Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials

Abstract

This paper is concerned with the investigation of the effect of material inhomogeneity on the decay of Saint–Venant end effects in functionally graded linear piezoelectric materials. Saint–Venant's principle and related results for elasticity theory have received considerable attention in the literature but it is only recently that analogous issues in piezoelectricity have been investigated. The current rapidly developing smart–structures technology provides motivation for the investigation of such problems. We examine the decay of Saint–Venant end effects in the context of anti–plane shear deformations for linear inhomogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations (PDEs) of equilibrium are shown to be a coupled system of second–order PDEs with variable coefficients for the mechanical displacement u and electric potential φ . The traction boundary–value problem with prescribed surface charge is formulated as an oblique derivative boundary–value problem for this elliptic system. The axial decay of solutions on a semi–infinite strip subjected to non–zero boundary conditions only at the near end is investigated. This analysis is carried out for a subclass of special graded materials, namely, laterally inhomogeneous solids. Two specific piezoceramics that have received widespread application are considered. For the first (corresponding to the hexagonal 6 mm crystal–symmetry class), it is shown that the mechanical and electrical problems essentially decouple, and that the decay rates for mechanical and electrical Saint–Venant end effects coincide and are equal to that of linear inhomogeneous isotropic anti–plane elasticity, for which results on the decay rate have been obtained previously. For the second class of materials (corresponding to the cubic 4∼3m symmetry) the situation is quite different. The boundary–value problem involves a full coupling of mechanical and electrical effects. Energy decay estimates using differential–inequality methods are used to obtain an explicit estimated decay rate (a lower bound for the actual decay rate) in terms of a single dimensionless piezoelectric coupling constant d and the decay rate for an associated linear inhomogeneous isotropic anti–plane shear elasticity problem. For fixed material inhomogeneity, the decay rate is shown to be monotone decreasing with increasing values of d In the limit as d→0 we recover the purely mechanical case. Thus, for this class of functionally graded materials, piezoelectric end effects are predicted to penetrate further into the strip than their functionally graded isotropic elastic counterparts , confirming recent results obtained in other contexts in linear piezoelectricity. For both classes of materials, the influence of material inhomogeneity on the decay rate is reflected solely by the shear modulus of an associated inhomogeneous isotropic anti–plane shear elasticity problem. It is shown that, for both material classes, material inhomogeneity has a significant influence on the decay of end effects.