Interaction of Two Coaxial Penny-Shaped Cracks Near an Arbitrarily Graded Interface in Functionally Graded Materials: Exact and Approximate Solutions

Abstract

Multiple cracks interaction is an important topic in fracture mechanics. The related solutions are helpful to understand the failure process and the toughening mechanism of brittle materials. Previous works on the topic were most for homogenous material. In this paper, we extend the analysis and examine the problem of interaction of two coaxial penny-shaped cracks near an arbitrarily graded interface in functionally graded materials (FGMs). The cracks are modelled as circular edge dislocation loops. An efficient dislocation solution for FGMs and Fredholm integral equation technique are used to solve the crack problem. Both exact solution using a system of integral equations and approximate solution by virtue of Kachanov’s method are presented. Unlike most existing analytical treatments to the crack problems in FGMs with the assumption of special gradation, i.e., graded shear modulus according to special functions and constant Poisson’s ratio, the present method is more flexible since it can consider arbitrarily graded shear modulus and Poisson’s ratio. The validity of the present solutions is checked by comparing to existing results in literatures for two stacked penny-shaped cracks in homogenous material and a penny-shaped crack near a graded interface with exponentially graded shear modulus. Finally, a practical example of double cracks interaction in a real epoxy-glass FGM with measured data of material properties is considered. The error due to the assumption of special gradation is also discussed.