# Crack Initiation in XFEM

**Introduction**

Predicting where a crack will initiate is a challenging area of computational mechanics. The most common approach is to place a crack at the location of maximum stress^{1}. However, it is well known that the stress fields from finite element simulations converge at a rate which is much slower than displacements. Therefore, it may be difficult, if not impossible to identify the exact location of maximum stress. It is possible for an optimization problem to be formulated to identify the initial crack location. While in general this optimization problem may be too expensive to consider with regards to a finite element simulation with a large number of degrees of freedom, the use of the proposed reanalysis algorithm^{2} makes the solution of the optimization problem possible.

**Optimization Formulation**

The energy release rate is used to identify the crack location corresponding to the location which will allow a structure to dissipate energy in the most efficient manner. The optimization problem takes the following general form:

where *h _{i}* is the

*i*

^{th}equality constraint and

*g*is the

_{j}*j*

^{th}inequality constraint.

**Example**

A simple example problem is that of a plate with a hole. The plate is fixed at the bottom left corner and has roller constraints along the bottom edge of the domain. A unit tensile and shear load is applied to the plate. It is assumed that the crack of known initial length a_{o} will be used. Here the only design variable is the angle of inclination, denoted theta. Therefore we have an unconstrained optimization problem of the form:

The optimization problem is solved using the *fminbnd *function within MATLAB. The reanalysis algorithm is used to modify only the enriched degrees of freedom and directly update an existing Cholesky factorization.

** References**

1. Edke, M.S., Chang, K.H., "Shape sensitivity analysis for 2-D mixed mode fractures using extended FEM (XFEM) and level set method (LSM)," *International Journal of Mechanics Based Design of Structures and Machines*, EISSN: 1539-7742.

2. Pais, M., Kim, N.H., Davis, T. (2010) "Reanalysis of the extended finite element method for crack initiation and propagation," *51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference*, Orlando, Florida.