5. XFEM: Introduction and formulation for cracks and discontinuities in FGM

XFEM or the extended finite element method is a numerical technique which allows crack modeling irrespective of the mesh, and eliminates the cumbersome process of remeshing in problems involving change in the crack geometry like crack growth. XFEM models a crack by enriching the standard finite element approximation with some functions, which are obtained from the theoretical background of the problem. Moving discontinuities are tracked by the level set method. XFEM is a numerical method, based on the finite element method (FEM) that is especially designed for treating discontinuities. The formulation is done as discussed in [35, 44]. The solution of FGM differs from homogeneous materials only in the spatial gradation in the material properties. After calculating the values of stress and strain, the SIF is determined.

### 6. Numerical examples and discussion

The FGM plate considered in all the numerical simulations has 100% aluminum alloy on one side and 100% alumina on the other side. The volume fraction of alumina changes from 0% on one side to 100% on the other side so as to produce an FGM. The equivalent composite is equivalent to the FGM in the sense that both the FGM and the composite plate contain the same amount of aluminum alloy and alumina. The fatigue crack growth analysis of alloy/ ceramic FGMs, aluminum alloy and equivalent composite is done by XFEM in the presence of multiple cracks, holes and inclusions under mode-I mechanical load and their fatigue life are compared. The constituents of the FGM plate are aluminum alloy and alumina. A major crack of large initial length is assumed to exist at the edge of the plate. The major crack is assumed to be in the direction of material gradation. The fatigue crack growth analyses of the FGM, the equivalent composite and the aluminum alloy plates have been carried out in the presence of minor cracks, holes and inclusions till the final failure of the plate under mode-I mechanical load. The effect of these small defects on the fatigue life as well as on the crack path has been investigated in detail.
