ENERGY PRINCIPLES IN DYNAMIC THEORY OF PIEZOELECTRIC MATERIALS WITH VOIDS
微孔压电弹性动力学的能量原理

The dynamic theory of piezoelectric materials with voids is intended for applications to natural and artifical materials with distributed voids, some bioengineering materials and intelligent materials. Therefore, it plays an important role in the development and application of modern new materials. But the energy principles in dynamic theory of piezoelectric materials with voids, which the principle of virtual work, the reciprocal theorem and various variational principles are not yet established systematically. According to the basic idea of classical yin-yang complementarity and modern dual- complementarity, in a simple and unified way proposed by Luo6,7], the energy principles in dynamic theory of piezoelectric materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of piezoelectric materials with voids, but also to derive systematically the complementary functionals for eleven-field, nine-field, six-field and three-field simplified Gurtin-type variational principles by the generalized Legendre transformations given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. In this paper first applying the principle of virtual work, one may obtain the reciprocal theorem in dynamic theory of piezoelectric materials with voids avoiding both the use of the Laplace transform and the incorporation of the initial conditions into the equations of motion. Obviously this method given in this paper is even simpler and directer than the conventional method. The energy principles (the principle of virtual work, the reciprocal theorem and various simplified Gurtin-type variational principles) established in this paper are an important part of dynamic theory of piezoelectric materials with voids. Obviously, the simplified Gurtin-type variational principle can fully characterize the initial-boundary-value problem of this dynamics. Consequently, the energy principles proposed in this paper will be of great value both in theoretical studies and in the establishment of various approximate methods and approximate engineering theories.