## 2 2-D Inhomogeneous Cracked Bodies  
### Linear Elasticity of 2-D Cracked Bodies  
Figure 1 represents a 2-D linearly elastic isotropic inhomogeneous material with an edge crack which is contained within a bounded domain \(\Omega\in\Re^{2}\) with the boundary \(\partial\Omega=\overline{\Gamma_{D}\cup\Gamma_{N}\cup\Gamma_{c}}\). Here, \(\Gamma_{D}\) and \(\Gamma_{N}\) indicate the displacement and force boundary regions, and \(\Gamma_{c}=\overline{\Gamma_{c}^{+}\cup\Gamma_{c}^{-}}\) denotes the traction-free crack surface. As a representative non-homogeneous material, FGMs are characterized by the spatially varying elastic modulus \(E\) and Poisson's ratio \(\
u\) over the bounded domain \(\Omega\). For the mathematical description purpose, two Cartesian co-ordinate systems are employed, \(\{x,y\}\) for the 2-D linear elasticity problem and \(\{x^{\prime},y^{\prime}\}\) for the SIF evaluation of crack respectively. Assuming the crack surface is traction-free and neglecting the body force \(\mathbf{b}\), then the displacement field \(\mathbf{u}(\mathbf{x})\) in the Cartesian coordinate system \(\{x,y\}\) is subjected to the equilibrium equations  
\[\
abla\cdot\mathbf{\sigma}=0\quad in\quad\Omega \tag{1}\]  
with the displacement constraint  
\[\mathbf{u}=\mathbf{\hat{u}}\quad on\quad\Gamma_{D} \tag{2}\]  
and the force boundary condition  
\[\mathbf{\sigma}\cdot\mathbf{n}=\left\{ \begin{array}{ccc}\mathbf{\hat{t}}&on&\Gamma_{N} \\ 0&on&\Gamma_{c}^{\pm}\end{array} \right. \tag{3}\]Here, \(\sigma\) are the Cauchy stresses, \(\mathbf{n}\) the outward unit vector normal to \(\partial\Omega\), and \(\hat{\mathbf{t}}\) the contour traction. When the displacement and strains are assumed to be small, the Cauchy strain \(\varepsilon\) is constituted to the Cauchy stress \(\sigma\) via the \((3\times 2)\) gradient-like operator \(\mathbf{L}\) such that  
\[\varepsilon=\varepsilon(\mathbf{u})=\mathbf{L}\mathbf{u} \tag{4}\]  
Letting \(\mathbf{D}\) be the constitutive tensor, the stresses and strains are constituted by  
\[\sigma=\mathbf{D}:\varepsilon \tag{5}\]  
Note that the displacement, strains, and stresses are calculated based on the co-ordinate system \(\{x,y\}\) and transformed into the co-ordinate system \(\{x^{\prime},y^{\prime}\}\). For a homogeneous cracked body, the energy release rate per unit crack propagation along the \(x^{\prime}-\)axis can be estimated by the path-independent \(J-\)integral defined by  
\[J=\int_{\Gamma}\!\!\left(W\delta_{1j}-\sigma_{ij}\frac{\partial u_{i}}{ \partial x^{\prime}{}_{1}}\right)n^{\prime}{}_{j}ds \tag{6}\]  
using the indicial notation (i.e., \(x^{\prime}{}_{1}=x^{\prime}\) and \(x^{\prime}{}_{2}=y^{\prime}\)), the Dirac delta function \(\delta_{1j}\), and the strain-energy density \(W=\sigma\cdot\varepsilon/2=\varepsilon_{ij}D_{ijkl}\varepsilon_{kl}/2\). Here, \(\Gamma\) indicates an arbitrary closed path, which surrounds the crack tip counterclockwise. As shown in Figure 2, it is expanded to \(C=\Gamma+\Gamma_{c}^{-}+\Gamma_{c}^{+}+\Gamma_{o}\) in order to generate a grayed donut-type region A, in which a smooth function \(q(\mathbf{x})(0\leq q\leq 1)\) is introduced to recast the integral into an equivalent domain form [33]. The function \(q\), called by weighting function, becomes unity on \(\Gamma\), zero on \(\Gamma_{o}\), and arbitrary value between \(0\) and \(1\) within the grayed donut region A. Then, the above Equation (6) can be expanded as following  
\[J=\int_{A}\!\!\left(\sigma_{ij}\frac{\partial u_{i}}{\partial x^{\prime}{}_{1 }}-W\delta_{1j}\right)\!\frac{\partial q}{\partial x^{\prime}{}_{j}}dA+\int_{A }\frac{\partial}{\partial x^{\prime}{}_{j}}\!\left(\sigma_{ij}\frac{\partial u _{i}}{\partial x^{\prime}{}_{1}}-W\delta_{1j}\right)\!qdA \tag{7}\]  
according to the divergence theorem, together with \(\mathbf{n}^{\prime}=-\mathbf{n}^{\prime}_{i}\) on \(\Gamma\) in \(C\).