Variable Kinematics and Advanced Variational Statements for Free Vibrations Analysis of Piezoelectric Plates and Shells

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1 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 Variable Kinematics and Advanced Variational Statements for Free Vibrations Analysis of Piezoelectric Plates and Shells E. Carrera, S. Brischetto and M. Cinefra 2 Abstract: This paper investigates the problem of free vibrations of multilayered plates and shells embedding anisotropic and thickness polarized piezoelectric layers. Carrera s Unified Formulation CUF has been employed to implement a large variety of electro-mechanical plate/shell theories. So-called Equivalent Single Layer and Layer Wise variable descriptions are employed for mechanical and electrical variables; linear to fourth order expansions are used in the thickness direction z in terms of power of z or Legendre polynomials. Various forms are considered for the Principle of Virtual Displacements PVD and Reissner s Mixed Variational Theorem RMVT to derive consistent differential electro-mechanical governing equations. The effect of electro-mechanical stiffness has been evaluated in both PVD and RMVT frameworks, while the effect of continuity of transverse variables transverse shear and normal stresses and transverse normal electric displacement has been addressed by comparing various forms of RMVT. According to CUF, governing equations related to a given variational statement have been written in terms of fundamental nuclei whose form is independent of the order of expansion and of the adopted variable description. The numerical results have been restricted to simply supported orthotropic plates and shells, for which exact three-dimensional solutions are available. A large numerical investigation has been conducted to compute fundamental and higher vibrations modes. An exhaustive numerical evaluation of assumptions, related to the various PVD and RMVT forms, is given. Classical, higher-order, layer-wise and mixed assumptions have been compared to available three-dimensional solutions. The convenience of hierarchical approaches based on CUF is shown, along with the suitability of the implemented RMVT forms to accurately trace the free vibration response of piezoelectric plates and shells. RMVT applications permit the vibration modes of transverse electro-mechanical variables Corresponding author: Salvatore Brischetto, Department of Aeronautics and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 029 Torino, Italy. Tel: , Fax: , salvatore.brischetto@polito.it. 2 Department of Aeronautics and Space Engineering, Politecnico di Torino, Italy

2 260 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 to be accurately evaluated in the thickness plate/shell direction. Keywords: multilayered plates; multilayered shells; thickness polarized piezoelectric layers; electro-mechanical coupling; Principle of Virtual Displacements; Reissner s Mixed Variational Theorem; layer wise; equivalent single layer; vibrations modes. Introduction Piezoelectric materials are one of the most suitable solutions for the design of smart structures Im and Atluri 989. These materials use the so-called piezoelectric effect, which consists of a linear energy conversion between the mechanical and electric field and viceversa, and this conversion leads to a direct or converse piezoelectric effect, respectively Ikeda 990. The main applications of smart structures are: vibration and noise damping, shape adaptation of aerodynamic surfaces, active aeroelastic control, shape control of optical and electro-magnetic devices and health monitoring. Exhaustive overviews on possible applications of smart structures have been given by Rao and Sunar 994, Crawley 994, Chopra 996, Tani et al. 998, Sunar and Rao 999, and more recently by Chopra 2002 and Yang In most applications, smart structures are multilayered anisotropic plates and shells with strong electro-mechanical coupling. The layers can be made of traditional metallic or advanced composite materials, as well as sandwich structures. Piezoelectric layers are embedded in the structures in various forms: one or more sensor layers or patches; one or more actuator layers or patches; a combination of one or more sensor/actuator layers. Several topics are of interest in the application of smart structures as well as in their computational simulation: material modelings, structural modelings and control algorithms. The attention of this work is restricted to advanced structural modelings for multilayered thickness polarized piezoelectric plates and shells with emphasis on the vibration response. In addition to the afore mentioned review papers, further overviews on modelling are those in Benjeddou 2000, Robbins and Chopra 2006 and Carrera and Boscolo As in any other multilayered structure, improved refined models for multilayered plates should account for: an accurate description of the interlaminar conditions, such as the Zig-Zag ZZ form of displacement in the thickness direction z rapid change in slope in correspondence to each layer interface and the Interlaminar Continuity IC of transverse stresses at each layer interface.

3 Variable Kinematics and Advanced Variational Statements 26 In Carrera 995 and Carrera 997b, ZZ and IC conditions are referred to as C 0 z -requirements. ZZ and IC can be introduced into Equivalent Single Layer ESL theories by implementing various techniques; according to Reddy 2004, the number of displacement variables should be kept independent of the number of constitutive layers in the ESL models, while the same variables depend on each layer in Layer Wise LW cases. A discussion on a historical review of so-called zig-zag theories has been given in Carrera Recent works about zigzag models for the analysis of multilayered piezoelectric structures have been proposed by Kapuria 2004a, Kapuria 2004b, Kapuria and Achary 2005a and Kapuria and Kulkarni In Carrera 200 it was established that the Reissner s Mixed Variational Theorem RMVT Reissner 984 should be considered as the natural extension of the Principle of Virtual Displacements PVD to multilayered structures in view of the fullfillment of C 0 z -requirements. RMVT in fact, permits the compatibility conditions of transverse shear and normal stress components to be enforced. The fulfillment of C 0 z -requirements remains a crucial point in the development of appropriate two-dimensional models for multilayered plates and shells embedding piezolectric layers. Figure shows the distribution of mechanical and electrical variables in layered plates made of piezoelectric layers. Displacement, in-plane stress components, electrical potential, transverse stress components and transverse normal electrical displacement are shown. An extended electrical form of RMVT permits the continuity of transverse normal electrical displacement to be fulfilled at each layer interface as well as the direct evaluation of the electrical charge Carrera et al. 2008; Carrera and Nali An alternative method has been proposed in Chen and Hwu 200, where the use of Green s function permits to exactly satisfy the interface continuity conditions and no meshes are needed along the interface, in this approach the materials can be any kinds of piezoelectric or anisotropic elastic materials. The Green s function is also employed by Wu and Chen 2007 to investigate the dynamic responses of several piezoelectric materials in order to yield the displacement or stress fields in the time domain directly. The attention of the present paper is restricted to a free vibration analysis of multilayered plates and shells embedding thickness polarized piezoelectric layers. The afore mentioned review papers discuss most of the available works on this topic. However, a short review of works which are relevant for this paper is given in the following. Three-dimensional exact solutions for the free vibration problem have been provided by Heyliger and Saravanos 995, where frequencies for the first three modes are given for both thick and thin multilayered piezoelectric plates. Kapuria and Achary 2005b proposed a three-dimensional piezoelasticity solution for hybrid cross-ply plates where a real mass density, different from the unit value suggested in Heyliger and Saravanos 995, was considered. Kapuria and Achary

4 262 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 z Displacements ux uy uz Transverse shear and In-plane stresses normal stresses xx yy xy z z z Electric potential and transverse normal electric displacement xz yz zz Dz x,y x,y x,y x,y Figure : C 0 z -requirements for multilayered piezoelectric plates. Zig-Zag form and Interlaminar Continuity for some mechanical and electrical variables. For shell geometry the reference system x,y,z is replaced by,,z. 2005b investigated also hybrid sandwich configurations where the different values of the mass density for the embedded layers were more relevant. Du et al investigated the thickness vibrations of a piezoelectric plate using an exact solution obtained for materials with general anisotropy; the effects of a uniform biasing acceleration were also considered. A three dimensional theory is presented in Zhu et al for the dynamic stability analysis of piezoelectric circular cylindrical shells. In this case, results indicate that piezoelectric effects and electric field have a minor effect on the unstable region with respect to the geometric parameters and the rigidity of constituent materials. A three dimensional solution for dynamic analysis of thick laminated shell panels is illustrated in Shakeri et al. 2006, direct and inverse effects of piezoelectric materials are considered. Applications of CLT and FSDT to piezoelectric plates have been given by Tiersten 969 and Mindlin 972. In He et al. 998, numerical and experimental results have been compared for free vibration analysis of thin plates embedding metallic and piezoceramic layers, but the transverse normal strain/stress effects were not taken into account. An annular plate has been considered in Duan et al 2005, where the free vibration analysis has been conducted using very simple models such as the Kirchhoff and Reissner-Mindlin plate models; different boundary conditions have been investigated and FEM solutions have been considered. In Heidary and Eslami 2006, the linear response of thermopiezoelectric plates is given using the Hamilton principle and the finite element method. Linear shape functions are used and the First order Shear deformation Theory FSDT of laminated plates is considered. Thermally induced vibration amplitudes are suppressed through application of electric potential differences across the piezoelectric layers attached to the surfaces of the composite plate. The numerical studies demonstrate the effectiveness of thermal environment, as well as the piezo-control of these thermal deformations

5 Variable Kinematics and Advanced Variational Statements 263 using piezoelectric structures. As example of a refined theory, the work by Yang and Yu 993, is mentioned. The electric field generated by stresses electrical stiffness was not considered in this work. Refined ESL models have been discussed by Benjeddou and Deü 200. ESL formulation, taking into account ZZ and IC, has been discussed by Ossadzow-David and Touratier Pan and Heyliger 2002 have shown natural frequencies and shape modes for sandwich piezoelectric/piezocomposite plates using analytical solutions. Mitchell and Reddy 995 introduced a Layer-Wise LW description of the electric potential, while an Equivalent Single Layer ESL description was retained for displacements. Han et al have considered the coupling between the elastic and electric field in each element when characteristics surface waves in hybrid multilayered piezoelectric plates have been investigated. Cupial 2005 has remarked that the capability of a two-dimensional model to predict vibration modes plays a fundamental role in both noise damping and health monitoring problems. Ramirez et al calculated the natural frequencies and through-thickness mode behavior of simply supported and cantilever laminates. An approximated solution for free vibration problems of two-dimensional magneto-electro-elastic laminates has been presented to determine their fundamental behavior. The solution for the elastic displacements, electric potential and magnetic potential is obtained by combining a discrete layer approach with the Ritz method. Free vibrations of multilayered piezoelectric composite plates can be also found in Zhang et al. 2006, where an analysis was performed using the differential quadratic DQ technique to solve three-dimensional piezoelasticity equations. Solutions for piezoelectric laminates are possible if the DQ layer-wise modelling technique is implemented. Becker et al have proposed a finite element modelling methodology which incorporates both piezoelectric coupling effects and the electrical dynamics of the employed passive electrical circuits. The effects of the electric boundary conditions and the influence of the direction of polarization are investigated in Dziatkiewicz and Fedelinski 2007 for the free vibrations of two-dimensional piezoelectric structures using the dual reciprocity boundary element method. Further works about shell geometries are listed in the following. In Wang et al the dynamic solution for a multilayered orthotropic piezoelectric hollow cylinder is obtained by means of a solution split in two parts: a quasi-static solution in addition to a dynamic one; displacements, stresses and electric potential are finally obtained. Numerical results for layered piezoelectric spherical caps and indication of their behavior are given in Wu and Heyliger 200. First, only elastic shells are examined to test the accuracy of the formulation, then solutions for piezoelectric shells are given, they could be a means of comparison for other techniques and methods. In Zheng et al a refined hybrid piezoelectric shell element formulation is developed for mechanical analysis and active vibration control of laminated structures bonded to piezoelectric sensors

6 264 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 and actuators. Benjeddou et al. 200a and Benjeddou et al. 200b consider shells of revolution, a finite element implementation is presented. Open circuit and closed circuit configurations are investigated and different types of modes are considered, such as bending, radial and torsion ones. Applications of CUF to piezolectric plates were first given in Carrera 997a, ZZ and IC were introduced in the First order Shear Deformation Theory FSDT. Ballhause et al gave closed-form solutions for the free vibration problem of multilayered piezoelectric plates; a quasi-3d formulation was obtained by employing Carrera s Unified Formulation CUF Carrera 995 and the Principle of Virtual Displacements PVD was extended to the electro-mechanical case. The closed form solution proposed by Ballhause et al for plates was extended to shell geometry by D Ottavio et al CUF is a variable kinematic framework which permits a large variety of electro-mechanical plate/shell theories to be implemented: Equivalent Single Layer ESL and Layer Wise LW variable descriptions are employed for mechanical and electrical variables; linear to fourth-order expansions are used in the thickness direction z, in terms of power of z and/or Legendre polynomials. According to CUF, all the governing equations related to a given variational statement have been written in terms of fundamental nuclei, and their form is independent of the order of expansion as well as of the adopted variable description. It can be concluded that CUF variable kinematic modelings as well as the use of mixed variational statements could be used to construct appropriate theories for the analysis of piezoelectric plates and shells in view of the fulfillment of the C 0 z - requirements. The present paper gives a complete discussion of available CUF modelings, and applications of various classical PVD and advanced RMVT statements for the free vibration problems of piezoelectric plates and shells. In particular, the CUF variable kinematics models are implemented according to the following variational statements: PV Du, the electrical stiffness is neglected u indicates displacement variables. RMV T u,σ n, the electrical stiffness is neglected and transverse shear/normal stresses IC are a priori fulfilled σ n denotes transverse shear/normal stresses. PV Du,Φ, the electrical stiffness is included Φ is the electric potential. RMV T u,φ,σ n, transverse shear/normal stresses IC are a priori fulfilled. RMV T u,φ,d n, transverse normal electric displacement IC is a priori fulfilled D n is the transverse normal electric displacement.

7 Variable Kinematics and Advanced Variational Statements 265 RMV T u,φ,σ n,d n, transverse shear/normal stresses and transverse normal electric displacement IC are a priori fulfilled. Other applications of RMVT and CUF to piezoelectric plates and shells have been given in Carrera and Boscolo 2007, Carrera et al. 2008, Carrera and Nali 2009, Carrera and Brischetto 2007a, Carrera and Brischetto 2007b and D Ottavio and Kröplin In Carrera and Boscolo 2007 the FEM static analysis of multilayered piezoelectric plates was given, in the proposed variational statements the case RMV T u,φ,d n was not considered. Carrera et al was an extension of Carrera and Boscolo 2007, where the magnetic field is also considered. In Carrera and Nali 2009 the RMV T u,φ,d n case was introduced but only static FEM analysis of multilayered piezoelectric plates was considered. In Carrera and Brischetto 2007a and Carrera and Brischetto 2007b RMV T u,φ,σ n and RMV T u,φ,σ n,d n were extended to the static and dynamic analysis of multilayered piezoelectric shells, respectively. D Ottavio and Kröplin 2006 extended the RMV T u,φ,σ n to piezoelectric laminates, this variational statement was employed for the free-vibration problem of multilayered piezoelectric shells. The present paper gives an exhaustive discussion about each possible extension of PVD and RMVT variational statement to electro-mechanical analysis of plates and shells, with emphasis to those variational statements not included in the above cited works. The governing differential equations related to the various formulations are given and their closed-form solutions are discussed. A numerical investigation is made to evaluate fundamental and higher-order modes. The in-house academic MUL2 code MUL has been used. Comparisons of classical, higher-order, layerwise and mixed assumptions are made. Evaluations of the effect of interlaminar continuity is given. The paper has been organized as follows. The various extensions of PVD and RMVT to electro-mechanical analysis are discussed in Section 2; the related consistent constitutive equations are derived in the same section. Geometrical relations for plates and shells are described in Section 3, while CUF is dealt with in detail in Section 4. The governing equations for the dynamic analysis of piezoelectric plates and shells are derived in Section 5. Closed-form solutions are given in Section 6. Governing eigenvalues problem can be found in Section 7. The results are discussed in Section 8 and the main conclusions are drawn in Section 9. Some appendices quote a few details of the considered formulations in order to see the main differences for fundamental nuclei in the case of plate geometry and those in the case of shell geometry both open spherical and cylindrical cases.

8 266 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , The considered variational statements The Principle of Virtual Displacements PVD is extended to the electro-mechanical case by simply adding the virtual internal electric work to the mechanical one. Constitutive equations are obtained from the quadratic form of the Gibbs free-energy function, written in the case of linear interaction between the mechanical and electrical field Ikeda 990; Rogacheva 994. Hooke s well-known law for the pure mechanical problem can be considered as a particular case of the more general constitutive equations written for the electro-mechanical case Carrera et al Three different extensions of Reissner s Mixed Variational Theorem RMVT to an electro-mechanical case are here discussed; for each proposed extension, the constitutive equations must be rearranged and written coherently with the employed variational statement Carrera et al Extended PVD cases The PVDu case For a multilayered plate or shell, including orthotropic layers, in the case of pure mechanical problem, the PVD states: PV Du : δε T pgσ pc + δε T ngσ nc dv = δl e + δl in, V subscripts C and G indicate the substitution of constitutive and geometrical relations, respectively. For plate geometry u = u x,u y,u z is the displacement vector; σ p = σ xx,σ yy,σ xy and σ n = σ xz,σ yz,σ zz are the in-plane and out-plane stress components, respectively. ε p = ε xx,ε yy,γ xy and ε n = γ xz,γ yz,ε zz are the in-plane and out-plane strain components, respectively. δl e is the virtual external work and δl in is the virtual inertial work. This form of PVD does not include electrical stiffness, so it is denoted as PVDu due to the fact that the displacements are the only primary variables. For shell geometry a curvilinear reference system,,z is employed in place of the rectilinear one x,y,z for the plate see also Figure 2. The PVDu,Φ case PVD is extended to the electro-mechanical case by simply adding the internal electric work: PV Du,Φ : δε T pgσ pc + δε T ngσ nc δe T pgd pc δe T ngd nc dv = δl e + δl in, V 2

9 Variable Kinematics and Advanced Variational Statements 267 a b Figure 2: Geometry and notation for a multilayered plate a and a multilayered shell b. where E p = E x,e y, E n = E z and D p = D x,d y, D n = D z are the in-plane and out-plane electric field and electric displacement components, respectively. Both displacement and electric potential are primary variables in PVDu,Φ. In case of electro-mechanical problem the constitutive equations are obtained from the Gibbs free-energy function G Ikeda 990; Rogacheva 994: Gε,E = 2 εt Cε 2 E T εe E T eε. 3 C is the matrix of elastic coefficients, e is the matrix of piezoelectric coefficients, ε is the matrix of dielectric coefficients. The 6 6 matrix C for an orthotropic material in the structural reference system assumes the following form Reddy 2004: C = C C 2 C C 3 C 2 C 22 C C 23 C 6 C 26 C C C 55 C C 45 C 44 0 C 3 C 23 C C 33 = C pp C np C pn C nn, 4 where C pp, C pn, C np and C nn are the 3 3 sub-matrices related to in-plane p and out-plane n strain/stress components. The matrices for the piezoelectric coupling and for the dielectric coefficients, when

10 268 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 the poling direction 3 coincides with the layer z direction, are: ε ε e 5 e 4 0 e = e 25 e 24 0, ε = ε 2 ε 22 0 e 3 e 32 e e ε Stresses and electric displacement are obtained upon direct differentiation of G as it follows: σ = G ε = ε 2 εt Cε 2 E T εe E T eε = Cε e T E, 6 D = G E = E 2 εt Cε 2 E T εe E T eε = eε + εe. 7 E = E x,e y,e z is the 3 electric field vector, ε = ε xx,ε yy,γ xy,γ xz,γ yz,ε zz is the 6 strain vector, σ = σ xx,σ yy,σ xy,σ xz,σ yz,σ zz is the 6 stress vector, D = D x,d y,d z is the 3 electric displacement vector. The correspondent constitutive equations are herein split in in-plane and out-plane components: σ pc = G ε p = C pp ε pg +C pn ε ng e T ppe pg e T npe ng, 8 σ nc = G = C np ε pg +C nn ε ng e T ε pne pg e T nne ng, n 9 D pc = G = e pp ε pg + e pn ε ng + ε pp E pg + ε pn E ng, E p 0 D nc = G E n = e np ε pg + e nn ε ng + ε np E pg + ε nn E ng, where e pp = , e pn = e 5 e 4 0 e 25 e 24 0 e np = e3 e 32 e 36, e nn = 0 0 e33, ε pp = ε ε 2 ε 2 ε 22, ε pn = 0, ε 0 np = 0 0, 2, ε nn = ε33.

11 Variable Kinematics and Advanced Variational Statements 269 From Eqs.8-, in case of pure mechanical problem, the classical Hooke law is obtained: σ p = C pp ε p +C pn ε n, 3 σ n = C np ε p +C nn ε n. 4 Eqs.8- are the consistent constitutive equations for the variational statement PVDu,Φ at Eq.2, while Eqs.3 and 4 are the constitutive equations consistent to the PVDu in Eq.. For shell geometry a curvilinear reference system,,z is employed in place of the rectilinear one x,y,z for the plate see also Figure Extended RMVT cases The RMVTu,Φ,σ n case Reissner s Mixed Variational Theorem RMVT for the pure mechanical case assumes transverse shear/normal stresses and displacement as independent variables Reissner 984: RMV T u,σ n : δε T pgσ pc + δε T ngσ nm + δσ T nmε ng ε nc dv = δl e + δl in, V a Lagrange multiplier δσ nm is added to permit to assume a priori interlaminar continuous transverse stresses σ nm subscript M means modelled variables. RMVT permits therefore the fulfillment a priori of C 0 z -requirements for transverse shear/normal stresses. A partial extension of RMVT to electro-mechanical problems is obtained simply adding the virtual internal electric work Carrera and Brischetto 2007a: RMV T u,φ,σ n : δε T pgσ pc + δε T ngσ nm + δσ T nmε ng ε nc 6 V δe T pgd pc δe T ngd nc dv = δl e + δl in, displacement u, electric potential Φ and transverse shear/normal stresses σ n are primary variables. The related constitutive equations are obtained from Eqs.8- by considering σ p, ε n, D p and D n : 5 σ pc = ĈC σp ε p ε pg +ĈC σp σ n σ nm +ĈC σp E p E pg +ĈC σp E n E ng, 7 ε nc = ĈC εn ε p ε pg +ĈC εn σ n σ nm +ĈC εn E p E pg +ĈC εn E n E ng, 8 D pc = ĈC Dp ε p ε pg +ĈC Dp σ n σ nm +ĈC Dp E p E pg +ĈC Dp E n E ng, 9

12 270 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 D nc = ĈC Dn ε p ε pg +ĈC Dn σ n σ nm +ĈC Dn E p E pg +ĈC Dn E n E ng, 20 where: ĈC σp ε p = C pp C pn C nn C np, ĈC σp E n = C pn C nn e T nn e T np, ĈC εn E p = C nn e np, ĈC Dp σ n = e pn C nn, ĈC Dp E n = e pn C nn e T nn + ε pn, ĈC Dn E p = e nn C nn e T pn + ε np, ĈC εn E n = C nn e nn, ĈC σp σ n = C pn C nn, ĈC εn ε p = C nn C np, ĈC Dp E p = e pn C nn e T pn + ε pp, ĈC σp E p = C pn C nn e T pn e T pp, ĈC εn σ n = C nn, ĈC Dp ε p = e pp e pn C nn C np, ĈC Dn ε p = e np e nn C nn C np, ĈC Dn E n = e nn C nn e T nn + ε nn. ĈC Dn σ n = e nn C nn, 2 RMVTu,σ n for the pure mechanical case can be considered as a particular case of the RMVTu,Φ,σ n. That is Eqs.7-20 degenerate for the pure mechanical case as: σ pc = ĈC σp ε p ε pg +ĈC σp σ n σ nm, 22 ε nc = ĈC εn ε p ε pg +ĈC εn σ n σ nm. 23 The RMVTu,Φ,D n case The second extension of RMVT to electro-mechanical case, implemented in this work, considers the transverse normal electric displacement D n as primary variable, for details readers can refer to Carrera and Nali 2009: RMV T u,φ,d n : δε T pgσ pc + δε T ngσ nc δe T pgd pc δe T ngd nm 24 V δd T nme ng E nc dv = δl e + δl in, the added Lagrange multiplier δd nm permits to assume an independent interlaminar continuous transverse normal electric displacement D n. The constitutive equations are obtained from Eqs.8- by expressing σ p, σ n, D p and E n : σ pc = C σp ε p ε pg + C σp ε n ε ng + C σp E p E pg + C σp D n D nm, 25 σ nc = C σn ε p ε pg + C σn ε n ε ng + C σn E p E pg + C σn D n D nm, 26 D pc = C Dp ε p ε pg + C Dp ε n ε ng + C Dp E p E pg + C Dp D n D nm, 27 E nc = C En ε p ε pg + C En ε n ε ng + C En E p E pg + C En D n D nm, 28 where: C σp ε p = C pp + e T npε nn e np, C σp ε n = C pn + e T npε nn e nn, C σp E p = e T npε nn ε np e T pp,

13 Variable Kinematics and Advanced Variational Statements 27 C σp D n = e T npε nn, C σn ε p = C np + e T nnε nn e np, C σn ε n = C nn + e T nnε nn e nn, C σn E p = e T nnε nn ε np e T pn, C σn D n = e T nnε nn, C Dp ε p = e pp ε pn ε nn e np, 29 C Dp ε n = e pn ε pn ε nn e nn, C Dp E p = ε pp ε pn ε nn ε np, C Dp D n = ε pn ε nn, C En ε p = ε nn e np, C En ε n = ε nn e nn, C En E p = ε nn ε np C En D n = ε nn. To be noticed that in this RMVT case, the constitutive equations for pure mechanical case cannot be derived as a particular case of Eqs because the modelled variables are different: transverse stresses in Eq.6 and transverse normal electric displacement in Eq.24, in fact, two different Lagrange multipliers are considered. The RMVTu,Φ,σ n,d n case The third, full extension case of RMVT to electro-mechanical problems considers both transverse shear/normal stresses σ n and transverse normal electrical displacement D n as primary variables: RMV T u,φ,σ n,d n : δε T pgσ pc + δε T ngσ nm δe T pgd pc δe T ngd nm + V δσ T nmε ng ε nc δd T nme ng E nc dv = δl e + δl in. 30 The full extension of RMVT permits the complete fulfillment of C 0 z -requirements for both transverse shear/normal stresses and transverse normal electric displacement. The constitutive equations are obtained from Eqs.8- by expressing σ p, ε n, D p and E n : σ pc = C σp ε p ε pg + C σp σ n σ nm + C σp E p E pg + C σp D n D nm, 3 ε nc = C εn ε p ε pg + C εn σ n σ nm + C εn E p E pg + C εn D n D nm, 32 D pc = C Dp ε p ε pg + C Dp σ n σ nm + C Dp E p E pg + C Dp D n D nm, 33 E nc = C En ε p ε pg + C En σ n σ nm + C En E p E pg + C En D n D nm. 34 The explicit form of the matrices in Eqs.3-34 is: C σp ε p = C pp C pn C nn C np C pn C nn e T nn e T npe nn C nn e T nn + ε nn e np e nn C nn C np, C σp σ n = C pn C nn C pn C nn e T nn e T npe nn C nn e T nn + ε nn e nn C nn, C σp E p = C pn C nn e T pn e T pp C pn C nn e T nn e T npe nn C nn e T nn + ε nn

14 272 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 e nn C nn e T pn + ε np, C σp D n = C pn C nn e T nn e T pne nn C nn e T nn + ε nn, C εn ε p = C nn C np C nn e T nne nn C nn e T nn + ε nn e nn C nn C np e np, C εn σ n = C nn C nn e T nne nn C nn e T nn + ε nn e nn C nn, C εn E p = C nn e T pn C nn e T nne nn C nn e T nn + ε nn e nn C nn e np + ε np, C εn D n = C nn e T nne nn C nn e T nn + ε nn, C Dp ε p = e pp e pn C nn C np e pn C nn e T nne nn C nn e T nn + ε nn e np e nn C nn C np ε pn e nn C nn e T nn + ε nn e np e nn C nn C np, 35 C Dp σ n = e pn C nn e pn C nn e T nne nn C nn e T nn + ε nn e nn C nn ε pn e nn C nn e T nn + ε nn e nn C nn, C Dp E p = ε pp + e pn C nn e T pn e pn C nn e T nne nn C nn e T nn + ε nn e nn C nn e np + ε np ε np e nn C nn e T nn + ε nn e nn C nn e T pn + ε np, C Dp D n = e pn C nn e T nne nn C nn e T nn + ε nn + ε pn e nn C nn e T nn + ε nn, C En ε p = e nn C nn e T nn + ε nn e T pn e nn C nn C np, C En σ n = e nn C nn e T nn + ε nn e nn C nn, C En E p = e nn C nn e T nn + ε nn e nn C nn e T pn + ε np, C En D n = e nn C nn e T nn + ε nn. 3 Geometrical relations Shells are bi-dimensional structures in which one dimension in general the thickness in z direction is negligible with respect to the other two in-plane dimensions. Geometry and the reference system are indicated in Figure 2. The square of an infinitesimal linear segment in the layer, the associated infinitesimal area and volume are given by: ds 2 k = H k 2 d 2 k + H k 2 d 2 k + Hz k 2 dz 2 k, 36 dω k = H k H k d k d k, 37 dv k = H k H k Hk z d k d k dz k, 38 where the metric coefficients are: H k = + z k /R k, H k = Bk + z k /R k, Hk z =. 39 k denotes the k-layer of the multilayered shell; R k and R k are the principal radii of curvature along the coordinates k and k, respectively. and B k are the coeffi-

15 Variable Kinematics and Advanced Variational Statements 273 cients of the first fundamental form of Ω k Γ k is the Ω k boundary. In this paper, the attention has been restricted to shells with constant radii of curvature cylindrical, open spherical panels, toroidal geometries for which = B k =. Details for shells are reported in Leissa 973. Eqs demonstrate as a LW representation of the curvature terms is employed for both ESL and LW kinematics models. Geometrical relations permit to express the in-plane ε p and out-plane ε n strains in terms of displacement u, and the in-plane components E p and out-plane components E n of electric field in terms of the electric potential Φ. The following relations hold: ε k pg = ε k,ε k,γk T = D k p + p u k, ε k ng = γ k z,γ k z,εk zz T = D k nω + Dk nz n u k, 40 E k pg = E k,e k T = D k eω Φk, E k ng = E k z T = D k en Φ k. The explicit form of the introduced arrays follows: 0 0 H k 0 0 D k p = 0 H 0 H k, Dk nω = k 0 0 H k, Dk nz = D k eω = H k H k H k H k, D k en = z, p = 0 0 H k R k 0 0 H k Rk 0 0 0, n = z z 0, z H k R k 0 H k Rk Geometrical relations for shells degenerate in those for plates when the radii of curvature R k and R k are infinite, and metric coefficients Hk and H k are equal to one see Eq.44. In the case of plate geometry the square of an infinitesimal linear segment, the associated infinitesimal area and volume of the generic k-layer are given by: ds 2 k = dx 2 k + dy 2 k + dz 2 k, dω k = dx k dy k, dv k = dx k dy k dz k, 42 x,y,z is the orthogonal cartesian reference system and k is the indicative of the layer, the following relations hold: ε k pg = ε k xx,ε k yy,γ k xy T = D p u k, ε k ng = γ k xz,γ k yz,ε k zz T = D nω + D nz u k 43 E k pg = E k x,e k y T = D eω Φ k, E k ng = E k z T = D en Φ k.

16 274 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 The explicit form of the introduced arrays follows: x x z 0 0 D p = 0 y 0, D nω = 0 0 y, D nz = 0 z 0, 44 y x z x D eω =, D en = z. y 4 Variable kinematics plate/shell modelings via Carrera s Unified Formulation Carrera s Unified Formulation CUF is a technique that permits to handle in a unified manner a large variety of bi-dimensional plate/shell models. CUF Carrera 995; Carrera 2002 permits to express the considered three-dimensional 3D variables in terms of a set of thickness functions depending only on the thickness coordinate z and two-dimensional variables depending on the in-plane coordinates x,y. According to CUF, the governing equations are written in terms of a few fundamental nuclei which do not formally depend on: - the order of expansion N used in the z-direction; - variables description Layer Wise LW or Equivalent Single Layer ESL. By expanding these fundamental nuclei in according to opportune indexes, the governing equations of the structure can be obtained. For a generic 3D variable a, the following expression is written: ax,y,z = F τ za τ x,y, 45 the same is done for its variation: δax,y,z = F s zδa s x,y. 46 The order of expansion ranges from first to fourth order, and depending on the used thickness functions, a model can be: ESL when the variable is assumed for the whole multilayer see Figure 3 and LW when the variable is considered for each layer see Figure 4. In case of an ESL theory, zig-zag forms of displacement variables can be accounted see Figure 5 by means of Murakami function Murakami 986; Carrera The expansion used in Eqs.45 and 46 are also employed for shell geometry when a curvilinear reference system,,z is assumed. 4. Equivalent Single Layer theories In case of electro-mechanical problem the assumed variables are the displacements u and the electric potential Φ. If Reissner s Mixed Variational Theorem RMVT is

17 Variable Kinematics and Advanced Variational Statements 275 ED ED3 a b Figure 3: Generic variable through the thickness direction z in Equivalent Single Layer form for plate a and shell b geometries. LD LD3 a b Figure 4: Generic variable through the thickness direction z in Layer Wise form for plate a and shell b geometries. zigzag ED3 EDZ3 a b Figure 5: Addition of Murakami zig-zag function to ESL models for plate a and shell b geometries.

18 276 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 used, other variables such as transverse shear/normal stresses σ n and/or transverse normal electric displacement D n can be also assumed. In this work only the displacement u can be modelled as ESL or LW, the other variables are always assumed in LW form. So a model is said ESL or LW depending on the choice made for the displacement variables. The expansion in the thickness direction z for ESL theories coincides to Taylor expansion: u = F 0 u 0 + F u F N u N = F τ u τ with τ = 0,,...,N, 47 N is the order of expansion and the thickness functions are: F 0 = z 0 =, F = z = z,..., F N = z N, 48 the order of expansion N ranges from to 4, that is from linear to fourth order. Typical zig-zag form of displacements can be recovered by means of Murakami zigzag function MZZF Murakami 986 as described in Figure 5. This function is Mz = k ζ k where ζ k = 2z k /h k is a non-dimensional layer coordinate z k is the physical coordinate of the k-layer whose thickness is h k which goes from to +: u = F 0 u 0 + F u F N u N + k ζ k u Z = F τ u τ + k ζ k u Z 49 with τ = 0,,...,N. The exponent k changes the sign of the zig-zag term in each layer. Such an artifice permits one to reproduce the discontinuity of the first derivative of the displacement variables in the z-direction, see Figure Layer Wise theories In the case of Layer Wise LW models, the considered variables are modelled in each layer k. For electro-mechanical problems the expansions in z direction are: u k,σ k nm,φ k,d k nm = F t u u k,σ k nm,φ k,d k nm t + F b u u k,σ k nm,φ k,d k nm b where + F r u u k,σ k nm,φ k,d k nm r 50 = F τ u u k,σ k nm,φ k,d k nm τ τ = t,b,r with r = 2,...,N, 5 t and b are the top and bottom values, and r the higher order terms of expansion. The thickness functions F τ ζ k have been defined at the k-layer level, they are a

19 Variable Kinematics and Advanced Variational Statements 277 linear combination of Legendre polynomials P j = P j ζ k of the j th -order defined in ζ k -domain ζ k. The first five Legendre polynomials are: P 0 =, P = ζ k, P 2 = 3ζ k 2, P 3 = 5ζ k ζ k and: F t = P 0 + P 2 2, P 4 = 35ζ 4 k 8 5ζ k , 52, F b = P 0 P, F r = P r P r 2 with r = 2,...,N The chosen functions have the following interesting properties: ζ k = : F t = ; F b = 0; F r = 0, 54 ζ k = : F t = 0; F b = ; F r = Eqs.54 and 55 permit to consider interface values of the variables as unknown variables, this fact permits to impose the compatibility and/or equilibrium conditions at each layer interface. 4.3 Acronyms system Several refined and advanced two-dimensional models can be obtained according to what in Sections 4. and 4.2. Depending on the used variational statement PVD or RMVT, variables description LW, ESL or ESL with MZZF, order of expansion N in z, a large variety of kinematics plate/shell theories is obtained. A system of acronyms is given in order to denote these models. The first letter indicates the multilayer approach which can be Equivalent Single Layer E or Layer Wise L. The second letter refers to the employed variational statement: D for Principle of Virtual Displacements and M for Reissner s Mixed Variational Theorem. The number N indicates the order of expansion used in the z-direction from to 4. In the case of ESL approach, a letter Z can be added if the zigzag effect of displacements is considered by means of MZZF. Summarizing, ED ED4 are ESL models based on PVD and EM EM4 are ESL models based on RMVT. If Murakami zigzag function is used, these equivalent single layer models are indicated as EDZ EDZ3 and EMZ EMZ3, respectively. In the case of layer wise approaches, the letter L is considered in place of E, so the acronyms are LD LD4 and LM LM4. Classical theories such as Classical Lamination Theory CLT and First order Shear Deformation Theory FSDT can be obtained as particular cases of ED theory simply imposing a constant value of u z through the thickness direction. An appropriate application of penalty technique to shear correction factor leads to CLT.

20 278 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , Governing equations The steps to obtain the consistent governing equations are: - choice of the opportune variational statement PVD or RMVT; - substitutions of consistent constitutive equations; - use of geometrical relations for plates and shells; - introduction of CUF for the two-dimensional approximation. 5. The classical PVD electro-mechanical case The complete procedure to obtain governing equations, boundary conditions and fundamental nuclei for PVD case extended to electro-mechanical problem is here discussed see Ballhause et al The variational statement, as obtained in Eq.2, and written for a multilayered structure is: PV Du,Φ : N l { k= Ω k N l = k= {δε kt pgσ k pc + δε kt ngσ k nc δe kt pgd k pc δe kt ngd k nc}dω k dz k } δl k in + δl k e, 56 where k denotes the layer and N l is the number of layers embedded in the multilayered structure. Ω k is the in-plane integration domain, denotes domain in the z direction. Upon substitution of the correspondent constitutive equations Eqs.8- in Eq.56 one has: N l { k= Ω k {δε kt pgc k ppε k pg +C k pnε k ng e kt ppe k pg e kt npe k ng+ δε kt ngc k npε k pg +C k nnε k ng e kt pne k pg e kt nn E k ng δe kt pge k ppε k pg + e k pnε k ng + ε k ppe k pg + ε k pne k ng δe kt nge k npε k pg + e k nnε k ng + ε k npe k pg + ε k nne k ng}dω k dz k } = N l k= δl k in + δl k e. 57

21 Variable Kinematics and Advanced Variational Statements Plate geometry If the geometrical relations for plate geometry Eqs.43 are substituted where the subscript G appears in Eq.57, the PVD becomes: N l { k= Ω k {D p δu k T C k ppd p u k +C k pnd nω + D nz u k + e kt ppd eω Φ k + e kt npd en Φ k + D nω + D nz δu k T C k npd p u k +C k nnd nω + D nz u k + e kt pnd eω Φ k + e kt nn D en Φ k D eω δφ k T e k ppd p u k + e k pnd nω + D nz u k ε k ppd eω Φ k ε k pnd en Φ k D en δφ k T e k npd p u k + e k nnd nω + D nz u k ε k npd eω Φ k ε k nnd en Φ k }dω k dz k } = N l k= δl k in + δl k e. 58 Upon substitution of two-dimensional approximation by means of CUF, the following form is obtained: N l { k= Ω k { D p δu k T s Fs C k ppd p F τ u k τ+ C k pnd nω + D nz F τ u k τ + e kt ppd eω F τ Φτ k + e kt npd en F τ Φτ k + DnΩ + D nz δu k T s Fs C k npd p F τ u k τ+ C k nnd nω + D nz F τ u k τ + e kt pnd eω F τ Φτ k + e kt nn D en F τ Φτ k DeΩ δφs k T Fs e k ppd p F τ u k τ + e k pnd nω + D nz F τ u k τ ε k ppd eω F τ Φ k τ ε k pnd en F τ Φ k τ Den δφ k s T Fs e k npd p F τ u k τ+ e k nnd nω + D nz F τ u k τ ε k npd eω F τ Φ k τ ε k nnd en F τ Φ k τ }dωk dz k } = N l k= δl k in + δl k e. In order to obtain a strong form of differential equations on the domain Ω k, as well as the correspondence boundary conditions on edge Γ k, the integration by parts must be employed. This latter permits to move the differential operator from the infinitesimal variation of the generic variable δa k to the finite quantity a k Carrera For a generic variable a k, the integration by parts states: Ωk D Ω δa k T a k dω k = 59

22 280 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 δa kt D Ω T a k dω k + δa kt I Ω T a k dγ k, 60 Ω k Γ k where Ω = p, nω, eω. The following additional arrays have been introduced to perform integration by parts: I p = 0 0 ;I nω = 0 0 ;I eω = After the integration by parts, the PVD assumes the following form: N l { k= Ω k {δu kt s D p T F s C k ppd p F τ u k τ+ C k pnd nω + D nz F τ u k τ + e kt ppd eω F τ Φ k τ + e kt npd en F τ Φ k τ + δu kt s D nω + D nz T F s C k npd p F τ u k τ+ C k nnd nω + D nz F τ u k τ + e kt pnd eω F τ Φ k τ + e kt nn D en F τ Φ k τ + δφs kt D eω T F s e k ppd p F τ u k τ + e k pnd nω + D nz F τ u k τ ε k ppd eω F τ Φτ k ε k pnd en F τ Φτ k + δφs kt D en T F s e k npd p F τ u k τ+ e k nnd nω + D nz F τ u k τ ε k npd eω F τ Φτ k ε k nnd en F τ Φτ k }dω k dz k } N l + { {δu kt s I T pf s C k ppd p F τ u k τ+ k= Γ k C k pnd nω + D nz F τ u k τ + e kt ppd eω F τ Φ k τ + e kt npd en F τ Φ k τ + δu kt s I T nωf s C k npd p F τ u k τ+ C k nnd nω + D nz F τ u k τ + e kt pnd eω F τ Φ τ + e kt nn D en F τ Φ k τ + δφs kt I T eωf s e k ppd p F τ u k τ + e k pnd nω + D nz F τ u k τ ε k ppd eω F τ Φτ k ε k pnd en F τ Φτ k }dγ k dz k } N l = { δu kt s ρ k F s F τ üüü k τdω k dz k } + k= Ω k N l k= The governing equations on the domain Ω k are: { δu kt s Ω k F s p k u + δφ kt s F s p k ΦdΩ k }. δu k s : u k τ + uφ Φk τ = P k us M kτs üüü k τ, 63 δφs k : Φu uk τ + ΦΦ Φk τ = P k Φs, 62

23 Variable Kinematics and Advanced Variational Statements 28 where üüü k τ denotes the second derivative with respect to the time of the displacement components. M kτs is the fundamental nucleus for the inertial array., uφ, Φu and Kkτs ΦΦ are the so-called fundamental nuclei of electro-mechanical stiffness array. P k us and P k Φs are the consistent variationally mechanical and electric loads, respectively. Boundary conditions of Dirichlet type are: u k τ = ūūū k τ, Φ k τ = Φ k τ, 64 Boundary conditions of Neumann type are: u k τ + uφ Φk τ = ūūū k τ + uφ Φ k τ, Φu uk τ + ΦΦ Φk τ = Φuūūūk τ + ΦΦ Φ k τ. 65 The fundamental nuclei are: Ak = D p T C k ppd p + D p T C k pnd nω + D nz + 66 D nω + D nz T C k npd p + D nω + D nz T C k nnd nω + D nz F τ F s dz, uφ = D p T e kt ppd eω + D p T e kt npd en + 67 D nω + D nz T e kt pnd eω + D nω + D nz T e kt nn D en F τ F s dz, Φu = D eω T e k ppd p + e k pnd nω + D nz + D en T e k npd p + 68 e k nnd nω + D nz F τ F s dz, ΦΦ = D eω T ε k ppd eω + D eω T ε k pnd en + 69 D en T ε k npd eω + D en T ε k nnd en F τ F s dz. The inertial array is: M kτs = I ρ k F τ F s dz, 70 where ρ k is the mass density for each layer k, and I is the identity matrix of dimension 3 3. The nuclei for the boundary conditions are:

24 282 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 = I Ak T p C k ppd p + I T p C k pnd nω + D nz + I T nωc k npd p + I T nω C k nnd nω + D nz F τ F s dz, 7 uφ = I T pe kt ppd eω + I T pe kt npd en + I T nωe kt pnd eω + I T nωe kt nn D en F τ F s dz, 72 Φu = I T eωe k ppd p + e k pnd nω + D nz F τ F s dz, 73 ΦΦ = I T eω ε k ppd eω + I T eω ε k pnd en F τ F s dz Shell geometry If the geometrical relations for shell geometry Eqs.40 are substituted where the subscript G appears in Eq.57, the PVD becomes: N l { k= Ω k {D k p + pδu k T C k ppd k p + pu k +C k pnd k nω + Dk nz nu k + e kt ppd k eω Φk + e kt npd k enφ k + D k nω + Dk nz nδu k T C k npd k p + pu k +C k nnd k nω + Dk nz nu k + e kt pnd k eω Φk + e kt nn D k enφ k D k eω δφk T e k ppd k p + pu k + e k pnd k nω + Dk nz nu k ε k ppd k eω Φk ε k pnd k enφ k D k enδφ k T e k npd k p + pu k + e k nnd k nω + Dk nz nu k ε k npd k eω Φk ε k nnd k enφ k }dω k dz k } = N l k= δl k in + δl k e. 75

25 Variable Kinematics and Advanced Variational Statements 283 Upon substitution of two-dimensional approximation, by means of CUF, the following form is obtained: N l { k= Ω k { D k p + pδu k T s Fs C k ppd k p + pf τ u k τ+ C k pnd k nω + Dk nz nf τ u k τ + e kt ppd k eω F τφτ k + e kt npd k enf τ Φτ k + D k nω + Dk nz nδu k T s Fs C k npd k p + pf τ u k τ+ C k nnd k nω + Dk nz nf τ u k τ + e kt pnd k eω F τφτ k + e kt nn D k enf τ Φτ k D k T eω δφk s Fs e k ppd k p + pf τ u k τ + e k pnd k nω + Dk nz nf τ u k τ 76 ε k ppd k eω F τφτ k ε k pnd k enf τ Φτ k D k enδφs k T Fs e k npd k p + pf τ u k τ+ e k nnd k nω + Dk nz nf τ u k τ ε k npd k eω F τφτ k ε k nnd k enf τ Φτ k }dωk dz k } = N l k= δl k in + δl k e. In order to obtain a strong form of differential equations on the domain Ω k, as well as the correspondence boundary conditions on edge Γ k, the integration by parts must be employed. For a generic variable a k, the integration by parts states as in Eq.60, the additional arrays introduced to perform integration by parts in the case of shell geometry change with respect to the plate case see Eq.6: I k p = H k 0 H k H k 0 0 H k 0 0 ; I k nω = 0 0 H k 0 0 H k ; Ik eω = H k H k. 77

26 284 Copyright 200 Tech Science Press CMES, vol.65, no.3, pp , 200 After the integration by parts, the PVD assumes the following form: N l { k= Ω k {δu kt s D k p + p T F s C k ppd k p + pf τ u k τ+ C k pnd k nω + Dk nz nf τ u k τ + e kt ppd k eω F τφ k τ + e kt npd k enf τ Φ k τ + δu kt s D k nω + Dk nz n T F s C k npd k p + pf τ u k τ+ C k nnd k nω + Dk nz nf τ u k τ + e kt pnd k eω F τφ k τ + e kt nn D k enf τ Φ k τ + δφs kt D k eω T F s e k ppd k p + pf τ u k τ + e k pnd k nω + Dk nz nf τ u k τ ε k ppd k eω F τφτ k ε k pnd k enf τ Φτ k + δφs kt D k en T F s e k npd k p + pf τ u k τ+ e k nnd k nω + Dk nz nf τ u k τ ε k npd k eω F τφτ k ε k nnd k enf τ Φτ k }dω k dz k } N l + { {δu kt s I kt p F s C k ppd k p + pf τ u k τ+ k= Γ k C k pnd k nω + Dk nz nf τ u k τ + e kt ppd k eω F τφ k τ + e kt npd k enf τ Φ k τ + δu kt s I kt nω F sc k npd k p + pf τ u k τ+ C k nnd k nω + Dk nz nf τ u k τ + e kt pnd k eω F τφ τ + e kt nn D k enf τ Φτ k + δφs kt I kt eω F se k ppd k p + pf τ u k τ + e k pnd k nω + Dk nz nf τ u k τ ε k ppd k eω F τφτ k ε k pnd k enf τ Φτ k }dγ k dz k } N l = { δu kt s ρ k F s F τ üüü k τdω k dz k }+ k= Ω k N l { δu kt s F s p k u + δφs kt F s p k ΦdΩ k }. Ω k k= 78 The governing equations on the domain Ω k and the relative boundary conditions are the same proposed in Eqs for plate geometry. The form of the fundamental nuclei changes as: = D Ak k p + p T C k ppd k p + p + D k p + p T C k pnd k nω + Dk nz n+ D k nω + Dk nz n T C k npd k p + p + D k nω + Dk nz n T C k nnd k nω + Dk nz n F τ F s H k H k dz, 79 uφ = D k p + p T e kt ppd k eω + Dk p + p T e kt npd k en+ 80

27 Variable Kinematics and Advanced Variational Statements 285 D k nω + Dk nz n T e kt pnd k eω + Dk nω + Dk nz n T e kt nn D k en F τ F s HH k k dz, Φu = D k eω T e k ppd k p + p + e k pnd k nω + Dk nz n + D k en T e k npd k p + p+ e k nnd k nω + Dk nz n F τ F s H k H k dz, ΦΦ = D k eω T ε k ppd k eω + Dk eω T ε k pnd k en+ 8 D k en T ε k npd k eω + Dk en T ε k nnd k en F τ F s HH k k dz. The inertial array is: M kτs = I ρ k F τ F s HH k k dz, 82 where ρ k is the mass density for each layer k, and I is the identity matrix of dimension 3 3. The nuclei for the boundary conditions are: = I Ak kt p C k ppd k p + p + I kt p C k pnd k nω + Dk nz n+ I kt nω Ck npd k p + p + I kt nω Ck nnd k nω + Dk nz n F τ F s HH k k dz, 83 uφ = I kt p e kt ppd k eω +IkT p e kt npd k en +I kt nω ekt pnd k eω +IkT nω ekt nn D k en F τ F s HH k k dz, 84 Φu = I kt eω ek ppd k p + p + e k pnd k nω + Dk nz n F τ F s HH k k dz, 85 ΦΦ = I kt eω εk ppd k eω + IkT eω εk pnd k en F τ F s HH k k dz The advanced RMVTu,Φ,σ n case The steps to obtain the governing equations for the partial extension of RMVT to electro-mechanical case are the same illustrated in Section 5. see D Ottavio and Kröplin 2006 and Carrera and Brischetto 2007a. The proposed variational statement considers as primary variables the displacements u, the electric potential Φ and the transverse shear/normal stresses σ n. Eq.6 written for a multilayered structure is: RMV T u,φ,σ n : N l { k= Ω k {δε kt pgσ k pc + δε kt ngσ k nm δe kt pgd k pc 87