Saint-Venant End Effects in Antiplane Shear for Functionally Graded Linearly Elastic Materials

Abstract

The purpose of this research is to investigate the influence of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. The work is motivated by the recent research activity on functionally graded materials (FGMs) i.e., materials with continuously varying properties tailored to satisfy specific engineering applications. In the framework of antiplane shear deformations of a semi-infinite strip, the mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to a Dirichlet boundary value problem for a second-order linear elliptic partial differential equation with variable coefficients. In previous work, the variable coefficient (the shear modulus) was assumed to be strictly positive on the closed semi-infinite strip. The authors relax this assumption to allow for the shear modulus to be zero on either long side of the strip. Thus, the results are applicable to FGMs that are graded continuously from zero shear modulus at one side of the strip. Lower bounds for the rate of decay of end effects are obtained, which allow for assessing the influence of material inhomogeneity. The results are illustrated using several specific material models.