## 2 Problem Formulation  
As shown in Figure 1, the fine-grained ceramic powder is prepared by high-efficiency mechanical ball milling, and then bonded with the substrate by plasma spraying technology to form the coating/substrate structure, while an open crack with a length of \(2l\) is along the interface between the coating and substrate. A fine-grained piezoelectric coating/substrate structure was obtained by polarization treatment of coating. The coating is polarized along the \(z\)-axis. Assumed that coating and substrate are transversely isotropic. During the heating process of coating preparation, crystal deflection or bending may be caused, resulting in dislocation. Therefore, we assume that the screw dislocation \(\mathbf{b}=[b_{z},b_{\phi}]\) with dislocation line parallel to the poling direction are situated at \(z_{0}\) in fine-grained piezoelectric coating, in which \(b_{z}\) and \(b_{\phi}\) are the classical elastic displacement jump and the electric potential jump, respectively. The thickness of coating and substrate are \(h_{1}\) and \(h_{2}\), respectively. It is assumed that the crack faces remain thermally and electrically insulated and \(T_{a}\), \(T_{b}\) are environment temperature. The control and constitutive equation of elastic field are [9]  
\[\frac{\partial\sigma_{xz}}{\partial x}+\frac{\partial\sigma_{yz}}{\partial y}= 0,\ \frac{\partial D_{x}}{\partial x}+\frac{\partial D_{y}}{\partial y}=0, \tag{1}\]  
\[\sigma_{xz}=c_{44}^{(n)}\frac{\partial W}{\partial x}+e_{15}^{(n)}\frac{ \partial\Phi}{\partial x}, \tag{2}\]  
\[\sigma_{yz}=c_{44}^{(n)}\frac{\partial W}{\partial y}+e_{15}^{(n)}\frac{ \partial\Phi}{\partial y}, \tag{3}\]  
\[D_{x}=e_{15}^{(n)}\frac{\partial W}{\partial x}-e_{11}^{(n)}\frac{\partial\Phi }{\partial x}+p_{1}^{(n)}T_{n}, \tag{4}\]  
Figure 1: Mechanical model of fine-grained piezoelectric coating/substrate structure under steady-state thermal loading. \[D_{y}=e_{15}^{(n)}\frac{\partial W}{\partial y}-e_{11}^{(n)}\frac{\partial\Phi}{ \partial y}+p_{1}^{(n)}T_{n}. \tag{5}\]  
In Equations (1)-(5) the \(W\), \(\Phi\) are anti plane displacement and the electric potential, \(c_{44}^{(n)}\) are elastic modulus, \(e_{15}^{(n)}\) and \(\varepsilon_{11}^{(n)}\) are piezoelectric and dielectric constants, respectively. \(T_{n}\) is the temperature change, \(\sigma_{xz}\), \(\sigma_{yz}\) are stress components, \(D_{x}\) and \(D_{y}\) are piezoelectric and dielectric constants, respectively. \(p_{1}^{(n)}\) is pyroelectric constant. the superscript \(n\) (\(n=1,2\)) stands for the fine-grained piezoelectric coating and piezoelectric substrate, respectively. Assume that the temperature satisfies the Fourier heat conduction equation, as follows:  
\[k_{n}^{2}\frac{\partial^{2}T_{n}}{\partial x^{2}}+\frac{\partial^{2}T_{n}}{ \partial y^{2}}=0 \tag{6}\]  
where \(k_{n}=\sqrt{\frac{k_{n}^{(n)}}{k_{y}^{(n)}}}\), \(k_{x}^{(n)}\), \(k_{y}^{(n)}\) are coefficients of thermal conductivity. Substituting Equations (2)-(5) into Equation (1), we obtained  
\[c_{44}^{(n)}\
abla^{2}W+e_{15}^{(n)}\
abla^{2}\Phi=0, \tag{7}\]  
\[e_{15}^{(n)}\
abla^{2}W-e_{11}^{(n)}\
abla^{2}\Phi=-p_{1}^{(n)}(\frac{ \partial T_{n}}{\partial x}+\frac{\partial T_{n}}{\partial y}),\ (n=1,2). \tag{8}\]  
where \(\
abla^{2}\) is the two-dimensional Laplacian operator. For the homogeneous form of Equations (7) and (8), let  
\[\mathbf{U}=[W,\Phi]^{T}\]  
The homogeneous form of Equations (7) and (8) can be rewritten as  
\[\sigma_{xz}=c_{44}^{(n)}\frac{\partial W}{\partial x}+e_{15}^{(n)}\frac{ \partial\Phi}{\partial x}, \tag{2}\]0  
Introduce the following variables  
\[\sigma_{xz}=c_{44}^{(n)}\frac{\partial W}{\partial x}+e_{15}^{(n)}\frac{ \partial\Phi}{\partial x}, \tag{2}\]1  
The Equation (9) can be written as follows  
\[\sigma_{xz}=c_{44}^{(n)}\frac{\partial W}{\partial x}+e_{15}^{(n)}\frac{ \partial\Phi}{\partial x}, \tag{2}\]2  
where  
\[\mathbf{L}^{(n)}=\left[ \begin{array}{cc}c_{44}^{(n)}&e_{15}^{(n)}\\ e_{15}^{(n)}&-e_{11}^{(n)}\end{array}