Saint-Venant's Principle in Linear Piezoelectricity

Toupin's version of Saint-Venant's principle in linear elasticity is generalized to the case of linear piezoelectricity. That is, it is shown that, for a straight prismatic bar made of a linear piezoelectric material and loaded by a self-equilibrated system at one end only, the internal energy stored in the portion of the bar which is beyond a distance s from the loaded end decreases exponentially with the distance s. Introduction Mathematical versions of Saint-Venant's principle in linear elasticity due to Stemberg, Knowles, Zanaboni, Robinson and Toupin have been discussed by Gurtin [ 1 ] in his monograph. Later developments of the principle for Laplace's equation, isotropic, anisotropic, and composite plane elasticity, three-dimensional problems, nonlinear problems, and time-dependent problems are summarized in the review articles by Horgan and Knowles [2] and by Horgan [3]. In this paper we prove an analogue of Toupin's version of Saint-Venant's principle for linear piezoelectricity. For a linear elastic homogeneous prismatic body of arbitrary length and crosssection loaded on one end only by an arbitrary system of self-equilibrated forces, Toupin [4] showed that the elastic energy U(s) stored in the part of the body which is beyond a distance s from the loaded end satisfies the inequality U(s) <<. U(O) e x p [ ( s l)/sc(1)]. (1) The characteristic decay length so(l) depends upon the maximum and the minimum elastic moduli of the material and the smallest nonzero characteristic frequency of the free vibration of a slice of the cylinder of length 1. By using an estimate due to Ericksen [5] for the norm of the stress tensor in terms of the strain energy density, one can show that so(1) depends on the maximum elastic modulus and not on the minimum elastic modulus. Inequalities similar to (1) have been obtained by Berglund [6] for linear elastic micropolar prismatic bodies and by Batra for non-polar and micropolar linear elastic helical bodies [7, 8] and prismatic bodies of linear elastic materials with microstructure [9]. Herein we prove a similar result for a straight prismatic body made of a linear piezoelectric material. 210 R.C. BATRA AND J. S. YANG We assume that the cross-sections are materially uniform in the sense that one cross-section can be obtained from the other by a rigid body motion. Thus the material properties are independent of the axial coordinate of the point. Ericksen [5] has discussed material uniformity in more general terms. Equations for Linear Piezoelectricity Let the finite spatial region occupied by the piezoelectric body be V, the boundary surface of V be S, the unit outward normal of S be nl, and S be partitioned as Su i..J S T = S E [..J SD -~ S, Su 0 S T = S E 0 S O ---0. (2) Physically, S~, ST are, respectively, parts of the boundary S on which mechanical displacements and tractions are prescribed. SE is the part of S which is in contact with a metal or electrode, hence the tangential electric field vanishes on it, and SD the part of S on which the surface electric charge is prescribed which is usually zero for dielectrics. Throughout this paper, a repeated index implies summation over the range of the index, and a comma followed by an index j stands for partial differentiation with respect to x j . The governing equations and boundary conditions for static linear piezoelectricity in rectangular Cartesian coordinates are [10] Tij,i = O, Di,i = O, ~i jkEk, j = 0 in V, OH Ti j -OSij -CijklSkl -e k i jEk in V, OH Di eiktSkl + e ikEk in V, (3) OE~ Si j = 1⁄2(uj,i + ul , j ) in V, ui = ui on S~, n iT i j = t j on ST, ~ i j k n j E k = 0 on SE, n i D i = O o n S D , where ui is the mechanical displacement, Tij the stress tensor, Sij the strain tensor, Ei the electric field vector, Di the electric displacement vector, and Cijk the permutation tensor, fii and t-i are the prescribed boundary mechanical displacement and traction vectors, respectively. H(S, E) is the electric enthalpy function given by H = 1⁄2CijkzSiiSkt 1⁄2 e i j E i E j e i j k E i S j k , (4) where cijkl are the elastic moduli, ~ij the electric permittivity, and eijk the piezoelectric moduli. The material constants have the following symmetry properties: cijkt = cjikt = cktlj, (5) eijk = eikj , Eij = Cji, SAINT-VENANT'S PRINCIPLE IN LINEAR PIEZOELECTRICITY 211 and Cijkl, eij are positive definite, i.e. for any nonzero symmetric tensor aij and vector bi cijklaijakl > O, eijbibj > 0. (6) Usually, in (3), a scalar potential ¢ is introduced by Ei = ¢ , i so that the e q u a t i o n EijkEk,j = 0 is identically satisfied and the boundary condition on SE becomes ¢ = c where c is a constant. It can be easily seen that H(0, E) ~< 0 and H(S, 0) 1> 0. Hence H is indefinite. Therefore, we introduce the internal energy density W by the following Legendre transform: W = W(S, D) = H + EiDi, (7) which generates the constitutive relations OW OW T i j OSi j ' E i ODi" (8) That W is positive definite can be seen from the following: W = H + E i D i ..~ lc i jk lSi jSkl -le l jEiEj -eijkEiSjk + Ei(eijEj + eijkSjk) : l c i j k lS i jSk l "[le l jEiEj >10. (9) Since we will use the internal energy W(S, D) which has Di as the electric variable, instead of a scalar potential ¢, it will be more convenient to use the vector potential Ck defined by 1 E Di = ~ ijkCk,j, (10) which satisfies the equation Did = 0 identically. For the uniqueness of the vector potential Ck, we may add the gauge condition ~bk,k = 0. Corresponding to the vector Di, we introduce an equivalent anti-symmetric electric displacement tensor 79ij by Di = leijkDjk, which, when substituted into (10), gives