Transient response of cracked nonhomogeneous substrate with piezoelectric coating by dislocation method

Abstract

The transient response of a functionally graded material (FGM) orthotropic strip with a piezoelectric coating weakened by multiple cracks is investigated. The system is subjected to out-of-plane mechanical and in-plane electrical loading. The properties of the nonhomogeneous substrate are assumed to vary exponentially along the thickness and the energy dissipation is modeled by viscous damping. In this study, the rate of the gradual change of the shear moduli, mass density, and damping constant are assumed to be same. At first, the transient response of a Volterra-type dislocation in an FGM orthotropic strip is obtained analytically. Imposing a distributed dislocation density on the crack surface and using the Fourier and Laplace integral transforms, the problem is reduced to a system of singular integral equations for a substrate weakened by multiple cracks in the form of Cauchy singularity; which are solved numerically to obtain the dynamic stress intensity factors at the crack tips. Finally, the effects of the geometrical parameters, material properties, viscous damping and cracks arrangement on the dynamic fracture behavior of the interacting cracks are studied.

Keywords

Nonhomogeneous layer viscous damping transient anti-plane loading several cracks piezoelectric coating

1. Introduction

In recent years, predicting the fracture behavior of the cracked structures such as smart material/composite coating-substrate systems during impact and crash events have attracted great attention in the engineering applications. Smart materials are often made as coated structures to improve the performance of the system. Great efforts have been made to study the static behavior of the smart materials by the researchers in recent years. Although most of the studies are related to the static or quasi-static conditions, the response of the cracks under dynamic loading is more important. Dynamic loadings are mainly categorized in two groups including harmonic loading and impact loading. Impact loads applied on the cracked structures lead to catastrophic failure of the system.

The dynamic response of a crack in a functionally graded material (FGM) layer between two dissimilar half planes under anti-plane shear impact loading was solved bay Babaei and Lukasiewicz [1]. They showed that the dynamic stress intensity factors (DSIFs) depend on the crack length and the material properties.

The transient DSIF of an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading was determined by Wei et al. [2]. They analyzed the DSIF during a small time-interval and the effects of the viscoelastic material parameters on the DSIF. Chen and Worswick [3] studied the transient response of two cracks in a half space under anti-plane shear impact loading. The effect of the geometry ratio and the depth of the crack on DSIFs were studied. Shul and Lee [4] considered the dynamic response of the subsurface interface crack in multi-layered orthotropic half-space subjected to an anti-plane shear impact loading. The effects of the geometric constants and material properties were discussed. Shul and Lee [5] also considered the dynamic response of a subsurface eccentric crack in a functionally graded coating layer on the layered half-space under an anti-plane shear impact loading. The nonhomogeneous parameter, geometric constants and the material properties were investigated on the normalized stress intensity factor (SIF). Zhang et al. [6] solved the anti-plane transient problem of a finite crack in an infinite FGM and explored the effects of the material gradients of the FGM on the transient DSIF and their dynamic overshoot corresponding with the static SIF. The transient in-plane problem was derived by Guo et al. [7] for a coating–substrate structure with a cracked functionally graded interfacial layer subjected to an impact load. By using the integral transform techniques, the boundary value problem reduced to a system of singular integral equations. Also, the influences of the material nonhomogeneity constant and the geometric parameters on the DSIFs were discussed. Chen [8] investigated the dynamic response of an electrically impermeable crack in a transversely isotropic piezoelectric material subjected to the pure electric load in mode III. The SIF, the mechanical energy release rate, and the total energy release rate were derived and expressed as a function of time for a given applied electric load. Yongdong et al. [9] studied the anti-plane transient analysis for two functionally gradient half-planes with a weak/infinitesimal-discontinuous interface. The effects of the nonhomogeneity parameters and the types of the discontinuity were studied on the SIF in mode III. A periodic array of cracks in an infinite FGM under transient mechanical loading was investigated by Wang and Mai [10]. The effect of the material nonhomogeneity on the crack tip intensity factors was discussed.

The above mentioned problems are restricted to one or two periodic cracks. However, to the best of the authors’ knowledge, the multiple cracks problem has not been studied in an FGM strip with piezoelectric coating under impact loadings. Following introduction in recent years of a powerful semi-analytical method called the distribution dislocation technique (DDT), the problem of the multiple cracks has been studied for out-of-plane and in-plane loadings based on the principle of the superposition. A brief review of relevant articles is given below.

Faal et al. [11] employed distributed dislocation technique to analyze an orthotropic strip having multiple cracks and cavities. The results were used to evaluate the SIFs of modes III. Mousavi and Fariborz [12] considered the elastodynamic analysis of multiple cracks in an FGM orthotropic layer with viscous damping. The effects of the excitation frequency and the material properties were studied on the DSIF. Monfared and Ayatollahi [13] considered the DSIF of multiple cracks in an orthotropic strip with FGM coatings under time-harmonic excitation. The influence of the angular frequency, crack lengths and material properties were investigated on the DSIF.

The DSIF associated with the crack tips was calculated by a numerical inverse Laplace scheme. Vafa et al. [14] used the Volterra-type screw dislocation technique and the Stehfest inversion method to simulate the dynamic response of FGM layers weakened by multiple cracks parallel to the boundary under anti-plane shear impact load.

According to the reviewed literature, there is not a promising investigation regarding the transient response of the several cracks in a nonhomogeneous substrate with piezoelectric coating. In this study, the transient problem of multiple cracks in a nonhomogeneous strip reinforced with piezoelectric coating under anti-plane mechanical and in-plane electrical loading is analyzed. The energy dissipation in the medium is exhibited by viscous damping. Using the distributed dislocation, Buckner’s principle and integral transform techniques in conjunction with the Stehfest inversion method, the integral equations are derived in the form of Cauchy singular kinds, and are solved numerically for the dislocation density on the cracks faces. These solutions are employed to determine the DSIFs of the multiple crack tips. Finally, the effects of the material properties, viscous damping, crack arrangements, and crack interactions on the DSIFs of the cracks are studied using different examples to demonstrate the advantage of this method.

2. Solution of dislocation

A piezoelectric coating of thickness $h_{2}$ is used to reinforce a functionally graded orthotropic strip with thickness $h_{1}$ , as depicted in Figure 1. A Volterra-type screw dislocation with Burgers vector $b_{z} (t)$ is situated at a point $(ξ, η)$ in the nonhomogeneous substrate. The x- and y-axes are in the directions of principal material of the orthotropic FGM substrate.

Figure 1.

A functionally graded orthotropic strip reinforced with piezoelectric coating with a screw dislocation in the strip.

For anti-plane elastic field coupled with the in-plane electric field, the constitutive equation of the problem in coordinate system $(x, y, z)$ is written as:

\begin{matrix} [\begin{matrix} τ_{zx} & τ_{zy} \\ D_{x} & D_{y} \end{matrix}] = [\begin{matrix} c_{44} & e_{15} \\ e_{15} & - d_{11} \end{matrix}] [\begin{matrix} \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} \\ \frac{\partial φ}{\partial x} & \frac{\partial φ}{\partial y} \end{matrix}] - h_{2} \leq y \leq 0 \end{matrix}

(1)

where $D_{x}$ , $D_{y}$ , and $φ$ are the components of the electric displacements and the electric potential, respectively. Also, $c_{44}$ is the shear modulus at a constant electric field, $e_{15}$ is the piezoelectric coefficient and $d_{11}$ is the dielectric parameter at a constant strain field. The governing equations of the functionally graded orthotropic substrate are given by

τ_{zx} = G_{zx} (y) \frac{\partial w}{\partial x} τ_{zy} = G_{zy} (y) \frac{\partial w}{\partial y} 0 \leq y \leq h_{1}

(2)

where $τ_{zx}, τ_{zy}$ are the shear stress components and $G_{zx} (y)$ , $G_{zy} (y)$ are the shear moduli of elasticity of the orthotropic FGM, respectively. The equilibrium equations and Maxwell’s equation of the piezoelectric coating and orthotropic FGM substrate with structural energy dissipation is expressed as follows:

\begin{matrix} \frac{\partial τ_{zx}}{\partial x} + \frac{\partial τ_{zy}}{\partial y} = ρ_{0} \frac{\partial^{2} w}{\partial t^{2}} + γ_{0} \frac{\partial w}{\partial t} - h_{2} \leq y \leq 0 \\ \frac{\partial D_{x}}{\partial x} + \frac{\partial D_{y}}{\partial y} = 0 - h_{2} \leq y \leq 0 \\ \frac{\partial τ_{zx}}{\partial x} + \frac{\partial τ_{zy}}{\partial y} = ρ (y) \frac{\partial^{2} w}{\partial t^{2}} + γ (y) \frac{\partial w}{\partial t} 0 \leq y \leq h_{1} \end{matrix}

(3)

where $ρ_{0}$ , $ρ (y)$ are the mass density and $γ_{0}$ , $γ (y)$ are the viscous damping coefficient per unit volume of the piezoelectric coating and the functionally graded orthotropic substrate, respectively. Substituting equations (1) and (2) into equations (3), the equilibrium equations and Maxwell’s equation will yield the following form:

\begin{matrix} c_{44} \frac{\partial^{2} w}{\partial x^{2}} + e_{15} \frac{\partial^{2} φ}{\partial x^{2}} + c_{44} \frac{\partial^{2} w}{\partial y^{2}} + e_{15} \frac{\partial^{2} φ}{\partial y^{2}} = ρ_{0} \frac{\partial^{2} w}{\partial t^{2}} + γ_{0} \frac{\partial w}{\partial t} - h_{2} \leq y \leq 0 \\ e_{15} \frac{\partial^{2} w}{\partial x^{2}} - d_{11} \frac{\partial^{2} φ}{\partial x^{2}} + e_{15} \frac{\partial^{2} w}{\partial y^{2}} - d_{11} \frac{\partial^{2} φ}{\partial y^{2}} = 0 - h_{2} \leq y \leq 0 \\ G_{zx} (y) \frac{\partial^{2} w}{\partial x^{2}} + G_{zy} (y) \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial G_{zy} (y)}{\partial y} \frac{\partial w}{\partial y} = ρ (y) \frac{\partial^{2} w}{\partial t^{2}} + γ (y) \frac{\partial w}{\partial t} 0 \leq y \leq h_{1} \end{matrix}

(4)

For the piezoelectric coating, all of the material properties are constant, which are denoted by the corresponding letter with subscript $0$ , i.e. $c_{44}$ is denoted as $c_{440}$ , while for the functionally graded orthotropic substrate, the material properties are supposed to change exponentially along the y-axis exponentially as

[G_{zx} (y), G_{zy} (y), ρ (y), γ (y)] = [G_{zx 0}, G_{zy 0}, ρ_{0}, γ_{0}] e^{2 λ y}

(5)

where $γ_{0}$ is a constant and $[G_{zx 0}, G_{zy 0}, ρ_{0}]$ are material properties at $y = 0$ . Also, $λ$ is the nonhomogeneity parameter of the substrate. By inserting equation (5) into equation (4), one may conclude that

\begin{matrix} c_{44} \nabla^{2} w + e_{15} \nabla^{2} φ = ρ_{0} \frac{\partial^{2} w}{\partial t^{2}} + γ_{0} \frac{\partial w}{\partial t} - h_{2} \leq y \leq 0 \\ e_{15} \nabla^{2} w - d_{11} \nabla^{2} φ = 0 - h_{2} \leq y \leq 0 \\ \frac{\partial^{2} w}{\partial y^{2}} + 2 λ \frac{\partial w}{\partial y} + g^{2} \frac{\partial^{2} w}{\partial x^{2}} = {s_{T 1}}^{2} \frac{\partial^{2} w}{\partial t^{2}} + m_{1} \frac{\partial w}{\partial t} 0 \leq y \leq h_{1} \end{matrix}

(6)

In the above equations, ${s_{T 1}}^{2} = ρ_{0} / G_{zy 0}$ , $m_{1} = γ_{0} / G_{zy 0}$ , $g^{2} = G_{zx 0} G_{zy 0}$ and $\nabla^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}$ is the two-dimensional Laplacian operator in the variable x and y. A simple approach for decoupling equations (6) is to introduce a new function $ψ$ as

ψ = φ - α w

(7)

in which $α = e_{15} / d_{11}$ . Hence, equations (6) can be written as decoupled equations as follows:

\begin{matrix} (c_{44} + α e_{15}) \nabla^{2} w = ρ_{0} \frac{\partial^{2} w}{\partial t^{2}} + γ_{0} \frac{\partial w}{\partial t} - h_{2} \leq y \leq 0 \\ \nabla^{2} ψ = 0 - h_{2} \leq y \leq 0 \\ \frac{\partial^{2} w}{\partial y^{2}} + 2 λ \frac{\partial w}{\partial y} + g^{2} \frac{\partial^{2} w}{\partial x^{2}} = {s_{T 1}}^{2} \frac{\partial^{2} w}{\partial t^{2}} + m_{1} \frac{\partial w}{\partial t} 0 \leq y \leq h_{1} \end{matrix}

(8)

The constitutive equation (1), in view of equation (7), leads to

\begin{matrix} [\begin{matrix} τ_{zx} & τ_{zy} \\ D_{x} & D_{y} \end{matrix}] = [\begin{matrix} c_{44} + α e_{15} & e_{15} \\ 0 & - d_{11} \end{matrix}] [\begin{matrix} \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} \\ \frac{\partial ψ}{\partial x} & \frac{\partial ψ}{\partial y} \end{matrix}] - h_{2} \leq y \leq 0 \end{matrix}

(9)

However, in this work, it is not necessary to consider the electrical boundary conditions on the crack surface, because the functionally graded orthotropic strip has no electric properties under the electrical loads. Therefore, the electrical loads are supposed to be applied only in the piezoelectric layer and are not perturbed by the cracks. The conditions representing the screw dislocation, the perfect bonding conditions at the interface of two materials, and the traction-free and electrical displacement-free conditions on the outer boundary of the layers are represented as follows:

\begin{matrix} w (x, η^{+}, t) - w (x, η^{-}, t) = b_{z} (t) H (x - ξ), \\ τ_{zy} (x, η^{+}, t) = τ_{zy} (x, η^{-}, t), \\ τ_{zy} (x, 0^{+}, t) = τ_{zy} (x, 0^{-}, t), \\ w (x, 0^{+}, t) = w (x, 0^{-}, t), \\ τ_{zy} (x, - h_{2}, t) = 0, \\ τ_{zy} (x, h_{1}, t) = 0, \\ D_{y} (x, - h_{2}, t) = 0, \\ D_{y} (x, h_{1}, t) = 0, \end{matrix}

(10)

where $b_{z} (t)$ stands for the dislocation Burgers vector and $H (.)$ is the Heaviside step function. Imposing the Laplace and complex Fourier transforms onto equation (8) by assuming that the piezoelectric coating and nonhomogeneous substrate are initially at rest, one may obtain the ordinary differential equations as follow:

\begin{matrix} \frac{d^{2} {\bar{w}}^{*} (p, y, s)}{d y^{2}} - (p^{2} + {s_{T 2}}^{2} s^{2} + m_{2} s) {\bar{w}}^{*} (p, y, s) = 0 - h_{2} \leq y \leq 0 \\ \frac{d^{2} {\bar{ψ}}^{*} (p, y, s)}{d y^{2}} - p^{2} {\bar{ψ}}^{*} (p, y, s) = 0 - h_{2} \leq y \leq 0 \\ \frac{d^{2} {\bar{w}}^{*} (p, y, s)}{d y^{2}} + 2 λ \frac{d {\bar{w}}^{*} (p, y, s)}{dy} - (g^{2} p^{2} + {s_{T 1}}^{2} s^{2} + m_{1} s) {\bar{w}}^{*} (p, y, s) = 0 0 \leq y \leq h_{1} \end{matrix}

(11)

where

{s_{T 2}}^{2} = \frac{ρ_{0}}{c_{44} + α e_{15}}, \frac{γ_{0}}{c_{44} + α e_{15}} = m_{2}

(12)

The general solutions of equation (11) for coating and substrate are derived as

\begin{matrix} {\bar{w}}^{*} (p, y, s) = A_{1} (p, s) e^{y β_{2}} + A_{2} (p, s) e^{- y β_{2}} - h_{2} \leq y \leq 0 \\ {\bar{ψ}}^{*} (p, y, s) = A_{3} (p, s) e^{y | p |} + A_{2} (p, s) e^{- y | p |} - h_{2} \leq y \leq 0 \\ {\bar{w}}^{*} (p, y, s) = A_{5} (p, s) e^{(- λ + β_{1}) y} + A_{6} (p, s) e^{- (λ + β_{1}) y} 0 \leq y \leq η \\ {\bar{w}}^{*} (p, y, s) = A_{7} (p, s) e^{(- λ + β_{1}) y} + A_{8} (p, s) e^{- (λ + β_{1}) y} η \leq y \leq h_{1} \end{matrix}

(13)

where

\begin{matrix} β_{1} = \sqrt{λ^{2} + g^{2} p^{2} + {s_{T 1}}^{2} s^{2} + m_{1} s} \\ β_{2} = \sqrt{p^{2} + {s_{T 2}}^{2} s^{2} + m_{2} s} \end{matrix}

(14)

The unknown coefficients in equation (13) are determined by applying the boundary conditions (10). The transformed anti-plane displacements $w (x, y, t)$ in the Laplace and Fourier domain are introduced as:

\begin{matrix} {\bar{w}}^{*} (p, y, s) = \frac{b_{z} (s) [π δ (p) - i / p] e^{- i ξ p} e^{λ (η - y)} M_{1} [M_{2} \sinh (β_{1} y) + M_{3} \cosh (β_{1} y)]}{M_{4}} 0 \leq y \leq η \\ {\bar{w}}^{*} (p, y, s) = \frac{b_{z} (s) [π δ (p) - i / p] e^{- i ξ p} e^{λ (η - y)} M_{5} [λ \sinh (β_{1} (y - h_{1})) + β_{1} \cosh (β_{1} (y - h_{1}))]}{M_{4}} η \leq y \leq h_{1} \end{matrix}

(15)

where

\begin{matrix} M_{1} = ({β_{1}}^{2} - λ^{2}) \sinh (β_{1} (η - h_{1})) \\ M_{2} = β_{2} (c_{44} + α e_{15}) \sinh (β_{2} h_{2}) + λ G_{zy 0} \cosh (β_{2} h_{2}) \\ M_{3} = β_{1} G_{zy 0} \cosh (β_{2} h_{2}) \\ M_{4} = β_{1} {β_{2} (c_{44} + α e_{15}) \sinh (β_{2} h_{2}) [β_{1} \cosh (β_{1} h_{1}) - λ \sinh (β_{1} h_{1})] + ({β_{1}}^{2} - λ^{2}) G_{zy 0} \cosh (β_{2} h_{2}) \sinh (β_{1} h_{1})} \\ M_{5} = {β_{2} (c_{44} + α e_{15}) \sinh (β_{2} h_{2}) [β_{1} \cosh (β_{1} η) - λ \sinh (β_{1} η)] + ({β_{1}}^{2} - λ^{2}) G_{zy 0} \cosh (β_{2} h_{2}) \sinh (β_{1} η)} \end{matrix}

(16)

According to the inverse complex Fourier transform, equations (15) are written as

\begin{matrix} \bar{w} (x, y, s) = \frac{- i b_{z} (s) e^{λ (η - y)}}{2 π} \int_{- \infty}^{+ \infty} \frac{M_{1} [M_{2} \sinh (β_{1} y) + M_{3} \cosh (β_{1} y)]}{p M_{4}} e^{ip (x - ξ)} dp 0 \leq y \leq η \\ \bar{w} (x, y, s) = \frac{- i b_{z} (s) e^{λ (η - y)}}{2 π} \int_{- \infty}^{+ \infty} \frac{M_{5} [λ \sinh (β_{1} (y - h_{1})) + β_{1} \cosh (β_{1} (y - h_{1}))]}{p M_{4}} e^{ip (x - ξ)} dp η \leq y \leq h_{1} \end{matrix}

(17)

Based on the constitutive equations and equation (17), the components of the stress in the Laplace domain are obtained as follows:

\begin{matrix} τ_{zy} = \frac{G_{zy 0} b_{z} (s) e^{λ (η + y)}}{π} \int_{0}^{\infty} \frac{({β_{1}}^{2} - λ^{2}) \sinh (β_{1} (η - h_{1})) M_{6}}{p M_{4}} \sin (p (x - ξ)) dp 0 \leq y \leq η \\ τ_{zy} = \frac{G_{zy 0} b_{z} (s) e^{λ (η + y)}}{π} \int_{0}^{\infty} \frac{({β_{1}}^{2} - λ^{2}) \sinh (β_{1} (y - h_{1})) M_{5}}{p M_{4}} \sin (p (x - ξ)) dp η \leq y \leq h_{1} \end{matrix}

(18)

where

M_{6} = {β_{2} (c_{44} + α e_{15}) \sinh (β_{2} h_{2}) [β_{1} \cosh (β_{1} y) - λ \sinh (β_{1} y)] + ({β_{1}}^{2} - λ^{2}) G_{zy 0} \cosh (β_{2} h_{2}) \sinh (β_{1} y)}

(19)

Because the kernels are expected to exhibit certain singular behavior, the singular behavior of the stress components is investigated by the asymptotic values of the integrands in equation (18) for $p \to \infty$ . The details of the analysis are not given in this paper. Therefore, equation (18) is recast to more suitable forms as below:

\begin{matrix} τ_{zy} = \frac{G_{zy 0} b_{z} (s) e^{λ (η + y)}}{π} {\int_{0}^{\infty} [\frac{({β_{1}}^{2} - λ^{2}) \sinh (β_{1} (η - h_{1})) M_{6}}{p M_{4}} + \frac{g}{2} e^{gp (y - η)}] \sin (p (x - ξ)) dp \\ - \frac{g}{2} \frac{(x - ξ)}{{(x - ξ)}^{2} + g^{2} {(y - η)}^{2}}} 0 \leq y \leq η \\ τ_{zy} = \frac{G_{zy 0} b_{z} (s) e^{λ (η + y)}}{π} {\int_{0}^{\infty} [\frac{({β_{1}}^{2} - λ^{2}) \sinh (β_{1} (y - h_{1})) M_{5}}{p M_{4}} + \frac{g}{2} e^{- gp (y - η)}] \sin (p (x - ξ)) dp \\ - \frac{g}{2} \frac{(x - ξ)}{{(x - ξ)}^{2} + g^{2} {(y - η)}^{2}}} η \leq y \leq h_{1} \end{matrix}

(20)

The stress component is obtained as Cauchy singularity kind at the dislocation location, which is a well-known characteristic of the stress fields caused by Volterra-type dislocations.

3. Nonhomogeneous substrate weakened by multiple cracks

Let N be the number of cracks in the nonhomogeneous substrate. The distributed dislocation technique has been used by several investigators for the analyses of the cracked bodies under mechanical loading [15]. The configuration of the cracks is expressed in a parametric form as

\begin{matrix} x_{i} (q) = x_{i} + l_{i} q \\ y_{i} (q) = y_{i} & i = 1, 2, \dots, N & - 1 \leq q \leq 1 \end{matrix}

(21)

where $(x_{i}, y_{i})$ are the coordinates of the center of the cracks and $l_{i}$ is the half-length of the cracks. By employing the superposition principle, the components of the stress on a given crack surface are obtained. Assume the screw dislocations with unknown density are distributed on the infinitesimal segment $l_{j} du$ of the surface of jth crack. So, the following integral equation is represented as

{\bar{σ}}_{y z_{i}} (x_{i} (q), y_{i} (q), s) = \sum_{j = 1}^{N} \int_{- 1}^{1} {\bar{K}}_{ij} (q, u, s) B_{zj} (u, s) l_{j} du

(22)

where $B_{zj} (u, s)$ is the Laplace transform of the dislocation density on the face of jth crack. From equation (20), the kernel of integral equation (22) is given by

\begin{matrix} {\bar{K}}_{ij} (q, u, s) = \frac{G_{zy 0} e^{λ (y_{j} + y_{i})}}{π} {\int_{0}^{\infty} [\frac{({β_{1}}^{2} - λ^{2}) \sinh (β_{1} (y_{j} - h_{1})) M_{7}}{p M_{4}} + \frac{g}{2} e^{gp (y_{i} - y_{j})}] \sin (p (x_{i} - x_{j} + l_{i} q - l_{j} u)) dp \\ - \frac{g}{2} \frac{(x_{i} - x_{j} + l_{i} q - l_{j} u)}{{(x_{i} - x_{j} + l_{i} q - l_{j} u)}^{2} + {(g (y_{i} - y_{j}))}^{2}}} 0 \leq y_{i} \leq y_{j} \\ {\bar{K}}_{ij} (q, u, s) = \frac{G_{zy 0} e^{λ (y_{j} + y_{i})}}{π} {\int_{0}^{\infty} [\frac{({β_{1}}^{2} - λ^{2}) \sinh (β_{1} (y_{i} - h_{1})) M_{8}}{p M_{4}} + \frac{g}{2} e^{- gp (y_{i} - y_{j})}] \sin (p (x_{i} - x_{j} + l_{i} q - l_{j} u)) dp \\ - \frac{g}{2} \frac{(x_{i} - x_{j} + l_{i} q - l_{j} u)}{{(x_{i} - x_{j} + l_{i} q - l_{j} u)}^{2} + {(g (y_{i} - y_{j}))}^{2}}} y_{j} \leq y_{i} \leq h_{1}} \end{matrix}

(23)

where

\begin{matrix} M_{7} = {β_{2} (c_{44} + α e_{15}) \sinh (β_{2} h_{2}) [β_{1} \cosh (β_{1} y_{i}) - λ \sinh (β_{1} y_{i})] + ({β_{1}}^{2} - λ^{2}) G_{zy 0} \cosh (β_{2} h_{2}) \sinh (β_{1} y_{i})} \\ M_{8} = {β_{2} (c_{44} + α e_{15}) \sinh (β_{2} h_{2}) [β_{1} \cosh (β_{1} y_{j}) - λ \sinh (β_{1} y_{j})] + ({β_{1}}^{2} - λ^{2}) G_{zy 0} \cosh (β_{2} h_{2}) \sinh (β_{1} y_{j})} \end{matrix}

(24)

The crack opening displacement across the jth crack is represented using the definition of dislocation as

{\bar{w}}_{j}^{-} (q, s) - {\bar{w}}_{j}^{+} (q, s) = \int_{- 1}^{s} l_{j} B_{zj} (u, s) du

(25)

The displacement fields must be a single-valued parameter; and so, the following closure conditions for embedded cracks is employed

l_{j} \int_{- 1}^{1} B_{zj} (u, s) du = 0

(26)

The numerical inversion of Laplace transform is carried out via Stehfest’s method [16] by introducing a time-dependent function $f (t)$ as

f (t) \approx \frac{l n 2}{t} \sum_{n = 1}^{M} υ_{n} F (\frac{l n 2}{t} n)

(27)

where $F (.)$ is the Laplace transform of $f (t)$ , M is a chosen even number, and the coefficients

υ_{n} = (- 1)^{\frac{M}{2} + n} \sum_{k = [0.5 (n + 1)]}^{min (\frac{M}{2}, n)} \frac{k^{\frac{M}{2}} (2 k)!}{(\frac{M}{2} - k)! k! (k - 1)! (n - k)! (2 k - n)!}

(28)

and [.] signifies the integral part of the quantity. From equation (27), the calculation of $f (t)$ at a fixed time t depends on the computation of $F (s)$ at M points $s = \frac{l n 2}{t} n, n \in {1, 2, \dots, M}$ . Applying the procedure into equations (22) and (26) leads to

\begin{matrix} {\bar{σ}}_{y z_{i}} (x_{i} (q), y_{i}, \frac{l n 2}{t} n) = \sum_{j = 1}^{N} \int_{- 1}^{1} {\bar{K}}_{ij} (q, u, \frac{l n 2}{t} n) B_{zj} (u, \frac{l n 2}{t} n) l_{j} du \\ l_{j} \int_{- 1}^{1} B_{zj} (u, \frac{l n 2}{t} n) du = 0 i \in {1, 2, \dots, N}, n \in {1, 2, \dots, M} \end{matrix}

(29)

The stress fields at the tip of an embedded crack behave like $1 / \sqrt{r}$ , where r is the distance from the crack tip. Consequently, the dislocation densities are given by

B_{zj} (u, \frac{l n 2}{t} n) = \frac{G_{zj} (u, \frac{l n 2}{t} n)}{\sqrt{1 - u^{2}}}, - 1 \leq u \leq 1,, j \in {1, 2, \dots, N}

(30)

Substituting equation (30) into equation (29) and applying the numerical technique expanded by Erdogan et al. [17] for the solution of singular integral equations, the resultant equations are solved. By using equation (27), the inverse Laplace transform of the solution yields to

g_{zj} (q, t) = \frac{l n 2}{t} \sum_{n = 1}^{M} υ_{n} G_{zi} (q, \frac{l n 2}{t} n), - 1 \leq q \leq 1, i \in {1, 2, \dots, N}

(31)

The SIFs of the embedded cracks derived by Bagheri et al. [18] are

{\begin{matrix} (K_{III})_{Li} \\ (K_{III})_{Ri} \end{matrix}} = \pm \frac{g \sqrt{l_{j} (\mp 1)}}{2} {\begin{matrix} G_{zy} (y_{Li}) \\ G_{zy} (y_{Ri}) \end{matrix}} g_{zi} (\mp 1, t)

(32)

where L and R designate the left and right tips of the crack, respectively; and $l_{j} (t) = \sqrt{{[{x'}_{j} (t)]}^{2} + {[{y'}_{j} (t)]}^{2}}$ . Finally, equation (31) is substituted into equation (32) to determine the SIFs.

4. Numerical solutions

In this section some examples are presented to demonstrate the applicability of the distributed dislocation technique to solve the problem of the orthotropic FGM strip with perfect piezoelectric coating with arbitrary number of cracks. This section is divided into two main parts. At first, the results are compared and verified and then, the effect of the parameters including the thickness of the piezoelectric coating, the orthotropic ratios, the damping coefficients and patterns of cracks on DSIFs are examined. In the all examples, the SIF is nondimensionalized by $K_{0} = τ_{0} \sqrt{l}$ where l is the half length of crack. For the PZT-4 piezoelectric ceramic considered in this work, the material properties are given as follows:

c_{44} = 2.56 \times 10^{10} N . m^{2}, e_{15} = 12.7 C . m^{- 2}, ε_{11} = 64.6 \times 10^{- 10} C / mV, ρ = 7.5 \times 10^{3} kg / m^{3}

The thickness of the substrate are assumed as $h_{1} = 0.1 (m)$ , and the number of points for the inversion of Laplace transform by employing Stehfest’s method is $M = 10$ , and the normalizing time is $t_{0} = l s_{T 1}$ , where $1 / s_{T 1}$ is the shear wave velocity at the orthotropic FG strip. To display the validity of the results, the cracks with length $2 l / h_{1} = 0.5$ under uniform shear traction $τ_{0}$ are considered. The quantity of interest in the fracture problems is the dimensionless SIFs, i.e. $K / K_{0}$ .

Figure 2 displays the effect of the thickness of the piezoelectric coating and damping coefficient on the non-dimensional SIFs versus $t / t_{0}$ . The effect of dimensionless time on the DSIFs in a homogenous layer and also in the FG layer is illustrated. It is interesting to note that the normalized SIF $K / K_{0}$ increases quickly at first with time to reach its maximum value, and then gets approximately stable. Finally, the DSIF approaches the corresponding static value.

Figure 2.

Dimensionless stress intensity factors for an embedded crack in an orthotropic FG layer without reinforcement and nonhomogeneous substrate with piezoelectric coating versus $t / t_{0}$ .

The SIFs attenuate as the damping effects enhance. For the analysis of the orthotropic FG strip, $h_{2}$ is set to zero [14]. The SIF of a crack in non-homogenous strip is in good agreement with that presented by Vafa et al. [14] for a crack in FGM strip. As shown in Figure 2, the SIFs decrease with increasing the thickness of the piezoelectric coating.

Furthermore, a central crack with length $2 l / h_{1} = 0.2$ at the time $t / t_{0} = 10$ is considered in FG layer with $λ = 10$ under uniform traction $τ_{0} H (t)$ . The effects of the damping coefficients and the thickness of the piezoelectric coating on the DSIFs are plotted in Figure 3. According to the results, by increasing the damping coefficients and the thickness of the piezoelectric coating, the SIFs decrease.

Figure 3.

The variations of the normalized stress intensity factor of a central crack in the substrate versus $t / t_{0}$ .

The effect of the dimensionless time on the behavior of the crack tips is plotted in Figure 4 for four orthotropic ratios (i.e. g). The material properties of the layers are $λ = 10$ and $γ_{0} = 0.01 G_{zy 0}$ . The thickness of the piezoelectric coating and the crack length are assumed as $h_{2} / h_{1} = 0.1$ and $l / h_{1} = 0.1$ respectively, while the center of the crack is set to ${y_{c} / h}_{1} = 0.5$ . As shown in Figure 4, by increasing g, the SIF peak of the non-homogenous substrate occurs in shorter time. One may conclude that for these patterns, the parameter ghas a great influence on DSIF.

Figure 4.

The variations of the normalized stress intensity factor of a crack versus $t / t_{0}$ for multiple orthotropic ratios.

In the next example, two equal-length cracks which are parallel to the FG layer edge with length ratio $2 l / h_{1} = 0.5$ and $h_{2} = 0.0$ are shown in Figure 5, where $γ_{0} = 0.0$ and FGM constant is $λ = 10$ . Crack $L_{1} R_{1}$ is located on the center-line of the strip, whereas crack $L_{2} R_{2}$ is situated on the vertical distance $(2 / 3) h_{1}$ from the x-axis. The variation of the normalized SIFs versus dimensionless $t / t_{0}$ is plotted in Figure 5. Obviously, increasing $λ$ and distance between two cracks tips lead to decreasing the SIFs.

Figure 5.

The variations of the normalized stress intensity factor for two cracks versus $t / t_{0}$ .

In the next example, two equal-length cracks which are parallel to the layer edges with length ratio $2 l / h_{1} = 0.5$ and $h_{2} / h_{1} = 0.2$ are shown. The FGM constant is $λ = 10$ and the distance between the cracks centers remains fixed, while the damping coefficient changes. Figure 6 depicts the dimensionless SIFs versus the dimensionless time when $γ_{0} / G_{zy 0} = 0.01, 0.02$ . The normalized SIFs increase with decreasing of the damping coefficient. Moreover, as expected, the variations of the SIFs for two approaching crack tips change rapidly.

Figure 6.

The variations of the normalized stress intensity factor for two cracks versus $t / t_{0}$ for two damping coefficients.

In the next example, three equal-length cracks with length ${2 l / h}_{1} = 0.5$ and ${h_{2} / h}_{1} = 0.2$ are studied (Figure 7). The material properties are $λ = 10$ and $γ_{0} = 0.02 G_{zy 0}$ . The variations of the DSIFs of the adjacent crack tips (i.e. $L_{2}$ ) are too large, whereas for other crack tips (i.e. $L_{1}$ and $L_{3}$ ) are not significant.

Figure 7.

The variations of the normalized stress intensity factor for three equal length cracks versus $t / t_{0}$ .

Figure 8 shows the variation of the normalized SIFs (i.e. $K / K_{0}$ ) versus $t / t_{0}$ for three positions of the $L_{2} R_{2}$ crack and two damping coefficients. The maximum SIF for the $L_{2}$ crack tips occurs when $γ_{0} = 0.0$ and the distance between the $L_{2} R_{2}$ crack and the two fixed cracks is minimal. Moreover, the SIFs decrease for bigger distances of crack from the interface perfect bonding.

Figure 8.

The variations of the normalized stress intensity factor for three equal length cracks versus $t / t_{0}$ .

5. Conclusion

The transient response of a functionally graded orthotropic substrate with a piezoelectric coating weakened by multiple cracks is studied. The system is subjected to anti-plane mechanical and in-plane electrical loadings. Using the Fourier and Laplace transform methods, the associated boundary value problem is reduced to a system of singular integral equations for the Volterra dislocation density. The results are verified by considering a single crack in an FGM strip. The examples of the multiple cracks show that the DSIF at the crack tips increases by growing the crack length, decreases by growing the damping coefficient, and decreases by increasing the thickness of the coating. The results are in excellent agreement with the analytical solutions obtained by Vafa et al. [14]

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Babaei

Lukasiewicz

SA.

Dynamic response of a crack in a functionally graded material between two dissimilar half planes under anti-plane shear impact load. Eng Fract Mech 1998; 60: 479-487.

Wei

Zhang

et al . Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading. Int J Fract 2000; 105: 127–136.

Chen

Worswiek

MJ.

Dynamic response of two coplanar cracks in a half space under antiplane shear. Mech Res Commun 2000; 27: 691-696.

Shul

Lee

KY.

Dynamic response of subsurface interface crack in multi-layered orthotropic half-space under anti-plane shear impact loading. Int J Solids Struct 2001; 38: 3563-3574.

Shul

Lee

KY.

A subsurface eccentric crack in a functionally graded coating layer on the layered half-space under an anti-plane shear impact load. Int J Solids Struct 2002; 39: 2019-2029.

Zhang

Sladek

Effects of material gradients on transient dynamic mode-III stress intensity factors in a FGM. Int J Solids Struct 2003; 40: 5251–5270.

Guo

Zeng

et al . The transient response of a coating–substrate structure with a crack in the functionally graded interfacial layer. Compos Struct 2005; 70: 109–119.

Chen

ZT.

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load. Int J Fract 2006; 141: 395-402.

Yongdong

Wei

Hongcai

Anti-plane transient fracture analysis of the functionally gradient elastic bi-material weak/infinitesimal-discontinuous interface. Int J Fract 2006; 142: 163-171.

10.

Wang

Mai

YW.

A periodic array of cracks in functionally graded materials subjected to transient loading. Int J Eng Sci 2006; 44: 351–364.

11.

Faal

Fariborz

Daghyani

. Antiplane deformation of orthotropic strips with multiple defects. J Mech Mater Struct 2006; 1: 1097-1114.

12.

Mousavi

Fariborz

SJ.

Anti-plane elastodynamic analysis of cracked graded orthotropic layers with viscous damping. Appl Math Model 2012; 36: 1626–1638.

13.

Monfared

Ayatollahi

Dynamic stress intensity factors of multiple cracks in an orthotropic strip with FGM coating. Eng Fract Mech 2013; 109: 45-57.

14.

Vafa

Baghestani

Fariborz

SJ.

Transient screw dislocation in exponentially graded FG layers. Arch Appl Mech 2015; 85: 1-11.

15.

Weertman

Dislocation based fracture mechanics. Singapore: World Scientific Publishing Co., 1996.

16.

Cohen

AM.

Numerical methods for Laplace transform inversion. Berlin: Springer, 2007.

17.

Erdogan

Gupta

Cook

. Numerical solution of singular integral equations. In: Sih

(ed.) Methods of analysis and solutions of crack problems. Dordrecht, the Netherlands: Springer, 1973, 368–425

18.

Bagheri

Ayatollahi

Mousavi

SM.

Analysis of cracked piezoelectric layer with imperfect non-homogeneous orthotropic coating. Int J Mech Sci 2015; 93: 93-101.