Numerically Efficient Finite Element Formulation for Modeling Active Composite Laminates

Transcription

1 Mechanics of Advanced Materials and Structures, 13: , 2006 Copyright c Taylor & Francis Group, LLC ISSN: print / online DOI: / 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 Address correspondence to Dragan Marinković, Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, Magdeburg, D-39106, Germany. Dragan. Marinkovic@Masch-Bau.Uni-Magdeburg.De 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

2 380 D. MARINKOVIĆ ETAL. 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 C 1 -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 Zemčík 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 ). 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 N i [17], the coordinates of a mid-surface point are given by: x x i y z = N i y i (1) with x i,y i and z i 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 e ai, 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 y z = N i x i y i z i + z i h i 2 N it e z i (2) where h i denotes the shell thickness at node i and 1 < t < 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 FIG. 1. Element geometry and coordinate systems.

3 NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION 381 about the local x - and y -axis through the mid-surface point, θ x and θ y : u v w = N i u i v i w i + h i 2 N it[ e y i e x i] { θx It is to be noted that the rotations are in the local coordinate system and upon transformation to the global coordinate system one gets: u v = w N i u i v i w i + h i 2 N it[ e y i e x i][t ] T θ y } θ x θ y θ z where [T ]isthe modified form of the transformation matrix relating the local and the global coordinate systems, [T], and is given in the form [T ] = e x e y. 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 e 31 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: (3) (4) x, r y, r z, r u, x v, x w, x [J] = x, s y, s z, s u, y v, y w, y x, t y, t z, t u, z v, z w, z u, r v, r w, r = [J] 1 u, s v, s w, s (5) u, t v, t w, t where, for example, u, x = u/ x. The global derivatives are afterwards transformed to the local derivatives by means of the transformation matrix [T] = e x e y e z : u, x v, x w, x u, x v, x w, x u, y v, y w, y = [T] T u, y v, y w, y [T] (6) u, z v, z w, z u, z v, z w, z 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: ( u ) x and ( u ) x R T = = [ ( Ni l 1 x l 1u i + N r x m 1v i + N ) i x n 1w i ( Ni + m 1 y l 1u i + N i m 1 v i + N i y y n 1w i + n 1 ( Ni z l 1u i + N i z m 1v i + N i z n 1w i ( h i [l 1 t N i 2 x + J 13 N i ( + n 1 t N )] i z + J 33 N i ) + m 1 ( ) )] (7) t N ) i y + J 23 N i (n 3i θ yi m 3i θ zi )l 1 + (l 3i θ zi n 3i θ xi )m 1 + (m 3i θ xi l 3i θ yi )n 1 (8) where l 1,m 1,n 1 and l 3,m 3,n 3 are the cosine directions of the unit vectors e x and e z with respect to the global coordinates, respectively, i.e., e x ={l 1 m 1 n 1 } T and e z ={l 3 m 3 n 3 } 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: N i x = N i J 11 r + N i J 12 s ; N i y = N i J 21 r + N i J 22 s ; N i z = N i J 31 r + N i J 32 s ; (9) FIG. 2. Mappings and transformations for the geometry and strain description.

4 382 D. MARINKOVIĆ ETAL. 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 x x ε y y u {ε }= γ x + v y y = x v γ y + z w z y γ x z u z u x v y + w x (10) The formulation adopts the plane-stress state assumption, i.e., σ z z = 0, and consequently, the normal transverse strain component ε z 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: {ε {ε mf } [B mf ] }= {ε s } = {d} [B s ] [B Tm ] t[b Rlf ] = + {d} [B Ts ] [B R0s ] + t[b Rls ] = [B u ]{d} (11) where {ε mf }={ε x x ε y y γ x y }T is the membrane-flexural (inplane) strain field, {ε s } = {γ y z γ x z }T comprises transverse shear strains, [B mf ] and [B s ] 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) 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} =[C E ]{ε} [e] T {E} {D} =[e]{ε}+[d ε ]{E} (12) where {σ} is the mechanical stress in vector (Voigt) notation, {D} is the electric displacement vector, [C E ]isthe piezoelectric material Hook s matrix at constant electric field E, [d ε ]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 e 31 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: E = ϕ z E k = k h k (13) where ϕ is the electric potential, k is the difference of the electric potentials between the electrodes of the kth layer and h k is the thickness of the piezolayer. The approximation in the Eq. (13) defines a diagonal electric field electric potential matrix [B φ ] with typical term 1/h k on the main diagonal. The diagonal form of the matrix [B φ ] 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 [C E ]{ε} {δε} T [e] T {E} v {δe} T [e]{ε} {δe} T [d ε ]{E}]dV

5 NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION 383 = {δu} T {δf v } dv + {δu} T{ } δf Si ds1 v S 1 +{δu} T {δf P } δφqds 2 δφq (14) S 2 -piezolayer electric charges: {Q ext }= {q} ds 2 (22) S 2 where F v,f S1 and F P are the external volume, surface (acting on surface S 1 ) and point loads, respectively, q and Q are the surface electric charge (acting on surface S 2 ) 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]{ d}+[c]{ḋ}+[k uu ]{d}+[k uφ {φ} ={F ext } (15) K φu {d}+ K φφ {φ} ={Q ext } (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: [M u ] = [N u ] T ρ[n u ]dv e (17) V e -element mechanical stiffness matrix: [K uu ] = [K mf ] + [K s ] + [K t ] -element piezoelectric coupling matrix: [K uφ ] = ([B u ] T [e] T [B φ ]) dv = [K φu ] T (18) V e -element dielectric stiffness matrix: [K φφ ] = [B φ ] T [d ε ][B φ ]dv e (19) V e -element damping matrix (Rayleigh): [C] = α[m] + β[k uu ] (20) -nodal mechanical loads: {F ext }= [N u ] T {F v } dv e + [N u ] T{ } F S1 ds1 + [N u ] T {F p } V e S 1 (21) where the matrix [N u ]isthe element shape-functions matrix, V e 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., N q 1). The element stiffness matrix comprises the contribution from the stiffness matrix related to the membrane-flexural (in-plane) strains [K mf ], the stiffness matrix related to the shear strains [K s ] and the so-called torsional stiffness [K t ]. The former two are determined by taking advantage of the developed strain field, hence: [B Tm ] T [K mf ] = [C m ][[B Tm ] t[b R1f ]] dv e (23) V e t[b R1f ] T [B Ts ] T [K s ] = [C s ][[B Ts ] [B R0s ] T V e [B R0s ] T + t[b R1s ] T + t[b R1s ]] dv e (24) where [C m ]isthe part of the Hook s matrix relating the inplane stresses and strains, while [C s ] 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, E t, that behaves as a penalty function forcing the local rotation θ z be approximately equal to 0.5 ( v / x u / y )atintegration points: E t = 1 2 α nyh n A [ θ z 1 ( v )] 2 u da, (25) 2 x y (r,s,o) where Y is the Young s elasticity module and α n is a fictitious elastic parameter (torsional coefficient) that can be determined

6 384 D. MARINKOVIĆ ETAL. 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 [B mf ] contributes in [K uφ ], which is therefore calculated by the following integral: [K uφ ] = V ae [B Tm ] T [e][b φ ]dv ae (26) t[b R1f ] T where V ae denotes that the integration in Eq. (26) involves only the active, i.e., piezoelectric layers of the element and the dimension of the matrix [K uφ ]is54 n pe, where n pe denotes the number of piezoelectric layers. Finally, the matrix [K φφ ] has the dimension n pe n pe 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 n pe 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 ε x term and the 3 2 rule for ε y 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 afew 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 T300/976 graphite/epoxy Elastic properties E 11 (GPa) E 22 (GPa) 63 9 ν G 12 (GPa) Piezoelectric properties e 31 (10 5 C/mm 2 ) e 32 (10 5 C/mm 2 )

7 NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION 385 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 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 = mm) has a cross-ply sequence of graphite 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 mm, and of the piezolayer 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). FIG. 4. Deformed shape of a composite plate under uniform pressure and piezoactuation.

8 386 D. MARINKOVIĆ ETAL. 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) Cantilevered Curved Composite Shell The second example considers a cylindrical composite shell of dimensions a b = 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 [30 2 /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 mm and of the PZT layers is 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 = N/mm (w 3 /b) 10 3 (w 2 /b) 10 3 (w 1 /b) 10 3 Mesh b = 254 mm Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof Int. rule ,894 6,957 6,319 5,154 6,740 5,484 5,780 7,945 5, ,095 6,598 6,520 6,110 6,512 6,232 7,268 7,701 7, ,386 6,594 6,528 6,370 6,506 6,378 7,461 7,697 7, ,520 6,571 6,535 6,453 6,486 6,457 7,618 7,671 7,638

9 NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION 387 FIG. 6. element. Cantilever composite shell under uniform edge force free edge transverse deflection, convergence of the fully (FI) and reduced (RI) integrated Shell9 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 (w 3 /b) 10 3 (w 2 /b) 10 3 (w 1 /b) 10 3 Mesh b = 254 mm Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof Shell9 Shell9 Semi-Loof Int. rule ,394 6,299 5,682 4,241 2,775 3,380 8,464 10,385 7, ,840 6,431 6,163 3,675 3,359 3,180 10,439 10,741 10, ,507 6,423 6,181 3,603 3,354 3,343 10,514 10,762 10, ,450 6,397 6,347 3,420 3,367 3,354 10,732 10,730 10,686

10 388 D. MARINKOVIĆ ETAL. FIG. 7. 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 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 FIG. 8. Cantilever composite shell under piezoelectric excitation free edge transverse deflection, convergence of the reduced integrated Shell9 element.

11 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

12 390 D. MARINKOVIĆ ETAL. 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 (w 2 /b) is observed, while the ratio (w 3 w 1 )/bis 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 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 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. FIG. 13. Radial deflection of the piezoelectric active composite arch convergence of the Shell9 element.

13 NUMERICALLY EFFICIENT FINITE ELEMENT FORMULATION 391 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 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 FIG. 14. Radial deflection of the piezoelectric active composite arch convergence of the Semi-Loof element.

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