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Multi-layer higher-order finite elements for the analysis of free-edge stresses in piezoelectric actuated laminates
Composite Structures 61 (2003) 271–278 www.elsevier.com/locate/compstruct
Multi-layer higher-order finite elements for the analysis of free-edge stresses in piezoelectric actuated laminates Alessandro Mannini, Paolo Gaudenzi
*
Dipartimento di Ingegneria Aerospaziale e Astronautica, Universit a di Roma La Sapienza, Via Eudossiana 16, 00184 Rome, Italy
Abstract The static interaction between a laminate and distributed piezoelectric actuators is considered. In particular the problem of the stress concentration at the free edge of the active elements of a piezoelectric composite is investigated. A finite element model for the laminated composite plate is developed using a multi-layer higher-order finite element approach. A typical configuration is considered, in which two active layers are bonded on the top and bottom of a passive substructure. A pure bending case is investigated. The obtained results provide useful information of the typical static response. A parametric study is also performed on the effects of the main geometrical characteristics on the intensity and the extension of edge effect. Ó 2002 Published by Elsevier Science Ltd.
1. Introduction During the last few years studies relevant to the use of distributed sensors and actuators integrated with the structures have focused on the possibility of building structures capable of changing their shape according to the environmental conditions and on the reduction of their dynamic response [1]. There are two characteristics of piezoelectric materials which allow them to be used as sensors or actuators. The first one is their direct piezoelectric effect which implies that the materials induce electric charge or potential when they are subjected to mechanical deformations. Conversely they are deformed if some electric charge or potential is imposed to them. Among the materials able to generate such effect, there are the piezoceramics (PZT). They are easily bonded to the surfaces or embedded within the structure. Structures with distributed piezoelectric sensors and actuators have been investigated by many researchers. Crawley and de Luis [2] proposed an analytical model for the static interaction between a beam and segmented piezoelectric actuators symmetrically bonded to the top and bottom surfaces. In their model a pure bending moment is considered; however the effects of the transverse shear and the axial force are neglected. *
Corresponding author. Tel.: +39-06-4458-5304; fax: +39-06-44585670. E-mail address: [email protected] (P. Gaudenzi).
Ray et al. [3] present an exact static analysis of simply supported rectangular-type plate. The intelligent structure proposed is treated as a laminated plate and the distributed piezoelectric sensor and actuator layers are considered to be as plies of the laminate intelligent structure and to be bonded perfectly to the surface of the substrate. The substrate is a laminate of graphite–epoxy composite. This study shows the capability of the actuator and sensor layers to cause and sense deformations of the substrate. The effectiveness of this control system significantly increases with the decrease in the length to the thickness ratio of the substrate. The same authors [4] developed a finite element model for the static analysis of a simply supported rectangular intelligent plate using the higher-order shear deformable displacement theory. The distributed actuator and sensor layer are bonded to the top and bottom surfaces of substrate. A two-dimensional (2D) eight-node isoparametric finite element is derived for modeling the structure. The static and dynamic interaction between a bonded piezoelectric actuator and an underlying beam structure is investigated by Robbins and Reddy [5], using an higher-order layer-wise displacement theory. A comparison between the layer-wise model and models based on the classical beam theory and shear deformation theory has been carried out. They concluded that the layer-wise model is more efficient and appropriate for thick composite beams than the other two models. However due to a substantial increase in the degrees of
0263-8223/03/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. doi:10.1016/S0263-8223(02)00040-5
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freedom, the layer-wise model becomes large and difficult to apply. An analysis of laminated composite plates forced into cylindrical bending by the application of voltages to piezoelectric actuators attached to the top and bottom surfaces is performed by Zhou and Tiersen [6] using the equation of linear elasticity. The piezoelectric actuator is modeled as a 2D surface film. The 2D equations for the laminated composite plate are obtained using the variational procedure of Mindlin. A fully non-linear theory for the dynamics and active control of elastic laminated plates with integrated piezoelectric actuators and sensors undergoing large rotation and small strain vibrations has been shown by Pai et al. [7]. An exact solution methodology for composite piezoelectric composite laminate in cylindrical bending has been used by Brooks and Heyliger [8], to consider several representative cases in which the laminate changes shape using distributed and patched actuators. Also embedded piezoelectric layers are taken into account. Heyliger [9] presented an exact solution for a hybrid laminate containing both piezoelectric and elastic layers under applied surface traction and surface potential. Alurajid et al. [10] developed a modified classical lamination theory to account for piezoelectric coupling terms under applied electric field. CLT was applied to predict the stress field and out-of-plane displacement of different types of piezoelectric actuators. The CLT results were verified using the finite element method. A first attempt to survey and discuss the advances and trends in the formulations and applications of the finite element modeling of adaptive structural elements was made by Benjeddou [11]. Gaudenzi [12] proposed the analysis of an adaptive beam composed by a passive layer and two surfaces bonded induced strain actuators by using a simple higher-order beam model. Exact solutions are obtained for a case of membrane actuation and for a pure bending case. The obtained solutions are then discussed in terms of the main geometrical parameters of the system and compared with the classical closed form solution based on Euler–Bernoulli models. Moreover this model allows the description of the edge effect which occurs close to the free boundary of the considered structure. An important aspect of the analysis of the active laminated structures concerns the stress determination of the 3D field stress, in particular the interlaminar components. The interlaminar stresses are mainly responsible of the delamination and consequently possible detachment of the piezoelectric layers. For this reason the existence of interlaminar stresses, although small with respect to the axial component, must be evaluated in the study of the laminated structures. In the present study, in order to evaluate those stresses, higher-order
theories have been developed in which the strain and the stresses can be evaluated as well as the global response of the structure. The evaluation of the 3D field stress becomes of particular interest when the stress concentration at the boundary of the active layers of the system is required. The numerical procedure here proposed to address this topic is based on the use of the above mentioned higherorder theories which results by a compromise between a fully 3D finite element solution and a bidimensional reduced model that has been developed by the authors in previous studies [13,14]. Several numerical applications and results are presented for the case of cylindrical bending. In particular the analysis of an active structure composed by a passive layer and two surface bonded induced strain actuators is considered.
2. Kinematic assumptions The basic assumption of higher-order theories is that each layer of the laminate can be first considered as a separate elastic body and its displacement field can be expressed in terms of a power series expansion along the thickness direction. By considering the most general formulation a rectangular laminated plate made of an arbitrary number of layers in a 3D Xðx; yÞZðh=2 6 z 6 h=2Þ of an Euclidean space ð0; x; y; zÞ is taken into account. Each layer is assumed to be transversally isotropic elastic continuum and all the layers are considered to be perfectly bonded to each other. The displacements of the qth layer are expressed as a series expansion along the thickness: uq ðx; y; zÞ ¼
M X
zmq uq;m ðx; yÞ
m¼0
vq ðx; y; zÞ ¼
N X
znq vq;n ðx; yÞ
n¼0
wq ðx; y; zÞ ¼
L X
zlq uq;l ðx; yÞ
l¼0
where zq represents the axis normal to the middle plane of qth layer in the local reference system. In that way the unknowns will depend only on the x and y variables. The layers can be thought as bonded along the top and bottom surfaces together with the other layers whose displacements have been treated in the same way. By enforcing the continuity of the displacements at some interfaces we may obtain in a global reference system a general expression for displacements valid throughout the thickness:
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278 M X
uðx; y; zÞ ¼
zm umð0Þ ðx; yÞ
m¼0 R X
þ F0 þ F0
M X m ðz hr Þ uðrÞ m ðx; yÞ
Fr
r¼1
m¼1
P X
M X
Fp
p¼1 N X
vð x; y; zÞ ¼
m ðpÞ
ðz hp Þ um ðx; yÞ
m¼1
zn vnð0Þ ð x; y Þ
273
from those uq;m ðx; yÞ, vq;n ðx; yÞ and wq;l ðx; yÞ as they are defined before, but can be easily derived from them. Such assumed displacement expressions are clearly continuous functions with respect of z, but their partial derivatives with respect to z, and consequently the interlaminar strains are not continuous. Therefore the equilibrium conditions on the interfaces can be satisfied also in presence of a mismatch of elastic properties of adjacent layers. For the details of the present kinematic approach see also [14].
n¼0
þ F0
R X
Fr
r¼1
þ F0
P X
N X
L X
Fp
N X n ðpÞ ðz hp Þ vn ð x; y Þ n¼1
ð0Þ
zl wl ð x; y Þ
l¼0
þ F0
R X
Fr
r¼1
þ F0
P X p¼1
3. Finite element formulation
n¼1
p¼1
wð x; y; zÞ ¼
n
ðz hr Þ vðrÞ n ð x; y Þ
L X
l
ðrÞ
ðz hr Þ wl ð x; y Þ
l¼1
Fp
L X
l
ðpÞ
ðz hp Þ wl ð x; y Þ
l¼1
where F 0 ¼ 1 when R 6¼ 0; F 0 ¼ 0 when R ¼ 0; F r ¼ 1 for z > hr ; F r ¼ 0 for z 6 hr ; F 0 ¼ 1 when P 6¼ 0; F 0 ¼ 0 when P ¼ 0; F p ¼ 1 for z < hp ; F p ¼ 0 for z P hp . hr represents the z coordinates of the interfaces on the positive side of the z-axis, where the continuity of displacements are enforced, hp are the z coordinates of the interfaces on the negative side of the z-axis (Fig. 1). The group of layers situated between two of such interfaces can be thought as macrolayer. In an analogous way uðrÞ m , ðrÞ ðrÞ ðrÞ ðrÞ vðrÞ , w , u , v , w represent the additional unknowns l n m l n relevant to the upper and lower macrolayers. The macrolayer containing the reference system is denoted by the label Ô0Õ. R represents the number of macrolayers placed on the positive side of the z-axis, P the number of macrolayers placed on the negative one. The expressions ðrÞ of umðrÞ ðx; yÞ, vðrÞ n ðx; yÞ and wl ðx; yÞ are in general different
Based on the kinematic assumptions described on the previous paragraph, a finite element approach has been consequently developed. The unknowns displacements uðx; y; zÞ, vðx; y; zÞ, wðx; y; zÞ and the generalized disðrÞ ðrÞ placements uðrÞ m ðx; yÞ, vn ðx; yÞ, wl ðx; yÞ can be rearranged as vectors, so that symbolically we may write as follows: T
fS ð x; y; zÞg ¼ fuð x; y; zÞ; vð x; y; zÞ; wð x; y; zÞg n ð1Þ fsðx; yÞg ¼ sð0Þ ; sð1Þ ; . . . ; sðrÞ ; . . . ; sðRÞ ; s ; . . . ; o ðpÞ ðP Þ T s ;...;s where ð0Þ ð0Þ ð0Þ ð0Þ T ¼ u ;v ;w s ðr Þ ðr Þ ðr Þ ðr Þ T s r ¼ 1; 2; . . . ; R ¼ u ;v ;w n o n oT ð pÞ ð pÞ ð pÞ ð pÞ s p ¼ 1; 2; . . . ; P ¼ u ;v ;w the displacement u can be written as oT ð0Þ n ð0Þ ð0Þ ð0Þ u ¼ u1 ; u2 ; . . . ; uðm0Þ ; . . . ; uM oT ðr Þ n ðr Þ ðr Þ ðr Þ u r ¼ 1; 2; . . . ; R ¼ u1 ; u2 ; . . . ; uðmrÞ ; . . . ; uM n o n o ð pÞ ð pÞ ð pÞ ð pÞ ð pÞ T u p ¼ 1; 2; . . . ; P ¼ u1 ; u2 ; . . . ; um ; . . . ; uM Similar expressions can be written for v and w. In general it is possible to write fS ð x; y; zÞg ¼ ½Z ð zÞ fsð x; y Þg where [ZðzÞ] can be expressed as h i ½Z ð zÞ ¼ Z ð0Þ ; Z; Z in which h i ð1Þ ð2 Þ ðr Þ ð RÞ Z ¼ F0 Z ;Z ;...;Z ;...;Z
h i ð1 Þ ð2Þ ð pÞ ðP Þ Z ¼ F0 Z ;Z ;...;Z ;...;Z
Fig. 1. Three layers model.
ðrÞ
ðpÞ
The matrices ½Z ð0Þ , ½Z and ½Z read
274
Z
h
Z
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
ð0Þ
ðr Þ
i
2
2 6 ¼ F r4
z hr
2
Z
ð pÞ
z2 0 0
1 z ¼ 40 0 0 0
6 6 ¼ F p6 6 4
zM 0 0
... 0 0
z hr
2
0 0 1 z 0 0
...
0 z2 0
0 ... 0
z hr
0
0
0
0
0
0
0
0
z hp
z hp
2
...
z hp
0
0
0
0
0
0
0
0
0 zN 0
M
0 0 1
0 z hr
M
0 0 0 0 z z2
3 0 05 zL
0 0 ...
0 z hr
2
0
...
0 z hr
0
0
0
0
0
0
0
0
z hp
0
Being [D] the partial differential operator which defines the strain components as a function of the displacements: 3 2 o 0 0 7 6 ox 7 6 o 7 6 6 0 07 7 6 oy 7 6 7 6 o 7 6 0 0 7 6 oz 7 6 ½ D ¼ 6 7 o o 7 6 0 7 6 oz oy 7 6 7 6 o o 7 6 0 7 6 ox 7 6 oz 5 4 o o 0 oy ox
z hp
2
0
... 0
z hp
N
0
0
0
0
0
0
0
z hr
N
0
0
z hr
2
...
z hr
0
0
0
0
0
0
0
0
z hp
z hp
2
...
z hp
3 L
7 5
3 7 7 7 7 L 5
T
fega ¼ ½d fEg where fEg is the vector of the electric field applied to the piezoelectric element, and ½d is the piezoelectric constant matrix. Their expressions are 8 9 < E1 = f E g ¼ E2 : ; E3 and 0 ½d ¼ 0 d13
0 0 d23
0 0 0
d15 0 0
0 d25 0
0 0 0
the expression for the strain components is ordered in general:
In the present formulation we assume fEg as a given quantity. In this way also the active strain will be given. For each unknown variable the shape function defined for the considered element can be used:
fegq ¼ ½ D fS ð x; y; zÞg ¼ ½ D ½Z fsð x; y Þg
fsð x; y Þg ¼ ½ N ð x; y Þ fdg
When some active layers are present in the laminate, the constitutive equations for the qth piezoelectric layer are in general: frgq ¼ ½Q q fegq fega in which ½Q q represents the elastic matrix of the qth layer and in which the second term at the right side is present when the qth layer is an active element. This last term represents the converse piezoelectric effect. In fact, for an the active layer we have
Since we consider the response of the structure to the piezoelectric action only, the finite element load vector will be generated by the actuation stress only. The stiffness matrix and the load vector due to the presence of the active layers can be consequently expressed as follows: Z T ½ K e ¼ f½ D ½Z ½ N ð x; y Þ g ½Q e ½ D ½Z ½ N ð x; y Þ dV Ve
fp e ¼
Z Ve
T
f½Z ½ N ð x; y Þ g frgp dV
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
where T
frgp ¼ ½Q ½d fEg By setting a convenient order for each series expansion together with a sufficient number of elements, different orders of approximation can be obtained for the relevant element. In an analogous way, the laminate can be subdivided in a certain number of sublaminates (that will be defined as ‘‘macrolayers’’). In this paper each sublaminate represents either an active (piezoelectric) layer or a passive one. In Appendix A a simple multilayer higher-order model used for the numerical simulations is explained in detail.
4. Numerical results In order to investigate about the numerical performance of the model here proposed for the purpose of evaluating the interlaminar stresses, a typical configuration has been considered, in which two active layers are bonded on the top and bottom of the laminate. The considered structure is composed by an aluminium substrate coupled with distributed actuators. In particular, the structure is a cantilever beam in which two actuating piezoelectric layers perfectly bonded to the surfaces of a passive substructure are present (Fig. 1). The following elastic parameters have been chosen: for the passive layer (aluminium): E ¼ 70 GPa;
m ¼ 0:3;
G ¼ 35 GPa
for the piezoelectric layers: E1 ¼ E2 ¼ 66 GPa; m ¼ 0:3;
E3 ¼ 26 GPa;
G1 ¼ G2 ¼ 33 GPa;
G3 ¼ 26 GPa
For each configuration the response variables (displacements, strains and stresses) have been calculated and the distribution both throughout the thickness and along the span have been evaluated; in this case the lower interface has been considered. The length-tothickness ratio of the whole structure has been assumed to be 64. Three-node C elements with a mesh refinement of 20 elements with 41 total nodes have been used for the numerical simulation. Due to the nature of the stress field, the mesh was more refined close to the free end of the beam and less refined close to the fixed end. In this way a more accurate analysis where high stress concentration is expected. By expanding the transverse displacement w at an order less than in-plane displacements u and v, that is the usual approach in the theory of bidimensional structures such as plates and shells, three sets of order of expansion were first tested: M ¼ N ¼ 3; L ¼ 2;
d15 ¼ 540 m=pV
The value of the electric field between the upper and lower surfaces of the piezoelectric layer has been stated as
M ¼ N ¼ 4; L ¼ 3;
M ¼ N ¼ 5; L ¼ 4 The same order of expansion was adopted for each macrolayer. As expected the value of Drz % decreases as the order of expansion increases. A good convergence is reached for higher-expansion order. Fig. 2 shows the jump of the interlaminar normal stress component evaluated at the top and bottom layer of the lower piezoelectric/structure interface. The results have been obtained corresponding to an abscissa x ¼ 63:9925, and with a 20 element model. A reasonable convergence is obtained with an expansion M ¼ N ¼ 4, L ¼ 3. Accordingly this last model for all macrolayers has been chosen in all simulation. The stresses have been non-dimensionalized according to
The piezoelectric constants have been assumed as d13 ¼ 180 m=pV;
275
r¼
rt ð EtÞAl þ ð EtÞPZT
s¼
st ðGtÞAl þ ðGtÞPZT
E3 ¼ 200 kV=m The following configurations of three layers (active/ passive/active) have been investigated: (a) Same thickness for each layer. (b) Piezoelectric layers with a thickness ratio related to the whole thickness of the beam of 1/10; the substructure layer has then a thickness ratio equal to 8/10. (c) Piezoelectric layers with a thickness ratio of 1/7. In all cases an electrical field of opposite sign for each piezoelectric layer has been applied, so a bending effect is obtained.
Fig. 2. Jump in percentage of interlaminar stress component evaluate at active/passive interface and x ¼ 63:9925, for a laminate of type b.
276
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
where t is the thickness ratio of the macro layer (respectively passive or active) to that of the beam. The stress fields predicted by the model are reported in Figs. 3–9. First the bending stresses rx are illustrated for several conditions. Fig. 3 shows respectively the distribution of the stresses at the top and bottom of the lower interface of the active/passive interface. These results have been obtained in the case of three equal layers. The stresses tend to zero when the free edge is reached. Figs. 4 and 5 show the through thickness distribution of stresses for different abscissas respectively in the case of structure with three equal layers and the one with a thickness ratio of 1/10. In both cases the removal of plane section hypothesis is evident in the area close to the edge. Fig. 6 illustrates a comparison of the results obtained for the structure of type c (thickness ratio of 1/7) with those obtained with an analytical model by Gaudenzi [12]. The solutions predicted at different value of x are in good agreement, in particular the ones corresponding to the passive structure. The shear stresses distributions have also been evaluated. Fig. 7 illustrates the interlaminar stresses for a structure of type a (three equal layers) at the lower interface of the active/passive layers. Fig. 8 shows the through thickness distribution of the s stresses for a laminate of type a. In the same figure the distribution of the interlaminar stresses predicted by the one layer model is reported.
Fig. 3. rx distribution along the active/passive interface, z=h ¼ 0:167, for a laminate of type a.
Fig. 4. Through thickness distribution of longitudinal stresses for different values of x and for the laminate of type b.
Fig. 5. Through thickness distribution of longitudinal stresses for different values of x (three equal layers model).
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
277
Fig. 6. A comparison of through thickness distribution of longitudinal stresses with the analytical model [12] for different values of x (analytical solution: j: x=l ¼ 0:95, : x=l ¼ 1:00; present solution: r: x=l ¼ 0:95 (an), M: x=l ¼ 1:00 (an)).
Fig. 7. sxz distribution along the active/passive interface, z=h ¼ 0:167 for the laminate of type a.
Fig. 9. rz distribution along the active/passive interface, z=h ¼ 0:4 for the laminate of type b.
Finally the rz distribution along the interface of the passive substructure for the laminate of type b is reported in Fig. 9.
5. Conclusions
Fig. 8. Through thickness distribution of sxz for the laminate of type a at the point x ¼ 63:9875.
An investigation that illustrates the behaviour of piezoelectric composite laminates in cylindrical bending under distributed forcing functions has been carried out. The stress and displacement distribution are presented. In particular the stress concentration at the free edge of the active elements has been calculated. A multi-layer higher-order finite element model has been formulated. The model, quite general, allows evaluating the presence of active layers, both bonded at the surface and
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embedded. The effectiveness of the approach has been evaluated referring to some typical cases, and a good agreement with similar other results has been obtained. Appendix A In the following the detailed expressions of the generalized displacements which are used for the implementation of a simple multi-layer model based on the general procedure are shown. The model consists of three layers; two interfaces are assumed for fulfill the discontinuity condition. As regards the power series expression, we assume M ¼ N ¼ 4 and L ¼ 3; consequently the degrees of freedom per node are 36.
n ð1Þ ð1Þ ð1Þ oT wð1Þ ¼ w1 ; w2 ; w3
n
ð1Þ
w
o
n o ð1Þ ð1Þ ð1Þ T ¼ w2 ; w2 ; w3
For the expression of [Z] we have
ð1Þ ð0Þ ð1Þ ð0Þ ½ZðzÞ ¼ Z ; F Z ; F 0 Z where
Z
ð0Þ
2
1 ¼ 40 0 2
z z2 0 0 0 0
z3 0 0
0 z2 0
0 z3 0
u0 ; u1 ; u2 ; u3 ; u4 ; v0 ; v1 ; v2 ; v3 ; v4 ; w0 ; w1 ; w2 ; w3 the following 11 are relative to the lower layer
being Dz ¼ ðz h1 Þ and Dz ¼ ðz h1 Þ.
the first 14 are relative to the central layer
ð1Þ
i
3
Dz 0 0
Dz4 0 0
0 0 1 z 0 0
Dz Dz2 Z ¼4 0 0 0 0 2 2
Dz Dz ð1Þ Z ¼4 0 0 0 0 h
Dz3 0 0
z4 0 0
4
Dz 0 0
0 z4 0
0 0 1
0 0 0 0 z z2
0 0 Dz Dz2 0 0
0 Dz3 0
0 Dz4 0
0 0 0 0 Dz Dz2
0 0 2 Dz Dz 0 0
0 3 Dz 0
0 4 Dz 0
0 0 0 0 2 Dz Dz
3 0 05 z3 3 0 0 5 Dz3 3 0 0 5 3 Dz
u1 ; u2 ; u3 ; u4 ; v1 ; v2 ; v3 ; v4 ; w1 ; w2 ; w3 and the last 11 are relative to the upper layer
References
u1 ; u2 ; u3 ; u4 ; v1 ; v2 ; v3 ; v4 ; w1 ; w2 ; w3
In matrix form the generalized displacements reads as follows: n o ð1 Þ T fsð x; y Þg ¼ sð0Þ ; sð1Þ ; s with ð0Þ ð0Þ ð0Þ ð0Þ T ¼ u ;v ;w s ð1Þ ð1Þ ð1Þ ð1Þ T ¼ u ;v ;w s n o n o ð1Þ ð1Þ ð1Þ ð1Þ T s ¼ u ;v ;w and ð0Þ n ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ oT ¼ u0 ; u1 ; u2 ; u3 ; u4 u ð1Þ n ð1Þ ð1Þ ð1Þ ð1Þ oT u ¼ u1 ; u2 ; u3 ; u4 n o n o ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ T u ¼ u2 ; u2 ; u3 ; u4 Similar expressions can be obtained for v and w: ð0Þ n ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ oT ¼ v0 ; v1 ; v2 ; v3 ; v4 v ð1Þ n ð1Þ ð1Þ ð1Þ ð1Þ oT ¼ v1 ; v2 ; v3 ; v4 v n o n o ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ T v ¼ v2 ; v2 ; v3 ; v4 ð0Þ n ð0Þ ð0Þ ð0Þ ð0Þ oT w ¼ w0 ; w1 ; w2 ; w3
[1] Rogers CA. Intelligent material systems––the dawn of a new materials age. J Intell Mater Syst Struct 1992;4(1). [2] Crawley EF, de Luis J. Use of piezoelectric actuators as elements of intelligent structures. AIAA J 1987;25(10). [3] Ray MC, Bhattacharya R, Samanta B. Exact solutions for static analysis of intelligent structures. AIAA J 1993;31(9). [4] Ray MC, Bhattacharya R, Samanta B. Static analysis of an intelligent structure by the finite element method. Comput Struct 1994;52(4):617–31. [5] Robbins DH, Reddy JN. Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Comput Struct 1991;41(2):265–79. [6] Zhou YS, Tiersen HF. An elastic analysis of laminated composite plates in cylindrical bending due to piezoelectric actuators. Smart Mater Struct 1994;3:255–65. [7] Pai PF, Nayfeh AH, Oh K, Mook DT. A refined nonlinear model of composite plates with integrated piezoelectric actuators and sensors. Int J Solids Struct 1993;30(12):1603–30. [8] Brooks S, Heyliger P. Static behavior of piezoelectric laminates with distributed and patched actuators. J Intell Mater Syst Struct 1994;5. [9] Heyliger P. Static behavior of laminated elastic/piezoelectric plates. AIAA J 1994;32(12). [10] Alurajid A, Toya M, Hudunt S. Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure. Int J Solids Struct 2001;38: 3377–91. [11] Benjeddou A. Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Comput Struct 2000;76: 347–63. [12] Gaudenzi P. Exact higher order solutions for a simple adaptive structure. Int J Solids Struct 1998;35(26-27):3595–610. [13] Gaudenzi P, Barboni R, Mannini A. A finite element evaluation of a single-layer and multi-layer theories for the analysis of laminated plates. Compos Struct 1995;30:427–40. [14] Gaudenzi P, Mannini A, Carbonaro R. Multi-layer higher-order finite elements for the analysis of free-edge stresses in composite laminates. Int J Numer Meth Eng 1998;41:851–73.
Multi-layer higher-order finite elements for the analysis of free-edge stresses in piezoelectric actuated laminates Alessandro Mannini, Paolo Gaudenzi
*
Dipartimento di Ingegneria Aerospaziale e Astronautica, Universit a di Roma La Sapienza, Via Eudossiana 16, 00184 Rome, Italy
Abstract The static interaction between a laminate and distributed piezoelectric actuators is considered. In particular the problem of the stress concentration at the free edge of the active elements of a piezoelectric composite is investigated. A finite element model for the laminated composite plate is developed using a multi-layer higher-order finite element approach. A typical configuration is considered, in which two active layers are bonded on the top and bottom of a passive substructure. A pure bending case is investigated. The obtained results provide useful information of the typical static response. A parametric study is also performed on the effects of the main geometrical characteristics on the intensity and the extension of edge effect. Ó 2002 Published by Elsevier Science Ltd.
1. Introduction During the last few years studies relevant to the use of distributed sensors and actuators integrated with the structures have focused on the possibility of building structures capable of changing their shape according to the environmental conditions and on the reduction of their dynamic response [1]. There are two characteristics of piezoelectric materials which allow them to be used as sensors or actuators. The first one is their direct piezoelectric effect which implies that the materials induce electric charge or potential when they are subjected to mechanical deformations. Conversely they are deformed if some electric charge or potential is imposed to them. Among the materials able to generate such effect, there are the piezoceramics (PZT). They are easily bonded to the surfaces or embedded within the structure. Structures with distributed piezoelectric sensors and actuators have been investigated by many researchers. Crawley and de Luis [2] proposed an analytical model for the static interaction between a beam and segmented piezoelectric actuators symmetrically bonded to the top and bottom surfaces. In their model a pure bending moment is considered; however the effects of the transverse shear and the axial force are neglected. *
Corresponding author. Tel.: +39-06-4458-5304; fax: +39-06-44585670. E-mail address: [email protected] (P. Gaudenzi).
Ray et al. [3] present an exact static analysis of simply supported rectangular-type plate. The intelligent structure proposed is treated as a laminated plate and the distributed piezoelectric sensor and actuator layers are considered to be as plies of the laminate intelligent structure and to be bonded perfectly to the surface of the substrate. The substrate is a laminate of graphite–epoxy composite. This study shows the capability of the actuator and sensor layers to cause and sense deformations of the substrate. The effectiveness of this control system significantly increases with the decrease in the length to the thickness ratio of the substrate. The same authors [4] developed a finite element model for the static analysis of a simply supported rectangular intelligent plate using the higher-order shear deformable displacement theory. The distributed actuator and sensor layer are bonded to the top and bottom surfaces of substrate. A two-dimensional (2D) eight-node isoparametric finite element is derived for modeling the structure. The static and dynamic interaction between a bonded piezoelectric actuator and an underlying beam structure is investigated by Robbins and Reddy [5], using an higher-order layer-wise displacement theory. A comparison between the layer-wise model and models based on the classical beam theory and shear deformation theory has been carried out. They concluded that the layer-wise model is more efficient and appropriate for thick composite beams than the other two models. However due to a substantial increase in the degrees of
0263-8223/03/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. doi:10.1016/S0263-8223(02)00040-5
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freedom, the layer-wise model becomes large and difficult to apply. An analysis of laminated composite plates forced into cylindrical bending by the application of voltages to piezoelectric actuators attached to the top and bottom surfaces is performed by Zhou and Tiersen [6] using the equation of linear elasticity. The piezoelectric actuator is modeled as a 2D surface film. The 2D equations for the laminated composite plate are obtained using the variational procedure of Mindlin. A fully non-linear theory for the dynamics and active control of elastic laminated plates with integrated piezoelectric actuators and sensors undergoing large rotation and small strain vibrations has been shown by Pai et al. [7]. An exact solution methodology for composite piezoelectric composite laminate in cylindrical bending has been used by Brooks and Heyliger [8], to consider several representative cases in which the laminate changes shape using distributed and patched actuators. Also embedded piezoelectric layers are taken into account. Heyliger [9] presented an exact solution for a hybrid laminate containing both piezoelectric and elastic layers under applied surface traction and surface potential. Alurajid et al. [10] developed a modified classical lamination theory to account for piezoelectric coupling terms under applied electric field. CLT was applied to predict the stress field and out-of-plane displacement of different types of piezoelectric actuators. The CLT results were verified using the finite element method. A first attempt to survey and discuss the advances and trends in the formulations and applications of the finite element modeling of adaptive structural elements was made by Benjeddou [11]. Gaudenzi [12] proposed the analysis of an adaptive beam composed by a passive layer and two surfaces bonded induced strain actuators by using a simple higher-order beam model. Exact solutions are obtained for a case of membrane actuation and for a pure bending case. The obtained solutions are then discussed in terms of the main geometrical parameters of the system and compared with the classical closed form solution based on Euler–Bernoulli models. Moreover this model allows the description of the edge effect which occurs close to the free boundary of the considered structure. An important aspect of the analysis of the active laminated structures concerns the stress determination of the 3D field stress, in particular the interlaminar components. The interlaminar stresses are mainly responsible of the delamination and consequently possible detachment of the piezoelectric layers. For this reason the existence of interlaminar stresses, although small with respect to the axial component, must be evaluated in the study of the laminated structures. In the present study, in order to evaluate those stresses, higher-order
theories have been developed in which the strain and the stresses can be evaluated as well as the global response of the structure. The evaluation of the 3D field stress becomes of particular interest when the stress concentration at the boundary of the active layers of the system is required. The numerical procedure here proposed to address this topic is based on the use of the above mentioned higherorder theories which results by a compromise between a fully 3D finite element solution and a bidimensional reduced model that has been developed by the authors in previous studies [13,14]. Several numerical applications and results are presented for the case of cylindrical bending. In particular the analysis of an active structure composed by a passive layer and two surface bonded induced strain actuators is considered.
2. Kinematic assumptions The basic assumption of higher-order theories is that each layer of the laminate can be first considered as a separate elastic body and its displacement field can be expressed in terms of a power series expansion along the thickness direction. By considering the most general formulation a rectangular laminated plate made of an arbitrary number of layers in a 3D Xðx; yÞZðh=2 6 z 6 h=2Þ of an Euclidean space ð0; x; y; zÞ is taken into account. Each layer is assumed to be transversally isotropic elastic continuum and all the layers are considered to be perfectly bonded to each other. The displacements of the qth layer are expressed as a series expansion along the thickness: uq ðx; y; zÞ ¼
M X
zmq uq;m ðx; yÞ
m¼0
vq ðx; y; zÞ ¼
N X
znq vq;n ðx; yÞ
n¼0
wq ðx; y; zÞ ¼
L X
zlq uq;l ðx; yÞ
l¼0
where zq represents the axis normal to the middle plane of qth layer in the local reference system. In that way the unknowns will depend only on the x and y variables. The layers can be thought as bonded along the top and bottom surfaces together with the other layers whose displacements have been treated in the same way. By enforcing the continuity of the displacements at some interfaces we may obtain in a global reference system a general expression for displacements valid throughout the thickness:
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278 M X
uðx; y; zÞ ¼
zm umð0Þ ðx; yÞ
m¼0 R X
þ F0 þ F0
M X m ðz hr Þ uðrÞ m ðx; yÞ
Fr
r¼1
m¼1
P X
M X
Fp
p¼1 N X
vð x; y; zÞ ¼
m ðpÞ
ðz hp Þ um ðx; yÞ
m¼1
zn vnð0Þ ð x; y Þ
273
from those uq;m ðx; yÞ, vq;n ðx; yÞ and wq;l ðx; yÞ as they are defined before, but can be easily derived from them. Such assumed displacement expressions are clearly continuous functions with respect of z, but their partial derivatives with respect to z, and consequently the interlaminar strains are not continuous. Therefore the equilibrium conditions on the interfaces can be satisfied also in presence of a mismatch of elastic properties of adjacent layers. For the details of the present kinematic approach see also [14].
n¼0
þ F0
R X
Fr
r¼1
þ F0
P X
N X
L X
Fp
N X n ðpÞ ðz hp Þ vn ð x; y Þ n¼1
ð0Þ
zl wl ð x; y Þ
l¼0
þ F0
R X
Fr
r¼1
þ F0
P X p¼1
3. Finite element formulation
n¼1
p¼1
wð x; y; zÞ ¼
n
ðz hr Þ vðrÞ n ð x; y Þ
L X
l
ðrÞ
ðz hr Þ wl ð x; y Þ
l¼1
Fp
L X
l
ðpÞ
ðz hp Þ wl ð x; y Þ
l¼1
where F 0 ¼ 1 when R 6¼ 0; F 0 ¼ 0 when R ¼ 0; F r ¼ 1 for z > hr ; F r ¼ 0 for z 6 hr ; F 0 ¼ 1 when P 6¼ 0; F 0 ¼ 0 when P ¼ 0; F p ¼ 1 for z < hp ; F p ¼ 0 for z P hp . hr represents the z coordinates of the interfaces on the positive side of the z-axis, where the continuity of displacements are enforced, hp are the z coordinates of the interfaces on the negative side of the z-axis (Fig. 1). The group of layers situated between two of such interfaces can be thought as macrolayer. In an analogous way uðrÞ m , ðrÞ ðrÞ ðrÞ ðrÞ vðrÞ , w , u , v , w represent the additional unknowns l n m l n relevant to the upper and lower macrolayers. The macrolayer containing the reference system is denoted by the label Ô0Õ. R represents the number of macrolayers placed on the positive side of the z-axis, P the number of macrolayers placed on the negative one. The expressions ðrÞ of umðrÞ ðx; yÞ, vðrÞ n ðx; yÞ and wl ðx; yÞ are in general different
Based on the kinematic assumptions described on the previous paragraph, a finite element approach has been consequently developed. The unknowns displacements uðx; y; zÞ, vðx; y; zÞ, wðx; y; zÞ and the generalized disðrÞ ðrÞ placements uðrÞ m ðx; yÞ, vn ðx; yÞ, wl ðx; yÞ can be rearranged as vectors, so that symbolically we may write as follows: T
fS ð x; y; zÞg ¼ fuð x; y; zÞ; vð x; y; zÞ; wð x; y; zÞg n ð1Þ fsðx; yÞg ¼ sð0Þ ; sð1Þ ; . . . ; sðrÞ ; . . . ; sðRÞ ; s ; . . . ; o ðpÞ ðP Þ T s ;...;s where ð0Þ ð0Þ ð0Þ ð0Þ T ¼ u ;v ;w s ðr Þ ðr Þ ðr Þ ðr Þ T s r ¼ 1; 2; . . . ; R ¼ u ;v ;w n o n oT ð pÞ ð pÞ ð pÞ ð pÞ s p ¼ 1; 2; . . . ; P ¼ u ;v ;w the displacement u can be written as oT ð0Þ n ð0Þ ð0Þ ð0Þ u ¼ u1 ; u2 ; . . . ; uðm0Þ ; . . . ; uM oT ðr Þ n ðr Þ ðr Þ ðr Þ u r ¼ 1; 2; . . . ; R ¼ u1 ; u2 ; . . . ; uðmrÞ ; . . . ; uM n o n o ð pÞ ð pÞ ð pÞ ð pÞ ð pÞ T u p ¼ 1; 2; . . . ; P ¼ u1 ; u2 ; . . . ; um ; . . . ; uM Similar expressions can be written for v and w. In general it is possible to write fS ð x; y; zÞg ¼ ½Z ð zÞ fsð x; y Þg where [ZðzÞ] can be expressed as h i ½Z ð zÞ ¼ Z ð0Þ ; Z; Z in which h i ð1Þ ð2 Þ ðr Þ ð RÞ Z ¼ F0 Z ;Z ;...;Z ;...;Z
h i ð1 Þ ð2Þ ð pÞ ðP Þ Z ¼ F0 Z ;Z ;...;Z ;...;Z
Fig. 1. Three layers model.
ðrÞ
ðpÞ
The matrices ½Z ð0Þ , ½Z and ½Z read
274
Z
h
Z
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
ð0Þ
ðr Þ
i
2
2 6 ¼ F r4
z hr
2
Z
ð pÞ
z2 0 0
1 z ¼ 40 0 0 0
6 6 ¼ F p6 6 4
zM 0 0
... 0 0
z hr
2
0 0 1 z 0 0
...
0 z2 0
0 ... 0
z hr
0
0
0
0
0
0
0
0
z hp
z hp
2
...
z hp
0
0
0
0
0
0
0
0
0 zN 0
M
0 0 1
0 z hr
M
0 0 0 0 z z2
3 0 05 zL
0 0 ...
0 z hr
2
0
...
0 z hr
0
0
0
0
0
0
0
0
z hp
0
Being [D] the partial differential operator which defines the strain components as a function of the displacements: 3 2 o 0 0 7 6 ox 7 6 o 7 6 6 0 07 7 6 oy 7 6 7 6 o 7 6 0 0 7 6 oz 7 6 ½ D ¼ 6 7 o o 7 6 0 7 6 oz oy 7 6 7 6 o o 7 6 0 7 6 ox 7 6 oz 5 4 o o 0 oy ox
z hp
2
0
... 0
z hp
N
0
0
0
0
0
0
0
z hr
N
0
0
z hr
2
...
z hr
0
0
0
0
0
0
0
0
z hp
z hp
2
...
z hp
3 L
7 5
3 7 7 7 7 L 5
T
fega ¼ ½d fEg where fEg is the vector of the electric field applied to the piezoelectric element, and ½d is the piezoelectric constant matrix. Their expressions are 8 9 < E1 = f E g ¼ E2 : ; E3 and 0 ½d ¼ 0 d13
0 0 d23
0 0 0
d15 0 0
0 d25 0
0 0 0
the expression for the strain components is ordered in general:
In the present formulation we assume fEg as a given quantity. In this way also the active strain will be given. For each unknown variable the shape function defined for the considered element can be used:
fegq ¼ ½ D fS ð x; y; zÞg ¼ ½ D ½Z fsð x; y Þg
fsð x; y Þg ¼ ½ N ð x; y Þ fdg
When some active layers are present in the laminate, the constitutive equations for the qth piezoelectric layer are in general: frgq ¼ ½Q q fegq fega in which ½Q q represents the elastic matrix of the qth layer and in which the second term at the right side is present when the qth layer is an active element. This last term represents the converse piezoelectric effect. In fact, for an the active layer we have
Since we consider the response of the structure to the piezoelectric action only, the finite element load vector will be generated by the actuation stress only. The stiffness matrix and the load vector due to the presence of the active layers can be consequently expressed as follows: Z T ½ K e ¼ f½ D ½Z ½ N ð x; y Þ g ½Q e ½ D ½Z ½ N ð x; y Þ dV Ve
fp e ¼
Z Ve
T
f½Z ½ N ð x; y Þ g frgp dV
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
where T
frgp ¼ ½Q ½d fEg By setting a convenient order for each series expansion together with a sufficient number of elements, different orders of approximation can be obtained for the relevant element. In an analogous way, the laminate can be subdivided in a certain number of sublaminates (that will be defined as ‘‘macrolayers’’). In this paper each sublaminate represents either an active (piezoelectric) layer or a passive one. In Appendix A a simple multilayer higher-order model used for the numerical simulations is explained in detail.
4. Numerical results In order to investigate about the numerical performance of the model here proposed for the purpose of evaluating the interlaminar stresses, a typical configuration has been considered, in which two active layers are bonded on the top and bottom of the laminate. The considered structure is composed by an aluminium substrate coupled with distributed actuators. In particular, the structure is a cantilever beam in which two actuating piezoelectric layers perfectly bonded to the surfaces of a passive substructure are present (Fig. 1). The following elastic parameters have been chosen: for the passive layer (aluminium): E ¼ 70 GPa;
m ¼ 0:3;
G ¼ 35 GPa
for the piezoelectric layers: E1 ¼ E2 ¼ 66 GPa; m ¼ 0:3;
E3 ¼ 26 GPa;
G1 ¼ G2 ¼ 33 GPa;
G3 ¼ 26 GPa
For each configuration the response variables (displacements, strains and stresses) have been calculated and the distribution both throughout the thickness and along the span have been evaluated; in this case the lower interface has been considered. The length-tothickness ratio of the whole structure has been assumed to be 64. Three-node C elements with a mesh refinement of 20 elements with 41 total nodes have been used for the numerical simulation. Due to the nature of the stress field, the mesh was more refined close to the free end of the beam and less refined close to the fixed end. In this way a more accurate analysis where high stress concentration is expected. By expanding the transverse displacement w at an order less than in-plane displacements u and v, that is the usual approach in the theory of bidimensional structures such as plates and shells, three sets of order of expansion were first tested: M ¼ N ¼ 3; L ¼ 2;
d15 ¼ 540 m=pV
The value of the electric field between the upper and lower surfaces of the piezoelectric layer has been stated as
M ¼ N ¼ 4; L ¼ 3;
M ¼ N ¼ 5; L ¼ 4 The same order of expansion was adopted for each macrolayer. As expected the value of Drz % decreases as the order of expansion increases. A good convergence is reached for higher-expansion order. Fig. 2 shows the jump of the interlaminar normal stress component evaluated at the top and bottom layer of the lower piezoelectric/structure interface. The results have been obtained corresponding to an abscissa x ¼ 63:9925, and with a 20 element model. A reasonable convergence is obtained with an expansion M ¼ N ¼ 4, L ¼ 3. Accordingly this last model for all macrolayers has been chosen in all simulation. The stresses have been non-dimensionalized according to
The piezoelectric constants have been assumed as d13 ¼ 180 m=pV;
275
r¼
rt ð EtÞAl þ ð EtÞPZT
s¼
st ðGtÞAl þ ðGtÞPZT
E3 ¼ 200 kV=m The following configurations of three layers (active/ passive/active) have been investigated: (a) Same thickness for each layer. (b) Piezoelectric layers with a thickness ratio related to the whole thickness of the beam of 1/10; the substructure layer has then a thickness ratio equal to 8/10. (c) Piezoelectric layers with a thickness ratio of 1/7. In all cases an electrical field of opposite sign for each piezoelectric layer has been applied, so a bending effect is obtained.
Fig. 2. Jump in percentage of interlaminar stress component evaluate at active/passive interface and x ¼ 63:9925, for a laminate of type b.
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A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
where t is the thickness ratio of the macro layer (respectively passive or active) to that of the beam. The stress fields predicted by the model are reported in Figs. 3–9. First the bending stresses rx are illustrated for several conditions. Fig. 3 shows respectively the distribution of the stresses at the top and bottom of the lower interface of the active/passive interface. These results have been obtained in the case of three equal layers. The stresses tend to zero when the free edge is reached. Figs. 4 and 5 show the through thickness distribution of stresses for different abscissas respectively in the case of structure with three equal layers and the one with a thickness ratio of 1/10. In both cases the removal of plane section hypothesis is evident in the area close to the edge. Fig. 6 illustrates a comparison of the results obtained for the structure of type c (thickness ratio of 1/7) with those obtained with an analytical model by Gaudenzi [12]. The solutions predicted at different value of x are in good agreement, in particular the ones corresponding to the passive structure. The shear stresses distributions have also been evaluated. Fig. 7 illustrates the interlaminar stresses for a structure of type a (three equal layers) at the lower interface of the active/passive layers. Fig. 8 shows the through thickness distribution of the s stresses for a laminate of type a. In the same figure the distribution of the interlaminar stresses predicted by the one layer model is reported.
Fig. 3. rx distribution along the active/passive interface, z=h ¼ 0:167, for a laminate of type a.
Fig. 4. Through thickness distribution of longitudinal stresses for different values of x and for the laminate of type b.
Fig. 5. Through thickness distribution of longitudinal stresses for different values of x (three equal layers model).
A. Mannini, P. Gaudenzi / Composite Structures 61 (2003) 271–278
277
Fig. 6. A comparison of through thickness distribution of longitudinal stresses with the analytical model [12] for different values of x (analytical solution: j: x=l ¼ 0:95, : x=l ¼ 1:00; present solution: r: x=l ¼ 0:95 (an), M: x=l ¼ 1:00 (an)).
Fig. 7. sxz distribution along the active/passive interface, z=h ¼ 0:167 for the laminate of type a.
Fig. 9. rz distribution along the active/passive interface, z=h ¼ 0:4 for the laminate of type b.
Finally the rz distribution along the interface of the passive substructure for the laminate of type b is reported in Fig. 9.
5. Conclusions
Fig. 8. Through thickness distribution of sxz for the laminate of type a at the point x ¼ 63:9875.
An investigation that illustrates the behaviour of piezoelectric composite laminates in cylindrical bending under distributed forcing functions has been carried out. The stress and displacement distribution are presented. In particular the stress concentration at the free edge of the active elements has been calculated. A multi-layer higher-order finite element model has been formulated. The model, quite general, allows evaluating the presence of active layers, both bonded at the surface and
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embedded. The effectiveness of the approach has been evaluated referring to some typical cases, and a good agreement with similar other results has been obtained. Appendix A In the following the detailed expressions of the generalized displacements which are used for the implementation of a simple multi-layer model based on the general procedure are shown. The model consists of three layers; two interfaces are assumed for fulfill the discontinuity condition. As regards the power series expression, we assume M ¼ N ¼ 4 and L ¼ 3; consequently the degrees of freedom per node are 36.
n ð1Þ ð1Þ ð1Þ oT wð1Þ ¼ w1 ; w2 ; w3
n
ð1Þ
w
o
n o ð1Þ ð1Þ ð1Þ T ¼ w2 ; w2 ; w3
For the expression of [Z] we have
ð1Þ ð0Þ ð1Þ ð0Þ ½ZðzÞ ¼ Z ; F Z ; F 0 Z where
Z
ð0Þ
2
1 ¼ 40 0 2
z z2 0 0 0 0
z3 0 0
0 z2 0
0 z3 0
u0 ; u1 ; u2 ; u3 ; u4 ; v0 ; v1 ; v2 ; v3 ; v4 ; w0 ; w1 ; w2 ; w3 the following 11 are relative to the lower layer
being Dz ¼ ðz h1 Þ and Dz ¼ ðz h1 Þ.
the first 14 are relative to the central layer
ð1Þ
i
3
Dz 0 0
Dz4 0 0
0 0 1 z 0 0
Dz Dz2 Z ¼4 0 0 0 0 2 2
Dz Dz ð1Þ Z ¼4 0 0 0 0 h
Dz3 0 0
z4 0 0
4
Dz 0 0
0 z4 0
0 0 1
0 0 0 0 z z2
0 0 Dz Dz2 0 0
0 Dz3 0
0 Dz4 0
0 0 0 0 Dz Dz2
0 0 2 Dz Dz 0 0
0 3 Dz 0
0 4 Dz 0
0 0 0 0 2 Dz Dz
3 0 05 z3 3 0 0 5 Dz3 3 0 0 5 3 Dz
u1 ; u2 ; u3 ; u4 ; v1 ; v2 ; v3 ; v4 ; w1 ; w2 ; w3 and the last 11 are relative to the upper layer
References
u1 ; u2 ; u3 ; u4 ; v1 ; v2 ; v3 ; v4 ; w1 ; w2 ; w3
In matrix form the generalized displacements reads as follows: n o ð1 Þ T fsð x; y Þg ¼ sð0Þ ; sð1Þ ; s with ð0Þ ð0Þ ð0Þ ð0Þ T ¼ u ;v ;w s ð1Þ ð1Þ ð1Þ ð1Þ T ¼ u ;v ;w s n o n o ð1Þ ð1Þ ð1Þ ð1Þ T s ¼ u ;v ;w and ð0Þ n ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ oT ¼ u0 ; u1 ; u2 ; u3 ; u4 u ð1Þ n ð1Þ ð1Þ ð1Þ ð1Þ oT u ¼ u1 ; u2 ; u3 ; u4 n o n o ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ T u ¼ u2 ; u2 ; u3 ; u4 Similar expressions can be obtained for v and w: ð0Þ n ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ oT ¼ v0 ; v1 ; v2 ; v3 ; v4 v ð1Þ n ð1Þ ð1Þ ð1Þ ð1Þ oT ¼ v1 ; v2 ; v3 ; v4 v n o n o ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ T v ¼ v2 ; v2 ; v3 ; v4 ð0Þ n ð0Þ ð0Þ ð0Þ ð0Þ oT w ¼ w0 ; w1 ; w2 ; w3
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