A unified mathematical treatment of interfacial edge dislocations in three-dimensional functionally graded materials

Abstract

Mechanical behavior of functionally graded materials (FGMs) with the presence of defects, such as dislocations and cracks, is of great scientific interest and practical importance. Existing methods for dislocation and crack problems are typically for homogenous materials or inhomogeneous materials with special kinds of variations in material properties, for instance variable shear modulus according to power-law or exponential functions and constant Poisson's ratio. Applicability of these methods to the practical problems with general inhomogeneity is questionable because variations of the real material properties are arbitrary and cannot be well approximated by special functions. To fill this gap, we present here a unified mathematical treatment of interfacial edge dislocation problems in three-dimensional FGMs. The basic idea is to use a multilayered system consisting of large number of dissimilar sublayers to approximate the arbitrary inhomogeneity in the FGMs. A novel solution approach is developed for calculating elastic field in the multilayered elastic medium due to interfacial edge dislocations of arbitrary shape and distribution profile. Comparing to the existing methods for multilayered elasticity, the present treatment is featured with high computational efficiency, accuracy, stability regardless the total number of sublayers. On the other hand, it is also mathematically consistent in that existing closed-form solutions for edge dislocations at bi-materials interface and homogeneous full space can be fully recovered. Moreover, it can accurately and clearly identify and isolate the singularities in the solutions, which enables the high-precision calculation of elastic field at the vicinity of the source plane. Numerical studies are also conducted on three examples associated with interfacial edge dislocation in multilayered coating and internal edge dislocation in bonded material with graded interfacial zone to show the applicability of the present treatment. It is shown that the present treatment can well address the dislocation problems in FGMs with arbitrary material inhomogeneity.