Transient Response of Fine-Grained Piezoelectric Coating Composite Structures with a Griffith-Type Interface Crack Under Thermal Impact Loading

Abstract

Transient behavior of a Griffith interface crack in fine-grained piezoelectric coating/substrate under thermal impact loading conditions is investigated. It is assumed that the crack faces are thermally and electrically insulated. By employing Fourie–Laplace integral transforms as well as the additivity of solutions, the theoretical expressions for the temperature field, displacement field, and electric field are constructed, thereby deriving specific expressions for the dynamic intensity factors of thermal stress and electric displacement. The boundary-value problem is reduced to singular integral equations solved numerically. Parametric studies quantify time-dependent effects of the coating elastic modulus, thickness, and crack length on dynamic responses. Numerical analyses demonstrate that variations in the elastic modulus ratio and coating thickness induce varied increases or decreases in peak dynamic stress intensity factor. Optimizing coating thickness and elastic modulus significantly enhances structural safety under thermal impact loading.

1. Introduction

Piezoelectric materials are pivotal across multidisciplinary fields. In smart structure manufacturing, piezoelectric coating sensors enable real-time vibration/stress monitoring on bridges and aerospace vehicles for damage early-warning. Industrially, they facilitate fluid status monitoring and advance adaptive structure development. Their low-cost scalability makes them essential smart devices. However, current technologies are prone to generating defects during the preparation of piezoelectric composites. When subjected to external loads, cracks tend to occur at the interfaces of composite structures, and more seriously, this can lead to structural failure. Consequently, the issue of interface fracture in piezoelectric composites has garnered significant research attention [1,2,3,4,5,6,7,8]. Ueda et al. [9,10] studied the crack problems in functionally graded piezoelectric material plates under thermal shock loading. Using Laplace and Fourier transforms, they reduced the thermal and electromechanical problem to a system of singular integral equations. Their analysis quantified temporal variations in stress/electric displacement intensity factors with material constants and geometric parameters. Additionally, they examined mixed-mode thermoelectromechanical fracture in FGPM strips containing penny-shaped cracks, reporting intensity factors for diverse crack sizes, positions, and material inhomogeneities. Wang et al. [11] investigated thermal shock fracture in piezoelectric materials exposed to varying-temperature ambient media. They considered two crack-face assumptions: electrically non-conducting versus conducting. Their work established a theoretical lower-bound solution for the maximum endurable thermal shock without catastrophic failure. Chen et al. [12] explored the dynamic response of cracked piezoelectric ceramic under in-plane electric field and anti-plane mechanical impact. They first assumed that the electric and mechanical loads are arbitrary functions of time, and then found, through numerical computations, that the dynamic crack-tip stress field and electric displacement field still exhibit a square-root singularity. Additionally, they discovered that the electric load has a significant influence on the dynamic response of the stress field, and appropriate external electromechanical loads can retard the propagation of cracks. Sladek et al. [13] proposed a meshless method based on the local Petrov–Galerkin approach for solving coupled thermomechanical problems in magnetoelectroelastic solids with functionally graded properties. Compared with the traditional BEM (Boundary Element Method), it does not require a fundamental solution, all integrands are regular, and no special numerical techniques are needed for integral computation. Liu et al. [14] analyzed the transient thermomechanical response of stationary cracks in functionally graded piezoelectric materials (FGPMs) using the extended finite element method. They considered two thermal shock scenarios: heating shock and cooling shock. Results revealed distinct dynamic crack behaviors between cooling and heating shocks, with thermal loads significantly amplifying dynamic intensity factors. Khatib et al. [15] researched thermoelectromechanical coupling effects on fracture behavior in ferroelectric ceramics. Utilizing the boundary layer modeling approach, they analyzed crack-tip thermal stress fields and quantified parametric influences on crack driving force energy.

Current research extensively explores interface fracture in piezoelectric materials under complex loading. Jani et al. [16,17] investigated transient symmetric responses of piezoelectric hollow spheres to thermal shock using Lord–Shulman′s thermoelastic framework, deriving dimensionless governing equations solved via GDQM spatial discretization and Newmark temporal integration to reveal finite-speed electrothermomechanical wave dynamics. Bayones et al. [18] developed magnetothermoelastic models incorporating two-temperature effects and hydrostatic stress, obtaining exact solutions that quantify magnetic field/relaxation time impacts on field variables. Tian et al. [19] established a transient frictionless contact model for thermoelectric devices, demonstrating elevated transient contact stresses via Laplace–Fourier transforms and collocation methods. Chadaram et al. [20] analyzed penny-shaped cracks under thermal loads using XFEM with Heaviside enrichment, validating thermal/electrical intensity factor dependencies on poling direction and crack geometry. However, these studies focus primarily on multi-domain large-grained piezoelectrics, leaving fine-grained systems—increasingly critical for industrial applications—largely unexplored. To date, constrained by material fabrication technologies, the mechanical, dielectric, and piezoelectric properties of fine-grained piezoelectric materials remain inadequately characterized. Current understanding of grain-size-dependent parameter relationships relies predominantly on experimental approaches [21,22]. Conversely, theoretical investigations into the interfacial mechanics of fine-grained (particularly nano-grained) piezoelectric composite structures remain scarce [23].

Therefore, building upon prior research investigating interface crack propagation in fine-grained piezoelectric composites under static loading, this study establishes a Griffith interface crack model for fine-grained piezoelectric coating/substrate systems subjected to thermal shock. By employing Laplace–Fourier transform techniques, the interface crack propagation problem is reduced to solving systems of singular integral equations. Furthermore, as a critical fracture parameter for structural safety assessment, the dynamic intensity factor is analyzed with quantitative evaluation of material parameter influences on its evolution.

2. Problem Formulation

The problem geometry is illustrated in Figure 1. A fine-grained ceramic powder coating is plasma-spray bonded to the substrate, featuring an interfacial crack of length $2l$ along the coating/substrate interface. The polarized coating/substrate structure has coating thickness $h_1$ and substrate thickness $h_2$. The crack surfaces are assumed to be electrically and thermally insulated. Boundary $z=-h_2$ is maintained at $T_1$, while boundary $z=h_1$ undergoes sudden temperature rise $T_{2}H(t)$, where $H(t)$ is the Heaviside step function and $t$ denotes time.

The constitutive equations governing the elastic field are as follows:

$${\sigma }_{xx}(x,y,t)=c_{11}^{(m)}{\displaystyle \frac{\partial w}{\partial x}}+c_{13}^{(m)}{\displaystyle \frac{\partial u}{\partial y}}+e_{31}^{(m)}{\displaystyle \frac{\partial \varphi }{\partial y}}-{\lambda }_{11}^{(m)}T(x,y,t)$$

(1)

$${\sigma }_{yy}(x,y,t)=c_{13}^{(m)}{\displaystyle \frac{\partial w}{\partial x}}+c_{33}^{(m)}{\displaystyle \frac{\partial u}{\partial y}}+e_{33}^{(m)}{\displaystyle \frac{\partial \varphi }{\partial y}}-{\lambda }_{33}^{(m)}T(x,y,t)$$

(2)

$${\sigma }_{xy}(x,y,t)=c_{44}^{(m)}\left({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial u}{\partial x}}\right)+e_{15}^{(m)}{\displaystyle \frac{\partial \varphi }{\partial x}}$$

(3)

$$D_x(x,y,t)=e_{15}^{(m)}\left({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial u}{\partial x}}\right)-{\epsilon }_{11}^{(m)}{\displaystyle \frac{\partial \varphi }{\partial x}}$$

(4)

$$D_y(x,y,t)=e_{31}^{(m)}{\displaystyle \frac{\partial w}{\partial x}}+e_{33}^{(m)}{\displaystyle \frac{\partial u}{\partial y}}-{\epsilon }_{33}^{(m)}{\displaystyle \frac{\partial \varphi }{\partial y}}+p_y^{(m)}T(x,y,t)$$

(5)

In Equations (1)–(5), $w(x,y,t)$ and $u(x,y,t)$ denote the displacement components; $\varphi (x,y,t)$ represents the electric potential; $T(x,y,t)$ is the temperature; ${\sigma }_{xx}(x,y,t)$, ${\sigma }_{yy}(x,y,t)$, and ${\sigma }_{xy}(x,y,t)$ are the stress components; $D_x(x,y,t)$ and $D_y(x,y,t)$ denote the electric displacement components; $c_{11}^{(m)}$, $c_{13}^{(m)}$, and $c_{44}^{(m)}$ are the elastic modulus; and $e_{31}^{(m)}$, $e_{33}^{(m)}$, $e_{15}^{(m)}$ and ${\epsilon }_{11}^{(m)}$, ${\epsilon }_{33}^{(m)}$ are the piezoelectric and dielectric constants, respectively. ${\lambda }_{11}^{(m)}$ and ${\lambda }_{33}^{(m)}$ are the stress–temperature coefficients and $p_y^{(m)}$ is the pyroelectric constant. The superscripts $m$($m=1,2$) stand for the fine-grained piezoelectric coating and piezoelectric substrate, respectively.

Assume that the temperature satisfies the Fourier heat conduction equation

$$k_m^2{\displaystyle \frac{{\partial }^{2}T(x,y,t)}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T(x,y,t)}{\partial y^2}}=0$$

(6)

where $k_m=\sqrt{{\displaystyle \frac{k_x^{(m)}}{k_y^{(m)}}}}$ and $k_x^{(m)}$ and $k_y^{(m)}$ are coefficients of thermal conductivity.

The equations of equilibrium are as follows:

$$c_{11}^{(m)}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+c_{44}^{(m)}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+\left(c_{13}^{(m)}+c_{44}^{(m)}\right){\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}}+\left(e_{31}^{(m)}+e_{15}^{(m)}\right){\displaystyle \frac{{\partial }^2\varphi }{\partial x\partial y}}={\lambda }_{11}^{(m)}{\displaystyle \frac{\partial T(x,y,t)}{\partial x}}$$

(7)

$$c_{44}^{(m)}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+c_{33}^{(m)}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+\left(c_{13}^{(m)}+c_{44}^{(m)}\right){\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}+e_{15}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}+e_{33}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial y^2}}={\lambda }_{33}^{(m)}{\displaystyle \frac{\partial T(x,y,t)}{\partial y}}$$

(8)

$$e_{15}^{(m)}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+e_{33}^{(m)}{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+\left(e_{15}^{(m)}+e_{31}^{(m)}\right){\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}-{\epsilon }_{11}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}-{\epsilon }_{33}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial y^2}}=-p_y^{(m)}{\displaystyle \frac{\partial T(x,y,t)}{\partial y}}$$

(9)

Without loss of generality, the boundary and initial conditions can be written as follows:

$$\left\{\begin{array}{c}w(x,0^{+},t)=w(x,0^{-},t),\left|x\right|\ge l\\ u(x,0^{+},t)=u(x,0^{-},t),\left|x\right|\ge l\\ \varphi (x,0^{+},t)=\varphi (x,0^{-},t),\left|x\right|\ge l\end{array}\right.$$

(10a)

$$\left\{\begin{array}{c}{\sigma }_{yy}(x,0^{+},t)={\sigma }_{yy}(x,0^{-},t)=0,\left|x\right|<l\\ {\sigma }_{xy}(x,0^{+},t)={\sigma }_{xy}(x,0^{-},t)=0,\left|x\right|<l\\ D_y(x,0^{+},t)=D_y(x,0^{-},t)=0,\left|x\right|<l\end{array}\right.$$

(10b)

$$\left\{\begin{array}{c}{\sigma }_{yy}(x,0^{+},t)={\sigma }_{yy}(x,0^{-},t),\left|x\right|\ge l\\ {\sigma }_{xy}(x,0^{+},t)={\sigma }_{xy}(x,0^{-},t),\left|x\right|\ge l\\ D_y(x,0^{+},t)=D_y(x,0^{-},t),\left|x\right|\ge l\end{array}\right.$$

(10c)

$$\left\{\begin{array}{c}{\sigma }_{yy}(x,h_1,t)={\sigma }_{yy}(x,-h_2,t)=0,\left|x\right|\ge l\\ {\sigma }_{xy}(x,h_1,t)={\sigma }_{xy}(x,-h_2,t)=0,\left|x\right|\ge l\\ D_y(x,h_1,t)=D_y(x,-h_2,t)=0,\left|x\right|\ge l\end{array}\right.$$

(10d)

The thermal boundary conditions are specified as follows:

$$\left\{\begin{array}{c}T(x,0^{+},t)=T(x,0^{-},t),\left|x\right|\ge l\\ T(x,h_1,t)=T_2(x)H(t),\left|x\right|\ge 0\\ T(x,h_2,t)=T_1(x),\left|x\right|\ge 0\end{array}\right.$$

(11a)

$$\left\{\begin{array}{c}{k}_y^{(1)}{\displaystyle \frac{\partial T(x,0^{+},t)}{\partial y}}=k_y^{(2)}{\displaystyle \frac{\partial T(x,0^{-},t)}{\partial y}}=0,\left|x\right|<l\\ k_y^{(1)}{\displaystyle \frac{\partial T(x,0^{+},t)}{\partial y}}=k_y^{(2)}{\displaystyle \frac{\partial T(x,0^{-},t)}{\partial y}},\left|x\right|\ge l\end{array}\right.$$

(11b)

where $k_y^{(1)}$ and $k_y^{(2)}$ are coefficients of thermal conductivity.

Since the temperature field is independent of the stress and displacement fields, it can be determined independently, after which the thermal stress and potential displacement fields can be derived from the governing equations.

3. Solution to the Problem

3.1. Temperature Field

By utilizing Laplace and Fourier integral transforms, the solution of the temperature field in the Laplace domain can be expressed as follows:

$$T^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[A_{1m}(s,\tau )e^{-\left|s\right|k_{m}y}+A_{2m}(s,\tau )e^{\left|s\right|k_{m}y}]}{e}^{-isx}ds$$

(12)

where $k_m=\sqrt{{\displaystyle \frac{k_x^{(m)}}{k_y^{(m)}}}}$ and $A_{1m}(s,\tau )$ and $A_{2m}(s,\tau )$ are the unknown functions $(m=1,2)$ and ‘*’ denotes the Laplace transform.

To facilitate the analysis, the density function is defined mathematically as follows:

$$G(x,\tau )={\displaystyle \frac{\partial }{\partial x}}[T^{*}(x,0^{+},\tau )-T^{*}(x,0^{-},\tau )]$$

(13)

Consequently, the thermal boundary conditions (11a) and (11b) in the Laplace domain are expressed as follows:

$$\left\{\begin{array}{c}{T}^{*}(x,0^{+},\tau )=T^{*}(x,0^{-},\tau ),\left|x\right|\ge l\\ T^{*}(x,h_1,\tau )={\displaystyle \frac{T_2(x)}{\tau }},\left|x\right|\ge 0\\ T^{*}(x,h_2,\tau )={\displaystyle \frac{T_1(x)}{\tau }},\left|x\right|\ge 0\end{array}\right.$$

(14a)

$$\left\{\begin{array}{c}{k}_y^{(1)}{\displaystyle \frac{\partial T^{*}(x,0^{+},\tau )}{\partial y}}=k_y^{(2)}{\displaystyle \frac{\partial T^{*}(x,0^{-},\tau )}{\partial y}}=0,\left|x\right|<l\\ k_y^{(1)}{\displaystyle \frac{\partial T^{*}(x,0^{+},\tau )}{\partial y}}=k_y^{(2)}{\displaystyle \frac{\partial T^{*}(x,0^{-},\tau )}{\partial y}},\left|x\right|\ge l\end{array}\right.$$

(14b)

According to Equation (14a), the dislocation density function must satisfy the following single-valuedness conditions:

$${\displaystyle {\int }_{-l}^{l}G(\xi ,\tau )}d\xi =0$$

(15)

Substituting Equation (12) into Equations (14a) and (14b), we obtain

$$\left\{\begin{array}{c}-\left|s\right|k_1{k}_y^{(1)}[A_{11}(s,\tau )-A_{21}(s,\tau )]=-\left|s\right|k_1{k}_y^{(2)}[A_{12}(s,\tau )-A_{21}(s,\tau )]\\ A_{11}(s,\tau )+A_{21}(s,\tau )-A_{12}(s,\tau )-A_{22}(s,\tau )={\displaystyle \frac{i}{\tau }}{\displaystyle {\int }_{-l}^{l}G(\xi ,\tau )}{e}^{i\tau s}d\xi \\ A_{11}(s,\tau )e^{-\left|s\right|k_1{h}_1}+A_{21}(s,\tau )e^{\left|s\right|k_1{h}_1}={\displaystyle {\int }_{-\infty }^{+\infty }{T}_2}(x)e^{itx}dx\\ A_{12}(s,\tau )e^{-\left|s\right|k_2{h}_2}+A_{22}(s,\tau )e^{\left|s\right|k_2{h}_2}={\displaystyle {\int }_{-\infty }^{+\infty }{T}_1}(x)e^{itx}dx\end{array}\right.$$

(16)

Equation (16) is expressed in matrix form as follows:

$$A\cdot {\rm X}=b$$

(17)

where

$$A=\left[\begin{array}{cccc}-\left|s\right|k_1{k}_y^{(1)}& \left|s\right|k_1{k}_y^{(1)}& \left|s\right|k_2{k}_y^{(2)}& -\left|s\right|k_2{k}_y^{(2)}\\ 1& 1& -1& -1\\ e^{-\left|s\right|k_1{h}_1}& e^{\left|s\right|k_1{h}_1}& 0& 0\\ 0& 0& e^{-\left|s\right|k_2{h}_2}& e^{\left|s\right|k_2{h}_2}\end{array}\right]$$

(18)

$${\rm X}={[A_{11}(s,\tau ),A_{21}(s,\tau ),A_{12}(s,\tau ),A_{22}(s,\tau )]}^T$$

(19)

$$b={[0,I_0,I_1,I_2]}^T$$

(20)

$$\left\{\begin{array}{c}{I}_0={\displaystyle \frac{i}{\tau }}{\displaystyle {\int }_{-l}^{l}G(\xi ,\tau )}{e}^{i\tau s}d\xi \\ I_1={\displaystyle {\int }_{-l}^l{T}_1}(x)e^{isx}dx\\ I_2={\displaystyle {\int }_{-l}^l{T}_2}(x)e^{isx}dx\end{array}\right.$$

(21)

Solving the linear Equation (17), we get

$$\left\{\begin{array}{c}{A}_{11}(s,\tau )={\omega }_1{I}_0-{\omega }_2{I}_1+{\omega }_3{I}_2\\ A_{12}(s,\tau )={\omega }_8{I}_0-{\omega }_9{I}_1+{\omega }_7{I}_2\\ A_{21}(s,\tau )=-{\omega }_5{I}_0+{\omega }_4{I}_1-{\omega }_6{I}_2\\ A_{22}(s,\tau )=-{\omega }_{12}{I}_0+{\omega }_{10}{I}_1-{\omega }_{11}{I}_2\end{array}\right.$$

The expressions for ${\omega }_q$, $q=1,2,\dots 12$, are provided in Supplementary Materials. Clearly, the unknown functions $A_{11}(s,\tau )$, $A_{21}(s,\tau )$, $A_{12}(s,\tau )$, and $A_{22}(s,\tau )$ depend on $I_0$. Once $I_0$ is determined, the temperature field in the Laplace domain can be derived. According to Reference [9], applying the inverse Laplace transform to Equation (12) yields the time-domain temperature field $T(x,y,t)$.

3.2. Thermal Stress and Electric Displacement Fields

The solutions to the equilibrium Equations (7)–(9) for displacement and electric potential can be derived using a method analogous to that for solving algebraic linear systems.

The Equations (7)–(9) can be written into homogeneous forms, as follows:

$$c_{11}^{(m)}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+c_{44}^{(m)}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+\left(c_{13}^{(m)}+c_{44}^{(m)}\right){\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}}+\left(e_{31}^{(m)}+e_{15}^{(m)}\right){\displaystyle \frac{{\partial }^2\varphi }{\partial x\partial y}}=0$$

(22)

$$c_{44}^{(m)}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+c_{33}^{(m)}{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}+\left(c_{13}^{(m)}+c_{44}^{(m)}\right){\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}+e_{15}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}+e_{33}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial y^2}}=0$$

(23)

$$e_{15}^{(m)}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+e_{33}^{(m)}{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+\left(e_{15}^{(m)}+e_{31}^{(m)}\right){\displaystyle \frac{{\partial }^{2}w}{\partial x\partial y}}-{\epsilon }_{11}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}-{\epsilon }_{33}^{(m)}{\displaystyle \frac{{\partial }^2\varphi }{\partial y^2}}=0$$

(24)

Using the Laplace transform, the solutions to Equations (22)–(24) can be expressed as

$$\left\{\begin{array}{c}{u}_p^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6{B}_{1k}^{(m)}(s,\tau )}}{e}^{s{\lambda }_{k}y}{f}_k(s,\tau )e^{-isx}ds\\ w_p^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6{B}_{2k}^{(m)}(s,\tau )}}{e}^{s{\lambda }_{k}y}{f}_k(s,\tau )e^{-isx}ds\\ {\varphi }_p^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6{B}_{3k}^{(m)}(s,\tau )}}{e}^{s{\lambda }_{k}y}{f}_k(s,\tau )e^{-isx}ds\end{array}\right.$$

(25)

where $B_{1k}^{(m)}(s,\tau )$, $B_{2k}^{(m)}(s,\tau )$, $B_{3k}^{(m)}(s,\tau )$, and $f_k(s,\tau )$,$k=1,\dots ,6$ are unknown functions to be determined.

Thus, in the Laplace transform domain, the mixed boundary conditions (10a)–(10d) are expressed as follows:

$$\left\{\begin{array}{c}{w}^{*}(x,0^{+},\tau )=w^{*}(x,0^{-},\tau ),\left|x\right|\ge l\\ u^{*}(x,0^{+},\tau )=u^{*}(x,0^{-},\tau ),\left|x\right|\ge l\\ {\varphi }^{*}(x,0^{+},\tau )={\varphi }^{*}(x,0^{-},\tau ),\left|x\right|\ge l\end{array}\right.$$

(26a)

$$\left\{\begin{array}{c}{\sigma }_{yy}^{*}(x,0^{+},\tau )={\sigma }_{yy}^{*}(x,0^{-},\tau )=0,\left|x\right|<l\\ {\sigma }_{xy}^{*}(x,0^{+},\tau )={\sigma }_{xy}^{*}(x,0^{-},\tau )=0,\left|x\right|<l\\ D_y^{*}(x,0^{+},\tau )=D_y^{*}(x,0^{-},\tau )=0,\left|x\right|<l\end{array}\right.$$

(26b)

$$\left\{\begin{array}{c}{\sigma }_{yy}^{*}(x,0^{+},\tau )={\sigma }_{yy}^{*}(x,0^{-},\tau ),\left|x\right|\ge l\\ {\sigma }_{xy}^{*}(x,0^{+},\tau )={\sigma }_{xy}^{*}(x,0^{-},\tau ),\left|x\right|\ge l\\ D_y^{*}(x,0^{+},\tau )=D_y^{*}(x,0^{-},\tau ),\left|x\right|\ge l\end{array}\right.$$

(26c)

$$\left\{\begin{array}{c}{\sigma }_{yy}^{*}(x,h_1,\tau )={\sigma }_{yy}^{*}(x,-h_2,\tau )=0,\left|x\right|\ge l\\ {\sigma }_{xy}^{*}(x,h_1,\tau )={\sigma }_{xy}^{*}(x,-h_2,\tau )=0,\left|x\right|\ge l\\ D_y^{*}(x,h_1,\tau )=D_y^{*}(x,-h_2,\tau )=0,\left|x\right|\ge l\end{array}\right.$$

(26d)

Meanwhile, the special solutions of Equations (7)–(9) are expressed as follows:

$$\left\{\begin{array}{c}{u}_q^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{1}{s}}(B_{1a}^{(m)}(s,\tau )A_{1m}(s,\tau )e^{-\left|s\right|k_{m}y}+B_{1b}^{(m)}(s,\tau )A_{2m}(s,\tau )e^{\left|s\right|k_{m}y})e^{-isx}ds\\ w_q^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{1}{s}}(B_{2a}^{(m)}(s,\tau )A_{1m}(s,\tau )e^{-\left|s\right|k_{m}y}+B_{2b}^{(m)}(s,\tau )A_{2m}(s,\tau )e^{\left|s\right|k_{m}y})e^{-isx}ds\\ {\varphi }_q^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{1}{s}}(B_{3a}^{(m)}(s,\tau )A_{1m}(s,\tau )e^{-\left|s\right|k_{m}y}+B_{3b}^{(m)}(s,\tau )A_{2m}(s,\tau )e^{\left|s\right|k_{m}y})e^{-isx}ds\end{array}\right.$$

(27)

where $B_{1a}^{(m)}(s,\tau )$, $B_{2a}^{(m)}(s,\tau )$, $B_{3a}^{(m)}(s,\tau )$, $B_{1b}^{(m)}(s,\tau )$, $B_{2b}^{(m)}(s,\tau )$, and $B_{3b}^{(m)}(s,\tau )$ satisfy the expression as follows:

$$\left({\delta }_2{k}_m^2-{\delta }_1{k}_m+{\delta }_0\right)\left[\begin{array}{c}{B}_{1a}^{(m)}(s,\tau )\\ B_{2a}^{(m)}(s,\tau )\\ B_{3a}^{(m)}(s,\tau )\end{array}\right]=\left[\begin{array}{c}-i{\lambda }_{11}^{(m)}\\ -is{\lambda }_{33}^{(m)}{k}_m\\ s{p}_y^{(m)}{k}_m\end{array}\right]$$

(28)

$$\left({\delta }_2{k}_m^2+{\delta }_1{k}_m+{\delta }_0\right)\left[\begin{array}{c}{B}_{1b}^{(m)}(s,\tau )\\ B_{2b}^{(m)}(s,\tau )\\ B_{3b}^{(m)}(s,\tau )\end{array}\right]=\left[\begin{array}{c}-i{\lambda }_{11}^{(m)}\\ -is{\lambda }_{33}^{(m)}{k}_m\\ s{p}_y^{(m)}{k}_m\end{array}\right]$$

(29)

where

$${\delta }_0=-\left[\begin{array}{ccc}{c}_{11}^{(m)}& 0& 0\\ 0& c_{44}^{(m)}& e_{15}^{(m)}\\ 0& e_{15}^{(m)}& {\epsilon }_{11}^{(m)}\end{array}\right]$$

$${\delta }_1=-\left[\begin{array}{ccc}0& is{c}_{44}^{(m)}& is(e_{31}^{(m)}+e_{15}^{(m)})\\ is{c}_{44}^{(m)}& 0& 0\\ is(e_{31}^{(m)}+e_{15}^{(m)})& 0& 0\end{array}\right]$$

$${\delta }_2=\left[\begin{array}{ccc}{c}_{44}^{(m)}& 0& 0\\ 0& c_{33}^{(m)}& e_{33}^{(m)}\\ 0& e_{33}^{(m)}& -{\epsilon }_{33}^{(m)}\end{array}\right]$$

Therefore, the solutions of Equations (7)–(9) take the following forms:

$$\left\{\begin{array}{c}{u}^{*}(x,y,\tau )=u_p^{*}(x,y,\tau )+u_q^{*}(x,y,\tau )\\ w^{*}(x,y,\tau )=w_p^{*}(x,y,\tau )+w_q^{*}(x,y,\tau )\\ {\varphi }^{*}(x,y)={\varphi }_p^{*}(x,y,\tau )+{\varphi }_q^{*}(x,y,\tau )\end{array}\right.$$

(30)

Substituting Equation (30) into Equations (2), (3) and (5) yields the thermal stress and electric displacement fields in the Laplace domain

$${\sigma }_{yy}^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\Phi }_{11}^{(m)}}{f}_{0}s{e}^{-isx}ds+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\Phi }_{12}^{(m)}}{A}_0{e}^{-isx}ds$$

(31)

$${\sigma }_{xy}^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\Phi }_{21}^{(m)}}{f}_{0}s{e}^{-isx}ds+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\Phi }_{22}^{(m)}}{A}_0{e}^{-isx}ds$$

(32)

$$D_y^{*}(x,y,\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\Phi }_{31}^{(m)}}{f}_{0}s{e}^{-isx}ds+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\Phi }_{32}^{(m)}}{A}_0{e}^{-isx}ds$$

(33)

where

$${\Phi }_{11}^{(m)}={\left[\begin{array}{c}{\lambda }_1{c}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{11}^{(m)}(s,\tau )-i{c}_{31}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{21}^{(m)}(s,\tau )+{\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{31}^{(m)}(s,\tau )\\ {\lambda }_1{c}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{12}^{(m)}(s,\tau )-i{c}_{32}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{22}^{(m)}(s,\tau )+{\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{32}^{(m)}(s,\tau )\\ {\lambda }_1{c}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{13}^{(m)}(s,\tau )-i{c}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{23}^{(m)}(s,\tau )+{\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{33}^{(m)}(s,\tau )\\ {\lambda }_1{c}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{14}^{(m)}(s,\tau )-i{c}_{34}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{24}^{(m)}(s,\tau )+{\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{34}^{(m)}(s,\tau )\\ {\lambda }_1{c}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{15}^{(m)}(s,\tau )-i{c}_{35}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{25}^{(m)}(s,\tau )+{\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{35}^{(m)}(s,\tau )\\ {\lambda }_1{c}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{16}^{(m)}(s,\tau )-i{c}_{36}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{26}^{(m)}(s,\tau )+{\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{36}^{(m)}(s,\tau )\end{array}\right]}^T$$

$${\Phi }_{21}^{(m)}={\left[\begin{array}{c}-i{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{11}^{(m)}(s,\tau )+{\lambda }_1{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{21}^{(m)}(s,\tau )-i{e}_{15}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{31}^{(m)}(s,\tau )\\ -i{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{12}^{(m)}(s,\tau )+{\lambda }_1{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{22}^{(m)}(s,\tau )-i{e}_{15}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{32}^{(m)}(s,\tau )\\ -i{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{13}^{(m)}(s,\tau )+{\lambda }_1{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{23}^{(m)}(s,\tau )-i{e}_{15}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{33}^{(m)}(s,\tau )\\ -i{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{14}^{(m)}(s,\tau )+{\lambda }_1{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{24}^{(m)}(s,\tau )-i{e}_{15}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{34}^{(m)}(s,\tau )\\ -i{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{15}^{(m)}(s,\tau )+{\lambda }_1{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{25}^{(m)}(s,\tau )-i{e}_{15}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{35}^{(m)}(s,\tau )\\ -i{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{16}^{(m)}(s,\tau )+{\lambda }_1{c}_{44}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{26}^{(m)}(s,\tau )-i{e}_{15}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{36}^{(m)}(s,\tau )\end{array}\right]}^T$$

$${\Phi }_{31}^{(m)}={\left[\begin{array}{c}{\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{11}^{(m)}(s,\tau )-i{e}_{31}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{21}^{(m)}(s,\tau )-{\lambda }_1{\epsilon }_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{31}^{(m)}(s,\tau )\\ {\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{12}^{(m)}(s,\tau )-i{e}_{31}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{22}^{(m)}(s,\tau )-{\lambda }_1{\epsilon }_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{32}^{(m)}(s,\tau )\\ {\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{13}^{(m)}(s,\tau )-i{e}_{31}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{23}^{(m)}(s,\tau )-{\lambda }_1{\epsilon }_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{33}^{(m)}(s,\tau )\\ {\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{14}^{(m)}(s,\tau )-i{e}_{31}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{24}^{(m)}(s,\tau )-{\lambda }_1{\epsilon }_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{34}^{(m)}(s,\tau )\\ {\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{15}^{(m)}(s,\tau )-i{e}_{31}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{25}^{(m)}(s,\tau )-{\lambda }_1{\epsilon }_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{35}^{(m)}(s,\tau )\\ {\lambda }_1{e}_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{16}^{(m)}(s,\tau )-i{e}_{31}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{26}^{(m)}(s,\tau )-{\lambda }_1{\epsilon }_{33}^{(m)}{e}^{s{\lambda }_{1}y}{B}_{36}^{(m)}(s,\tau )\end{array}\right]}^T$$

$${\Phi }_{11}^{(m)}={\left[\begin{array}{c}-(i{c}_{13}^{(m)}{B}_{2a}^{(m)}(s,\tau )+s{k}_m{c}_{33}^{(m)}{B}_{1a}^{(m)}(s,\tau )+{\lambda }_{11}^{(m)})e^{-\left|s\right|k_{m}y}\\ (s{k}_m{c}_{33}^{(m)}{B}_{1b}^{(m)}(s,\tau )-i{c}_{13}^{(m)}{B}_{2b}^{(m)}(s,\tau )-{\lambda }_{11}^{(m)})e^{-\left|s\right|k_{m}y}\end{array}\right]}^T$$

$${\Phi }_{21}^{(m)}={\left[\begin{array}{c}-(s{k}_m{c}_{44}^{(m)}{B}_{2a}^{(m)}(s,\tau )+i{c}_{44}^{(m)}{B}_{1a}^{(m)}(s,\tau )+i{e}_{15}^{(m)}{B}_{3a}^{(m)}(s,\tau ))e^{-\left|s\right|k_{m}y}\\ (s{k}_m{c}_{44}^{(m)}{B}_{2b}^{(m)}(s,\tau )-i{c}_{44}^{(m)}{B}_{1b}^{(m)}(s,\tau )-i{e}_{15}^{(m)}{B}_{3b}^{(m)}(s,\tau ))e^{-\left|s\right|k_{m}y}\end{array}\right]}^T$$

$${\Phi }_{31}^{(m)}={\left[\begin{array}{c}(-s{k}_m{e}_{33}^{(m)}{B}_{1a}^{(m)}(s,\tau )+s{k}_m{\epsilon }_{33}^{(m)}{B}_{3a}^{(m)}(s,\tau )+P_y^{(m)})e^{-\left|s\right|k_{m}y}\\ (s{k}_m{e}_{33}^{(m)}{B}_{1b}^{(m)}(s,\tau )-s{k}_m{\epsilon }_{33}^{(m)}{B}_{3b}^{(m)}(s,\tau )+P_y^{(m)})e^{-\left|s\right|k_{m}y}\end{array}\right]}^T$$

$$f_0={\left[\begin{array}{cccccc}{f}_1(s,\tau )& f_2(s,\tau )& f_3(s,\tau )& f_4(s,\tau )& f_5(s,\tau )& f_6(s,\tau )\end{array}\right]}^T$$

$$A_0={\left[\begin{array}{cc}{A}_{1m}(s,\tau )& A_{2m}(s,\tau )\end{array}\right]}^T$$

Similar to Equation (13), the thermal stress and electric displacement fields can be expressed by defining the dislocation density functions as follows:

$$d_1(x,\tau )={\displaystyle \frac{\partial }{\partial x}}[u^{*}(x,0^{+},\tau )-u^{*}(x,0^{-},\tau )]$$

(34)

$$d_2(x,\tau )={\displaystyle \frac{\partial }{\partial x}}[w^{*}(x,0^{+},\tau )-w^{*}(x,0^{-},\tau )]$$

(35)

$$d_3(x,\tau )={\displaystyle \frac{\partial }{\partial x}}[{\varphi }^{*}(x,0^{+},\tau )-{\varphi }^{*}(x,0^{-},\tau )]$$

(36)

Substituting Equations (34)–(36) into Equations (26a)–(26d), we obtain

$${\displaystyle \sum _{k=1}^6{B}_{1k}^{(1)}(s,\tau )}{f}_k(s,\tau )={\displaystyle \frac{{\eta }_1{\overline{I}}_1-{\eta }_2{\overline{I}}_2-{\eta }_3{\overline{I}}_3+{\overline{a}}_1}{\eta }}$$

(37)

$${\displaystyle \sum _{k=1}^6{B}_{1k}^{(2)}}(s,\tau )f_k(s,\tau )={\displaystyle \frac{{\eta }_4{\overline{I}}_1-{\eta }_5{\overline{I}}_2-{\eta }_6{\overline{I}}_3+{\overline{a}}_2}{\eta }}$$

(38)

$${\displaystyle \sum _{k=1}^6{B}_{2k}^{(1)}}(s,\tau )f_k(s,\tau )={\displaystyle \frac{{\eta }_7{\overline{I}}_1-{\eta }_8{\overline{I}}_2+{\eta }_9{\overline{I}}_3+{\overline{a}}_3}{\eta }}$$

(39)

$${\displaystyle \sum _{k=1}^6{B}_{2k}^{(2)}}(s,\tau )f_k(s,\tau )={\displaystyle \frac{{\eta }_{10}{\overline{I}}_1-{\eta }_{11}{\overline{I}}_2-{\eta }_{12}{\overline{I}}_3+{\overline{a}}_4}{\eta }}$$

(40)

$${\displaystyle \sum _{k=1}^6{B}_{3k}^{(1)}}(s,\tau )f_k(s,\tau )={\displaystyle \frac{{\eta }_{13}{\overline{I}}_1-{\eta }_{14}{\overline{I}}_2-{\eta }_{15}{\overline{I}}_3+{\overline{a}}_5}{\eta }}$$

(41)

$${\displaystyle \sum _{k=1}^6{B}_{3k}^{(2)}}(s,\tau )f_k(s,\tau )={\displaystyle \frac{{\eta }_{16}{\overline{I}}_1-{\eta }_{17}{\overline{I}}_2-{\eta }_{18}{\overline{I}}_3+{\overline{a}}_6}{\eta }}$$

(42)

where

$$\left\{\begin{array}{c}{\overline{I}}_1={\displaystyle \frac{i}{\tau }}{\displaystyle {\int }_{-l}^l{d}_1(\xi ,\tau )}{e}^{i\tau s}d\xi \\ {\overline{I}}_2={\displaystyle \frac{i}{\tau }}{\displaystyle {\int }_{-l}^l{d}_2(\xi ,\tau )}{e}^{i\tau s}d\xi \\ {\overline{I}}_3={\displaystyle \frac{i}{\tau }}{\displaystyle {\int }_{-l}^l{d}_3(\xi ,\tau )}{e}^{i\tau s}d\xi \end{array}\right.$$

(43)

The expressions of $\eta$, ${\overline{a}}_p$, and ${\eta }_q$, $p=1,2,\dots ,6$, $q=1,2,\dots ,18$, are given in the Supplementary Materials. Analogous to the temperature field solution, once ${\overline{I}}_1$, ${\overline{I}}_2$, and ${\overline{I}}_3$ are determined, the thermal stress and electric displacement fields can be obtained.

Substituting Equations (37)–(42) into Equations (31)–(33), we get

$$\begin{array}{l}{\sigma }_{yy}^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6\frac{s{e}^{s{\lambda }_{k}y}}{\eta }}}[(-i{c}_{13}^{(2)}{\eta }_{10}+{\lambda }_k{c}_{33}^{(2)}{\eta }_4+{\lambda }_k{e}_{33}^{(2)}{\eta }_{16}){\overline{I}}_1\\ \begin{array}{ccc}& & \end{array}+(-i{c}_{13}^{(2)}{\eta }_{11}-{\lambda }_k{c}_{33}^{(2)}{\eta }_5-{\lambda }_k{e}_{33}^{(2)}{\eta }_{17}){\overline{I}}_2+(i{c}_{13}^{(2)}{\eta }_{12}-{\lambda }_k{c}_{33}^{(2)}{\eta }_6-{\lambda }_k{e}_{33}^{(2)}{\eta }_{18}){\overline{I}}_3]e^{-isx}ds\\ \begin{array}{ccc}& & \end{array}+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{c}_{13}^{(2)}(B_{2a}{A}_{12}+B_{2b}{A}_{22})}-{\lambda }_{11}^{(2)}(A_{12}+A_{22})+s{k}_2{e}_{33}^{(2)}\cdot \\ \begin{array}{ccc}& & \end{array}(-B_{3a}{A}_{12}+B_{3b}{A}_{22})+s{k}_2{c}_{33}^{(2)}(-B_{1a}{A}_{12}+B_{1b}{A}_{22})]e^{-isx}ds=0\end{array}$$

(44)

$$\begin{array}{l}{\sigma }_{xy}^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6\frac{s{e}^{s{\lambda }_{\mathrm{k}}y}}{\eta }}}[({\lambda }_k{c}_{44}^{(2)}{\eta }_{10}-i{\lambda }_k{c}_{44}^{(2)}{\eta }_4-i{e}_{15}^{(2)}{\eta }_{16}){\overline{I}}_1\\ \begin{array}{ccc}& & \end{array}+(i{c}_{44}^{(2)}{\eta }_5-{\lambda }_k{c}_{44}^{(2)}{\eta }_{11}+i{e}_{15}^{(2)}{\eta }_{17}){\overline{I}}_2+(i{c}_{44}^{(2)}{\eta }_6-{\lambda }_k{c}_{44}^{(2)}{\eta }_{12}+i{e}_{15}^{(2)}{\eta }_{18}){\overline{I}}_3]e^{-isx}ds\\ \begin{array}{ccc}& & \end{array}+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{c}_{44}^{(2)}(B_{1a}{A}_{12}+B_{1b}{A}_{22})}+s{k}_2{c}_{44}^{(2)}(-B_{2a}{A}_{12}+B_{2b}{A}_{22})\\ \begin{array}{ccc}& & \end{array}-i{e}_{15}^{(2)}(-B_{3a}{A}_{12}+B_{3b}{A}_{22})]e^{-isx}ds=0\end{array}$$

(45)

$$\begin{array}{l}{D}_y^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6\frac{s{e}^{s{\lambda }_{k}y}}{\eta }}}[({\lambda }_k{e}_{33}^{(2)}{\eta }_4-i{e}_{31}^{(2)}{\eta }_{10}-{\lambda }_k{\epsilon }_{33}^{(2)}{\eta }_{16}){\overline{I}}_1\\ \begin{array}{ccc}& & \end{array}+(i{e}_{31}^{(2)}{\eta }_{11}-{\lambda }_k{e}_{33}^{(2)}{\eta }_5+{\lambda }_k{\epsilon }_{33}^{(2)}{\eta }_{17}){\overline{I}}_2+(i{e}_{31}^{(2)}{\eta }_{12}-{\lambda }_k{e}_{33}^{(2)}{\eta }_6+{\lambda }_k{\epsilon }_{33}^{(2)}{\eta }_{18}){\overline{I}}_3]e^{-isx}ds\\ \begin{array}{ccc}& & \end{array}+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{e}_{31}^{(2)}(B_{2a}{A}_{12}+B_{2b}{A}_{22})}-p_{\mathrm{y}}^{(2)}(A_{12}+A_{22})-s{k}_2{\epsilon }_{33}^{(2)}\cdot \\ \begin{array}{ccc}& & \end{array}(-B_{3a}{A}_{12}+B_{3b}{A}_{22})+s{k}_2{e}_{33}^{(2)}(-B_{1a}{A}_{12}+B_{1b}{A}_{22})]e^{-isx}ds=0\end{array}$$

(46)

Equations (44)–(46) are divided into two parts, as follows:

$${\sigma }_{yy}^{*}(x,0^{-},\tau )={\sigma }_{yy1}^{*}(x,0^{-},\tau )+{\sigma }_{yy2}^{*}(x,0^{-},\tau )$$

(47)

$${\sigma }_{xy}^{*}(x,0^{-},\tau )={\sigma }_{xy1}^{*}(x,0^{-},\tau )+{\sigma }_{xy2}^{*}(x,0^{-},\tau )$$

(48)

$$D_y^{*}(x,0^{-},\tau )=D_{y1}^{*}(x,0^{-},\tau )+D_{y2}^{*}(x,0^{-},\tau )$$

(49)

where

$$\begin{array}{l}{\sigma }_{yy1}^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6\frac{s{e}^{s{\lambda }_{k}y}}{H}}}[(-i{c}_{13}^{(2)}{\eta }_{10}+{\lambda }_k{c}_{33}^{(2)}{\eta }_4+{\lambda }_k{e}_{33}^{(2)}{\eta }_{16}){\overline{I}}_1\\ +(-i{c}_{13}^{(2)}{\eta }_{11}-{\lambda }_k{c}_{33}^{(2)}{\eta }_5-{\lambda }_k{e}_{33}^{(2)}{\eta }_{17}){\overline{I}}_2+(i{c}_{13}^{(2)}{\eta }_{12}-{\lambda }_k{c}_{33}^{(2)}{\eta }_6-{\lambda }_k{e}_{33}^{(2)}{\eta }_{18}){\overline{I}}_3]e^{-isx}ds\end{array}$$

(50)

$$\begin{array}{l}{\sigma }_{xy1}^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6\frac{s{e}^{s{\lambda }_{\mathrm{k}}y}}{\eta }}}[({\lambda }_k{c}_{44}^{(2)}{\eta }_{10}-i{\lambda }_k{c}_{44}^{(2)}{\eta }_4-i{e}_{15}^{(2)}{\eta }_{16}){\overline{I}}_1\\ +(i{c}_{44}^{(2)}{\eta }_5-{\lambda }_k{c}_{44}^{(2)}{\eta }_{11}+i{e}_{15}^{(2)}{\eta }_{17}){\overline{I}}_2+(i{c}_{44}^{(2)}{\eta }_6-{\lambda }_k{c}_{44}^{(2)}{\eta }_{12}+i{e}_{15}^{(2)}{\eta }_{18}){\overline{I}}_3]e^{-isx}ds\end{array}$$

(51)

$$\begin{array}{l}{D}_{y1}^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{k=1}^6\frac{s{e}^{s{\lambda }_{k}y}}{\eta }}}[({\lambda }_k{e}_{33}^{(2)}{\eta }_4-i{e}_{31}^{(2)}{\eta }_{10}-{\lambda }_k{\epsilon }_{33}^{(2)}{\eta }_{16}){\overline{I}}_1\\ +(i{e}_{31}^{(2)}{\eta }_{11}-{\lambda }_k{e}_{33}^{(2)}{\eta }_5+{\lambda }_k{\epsilon }_{33}^{(2)}{\eta }_{17}){\overline{I}}_2+(i{e}_{31}^{(2)}{\eta }_{12}-{\lambda }_k{e}_{33}^{(2)}{\eta }_6+{\lambda }_k{\epsilon }_{33}^{(2)}{\eta }_{18}){\overline{I}}_3]e^{-isx}ds\end{array}$$

(52)

$$\begin{array}{l}{\sigma }_{yy2}^{*}(x,0^{-},\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{c}_{13}^{(2)}(B_{2a}{A}_{12}+B_{2b}{A}_{22})}-{\lambda }_{11}^{(2)}(A_{12}+A_{22})+s{k}_2{e}_{33}^{(2)}\cdot \\ (-B_{3a}{A}_{12}+B_{3b}{A}_{22})+s{k}_2{c}_{33}^{(2)}(-B_{1a}{A}_{12}+B_{1b}{A}_{22})]e^{-isx}ds\end{array}$$

(53)

$$\begin{array}{l}{\sigma }_{xy2}^{*}(x,0^{-},\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{c}_{44}^{(2)}(B_{1a}{A}_{12}+B_{1b}{A}_{22})}+s{k}_2{c}_{44}^{(2)}(-B_{2a}{A}_{12}+B_{2b}{A}_{22})\\ -i{e}_{15}^{(2)}(-B_{3a}{A}_{12}+B_{3b}{A}_{22})]e^{-isx}ds\end{array}$$

(54)

$$\begin{array}{l}{D}_{y2}^{*}(x,0^{-},\tau )={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{e}_{31}^{(2)}(B_{2a}{A}_{12}+B_{2b}{A}_{22})}-p_{\mathrm{y}}^{(2)}(A_{12}+A_{22})-s{k}_2{\epsilon }_{33}^{(2)}\cdot \\ (-B_{3a}{A}_{12}+B_{3b}{A}_{22})+s{k}_2{e}_{33}^{(2)}(-B_{1a}{A}_{12}+B_{1b}{A}_{22})]e^{-isx}ds\end{array}$$

(55)

Clearly, Equations (53)–(55) can be readily solved by following Equation (27). Subsequently, Equations (66)–(68) must be transformed into first-kind singular integral equations with a Cauchy kernel.

In an alternative formulation, Equations (50)–(52) can be expressed as follows:

$${\sigma }_{yy1}^{*}(x,0^{-},t)=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{s{e}^{s\lambda y}}{\eta }}(Z_1{\overline{I}}_1+Z_2{\overline{I}}_2+Z_3{\overline{I}}_3)e^{-isx}ds$$

(56)

$${\sigma }_{xy1}^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{s{e}^{s\lambda y}}{\eta }}(Z_4{\overline{I}}_1+Z_5{\overline{I}}_2+Z_6{\overline{I}}_3)e^{-isx}ds$$

(57)

$$D_{y1}^{*}(x,0^{-},\tau )=\underset{y\to 0^{-}}{\lim }{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{s{e}^{s\lambda y}}{\eta }}(Z_7{\overline{I}}_1+Z_8{\overline{I}}_2+Z_9{\overline{I}}_3)e^{-isx}ds$$

(58)

The expressions of $e^{s\lambda y}$ and $Z_r$, $r=1,2,\dots ,9$ are provided in the Supplementary Materials.

Substituting Equation (43) into Equations (56)–(58), we get

$$\begin{array}{l}{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_1{e}^{is(\tau -x)}ds)d_1(t,\tau )dt+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_2{e}^{is(\tau -x)}ds)d_2(t,\tau )dt\\ +{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_3{e}^{is(\tau -x)}ds)d_3(t,\tau )dt={\sigma }_1\end{array}$$

(59)

$$\begin{array}{l}{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_4{e}^{is(\tau -x)}ds)d_4(t,\tau )dt+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_5{e}^{is(\tau -x)}ds)d_5(t,\tau )dt\\ +{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_6{e}^{is(\tau -x)}ds)d_6(t,\tau )dt={\sigma }_2\end{array}$$

(60)

$$\begin{array}{l}{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_7{e}^{is(\tau -x)}ds)d_7(t,\tau )dt+{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_8{e}^{is(\tau -x)}ds)d_8(t,\tau )dt\\ +{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-l}^l({\displaystyle {\int }_{-\infty }^{+\infty }\frac{i{e}^{s\lambda y}}{\eta }}}{Z}_9{e}^{is(\tau -x)}ds)d_9(t,\tau )dt={\sigma }_3\end{array}$$

(61)

where

$$\begin{array}{cc}\hfill {\sigma }_1=& -{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{c}_{13}^{(2)}(B_{2a}{A}_{12}{e}^{-\left|s\right|k_{2}y}+B_{2b}{A}_{22}{e}^{\left|s\right|k_{2}y})}-{\lambda }_{11}^{(2)}(A_{12}+A_{22})+s{k}_2{e}_{33}^{(2)}\cdot \hfill \\ & (-B_{3a}{A}_{12}{e}^{-\left|s\right|k_{2}y}+B_{3b}{A}_{22}{e}^{\left|s\right|k_{2}y})+s{k}_2{c}_{33}^{(2)}(-B_{1a}{A}_{12}{e}^{-\left|s\right|k_{2}y}+B_{1b}{A}_{22}{e}^{\left|s\right|k_{2}y})]e^{-isx}ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\sigma }_2=& -{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{c}_{44}^{(2)}(B_{1a}{A}_{12}+B_{1b}{A}_{22})}+s{k}_2{c}_{44}^{(2)}(-B_{2a}{A}_{12}+B_{2b}{A}_{22})\hfill \\ & -i{e}_{15}^{(2)}(-B_{3a}{A}_{12}+B_{3b}{A}_{22})]e^{-isx}ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\sigma }_3=& -{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{+\infty }[-i{e}_{31}^{(2)}(B_{2a}{A}_{12}{e}^{-\left|s\right|k_{2}y}+B_{2b}{A}_{22}{e}^{\left|s\right|k_{2}y})}-p_{\mathrm{y}}^{(2)}(A_{12}+A_{22})-s{k}_2{\epsilon }_{33}^{(2)}\cdot \hfill \\ & (-B_{3a}{A}_{12}{e}^{-\left|s\right|k_{2}y}+B_{3b}{A}_{22}{e}^{\left|s\right|k_{2}y})+s{k}_2{e}_{33}^{(2)}(-B_{1a}{A}_{12}{e}^{-\left|s\right|k_{2}y}+B_{1b}{A}_{22}{e}^{\left|s\right|k_{2}y})]e^{-isx}ds\hfill \end{array}$$

On the crack surfaces

$$\begin{array}{l}{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_1}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_1(x,t,\tau ))d_1(t,\tau )dt+{\displaystyle {\int }_{-l}^l\frac{{\overline{Z}}_2}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_2(x,t,\tau ))d_2(t,\tau )dt\\ +{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_3}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_3(x,t,\tau ))d_3(t,\tau )dt=\pi {\sigma }_1\end{array}$$

(62)

$$\begin{array}{l}{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_4}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_4(x,t,\tau ))d_1(t,\tau )dt+{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_5}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_5(x,t,\tau ))d_2(t,\tau )dt\\ +{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_6}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_6(x,t,\tau ))d_3(t,\tau )dt=\pi {\sigma }_2\end{array}$$

(63)

$$\begin{array}{l}{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_7}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_7(x,t,\tau ))d_1(t,\tau )dt+{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_8}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_8(x,t,\tau ))d_2(t,\tau )dt\\ +{\displaystyle {\int }_{-l}^l(\frac{{\overline{Z}}_9}{\overline{\eta }}}{\displaystyle \frac{1}{t-x}}+{\kappa }_9(x,t,\tau ))d_3(t,\tau )dt=\pi {\sigma }_3\end{array}$$

(64)

where

$${\kappa }_1(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_1(s)}{\eta }-\frac{{\tilde{Z}}_1}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_2(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_2(s)}{\eta }-\frac{{\overline{Z}}_2}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_3(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_3(s)}{\eta }-\frac{{\overline{Z}}_3}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_4(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_4(s)}{\eta }-\frac{{\overline{Z}}_4}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_5(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_5(s)}{\eta }-\frac{{\overline{Z}}_5}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_6(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_6(s)}{\eta }-\frac{{\overline{Z}}_6}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_7(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_7(s)}{\eta }-\frac{{\overline{Z}}_7}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_8(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_8(s)}{\eta }-\frac{{\overline{Z}}_8}{\overline{\eta }})}\sin (s(t-x))ds$$

$${\kappa }_9(x,t,\tau )={\displaystyle {\int }_0^{\infty }(\frac{Z_9(s)}{\eta }-\frac{{\overline{Z}}_9}{\overline{\eta }})}\sin (s(t-x))ds$$

The expressions of $\overline{\eta }$ and ${\overline{Z}}_r$ are provided in the Supplementary Materials.

In Equations (34)–(36), the unknown functions $d_1(x,\tau )$, $d_2(x,\tau )$, and $d_3(x,\tau )$ satisfy the following conditions:

$$\left\{\begin{array}{c}{\displaystyle {\int }_{-l}^l{d}_1(\xi ,\tau )}d\xi =0\\ {\displaystyle {\int }_{-l}^l{d}_2(\xi ,\tau )}d\xi =0\\ {\displaystyle {\int }_{-l}^l{d}_3(\xi ,\tau )}d\zeta =0\end{array}\right.$$

(65)

To solve the singular integral Equations (62)–(64), the unknown functions $d_n(\xi ,\tau )$, $n=1,2,3$ exhibit a square root singularity:

$$d_n(\xi ,\tau )={\displaystyle \frac{1}{\sqrt{l^2-{\xi }^2}}}{g}_n(\xi ,\tau )$$

(66)

where $g_n(\xi ,\tau )$ are continuous functions defined in the interval $[-l,l]$.

The intensity factor in the Laplace transform plane can be expressed as follows:

$$K_I^{*}=\underset{x\to l^{+}}{\lim }\sqrt{2\pi (l-x)}{\sigma }_{yy}^{*}(x,0,\tau )$$

(67)

$$K_{II}^{*}=\underset{x\to l^{+}}{\lim }\sqrt{2\pi (l-x)}{\sigma }_{xy}^{*}(x,0,\tau )$$

(68)

$$K_D^{*}=\underset{x\to l^{+}}{\lim }\sqrt{2\pi (l-x)}{D}_y^{*}(x,0,\tau )$$

(69)

The intensity factors in the time domain are given by the inverse Laplace transform as follows:

$$K_I=\sqrt{\pi l}[{\displaystyle \frac{{\overline{Z}}_1}{\overline{\eta }}}{g}_1(-l,\tau )+{\displaystyle \frac{{\overline{Z}}_2}{\overline{\eta }}}{g}_2(-l,\tau )+{\displaystyle \frac{{\overline{Z}}_3}{\overline{\eta }}}{g}_3(-l,\tau )]$$

(70)

$$K_{II}=\sqrt{\pi l}[{\displaystyle \frac{{\overline{Z}}_4}{\overline{\eta }}}{g}_1(-l,\tau )+{\displaystyle \frac{{\overline{Z}}_5}{\overline{\eta }}}{g}_2(-l,\tau )+{\displaystyle \frac{{\overline{Z}}_6}{\overline{\eta }}}{g}_3(-l,\tau )]$$

(71)

$$K_D=-\sqrt{\pi l}[{\displaystyle \frac{{\overline{Z}}_7}{\overline{\eta }}}{g}_1(l,\tau )+{\displaystyle \frac{{\overline{Z}}_8}{\overline{\eta }}}{g}_2(l,\tau )+{\displaystyle \frac{{\overline{Z}}_9}{\overline{\eta }}}{g}_3(l,\tau )]$$

(72)

Based on the numerical method of Hu et al. [24], the stress intensity factors and the electric displacement intensity factors in the time domain can finally be calculated.

4. Numerical Examples and Discussion

To verify the correctness of the derivation, we select PZT-5H piezoelectric ceramic as the substrate material, with its material parameters as follows:

$$c_{11}^{(2)}=12.6\times {10}^{10}{ \mathrm{N}/\mathrm{m}}^2, e_{15}^{(2)}=17.44{ \mathrm{C}/\mathrm{m}}^2$$

$$c_{13}^{(2)}=8.41\times {10}^{10}{ \mathrm{N}/\mathrm{m}}^2, {\epsilon }_{11}^{(2)}=150.3\times {10}^{-10} \mathrm{C}/\mathrm{Vm}$$

$$c_{44}^{(2)}=2.3\times {10}^{10}{ \mathrm{N}/\mathrm{m}}^2, {\epsilon }_{33}^{(2)}=130.0\times {10}^{-10} \mathrm{C}/\mathrm{Vm}$$

(73)

$$e_{31}^{(2)}=-6.5{ \mathrm{C}/\mathrm{m}}^2, {\lambda }_{11}^{(2)}=0.594\times {10}^6{ \mathrm{N}/\mathrm{m}}^2\mathrm{K}$$

$$e_{33}^{(2)}=23.3{ \mathrm{C}/\mathrm{m}}^2, {\lambda }_{33}^{(2)}=0.468\times {10}^6{ \mathrm{N}/\mathrm{m}}^2\mathrm{K}$$

where superscript 2 represents the substrate material.

4.1. Verification

To validate the derivation accuracy, we simplified the fine-grained coating/substrate structure in Figure 1 to a single PZT-5H piezoelectric ceramic plate with a central crack, subjected to a thermal shock load of $T_{2}H(t)$.

According to Ueda et al. [9], $K_{I0}$,$K_{II0}$,$K_{D0}$, and $t_0$ are given by the following expressions:

$$\left\{\begin{array}{c}{K}_{I0}={\displaystyle \frac{T_2(c_{13}^{(2)}{\lambda }_{11}^{(2)}-c_{11}^{(2)}{\lambda }_{33}^{(2)})\sqrt{\pi l}}{c_{11}^{(2)}}}\\ K_{II0}={\displaystyle \frac{T_2(c_{13}^{(2)}{\lambda }_{11}^{(2)}-c_{11}^{(2)}{\lambda }_{33}^{(2)})\sqrt{\pi l}}{c_{11}^{(2)}}}\\ K_{D0}={\displaystyle \frac{T_2(c_{13}^{(2)}{\lambda }_{11}^{(2)}-c_{11}^{(2)}{\lambda }_{33}^{(2)})\sqrt{\pi l}}{e_{33}^{(2)}}}\\ t_0=\sqrt{{\displaystyle \frac{l^2(c_{44}^{(2)}{\epsilon }_{11}^{(2)}+{(e_{12}^{(2)})}^2)}{{\rho }^{(2)}{\epsilon }_{11}^{(2)}}}}\end{array}\right.$$

(74)

The temporal variation in the mode I dynamic stress intensity factor is presented in Figure 2.

Figure 2 shows that the mode I thermal stress intensity factor increases to a peak at time $t/t_0=0.02$, then decreases and converges to a steady-state value. At time $t/t_0=0.10$, it declines to 0.8, consistent with Reference [11].

4.2. Dynamic Intensity Factors of Fine-Grained Piezoelectric Coating/Substrate Structure

This section investigates the influence of material parameters on the dynamic stress intensity factor. Material 1 is defined as a fine-grained PZT-5H piezoelectric coating, while Material 2 is the PZT-5H substrate with conventional grain structure. Here, 1 and 2 denote the fine-grained coating and substrate, respectively.

4.2.1. Discussing the Effect of Material Parameters on Dynamic Intensity Factor

Figure 3 shows the effect of different elastic moduli on the dynamic intensity factor $K_I$ under various non-dimensional parameters $t/t_0$ (where $h_1/h_2=1/500$, $h_1=0.5\mathrm{mm}$, and $l=2\mathrm{mm}$). The $K_I$ value increases to a peak, then decreases and converges to a steady state. As the elastic modulus increases, the peak $K_I$ rises, and the time to reach steady state shortens. These results demonstrate the significant influence of elastic modulus on $K_I$ under thermal shock loading. It is also found that as the elastic modulus ratio increases from 1.5 to 2.5 and the peak $K_I/K_{I0}$ rises by 13%–35%, while the steady-state time decreases from 0.8 to 0.6.

Figure 4 illustrates the effect of the coating thickness on the dynamic intensity factor under various non-dimensional parameters $t/t_0$ (where $h_2=250\mathrm{mm}$, $c_{44}^{(1)}/c_{44}^{(2)}=1.5$, $l=2\mathrm{mm}$). As time progresses, the dynamic intensity factor initially increases, then decreases, and finally stabilizes. However, increasing coating thickness reduces its peak value—contrary to the effect of elastic modulus. It is also found that increasing the coating thickness ratio from 0.01 to 0.15 results in a reduction in the peak intensity factor by approximately 22% to 32%.

Figure 5 reveals the effects of the crack length on the dynamic intensity factor for different $t/t_0$ (where $h_1/h_2=1/500$, $h_1=0.5mm$, and $c_{44}^{(1)}/c_{44}^{(2)}=1.5$). It is shown that as time increases, the variation law of the dynamic intensity factor is similar to that in Figure 4 and Figure 5. However, it is further found that as the crack length increases, the peak value of the dynamic intensity factor increases more significantly (it decreases by up to approximately 60%) than in the previous two cases, which indicates that the crack size exerts a more pronounced influence on the dynamic intensity factor.

4.2.2. Discussing the Effect of Material Parameters on the Dynamic Intensity Factor $K_{II}$

Figure 6 illustrates the effect of coating thickness on the dynamic stress intensity factor $K_{II}$ under non-dimensional parameters $t/t_0$ (where $h_1/h_2=1/500$, $h_1=0.5\mathrm{mm}$, $l=2\mathrm{mm}$). As time increases, the dynamic intensity factor initially increases, then decreases, but exhibits a transient rebound near $t/t_0=0.5$, before stabilizing—distinct from the monotonic evolution observed in the dynamic intensity factor $K_I$. Furthermore, increasing the elastic modulus ratio elevates the dynamic stress intensity factor peak value by 26%–37%. These results demonstrate that material elasticity critically governs dynamic fracture behavior, and structural safety can be enhanced by optimizing elastic modulus matching.

Figure 7 demonstrates the influence of the coating thickness on the dynamic intensity factor $K_{II}$ under varying non-dimensional parameters $t/t_0$ (where $h_2=250\mathrm{mm}$, $c_{44}^{(1)}/c_{44}^{(2)}=1.5$, and $l=2\mathrm{mm}$). As time progresses, the dynamic intensity factor exhibits a trend similar to that in Figure 6, but with amplified fluctuation amplitude as coating thickness increases. Specifically, a 15-fold increase in thickness attenuates the dynamic intensity factor peak value by 22%–36%. These phenomena also indicate that the coating thickness exerts a significant influence on the dynamic intensity factor $K_{II}$.This attenuation may also be related to the enhanced stress redistribution capacity in thick coatings.

Figure 8 depicts the influence of the crack length on the dynamic intensity factor $K_{II}$ under non-dimensional parameters $t/t_0$ (where $h_1/h_2=1/500$, $h_1=0.5\mathrm{mm}$, and $c_{44}^{(1)}/c_{44}^{(2)}=1.5$). As time increases, the dynamic intensity factor $K_{II}$ exhibits a trend similar to that in Figure 5, but with a 23%–56% lower peak value. Notably, with the increase in crack length, the time for the dynamic intensity factor $K_{II}$ curve to tend to a steady state lags behind relative to that in Figure 5.

4.2.3. Discussing the Effect of Material Parameters on the Electric Displacement Intensity Factor $K_D$

Figure 9 shows the effect of elastic moduli on the electric displacement intensity factor $K_D$ as under non-dimensional parameters $t/t_0$ (where $h_1/h_2=1/500$, $h_1=0.5\mathrm{mm}$, and $l=2\mathrm{mm}$). As time progresses, the absolute electric displacement intensity factor initially increases, peaks, then declines before stabilizing—distinct from the rapid stabilization observed for dynamic stress intensity factors ($K_I$ and $K_{II}$). Notably, increasing the elastic modulus ratio elevates the electric displacement intensity factor peak magnitude by 31%–50%, confirming that the elastic modulus of the material also exerts a significant influence on the electric displacement intensity factor.

Figure 10 demonstrates the influence of the coating thickness on the electric displacement intensity factor $K_{II}$ under varying non-dimensional parameters $t/t_0$ (where $h_2=250\mathrm{mm}$, $c_{44}^{(1)}/c_{44}^{(2)}=1.5$, and $l=2\mathrm{mm}$). As time increases, the electric displacement intensity factor $K_D$ exhibits a trend similar to that in Figure 9, but with a 19%–48% reduction in peak magnitude. This attenuation suggests that coating thickness exerts a weaker influence on electric displacement intensity factor than elastic modulus, which elevates the electric displacement intensity factor peak. Nevertheless, increasing coating thickness alters the electric displacement intensity factor peak value by 22%–41%.

Figure 11 reveals the effects of the crack length on the electric displacement intensity factor $K_{II}$ under non-dimensional parameters $t/t_0$ (where $h_1/h_2=1/500$, $h_1=0.5\mathrm{mm}$, and $c_{44}^{(1)}/c_{44}^{(2)}=1.5$). As time progresses, the peak magnitude of the electric displacement intensity factor $K_D$ increases by 16%–40% with crack length extension. This fully demonstrates that the crack length exerts a very significant influence on the electric displacement intensity factor $K_D$.

5. Conclusions

This study investigates the dynamic fracture behavior of a Griffith crack in a fine-grained piezoelectric coating/substrate system under thermal impact loading. Utilizing integral transformation, the dynamic response problem is reduced to a system of singular integral equations, and expressions for thermal stress and dynamic intensity factors are derived. Subsequently, the effects of crack size, coating thickness, and material elastic modulus on the dynamic fracture behavior of the crack under thermal shock loading are analyzed. The results are as follows:

Increasing the elastic modulus ratio elevates dynamic stress intensity factors peaks by 13%–35% (Figure 3) and electric displacement intensity factors peaks by 31%–50% (Figure 9), while reducing dynamic stress intensity factors stabilization time by 25% (from 0.8 to 0, Figure 3). This confirms material elasticity as the primary driver for intensifying crack-tip stress/charge concentration.

Increasing coating thickness ratio reduces dynamic stress intensity factors peaks by 22%–32% (Figure 4) and electric displacement intensity factors peaks by 19%–48% (Figure 10), with thicker coatings (15 × baseline) attenuating dynamic stress intensity factors fluctuations by 22%–36% (Figure 7). This demonstrates thickness-enhanced stress redistribution capacity.

Crack extension increases dynamic stress intensity factors peaks by up to 60% (Figure 5) and electric displacement intensity factors peaks by 16%–40% (Figure 11). This highlights crack length as the most potent geometric parameter governing fracture severity.