# Crack Growth Model

**Introduction**

In order to predict crack growth, it is necessary to predict the direction and magnitude of crack growth at each iteration. An example of crack growth in the presence of both a hard and soft inclusion is given below.

**Crack Growth Direction**

The maximum circumferential stress criterion^{1}, which is based on the assumption that the crack will grow in the direction such the Mode I stress intensity factor is maximized. This requires the calculation of mixed-mode stress intensity factors, which are calculated using the domain form of the interaction integrals^{1,2}. Once the mixed-mode stress intensity factors have been found the following equation can be used to find the angle of crack growth.

**Crack Growth Magnitude**

Two common approaches have been used when modeling quasi-static crack growth within the XFEM framework. The first approach is to assume a constant crack growth increment^{3} and simply update the crack geometry in a constant manner. The crack growth increment commonly used in literature is 0.1. The second option is to use an external criteria to predict the increment of crack growth. Paris Law^{4} is used in our case where we can find the increment of crack growth to take the form given below where C is the Paris Law constant, m is the Paris Law exponent, N is the number of elapsed cycles. The mixed mode correction for Paris Law originally introduced by Tanaka^{5} is used here.

**References**

1. Shih, C., Asaro, R. (1988) "Elastic-plastic analysis of cracks on bimaterial interfaces: Part I - small scale yielding," *Journal of Applied Mechanics*, 55, 299-316.

2. Yau, J., Wang, S., Corten, H. (1980) "A mixed-mode crack analysis of isotropic solids using conservations laws of elasticity," *Journal of Applied Mechanics*, 47, 335-341.

3. Moes, N., Dolbow, J., Belytschko, T. (1999) "A finite element method for crack growth without remeshing," *International Journal for Numerical Methods in Engineering*, 46, 131-150.

4. Paris, P., Gomez, M., Anderson, W. (1961) "A rational analytic theory of fatigue," *The Trend in Engineering*, 13, 9-14.

5. Tanaka, K (1974) "Fatigue crack propagation from a crack inclined to the cyclic tension axis," *Engineering Fracture Mechanics*, 6, 493-507.