# A Symmetric Three-Layer Plate with Two Coaxial Cracks under Pure Bending
## 2 Materials and Methods  
Consider an infinite, thin, three-layer plate of thickness \(h\). The plate has physical and geometrical symmetry about its middle plane. The origin \(O\) of a chosen global Cartesian coordinate system \(Oxyz\) is placed at the center of symmetry, the \(xy\)-plane coincides with the middle plane of the plate and the \(\widetilde{z}-\)axis is directed along the downward vertical plane. The transverse isotropy of each layer is assumed (the planes of isotropy are parallel to the middle plane). The material of the outer layers is characterized by [PERSON]'s modulus \begin{table}
\begin{tabular}{c c} \hline
**Mark** & **Mark Meaning** \\ \hline \(h\) & plate thickness \\ \hline \(E_{1}\) & [PERSON]’s modulus of outer layers in the plane of isotropy \\ \hline \(E_{2}\) & [PERSON]’s modulus of inner layer in the plane of isotropy \\ \hline \(\
u_{1}\) & [PERSON]’s ratio of outer layers in the plane of isotropy \\ \hline \(\
u_{2}\) & [PERSON]’s ratio of inner layer in the plane of isotropy \\ \hline \(G_{1}^{\prime}\) & shear modulus of outer layers in the transverse direction \\ \hline \(G_{2}^{\prime}\) & shear modulus of inner layer in the transverse direction \\ \hline \(2l\) & crack length \\ \hline \(x_{j}\) & coordinates of the center of \(j\)-th crack \\ \hline \(2c\) & distance between inner tips of cracks \\ \hline \(M\) & bending loads at infinity \\ \hline \(M_{y}\) & bending moments \\ \hline \(H_{yy}\) & twisting moments \\ \hline \(Q_{x}\) & \(Q_{y}\) & shear forces \\ \hline \(u^{(k)}\), \(v^{(k)}\), \(w^{(k)}\) & components of displacement vector in the \(k\)-th layer \\ \hline \(\sigma_{x}^{(k)}\), \(\sigma_{y}^{(k)}\), \(\tau_{xy}^{(k)}\), \(\tau_{zx}^{(k)}\), \(\tau_{zy}^{(k)}\) & components of stress tensor in the \(k\)-th layer \\ \hline \(W(x,y)\), \(F(x,y)\), \(\Omega(x,y)\) & unknown functions \\ \hline \(z\) & complex variable \\ \hline \(\Phi(z)\), \(\Psi(z)\) & complex potentials \\ \hline \(K_{i}(x)\) & Macdonald function of \(i\)-th order \\ \hline \(K_{M}^{\pm}\) & bending moment intensity factor \\ \hline \(K\) & reduced moment intensity factor \\ \hline \end{tabular}
\end{table} 
Table 1: Mark meanings. \(E_{1}\), [PERSON]'s ratio \(\
u_{1}\) in the plane of isotropy and the shear modulus \(G_{1}^{\prime}\) in the transverse direction, the material of the inner layer--by \(E_{2}\), \(\
u_{2}\) and \(G_{2}^{\prime}\), respectively. We also require the conditions of an ideal mechanical contact at the interfaces \(\widetilde{z}=\pm h_{1}/2\). The plate contains two equal through coaxial cracks of length \(2l\). The centers \(O_{j}\) (hereinafter \(j=1,2\)) of the cracks have the coordinates \(x_{j}=(-1)^{j}(c+l)\), \(y_{j}=\widetilde{z}_{j}=0\) and are the origins of local Cartesian coordinate systems \(O_{j}x_{j}y_{j}\widetilde{z}_{j}\) (see Figure 1). Let us determine and compute the moment intensity factor under pure bending (bending moments \(M_{y}^{\infty}=M\) are applied at infinity and there is no contact between the faces of cracks). The absence of contact can be ensured by tensile loads at infinity, the contribution of which to the stress state of the plate can be obtained from the known solution of the plane problem of the elasticity theory [18]. The solution to the given problem is a superposition of the solutions to the following two problems: bending of the solid plate and of the plate with cracks under the corresponding load applied to their faces. We restrict ourselves to the second one, the boundary conditions of which in local coordinate systems are as follows:  
\[M_{y}^{+}-M_{y}^{-}=H_{xy}^{+}-H_{xy}^{-}=Q_{y}^{+}-Q_{y}^{-}=0,\,\big{|}x_{j} \big{|}<l, \tag{1}\]  
\[M_{y}^{+}-M_{y}^{-}=2M,\,H_{xy}^{+}+H_{xy}^{-}=Q_{y}^{+}+Q_{y}^{-}=0,\,\big{|} x_{j}\big{|}<l, \tag{2}\]  
where \(M_{y}\), \(H_{xy}\), and \(Q_{y}\) are bending moments, twisting moments and shear forces, respectively; superscripts \"\(+\)\" and \"\(-\)\" stand for limits at \(y\to\pm 0\).