A general modelling method for functionally graded materials with an arbitrarily oriented crack

Abstract

In order to analytically solve crack problems regarding functionally graded materials (FGMs), some ideal assumptions are often made. They are: (1) the properties of FGMs are usually assumed to be described by very particular functions; (2) the crack is assumed to be vertical to (or parallel to) the gradient of FGMs. However, these assumptions may not be practical for actual FGMs. Since the controlling differential equations with general mechanical properties are very difficult to solve and the arbitrarily oriented crack causes great trouble in the analytical procedure, a general piecewise-exponential model (GPE model) is proposed to investigate the fracture behaviour of FGMs with general mechanical properties and an arbitrarily oriented crack. “General mechanical properties” means that the mechanical properties in the GPE model are not required to be very particularly pre-defined functions but arbitrary functions determined by fitting the experimental results of FGMs. The studied FGMs are divided into some sub-layers with each layer’s properties varying exponentially so that the general mechanical properties can be approximated by a series of exponential functions and hence the stresses and displacements of each layer which may contain a mixed-mode crack can be solved analytically. By use of integral transform methods, principle of superposition, residual theorem and theory of singular integral equations, the mixed-mode crack problem can be turned into solving a group of singular integral equations from which mixed-mode stress intensity factors (SIFs) can be obtained. Finally, the influences of the nonhomogeneous and geometric parameters on the mixed-mode SIFs are analysed.