Singularity analysis and fracture energy-release rate for composites: Piecewise homogeneous-anisotropic materials

We present a formulation for the computation of singularity solution and the fracture energy release rate for piecewise homogeneous-anisotropic materials. A new approach of matched asymptotic analysis is proposed to derive the expression of the fracture energy release rate in terms of the critical singularity exponent and the generalized intensity factors. For the computation of the singular exponents, 2-D elasticity and conductivity problems are first transformed into polar coordinates, then reduced to 1-D problems in terms of the angular variable via the Mellin transform, prior to a finite-element discretization. The resulting 1-D problem is a quadratic eigenvalue problem, which can be transformed into a linear eigenvalue problem for the computation of the eigenpairs. The generalized intensity factors, which are obtainable using contour-independent integrals, are given in terms of equivalent domain integrals, which are numerically easier to implement, more accurate and stable than the contour integrals. A better and concise proof of a fundamental proposition concerning the refracted flux across a bi-material interface in the conductivity problem is also provided. The present work provides a new and clearer exposition of the asymptotic analysis presented in [D. Leguillon, E. Sanchez-Palencia, Crack phenomena in heterogenous media, in: B.W. Schulze, H. Triebel (Eds.), Proceedings of Symposium on Analysis on Manifolds with Singularities, Breitenbrunn, 1990, B.G. Teubner, Stuttgart, 1992], and is thus more accessible to the general engineering readers, e.g., by drawing a parallel between the present formulation with traditional practices in structural dynamics. Further, the present formulation—which is more amenable to a direct implementation in a finite element code—sets the stage for a future generalization to functionally-graded materials. Several numerical examples involving bi-material and tri-material interfaces, including carbon-nanotube reinforced composites, together with validation with available analytical results, are given to illustrate the accuracy and effectiveness of the methodology.