# 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 criterion1, 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 integrals1,2. 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 increment3 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 Law4 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 Tanaka5 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.