Dynamic non-local theory analysis of multiple cracks in a FGPM under anti-plane shear waves

Abstract

This paper investigates the dynamic interaction of multiple cracks in a functionally graded piezoelectric material (FGPM) plane subject to an anti-plane shear stress wave. The non-local theory and Schmidt method are used. Employing the Fourier transform techniques, the problem is converted to dual integral equations, which the unknown variables are jumps of displacements across the crack surfaces. The present solution exhibits no stress and electric displacement singularities at the crack tips and the stress and electric displacement fields near the crack tips are obtained. The non-local elastic solutions yield a finite hoop stress and electric displacement at the crack tips. This research will give us ideas on material selection for evaluating strength and preventing material failure of FGPM with multiple cracks.

Keywords

Anti-plane shear waves functionally graded piezoelectric material multiple cracks non-local theory solid mechanics

1. Introduction

The functionally graded piezoelectric material (FGPM) has been applied in several environments with extremely high temperature. Therefore, the knowledge of cracks growth in FGPM is important in designing components of FGPM. Recently, the fracture analysis of FGPM has received much attention [1–4]. In engineering, materials are occasionally weakened by several cracks and this not only concerns single crack or two parallel cracks. Interaction effect takes place between multiple cracks. The static and dynamic of multiple cracks fracture problems have been reported in several works [5–9]. For example, Zhou and Wang [6] studied two parallel limited-permeable Mode-I cracks or four parallel limited-permeable Mode-I cracks in the piezoelectric materials. The problem of a homogeneous linear elastic body containing multiple non-collinear cracks under anti-plane dynamic loading has been considered by Wu et al. [8].

All the above investigations have a common conclusion that the stress field or electric displacement field is singular near the crack tips. To overcome the stress singularity, the stresses near the sharp crack tips in an isotropic elastic plate under uniform tension, shear and anti-plane shear have been investigated by Eringen et al. [10] and Eringen [11,12]. Several nonlocal theories have been formulated to address strain-gradient and size effects [13]. In recent years, Zhou and Wang [14] analyzed the non-local theory solution of two collinear cracks in functionally graded materials. The dynamic non-local theory solution of two parallel symmetric cracks in piezoelectric materials was solved by Zhou et al. [15]. The non-local theory solution of two collinear Mode-I cracks in piezoelectric materials and piezoelectric/piezomagnetic material have been solved by Liang [16] and Zhou et al. [17], respectively. Chen et al. [18] considered anti-plane transverse wave propagation in nanoscale periodic layered piezoelectric structures. A novel non-local lattice particle method for three-dimensional elasticity and fracture simulation of isotropic solids was proposed by Chen and Liu [19].

To the best knowledge of the authors, no results on dynamic non-local solution of multiple cracks in FGPM have been reported in any open literature. This problem is solved by the Schmidt method [20] in the paper. Using Fourier transform and a mixed boundary value problem is reduced to solving dual integral equations. The present solutions do not contain the stress and electric displacement singularities near the crack tips.

2. Formulation of the problem

2.1. Boundary conditions

As shown in Fig. 1, consider four parallel cracks of length 1 − b in a FGPM plane and h is the distance between parallel cracks with respect to the rectangular coordinates (x, y). Owing to the incident wave is a harmonic anti-plane shear stress wave, the quantity $w_{0}^{(j)}(x,y,t)$ , ${\phi}_{0}^{(j)}(x,y,t)$ , ${\tau}_{zk0}^{(j)}(x,y,t)$ and $D_{k0}^{(j)}(x,y,t)$ can be assumed as $\begin{eqnarray}\displaystyle [w_{0}^{(j)}(x,y,t),{\phi}_{0}^{(j)}(x,y,t),{\tau}_{zk0}^{(j)}(x,y,t),D_{k0}^{(j)}(x,y,t)] & & \displaystyle \nonumber\\ \displaystyle =[w^{(j)}(x,y),{\phi}^{(j)}(x,y),{\tau}_{zk}^{(j)}(x,y),D_{k}^{(j)}(x,y)]e^{-i{\omega}t} & & \displaystyle\end{eqnarray}$ (1) where 𝜔 be the circular frequency of the incident wave, $w_{0}^{(j)}(x,y,t)$ is the mechanical displacement and ${\phi}_{0}^{(j)}(x,y,t)$ is electric potential; ${\tau}_{zk0}^{(j)}(x,y,t)$ is the non-local shear stress field and $D_{k0}^{(j)}(x,y,t)$ is non-local electric displacement field. All quantities with superscript j (j =1,2,3) represent layer 1, layer 2 and layer 3. The time dependence of e ^−i𝜔t will be suppressed but understood. As discussed in Soh et al. [21], the permeable condition is enforced and the boundary conditions can be given as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\tau}_{yz}^{(1)}(x,h^{+})={\tau}_{yz}^{(2)}(x,h^{-})=-{\tau}_{0},\quad b\leq \left|x\right|\leq 1\\[5pt] {\tau}_{yz}^{(1)}(x,h^{+})={\tau}_{yz}^{(2)}(x,h^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\\[5pt] D_{y}^{(1)}(x,h^{+})=D_{y}^{(2)}(x,h^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\\[5pt] {\phi}^{(1)}(x,h^{+})={\phi}^{(2)}(x,h^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\\[5pt] w^{(1)}(x,h^{+})=w^{(2)}(x,h^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\end{array}\right. & & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\tau}_{yz}^{(2)}(x,0^{+})={\tau}_{yz}^{(3)}(x,0^{-})=-{\tau}_{0},\quad r\leq \left|x\right|\leq 1\\ {\tau}_{yz}^{(2)}(x,0^{+})={\tau}_{yz}^{(3)}(x,0^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\\ D_{y}^{(2)}(x,0^{+})=D_{y}^{(3)}(x,0^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\\ {\phi}^{(2)}(x,0^{+})={\phi}^{(3)}(x,0^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\\ w^{(2)}(x,0^{+})=w^{(3)}(x,0^{-}),\quad \left|x\right|>1\quad \text{and}\quad \left|x\right|\lt b\end{array}\right. & & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle w^{(j)}(x,y)={\phi}^{(j)}(x,y)=0,(x^{2}+y^{2})^{1/2}\rightarrow \infty (j=1,2,3) & & \displaystyle\end{eqnarray}$ (4) in which 𝜏₀ is a magnitude of the incident wave.

2.2. Basic equations of non-local FGPM

The equilibrium equations of non-local FGPM with the absence of body force and free charge are given as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\frac{\partial {\tau}_{xz}^{(j)}(x,y)}{\partial x}+\frac{\partial {\tau}_{yz}^{(j)}(x,y)}{\partial y}=-{\rho}(y){\omega}^{2}w^{(j)}(x,y)\\[4.79993pt] \frac{\partial D_{x}^{(j)}(x,y)}{\partial x}+\frac{\partial D_{y}^{(j)}(x,y)}{\partial y}=0\end{array}\right. & & \displaystyle\end{eqnarray}$ (5) where 𝜌(y) is the material density.

The constitutive equations of non-local FGPM are stated as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\tau}_{kz}^{(j)}(X)=\displaystyle \int _{V}[c_{44}^{\prime }(\left|X^{\prime }-X\right|)w_{,k}^{(j)}(X^{\prime })+e_{15}^{\prime }(\left|X^{\prime }-X\right|){\phi}_{,k}^{(j)}(X^{\prime })]dV(X^{\prime })\\[4.79993pt] D_{k}^{(j)}(X)=\displaystyle \int _{V}[e_{15}^{\prime }(\left|X^{\prime }-X\right|)w_{,k}^{(j)}(X^{\prime })-{\varepsilon}_{11}^{\prime }(\left|X^{\prime }-X\right|){\phi}_{,k}^{(j)}(X^{\prime })]dV(X^{\prime })\end{array}\right.,\quad (k=x,y) & & \displaystyle\end{eqnarray}$ (6)

The stress non-local ${\tau}_{zk}^{(j)}(X)$ and the non-local electric displacement $D_{k}^{(j)}(X)$ at a point X depends on $w_{,k}^{(j)}(X)$ and ${\phi}_{,k}^{(j)}(X)$ , at all points X ^′ of the body. The integrals in equation (6) are over the volume V of the body enclosed within a surface ∂V . As stated in Eringen [22], the forms of material parameters $c_{44}^{\prime }(\left|X^{\prime }-X\right|)$ , $e_{15}^{\prime }(\left|X^{\prime }-X\right|)$ and ${\varepsilon}_{11}^{\prime }(\left|X^{\prime }-X\right|)$ can be assumed as $\begin{eqnarray}\displaystyle [c_{44}^{\prime }(\left|X^{\prime }-X\right|),e_{15}^{\prime }(\left|X^{\prime }-X\right|),{\varepsilon}_{11}^{\prime }(\left|X^{\prime }-X\right|)]=[c_{44}(y),e_{15}(y),{\varepsilon}_{11}(y)]{\alpha}(\left|X^{\prime }-X\right|) & & \displaystyle\end{eqnarray}$ (7) where 𝛼(|X ^′− X |) is the influence function.

For making the analysis tractable, the material properties of the FGPM are supposed as $\begin{eqnarray}\displaystyle [c_{44}(y),e_{15}(y),{\varepsilon}_{11}(y),{\rho}(y)]=[c_{440},e_{150},{\varepsilon}_{110},{\rho}_{0}]e^{{\lambda}y} & & \displaystyle\end{eqnarray}$ (8) where 𝜆 is the functionally graded parameter and is a positive or negative constant.

Substituting equations (7) and (8) into equation (6) yield $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\tau}_{kz}^{(j)}(X)=\displaystyle \int _{\,V}{\alpha}(\left|X^{\prime }-X\right|){\sigma}_{kz}^{(j)}(X^{\prime })dV(X^{\prime })\\[6.0pt] D_{k}^{(j)}(X)=\displaystyle \int _{\,V}{\alpha}(\left|X^{\prime }-X\right|)D_{k}^{c(j)}(X^{\prime })dV(X^{\prime })\end{array}\right.,\quad (k=x,y) & & \displaystyle\end{eqnarray}$ (9) where $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\sigma}_{kz}^{(j)}=e^{{\lambda}y}(c_{440}w_{,k}^{(j)}+e_{150}{\phi}_{,k}^{(j)})\\[6.0pt] D_{k}^{c(j)}=e^{{\lambda}y}(e_{150}w_{,k}^{(j)}-{\varepsilon}_{110}{\phi}_{,k}^{(j)})\end{array}\right. & & \displaystyle\end{eqnarray}$ (10)

The expression (10) is the classical constitutive equations of FGPM.

3. The dual integral equation

Substituting equation (9) into equation (5) and using the Green--Gauss theorem leads to $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }{\alpha}(\left|x^{\prime }-x\right|,\left|y^{\prime }-y\right|)\left\{c_{440}\left[{\nabla}^{2}w^{(j)}(x^{\prime },y^{\prime })+{\lambda}\frac{\partial w^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\right]+e_{150}\bigg[{\nabla}^{2}{\phi}^{(j)}(x^{\prime },y^{\prime })\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +\left.∼{\lambda}\frac{\partial {\phi}^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\bigg]\right\}dx^{\prime }dy^{\prime }-\left(\int _{-1}^{-b}+\int _{\,b}^{1}\right){\alpha}(\left|x^{\prime }-x\right|,h)[{\sigma}_{yz}^{(1)}(x,h^{+})-{\sigma}_{yz}^{(2)}(x,h^{-})]dx^{\prime }\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -\left(\int _{-1}^{-b}+\int _{\,b}^{1}\right){\alpha}(\left|x^{\prime }-x\right|,0)[{\sigma}_{yz}^{(2)}(x,0^{+})-{\sigma}_{yz}^{(3)}(x,0^{-})]dx^{\prime }=-{\rho}_{0}{\omega}^{2}w^{(j)}(x,y)\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }{\alpha}(\left|x^{\prime }-x\right|,\left|y^{\prime }-y\right|)\left\{e_{150}\left[{\nabla}^{2}w^{(j)}(x^{\prime },y^{\prime })+{\lambda}\frac{\partial w^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\right]-{\varepsilon}_{110}\bigg[{\nabla}^{2}{\phi}^{(j)}(x^{\prime },y^{\prime })\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \left.+∼{\lambda}\frac{\partial {\phi}^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\bigg]\right\}dx^{\prime }dy^{\prime }-\left(\int _{-1}^{-b}+\int _{\,b}^{1}\right){\alpha}(\left|x^{\prime }-x\right|,h)[D_{y}^{c(1)}(x,h^{+})-D_{y}^{c(2)}(x,h^{-})]dx^{\prime }\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -\left(\int _{-1}^{-b}+\int _{\,b}^{1}\right){\alpha}(\left|x^{\prime }-x\right|,0)[D_{y}^{c(2)}(x,0^{+})-D_{y}^{c(3)}(x,0^{-})]dx^{\prime }=0\end{eqnarray}$ (12) where 𝛻² = ∂ ²∕∂x ² + ∂ ²∕∂y ² is the two-dimensional Laplace operator.

As stated in Eringen et al. [10], we have $[{\sigma}_{yz}^{(1)}(x,h^{+})-{\sigma}_{yz}^{(2)}(x,h^{-})]=0$ , $[{\sigma}_{yz}^{(2)}(x,0^{+})-{\sigma}_{yz}^{(3)}(x,0^{-})]=0$ , $[D_{y}^{c(1)}(x,h^{+})-D_{y}^{c(2)}(x,h^{-})]=0$ and $[D_{y}^{c(2)}(x,0^{+})-D_{y}^{c(3)}(x,0^{-})]=0$ . Hence, the line integrals in equations (11) and (12) are vanishing and the general solutions of equations (11) and (12) are identical to $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }{\alpha}(\left|x^{\prime }-x\right|,\left|y^{\prime }-y\right|)\bigg\{c_{440}\left[{\nabla}^{2}w^{(j)}(x^{\prime },y^{\prime })+{\lambda}\frac{\partial w^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\right]+e_{150}\bigg[{\nabla}^{2}{\phi}^{(j)}(x^{\prime },y^{\prime })\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼{\lambda}\frac{\partial {\phi}^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\bigg]\bigg\}dx^{\prime }dy^{\prime }=-{\rho}_{0}{\omega}^{2}w^{(j)}(x,y)\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }{\alpha}(\left|x^{\prime }-x\right|,\left|y^{\prime }-y\right|)\left\{e_{150}\left[{\nabla}^{2}w^{(j)}(x^{\prime },y^{\prime })+{\lambda}\frac{\partial w^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\right]\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼\left.{\varepsilon}_{110}\left[{\nabla}^{2}{\phi}^{(j)}(x^{\prime },y^{\prime })+{\lambda}\frac{\partial {\phi}^{(j)}(x^{\prime },y^{\prime })}{\partial y^{\prime }}\right]\right\}dx^{\prime }dy^{\prime }=0\end{eqnarray}$ (14)

In order to solve w ^(j)(x, y) and 𝜙^(j)(x, y), as discussed in Nowinski [23,24], the non-local interaction in the y-direction is ignored. In view of our assumption, we have $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\alpha}(\left|x^{\prime }-x\right|,\left|y^{\prime }-y\right|)={\alpha}_{0}(\left|x^{\prime }-x\right|){\delta}(y^{\prime }-y)\\[12.0pt] {\alpha}_{0}(\left|x^{\prime }-x\right|)=\frac{1}{\sqrt{{\pi}}}({\beta}/a)\exp [-({\beta}/a)^{2}(x^{\prime }-x)^{2}]\end{array}\right. & & \displaystyle\end{eqnarray}$ (15) where 𝛽 is a constant and a is the characteristic length. In this paper, a is the lattice parameter.

Taking equation (15) into equations (13) and (14), and using the Fourier transform, we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \bar{{\alpha}}_{0}(s)\left\{c_{440}\left[\frac{d^{2}\bar{w}^{(j)}(s,y)}{dy^{2}}-s^{2}\bar{w}^{(j)}(s,y)+{\lambda}\frac{\partial \bar{w}^{(j)}(s,y^{\prime })}{\partial y^{\prime }}\right]\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +\left.∼e_{150}\left[\frac{\partial ^{2}\bar{{\phi}}^{(j)}(s,y)}{\partial y^{2}}-s^{2}\bar{{\phi}}^{(j)}(s,y)+{\lambda}\frac{\partial \bar{{\phi}}^{(j)}(s,y^{\prime })}{\partial y^{\prime }}\right]\right\}=-{\rho}_{0}{\omega}^{2}\bar{w}^{(j)}(s,y)\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle e_{150}\left[\frac{\partial ^{2}\bar{w}^{(j)}(s,y)}{\partial y^{2}}-s^{2}\bar{w}^{(j)}(s,y)+{\lambda}\frac{\partial \bar{w}^{(j)}(s,y^{\prime })}{\partial y^{\prime }}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼{\varepsilon}_{110}\left[\frac{\partial ^{2}\bar{{\phi}}^{(j)}(s,y)}{\partial y^{2}}-s^{2}\bar{{\phi}}^{(j)}(s,y)+{\lambda}\frac{\partial \bar{{\phi}}^{(j)}(s,y^{\prime })}{\partial y^{\prime }}\right]=0\end{eqnarray}$ (17)

The general solution of equations (16) and (17) satisfying equation (4) are given as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}w^{(1)}(x,y)=\displaystyle\frac{2}{{\pi}}\int _{0}^{\infty }A_{1}(s)e^{-{\gamma}_{11}y}\cos (sx)ds\\[12.0pt] {\phi}^{(1)}(x,y)=\displaystyle\frac{e_{150}}{{\varepsilon}_{110}}w^{(1)}(x,y)+\frac{2}{{\pi}}\int _{0}^{\infty }B_{1}(s)e^{-{\gamma}_{21}y}\cos (sx)ds\end{array}\right.,\quad y\geq 0 & & \displaystyle\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}w^{(2)}(x,y)=\displaystyle\frac{2}{{\pi}}\int _{0}^{\infty }[A_{2}(s)e^{-{\gamma}_{11}y}+B_{2}(s)e^{-{\gamma}_{12}y}]\cos (sx)ds\\[12.0pt] {\phi}^{(2)}(x,y)=\displaystyle\frac{e_{150}}{{\varepsilon}_{110}}w^{(2)}(x,y)+\frac{2}{{\pi}}\int _{0}^{\infty }[C_{2}(s)e^{-{\gamma}_{21}y}+D_{2}(s)e^{-{\gamma}_{22}y}]\cos (sx)ds\end{array}\right.,\quad 0\leq y\leq h & & \displaystyle\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}w^{(3)}(x,y)=\displaystyle\frac{2}{{\pi}}\int _{0}^{\infty }A_{3}(s)e^{-{\gamma}_{12}y}\cos (sx)ds\\[12.0pt] {\phi}^{(3)}(x,y)=\displaystyle\frac{e_{150}}{{\varepsilon}_{110}}w^{(3)}(x,y)+\frac{2}{{\pi}}\int _{0}^{\infty }B_{3}(s)e^{-{\gamma}_{22}y}\cos (sx)ds\end{array}\right.,\quad y\leq 0 & & \displaystyle\end{eqnarray}$ (20) where A ₁(s), B ₁(s), A ₂(s), B ₂(s), C ₂(s), D ₂(s), A ₃(s) and B ₃(s) are to be determined from the boundary conditions, ${\gamma}_{11}=\frac{{\lambda}+\sqrt{{\lambda}^{2}+4[s^{2}-{\omega}^{2}/c_{1}^{2}\bar{{\alpha}}_{0}(s)]}}{2}$ , ${\gamma}_{12}=\frac{{\lambda}-\sqrt{{\lambda}^{2}+4[s^{2}-{\omega}^{2}/c_{1}^{2}\bar{{\alpha}}_{0}(s)]}}{2}$ , ${\gamma}_{21}=\frac{{\lambda}+\sqrt{{\lambda}^{2}+4s^{2}}}{2}$ , ${\gamma}_{22}=\frac{{\lambda}-\sqrt{{\lambda}^{2}+4s^{2}}}{2}$ , $c_{1}=\sqrt{{\mu}_{0}/{\rho}_{0}}$ , ${\mu}_{0}=c_{440}+\frac{e_{150}^{2}}{{\varepsilon}_{110}}$ .

Taking equations (18)–(20) into equation (10), we get $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\sigma}_{yz}^{(1)}(x,y)=\displaystyle-\frac{2e^{{\lambda}y}}{{\pi}}\int _{0}^{\infty }[{\mu}_{0}{\gamma}_{11}A_{1}(s)e^{-{\gamma}_{11}y}+e_{150}{\gamma}_{21}(s)B_{1}(s)e^{-{\gamma}_{21}y}]\cos (sx)ds\\[1pc] D_{y}^{c(1)}(x,y)=\displaystyle\frac{2{\varepsilon}_{110}e^{{\lambda}y}}{{\pi}}\int _{0}^{\infty }{\gamma}_{21}B_{1}(s)e^{-{\gamma}_{21}y}\cos (sx)ds\end{array}\right. & & \displaystyle\end{eqnarray}$ (21) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\sigma}_{yz}^{(2)}(x,y)=\displaystyle-\frac{2e^{{\lambda}y}}{{\pi}}\int _{0}^{\infty }\big\{{\mu}_{0}[{\gamma}_{11}A_{2}(s)e^{-{\gamma}_{11}y}+{\gamma}_{12}B_{2}(s)e^{-{\gamma}_{12}y}]\\[1pc] +e_{150}[{\gamma}_{21}C_{2}(s)e^{-{\gamma}_{21}y}+{\gamma}_{22}D_{2}(s)e^{-{\gamma}_{22}y}]\big\}\cos (sx)ds\\[1pc] D_{y}^{c(2)}(x,y)=\displaystyle\frac{2{\varepsilon}_{110}e^{{\lambda}y}}{{\pi}}\int _{0}^{\infty }[{\gamma}_{21}C_{2}(s)e^{-{\gamma}_{21}y}+{\gamma}_{22}D_{2}(s)e^{-{\gamma}_{22}y}]\cos (sx)ds\end{array}\right. & & \displaystyle\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\sigma}_{yz}^{(3)}(x,y)=\displaystyle-\frac{2e^{{\lambda}y}}{{\pi}}\int _{0}^{\infty }[{\mu}_{0}{\gamma}_{12}A_{3}(s)e^{-{\gamma}_{12}y}+e_{150}{\gamma}_{22}B_{3}(s)e^{-{\gamma}_{22}y}]\cos (sx)ds\\[1pc] D_{y}^{c(3)}(x,y)=\displaystyle\frac{2{\varepsilon}_{110}e^{{\lambda}y}}{{\pi}}\int _{0}^{\infty }{\gamma}_{22}B_{3}(s)e^{-{\gamma}_{22}y}\cos (sx)ds\end{array}\right. & & \displaystyle\end{eqnarray}$ (23)

According to the boundary conditions (2) and (3) lead to ${\sigma}_{yz}^{(1)}(x,h^{+})={\sigma}_{yz}^{(2)}(x,h^{-})$ , ${\sigma}_{yz}^{(2)}(x,0^{+})={\sigma}_{yz}^{(3)}(x,0^{-})$ , $D_{y}^{c(1)}(x,h^{+})=D_{y}^{c(2)}(x,h^{-})$ and $D_{y}^{c(2)}(x,0^{+})=D_{y}^{c(3)}(x,0^{-})$ . So we obtain $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle -{\mu}_{0}\{{\gamma}_{11}[A_{1}(s)-A_{2}(s)]e^{-{\gamma}_{11}h}-{\gamma}_{12}B_{2}(s)e^{-{\gamma}_{12}h}\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -e_{150}\{{\gamma}_{21}[B_{1}(s)-C_{2}(s)]e^{-{\gamma}_{21}h}-{\gamma}_{22}D_{2}(s)e^{-{\gamma}_{22}h}\}=0\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle -{\mu}_{0}[{\gamma}_{11}A_{2}(s)+{\gamma}_{12}B_{2}(s)-{\gamma}_{12}A_{3}(s)]-e_{150}[{\gamma}_{21}C_{2}(s)+{\gamma}_{22}D_{2}(s)-{\gamma}_{22}B_{3}(s)]=0\end{eqnarray}$ (25) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}_{21}[B_{1}(s)-C_{2}(s)]e^{-{\gamma}_{21}h}-{\gamma}_{22}D_{2}(s)e^{-{\gamma}_{22}h}=0\end{eqnarray}$ (26) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\gamma}_{21}C_{2}(s)+{\gamma}_{22}D_{2}(s)-{\gamma}_{22}B_{3}(s)=0\end{eqnarray}$ (27)

The jumps of displacements across the crack surfaces are defined as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}f_{1}(x)=w^{(1)}(x,h^{+})-w^{(2)}(x,h^{-})\\ f_{2}(x)=w^{(2)}(x,0^{+})-w^{(3)}(x,0^{-})\end{array}\right. & & \displaystyle\end{eqnarray}$ (28) Substituting equations (18) and (20) into equation (28), applying the boundary conditions (2) and (3) and the Fourier transform, we obtain $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \bar{f}_{1}(s)=A_{1}(s)e^{-{\gamma}_{11}h}-A_{2}(s)e^{-{\gamma}_{11}h}-B_{2}(s)e^{-{\gamma}_{12}h}\end{eqnarray}$ (29) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \bar{f}_{2}(s)=A_{2}(s)+B_{2}(s)-A_{3}(s)\end{eqnarray}$ (30) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{e_{150}}{{\varepsilon}_{110}}\bar{f}_{1}(s)+B_{1}(s)e^{-{\gamma}_{21}h}-C_{2}(s)e^{-{\gamma}_{21}h}-D_{2}(s)e^{-{\gamma}_{22}h}=0\end{eqnarray}$ (31) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{e_{150}}{{\varepsilon}_{110}}\bar{f}_{2}(s)+C_{2}(s)+D_{2}(s)-B_{3}(s)=0\end{eqnarray}$ (32)

By solving equations (24)–(27) and (29)–(32) with unknown functions A ₁(s), B ₁(s), A ₂(s), B ₂(s), C ₂(s), D ₂(s), A ₃(s) and B ₃(s), substituting the solutions into equation (22), applying 𝛼 from equation (15), the boundary conditions (2) and (3) and the Fourier transform, the dual integral equations are obtained as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\tau}_{yz}^{(2)}(x,h)=\displaystyle\frac{2e^{{\lambda}h}}{{\pi}}\int _{0}^{\infty }e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}[g_{1}(s)\bar{f}_{1}(s)+g_{2}(s)\bar{f}_{2}(s)]\cos (sx)ds=-{\tau}_{0}\\[1pc] {\tau}_{yz}^{(2)}(x,0)=\displaystyle\frac{2}{{\pi}}\int _{0}^{\infty }e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}[g_{3}(s)\bar{f}_{1}(s)+g_{4}(s)\bar{f}_{2}(s)]\cos (sx)ds=-{\tau}_{0}\end{array}\right. & & \displaystyle\end{eqnarray}$ (33) $\begin{eqnarray}\displaystyle \frac{2}{{\pi}}\int _{0}^{\infty }\bar{f}_{1}(s)\cos (sx)ds=0,\frac{2}{{\pi}}\int _{0}^{\infty }\bar{f}_{2}(s)\cos (sx)ds=0,x>1\quad \text{and}\quad x\lt b & & \displaystyle\end{eqnarray}$ (34) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}D_{y}^{(2)}(x,h)=\displaystyle\frac{2e_{150}e^{{\lambda}h}}{{\pi}}\int _{0}^{\infty }e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}\frac{{\gamma}_{21}{\gamma}_{22}}{{\gamma}_{21}-{\gamma}_{22}}[\bar{f}_{1}(s)+e^{-{\gamma}_{21}h}\bar{f}_{2}(s)]\cos (sx)ds\\[1pc] D_{y}^{(2)}(x,0)=\displaystyle\frac{2e_{150}}{{\pi}}\int _{0}^{\infty }e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}\frac{{\gamma}_{21}{\gamma}_{22}}{{\gamma}_{21}-{\gamma}_{22}}[e^{{\gamma}_{22}h}\bar{f}_{1}(s)+\bar{f}_{2}(s)]\cos (sx)ds\end{array}\right. & & \displaystyle\end{eqnarray}$ (35) where $g_{1}(s)=\frac{{\mu}_{0}{\gamma}_{11}{\gamma}_{12}}{{\gamma}_{11}-{\gamma}_{12}}-\frac{{\gamma}_{21}{\gamma}_{22}e_{15}^{2}}{({\gamma}_{21}-{\gamma}_{22}){\varepsilon}_{11}}$ , $g_{2}(s)=\frac{{\mu}_{0}{\gamma}_{11}{\gamma}_{12}e^{-r_{11}h}}{{\gamma}_{11}-{\gamma}_{12}}-\frac{{\gamma}_{21}{\gamma}_{22}e^{-r_{21}h}e_{15}^{2}}{({\gamma}_{21}-{\gamma}_{22}){\varepsilon}_{11}}$ , $g_{3}(s)=\frac{{\mu}_{0}{\gamma}_{11}{\gamma}_{12}e^{r_{12}h}}{{\gamma}_{11}-{\gamma}_{12}}-\frac{{\gamma}_{21}{\gamma}_{22}e^{r_{22}h}e_{15}^{2}}{({\gamma}_{21}-{\gamma}_{22}){\varepsilon}_{11}}$ , $g_{4}(s)=\frac{{\mu}_{0}{\gamma}_{11}{\gamma}_{12}}{{\gamma}_{11}-{\gamma}_{12}}-\frac{{\gamma}_{21}{\gamma}_{22}e_{15}^{2}}{({\gamma}_{21}-{\gamma}_{22}){\varepsilon}_{11}}$ .

The dual-integral equations (33)--(34) must be solved to determine the unknown functions $\bar{f}_{1}(s)$ and $\bar{f}_{2}(s)$ .

4. Solution of the dual integral equations

To solve the dual integral equations (33)–(34) and the Schmidt method [20] is used. The jumps of displacements across the crack surfaces are represented as the form series $\begin{eqnarray}\displaystyle f_{1}(x)=\left\{\begin{array}{@{}l@{}}\displaystyle \mathop{\sum }_{n=0}^{\infty }a_{n}P_{n}^{\Big(\frac{1}{2},\frac{1}{2}\Big)}\left(\displaystyle \frac{x-\displaystyle \frac{1+b}{2}}{\displaystyle \frac{1-b}{2}}\right)\left[1-\displaystyle \frac{\left(x-\displaystyle \frac{1+b}{2}\right)^{2}}{\left(\frac{1-b}{2}\right)^{2}}\right]^{\frac{1}{2}},\quad b\leq x\leq 1\\ 0,\quad 0\leq x\lt b,\,x>1\end{array}\right. & & \displaystyle\end{eqnarray}$ (36) $\begin{eqnarray}\displaystyle f_{2}(x)=\left\{\begin{array}{@{}l@{}}\displaystyle \mathop{\sum }_{n=0}^{\infty }b_{n}P_{n}^{\Big(\frac{1}{2}, \frac{1}{2}\Big)}\left(\displaystyle \frac{x-\displaystyle \frac{1+b}{2}}{\frac{1-b}{2}}\right)\left[1-\displaystyle \displaystyle \frac{\left(x-\frac{1+b}{2}\right)^{2}}{\left(\displaystyle \frac{1-b}{2}\right)^{2}}\right]^{\frac{1}{2}},\quad b\leq x\leq 1\\ 0,\quad 0\leq x\lt b,x>1\end{array}\right. & & \displaystyle\end{eqnarray}$ (37) where a _n and b _n are unknown coefficients and $P_{n}^{(1/2,1/2)}(x)$ is a Jacobi polynomial [25]. Application of the Fourier transforms [26] into equations (36)–(37) are $\begin{eqnarray}\displaystyle \bar{f}_{1}(s)=\mathop{\sum }_{n=0}^{\infty }a_{n}F_{n}G_{n}^{(1)}(s)\frac{1}{s}J_{n+1}\left(s\frac{1-b}{2}\right),G_{n}^{(1)}(s)=\left\{\begin{array}{@{}l@{}}(-1)^{\frac{n}{2}}\sin \left(s\frac{1+b}{2}\right),\quad n=0,2,4,6,\ldots \\ (-1)^{\frac{n-1}{2}}\cos \left(s\frac{1+b}{2}\right),\quad n=1,3,5,7,\ldots \end{array}\right. & & \displaystyle\end{eqnarray}$ (38) $\begin{eqnarray}\displaystyle \bar{f}_{2}(s)=\mathop{\sum }_{n=0}^{\infty }b_{n}F_{n}G_{n}^{(2)}(s)\frac{1}{s}J_{n+1}\left(s\frac{1-b}{2}\right),G_{n}^{(2)}(s)=\left\{\begin{array}{@{}l@{}}(-1)^{\frac{n}{2}}\cos \left(s\frac{1+b}{2}\right),\quad n=0,2,4,6,\ldots \\ (-1)^{\frac{n+1}{2}}\sin \left(s\frac{1+b}{2}\right),\quad n=1,3,5,7,\ldots \end{array}\right. & & \displaystyle\end{eqnarray}$ (39) where $F_{n}=2\sqrt{{\pi}}\frac{{\Gamma}(n+1+\frac{1}{2})}{n!}$ , 𝛤(x) and J _n(x) are the Gamma and Bessel functions of order n, respectively.

Substituting equations (38) and (39) into equations (33) and (34), and equation (34) is automatically satisfied. Then the equation (33) is reduced as $\begin{eqnarray}\displaystyle \mathop{\sum }_{n=1}^{\infty }F_{n}\int _{0}^{\infty }\frac{1}{s}e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}[g_{1}(s)a_{n}G_{n}^{(1)}(s)+g_{2}(s)b_{n}G_{n}^{(2)}(s)]J_{n+1}\left(s\frac{1-b}{2}\right)\cos (sx)ds={\pi}{\tau}_{0}e^{-{\lambda}h}/2 & & \displaystyle\end{eqnarray}$ (40) $\begin{eqnarray}\displaystyle \mathop{\sum }_{n=1}^{\infty }F_{n}\int _{0}^{\infty }\frac{1}{s}e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}[g_{3}(s)a_{n}G_{n}^{(1)}(s)+g_{4}(s)b_{n}G_{n}^{(1)}(s)]J_{n+1}\left(s\frac{1-b}{2}\right)\cos (sx)ds={\pi}{\tau}_{0}/2 & & \displaystyle\end{eqnarray}$ (41)

Briefly, equations (40) and (41) can be rewritten as $\begin{eqnarray}\displaystyle \mathop{\sum }_{n=1}^{\infty }a_{n}E_{n}^{\ast }(x)+\mathop{\sum }_{n=1}^{\infty }b_{n}F_{n}^{\ast }(x)=U_{0}(x),\quad b\leq x\leq 1 & & \displaystyle\end{eqnarray}$ (42) $\begin{eqnarray}\displaystyle \mathop{\sum }_{n=1}^{\infty }a_{n}G_{n}^{\ast }(x)+\mathop{\sum }_{n=1}^{\infty }b_{n}H_{n}^{\ast }(x)=V_{0}(x),\quad b\leq x\leq 1 & & \displaystyle\end{eqnarray}$ (43) where $E_{n}^{\ast }(x)$ , $F_{n}^{\ast }(x)$ , $G_{n}^{\ast }(x)$ , $H_{n}^{\ast }(x)$ , U ₀(x) and V ₀(x) are known functions. Equations (34) and (35) can be solved for coefficients a _n and b _n by the Schmidt method [20].

5. Numerical results and discussion

When coefficients a _n and b _n are known, we obtained entire stress and the electric displacement fields. In the case of the present study, ${\tau}_{yz}^{(2)}(x,h)$ , ${\tau}_{yz}^{(2)}(x,0)$ , $D_{y}^{(2)}(x,h)$ and $D_{y}^{(2)}(x,0)$ along the crack line are $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\tau}_{yz1}={\tau}_{yz}^{(2)}(x,h)=\frac{2e^{{\lambda}h}}{{\pi}}\displaystyle \mathop{\sum }_{n=1}^{\infty }F_{n}\displaystyle \int _{0}^{\infty }\frac{1}{s}e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}[g_{1}(s)a_{n}G_{n}^{(1)}(s)+g_{2}(s)b_{n}G_{n}^{(2)}(s)]\\ J_{n+1}(s\frac{1-b}{2})\cos (sx)ds\\ {\tau}_{yz2}={\tau}_{yz}^{(2)}(x,0)=\frac{2}{{\pi}}\displaystyle \mathop{\sum }_{n=1}^{\infty }F_{n}\displaystyle \int _{0}^{\infty }\frac{1}{s}e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}[g_{3}(s)a_{n}G_{n}^{(1)}(s)+g_{4}(s)b_{n}G_{n}^{(1)}(s)]\\ J_{n+1}(s\frac{1-b}{2})\cos (sx)ds\end{array}\right. & & \displaystyle\end{eqnarray}$ (44) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}D_{y1}=D_{y}^{(2)}(x,h)=\frac{2e_{150}e^{{\lambda}h}}{{\pi}}\displaystyle \mathop{\sum }_{n=1}^{\infty }F_{n}\int _{0}^{\infty }e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}\frac{{\gamma}_{21}{\gamma}_{22}}{s({\gamma}_{21}-{\gamma}_{22})}[a_{n}G_{n}^{(1)}(s)+e^{-{\gamma}_{21}h}b_{n}G_{n}^{(2)}(s)]\\ J_{n+1}(s\frac{1-b}{2})\cos (sx)ds\\ D_{y2}=D_{y}^{(2)}(x,0)=\frac{e_{150}}{{\pi}}\displaystyle \mathop{\sum }_{n=1}^{\infty }F_{n}\displaystyle \int _{0}^{\infty }e^{-\frac{a^{2}s^{2}}{4{\beta}^{2}}}\frac{{\gamma}_{21}{\gamma}_{22}}{s({\gamma}_{21}-{\gamma}_{22})}[e^{{\gamma}_{22}h}a_{n}G_{n}^{(1)}(s)+b_{n}G_{n}^{(2)}(s)]\\ J_{n+1}(s\frac{1-b}{2})\cos (sx)ds\end{array}\right. & & \displaystyle\end{eqnarray}$ (45) The material properties of PZT-4 are used by c ₄₄₀ = 2.56( × 10¹⁰N∕m²), e ₁₅₀ = 12.7 (C/m²) and 𝜀₁₁₀ = 64.6( × 10⁻¹⁰C/Vm) in the calculation.

(1) The dynamic non-local stress field (DNSF) and dynamic non-local electric displacement field (DNEDF) are obtained at the crack tips in Figs. 2–7. Figures 6 and 7 indicate the maximum stress does not occur at the crack tips, and slightly away from it. Meanwhile, it is noted that the distance between the crack tip and the maximum stress point is very small which depends on the crack length, the distance between parallel cracks, the frequency of the incident wave, the functionally graded parameter (FGP) and the lattice parameter. The phenomenon has been testified in Eringen [27].

(2) As shown in Figs. 8 and 9, the values of DNSF and DNEDF at the crack tips tend to decrease with the increase of the lattice parameter. We can be found that the amplitude value of the DNEDF is very small as shown in Fig. 9.

(3) Figures 10 and 11 show the values of DNSF and DNEDF at the crack tips increase with increase of the crack length. It noted fact that experiments indicate that materials with smaller crack are more resistant to facture than those with larger crack.

(4) The values of DNSF and DNEDF at the upper crack tips increase with increase of the distance between parallel cracks h, as shown in Fig. 12. Meanwhile, Fig. 13 displays the values of DNSF and DNEDF at the lower crack tips decrease with increase of the distance between parallel cracks h.

(5) From Figs. 14 and 15, for 𝜆 ≤ 0, the values of DNSF and DNEDF at the crack tips increase with increase of the FGP 𝜆 reach a maximum at 𝜆 = −1.4 and then decrease in magnitude; for 𝜆 ≥ 0, the values of DNSF and DNEDF at the crack tips decrease with increase of the FGP 𝜆 reach a maximum at 𝜆 = 1.4 and then decrease in magnitude. When 𝜆 ≤ 0, the values of DNSF and DNEDF near the upper crack tips are smaller than the values of DNSF and DNEDF near the lower crack tips, however, when 𝜆 ≥ 0, the values of DNSF and DNEDF near the upper crack tips are larger than the values of DNSF and DNEDF near the lower crack tips, as shown in Figs. 14 and 15. It is implied that the dynamic stress and dynamic electric displacement fields near the crack tips can be reduced by adjusting the FGP of the FGPM.

(6) The values of DNSF and DNEDF at the upper crack tips tend to increase with the increase of the frequency of the incident waves 𝜔∕c ₁, reach a maximum at 𝜔∕c ₁ = 0.4 and then decrease in magnitude, as shown in Figs. 16 and 17. At the same time, Figs. 16 and 17 show the values of DNSF and DNEDF at the lower crack tips tend to increase with the increase of the frequency of the incident waves 𝜔∕c ₁, reach a maximum at 𝜔∕c ₁ = 0.4 and then decrease in magnitude reach a minimum at 𝜔∕c ₁ = 1.2.

6. Conclusion

This paper presents the traditional concepts of the non-local theory to solve the dynamic behavior of multiple cracks in a FGPM. With the Fourier transform and the Schmidt method, the dynamic non-local stress and dynamic non-local electric displacement fields are obtained at the crack tips. The magnitude of the finite stress and electric displacement fields depend on relevant parameters, such as the distance between four parallel cracks, the functionally graded parameter, the frequency of the incident wave and the lattice parameter in a FGPM. It can be obtained that the solution of the present paper yields a finite hoop stress and electric displacement at the crack tips, thus the maximum stress and electric displacement as a fracture criterion can be used. For four cracks model, the dynamic interaction behaviors of the two parallel cracks and the two collinear cracks are all investigated in the present paper. Meanwhile, theoretical derivation is also more complex than two cracks. These results can be used as theoretical references helpful for evaluating strength and preventing material failure of a FGPM with multiple cracks in engineering.

Footnotes

Acknowledgements

This work is supported by the Hebei Higher School of Young Talents Program (BJ2016014), the National Natural Science Foundation of China (11702079) and the Hebei Excellent Youth Science Fund (A2017202107).

Conflict of interest

None to report.

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