# Functionally Graded Plate Fracture Analysis Using the Field Boundary Element Method
## 2 Materials and Methods  
This section aims to introduce the fundamentals of integral equations applied to FGM through a mixed field and boundary approach. After a brief description of the mechanical behavior of FGM within the framework of continuum mechanics, the integral formulation of equilibrium is described. The proof originates from the Somigliana identity specifically for materials for which there is no analytic fundamental solution. [PERSON]'s reciprocity theorem is transformed using a sample material whose fundamental equation is known. The resultant equation includes internal variables that cannot be simplified in addition to the ordinary boundary variables. This yields a combined field and boundary integral equation method. The structure under consideration is constituted by a solid occupying a domain V. The points of the solid undergoes a deformation described by the displacement vector field \(u_{i}(x)\) which is assumed to be infinitesimal. In the following, the indicial tensor notation is used where subscripts denote the vector and the tensor components to a reference frame. Moreover, a comma followed by a subscript denotes the partial derivative with respect to the component direction of the index. Finally, the sum on repeated indexes on a monomial is subtended. The structure is subjected to body forces \(b_{i}(x)\) and traction \(t_{i}(x)\) on the boundary. The traction is prescribed on the loaded boundary \(\partial V_{f}\) where displacement is unknown, on the constrained boundary \(\partial V_{u}=\partial V\backslash\partial V_{f}\), appropriate description of the mechanical problem requires that the displacement be prescribed and traction is unknown. The elastic constitutive law of the material is governed by a fourth-order elastic tensor that has the form:  
\[\sigma_{ij}(x)=C_{ijhk}(x)\ \varepsilon_{hk}(x) \tag{1}\]where \(\sigma_{ij}\)\((x)\) is the Cauchy stress tensor and \(\varepsilon_{hk}(x)=\frac{u_{hk}(x)+u_{hk}(x)}{2}\) is the infinitesimal Green-Lagrange strain tensor. The material constituting the structure is assumed to be hyperelastic, hence the constitutive tensor in Equation (1) has the usual symmetries that ensure that the strain energy be conservative and that the stress and the strain be symmetrical:  
\[C_{ijhk}(x)=C_{hkij}(x)=C_{jhk}(x)=C_{ijkh}(x) \tag{2}\]  
The equilibrium equation of the structure, in terms of displacement, \(u_{i}(x)\) is:  
\[\left(C_{ijhk}(x)u_{h,k}(x)\right)_{j}+b_{i}(x)=0 \tag{3}\]  
Let us consider \(u_{li}^{*}(x,y)\) the displacement at a point x of the elastic infinite space where elasticity is described by a constant elastic tensor \(C_{ijhk}^{0}\), due to a one-point force acting at point y. Let us introduce the displacement field \(u_{li}^{*}(x,y)\) resulting from [PERSON]'s solution of the elastic space at point \(x\) when the unit point force acts at point \(y\). Considering the weak form of Equation (3) obtained through weighted residuals and using as weight function the \(u_{li}^{*}(x,y)\) of [PERSON]'s solution as previously mentioned, the following Field Boundary Integral equation results. A detailed derivation of the equation can be found in [17]:  
\[ \begin{split}(\gamma\delta_{lh}+A_{lh})u_{h}&=\int _{\partial V}t_{h}u_{lh}^{*}dS-PV\int_{\partial V}\gamma t_{lh}^{*}u_{h}dS+ \int_{V}b_{h}u_{lh}^{*}dV+\int_{V}\gamma\sigma_{lh}^{*}\varepsilon_{hk}^{p} dV\\ &+\int_{V}\sigma_{lh}^{*}\gamma_{rh}\,u_{k}dV-\int_{\partial V} \tilde{t}_{lh}^{*}u_{h}dS+PV\int_{V}\tilde{b}_{lh}^{*}u_{h}dV\end{split}  \tag{4}\]  
where  
\[ \begin{split} A_{lh}&=L_{ijhk}(y)\tilde{t}_{lijk}^{ \sigma}\\ \tilde{\tilde{t}}_{lh}^{*}&=L_{ijhk}\varepsilon_{ij}^{ *}n_{k}\\ \tilde{b}_{lh}^{*}&=\left(L_{ijhk}\varepsilon_{ij}^{ *}\right)_{k}\end{split}