13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- structure. By using Fourier transforms to reduce electro-elastic boundary value problem to the solutions of integral equations, Ye and He [12] solved the problem of electric field concentrations of a pair of parallel electrodes arrayed in one plane. Finite element method was chosen to show the effect of piezoelectric coupling on interlaminar stresses and electric field strengths near the free edge by Artel and Becker [13]. In addition, an analytical solution was also developed to determine the state variables of piezoelectric elasticity in the vicinity of free edges using a layerwise displacement theory by Mirzababaee and Tahani [14]. In this paper, on the basis of 3D elasticity and piezoelectricity, an exact analytical solution that satisfies both mechanical and electric boundary conditions is established by considering continuity of displacements, transverse stresses, electric potentials and vertical electric displacements across interfaces between different materials. 2. Fundamental State Space Method Formulation 2.1. State space equations for cross-ply piezoelectric plate Figure 1. Geometry of a piezoelectric laminated plate The rectangular piezoelectric laminated plate is subjected to a uniform constant axial strain ε0 and it is assumed to have length a, width b, and uniform thickness h (Fig. 1). The principle elastic directions of plate coincide with the axes of the chosen rectangular coordinate system and full coupled three-dimensional piezoelectric-elastic constitutive relations of orthotropic piezoelectric lamina are given { } [ ]{ } [ ] { }, T C e E σ ε = − (1) { } [ ]{ } [ ]{ }, D e E ε = + ∈ (2) Where {σ}, {ε}, {E} and {D} are, respectively, stress, strain, electric field, and electric displacement vectors. [C], [e] and [є] are elastic, piezoelectric and electric permittivity constants, respectively. Explicit forms of Eqs. (1) and (2) are given as follows:
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