Abstract
This article provides a detail derivation of a singular Fredholm integral equation for the solution of a mixed mode crack problem in a nonhomogeneous medium. The integral equation derived here has already been addressed by F. Delale and F. Erdogan (Delale & Erdogan 1983), one of the most cited and pioneer papers in fracture mechanics that uses singulalr integral equation method (SIEM) to solve crack problems. However, probably due to its limit of paper length, some mathematical details are not provided to bring this powerful method, SIEM, to its full strength. In this paper we fill in the mathematical gaps, and both analytical and numerical parts are addressed in details. Some discussions from the view point of differential equations are given, and new numerical outcomes under different loading functions are provided.
Highlights
1.1 Motivation and BackgroundRigorously speaking, most numerical approximation methods in engineering applications can be considered as “hybrid” methods
These six steps of formulation of the crack problem is pretty uniform in solving problems in linear elasticity fracture mechanics (LEFM), and it can be considered as a standard solution technique to the partial differential equations (PDEs) that arise in LEFM
Our contribution of this paper falls into the review category, but singulalr integral equation method (SIEM) is a method that a lot of original research can take advantage of it; just like finite element method (FEM) has been so widely used in fracture mechanics
Summary
Speaking, most numerical approximation methods (including finite elements, finite difference, and finite volume) in engineering applications can be considered as “hybrid” methods. The example used here is a mode-I crack problem in nonhomogeneous materials, well known as functionally graded materials, FGMs (Chan, Paulino, & Fannjiang, 2001; Erdogan, 1978; Erdogan, 1995; Erdogan & Ozturk, 1992; Jin & Noda, 1994; Konda & Erdogan, 1994). This is another SIEM advantage—in addition to its accuracy and the ability to incorporate the crack-tip singularity, it can be very flexible to model crack problems in nonhomogeneous materials. We have filled in every theoretical step with mathematical details
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