Numerical distributed dislocation modeling of multiple cracks in piezoelectric media considering different crack-face boundary conditions and finite size effects

Abstract

Piezoelectric materials primarily consist in smart composite materials have been extensively used as sensors/actuators in smart adaptive structures due to their excellent electromechanical properties. In this paper, the generalized fracture parameters of multiple semi-permeable and impermeable cracks in two-dimensional (2-D) piezoelectric materials related to multilayered/multiferroic composite materials are numerically investigated based on distributed dislocation method (DDM), emphasizing on the finite-specimen-size effects. The Lobatto–Chebychev quadrature method is used to solve the simultaneous singular integral equations, which are obtained after distributed dislocation modeling of the problem. Different configurations of multiple cracks under impermeable and semi-permeable crack-face boundary conditions are performed. The numerical results of generalized intensity factors obtained by the DDM are validated against reference solutions derived from the extended finite element method. The present results show great effects of finite size of a specimen, mutual influence of multiple cracks and crack-face boundary conditions on the generalized fracture parameters. In addition, numerical results also reveal the accuracy and efficacy of the DDM in studying the crack problems in 2-D finite piezoelectric media. The method of distributed dislocation which employed here to analyze the fracture parameters in finite piezoelectric media can be extended to finite multiferroic composites or multilayered piezoelectric/piezomagnetic media under various electrical boundary conditions.

Keywords

Smart materials fracture computational modeling finite element analysis (FEA)piezoelectric distributed dislocation

1. Introduction

In recent years, due to the development of stable and high performance sensors and actuators, smart composite materials have been extensively used in smart adaptive structures technology and other micro-electromechanical systems to actively monitor or control the systems according to their environmental change. This could be done by modifying the system characteristics or the system response in a controlled fashion. Piezoelectric materials due to their excellent electromechanical properties have been extensively used in smart composite adaptive structures. Even due to their easy fabrication, design flexibility and low cost, these materials have been easily integrated into multilayered composite structures by combining with polymers, piezomagnetic and ferroelectric ceramics, etc. In fact, piezoelectric composites composed of a piezoelectric ceramic and a polymer are promising materials because of their high coupling factors, low acoustic impedance, good matching to water or human tissue, mechanical flexibility, broad bandwidth in combination with a low mechanical quality factor and the possibility of making undiced arrays by simply patterning the electrodes. Among smart composite materials, multiferroic composites which consist of alternate piezoelectric and piezomagnetic layers have enormous potential in high performance engineering applications. So, accurate material models capable of predicting their properties and response are of great interest even at the level of materials used in layer wise fabrication. Since, the presence of cracks, holes and voids in piezoelectric materials resulting from manufacturing process or/and created during fabrication of adaptive structures, affect the performance of smart adaptive structures. In particular, the study of defects of order equal to the size of specimen is crucial and imperative to thoroughly investigate the fracture behaviors of such materials, and for the same computational approach is required. Therefore, it is of great importance to study the fracture of finite piezoelectric media under different crack-face boundary conditions using DDM.

Historically, intensive research on piezoelectric fracture mechanics started in the 1980s motivated by the failure work in piezoelectric devices [1]. Rajapakse [2] applied the Fourier integral transforms to study the stress analysis of piezoelectric solids and mechanical problems related to smart composite elements. Xu and Rajapakse [3] applied the boundary integral equation method to study the arbitrary shaped defects in 2-D piezoelectric media under mechanical and electrical loads. Yu [4] developed a micromechanics model based on periodic microstructure to predict the overall properties of piezoelectric composites. Analytic estimates of the overall elastic, dielectric, and piezoelectric properties of two-phase piezoelectric composites were obtained with the aid of equivalent inclusion method and Fourier series expansion. Denda and Lua [5] extended a physical interpretation of Somigliana’s identity to piezoelectric media and gave a direct formulation of the BEM in terms of continuous distributions of point forces/charges and displacement/electric potential discontinuities in the infinite piezoelectric domain. Zeng and Rajapakse [6] analyzed the amplification and shielding effects for the main crack due the presence of a micro crack in piezoelectric media. Numerical results for stress intensity factors and mechanical strain energy release rate at the crack tips of a main crack were presented under various loading conditions and orientation of a micro crack. Sapsathiarn et al. [7] studied the electro-mechanical interaction between a fiber and a matrix material in a 1–3 piezocomposite due to an axial load and electric charge applied to the fiber using Fourier integral transforms. Malakooti and Sodano [8] extended the multi-Inclusion models to predict the effective electroelastic properties of multiphase piezoelectric composites. To evaluate the micromechanics modeling results, a three dimensional finite element model of a four-phase piezoelectric composite was created in the commercial finite element software ABAQUS with different volume fractions and aspect ratios. Wan et al. [9] studied the multilayered piezomagnetic/piezoelectric composite with periodic interface cracks, subjected to in-plane magnetic or electric fields using Cauchy singular integration method. Several researchers, e.g., see [10–20], initiated working on crack problems in such smart materials by applying complex variable & Fourier transforms approach subjected to a far-field uniform tension and in-plane electric displacement but the domain of their analysis is infinite. To study the fracture parameters in 2-D cracked finite piezoelectric media, various numerical techniques such as boundary element method (BEM) [21–23], finite element method (FEM) [24,25], extended finite element method [26–32], etc. have been extensively applied to these problems. However, most of the previous works, e.g., see [21–32], are studied under impermeable crack-face boundary conditions. Hao and Shen [33] proposed a new electric boundary condition formed as semi-permeable crack-face condition in which the electric permeability of air in a crack gap was considered. An exact solution to this problem was described and some numerical results were gained and investigated. Importantly, it was found that the electric permeability of air in a crack gap leads to a value of the electric displacement intensity factor which was less than that of an impermeable crack boundary condition. Later on the semi-permeable boundary conditions in 2-D piezoelectric media is applied to investigate different crack configurations [34–44].

The distributed dislocation method (DDM) is one of the elegant techniques developed for analyzing the crack problems in an infinite and finite domain of isotropic and anisotropic materials [1,45]. Recently, Zhang et al. [46] showed that applying of this method has significant computational advantages than other computational techniques including the FEM, X-FEM and BEM to examine the fracture problems in a finite specimen. The underlying idea of this method is to construct integral equations for a specimen weakened by crack(s). These equations are of Cauchy singular type at the location of dislocation and solved numerically to obtain the dislocation density on the crack-faces. The dislocation densities are further used to determine the stress intensity factors for crack(s) located in the domain. Zhang et al. [46] presented a numerical solution of interaction between cracks and a circular inclusion in a finite plate using DDM. Furthermore, Han and Dhanasekar [47] presented an analytical approach based on the principle of continuous distribution of dislocation to model curved cracks in solids of arbitrarily shaped finite geometries. Hejazi et al. [48] studied the transient response of multiple cracks subjected to shear impact load in a half-plane employing DDM. Zhang et al. [49] reported a numerical solution with the aid of the DDM to model multiple cracks in a finite plate of an elastic isotropic material. Deeg [1] analyzed the dislocation, cracks and inclusion problems in piezoelectric solids. Various crack problems in 2-D piezoelectric infinite domain [50] and half-plane [51,52] are demonstrated by the DDM.

However, it is very important to note that most of the previous studies in 2-D piezoelectric media are reported for the impermeable crack problems present in half-plane and infinite domains using DDM. Based on the best knowledge of the authors, this method has not been so far explored for studying the interaction among the cracks (or multiple cracks) in 2-D finite piezoelectric specimen subjected to different crack-face boundary conditions. Hence, to attempt this paucity, in this paper the authors applied the DDM to present the specimen size effects on multiple cracks in 2-D piezoelectric media subjected to both the impermeable and semi-permeable crack-face boundary conditions, and that is thought as the main objective of the present work. By accomplishing that, a numerical approach based on the DDM to model multiple cracks is developed in which the Lobatto–Chebychev quadrature method [45] is employed to solve the simultaneous singular integral equations. The singular integral equations are obtained after distributed dislocation modeling of the problems. Different configurations involving multiple cracks, e.g., two collinear equal cracks, two horizontal parallel equal cracks, two inclined parallel equal cracks, three collinear cracks, and so on are considered. To show the advantages and the accuracy of the present approach, the numerical results computed by using the present method for both crack-face boundary conditions are verified against the conventional X-FEM. A finite computational domain is taken throughout the study and the finite size effect on the solutions is hence explored. The method of distributed dislocation which employed here to analyze the fracture parameters in finite piezoelectric media under different crack-face boundary conditions can also be extended to finite multiferroic composites or multilayered piezoelectric/piezomagnetic media.

The body of the paper is structured as follows: Section 2 briefly presents the basis equations for piezoelectric materials, while the distributed dislocation method in 2-D piezoelectric media is given in Section 3. We then present the problem formulation which is being studied in the present work. Numerical validation with several numerical examples of benchmarks to show the accuracy of the present method is provided in Section 5. In Section 6, multiple semi-permeable cracks in 2-D finite piezoelectric media is presented, discussed and investigated. Some conclusions drawn from the study are ended in Section 7.

2. Fundamental equations for piezoelectricity

The fundamental equations and the boundary conditions for two-dimensional linear piezoelectric materials can be written below [27–29].

2.1. Field equations

Basically, the field equations for a linear piezoelectric medium in a fixed rectangular coordinate system $x_{j}$ ( $j = 1, 2, 3$ ) can be expressed as

Constitutive equations $\begin{matrix} (1) & σ_{i j} = C_{i j k s} ε_{k s} - e_{s i j} E_{s}; D_{i} = e_{i k s} ε_{k s} + κ_{i s} E_{s} \end{matrix}$

Kinematic equations $\begin{matrix} (2) & ε_{i j} = \frac{1}{2} (u_{i, j} + u_{j, i}); E_{i} = - φ_{, i} \end{matrix}$

In the absence of body forces and charges, the equilibrium equations for the stresses and electric displacements can be described as $\begin{matrix} (3) & σ_{i j, j} = 0; D_{i, i} = 0 \end{matrix}$

where

C_{i j k s}

and

e_{i k s}

are the mechanical elastic and piezoelectric constants, respectively;

σ_{i j}

ε_{i j}

D_{i}

and

E_{i}

are the components of the stress, strain, electric displacement, and electric field, respectively; while

κ_{i s}

defines the dielectric permittivity. The comma denoted in Eqs (2) and (3) represents the partial differentiation against the argument following it;

u_{i}

is the component of the elastic displacement vector u; while φ is the electric potential; where i, j, k and

s = 1, 2, 3

2.2. Boundary conditions

Let us consider a piezoelectric body occupying the space Ω enclosed by the surface S. The resultant of the stresses and electric displacements on the boundaries $S_{σ}$ and $S_{D}$ are given by $\begin{array}{l} (4) & σ_{i j} n_{j} = t_{j}^{0}, on S_{σ}, \\ (5) & D_{j} n_{j} = ω^{0}, on S_{D}, \end{array}$ where $t_{j}^{0}$ represents the prescribed value on the surface $S_{σ}$ and $ω^{0}$ is given on surface $S_{D}$ while n being the outward drawn unit normal vector on S.

2.3. Crack-face boundary conditions

There are mainly three boundary conditions on crack-face taken in literature [1,33–44] namely impermeable, permeable and semi-permeable. These crack face boundary conditions are represented mathematically as

Impermeable boundary conditions

Crack faces $Γ_{C}$ are assumed to be traction-free and electrically impermeable $\begin{matrix} (6) & σ_{i j} n_{j} = 0 and D_{j} n_{j} = 0 on Γ_{C} . \end{matrix}$

Permeable boundary conditions

In this case, crack is traction-free and does not obstruct any electric field $\begin{matrix} (7) & σ_{i j} n_{j} = 0 and φ^{+} = φ^{-}, D_{2}^{+} = D_{2}^{-} \neq 0 \end{matrix}$

Semi-permeable boundary conditions

Semi-permeable boundary conditions proposed by Hao and Shen [33–44] for piezoelectric ceramics are more realistic boundary conditions and are given by $\begin{matrix} (8) & σ_{i j} n_{j} = 0 and D_{2}^{+} = D_{2}^{-} = D_{2}^{c} = - κ_{c} \frac{Δ φ (x_{1})}{Δ u_{2} (x_{1})} \end{matrix}$ where “+” and “−” represent the upper and lower crack surfaces, $Δ φ (x_{1})$ is the electrical potential jump, and $Δ u_{2} (x_{1})$ is the crack opening displacement; while $κ_{c}$ is the permittivity of medium between crack faces.

It is interesting to see that one can reduce semi-permeable boundary conditions to impermeable one when the term $κ_{c}$ in Eq. (8) is set to be equal to zero, and to permeable one when the jump in electric potential vanishes.

3. Distributed dislocation method in 2-D piezoelectric media

The crack problems in 2-D piezoelectric media can be solved by using the distributed dislocation method, which essentially is based on modeling a crack with a continuous distribution of dislocations.

For two dimensional deformations, the generalized displacement vector $U = {[u_{1}, u_{2}, u_{3}, \emptyset]}^{T}$ , and the generalized stress vector $φ = {[φ_{1}, φ_{2}, φ_{3}, φ_{4}]}^{T}$ which depend on the coordinate system $(x, y)$ can be expressed as $\begin{matrix} (9) & U = 2 Re (A f (z)) and φ = 2 Re (B f (z)) \end{matrix}$ where $z = x + p y$ . The eigenvalues p and eigenvectors A are obtained after solving the following characteristic equation $\begin{matrix} (10) & ([\begin{matrix} C_{i 1 k 1} & e_{i 11} \\ e_{1 k 1} & - κ_{11} \end{matrix}] + [\begin{matrix} C_{i 2 k 1} + C_{i 1 k 2} & e_{i 21} + e_{i 12} \\ e_{2 k 1} + e_{1 k 2} & - κ_{12} - κ_{21} \end{matrix}] p + [\begin{matrix} C_{i 2 k 2} & e_{i 22} \\ e_{2 k 2} & - κ_{22} \end{matrix}] p^{2}) [\begin{matrix} A_{i} \\ A_{4} \end{matrix}] = 0 \end{matrix}$ whereas the eigenvectors B are determined by the following relation: $\begin{matrix} (11) & [\begin{matrix} B_{i} \\ B_{4} \end{matrix}] = \{[\begin{matrix} C_{i 1 k 2} & e_{i 12} \\ e_{1 k 2} & - κ_{21} \end{matrix}] + p [\begin{matrix} C_{i 2 k 2} & e_{i 22} \\ e_{2 k 2} & - κ_{22} \end{matrix}]\} [\begin{matrix} A_{i} \\ A_{4} \end{matrix}] . \end{matrix}$ The stress and electric displacement can be evaluated by using the following relations as $\begin{matrix} (12) & [\begin{matrix} t_{1} \\ t_{2} \end{matrix}] = [\begin{matrix} - φ_{, 2} \\ φ_{, 1} \end{matrix}] = [\begin{matrix} {σ_{11}, σ_{12}, σ_{13}, D_{1}} \\ {σ_{21}, σ_{22}, σ_{23}, D_{2}} \end{matrix}] . \end{matrix}$

In fact, Green’s function is a solution due to a point load or a dislocation located in a physical solid of infinite domain. By determining the appropriate distribution of slipping and climbing edge dislocations and screw dislocations, all three modes of fracture can be simulated. Barnett and Lothe [10] extended this concept to piezoelectric media. Later, Deeg [1], Pak [12], Suo et al. [15] and others [50–52] presented a more comprehensive discussion on this topic. As a result, the solution for an infinite domain subjected to a line dislocation b located at $(x_{1}, x_{2})$ as $\begin{matrix} (13) & \begin{matrix} U = \frac{1}{π} Im {A ⟨ ln (z_{k} - z_{k 0}) ⟩ B^{T}} b \\ φ = \frac{1}{π} Im {B ⟨ ln (z_{k} - z_{k 0}) ⟩ B^{T}} b \end{matrix} \end{matrix}$ where $z_{k} = z_{k 0} + ξ z_{k}^{*}$ with $z_{k 0} = x_{1} + p x_{2}$ and $z_{k}^{*} = cos α + p_{k} sin α$ .

For a given problem, b can be determined by imposing it to be satisfied the related loading and boundary conditions. Having applied the dislocation density $b (x)$ , the fracture parameters such as intensity factors and energy release rate can then be determined.

4. Problem formulation

In this work, a finite computational domain by a rectangular specimen of size $2 W \times 2 H$ is considered, which is cut-out by an infinite piezoelectric domain. Figure 1 schematically shows the geometrical configuration of multiple cracks inclined at angles $α_{i}$ and length $2 c_{i}$ presenting in the finite specimen cut-out from half-plane. For convenience in the implementation, a local coordinate system $(ξ, t)$ is defined to handle multiple oriented cracks situated in the specimen.

Fig. 1.

Schematic notation of a finite specimen cut-out from an infinite domain used for the implementation.

The cracks and boundaries of the specimen are modeled as a continuous distribution of dislocations with generalized Burgers vectors $b_{h}^{i} = {[b_{1}^{i} b_{2}^{i} b_{3}^{i} b_{4}^{i}]}^{T}$ , where $b_{h}^{i}$ ( $h = 1, 2, 3$ ) and $b_{4}^{i}$ are mechanical displacement jump and an electrical potential jump in the plane at points $(x_{1}^{i}, x_{2}^{i})$ , respectively. It is equivalent to a multiple cracks problem in the infinite solid with prescribed loads on the actual cracks faces and traction/induction free conditions on the boundary of the specimen [45,47,49]. Expressing the cracks lines as $z_{k}^{i} = z_{k 0}^{i} + ξ z_{k}^{* i}$ , where $z_{k 0}^{i} = x_{1}^{i} + p_{k} x_{2}^{i}$ and $z_{k}^{* i} = cos α^{i} + p_{k} sin α^{i}$ , the dislocation $b_{h}^{i}$ becomes $b_{h}^{i} (ξ)$ . Using the principle of superposition, n multiple cracks in 2-D piezoelectric finite specimen is decomposed into $n + 4$ sub-problems i.e., n cracks and four traction and induction free cracks corresponding to the boundaries of the rectangular specimen [45,47,48]. Enforcing the applied surface traction charge conditions on each crack face, a system of singular integral equations for dislocation density $b_{h}^{i} (ξ)$ is obtained by $\begin{matrix} (14) & \begin{matrix} \frac{L}{2 π} \int_{- c_{i}}^{c_{i}} \frac{b_{h}^{i} (ξ_{i}) d ξ_{i}}{t_{i} - ξ_{i}} + \frac{1}{π} {\sum_{\begin{array}{c} j = 1 \\ j \neq i \end{array}}^{n} \int_{- c_{j}}^{c_{j}} K_{i j} (t_{i}, ξ_{j}) b_{h}^{j} (ξ_{j}) d ξ_{j}} \\ = - t_{η_{i}} (t_{i}), | t_{i} | < c_{i}, i = 1, 2, \dots, n \end{matrix} \end{matrix}$ where $K_{i j} (t_{i}, ξ_{j}) = Im (B ⟨ z_{k i}^{*} {(z_{k i}^{0} + t_{i} z_{k i}^{*} - z_{k j}^{0} - ξ_{j} z_{k j}^{*})}^{- 1} ⟩ B^{T})$ is a kernel function of the singular integral equations and are Holder-continuous along $- c_{i} ⩽ ξ ⩽ c_{i}$ .

The boundary conditions on each ith crack: $\begin{matrix} (15) & t_{n i} = [\begin{matrix} - σ_{x x} \\ - σ_{x y} \\ - σ_{x z} \\ - D_{x} \end{matrix}] sin θ_{i} + [\begin{matrix} - σ_{y x} \\ - σ_{y y} \\ - σ_{y z} \\ - (D_{y} - D_{y}^{c}) \end{matrix}] cos θ_{i}, \end{matrix}$ where $\begin{matrix} (16) & D_{y}^{c} = - κ_{c} \frac{H_{41} σ_{y x} + H_{42} σ_{y y} + H_{43} σ_{y z} + H_{44} (D_{y} - D_{y}^{c})}{H_{21} σ_{y x} + H_{22} σ_{y y} + H_{23} σ_{y z} + H_{24} (D_{y} - D_{y}^{c})} and H = - 2 Im (A L^{- 1}) . \end{matrix}$ On the boundaries of specimen: $\begin{matrix} (17) & t_{n i} = 0 . \end{matrix}$ For single valued displacements and electric potential around a closed contour surrounding the actual cracks, the following conditions must also be satisfied $\begin{matrix} (18) & \int_{- c_{i}}^{c_{i}} b_{h}^{i} (ξ) d ξ = 0 for i = 1, 2, \dots, n . \end{matrix}$

Furthermore, the dislocation densities along ith boundary and along $(i + 1)$ th boundary must equal at the corner points of the boundary of finite specimen i.e., $\begin{matrix} (19) & b_{h}^{i th boundary} (c_{i}) = [\begin{matrix} cos α & sin α & 0 & 0 \\ - sin α & cos α & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] b_{h}^{(i + 1) th boundary} (- c_{i}); for i = n + 1 to n + 4 . \end{matrix}$ The numerical solution of the above system of singular integral equations is obtained by using closed type Lobatto–Chebychev quadrature method [45] so that the end points are also included in the set of integration points, and the discretized form of equations (15) and (16) can be written as $\begin{array}{l} (20) & \sum_{k = 1}^{m} \frac{1}{m} [\frac{L}{2 (s_{r 0} - s_{k})} + \sum_{\begin{array}{c} j = 1 \\ j \neq i \end{array}}^{j = n} K (s_{r 0}, s_{k})] ϑ_{h}^{i} (s_{k}) = - t_{n i} (s_{r 0}) for i = 1, 2, \dots, n \\ (21) & \sum_{k = 1}^{m} ϑ_{h}^{i} (s_{k}) = 0 for i = 1, 2, \dots, n \end{array}$ and $\begin{matrix} (22) & ϑ_{h}^{i th boundary} (1) = Ω (α) ϑ_{h}^{(i + 1) th boundary} (- 1) for i = n + 1 to n + 4, \end{matrix}$ where $Ω (α)$ is a rotation matrix, $b (ξ) = \frac{\sum_{k = 1}^{m} w_{k} ϑ (s_{k})}{\sqrt{} (1 - s_{k}^{2})}$ , $s_{1} = 1$ , $s_{m} = - 1$ , $s_{k} = cos (\frac{π (k - 1)}{m - 1})$ , $w_{1} = w_{m} = 0.5$ , $w_{k} = 1$ for $k = 2, 3, \dots, m - 1$ and $s_{r 0} = cos (\frac{π (2 r - 1)}{2 (m - 1)})$ , $r = 1, 2, \dots, m - 1$ .

Once the generalized dislocation densities have been obtained, the intensity factors (IFs) at the crack tips ( $\pm c_{i}$ ) are given by $\begin{matrix} (23) & \begin{matrix} {[\begin{matrix} K_{II} & K_{I} & K_{III} & K_{IV} \end{matrix}]}^{T} \\ = \sqrt{\frac{π}{4 c_{i}}} [\begin{matrix} cos α & sin α & 0 & 0 \\ - sin α & cos α & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] L {[\begin{matrix} ϑ_{1}^{i} (\pm 1) & ϑ_{2}^{i} (\pm 1) & ϑ_{3}^{i} (\pm 1) & ϑ_{4}^{i} (\pm 1) \end{matrix}]}^{T} \end{matrix} \end{matrix}$ where $L = - 2 i B B^{T}$ .

5. Numerical validation

This section is devoted to the numerical validation of the distributed dislocation method applied to 2-D piezoelectric cracks problems under impermeable crack-face boundary conditions. To verify the accuracy of the present formulation, three examples are thus considered to serve the validation purpose. The present results are compared with the reference solutions derived from the standard X-FEM. The X-FEM associated with the six fold crack-tip enrichment functions defined for 2-D piezoelectric domain problems [18–23] have been used for the analysis. The plain strain assumption is discussed once a tensile loading $σ_{y y} = 5 MPa$ and an electric-displacement $D_{y} = 0.003 C / m^{2}$ prescribed on the remote boundary of a 2-D finite piezoelectric body. The polarization is directed along the $x_{2}$ -axis. It must be noted here that the implementation of the DDM is fulfilled by taking a finite specimen cut-out from infinite domain, and modeling the boundaries of the finite specimen and cracks as a continuous distribution of dislocations as discussed in Section 4. In this work, the piezoelectric material named as PZT-5H is taken for the present investigation. The material constants of PZT-5H is tabulated in Table 1 [28,29,40]. The dielectric constant of the material in the crack cavity is taken to be $κ_{c} = 8.85 \times 10^{- 12} {Fm}^{- 1}$ , and $D_{2}^{c}$ mentioned in Eq. (8) is calculated from Eq. (16). For convenience in the representation of the numerical results, the mechanical stress intensity factors $K_{I}$ , $K_{II}$ and the electrical displacement intensity factor $K_{IV}$ are then normalized by the following relations $\begin{matrix} (24) & K_{I}^{*} = \frac{K_{I}}{σ_{y y} \sqrt{π c}}; K_{II}^{*} = \frac{K_{II}}{σ_{y y} \sqrt{π c}} and K_{IV}^{*} = \frac{K_{IV}}{D_{y} \sqrt{π c}} \end{matrix}$ where c is the half-crack length.

Table 1
The material constants of piezoelectric PZT-5H used for the analysis

Properties PZT-5H

Elastic constants $c_{11} = 126$ GPa

$c_{12} = 55$ GPa

$c_{13} = 53$ GPa

$c_{33} = 117$ GPa

$c_{44} = 35.3$ GPa

Piezoelectric constants $e_{15} = 17$ C/m²

$e_{31} = - 6.5$ C/m²

$e_{33} = 23.3$ C/m²

Permittivity $κ_{11} = 15.1$ nC/Vm

$κ_{33} = 13.1$ nC/Vm

Properties	PZT-5H
Elastic constants	$c_{11} = 126$ GPa
$c_{12} = 55$ GPa
$c_{13} = 53$ GPa
$c_{33} = 117$ GPa
$c_{44} = 35.3$ GPa
Piezoelectric constants	$e_{15} = 17$ C/m²
$e_{31} = - 6.5$ C/m²
$e_{33} = 23.3$ C/m²
Permittivity	$κ_{11} = 15.1$ nC/Vm
$κ_{33} = 13.1$ nC/Vm

For multiple cracks problems, the normalized generalized IFs are also given by $\begin{matrix} (25) & K_{I}^{*} = \frac{K_{I}}{σ_{y y} \sqrt{π c_{01}}}; K_{II}^{*} = \frac{K_{II}}{σ_{y y} \sqrt{π c_{01}}} and K_{IV}^{*} = \frac{K_{IV}}{D_{y} \sqrt{π c_{01}}} \end{matrix}$ where $c_{01}$ is the half-crack length of the shorter crack.

5.1. A central-crack problem

We start by considering a finite piezoelectric plate of width $2 W$ and height $2 H$ subjected to impermeable crack face boundary conditions. The plate is weakened by a horizontal central crack of length $2 c$ , which occupies the interval $[- c, c]$ on $x_{1}$ -axis. The DDM is applied to solve the given problem. Reference numerical solutions calculated by the X-FEM are however employed for the comparison purpose. Figures 2–4 depict the model geometry and aspect ratio analysis of the central crack problem. Figure 2 shows the behavior of the normalized Ifs, $K_{I}^{*}$ and $K_{IV}^{*}$ , with respect to $H / W$ ratio, respectively. The ratio of $c / W = 0.1$ is taken for the analysis. The present numerical results are very interesting and it is observed in the figure that the amplitudes of the $K_{I}^{*}$ and $K_{IV}^{*}$ decrease significantly with increasing in the aspect ratio $H / W$ , and they approach to 1.0 for larger values of $H / W$ . It essentially depicts that the fracture parameters considerably depend on the specimen size of the domain.

Fig. 2.

Variation of $K_{I}^{*}$ and $K_{IV}^{*}$ versus $H / W$ obtained by the present DDM and the X-FEM for a center crack problem.

Fig. 3.

Variation of $K_{I}^{*}$ and $K_{IV}^{*}$ versus $W / c$ obtained by the present DDM and the X-FEM for a center crack problem.

Fig. 4.

Variation of $K_{I}^{*}$ and $K_{IV}^{*}$ versus $c / W$ for different aspect ratios $H / W$ obtained by the DDM and the X-FEM for a center crack problem.

Now, to further compare it with the infinite domain problem, both H and W equally approach to large values in respect of crack length. Consequently, the computed results of the $K_{I}^{*}$ and $K_{IV}^{*}$ versus $W / c$ ratio are thus plotted in Fig. 3. The present results of the normalized generalized IFs show that whenever $W / c$ ratio is equal or greater than 20, then the center crack problem in a finite specimen could be considered to be an infinite domain center crack problem. Moreover, to analyze the specimen size effects, the results of normalized generalized IFs versus $c / W$ ratio are compared in Fig. 4 for different aspect ratios $H / W$ . It is found that both the $K_{I}^{*}$ and $K_{IV}^{*}$ approach to 1.0 as the $c / W$ ratio decreases, implying that the solutions approach to an infinite domain results. The finite specimen size effects on the fracture parameters are hence confirmed as the values of the normalized generalized IFs decrease with increasing the aspect ratio. In Figs 2–4, the present results of the $K_{I}^{*}$ and $K_{IV}^{*}$ obtained through the DDM are also compared with those derived from the X-FEM, and a good agreement between the results validates and shows the efficacy of the DDM applied for 2-D finite cracked piezoelectric domain problems. One important thing is that the amplitudes of the mechanical stress intensity factors are larger than that of the electric displacement intensity factor in all cases, which can be found in all the numerical results in Figs 2–4.

Additionally, Fig. 5 highlights the computational time taken by the DDM and X-FEM with respect to the aspect ratio. Interestingly, it is obvious that the DDM has a good computational advantage than the X-FEM, and this conclusion is in agreement to the computational analysis done in [46]. In further subsections, study of different set of multiple cracks in 2-D finite piezoelectric specimen is attempted and the schematic representation of these problems is shown in Fig. 6.

Fig. 5.

Computational comparison of the DDM and X-FEM with respect to $H / W$ for a center crack problem. It is obvious that the DDM dominates over the X-FEM in terms of the computational time.

Fig. 6.

Schematic representation of different multiple cracks in 2-D piezoelectric media.

5.2. A two collinear equal cracks problem

We next study a specimen of width $2 W$ , length $2 H$ containing two equal cracks. Each crack has its length equal to $2 c_{01}$ with inter crack space distance equal to $2 d$ . The prescribed loading and other conditions are the same as discussed early in Section 5. The variations in the normalized generalized IFs with respect to $d / c_{01}$ and the model geometry of plate are shown in Fig. 7. Here, two different aspect ratios, $H / W = 0.5$ and 1.0 for instance, are considered for the analysis. The higher values of normalized generalized IFs at inner and outer tips for $H / W = 0.5$ clearly depict the effects of specimen size on fracture parameters. In addition, the interaction between the cracks can be noticed as the values of the $K_{I}^{*}$ and $K_{IV}^{*}$ are more at the inner tips than outer tips of the cracks, and coincide at each tip as the inter crack space distance increases. A good agreement against the X-FEM results is also found in these numerical results and it essentially reveals and validates the accuracy of the computer codes developed for solving the multiple cracks problems using the DDM.

Fig. 7.

Variation of $K_{I}^{*}$ and $K_{IV}^{*}$ with respect to $d / c_{01}$ for different aspect ratios $H / W$ for a two collinear equal crack problem under impermeable crack face boundary conditions.

Fig. 8.

Variation of $K_{I}^{*}$ and $K_{IV}^{*}$ with respect to $d t / c_{01}$ for different aspect ratios $H / W$ for a two horizontal parallel equal crack problem under impermeable crack face boundary conditions.

5.3. A two horizontal parallel equal cracks problem

To further validate the accuracy of the DDM for in modeling multiple parallel cracks problems in 2-D finite piezoelectric media, a two parallel equal cracks problem is considered. Each crack is set up to have a length of $2 c_{01}$ symmetrically placed at a distance $d t$ from the center of the specimen. The model geometry of plate and the variations in the normalized generalized IFs against $d t / c_{01}$ are sketched in Fig. 8. The numerical results of the $K_{I}^{*}$ and $K_{IV}^{*}$ are also obtained for the same aspect ratios as taken in the previous example addressed in Section 5.2. It is found that the normalized generalized IFs increase with the increase in distance $d t$ . The reason may be due to the fact that the cracks approach to the boundary of the specimen due to the increase in distance between the cracks and the applied loads can no longer be considered as far-field loadings. The higher values of fracture parameters obtained for smaller aspect ratio have also ensured the effects of specimen size on the generalized IFs. The comparison of the obtained DDM results with X-FEM results for parallel equal cracks problem further validate the code and approach of this method.

Fig. 9.

Variation of $K_{I}^{*}$ and $K_{IV}^{*}$ with respect to $d / c_{01}$ for different aspect ratios $H / W$ for a two collinear equal crack problem under semi-permeable crack face boundary conditions.

Fig. 10.

Variation of $K_{I}^{*}$ and $K_{IV}^{*}$ with respect to $d t / c_{01}$ for different aspect ratios $H / W$ for a two horizontal parallel equal crack problem under semi-permeable crack face boundary conditions.

Fig. 11.

Comparison of $K_{I}^{*}$ , $K_{II}^{*}$ and $K_{IV}^{*}$ obtained under impermeable and semi-permeable crack face boundary conditions with respect to $d t / c_{01}$ for different aspect ratios $H / W$ for a two inclined parallel equal crack problem.

Fig. 12.

Fig. 13.

Fig. 14.

Comparison of $K_{I}^{*}$ and $K_{IV}^{*}$ obtained under impermeable and semi-permeable crack face boundary conditions against $d / c_{01}$ for different aspect ratios $H / W$ for a three collinear cracks problem.

6. Multiple semi-permeable cracks in 2-D finite piezoelectric media

As the obtained DDM results are in good agreement with the X-FEM reference solutions for the aforementioned examples under impermeable crack face boundary conditions, therefore, in this section, five different multiple cracks problems in 2-D piezoelectric finite specimen are discussed under realistic boundary conditions or semi-permeable crack face boundary conditions using DDM. One of the straightway advantage of implementing this technique, for the study of 2-D piezoelectric semi-permeable multiple cracks, is the low computational cost [46,49] involved in this technique than other numerical techniques such as FEM, BEM, X-FEM, etc., applied for non-linear problems. Here, the obtained results are also compared with the solutions computed for the impermeable crack-face boundary conditions.

6.1. A two collinear equal cracks problems

The model geometry and all other conditions are the same as discussed in Section 5.2 except the crack face boundary conditions. In this section, the normalized generalized IFs are calculated by using the present method but for the semi-permeable crack face boundary conditions. Figure 9 depicts the variations in the $K_{I}^{*}$ and $K_{IV}^{*}$ versus $d / c_{01}$ for different aspect ratios such as $H / W = 0.5$ , 1.0 and 2.0. It is very interesting to see that both the figures show the same behaviors of the normalized generalized IFs with respect to $d t / c_{01}$ and $H / W$ , as concluded in Section 5.2. It also confirms the interaction between the cracks and specimen size effect. But the comparison between the results shown in Fig. 7 and Fig. 9 emphasize the effects of crack face boundary conditions on fracture parameters. It is found that the values of the normalized electric displacement IF, $K_{IV}^{*}$ in case of semi-permeable boundary conditions are approximately half than the values of which for the impermeable case, whereas the values of $K_{I}^{*}$ are almost same in both the cases.

6.2. A two horizontal parallel equal cracks problem

This is similar to the problem as discussed in Section 5.3 except the crack face boundary conditions, which are taken here as semi-permeable for the study of finite specimen size effects and crack face boundary conditions in 2-D cracked finite piezoelectric media. Figure 10 shows the model geometry and the plots of the normalized generalized IFs versus $d t / c_{01}$ under semi-permeable boundary conditions. The finite specimen size analysis is carried out by varying the aspect ratios $H / W = 0.5$ , 1.0 and 2.0. The obtained results of semi-permeable case shown in Fig. 10 are compared with impermeable case as shown in Fig. 8, and it is evident that the behaviors of the $K_{I}^{*}$ and $K_{IV}^{*}$ in both the cases behave the same. It essentially ensures the specimen size effects and interaction between the parallel cracks, even in semi-permeable crack face boundary conditions.

Similar to Section 6.1, here also the values of $K_{IV}^{*}$ in semi-permeable boundary conditions are almost half than the impermeable ones but there is no significant change observed in the numerical values of $K_{I}^{*}$ .

6.3. A two inclined parallel equal cracks problem – a mixed-mode example

This example is devoted to study the mixed-mode problem. A generalized case of two inclined equal parallel cracks is attempted in this section. Two inclined cracks with equal crack lengths $2 c_{01}$ and inclination angles each equal to $α = π / 6$ is considered within a finite specimen. The specimen size and prescribed loading conditions are taken the same as mentioned in Section 6.2. For this problem, the numerical results of the normalized generalized IFs $K_{I}^{*}$ , $K_{II}^{*}$ and $K_{IV}^{*}$ are however visualized against $d t / c_{01}$ , i.e., ratio of vertical half inter crack space distance to half crack length, for both impermeable and semi-permeable crack face boundary conditions. The effects of finite specimen size are highlighted by considering the problem for aspect ratio 0.5; 1.0 and 2.0. Figure 11 shows a comparison of the numerical results of the normalized generalized IFs between the impermeable and semi-permeable crack face boundary conditions. In both the crack face boundary conditions, specimen size effect is found on the $K_{I}^{*}$ , $K_{II}^{*}$ and $K_{IV}^{*}$ , as their values are more in case of lesser $H / W$ . For higher values of $H / W$ , the $K_{I}^{*}$ , $K_{II}^{*}$ and $K_{IV}^{*}$ approach to infinite domain results as the distance $d t$ increases. Similar to the previous Sections 6.1 and 6.2, the effect of the semi-permeable crack face boundary conditions is observed on $K_{IV}^{*}$ whereas almost no effect on $K_{I}^{*}$ and $K_{II}^{*}$

6.4. A one inclined and second horizontal cracks problem

This numerical example is also a mixed-mode problem. Two cracks each equal to crack length $2 c_{01}$ is taken in a 2-D piezoelectric finite specimen with one inclined crack positioned above center with inclination angle $α = π / 6$ and second horizontal crack below the center of the specimen. The model geometry of the specimen and prescribed loading conditions are taken the same as discussed in Section 6.2. The results of the normalized generalized IFs $K_{I}^{*}$ , $K_{II}^{*}$ and $K_{IV}^{*}$ versus $d t / c_{01}$ are shown in Figs 12–13, and both the impermeable and semi-permeable crack face boundary conditions are compared with each other. The analysis is investigated for different aspect ratios i.e. $H / W = 0.5$ , 1.0 and 2.0. The behavior of $K_{I}^{*}$ , $K_{II}^{*}$ and $K_{IV}^{*}$ is same subjected to both the applied crack face boundary conditions. It is observed in the numerical values that for smaller $H / W$ , the values of the $K_{I}^{*}$ , $K_{II}^{*}$ and $K_{IV}^{*}$ increase with increasing in inter crack space distance $d t$ . Once again, it can be found that the higher values at low aspect ratio are most likely due to the fact that the applied loads is no longer considered as the far-field loadings. The deviations from impermeable case obtained only in the numerical values of $K_{IV}^{*}$ for semi-permeable boundary conditions, show the effect of crack face boundary conditions on electrical displacement intensity factors and hence on total energy release rate.

6.5. A three collinear cracks problem

The problem of three collinear cracks in a 2-D finite piezoelectric media is studied using DDM. The plate of finite specimen with three collinear cracks, one bigger crack of length $2 c_{01}$ situated at center and two smaller cracks each of length $2 c_{01}$ (equal to $c_{02}$ ) symmetrically placed on both sides of the bigger crack, is considered. The size of the specimen and loadings are taken the same as considered in Section 5.2 for the numerical results. The study is performed for different aspect ratios under both the crack face conditions, and thus, the obtained normalized generalized IFs are compared to investigate the crack face boundary conditions. Figure 14 depicts the variations in the $K_{I}^{*}$ and $K_{IV}^{*}$ at the tips of bigger and smaller cracks versus $d / c_{01}$ . Obviously, the mutual effect of two cracks on each other is observed in the results as the $K_{I}^{*}$ and $K_{IV}^{*}$ are more in case of inner tips than the outer tips for all three cracks. It is also noticed that the finite domain approaches to infinite one as aspect ratio $H / W$ increases, and even for higher values of $H / W$ and $d / c_{01}$ , the results of the $K_{I}^{*}$ and $K_{IV}^{*}$ matched at both inner and outer tips. On comparing the graphs shown in Fig. 14, it is again found here that the $K_{IV}^{*}$ values obtained under semi-permeable conditions are almost half than the values of $K_{IV}^{*}$ corresponding to impermeable boundary conditions. The crack face boundary conditions has almost no influence on mechanical intensity factor $K_{I}^{*}$ and hence on mechanical energy release rate.

7. Conclusions

We present a detailed numerical study of finite-specimen-size effects on the cracked 2-D piezoelectric materials using the developed DDM. Different crack face boundary conditions such as semi-impermeable and impermeable are taken into account. A number of numerical results with multiple cracks have been considered and their computed results of the normalized generalized IFs with mixed-mode problems are analyzed in detail. This approach can be used to study the fracture parameters in a finite piezoelectric composite or multi-layered piezoelectric/piezomagnetic composites under various electrical crack-face boundary conditions. Some conclusions derived from the present work are as follows:

The results obtained from the DDM show the finite-specimen-size effects. It is observed that the normalized generalized IFs have significantly higher values for low aspect ratio ( $H / W$ ) than infinite domain results for all the cracks configurations under both impermeable and semi-permeable boundary conditions. Even a finite cracked specimen with dimensions $c / W$ or $c_{01} / W = 0.1$ and an aspect ratio $H / W = 1$ may be considered as an infinite domain cracked problem.

The mutual influence of multiple cracks is also found on the IFs at the tips of the cracks but their behavior is independent of the crack-face boundary conditions.

The crack face boundary conditions effect only the electric displacement intensity factor whereas no effect on mode-I and mode-II intensity factors. In case of semi-permeable crack face boundary conditions, it is found that the normalized IF $K_{IV}^{*}$ has almost half the value of impermeable one, and the same is true for all the crack configurations discussed in the manuscript.

A good agreement of the normalized generalized IFs calculated by the developed DDM with the reference X-FEM solutions is found, which essentially shows the efficiency and accuracy of the method in simulating 2-D finite cracked piezoelectric specimen under different crack face boundary conditions, and with significant computational advantage.

Conflict of interest

The authors have no conflict of interest to report.

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