XFEM_Fracture2D (supplementary)

Resolving the crack tip extension lengths

This is a supplementary (and separate) piece of code to the Matlab program XFEM_Fracture2D that attempts to justify the adequacy/inadequacy of the gradient-based energy minimization algorithm to solve for the critical crack tip extensions under the assumption (or the discretization constraint) of fixed-length crack tip extensions that is adopted by XFEM discretization in XFEM_Fracture2D

More specifically, the code included herein attempts to verify that the critical crack tip extension lengths can be determined provided: the cracks are non-competing, or the cracks are competing but the energy function is convex. Conversely, it is shown that the critical crack tip extension lengths can not be resolved robustly if the cracks are competing and the energy function is non-convex or concave.

Note that sufficiently small crack tip extensions are assumed in all cases.

Crack tip competition is defined as the fracture configuration where more than one crack tip has the same energy release rate. A convex energy function can be visualized as one where for any crack tip extension it's curvature is positive whereas a concave energy function is one where for any crack tip extension it's curvature is negative. There also exists the case of a non-convex or a partially-convex energy function which means that the curvature of the function is positive for certain crack tip extensions while negative for others; note that the non-convex case can not be resolved robustly either.

In order to resolve the cases of competing unstable crack growth while satisfying the XFEM discretization constraint of fixed-length crack tip extensions, it is necessary to:

1) perform a multi-trial energy minimization considering an explicit approximation of the total energy function via Tailor's series expansion so as to first determine the "ideal" crack tip extension lengths. This step (called the off-line solution step) serves to identify the critical crack tips exactly.
2) Given that the "ideal" crack tip extension lengths can not be represented in the XFEM discretization, the solution is to coarse the "ideal" solution by rounding it off to fit the XFEM discretization.

This two step procedure in combination with the gradient-based energy minimization algorithm proves to be effective in resolving the case of competing crack growth and a non-convex energy function, as demonstrated by the solutions to the fabricated cases that can be executed by running the attached Matlab code.

However, the proposed algorithm can not be implemented successfully in XFEM_Fracture2D because it is not possible to compute the curvature of the energy function with respect to the crack tip extension lengths sufficiently accurately. So, in the end, XFEM_Fracture2D can not be used to robustly resolve the case of competing crack growth and a non-convex energy function.