**5. Numerical examples**

#### **5.1 Example 1**

The first example is that there is a crack in the FGM beam that undergoes a three-point bend as shown in **Figure 4**, and in this problem, the beams are homogeneous and gradient (along with the X2 direction). **Figure 4a** and **b** show the geometry of the sample and BCs for two different boundary conditions: the states (a) and (b), respectively. Also, this figure shows the complete nodal distributions, and the adaptive background cells visualize the distribution of Gauss points and fertilization nodes around the crack for Case (a). Note that the nodal and background cell distributions are valid for both conditions. The material properties (**Table 1**) of the monolithic beam used are as follows:

$$E = 2890 \text{ MPa}, \nu = 0.4, K\_{lc} = 1.09 \text{ MPa} \sqrt{m}$$

A 64 x 28 back grid and 1856 non-uniform distribution nodes are adopted in this case (**Figure 5**). 2875 nodes in finite element method was previously used by Kim and Paulino [3]. **Figure 6** depicts the comparison of the crack path of a homogeneous case (b) beam obtained by current work with the experimental results reported by Galvez et al. [32] With numerical simulation by [3]. The reasonably well output between the numerical and experimental results are obtained. Note that in this case, the gradient of the material does not affect the path of the crack. **Figure 7** shows the effect of increasing the slit length by current numerical simulations of (b) condition on the slit path compared to the experiment available for a

*XEFGM Fracture Analysis of Functionally Graded Materials under Mixed Mode… DOI: http://dx.doi.org/10.5772/intechopen.98765*

#### **Figure 4.**

*Three-point bending cracked beam. (a) Case (a). (b) Case (b). (c) The nodal allocation. (d) back grid structure. (e) Gauss points allocation.*

gradient beam. **Figures 8** and **9** depict results of KI and KII for the different relative size of the J-integral domain (rJ) respectively. The results of the proposed method remain accurate for a wide range of rJ values and the integral field size J (rJ) does not significantly influence the values of SIFs.

### *Advances in Fatigue and Fracture Testing and Modelling*


#### **Table 1.**

*Material characteristics of the graduated beam.*


#### **Figure 5.**

*Nodes of enrichment around the crack prior to the final step of crack propagation.*

**Figure 6.** *Comparison of crack paths for a homogeneous beam (Case b).*

### **5.2 Example 2**

**Figure 10** depicts the configuration and mechanical properties (**Table 2**) of the case study (2) that is bending four points with vertical cracks that perpendicular on the gradient of material.

*XEFGM Fracture Analysis of Functionally Graded Materials under Mixed Mode… DOI: http://dx.doi.org/10.5772/intechopen.98765*

**Figure 7.**

*The effect of increasing the crack length by current numerical simulations of case (b) on the crack path in a graded beam.*

**Figure 8.** *KI values for case (b) with different relative rJ.*

Rousseau and Tippur [33] applied ξ that is zero on the left side of the stepping part, and one on the right side (**Figure 10**). In current work, A 64 x 28 back grid and 2070 non-uniform distribution nodes are adopted in this case (**Figures 11** and **12**), while more than 10,000 element and 30,000 nodes were adopted by [33] to study this case.

The results of the current research work give high accuracy with related references as depicted in **Tables 3** and **4**. The consistent of present research can be depicted in **Tables 3**–**5**. **Figure 13** gives a comparison of the effect of increasing the slit length on the slit path of the current work with experimental work [33] at

**Figure 9.** *KII values for case (b) with different relative rJ.*

#### **Figure 10.**

*Bending four points with vertical cracks on the physical gradient.*


#### **Table 2.**

*Material characteristics of the graduated beam.*

ξ = 0.17, 0.58, and ξ = 1.00. Finally, **Figures 14** and **15** appear the data of KI and KII for different relative rJ respectively. It is clear in this example that the growth of the crack is moving towards the soft side.

*XEFGM Fracture Analysis of Functionally Graded Materials under Mixed Mode… DOI: http://dx.doi.org/10.5772/intechopen.98765*

**Figure 11.**

*(a) Distribution of 2070 irregular nodes, (b) Back grid structure, (c-d) Sub-triangulation technique upon initial fracture propagation.*


**Figure 12.** *Nodes of enrichment around the crack.*
