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.

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.

The local cyclic plasticity of the interface around a broken fiber in ductile matrix composites under the in-phase and out-of-phase thermomechanical fatigue (TMF) loads is analyzed by using the single-fiber shear-lag model. The elastic, perfectly-plastic shear stress-strain relation is used to model the thermomechanical behavior of the fiber/matrix interface. It is shown that the alternating plastic shearing of the interface takes place under an appropriate combination of mechanical stress and thermal load. In the stress versus temperature diagram the so-called cyclic plasticity zone is identified. A new parameter, i.e. the cyclic plasticity length, L s, is found which is smaller than the yield length, Ls, caused by monotonic loading. The closed-form solutions for L s, Ls, the fiber stress profiles and the cyclic plastic shear strain range, p, are obtained. L s and p increase for both the in-phase and out-of-phase TMF conditions as the mechanical load and/or thermal load increase. The in-phase condition produces a higher plastic shear strain range than the out-of-phase condition does. The solutions obtained may be used for modeling fiber/matrix debonding caused by the fiber breakage under TMF fatigue loading.