6.1. Example 1

For simplicity we take <sup>∂</sup><sup>F</sup>

182 Contact and Fracture Mechanics

in FGM

<sup>∂</sup><sup>σ</sup> <sup>¼</sup> <sup>a</sup>, <sup>∂</sup><sup>f</sup>

6. Numerical examples and discussion

investigated in detail.

<sup>∂</sup><sup>σ</sup> ¼ a

<sup>a</sup>:d<sup>σ</sup> <sup>¼</sup> <sup>a</sup>:Η: <sup>1</sup>

By substituting dλ in Eq. (27), the final form of material matrix is obtained as [43]

σ 

dλ is calculated by omitting dσ between Eqs. (30) and (31) and substituting dε<sup>p</sup> from Eq. (27).

where Dp <sup>¼</sup> Da aTD a

5. XFEM: Introduction and formulation for cracks and discontinuities

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.

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

σ:dε<sup>p</sup> (32)

Dep ¼ De � Dp (33)

<sup>σ</sup> Ησ<sup>T</sup> <sup>þ</sup> aTDa (34)

A rectangular FGM plate of length (L) 100 mm. and height (D) 200 mm. with 100% aluminum alloy on left side and 100% ceramic (alumina) on right side is considered. Property variation is taken in x-direction, where x = 0 to x = 100 mm. The plate with a major edge crack of length a = 20 mm is analyzed under plane strain condition in the presence of multiple discontinuities. In all simulations, the plate dimensions, initial crack length and material properties are taken to be same. The properties of FGM, composites and aluminum alloy are already described in Table 1. The material properties of the inclusions are taken as Ε ¼ 20 GPa and ν ¼ 0:2. The plate domain is discretized using uniformly distributed 117 nodes in x-direction and 235 nodes in y-direction. The fatigue crack growth analysis is performed by taking a crack increment of <sup>Δ</sup><sup>a</sup> <sup>¼</sup> <sup>a</sup> 10= 2 mm. A cyclic tensile load varying from σmax ¼ 70 MPa to σmin ¼ 0 MPa is applied in all the simulations. The geometric discontinuities like holes, inclusions and minor cracks are added in the plate in addition to the major edge or center crack to analyze their effect on the fatigue life of the material. The fatigue life of the FGM, equivalent composite and aluminum alloy are obtained under mode-I loading, and are compared with each other.

#### 6.2. Plate with a major edge crack under linear elastic condition

Figures 9 and 10 show a plate with a major edge crack of length a = 20 mm at the left and right edge respectively. These plates have been analyzed under plane strain condition using a uniform mesh of 117 by 235 nodes. The plots of the fatigue life for different materials are shown in Figure 11. From these figures, it is seen that the equivalent composite withstands 7885 cycles before it fails while the FGM with crack on alloy side undergoes 15,561 cycles and

Figure 9. Plate with an edge crack on the alloy rich side under mode-I loading.

Figure 10. Plate with an edge crack on the ceramic rich side under mode-I loading.

pure aluminum alloy undergoes 19,145 cycles before failure. It is also observed that when a major crack initiates from the ceramic (alumina) rich side then it fails much earlier (4872 cycles) as compared to when the crack initiates from the aluminum alloy side.

These plots show that when a crack is present on the ceramic rich side, the life diminishes by a considerable extent as compared to when a crack is present on the alloy rich side. The equivalent composite shows the minimum life except in case when a crack is present on the ceramic side. It is also observed that the crack follows nearly a straight path in all the materials.
