Wrinkle surface instability of an inhomogeneous elastic block with graded stiffness

Abstract

Surface instabilities have been studied extensively for both homogeneous materials and film/substrate structures but relatively less for materials with continuously varying properties. This paper studies wrinkle surface instability of a graded neo-Hookean block with exponentially varying modulus under plane strain by using the linear bifurcation analysis. We derive the first variation condition for minimizing the potential energy functional and solve the linearized equations of equilibrium to find the necessary conditions for surface instability. It is found that for a homogeneous block or an inhomogeneous block with increasing modulus from the surface, the critical stretch for surface instability is 0.544 (0.456 strain), which is independent of the geometry and the elastic modulus on the surface of the block. This critical stretch coincides with that reported by Biot (1963 Appl. Sci. Res. 12 , 168–182. ( doi:10.1007/BF03184638 )) 53 years ago for the onset of wrinkle instabilities in a half-space of homogeneous neo-Hookean materials. On the other hand, for an inhomogeneous block with decreasing modulus from the surface, the critical stretch for surface instability ranges from 0.544 to 1 (0–0.456 strain), depending on the modulus gradient, and the length and height of the block. This sheds light on the effects of the material inhomogeneity and structural geometry on surface instability.