Interfacial fracture of a radially inhomogeneous elastic bimaterial

Abstract

Anti-plane strain crack problems in radially inhomogeneous materials are considered; the model is used as a simple analogy with a non-linear power law material. Both an asymptotic and a rigorous analysis are used to determine the crack tip behaviour. A combined Mellin transform-difference equation method is used to solve the problem of a crack at an interface between a uniform elastic medium and a radially inhomogeneous material. These rigorous results are compared with asymptotic non-linear bimaterial results of Champion and Atkinson [Proc. R. Soc. Land.A429, 247–257 (1990)]. Some of the crack tip behaviour associated with the non-linear power law material is shown to be due to the apparent modulus tending to zero or infinity at the crack tip. Problems of cracks at an arbitrary angle to the interface can also be considered using this approach. We briefly consider such cases. To emphasize further similarities between the radially inhomogeneous and the non-linear material we derive some integral invariants.