In 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 differential 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.

In this paper, structural configuration of layered Li-ion battery electrode plates is evaluated by analytically formulating the diffusion induced stress. Both symmetric electrode and asymmetric bilayer electrode are discussed. The thickness ratio and the modulus ratio of current collector to active plate are analytically identified to be the important influence parameters on the stress. Applying a material with smaller elastic modulus for current collector could reduce the peak stresses in both current collector and active plate. Increasing the thickness of current collector would reduce the stress in itself while promote the stress in active plate. Therefore, from mechanical viewpoint on designing an electrode, the material for current collector should be as soft and flexible as possible. And the thickness of current collector should be in an appropriate range. Basically, it should be as small as possible on the precondition that the mechanical strength is satisfied. Finally, effects of three charging conditions, i.e. uniform, galvanostatic, and potentiostatic, on the diffusion induced stress is discussed. It is found the maximum stresses for three cases are linear to the total amount of intercalated lithium ions. Based on the stresses, an optimized charging operation, i.e. first galvanostatic followed by potentiostatic, is suggested.