Journal of Elasticity 38: 209-218, 1995. 209 © 1995 Kluwer Academic Publishers. Printed in the Netherlands. Saint-Venant's Principle in Linear Piezoelectricity R. C. BATRA 1 and J. S. YANG 2 1Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, USA 2Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, iVY 12180-3590, USA Received 28 July 1994 Abstract. Toupin's version of Saint-Venant's principle in linear elasticityis generalized to the case of linear piezoelectricity.That is, it is shownthat, 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 Stem- berg, 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 cross- section 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) exp[-(s - l)/sc(1)]. (1) The characteristic decay length so(l) depends upon the maximum and the min- imum elastic moduli of the material and the smallest nonzero characteristic fre- quency 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.