Modelling of multilayered piezoelectric composites

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1 Modelling of multilayered piezoelectric composites Claire David Université Paris 6 - Pierre-et-Marie-Curie Institut Jean Le Rond d Alembert, UMR CNRS 7190 Boîte courrier n 0 162, 4 place Jussieu, Paris, cedex 05, France david@lmm.jussieu.fr This chapter is the continuation of my Ph. D. [12], [5], [3]. It generalizes the multilayered shell model to piezoelectric structures. In a first time, it was natural to consider the related plate model; thanks to satisfying results, we then considered the shell model. 1 Multilayered piezoelectric plates We hereafter study the modelling of composite multilayered piezoelectric plates. Our theory is based on an hybrid approach, where the mechanical and electrical continuity conditions are satisfied, as well as the boundary conditions on the top and bottom surfaces of the structure. The accuracy of the proposed theory is assessed through investigation of significant problems, for which an exact three-dimensional solution is known. 1.1 Introduction The development of the so-called "smart-structures", e.g. made of piezoelectric composites, require nowadays more and more precision in their design and sizing. The importance of efficient models has, so far, led to numerous theories. The first kind of approach is generally based on the assumption according to which the multilayered piezoelectric structure behaves as a single-layered one. Tiersten [47], Mindlin [33], Lee et al. [31], [32], have applied Kirchhoff-Love s theory to piezoelectricity. Yet, only thin structures are concerned. First order theories can be found, for single-layered structures, for instance, in Chandrashekhara et al. [15]. However, the electric field induced by mechanical efforts is, generally, not taken into account. 1

2 Vatal yan et al [56] devoted themselves to bilayered ceramic plates. Thanks to expansions into series of the thickness coordinate, Lee et al. [32] obtained the two dimensional equations of motion, for plates made of piezoelectric crystals. Yang et al. [58], Yong et al. [59] have generalized these results to multilayered plates. The problem is that those models do not generally take into account the coupling that occurs in the equations of motion. Hybrid theories, with a " single-layered type " approach, and a modelling " per layer " of the electrostatic potential, have been proposed: Mitchell et al. [34] used a third-order theory for the displacement field, while taking into account the variations of the electrostatic potential through the thickness of the plate; Fernandes et al. [21], [22] developed a model that takes into account refinements of the shear terms, as in Touratier [51]. Other " layerwise models " have been developed, in a first time by Pauley [42]. Pai et al. [39] have proposed an induced-strain model of multilayered piezoelectric plate model. Heyliger et al. [25], [26] developed an exact three-dimensional theory, for composite multilayered plates. Yet, those models do not simultaneously take into account the continuity conditions for the mechanical and electrical quantities. One can of course find asymptotic theories, which enable us to satisfy both kinds of conditions, but they only apply to thin structures [17]. We propose in the following, a two-dimensional theory, more accurate, that enables us to model thick multilayered piezoelectric structures. This theory generalizes to piezoelectricity our composite multilayered model, developed in [12], [3], [5], relating our displacement approach, to a " singlelayered " type approach, continuous at layer interfaces, to quadratic variations through the thickness of the considered structure, of the electrostatic potential, which is also continuous. The transverse shear stresses, as well as the electric displacement, under a constant strain, are continuous. Refinements of the shear and membrane terms are taken into account, by means of trigonometric functions. Eventually, the conditions on the frontier of the structure, or at layer interfaces, are satisfied. The accuracy of the plate model thus obtained is assessed through investigation of significant problems, for which a three-dimensional solution is known [25], [26]. 2

3 Consider a multilayered piezoelectric plate, made of an arrangement of N layers. Denote by a the length of the plate, by b its width, by h its thickness, and by V the volume occupied by the plate (see figure 1). z S h z N Layer N h z N 1 x 2 z 2 z 1 Layer 2 Layer 1 a b x 1 Figure 1: The multilayered piezoelectric plate Notation. The frontier of the plate is made of the reunion of its bottom surface S 0, its top surface S h, and its lateral surface A. Denote by S i the interface between the i th and (i + 1) th layers, and by z i the distance between S 0 and S i. The reference surface coïncides with the bottom surface S 0. 3

4 Assumptions 1.1. The transverse shear stresses are supposed equal to zero on the top and bottom surfaces of the plate. Notation. The Einsteinian summation convention applies to repeated indices, where Latin indices range from 1 to 3 while Greek indices range from 1 to 2. 4

5 Notation. V h S 0 S h S i z i z i0 A Volume occupied by the plate Total thickness of the plate Bottom surface of the plate Top surface of the plate Bottom surface of the i th layer Distance between S 0 et S i Distance between S 0 and the mid-surface of the i th layer Lateral surface of the plate (x i ) Cartesian coordinates L i Lamé coefficients s k, k = 1,..., 6 Strains σ k, k = 1,..., 6 Stresses C (i) (i) mnpq, or C KL Components of the elastic strain tensor under a constant electric field constant Differentiation with respect to z,i Differentiation with respect to x i. Differentiation with respect to time t δ Variational operator ϕ Electrostatic potential o ϕ 1B Electrostatic potential on S 0 ϕ N+1,B Electrostatic potential on S h ϕ ib Electrostatic potential on S i ϕ im Electrostatic potential on the midsurface of the i th layer ϕ it Electrostatic potential on S i+1 E l, l = 1,..., 3 Components of the electrical field E D k, k = 1,..., 3 Components of the electric field (i) e kl piezoelectric constants, under a constant strain, of the i th layer (i) ε kl dielectric constants, under a constant strain, of the i th layer (components of the dielectric tensor of the i th layer) ρ Mass density ε 0 Permittivity of vacuum 1 5

6 1 ε 0 = F / m 1.2 Mechanical study Kinematic assumptions Assumptions 1.2. The displacement field U of any point M(x α, z) of the structure is determined by its components (U α, U z ) in the basis ( ) i, j, k, which are approximated under the following form: { Uα = u α + z η α + f(z) ψ α + g(z) γα 0 + N 1 m=1 (z z m) u mα H(z z m ) U z = w (1) where f(z) = h π sin ( π z ) h, g(z) = h π cos ( π z ) h (2) and where H denotes the Heaviside step function. Remark 1.1. i. The Heaviside step function has been previously used, among others, by Di Sciuva [18] and He [24]. ii. The use of the sine and cosine functions can be justified as in Touratier [49], by a discrete layer theory from the three-dimensional modelling of Cheng [16] for thick plates. Definition 1.1. The u α are membrane displacements, the γ 0 α are the components of the transverse shear stress at z = 0, w is the deflection, the ψ α and u mα are a priori unknown functions, which are to be determined using the boundary conditions on the top and bottom surfaces, as well as at layer interfaces. 6

7 1.2.2 Voigt notation The anisotropy of the piezoelectric materials requires using the two indices Voigt notation; we thus use the correspondance: ( 1, 1 ) 1, ( 2, 2 ) 2, ( 3, 3 ) 3 ( 2, 3 ) 4, ( 3, 2 ) 4 ( 1, 3 ) 5, ( 3, 1 ) 5 ( 1, 2 ) 6, ( 2, 1 ) 6 Notation. In the following, capital letters denote a couple of latin indices Uncoupled constitutive equations Denote by C mnkl (i) (or C KL (i), Voigt notation) the components of the elastic strain tensor, under a electric field, related to the i th layer, and by s kl the components of the strain tensor (or s I, Voigt notation). Assumptions 1.3. We use, in the following, the assumption of small strains, which yields: s kl = U k,l + U l,k 2 (3) Remark 1.2. The piezoelectric theory thus obtained is linear. The uncoupled constitutive equations are: or, Voigt notation: σ mn (i) = C mnkl (i) s kl (4) σ (i) d J = (i) CJK s K (5) σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 = C s 1 s 2 s 3 s 4 s 5 s 6 (6) 7

8 1.3 Boundary conditions i. Conditions on the top and bottom surfaces of the plate: The nullity of the uncoupled transverse shear stress on S 0 and S h yields: The latter system yields: { (1) σ d 6 α = 0 (N) σ d 6 α = 0 (7) { (1) C6 α {U α,3 (x α, 0; t) + w,α } = 0 C (N) 6 α {U α,3 (x α, h; t) + w,α } = 0 (8) i. e.: ii. Layer interfaces conditions: { ηα = w,α N 1 ψ α = 1 N 1 2 m=1 u mα m=1 u mα The continuity of the uncoupled shear stress at layer interface between i th and (i + 1) th layers can be written as: (9) i. e.: C 6 α,6 α (i) [ g (z i ) γ 0 α+ 1 2 σ 6 α (i) d (x α, z i ) = σ 6 α (i+1) d (x α, z i ) (10) i 1 (f ] (z i ) 1) u mα = (i+1) [ C6 α,6 α g (z i ) γα m=1 The linear system thus obtained enables us to express the u mα, m = 1,..., N, as functions of the transverse shears γ 0 α: i (f ] (z i ) 1) u mα m=1 (11) u mα = λ mα γ 0 α (12) where the λ mα are real constants, given by the resolution of the above system. 8

9 1.4 Final form of the kinematic field Proposition 1.4. The kinematic field U of a point M(x α, z) of the plate is given by: { Uα = u α z w,α + f(z) ψ α + N 1 m=1 u mα h α (z) (13) U z = w where the h α are functions of the global thickness parameter z, given by: h α (z) = g(z) + N 1 m=1 { } 1 2 (f(z) z) + (z z m) H(z z m ) λ mα (14) This kinematic field has been developed in [12], [3], [5]. 1.5 Electrical study Electrical assumptions Assumptions 1.5. The electrostatic potential is approximated under the form: φ(x α, z; t) = N 1 i=1 ϕ i (x α, z; t) χ i (z) (15) where the ϕ i denote the potentials per layer, and the χ i the characteristic functions per layer: { 1 si z χ i [zi, z (z) = i+1 [ 0 else (16) Notation. Let us introduce, for each layer: i. the thickness coordinate ξ i, defined by: (see figure 2). ξ i = 2 (z i z 0 i ) h i (17) 9

10 ii. the electrostatic potential on the bottom face, S i : ϕ ib = ϕ i (x α, z i ; t); iii. the electrostatic potential on the mid-surface: ϕ im = ϕ i (x α, z i+1 z i 2 ; t); iv. the electrostatic potential on the top face, S i+1 : ϕ it = ϕ i (x α, z i+1 ; t); ξ i ϕ it S i+1 h i O i ϕ im S i ϕ ib Figure 2: The i th layer Assumptions 1.6. The " potentials per layer " are assumed of the following form: ϕ i (x α, z; t) = 1 2 ξ i (ξ i 1) ϕ ib (x α ; t)+(1 ξ i 2 ) ϕ im (x α ; t)+ 1 2 ξ i (ξ i +1) ϕ it (x α ; t) (18) Remark 1.3. In so far as: ϕ ib (x α, z i+1 ; t) = ϕ it (x α, z i ; t), the continuity of the electrostatic potential at layer interfaces of the potential is thus automatically satisfied. For this reason, we choose to keep, as unknowns quantities, the ϕ ib and ϕ im. For future applications, we shall suppose that p values of the ϕ ib are known. 10

11 1.5.2 Uncoupled piezoelectric constitutive law The uncoupled piezoelectric constitutive law takes the form: D 3 = ε 33 ϕ,3 + e 3j s j (19) Electric boundary conditions Proposition 1.7. The electrostatic potential can be written as: ϕ(x α, z; t) = Q ik (z; t) ϕ i k B + Q jl (z; t) ϕ j l M (20) i k I j l J where the Q ik, Q jl z. are polynomial functions of the global thickness coordinate Proof. The continuity of D 3 d on the top and bottom surfaces, and at the p layer interfaces where the value of electrostatic potential is known, lead to a linear system of N + p 1 equations, which enable us to eliminate some of the ϕ ib and ϕ im. Notation. Denote by ϕ i k B, i k I, ϕ j l M, j l J, I and J being finite subsets IN, the remaining unknowns quantities. 1.6 How to take the coupling into account The need of some correction factors The resolution of the boundary problem enables us to obtain the values of the generalized mechanical and electrical unknowns. The sole quantity that cannot be obtained without an a posteriori treatment is the coupled electric displacement. We recall the coupled constitutive piezoelectric law: D 3 = ε 33 ϕ,3 + e 3j s j (21) The dielectric constants ε 33 ϕ,3 are very small compared to the mechanical ones e 3j s j. 11

12 So, if the final value of the electric displacement is not corrected a posteriori, mechanical quantities e 3j s j are going to be predominant. Those terms are not continued at layer interfaces (see former section). Proposition 1.8. The final electrical displacement is obtained, after correction, as: N D final 3 (x α, z; t) = ε 33 ϕ,3 +e 3j s j [e (k) 3j s (k) j (z k ) χ k (z) e (k+1) 3j s (k+1) j (z k ) χ k+1 (z)] k=1 (22) 1.7 The linear piezoelectric constitutive law Tiersten equations Notation. We recall that s denotes the strain tensor, E the electrical field, C the elastic strain tensor, under a constant electrical field, e the tensor of piezoelectric coefficients, under constant strain, and ε the tensor of dielectric constants, under a constant strain. i, j vary from 1 to 6, k, l from 1 to 3. Properties. Tensors C, e, ε satisfy the following symmetry properties: C ijkl = C klij = C jikl = C ijlk (23) e ijk = e ikj (24) ε ij = ε ji (25) Proposition 1.9. The linear piezoelectric constitutive law is given by: { σij = C ijkl s kl e kij E k D k = ε ij E j + e ijk s jk (26) or, under double indexation: { σi = C IJ s J e ki E k D k = ε kl E l + e ki s I (27) 12

13 Remark 1.4. For orthotropic piezoelectric materials, as PVDF polymers, part of the piezoelectric coefficients vanish; the tensors e and ε can thus be written as: e = ε = 0 0 e e e 33 0 e 24 0 e ε ε ε33 (28) (29) Remark 1.5. Lots of piezoelectric materials, as PZT ceramics, or ZnO crystals,..., are transversally isotropic. By choosing the orthotropy axis parallel to the vertical axis, the resulting simplifications occur: e 31 = e 32, e 24 = e 15, ε 11 = ε 22 (30) Two-dimensional coefficients Assumptions In the following, the normal stress is assumed to be equal to zero. Proposition Under the assumption of nullity of the normal stress, the linear piezoelectric constitutive law is given by: { σi = C IJ 2D s J e ki 2D E k D k = ε kl 2D e l + e ki 2D s I (31) where: C 2D C IJ = C IJ C 3J I3 C 33 e 2D e ki = e ki C k3 I3 C 33 (32) ε 2D e kl = ε kl + e k3 l3 C 33 13

14 Proof The assumption of nullity of the normal stress: can be written as: which enables us to eliminate the normal strain s 3 : σ 3 = 0 (33) C 3J s J e k3 E k = 0 (34) s 3 = e k3 C 33 E k C 3α C 33 s α C 3,6 α C 33 s 6 α C 36 C 33 s 6 (35) Substituting this expressions into (27), we obtain: σ I = C IJ s J e ki E k = C Iα s α + C I,6 α s 6 α + C I,6 s 6 + C I3 s 3 e ki E k { ek3 = C Iα s α + C I3 E k C 3,6 α s 6 α C } 36 s 6 e ki E k C 33 C 33 C { } { 33 } C 3α C 3,6 α = C Iα C I3 s α + C I,6 α C I3 s 6 α C 33 C { } { } 33 C 3,6 e k3 + C I,6 C I3 s 6 e ki C I3 E k C 33 C 33 (36) 1.8 The two-dimensional boundary problem Variational formulation Hamilton s Principle yields: t 0 { {σ i δs i + D i δϕ i }dv + V V { f v δ U + W δϕ} dv + fs δ } U da + (p 0 p h ) ds A S 0 (37) the density of external volumic where δ denotes a variational operator, f v forces, f s density of external lateral forces, p 0 and p h the pressures respectively acting on the top and bottom surfaces of the structure, W the density of external surface electrical forces. dt = 0 14

15 Notation. Let us introduce: i. the generalized forces: N α 1 = h {C 0 αj s j e 3α E 3 } dz = h C 0 66 s 6 (1 δ αβ ) dz N α 3 = h {C 0 αj s j e 3α E 3 } h α (z) dz = h C 0 66 s 6 (1 δ αβ ) h α (z) dz N α 5 = h 1 {C α,6 α s 6 α e 2,6 α E 2 }dz N 2 αβ N 4 αβ (38) { ii. the generalized momentums: N ib = h 0 E 3 ε 33 Q ib (z)dz, i I N jm = h 0 E 3 ε 33 Q jm (z) dz, j J { M α 1 = h {C 0 αj s j e 3α E 3 } z dz = h C 0 66 s 6 (1 δ αβ ) zdz M 2 αβ (39) (40) { M ibα = h {E 0 α ε αα + e kα s α } Q ib (z) dz, i I M jm α = h {E 0 α ε αα + e kα s α } Q im (z) dz, j J (41) iii. the generalized mechanical forces: iv. the generalized electrical forces: { 1 F ν α = h f 0 ν α dz 2 F ν α = h f 0 ν α z dz 3 F ν α = h f 0 ν α h α (z) dz 3 F ν = h f 0 ν α dz P = p 0 p h W ib = h W 0 QiB (z) dz, i I W jm = h W 0 QiB (z) dz, j J (42) (43) 15

16 iv. the inertia terms: I 1 = h ρ dz 0 I 2 = h ρ z dz 0 I 3 = h ρ h 0 α(z) dz I 4 = h ρ 0 z2 dz I 5 = h ρ z h 0 α(z) dz I 6 = h ρ z h 0 α 2 (z) dz (44) Proposition The movement equations are given by: α αβ N 1,α + N 2,β = I 1α ü α I 2 ẅ,α + I 3α γ α 0 α αβ M 1,αα + M 2,αβ = I 2α ü α,α I 4 ẅ,αα + I 5α γ α,α 0 + I 1 ẅ α αβ N 3,α + N 4,β + N α 5 = I 3α ü α I 5 ẅ,α + I 6α γ α 0 N ib + M ibα,α = 0, i I N jm + M jmα,α = 0, j J (45) Proof The movement equations are deduced from Hamilton s Principle, in conjunction with the kinematic (13), and the constitutive equations (27), by integration through the thickness of the plate. Proposition The boundary conditions leading to a regular problem are given by: N α 1 n α + N αβ 1 2 n αβ = F ν α ou δu α = 0 α αβ 2 M 1,α + M 2,β = F ν α ou δw = 0 N αβ 3 4 n β = F ν α ou δγ 0 α = 0 M α 1 n α + M αβ (46) 2 nβ = F ν3 ou δw,α = 0 N ibα n α = W ib ou δϕ ib = 0, i I M jm α n α = W jm ou δϕ jm = 0, j J 16

17 1.9 Numerical validation of the piezoelectric plate model Free vibrations of a single-layered plate Consider, in the following, a single-layered plate (N = 1), made of PZT4 ceramic, simply supported, in closed circuit (which means that the electrostatic potential on its top and bottom surfaces is equal to zero: ϕ 1B = ϕ 1T = 0) (see figure 3). z h O PZT 4 0 V Figure 3: Single-layered piezoelectric plate, in closed circuit Data Material constants of the ceramic PZT4 can be found in tables 1, 2. C 11 C 22 C 33 C 12 C 13 C 23 C 44 C 55 C 66 PZT Table 1: Independent elastic constants of the PZT4 ceramic (in GP a) 17

18 e 31 e 32 e 33 e 15 ε 11 ε 22 ε 33 PZT Table 2: Independent piezoelectric and dielectric constants of the PZT4 ceramic (e ij in C/m 2, ε ii in nf/m) Proposition The simply supported boundary conditions yield: w(x α = 0, z; t) = w(x α = a, z; t) = 0 (47) Proposition The electrostatic potential (20) is approximated as (18) ϕ 1 (x α, z; t) = (1 ξ 1 2 ) ϕ 1M (x α ; t) (48) The generalized mechanical unknowns are the membrane displacements u α, the deflection w, and the transverse shear stresses γ α 0. The generalized electrical unknown is ϕ 1M. Assumptions The solution is searched under the following form, which characterizes the propagation of two-dimensional plane waves: u 1 = A 1 e j ω t cos( π x 1 ) sin( π x 2 ) a b u 2 = A 2 e j ω t sin( π x 1 ) cos( π x 2 ) a b w = B e j ω t sin( π x 1 ) sin( π x 2 ), ϕ 1M = Φ a b 1 e j ω t sin( π x 1 0 γ 1 = C 1 e j ω t cos( π x 1 ) sin( π x 2 ) a ) sin(π x 2 b ) a a 0 γ 2 = C 2 e j ω t sin( π x 1 ) cos( π x 2 ) a a (49) and which enables us to satisfy the simply supported boundary conditions (47). By substituting these expressions into the equations of motion given by equations (13), in conjunction with the boundary conditions (45), the constitutive law (31) and the displacement field (13), we obtain a linear system in A α, B, C α, Φ 1, under the form: [K 1 ω 2 M 1 ] 18 A 1 A 2 B C 1 C 2 Φ 1 = 0 (50)

19 The detail of coefficients of the matrices K 1, M 1 is given in [7]. The system admits a non trivial solution if its determinant vanishes: det[k 1 ω 2 M 1 ] = 0 (51) A symbolic calculus tool (Mathematica or Maple) enables us to calculate the free pulsations of the considered plate, and to compare them to Heyliger s analytical solution (see Table 3). Model Exact solution a h a h a h = = Table 3: Free pulsations of the single-layered piezoelectric plate (in rad.s 1 ) Figures 4, 5 display the variations, as functions of the global thickness coordinate, of the normalized electrostatic potential: ϕ(x α, z; t) ϕ(z; t) = ϕ(x α = a, z = h; t) (52) 2 2 for significative values of the plate s thickness: a = 4 h, and a = 10 h. Results perfectly fit the exact solution Free vibrations of a 5-layered plate Consider, in the following a 5-layered plate (N = 5), the external layers of which are made of PZT 4 ceramic, with a core made of an arrangement 0 / 90 / 0 of epoxy resin, simply supported, in closed circuit (which means that the electrostatic potential on its top and bottom surfaces is equal to zero: ϕ 1B = ϕ 3T = 0) (see figure 6). Data The respective thicknesses of the external layers are h 1 = h 5 = h, 10 those of each of the epoxy resin, h 2 = h 3 = h 4 = 1 8 h

20 φ z h Figure 4: Electrostatic normalized potential, single-layered plate, a = 4 h. In gray: exact solution. In black: our model. z h φ z h z h Figure 5: Electrostatic normalized potential, single-layered plate, a = 10 h. black: our model. In Data Material constants of the ceramic PZT4 and the epoxy resin are given in tables 4, 5. C 11 C 22 C 33 C 12 C 13 C 23 C 44 C 55 C 66 PZT Epoxy Table 4: Independent elastic constants of PZT4 and epoxy resin (in GP a) Proposition The simply supported boundary conditions yield: w(x α = 0, z; t) = w(x α = a, z; t) = 0 (53) 20

21 z PZT 4 Epoxy 0 h Epoxy 90 Epoxy 0 0 V O PZT 4 Figure 6: 5-layered piezoelectric plate e 31 e 32 e 33 e 15 ε 11 ε 22 ε 33 PZT Epoxy Table 5: Independent piezoelectric and dielectric constants of PZT4 and epoxy resin (e ij in C/m 2, ε ii in nf/m) Lemma Electrical and dielectric properties of the 3 layers of the core of the plate being identical, they behave, under an electrical point of view, as a single layer. Lemma We thus take into account, for the 5-layered plate, a 3-layered type modelling, with an elastic core of epoxy resin. Proposition The electrostatic potential (20) is approximated as: ϕ(x α, z; t) = Q 2M (z) ϕ 2 M (54) 21

22 where: Q 2M (z) = [ (1 ξ 2 1 ) λ 1M,2M M + 1 ξ 2 1 (ξ 1 + 1) λ 2B,2M] χ 1 (z) + [ 1 ξ 2 2 (ξ 2 1) λ 2B,2M + (1 ξ 2 2 ) + 1 ξ 2 2 (ξ 2 + 1) λ 2B,2M] χ 2 (z) + [ 1 ξ 2 3 (ξ 3 1) λ 2B,2M + (1 ξ 2 3 ) λ 1M,2M] χ 3 (z) (55) Proof Symmetries of the problem yield: ϕ 2B (x α ; t) = ϕ 3B (x α ; t), ϕ 1M (x α ; t) = ϕ 3M (x α ; t) (56) The approximation of the electrostatic potential (20), (18) becomes thus: ϕ(x α, z; t) = [ (1 ξ 2 1 ) ϕ 1M (x α ; t) ξ 1 (ξ 1 + 1) ϕ 1T (x α ; t) ] χ 1 (z) + [ 1 2 ξ 2 (ξ 2 1) ϕ 2B (x α ; t) + (1 ξ 2 2 ) ϕ 2M (x α ; t) ξ 2 (ξ 2 + 1) ϕ 2B (x α ; t) ] χ 2 (z) + [ 1 2 ξ 3 (ξ 3 1) ϕ 2B (x α ; t) + (1 ξ 2 3 ) ϕ 1M (x α ; t) ] χ 3 (z) (57) The continuity of the uncoupled electrical displacement at layer interfaces can be written as: { 1 ε33,1 ϕ 1,3 (x α, z 1 ; t) = ε 2 33,1 ϕ 2,3 (x α, z 1 ; t) ε 2 33,1 ϕ 1,3 (x α, z 2 ; t) = ε 1 33,1 ϕ 3 (58),3 (x α, z 2 ; t) We thus have a linear system, which enables us to express ϕ 1M and ϕ 2B as functions of ϕ 2M, under the following form: { ϕ 1M = λ 1M,2M M ϕ 2 M ϕ 2B = λ 2B,2M M ϕ 2 M (59) The generalized mechanical unknowns are membrane displacements u α, the deflection w, and the transverse shear stresses γ α 0. The generalized electrical unknown is ϕ 2M. Assumptions In the same way as for the single-layered plate, the solution is searched under the form (167): 22

23 u 1 = A 1 e j ω t cos( π x 1 ) sin( π x 2 a u 2 = A 2 e j ω t sin( π x 1 ) cos( π x 2 a w = B e j ω t sin( π x 1 ) sin( π x 2 ) a b 0 γ 1 = C 1 e j ω t cos( π x 1 ) sin( π x 2 a 0 γ 2 = C 2 e j ω t sin( π x 1 ) cos( π x 2 a ) b ) b ) a ) a, ϕ 2M = Φ 2 e j ω t sin( π x 1 a ) sin(π x 2 b ) (60) which enables us to satisfy the simply supported boundary conditions (53). By substituting these expressions into the equations of motion given by equations (13), in conjunction with the boundary conditions (45), the constitutive law (31) and the displacement field (13), we obtain a linear system in A α, B, C α, Φ 1 : [K 5 ω 2 M 5 ] A 1 B C 1 Φ 2 Detail of coefficients of matrices K 5 and M 5 is given in [7]. = 0 (61) The system admits a non trivial solution if its determinant vanishes: det[k 5 ω 2 M 5 ] = 0 (62) A symbolic calculus tool (Mathematica or Maple) enables then us to calculate the free pulsations of the considered plate, and to compare them with Heyliger s analytical solution [25] (see 6). Model Exact solution a h a h = Table 6: Free pulsations of the 5-layered piezoelectric (in rad.s 1 ) 23

24 Figure 7 displays the variations, as a function of the global thickness coordinate z, of the normalized electrostatic potential: ϕ(z; t) = ϕ(x α, z; t) ϕ(x α = a 2, z = h 2 ; t) (63) 1 φ z h Figure 7: Normalized electrostatic potential, 5-layered plate, a = 4 h In gray: the exact solution. In black: our model. z h Figure 8 displays the variations, as a function of the global thickness coordinate z, of the normalized electric displacement: D 3 (z; t) = D 3 (x α, z; t) D 3 (x 1 = 0, x 2 = a 2, z = h; t) (64) z h D Figure 8: Normalized electric displacement, 5-layered plate, a = 4h. In gray: the exact solution. In black: our model. Figure 9 displays the variations, as a function of the global thickness coordinate z, of the normalized transverse shear stress: 24 z h

25 σ 13 (z; t) = σ 13 (x α, z; t) σ 13 (x 1 = 0, x 2 = a 2, z = h 2 ; t) (65) Σ z h Figure 9: Normalized transverse shear stress, 5-layered plate, a = 4 h. In gray: the exact solution. In black: our model. Figure 10 displays the variations, as a function of the global thickness coordinate z, of the normalized longitudinal shear stress: z h σ 11 (z; t) = σ 11 (x α, z; t) σ 11 (x α = a 2, z = 0; t) (66) z h Σ Figure 10: Normalized longitudinal shear stress, 5-layered plate, a = 4 h. In gray: the exact solution. In black: our model. z h 25

26 1.10 Applications of the piezoelectric plate model In the following, we present applications of the piezoelectric plate model; more results can be found in [7] Bimorph plate under mechanical loading Consider a rectangular plate, of width a, bimorph 1, supposed of infinite length, made of PZT4 ceramic, under cylindrical bending, simply supported, submitted to a force density p on its top face, in closed circuit (which means that the electrostatic potential on the top and bottom surfaces is equal zero): ϕ 1B = ϕ 3T = 0) (see figure 11). z p h 0 V O Figure 11: Bimorph plate under mechanical loading Proposition The plate being supposed of infinite length, mechanical and electrical quantities (stresses, strains, displacements, electric field, electrostatic potential) do not depend on x 2. 26

27 Assumptions Since the component U 2 of the displacement field does not play any part, we shall take: U 2 = 0 Data Material constants of PZT4 ceramic are given in tables 1, 2. Proposition The simply supported boundary conditions can be written as: w(x 1 = 0, z; t) = w(x 1 = a, z; t) = 0 (67) 1 A bimorph plate is a structure made of two identical materials, but with opposed polarization axes: e 3α 2 = e 3α 1, e 15 2 = e 15 1, e 24 2 = e 24 1 Assumptions The force density p is assumed to be simply sinusoidal: where p 0 = 0.05MP a. p (x α, z; t) = p 0 e j ω t sin( π x 1 a ) (68) Proposition The electrostatic potential (20) can be written as: where: ϕ(x α, z; t) = Q 1M (z) ϕ 1 M (69) Q 1M (z) = (1 ξ 1 2 ) χ 1 (z) (1 ξ 2 2 ) χ 2 (z) (70) Proof The antisymmetry of the problem yields: ϕ 2M = ϕ 1 M (71) Remark 1.6. The piezoelectric coefficients of both layers being identical, no continuity conditions are required for the electrical displacement. 27

28 The generalized mechanical unknowns are the membrane displacements u α, the deflection w, and the transverse shear stresses γ α 0. The generalized electrostatic unknown is ϕ 1M. Assumptions In the same way as for the single layered plate, the solution is searched as (167): u 1 = A 1 e j ω t cos( π x 1 ) sin( π x 2 a w = B e j ω t sin( π x 1 ) sin( π x 2 ) a b 0 γ 1 = C 1 e j ω t cos( π x 1 ) sin( π x 2 a b ) π x 1 a ) sin(π x 2 b ), ϕ 1M = Φ 1 e j ω t sin( ) a (72) which enables us to satisfy the simply supported boundary conditions (53). By substituting these expressions into the equations of motion given by equations (13), in conjunction with the boundary conditions (45), the constitutive law (31) and the displacement field (13), we obtain a linear system in A 1, B, C 1, Φ 1, of the form: K 2 A 1 B C 1 Φ 1 = B 2 (73) Detail of the coefficients of the matrices K 2 and of the vector B 2 is given in [7]. Figure 12 displays the variations, as a function of the global thickness parameter z, of the normalized electrostatic potential: ϕ(z; t) = ϕ(x α, z; t) ϕ(x α = a 2, z = h 2 ; t) (74) Figure 13 displays the variations, as a function of the global thickness parameter z, of the normalized transverse shear stress: σ 13 (z; t) = σ 13 (x α, z; t) σ 13 (x 1 = 0, x 2 = a 2, z = h 2 ; t) (75) 28

29 z h φ 1-1 Figure 12: Normalized electrostatic potential, bimorph plate under mechanical loading, a = 5 h. In gray: exact solutio. In black: our model. Σ z h Figure 13: Normalized transverse shear stress, bimorph plate under mechanical loading, a = 5 h. Figure 14 displays the variations, as a function of the global thickness parameter z, of the normalized longitudinal stress: σ 11 (z; t) = σ 11 (x α, z; t) σ 11 (x α = a 2, z = 0; t) (76) Figure 15 displays the variations, as a function of the global thickness parameter z, of the normalized longitudinal displacement: U 1 (z; t) = U 1 (x α, z; t) U 1 (x α = a 2, z = 0; t) (77) Bimorph plate with imposed potentials Consider a rectangular plate, of width a, bimorph, supposed of infinite length, made of PZT4 ceramic, under cylindrical bending, simply supported, submitted, on its top face, to an electrostatic potential +V, and, on its bottom face, to an electrostatic potential V (see figure 16). 29

30 z h Σ Figure 14: Normalized longitudinal stress, bimorph plate under mechanical loading, a = 5 h. In gray: exact solution. In black: our model z h U Figure 15: Normalized longitudinal displacement, bimorph plate under mechanical loading, a = 5 h. In gray: exact solution. In black: our model. Proposition The plate being supposed of infinite length, the mechanical and electrical quantities (stresses, strains, displacements, electric field, electrostatic potential), do not depend on x 2. Assumptions Since the component U 2 of the kinematic field does not play any part, we shall take: U 2 = 0 Data Material constants of the PZT4 ceramic are given in tables 1, 2. Proposition The simply supported boundary conditions yield: w(x 1 = 0, z; t) = w(x 1 = a, z; t) = 0 (78) Assumptions The potential V is assumed to be simply sinusoidal: V (x α, z; t) = V 0 e j ω t sin ( π x 1 ) a 30 (79)

31 z +V h O V Figure 16: Bimorph plate, with imposed potentials Proposition The electrostatic potential (20) can be written as: ϕ(x α, z; t) = [ 1 2 ξ 1 (ξ 1 1) V +(1 ξ 1 2 ) ϕ 1M (x α ; t)+ ] χ 1 (z) [ 1 2 ξ 2 (ξ 2 +1) V (1 ξ 2 2 ) ϕ 1M (x α ; t) ] χ 2 (z) (80) Proof The antisymmetry of the problem yields: ϕ 2M = ϕ 1 M (81) Remark 1.7. The dielectric coefficients of both layers being identical, no continuity conditions are required for the electrical displacement. The generalized mechanical unknowns are membrane displacements u α, w, and the transverse shears γ α 0. The generalized electrical unknown is ϕ 1M. 31

32 Assumptions As well as for the single-layered plate, the solution is searched as (167): u 1 = A 1 e j ω t cos( π x 1 ) sin( π x 2 a w = B e j ω t sin( π x 1 ) sin( π x 2 ) a b 1 γ 0 = C 1 e j ω t cos( π x 1 ) sin( π x 2 a b ) π x 1 a ) sin(π x 2 b ), ϕ 1M = Φ 1 e j ω t sin( ) a (82) which enables us to satisfy the simply supported boundary conditions (??). By substituting these expressions into the equations of motion given by equations (13), in conjunction with the boundary conditions (45), the constitutive law (31) and the displacement field (13), we obtain a linear system in A 1, B, C 1, Φ 1 : K 2 A 1 B C 1 Φ 1 = B 2 (83) The detail of coefficients of the matrices K 2 and of the vector B 2 is given in [7]. Figure 17 displays the variations, as a function of the global thickness parameter z, of the normalized electrostatic potential: ϕ(z; t) = ϕ(x α, z; t) ϕ(x α = a 2, z = h 2 ; t) (84) φ z h Figure 17: Normalized electrostatic potential, bimorph plate with imposed potentials, a = 4 h. In gray: exact solution. In black: our model. 32

33 Figure 18 displays the variations, as a function of the global thickness parameter z, of the normalized transverse shear stress: σ 13 (z; t) = σ 13 (x α, z; t) σ 13 (x 1 = 0, x 2 = a 2, z = h 2 ; t) (85) Σ z h Figure 18: Normalized transverse shear stress, bimorph plate with imposed potentials, a = 5 h. Figure 19 displays the variations, as a function of the global thickness parameter z, of the normalized longitudinal stress: σ 11 (z; t) = σ 11 (x α, z; t) σ 11 (x α = a 2, z = 0; t) (86) z h Σ Figure 19: Normalized longitudinal stress, bimorph plate with imposed potentials, a = 5 h. 33

34 2 Multilayered piezoelectric shells modelling We hereafter study the modelling of multilayered piezoelectric shells. Our theory is based on the same hybrid approach as previously: the electrical and mechanical continuity conditions at layer interfaces are satisfied, as well as the boundary conditions on the top and bottom surfaces of the shell. The accuracy of our theory is assessed through investigation of significant problems, for which an exact three-dimensional solution is known. 2.1 Introduction The modeling of piezoelectric shells mostly concerns cases attached to specific geometries (cylindrical, spherical). Toupin [48] studied the static response of a radially polarized spherical piezoelectric shell. Adelman et al.[13], [14] examined cases involving hollow piezoelectric cylinders. Sun et al. [45], Karlash [27] studied wave propagation in layered piezoelectric cylinders. Paul et al. [40], [41] examined free vibration problems. Siao et al. [44] ] proposed a semi-analytic model for layered piezoelectric cylinders taking into account a layerwise behavior of the composite. Analytic solutions for laminated piezoelectric cylinders were proposed by Mitchell et al. [35], Xu et al. [57], Heyliger [26], Dumir et al. [20], Drumheller et al. [19]. For this purpose, Drumheller [19] used classical shell theory for free vibrations of shells of revolution. Haskins et al. [23] proposed the development of electrical and mechanical quantities as expansions of the thickness variable. Tzou et al. [52] proposed the development of electrical and mechanical quantities as expansions of the thickness variable. It was done by Tzou et al. [53], this time with a shear-deformation theory. Other piezoelectric shell models and finite element approximations, based on single-layer models were also developped by Tzou et al. [54]. A Reissner-Mindlin shear-deformation shell finite element with surface bonded piezoelectric layers was developped by Lammering [29]. Koconis et al. [28] used a Ritz method for three-layered shells with embedded piezoelectric actuators. Tzou et al. [54] proposed a coupled theory where the piezoelectric shells are considered as a layerwise assembly of curvilinear solid piezoelectric triangular elements. 34

35 Heyliger et al. [26] developed a finite-element for laminated piezoelectric shells. Saravanos [43] used a coupled mixed theory for curvilinear composite piezoelectric laminates with the first-order shear deformation theory hypothesis and a layerwise approximation of the electrostatic potential, along with the corresponding finite element for piezoelectric shells. We presently extend our piezoelectric plate model to shells. As previously, we associate our displacement type approach, which is a " singlelayered " one, continuous at layer interfaces, to quadratic variations through the thickness of the electrostatic potential, also continuous. The transverse shear stresses, under a constant electrical field, as well as the electrical displacement, under a constant strain, are also continuous. Refinements of the membrane and transverse shear stresses are taken into account by means of trigonometric functions Also, the conditions at layer interfaces, where values of the electrostatic potential can be imposed, are satisfied. Finally, the piezoelectric boundary value problem is constructed using the consistant coupled constitutive law, in conjunction with the above displacements and electrostatic potential fields. The proposed piezoelectric shell model is evaluated for significant problems, for which the exact three-dimensional solution is known [26]. 2.2 Mechanical study Consider an undeformed laminated shell of constant thickness h. The space occupied by the shell will be denoted V. The boundary of the shell is the reunion of the upper surface S h, the lower surface S 0, and the edge faces A. consisting of an arrangement of a finite number N of piezoelectric layers. a denotes the length, b the width, h its thickness, and V the volume occupied by the shell (see figure 2.2). Notation. The frontier of the shell is constituted by the reunion of its bottom surface S 0, its top surface S h, and its lateral surface A. S i denotes the interface between the i th and (i + 1) th layers, and z i the distance between S 0 and S i. The reference surface coïncides with the bottom surface S 0. 35

36 Figure 20: The multilayered piezoelectric shell. Notation. The Einsteinian summation convention applies to repeated indices, where Latin indices range from 1 to 3 while Greek indices range from 1 to 2. 36

37 Notation. V h R R 1, R 2 S 0 S h S i z i z i0 A ( a i ) Covariant basis ( g i ) Covariant basis ( a i ) Contravariant basis Volume occupied by the shell Total thickness of the shell Radiius of curvature of the shell Main radii of curvature of the reference surface Bottom surface of the shell Top surface of the shell Bottom surface of the i th layer Distance between S 0 and S i Distance between S 0 and the mid surface of the i th layer Lateral surface of the shell ( g i ) Contravariant basis β δ α Kronecker symbol b α β Covariant components of the curvature tensor β b α Mix components of the curvature tensor (x i ) Curvilinear coordinatess L i Lamé coefficients s k, k = 1,..., 6 Strains σ k, k = 1,..., 6 Stresses C (i) (i) mnpq, or C KL Components of the elastic stiffness tensor under a constant electrical field Differentiation with respect to z i Covariant derivation with respect to x i. Differentiation with respect to time t δ Variational operator ϕ Electrostatic potential ϕ 1 B Electrostatic potential on S 0 ϕ N+1B Electrostatic potential on S h ϕ ib Electrostatic potential on S i ϕ im Electrostatic potential on the midsurface of the i th layer ϕ it Electrostatic potential sur S i+1 E l, l = 1,..., 3 Components of the electric field E D k, k = 1,..., 3 Components of the electric displacement (i) e kl Piezoelectric constants, under a constant strain, of the i th layer (i) ε kl Dielectric constants, 37 under a constant strain, of the i th layer (or components of the tensor of permittivities of the i th layer) ρ Mass density ε 0 Permittivity of vacuum 1

38 1 ε 0 = F / m Geometric considerations for shells A point M out of the reference surface being given, let us denote by P the point of the reference surface closest to M. Covariant base vectors ( a i ), ( g i ), and contravariant base vectors ( a i ), ( g i ), are defined by: a α = P,α, a3 = a1 a 2 a 1 a 2, ( a1 a 2 ) a3 > 0 (87) { aα a β = δ α β a 3 = a 3 (88) gi = M,i, ( g1 g 2 ) g3 > 0 (89) { gα g β = δ α β g 3 = g 3 (90) Thus: P = M + z a 3 (91) Notation. Let us introduce: a αβ = a α a β, g αβ = g α g β (92) Proposition 2.1. aβ = a αβ a β, a β = a αβ a β (93) gα = µ β α a β = g αβ g β, g α = µ α β 1 a β = g αβ g β, g 3 = a 3 = a 3 (94) 38

39 Definition 2.1. The mixed components of the shifter tensor are given by: µ α β = δ α β z b α β (95) The covariant components of the curvature tensor are given by: b αβ = a α,β a 3 (96) The mixed components of the curvature tensor are given by: b α β = a 3,β a α (97) Assumptions 2.2. In the following, the curvilinear coordinates (or shell coordinates) are assumed to be orthogonal, and are such that the curves x 1 = constant, x 2 = constant are lines of curvature on the reference surface. The curves z = constant are straight lines perpendicular to the surface S 0. R 1, R 2 denote the principal radii of curvature of the reference surface. Proposition 2.3. The distance between two points P (x 1, x 2, 0) and P (x 1 + dx 1, x 2 + dx 2, 0) of the reference surface is given by: ds 2 = α 1 2 dx α 2 2 dx 2 2 where α 1 and α 2 are the coefficients of metrics, given by: (98) α l 2 = ( P x l ) ( P x l ) l = 1, 2 (99) Proposition 2.4. The distance between two points M(x 1, x 2, x 3 ) and M (x 1 + dx 1, x 2 + dx 2, x 3 + dx 3 ), out of the reference surface, is given by: ds 2 = L 1 2 dx L 2 2 dx L 3 2 dx 3 2 where L 1, L 2 and L 3 denote the Lamé coefficients, given by: (100) L 1 = α l ( 1 + z R 1 ), L2 = α l ( 1 + z R 2 ), L3 = 1 (101) 39

40 2.3 Kinematic assumptions Assumptions 2.5. The displacement field U of a point M(x α, z) of the shell, is defined by its components (U α, U z ) in the covariant basis ( g α, g 3), approximated under the form: { Uα = u α + z η α + f(z) ψ α + g(z) γα 0 + N 1 m=1 (z z m) u mα H(z z m ) U z = w (102) where: f(z) = h π sin ( π z ) h, g(z) = h π cos ( π z ) h (103) where H denotes the Heaviside step function. As in the previous part, the use of the sine and cosine functions can be justified as in Touratier [49], by a discrete-layer approach, from the three-dimensional modelling of Cheng [16] for thick plates. The u α are membrane displacements, the γ 0 α the components of the transverse shear stress at z = 0, w the deflection, ψ α and u mα a priori unknown functions, which will be determined thanks to the conditions at layer interfaces as well as on the top and bottom surfaces. 2.4 Uncoupled constitutive law Let us denote by C mnkl (i) (or C KL (i), double indexation) the components of the elastic stiffness tensor, under a constant electric field, of the i th layer, and by s kl the components of the stress tensor (or s I, double indexation). Assumptions 2.6. We use, in the following, the hypothesis of small pertubations,which yields: where: s kl = V k l + V l k 2 (104) V α β = µ ν α [ U ν β b νβ w ] V α 3 = µ ν α U ν,3 (105) V 3 3 = U 3,3 40

41 Remark 2.1. Equations (105) yield: V α β = µ ν α [ u ν β + z η ν β + f(z) ψ ν β + g(z) γ 0 ν β + N 1 m=1 (z z m) u mν β H(z z m ) b νβ w ] V α 3 = µ ν α [η ν + f (z) ψ ν + g (z) γαν 0 + N 1 m=1 u mν H(z z m ) ] V 3 α = w α + b ν α [u ν + z η ν + f(z) ψ ν + g(z) γ 0 ν + N 1 m=1 (z z m) u mν H(z z m )] V 3 3 = w,3 (106) Especially: s α3 = 1 [ V 2 α 3 { + V 3 α ] = 1 2 µ ν α [η ν + f (z) ψ ν + g (z) γαν 0 + N 1 m=1 u mν H(z z m ) ] + w α + b ν α [u ν + z η ν + f(z) ψ ν + g(z) γ 0 ν + N 1 m=1 (z z m) u mν H(z z m )] } The uncoupled constitutive law yields: i.e., under double indexation: (107) σ mn (i) = C mnkl (i) s kl (108) σ (i) d J = (i) CJK s K (109) σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 = C s 1 s 2 s 3 s 4 s 5 s 6 (110) 2.5 Boundary conditions i. Conditions on the top and bottom surfaces: Proposition 2.7. The zero value of the transverse shear stress on S 0 and S h, under a constant electric field, yields: and: η α = ψ α w α b ν α [u ν + h π γ0 ν] (111) 41

42 N 1 ψ α = d β α γ 0 β + f m βα u mβ (112) m=1 where: { [ d β α ] = [ h b β α 2 δ β α ] 1 [ 2 h [ fm β π bβ α ] α ] = [h b β α 2 δ β α] 1 [δ β α b β α z m ] (113) Proof. The nullity of the transverse shear stress on S 0 and S h, under a constant electric field, can be written as: which yields: { (1) σ d 6 α = 0 (N) σ d 6 α = 0 (114) { sα3 (z = 0) = 0 s α3 (z = h) = 0 (115) i.e., according to (107): and: s α3 (z = 0) = 1 2 [ δ ν α [η ν + h π ψ ν ] + w α + b ν α [u ν + h π γ0 ν] ] (116) s α3 (z = h) = 1 2 since: N 1 [ µ ν α [η ν ψ ν + m=1 u mν ]+w α +b ν α [u ν +h η ν h N 1 π γ0 ν+ m=1 (h z m ) u mν ] ] (117) f(0) = g (0) = f(h) = g (h) = 0, f (0) = f (h) = 1, g(0) = g(h) = h π (118) µ ν α = δ ν α z b ν α (119) 42

43 we thus have: and: s α3 (z = 0) = 1 2 [ ηα + ψ α + w α + b ν α [u ν + h π γ0 ν] ] (120) s α3 (z = h) = 1 2 N 1 [ µ ν α [η ν ψ ν + m=1 u mν ]+w α +b ν α [u ν +h η ν h N 1 π γ0 ν+ m=1 (h z m ) u mν ] ] (121) s α3 = 1 2 [ V α 3 + V 3 α ] { = 1 2 µ ν α [η ν + f (z) ψ ν + g (z) γν 0 + N 1 m=1 u mν H(z z m ) ] + w α + b ν α [u ν + z η ν + f(z) ψ ν + g(z) γ 0 ν + N 1 m=1 (z z m) u mν H(z z m )] } (122) Hence: η α = ψ α w α b ν α [u ν + h π γ0 ν] (123) and: 2 ψ α + b ν α ψ ν n 2 h N 1 π γ0 ν + (δ ν α b ν α z m ) u mν = 0 (124) which can be written as: m=1 where: { ψ α = d β α γ 0 β + N 1 m=1 f m β α u mβ (125) [ d β α ] = [ h b β α 2 δ β α ] 1 [ 2 h π bβ α ] [ fm β α ] = [h b β α 2 δ β α] 1 [δ β α b β α z m ] (126) By substituting these expressions in (123), we deduce: η α = d β α γ 0 β N 1 m=1 f m β α u mβ w α b ν α [u ν + h π γ0 ν] (127) 43

44 We then have, thanks to (122): { s α3 = 1 2 [ µ ν α + z b ν α ] η ν + [ µ ν α f (z) + b ν α f(z) ] ψ ν + [µ ν α g (z)b ν α g(z) ] γαν 0 + N 1 m=1 [µν α + } b ν α (z z m ) ] u mν H(z z m ) + w α + b ν α u ν (128) i.e.: { s α3 = 1 2 [ µ ν α + z b ν α ] { d β ν γ 0 β N 1 m=1 f m β ν u mβ w ν b λ ν [u λ + h π γ0 λ]} + [ µ ν α f (z) + b ν α f(z) ] { d λ ν γ 0 λ + N 1 m=1 f m λ ν u mβ } + [µ ν α g (z)b ν α g(z) ] γν 0 + N 1 m=1 [µν α + } b ν α (z z m ) ] u mν H(z z m ) + w α + b ν α u ν ii. Conditions at layer interfaces: (129) The continuity of the uncoupled transverse shear stress between the i th and (i + 1) th layer can be written as: σ 6 α (i) d (x α, z i ) = σ 6 α (i+1) d (x α, z i ) (130) Thanks to (129), we then obtain a linear system of N 1 equations, which enables us to express the u mα, m = 1,..., N 1 as functions of the transverse shears γ 0 α: u mα = λ mα γ 0 α (131) where the λ mα are real constants, given by the resolution of the latter system. 2.6 Final form of the displacement field Proposition 2.8. The displacement field U of any point M(x α, z) of the structure is given by: { Uα = µ β α u β z w α + h β α γ 0 β (132) U z = w 44

45 where the h β α are functions of the global thickness variable z, given by: h β α(z) = g(z) δ β α z h N 1 { } π bβ α +[f(z) z] d β α + f (m) βα + (z z m) H(z z m ) δ β α λ mβ m=1 (133) This displacement field has been developed in [12], [5], [3]. 2.7 Electrical study; the linear piezoelectric constitutive law This section refers to the same results as in the case of the plate (see above) 2.8 The two-dimensional boundary-value problem Variational formulation Hamilton s Principle yields: t 0 { {σ i δs i + D i δϕ i }dv + V V { f v δ U + W δϕ} dv + fs δ U da + A } (p 0 p h ) ds dt = 0 S 0 (134) δ being a variational operator, f v the volumic density of body forces, f s the surface density of body forces on the lateral surface of the shell, p 0 and p h the prescribed components of traction on the top and bottom surfaces, and W the density of electric forces. Notation. µ denotes the value of the determinant of shifter tensor [µ β α] at z = 0. Let us ntroduce: i. the generalized stresses: αβ N 1 αβ N 2 N 3 = h 0 α N 4 α N 5 α N 6 = h 0 {C λj s j e 3α E 3 } µ ν λ µ α ν µ dz = h 0 C 66 s 6 (1 δ λβ ) µ ν λ µ α ν µ dz [ ] {Cαj s j e 3α E 3 } µ ν α b να + C 66 s 6 (1 δ αβ ) µ ν α b νβ µ dz = h [ ] C6 λ,6 λ s 0 6 λ,6 λ e k,6 λ E k { µ ν λ h α ν,3 + b ν λ h α ν } µ dz = h 0 {C λj s j e 3α E 3 } µ ν λ h α ν(z) µ dz = h 0 C 66 s 6 (1 δ λβ ) µ ν λ h α ν(z) µ dz (135) 45

46 { ii. the generalized momentums: N ib = h 0 E 3 ε 33 Q ib (z) µ dz, i I N jm = h 0 E 3 ε 33 Q jm (z) µ dz, j J (136) { αβ M 1 αβ M 2 = h [ ] 0 { Cαj s j e 3α E 3 } µ ν α + C 66 s 6 νβ (1 δ αβ ) µ ν α z µ dz = h C 0 66 s 6 (1 δ αβ ) z µ dz (137) { M ibα = h {E 0 α ε αα + e kα s α } Q ib (z) µ dz, i I M jm α = h {E 0 α ε αα + e kα s α } Q im (z) µ dz, j J (138) iii. the generalized external mechanical forces : 1β F ν α = h f 0 ν α µ β α µ dz 2 F ν α = h f 0 ν α z µ dz 3β F ν α = h f 0 ν α h β α(z) µ dz 3 F ν = h f 0 ν α µ dz P = p 0 p h iv. the generalized external electrostatic forces : { W ib = h W 0 QiB (z) µ dz, i I W jm = h W 0 QiB (z) µ dz, j J (139) (140) iv. the inertia terms: I 1 = h ρ µ dz 0 I 2 = h ρ z µ dz 0 I 3 = h ρ h 0 α(z) µ dz I 4 = h ρ 0 z2 µ dz I 5 = h ρ z h 0 α(z) µ dz I 6 = h ρ z h 0 α 2 (z) µ dz (141) 46

47 Proposition 2.9. The equations of motion are given by: αβ αβ N 1 β + N 2 β = I 1α ü α I 2 ẅ α + I 3α γ α 0 αβ αβ M 1 αβ + M 2 αβ = I 2α ü α α I 4 ẅ αα + I 5α γ α α 0 + I 1 ẅ α αβ N 5 α + N 6 β + N α 4 = I 3α ü α I 5 ẅ α + I 6α γ α 0 N ib + M ibα α = 0, i I N jm + M jmα α = 0, j J (142) Proof The equations of motion are deduced from Hamilton s Principle, in conjunction with the kinematics (132), including the constitutive law given by equations (27), by integration through the thickness of the shell. Proposition The boundary conditions leading to a " regular problem " are: N αβ 1 n β + N αβ 2 n β = F ν 1αβ ou δu α = 0 αβ αβ 2 M 1 β + M 2 β = F ν α ou δw = 0 N αβ 4 n β = F ν 3αβ or δγ 0 α = 0 M αβ 1 n β + M αβ 2 n β = F ν3 ou δw α = 0 N ibα n α = W ib ou δϕ ib = 0, i I M jm α n α = W jm ou δϕ jm = 0, j J (143) Proof The equations of motions can be derived from Principe de Hamilton s Principle, the kinematics (132), and the constitutive law (27), through integration on the thickness of the shell. 2.9 Numerical validation of the piezoelectric shell model Associated plate model The plate being a degenerated shell, our shell model is, in a first time, validated by the related plate model Free vibrations of an orthotropic cylindrical panel Consider an orthotropic cylindrical panel, supposed of infinite length, made of PZT4 ceramic, under cylindrical bending, simply supported, submitted to a 47