and are not given in the table. It should be noted that the
sequence and the thickness of layers differs in the considered
examples and will be specified separately.
5.1. Shape Control of Adaptive Composite Plate
The first example demonstrates the shape control of structures
by means of piezoelectric actuation. A simply supported plate
(a × b = 254 × 254 mm) has a cross-ply sequence of graphite
FIG. 4.
385
epoxy [0/90/0]s and two piezolayers bonded to the top and bottom surfaces (material properties summarized in Table 1). It is
initially subjected to a uniformly distributed load of 200 Nm−2 .
The thickness of each graphite/epoxy layer is 0.138 mm, and of
the piezolayer 0.254 mm. The piezolayers are oppositely polarized and a constant voltage is supplied to both layers resulting in
bending moments which tend to recover the initial flat shape of
the plate. The voltage giving the shape that corresponds mostly
to the initial undeformed shape is found. This example is originally proposed by Kioua and Mirza [28] whose approximate
solution is obtained using the conventional Ritz analysis based
on the shallow-shell theory.
The structure was discretized by an 8 × 8 finite element mesh
and the normalized centerline deflection (deflection/plate side)
was calculated for the same voltages as given in the previous
reference, i.e., 0 V (initial deformed shape under distributed
load), 15 V and 27 V. Only the full integration of the elements
was used in this case, since no locking effects were excited.
Figure 3 shows the results obtained with the Shell9 element and
the 8-node Semi-Loof element (solid lines, nearly congruent due
to the high agreement of the results) and the results from Kioua
and Mirza (dashed lines).
Deformed shape of a composite plate under uniform pressure and piezoactuation.
386
D. MARINKOVIĆ ET AL.
FIG. 5.
Cantilevered composite shell—geometry and finite element meshes.
According to the Ritz solution, the structure recovers the flat
shape for the last applied voltage (Figure 3, 27 V). However,
the finite element results show that the structure subjected to the
voltage of 27 V (and the uniform load) is not exactly flat, although very close to it (Figures 3 and 4b). It should be noted that
the action of the moments uniformly distributed over the plate
edges, such as those obtained by the actuation of the piezolayers,
certainly cannot recover the original (unloaded) flat geometry in
this case. This points out the deficiency of the assumed shape
functions within the Ritz solution, which are obviously not of a
high enough order (only up to the 2nd order polynomials used,
while at least the 4th order polynomials would be required in
this case). Actually, the Ritz solution represents the best fit of
the actual solution yielding a flat line in the case of the above
considered plate for the voltage of 27 V.
The here considered example was also calculated by a number of authors. The here presented results were compared with
the results published by Lee et al. [29] and Lim et al. [30].
The curves of the graphical representations published in both
papers coincide with the results in Figure 3 in the limit of accuracy of the pictures. Therefore these curves are not included
in Figure 3. Lee et al. [29] used their 9-node assumed strain
shell element and also an 8 × 8 mesh. Lim at al. [30] applied a
piezoelectric 18-node assumed strain solid element and, additionally, the NASTRAN HEXA8 element with the piezoelectric
strain equivalently replaced by thermally induced strain (thermal
analogy).
5.2. Cantilevered Curved Composite Shell
The second example considers a cylindrical composite shell
of dimensions a × b = 254 × 254 mm (Figure 5). The radius
of the shell will vary throughout the considered cases. Material
of the shell is defined by the following stacking sequence of
the graphite/epoxy [302 /0]s and two PZT layers are bonded to
the top and bottom surfaces. The structure reference direction is
taken to be the global x-axis. The thickness of the graphite/epoxy
layers is 0.138 mm and of the PZT layers is 0.254 mm.
At first this structure is subjected to a uniformly distributed
transverse force over the free edge of the shell (Figure 5). The
PZT layers are short-circuited so that a zero voltage is imposed and the layers act only as passive layers. The radius is
taken to be R = 10 × b. Due to the unbalanced stacking sequence of the shell, the load causes both bending and twisting of
the shell. Therefore, the transverse deflection is observed at three
characteristic points, two end-points and the mid-point of the
free edge, denoted in the Figure 5 as 1, 3 and 2, respectively.
The structure is discretized by 4 different meshes given in
Figure 5 and the results for the above mentioned characteristic
points are summarized in Table 2. The Shell9 element is used
with both full and reduced integrations, while for the application of the Semi-Loof element the full integration is available
only. Due to the large span to thickness ratio of approximately
190, the locking effects are obvious when a rough mesh is used
with exactly integrated Shell9 element. The diagram in Figure 6
shows the results for the shell free edge transverse deflection
TABLE 2
Transverse deflection of the characteristic points of the composite shell subjected to a uniformly distributed edge force
q (R = 10 × b)
Edge Force q = 2.0 10−2 N/mm
(w3 /b) × 10−3
(w2 /b) × 10−3
(w1 /b) × 10−3
Mesh
b = 254 mm
Shell9
Shell9
Semi-Loof
Shell9
Shell9
Semi-Loof
Shell9
Shell9
Semi-Loof
Int. rule
1×2
2×4
4×4
8×8
3×3
−4,894
−6,095
−6,386
−6,520
2×2
−6,957
−6,598
−6,594
−6,571
3×3
−6,319
−6,520
−6,528
−6,535
3×3
−5,154
−6,110
−6,370
−6,453
2×2
−6,740
−6,512
−6,506
−6,486
3×3
−5,484
−6,232
−6,378
−6,457
3×3
−5,780
−7,268
−7,461
−7,618
2×2
−7,945
−7,701
−7,697
−7,671
3×3
−5,556
−7,260
−7,492
−7,638
387
NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION
FIG. 6. Cantilever composite shell under uniform edge force—free edge transverse deflection, convergence of the fully (FI) and reduced (RI) integrated Shell9
element.
obtained with the fully and reduced integrated Shell9 element.
As the diagram and Table 2 show, the refinement of the mesh with
the fully integrated Shell9 element (dotted lines on the diagram,
in the legend denoted with FI) leads to a converged solution. On
the other hand, a much better convergence rate may be noticed
with the uniformly reduced Shell9 element (solid lines on the
diagram, in the legend denoted with RI), pointing out that it is
free of locking effects. It should also be noticed that the convergence is monotonic, but the results of the fully integrated element
converge from below, while those from the reduced integrated
element from above.
In the second case the same structure under the influence of
the electrical load is observed. The two piezolayers are oppositely polarized and subjected to the same voltage of 100 V. This
results in bending moments uniformly distributed over the shell
edges. This case is also proposed by Kioua and Mirza [28] and
solved by already mentioned conventional Ritz analysis based
on the shallow-shell theory, whereby variable radius of the shell
was considered. The results obtained with the Shell9 and the
Semi-Loof element for the cylindrical shell with R = 10 × b are
compared in Table 3, which has the same structure as Table 2.
As in the previous case, the diagrams in Figures 7 and 8 give
the results for the shell free edge transverse deflection obtained
with the fully and reduced integrated Shell9 element, respectively. The faster convergence rate of the reduced integrated
Shell9 element is obvious once again and except for the coarsest
mesh, the results are nearly congruent in Figure 8. It should also
be noticed that the convergence progresses in a different manner
TABLE 3
Transverse deflection of the characteristic points of the active composite shell subjected to electric voltage ϕ(R = 10 × b)
Electric Potential ϕ = 100 V
(w3 /b) × 10−3
(w2 /b) × 10−3
(w1 /b) × 10−3
Mesh
b = 254 mm
Shell9
Shell9
Semi-Loof
Shell9
Shell9
Semi-Loof
Shell9
Shell9
Semi-Loof
Int. rule
1×2
2×4
4×4
8×8
3×3
−6,394
−6,840
−6,507
−6,450
2×2
−6,299
−6,431
−6,423
−6,397
3×3
−5,682
−6,163
−6,181
−6,347
3×3
−4,241
−3,675
−3,603
−3,420
2×2
−2,775
−3,359
−3,354
−3,367
3×3
−3,380
−3,180
−3,343
−3,354
3×3
−8,464
−10,439
−10,514
−10,732
2×2
−10,385
−10,741
−10,762
−10,730
3×3
−7,287
−10,193
−10,394
−10,686
388
FIG. 7.
D. MARINKOVIĆ ET AL.
Cantilever composite shell under piezoelectric excitation—free edge transverse deflection, convergence of the fully integrated Shell9 element.
in comparison to the case of the purely mechanical field. Actually, it is non-monotonic. This is due to the loads originating
from the piezoelectric coupling effect. The refinement of the
mesh does not affect the mechanical stiffness matrix only, but
also the piezoelectric stiffness matrix, and, in further instance,
the mechanical loads originating from the piezoelectric coupling. Furthermore, the description of the piezoelectric coupling
FIG. 8.
is directly affected by the locking effects and their alleviating
by means of the reduced integration technique since it couples
a part of the strain field affected by the locking effects to the
electric field.
The diagram in Figure 9 compares the solution from the
Shell9 and Semi-Loof element obtained by the 8 × 8 mesh (full
integration) to the above mentioned approximate Ritz solution
Cantilever composite shell under piezoelectric excitation—free edge transverse deflection, convergence of the reduced integrated Shell9 element.
NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION
FIG. 9.
Bending and twisting of active cantilevered composite shell.
FIG. 10.
Simply supported composite cylindrical arch subjected to uniformly distributed vertical force.
FIG. 11.
Radial deflection of the composite arch, force excitation—convergence of the Shell9 element.
389
390
D. MARINKOVIĆ ET AL.
FIG. 12.
Radial deflection of the composite arch, force excitation—convergence of the Semi-Loof element.
obtained by the shallow shell assumptions [28]. The indicator
of bending deformation is taken similarly to the previous case,
i.e., the ratio (w2 /b) is observed, while the ratio (w3 − w1 )/b is
taken as an indicator of twisting. The solutions from both elements show very high agreement. Also, a good agreement with
the Ritz solution can be noticed, especially for higher values
of radius to span ratio, which is expected. As this ratio takes
lower values, the deformation of the shell becomes more complex and the previously mentioned deficiency of the assumed
shape functions of the Ritz solution becomes more pronounced
FIG. 13.
resulting in higher discrepancy between the finite element and
Ritz results observable in the Figure 9. It should also be noted
that the shallow shell assumptions have higher (negative) impact
on the accuracy of the Ritz solution in the area of smaller radii,
which is, however, not expected to be crucial for the in this case
considered radius to span ratios.
5.3. Simply Supported Composite Cylindrical Arch
In the third example a simply supported cylindrical arch
of radius R = 100 mm and width b = 60 mm is considered.
Radial deflection of the piezoelectric active composite arch—convergence of the Shell9 element.
NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION
The stacking sequence of the composite is now [45/–45/0]s
and two PZT layers are also added to the top and bottom surfaces. The thickness of the graphite/epoxy is 0.12 mm and of
the PZT layer 0.24 mm. It should be noted that the stacking
sequence is now balanced and results in orthotropic material
properties.
Following the same pattern from the first example, the structure is at first subjected to the vertical force uniformly distributed over the width line according to Figure 10, while the
piezoelectric layers are short-circuited imposing zero electric
voltage.
The structure is discretized by 3 different meshes. Each mesh
has 4 elements in the axial direction (width), but the number of
elements in the circumferential direction varies from 10, 20 to
40 elements. The calculated radial deflection of the width midline is presented in Figures 11 and 12. Again, the Shell9 element
in combination with the reduced integration technique yields
satisfactory results already with a very rough mesh (Figure 11).
The refinement of the mesh with fully integrated Shell9 element
leads to the converged solution, but a much larger number of
elements is needed for a satisfying accuracy.
The conclusions from the purely mechanical case extend as
well to the case when the considered cylindrical arch is excited
by the actuation of the piezolayers. The oppositely polarized
layers are subjected to the voltage of 100 V. As can be seen in
Figure 13, already the mesh with 10 uniformly reduced Shell9
elements in the circumferential direction yields very accurate
results. Fully integrated element needed a mesh with 40 elements
in the circumferential direction to provide the same accuracy. A
similar behavior is observed with the fully integrated Semi-Loof
element in the Figure 14.
FIG. 14.
391
6. CONCLUSIONS
Development of active structures requires adequate modeling
tools in order to simulate and analyze their behavior. Such a
modeling tool should provide a satisfying accuracy, but at the
same time the numerical effort is supposed to be acceptable—
the two requirements that are not easy to conciliate. The present
paper tried to achieve that by extending the degenerate shell
approach to modeling thin-walled piezoelectric active composite
laminates.
The originally formulated 9-node degenerate shell element,
although numerically more efficient than a 3D solid element,
still exhibits numerical problems recognized as locking effects.
It can be argued that the element is reliable since the locking
is not severe, meaning that the refinement of the mesh leads to
a converged solution but at a suboptimal rate. Resolving this
problem in a variationally consistent manner still remains an
open and attractive task. Among several empirical solutions developed for the “passive” structures, the authors have chosen the
uniformly reduced integration technique as a numerically very
efficient one. The use of this technique is successfully demonstrated through several numerical tests, which prove that a satisfying accuracy may be achieved with a relatively rough mesh
and by using only 4 (2 × 2 integration rule) instead of 9 integration points (3 × 3 integration rule). A multiple reduction of the
numerical effort is obvious, which offers a good basis to handle large-scale problems, geometrically nonlinear analysis, etc.
The most important drawback of the method, although only in
rare cases, is the possible appearance of the zero-energy modes.
The possible solution to this problem could be an hour-glass
control [31]. The presented Shell9 element has noteworthy advantages in geometrically non-linear applications and also to
Radial deflection of the piezoelectric active composite arch—convergence of the Semi-Loof element.
392
D. MARINKOVIĆ ET AL.
include a material non-linear behavior. These extensions of the
element are under progress.
ACKNOWLEDGEMENTS
This work has been partially supported by the postgraduate
program of the German Federal State of Saxony-Anhalt. This
support is gratefully acknowledged.
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