This two-part contribution presents a novel and efficient method to analyze the two-dimensional (2-D) electromechanical fields of a piezoelectric layer bonded to an elastic substrate, which takes into account the fully coupled electromechanical behavior. In Part I, Hellinger-Reissner variational principle for elasticity is extended to electromechanical problems of the bimaterial, and is utilized to obtain the governing equations for the problems concerned. The 2-D electromechanical field quantities in the piezoelectric layer are expanded in the thickness-coordinate with seven one-dimensional (1-D) unknown functions. Such an expansion satisfies exactly the mechanical equilibrium equations, Gauss law, the constitutive equations, two of the three displacement-strain relations as well as one of the two electric field-electric potential relations. For the substrate the fundamental solutions of a half-plane subjected to a vertical or horizontal concentrated force on the surface are used. Two differential equations and two singular integro-differential equations of four unknown functions, the axial force, N, the moment, M, the average and the first moment of electric displacement, D0 and D1, as well as the associated boundary conditions have been derived rigorously from the stationary conditions of Hellinger–Reissner variational functional. In contrast to the thin film/substrate theory that ignores the interfacial normal stress the present one can predict both the interfacial shear and normal stresses, the latter one is believed to control the delamination initiation.

A theory is developed to predict the evolution of transverse ply cracking in a composite laminate as a function of the underlying statistical fracture toughness and the applied load. Th e instantaneous formation of a matrix crack spanning both the ply thickness and the ply width is assumed to be governed by the energy criterion associated with the material fracture oughness, Γ, at the ply level. Assume multiple matrix fractures occur quasi statically and sequentially such that the ply cracks form one after another under the constant external load imposed on the specimen. Th e number of cracks, n, within the gauge length, 2L, is a discrete random variable for a given applied load, σ, because the fracture toughness varies with the location of fractures in a given specimen as well as from specimen to specimen. Th e probability function f (n, σ, L) of the discrete random variable, n, is determined from the fracture toughness distribution and the solution for the potential energy release rate. Consequently, the distribution of the crack density, dn-n/2L, is obtained. Finally, the mean crack density is formulated as a function of the applied load.

This two-part contribution presents a novel and efficient method to analyze the two-dimensional (2-D) electromechanical fields of a piezoelectric layer bonded to an elastic substrate, which takes into account the fully coupled electric and mechanical behaviors. In Part I, we have obtained a system of governing integro-differential equations for the structure via a variational principle. This part presents a numerical solution algorithm of the integro-differential equations and the numerical results of some applications. A numerical algorithm for solving the system of four integrodi erential equations with strongly singular kernels is developed. The convergence of the numerical algorithm is discussed. The numerical results suggest that the fully coupled electromechanical analysis is helpful for a better understanding of the performance of the piezoelectric sensor and actuator. The interfacial normal stress is much higher than the interfacial shear stress, suggesting that the interfacial normal stress causes a delamination initiation.

this paper, the two-dimensional electromechanical coupling problems that a piezoelectric patch of finite size bonded to an elastic substrate are considered. A subdivision model that the single physical piezoelectric layer is mathematically divided into a number of thinner layers is proposed to analyze the electromechanical responses of the structures. Within each virtual sub-layer of the piezoelectric patch the electric displacement and normal stress in the axial direction are assumed to be linear functions of the thickness coordinate. Hellinger-Reissner variational principle for elasticity is extended to the systems of piezoelectric multi-materials. The governing equations that comprise one-dimensional di erential equations and integro-differential equations are rigorously derived from the stationary conditions of the variational functional along with substitution of the assumed electromechanical fields. The subdivision model satisfies all mechanical and electric continuity conditions across the virtual interfaces and the physical interface of piezoelectrics/substrate. The numerical solutions of the governing equations are conducted, and the convergence of the subdivision model is demonstrated.

The elastic properties of symmetric laminates containing a cracked mid-layer are predicted by using the equivalent constraint model (ECM) proposed recently by the authors. In accordance with the ECM methodology the symmetric laminates subjected to an in-plane tension/shear load are subdivided into such sublaminates as the cracked centrallayer, two nearest laminae and two homogeneous layers smeared from the remaining plies (if applicable). The