**3. XEFGM Structure**

In the two-dimensional solid elasticodynamics as depicted in **Figure 2**, the equilibrium equation that gives the relationship between the stresses and traction load *t* can be depicted in matrix form as:

$$\mathbf{L}^T \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0} \quad \text{in problemd } d \text{main } \Omega \tag{13}$$

Where the following Eqs. (15) to (17) represent boundary equations.

$$
\sigma \mathfrak{m} = \mathfrak{F} \qquad \qquad \qquad \qquad \mathfrak{m} \quad \Gamma\_t \tag{14}
$$

$$
\overline{u} = \overline{u} \qquad \qquad \qquad \text{on} \quad \Gamma\_u \tag{15}
$$

$$
\sigma \mathfrak{m} = 0 \qquad\qquad\qquad on \quad \Gamma\_{\varepsilon} \tag{16}
$$

And the differential operator matrix *L* is

$$L = \begin{bmatrix} \frac{\partial}{\partial \mathcal{X}} & 0\\ 0 & \frac{\partial}{\partial \mathcal{Y}}\\ \frac{\partial}{\partial \mathcal{Y}} & \frac{\partial}{\partial \mathcal{X}} \end{bmatrix} \tag{17}$$

EFGM adopted the moving least squares (MLS) approximation [19] to find shape function of the numerical method. The governing equation of the relevant problem:

**Figure 2.** *2D cracked body.*

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

$$\int\_{\Omega} (\mathbf{L}\delta\mathbf{u})^T (\mathbf{D}\mathbf{L}\mathbf{u}) d\Omega - \int\_{\Omega} \delta\mathbf{u}^T \mathbf{b} d\Omega - \int\_{\Gamma\_t} \delta\mathbf{u}^T \mathbf{\tilde{f}} d\Gamma - \int\_{\Gamma\_u} \delta\mathbf{\tilde{\lambda}}^T (\mathbf{u} - \mathbf{\overline{u}}) d\Gamma - \int\_{\Gamma\_u} \delta\mathbf{\tilde{u}}^T \lambda d\Gamma = \mathbf{0} \tag{18}$$

where *λ* represents the Lagrange multiplier variable. The MLS shape function *ϕ<sup>i</sup>* at node *i* a point *x* can be defined as [20]:

$$\phi\_i(\mathbf{x}) = \mathbf{p}^T(\mathbf{x})[\mathbf{A}(\mathbf{x})]^{-1}w(\mathbf{x} - \mathbf{x}\_i)\mathbf{p}(\mathbf{x}\_i) \tag{19}$$

*p x*ð Þ is the basis function

$$\boldsymbol{p}^T(\boldsymbol{\omega}) = [\mathbf{1} \ \boldsymbol{\omega} \boldsymbol{\uprho}] \tag{20}$$

And *A* can be extracted as

$$\mathbf{A}(\mathbf{x}) = \sum\_{i=1}^{n} w(\mathbf{x} - \mathbf{x}\_{i}) \mathbf{p}(\mathbf{x}\_{i}) \mathbf{p}^{T}(\mathbf{x}\_{i}) \tag{21}$$

And

$$w(r) = \begin{cases} \frac{2}{3} - 4r\_s^{-2} + 4r\_s^{-3} & r\_s \le \frac{1}{2} \\\\ \frac{4}{3} - 4r\_s + 4r\_s^{-2} - \frac{4}{3}r\_s^{-3} & \frac{1}{2} < r\_s \le 1 \\\\ 0 & r\_s > 1 \end{cases} \tag{22}$$

Where *rs* is the size of the influence domain for node *i*.

The externally enriched displacement approximation *u<sup>h</sup>* of a model point x [21–23]:

$$\mathfrak{u}^h(\mathbf{x}) = \sum\_{i=1}^n \phi\_i(\mathbf{x})\mathfrak{u}\_i + \sum\_{k=1}^{m\_l} \phi\_k \sum\_{a=1}^4 \mathcal{Q}\_a(\mathbf{x})\mathfrak{b}\_k \tag{23}$$

For isotropic FGMs, enrichment functions are applied to enrich the MLS formulation [24, 25]:

$$Q(r,\theta) = \left(\sqrt{r}\cos\left(\frac{\theta}{2}\right), \sqrt{r}\sin\left(\frac{\theta}{2}\right), \sqrt{r}\sin\left(\frac{\theta}{2}\right)\sin\left(\theta\right), \sqrt{r}\cos\left(\frac{\theta}{2}\right)\sin\left(\theta\right)\right) \tag{24}$$

Therefore the final discretized system equations,

$$
\begin{bmatrix}
\mathbf{K} & \mathbf{Q} \\
\mathbf{Q}^T & \mathbf{0}
\end{bmatrix}
\begin{bmatrix}
\mathbf{U} \\
\mathbf{A}
\end{bmatrix} = \begin{bmatrix}
\mathbf{F} \\
\mathbf{q}
\end{bmatrix} \tag{25}
$$

The vectors in Eq. (26)

$$\mathbf{Q} = -\oint\_{\Gamma\_{\rm u}} \mathbf{N}^T \boldsymbol{\phi} d\Gamma \tag{26}$$

*Advances in Fatigue and Fracture Testing and Modelling*

$$\boldsymbol{q} = -\oint\_{\Gamma\_u} \mathbf{N}^T \overline{\boldsymbol{u}} \tag{27}$$

$$\lambda(\mathbf{x}) = \sum\_{i=1}^{n\_k} N\_i(\mathbf{x}) \lambda\_i \tag{28}$$

*U* is the global displacement vector can be defined as:

$$U = \begin{pmatrix} \mathfrak{u} & \mathfrak{b}\_1 \ \mathfrak{b}\_2 \ \mathfrak{b}\_3 \ \mathfrak{b}\_4 \end{pmatrix}^T \tag{29}$$

Where b1 to b4 represent the enrichment, function terms. And

$$\mathbf{K}\_{\vec{ij}}^{u} = \begin{bmatrix} \mathbf{K}\_{\vec{ij}}^{uu} & \mathbf{K}\_{\vec{ij}}^{ub} \\ \mathbf{K}\_{\vec{ij}}^{bu} & \mathbf{K}\_{\vec{ij}}^{bb} \end{bmatrix} \tag{30}$$

$$F\_i^a = \left\{ F\_i^a \quad F\_i^{b\_1} \quad F\_i^{b\_2} \quad F\_i^{b\_3} \quad F\_i^{b\_4} \right\}^T \tag{31}$$

where

$$\mathbf{K}\_{ij}^{r} = \int\_{\Omega} \left(\mathbf{B}\_{i}^{r}\right)^{T} \mathbf{D} \mathbf{B}\_{j}^{s} d\Omega \quad (r, s = u, b) \tag{32}$$

$$\mathbf{F}\_i^u = \int\_{\Omega} \phi\_i^t \mathbf{b} d\Omega + \int\_{\Gamma\_t} \phi\_i^T \overline{\mathbf{t}} d\Gamma \tag{33}$$

$$\mathbf{F}\_i^{b\_a} = \oint\_{\Omega} \boldsymbol{\phi}\_i^T \mathbf{Q}\_a \mathbf{b} d\Omega + \int\_{\Gamma\_t} \boldsymbol{\phi}\_i^T \mathbf{Q}\_a \mathbf{\tilde{t}} d\Gamma \ \ (a = 1, 2, 3, 4) \tag{34}$$

*Bu <sup>i</sup>* and *B<sup>b</sup> <sup>i</sup>* are matrices of shape function derivatives:

$$\mathbf{B}\_{i}^{u} = \begin{bmatrix} \phi\_{i,\mathbf{x}} & \mathbf{0} \\ \mathbf{0} & \phi\_{i,\mathbf{y}} \\ \phi\_{i,\mathbf{y}} & \phi\_{i,\mathbf{x}} \end{bmatrix} \tag{35}$$

$$\mathbf{B}\_i^b = \begin{bmatrix} \mathbf{B}\_i^{b\_1} & \mathbf{B}\_i^{b\_2} & \mathbf{B}\_i^{b\_3} & \mathbf{B}\_i^{b\_4} \end{bmatrix} \tag{36}$$

$$\mathbf{B}\_i^u = \begin{bmatrix} (\phi\_i \mathbf{Q}\_a)\_{,x} & \mathbf{0} \\ \mathbf{0} & (\phi\_i \mathbf{Q}\_a)\_{,y} \\ (\phi\_i \mathbf{Q}\_a)\_{,y} & (\phi\_i \mathbf{Q}\_a)\_{,x} \end{bmatrix} \tag{37}$$

$$a = \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4} \tag{38}$$

$$\mathbf{e} = L\mathbf{u}^{\mathbf{h}} \tag{39}$$

$$
\sigma = \mathsf{D}\mathfrak{e} \tag{40}
$$

Eq. (40) and (41) represent stress and strain for whole body of FG material calculated by numerical method.

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