Fracture analysis of two unequal collinear cracks weakening a piezo-electromagnetic media via complex variable approach

Abstract

In this paper, the problem of two unequal collinear semi-permeable cracks in a piezo-electromagnetic media is proposed and addressed mathematically using a complex variable technique. The problem is mathematically modeled as a non-homogeneous Riemann–Hilbert problem in terms of unknown complex potential functions. The solutions to the fracture parameters are obtained in explicit forms by solving the Hilbert problem. An illustrative numerical case study is presented for poled BaTiO₃-CoFe₂O₄ ceramic cracked plate to show the effect of volume fractions, prescribed loads and inter-crack distance on fracture parameters.

Keywords

Complex variable intensity factor piezo-electromagnetic ceramic semi-permeable cracks

1. Introduction

Piezo-electro-elastic/Magneto-electro-elastic (MEE) composite materials are widely used in e.g. magnetic field probes, acoustic, medical ultrasonic imaging, hydrophones, electronic packaging, electromagnetic sensors, actuators and transducers due to their multifield-coupled effects. MEE ceramics are brittle in nature and have low fracture toughness. The presence of defects such as cracks and voids lead to the premature failure of these materials under mechanical/electrical/magnetic loadings. Thus, fracture study becomes essential for such materials to predict structural integrity and to advance the design of MEE devices.

This article reviews extensive work that has been done to better understand the mechanics of MEE materials in the presence of defects such as cracks. As compared to piezo-electric or anisotropic cases, relatively limited work has been done so far in MEE fracture analysis. A large number of publications for a single crack in MEE materials are available in the literature [4,6,7,13–16,20,21,23]. However, few works related to multiple cracks in MEE media are available in the literature. It is also worth noting that problems of collinear cracks have been a typical and active topic in fracture mechanics. With the application of MEE ceramics, the collinear-crack problems in them have drawn the attention of many researchers [1,8,24]. The static and dynamic problems of two collinear interfacial cracks in MEE composites [25–28] have been solved by Zhou et al., who used the Schmidt method. Exact solutions for anti-plane collinear cracks in a MEE strip or layer have been derived by Wang et al. [19], Wang and Mai [18], and Singh et al. [11] under different conditions. Most recently Jangid and Bharagva [5] have derived an analytical solution for two collinear semi-permeable cracks in MEE media using Stroh’s formalism and complex variable technique.

The main objective of this article is to show the effect of volume fraction, inter-crack distance and prescribed loadings on the collinear semi-permeable cracks. For this, the problem of two unequal collinear semi-permeable cracks weakening a MEE media is studied. Only in-plane electromagnetic and mechanical loading conditions are considered. The problem is formulated employing Stroh’s formalism and solved using a complex variable technique. Closed form analytical expressions are derived for various fracture parameters.

2. Basic equations for piezo-electromagnetic media

The fundamental equations and the boundary conditions for linear piezo-electromagnetic media are defined below.

Constitutive equations $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\sigma}_{ij}=C_{ijks}{\varepsilon}_{ks}-e_{sij}E_{s}-h_{sij}H_{s},\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{i}=e_{kis}{\varepsilon}_{ks}+{\kappa}_{is}E_{s}+{\beta}_{is}H_{s},\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{i}=h_{iks}{\varepsilon}_{ks}+{\beta}_{is}E_{s}+{\gamma}_{is}H_{s}.\end{eqnarray}$ (3)

Kinematic equations $\begin{eqnarray}\displaystyle {\varepsilon}_{ij}={\displaystyle \frac{1}{2}}(u_{i,j}+u_{j,i}),\quad E_{i}={\phi}_{,i},\quad H_{i}={\varphi}_{,i} & & \displaystyle\end{eqnarray}$ (4)

Equilibrium equations

Equilibrium equations for stresses, electric displacements and magnetic inductions in the absence of body forces, free electric charges and free magnetic currents, may, respectively, be written as $\begin{eqnarray}\displaystyle {\sigma}_{ij,j}=0,D_{i,i}=0\quad \text{and}\quad B_{i,i}=0, & & \displaystyle\end{eqnarray}$ (5) where, 𝜎_ij, 𝜀_ij, D _i, E _i, B _i and H _i denote the components of the stress, strain, electric displacement, electric field, magnetic induction and magnetic field, respectively; C _ijks, e _iks, h _iks and 𝛽_is denote the elastic, piezo-electric, piezo-magnetic and electromagnetic constants; 𝜅_is and 𝛾_is denote the dielectric permittivities and magnetic permeabilities, respectively. A comma denotes partial differentiation with respect to the argument that follows it; 𝜙 is the electric potential; and 𝜑 is the magnetic potential; where i, j, k and s = 1,2,3.

3. Crack face boundary conditions

In the literature, mainly three crack face boundary conditions for MEE ceramics are available. These are represented mathematically as:

Impermeable boundary conditions (proposed by Deeg [2])

The crack faces are assumed to be traction free, electrically and magnetically impermeable $\begin{eqnarray}\displaystyle {\sigma}_{ij}n_{j}=0;∼∼D_{2}^{+}=D_{2}^{-}=0∼∼\text{and}∼∼B_{2}^{+}=B_{2}^{-}=0. & & \displaystyle\end{eqnarray}$ (6)

Permeable boundary conditions (proposed by Parton [10])

In this case, the crack is traction free and does not obstruct any electric field from conduction $\begin{eqnarray}\displaystyle {\sigma}_{ij}n_{j}=0;\quad {\phi}^{+}={\phi}^{-};\quad {\varphi}^{+}={\varphi}^{-};\quad D_{2}^{+}=D_{2}^{-}\neq 0\quad \text{and}\quad B_{2}^{+}=B_{2}^{-}\neq 0. & & \displaystyle\end{eqnarray}$ (7)

Semi-permeable boundary conditions

This condition gives a more realistic boundary condition for open cracks. Its modification is proposed by Hao and Shen [3] for piezo-electric solids. These assumptions establish that the medium between the crack surfaces partially conducts the electric and magnetic fields and can be expressed as $\begin{eqnarray}\displaystyle {\sigma}_{ij}n_{j}=0;\quad D_{2}^{+}=D_{2}^{-}=D_{2}^{c}=-{\kappa}_{c}{\displaystyle \frac{{\Delta}{\phi}(x_{1})}{{\Delta}u(x_{1})}}∼∼\text{and}\quad B_{2}^{+}=B_{2}^{-}=B_{2}^{c}=-{\gamma}_{c}{\displaystyle \frac{{\Delta}{\varphi}(x_{1})}{{\Delta}u(x_{1})}}, & & \displaystyle\end{eqnarray}$ (8) where superscripts + and − represent, respective, values on the upper and lower crack surfaces, considering crack along x ₁-axis; 𝜅_c = 𝜅_r𝜅_o(𝜅_o = 8.85 ×10⁻¹² F∕m) is electric permittivity and 𝛾_c = 𝛾_r𝛾_o(𝛾_o =1.26 ×10⁻⁶ Ns ²∕C ²) is magnetic permeability of the medium between the crack faces, respectively; 𝛥𝜙, 𝛥𝜑 and 𝛥u are the jumps of electric potential, magnetic potential and crack opening displacement across the crack, respectively.

4. Fundamental formulation and solution methodology

For a two-dimensional problem all the field variables depend on x ₁ and x ₂.Therefore, introduce a generalized displacement vector u as $\begin{eqnarray}\displaystyle \mathbf{u}=[u_{1},u_{2},u_{3},{\phi},{\varphi}]^{T}=\mathbf{a}f(x_{1}+px_{2}), & & \displaystyle\end{eqnarray}$ (9) where superscript T denotes the transpose of the matrix, f (x ₁ + px ₂) is an analytic function, p is a complex number and a is a constant five element column vector. Equations (1) to (5) satisfy Eq. (9) for an arbitrary function f (x ₁ + px ₂) if $\begin{eqnarray}\displaystyle [\mathbf{W}+p(\mathbf{R}+\mathbf{R}^{T})+p^{2}\mathbf{Q}]\mathbf{a}=0, & & \displaystyle\end{eqnarray}$ (10) which has non-trivial solution only if $\begin{eqnarray}\displaystyle |\mathbf{W}+p\left(\mathbf{R}+\mathbf{R}^{T}\right)+p^{2}\mathbf{Q}|=0. & & \displaystyle\end{eqnarray}$ (11) The matrices W, R and Q are given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{W}=\left[\begin{array}{@{}ccc@{}}C_{1jk1} & e_{1j1} & h_{1j1}\\[5.0pt] e_{1k1}^{T} & -{\kappa}_{11} & -{\beta}_{11}\\[5.0pt] h_{1k1}^{T} & -{\beta}_{11} & -{\gamma}_{11}\end{array}\right],\quad \mathbf{R}=\left[\begin{array}{@{}ccc@{}}C_{1jk1} & e_{2j1} & h_{2j1}\\[5.0pt] e_{1k2}^{T} & -{\kappa}_{12} & -{\beta}_{12}\\[5.0pt] h_{1k2}^{T} & -{\beta}_{12} & -{\gamma}_{12}\end{array}\right],\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \mathbf{Q}=\left[\begin{array}{@{}ccc@{}}C_{2jk2} & e_{2j2} & h_{2j2}\\[5.0pt] e_{2k2}^{T} & -{\kappa}_{22} & -{\beta}_{22}\\[5.0pt] h_{2k2}^{T} & -{\beta}_{22} & -{\gamma}_{22}\end{array}\right],\quad i,k=1,2,3.\end{eqnarray}$ (12) Let the ten roots of Eq. (11) be denoted by p _𝛼 and p _𝛼 . The conditions of positive definiteness of the strain energy and electrical energy densities give that p _𝛼 is not real. To determine p _𝛼 the following standard Eigen-equations are solved $\begin{eqnarray}\displaystyle \mathbf{N}\boldsymbol{{\zeta}}=p\boldsymbol{{\zeta}}, & & \displaystyle\end{eqnarray}$ (13) where $\begin{eqnarray}\displaystyle \mathbf{N}=\left(\begin{array}{@{}cc@{}}\mathbf{N}_{1} & \mathbf{N}_{2}\\ \mathbf{N}_{3} & \mathbf{N}_{1}^{T}\end{array}\right),\quad {\zeta}=\left(\begin{array}{@{}c@{}}\mathbf{a}\\ \boldsymbol{ b}\end{array}\right),\quad \mathbf{N}_{1}=-\mathbf{Q}^{-1}\mathbf{R}^{T},\quad \mathbf{N}_{2}=\mathbf{Q}^{-1}=\mathbf{N}_{2}^{T},\quad \mathbf{N}_{3}=\mathbf{R}\mathbf{Q}^{-1}\mathbf{R}^{T}-\mathbf{W}=\mathbf{N}_{3}^{T}. & & \displaystyle \nonumber\end{eqnarray}$ According to Stroh’s formulation [12], the general solution satisfying Eqs (1) to (5) may be written as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{u}_{,1}=\mathbf{AF}(z)+\overline{\mathbf{A}}\overline{\mathbf{F}(z)},\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{{\Phi}}_{,1}=\mathbf{BF}(z)+\overline{\mathbf{B}}\overline{\mathbf{F}(z)},\end{eqnarray}$ (15) where A = (a ₁, a ₂, a ₃, a ₄, a ₅), B = (b ₁, b ₂, b ₃, b ₄, b ₅), F(z) = d f(z)∕dz f(z _𝛼) = [f ₁(z ₁), f ₂(z ₂), f ₃(z ₃), f ₄(z ₄), f ₅(z ₅)]^T and z _𝛼 = x ₁ + p _𝛼 x ₂.

The column vectors of matrix B = (b ₁, b ₂, b ₃, b ₄, b ₅) are related to the column vectors of matrix A = (a ₁, a ₂, a ₃, a ₄, a ₅) in the following form $\begin{eqnarray}\displaystyle \mathbf{b}_{k}=(\mathbf{R}^{T}+p_{k}\mathbf{Q})\mathbf{a}_{k},\quad k=1,2,3,4,5, & & \displaystyle \nonumber\end{eqnarray}$ and 𝛷 is the generalized stress function such that $\begin{eqnarray}\displaystyle \boldsymbol{{\sigma}}_{2}=[{\sigma}_{2j},D_{2},B_{2}]^{T}=\boldsymbol{{\Phi}}_{,1},\quad \boldsymbol{{\sigma}}_{1}=[{\sigma}_{1j},D_{1},B_{1}]^{T}=-\boldsymbol{{\Phi}}_{,2}. & & \displaystyle\end{eqnarray}$ (16)

5. Statement of the problem

An infinite transversely isotropic piezo-electromagnetic 2D domain is considered for the analysis in the ox ₁ x ₂-plane. Two unequal collinear cracks L ₁ and L ₂ are taken along the x ₁-axis occupying the intervals [d, c] and [b, a] respectively. The traction free crack face and semi-permeable boundary condition are taken for the analysis. The remote boundary of the plate is prescribed in-plane mechanical load ${\sigma}_{22}^{\infty }$ , electric displacement $D_{2}^{\infty }$ and magnetic induction $B_{2}^{\infty }$ . The entire configuration is schematically presented in Fig. 1. The physical boundary conditions stated above may be written as

${\sigma}_{2j}^{+}={\sigma}_{2j}^{-}=0,D_{2}=D^{c},B_{2}=B^{c}$ on L = L ₁ ∪ L ₂

${\sigma}_{22}={\sigma}_{22}^{\infty },D_{2}=D_{2}^{\infty },B_{2}=B_{2}^{\infty }$ for |x ₂|→∞

$u_{j}^{+}=u_{j}^{-}$ , ${\sigma}_{2j}^{+}={\sigma}_{2j}^{-}$ , $D_{2}^{+}=D_{2}^{-}$ , $B_{2}^{+}=B_{2}^{-}$ , 𝜙⁺ = 𝜙⁻, 𝜑⁺ = 𝜑⁻ for |x ₁| < d, c < |x ₁| < b, |x ₁| > a

$\boldsymbol{{\Phi}}_{,1}^{+}=\boldsymbol{{\Phi}}_{,1}^{-}=-\mathbf{V }$ , V $=[\begin{array}{@{}ccccc@{}}0 & {\sigma}_{22}^{\infty } & 0 & D_{2}^{\infty } & B_{2}^{\infty }\end{array}]^{T}$ for d < |x ₁| < c, b < |x ₁| < a.

where, D ^c and B ^c are the electric and magnetic fluxes through the crack regions (d, c) and (b, a), which can be determined with the help of Eq. (8).

Fig. 1.

Schematic representation of the problem.

6. Solution of the problem

The continuity of 𝛷 _,1(x ₁) on the whole real axis implies that $\begin{eqnarray}\displaystyle [\mathbf{BF}(x_{1})-\overline{\mathbf{B}}\overline{\mathbf{F}}(x_{1})]^{+}-[\mathbf{BF}(x_{1})-\overline{\mathbf{B}}\overline{\mathbf{F}}(x_{1})]^{-}=\mathbf{0}. & & \displaystyle\end{eqnarray}$ (17) According to Muskhelishvili [9] its solution may be written as $\begin{eqnarray}\displaystyle \mathbf{BF}(\mathbf{z})=\overline{\mathbf{B}}\overline{\mathbf{F}}(\mathbf{z})=\mathbf{h}(\mathbf{z})(\mathbf{say}). & & \displaystyle\end{eqnarray}$ (18)The boundary condition (iv) together with Eqs (17) and (18) yield the following vector Hilbert problem $\begin{eqnarray}\displaystyle \mathbf{h}^{+}(x_{1})+\mathbf{h}^{-}(x_{1})=\mathbf{V }^{0}-\mathbf{V },\quad \mathbf{V }^{0}=[0,0,0,D^{c},B^{c}]^{T}\quad \text{on}∼L. & & \displaystyle\end{eqnarray}$ (19) Introducing a new complex function vector 𝛺(z) = [𝛺₁(z), 𝛺₂(z), 𝛺₃(z), 𝛺₄(z), 𝛺₅(z)]^T as $\begin{eqnarray}\displaystyle \boldsymbol{{\Omega}}(z)=\mathbf{H}^{R}\mathbf{BF}(z), & & \displaystyle \nonumber\end{eqnarray}$ which, when using Eq. (18), gives the relation $\begin{eqnarray}\displaystyle \mathbf{h}(z)=\boldsymbol{{\Lambda}}\boldsymbol{{\Omega}}(z), & & \displaystyle\end{eqnarray}$ (20) where 𝛬 = [H ^R]⁻¹, H ^R = 2Re Y, Y = i AB ⁻¹.

Consequently Eq. (19) may be written in component form for 𝛺₂(z), 𝛺₄(z) and 𝛺₅(z), which yield the following scalar Hilbert problems $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Lambda}_{22}[{\Omega}_{2}^{+}(x_{1})+{\Omega}_{2}^{-}(x_{1})]+{\Lambda}_{24}[{\Omega}_{4}^{+}(x_{1})+{\Omega}_{4}^{-}(x_{1})]+{\Lambda}_{25}[{\Omega}_{5}^{+}(x_{1})+{\Omega}_{5}^{-}(x_{1})]=-{\sigma}_{22}^{\infty },\end{eqnarray}$ (21) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Lambda}_{42}[{\Omega}_{2}^{+}(x_{1})+{\Omega}_{2}^{-}(x_{1})]+{\Lambda}_{44}[{\Omega}_{4}^{+}(x_{1})+{\Omega}_{4}^{-}(x_{1})]+{\Lambda}_{45}[{\Omega}_{5}^{+}(x_{1})+{\Omega}_{5}^{-}(x_{1})]=D^{c}-D_{2}^{\infty },\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Lambda}_{52}[{\Omega}_{2}^{+}(x_{1})+{\Omega}_{2}^{-}(x_{1})]+{\Lambda}_{54}[{\Omega}_{4}^{+}(x_{1})+{\Omega}_{4}^{-}(x_{1})]+{\Lambda}_{55}[{\Omega}_{5}^{+}(x_{1})+{\Omega}_{5}^{-}(x_{1})]=B^{c}-B_{2}^{\infty }.\end{eqnarray}$ (23) According to Muskhelishvili [9] the solutions to the above Hilbert problems are written as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Omega}_{2}(z)={\displaystyle \frac{{\Delta}_{1}}{2{\Delta}}}\left\{{\displaystyle \frac{P_{1}(z)}{(a_{11}a_{22}-a_{12}a_{21})X_{1}(z)}}-1\right\},\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Omega}_{4}(z)={\displaystyle \frac{{\Delta}_{2}}{2{\Delta}}}\left\{1-{\displaystyle \frac{P_{1}(z)}{(a_{11}a_{22}-a_{12}a_{21})X_{1}(z)}}\right\},\end{eqnarray}$ (25) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Omega}_{5}(z)={\displaystyle \frac{{\Delta}_{3}}{2{\Delta}}}\left\{1-{\displaystyle \frac{P_{1}(z)}{(a_{11}a_{22}-a_{12}a_{21})X_{1}(z)}}\right\}.\end{eqnarray}$ (26)

X ₁(z), P ₁(z) etc. are listed in Appendix A.

7. Applications

In this section, closed form analytical expressions are derived for crack opening displacement (COD), crack opening potential drop (COPD), crack opening induction drop (COID), stress intensity factor (SIF), electric displacement intensity factor (EDIF), and magnetic induction intensity factor (MIIF).

7.1. Crack opening displacement (COD)

The jump displacement vector 𝛥u _{, 1} may be given as $\begin{eqnarray}\displaystyle i{\Delta}\mathbf{u}_{,1}=\boldsymbol{{\Omega}}^{+}(x_{1})-\boldsymbol{{\Omega}}^{-}(x_{1}). & & \displaystyle\end{eqnarray}$ (27) Taking the second component of the above equation, we get $\begin{eqnarray}\displaystyle {\Delta}u_{2,1}(x_{1})=-i[{\Omega}_{2}^{+}(x_{1})-{\Omega}_{2}^{-}(x_{1})]. & & \displaystyle\end{eqnarray}$ (28) Substituting value of 𝛺₂(z) from Eq. (24) and integrating one obtains $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}u_{2}(x_{1})=\frac{{\Delta}_{1}}{(a_{11}a_{22}-a_{12}a_{21}){\Delta}}\{C_{0}S_{3}+C_{1}S_{4}+C_{2}S_{5}\},\quad \text{on}∼d\lt |x_{1}|\lt c\end{eqnarray}$ (29) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}u_{2}(x_{1})=\frac{-{\Delta}_{1}}{(a_{11}a_{22}-a_{12}a_{21}){\Delta}}\{C_{0}S_{6}+C_{1}S_{7}+C_{2}S_{8}\},\quad \text{on}∼b\lt |x_{1}|\lt a,\end{eqnarray}$ (30) where 𝛥 indicates the difference between the values on the upper and lower crack surfaces. S ₃, S ₄ etc. are listed in Appendix B.

7.2. Crack opening potential drop (COPD)

Comparing the fourth component from Eq. (27) and using the value of 𝛺₄(x ₁) from Eq. (25) and integrating one obtains the COP drop, 𝛥𝜙(x ₁), between the two faces of the crack as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}u_{4}(x_{1})=\frac{-{\Delta}_{2}}{(a_{11}a_{22}-a_{12}a_{21}){\Delta}}\{C_{0}S_{3}+C_{1}S_{4}+C_{2}S_{5}\},\quad \text{on}∼d<|x_{1}|<c\end{eqnarray}$ (31) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}u_{4}(x_{1})=\frac{{\Delta}_{2}}{(a_{11}a_{22}-a_{12}a_{21}){\Delta}}\{C_{0}S_{6}+C_{1}S_{7}+C_{2}S_{8}\},\quad \text{on}∼b<|x_{1}|<a.\end{eqnarray}$ (32)

7.3. Crack opening induction drop (COID)

Comparing the fifth component from Eq. (27) and using the value of 𝛺₅(x ₁) from Eq. (26) and integrating one obtains the COI drop, 𝛥𝜑(x ₁), between the two faces of the crack as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}u_{5}(x_{1})={\displaystyle \frac{-{\Delta}_{3}}{(a_{11}a_{22}-a_{12}a_{21}){\Delta}}}\{C_{0}S_{3}+C_{1}S_{4}+C_{2}S_{5}\},\quad \text{on}∼d\lt |x_{1}|\lt c\end{eqnarray}$ (33) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}u_{5}(x_{1})=\frac{{\Delta}_{3}}{(a_{11}a_{22}-a_{12}a_{21}){\Delta}}\{C_{0}S_{6}+C_{1}S_{7}+C_{2}S_{8}\},\quad \text{on}∼b\lt |x_{1}|\lt a.\end{eqnarray}$ (34) The values of electric and magnetic fluxes, D ^c and B ^c respectively, are obtained by substituting the required values from Eqs (29), (31) and (33) into Eq. (8) simplifying and solving the system of non-linear equations $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle m_{1}D^{c2}+D^{c}(m_{4}{\sigma}_{22}^{\infty }-m_{1}D_{2}^{\infty }-m_{5}B_{2}^{\infty }+m_{2}{\kappa}_{c})+B^{c}D^{c}m_{5}+B^{c}m_{3}{\kappa}_{c}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =-{\kappa}_{c}(m_{1}{\sigma}_{22}^{\infty }-m_{2}D_{2}^{\infty }-m_{3}B_{2}^{\infty }),\end{eqnarray}$ (35) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle m_{5}B^{c2}+B^{c}(m_{4}{\sigma}_{22}^{\infty }-m_{1}D_{2}^{\infty }-m_{5}B_{2}^{\infty }+m_{6}{\gamma}_{c})+B^{c}D^{c}m_{1}+D^{c}m_{3}{\gamma}_{c}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =-{\gamma}_{c}(m_{5}{\sigma}_{22}^{\infty }-m_{3}D_{2}^{\infty }-m_{6}B_{2}^{\infty }),\end{eqnarray}$ (36) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle m_{1}={\Lambda}_{42}{\Lambda}_{55}-{\Lambda}_{45}{\Lambda}_{52},\quad m_{2}={\Lambda}_{22}{\Lambda}_{55}-{\Lambda}_{25}{\Lambda}_{52},\quad m_{3}={\Lambda}_{25}{\Lambda}_{42}-{\Lambda}_{22}{\Lambda}_{45},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle m_{4}={\Lambda}_{44}{\Lambda}_{55}-{\Lambda}_{45}{\Lambda}_{54},\quad m_{5}={\Lambda}_{25}{\Lambda}_{44}-{\Lambda}_{24}{\Lambda}_{45},\quad m_{6}={\Lambda}_{22}{\Lambda}_{44}-{\Lambda}_{24}{\Lambda}_{42}.\nonumber\end{eqnarray}$

7.4. Stress intensity factor (SIF)

Open mode stress intensity factor K _I at the crack tips x ₁ = d, b, c and a is obtained using the following formulae $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(d)=\lim _{x_{1}\rightarrow d^{-}}\sqrt{2{\pi}(d-x_{1})}{\sigma}_{22}(x_{1}),\end{eqnarray}$ (37) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(c)=\lim _{x_{1}\rightarrow c^{+}}\sqrt{2{\pi}(x_{1}-c)}{\sigma}_{22}(x_{1}),\end{eqnarray}$ (38) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(b)=\lim _{x_{1}\rightarrow b^{-}}\sqrt{2{\pi}(b-x_{1})}{\sigma}_{22}(x_{1}),\end{eqnarray}$ (39) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(a)=\lim _{x_{1}\rightarrow a^{+}}\sqrt{2{\pi}(x_{1}-a)}{\sigma}_{22}(x_{1}).\end{eqnarray}$ (40) Substituting 𝜎₂₂(x ₁) obtained from Eqs (15), (16), (18) and (23) into the above equations and simplifying them we obtain $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(d)={\displaystyle \frac{-\sqrt{2{\pi}}({\Lambda}_{25}{\Delta}_{3}+{\Lambda}_{24}{\Delta}_{2}-{\Lambda}_{22}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}d^{2}+C_{1}d+C_{2}}{\sqrt{(a-d)(b-d)(c-d)}}}\right\},\end{eqnarray}$ (41) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(c)={\displaystyle \frac{\sqrt{2{\pi}}({\Lambda}_{25}{\Delta}_{3}+{\Lambda}_{24}{\Delta}_{2}-{\Lambda}_{22}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}c^{2}+C_{1}c+C_{2}}{\sqrt{(a-c)(b-c)(c-d)}}}\right\},\end{eqnarray}$ (42) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(b)={\displaystyle \frac{\sqrt{2{\pi}}({\Lambda}_{25}{\Delta}_{3}+{\Lambda}_{24}{\Delta}_{2}-{\Lambda}_{22}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}b^{2}+C_{1}b+C_{2}}{\sqrt{(a-b)(b-c)(b-d)}}}\right\},\end{eqnarray}$ (43) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{I}(a)={\displaystyle \frac{-\sqrt{2{\pi}}({\Lambda}_{25}{\Delta}_{3}+{\Lambda}_{24}{\Delta}_{2}-{\Lambda}_{22}{\Delta}_{1})}{{\Delta}\left(a_{11}a_{22}-a_{12}a_{21}\right)}}\left\{{\displaystyle \frac{C_{0}a^{2}+C_{1}a+C_{2}}{\sqrt{(a-b)(a-c)(a-d)}}}\right\}.\end{eqnarray}$ (44)

7.5. Electric displacement intensity factor (EDIF)

Similarly, open mode EDIF, K _IV, at the crack tips x ₁ = d, c, b and a may be obtained as $\begin{eqnarray}\displaystyle \text{} & \text{}K_{IV}(d)={\displaystyle \frac{-\sqrt{2{\pi}}\left({\Lambda}_{45}{\Delta}_{3}+{\Lambda}_{44}{\Delta}_{2}-{\Lambda}_{42}{\Delta}_{1}\right)}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}d^{2}+C_{1}d+C_{2}}{\sqrt{(a-d)(b-d)(c-d)}}}\right\}, & \displaystyle\end{eqnarray}$ (45) $\begin{eqnarray}\displaystyle \text{} & \text{}K_{IV}(c)={\displaystyle \frac{\sqrt{2{\pi}}({\Lambda}_{45}{\Delta}_{3}+{\Lambda}_{44}{\Delta}_{2}-{\Lambda}_{42}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}c^{2}+C_{1}c+C_{2}}{\sqrt{(a-c)(b-c)(c-d)}}}\right\}, & \displaystyle\end{eqnarray}$ (46) $\begin{eqnarray}\displaystyle \text{} & \text{}K_{IV}(b)={\displaystyle \frac{\sqrt{2{\pi}}({\Lambda}_{45}{\Delta}_{3}+{\Lambda}_{44}{\Delta}_{2}-{\Lambda}_{42}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}b^{2}+C_{1}b+C_{2}}{\sqrt{(a-b)(b-c)(b-d)}}}\right\}, & \displaystyle\end{eqnarray}$ (47) $\begin{eqnarray}\displaystyle \text{} & \text{}K_{IV}(a)={\displaystyle \frac{-\sqrt{2{\pi}}({\Lambda}_{45}{\Delta}_{3}+{\Lambda}_{44}{\Delta}_{2}-{\Lambda}_{42}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}a^{2}+C_{1}a+C_{2}}{\sqrt{(a-b)(a-c)(a-d)}}}\right\}. & \displaystyle\end{eqnarray}$ (48)

7.6. Magnetic induction intensity factor (MIIF)

Analogously, MIIF, K _V, at the crack tips x ₁ = d, c, b and a may be obtained as $\begin{eqnarray}\displaystyle \text{} & \text{}K_{V}(d)={\displaystyle \frac{-\sqrt{2{\pi}}({\Lambda}_{55}{\Delta}_{3}+{\Lambda}_{54}{\Delta}_{2}-{\Lambda}_{52}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}d^{2}+C_{1}d+C_{2}}{\sqrt{(a-d)(b-d)(c-d)}}}\right\}, & \displaystyle\end{eqnarray}$ (49) $\begin{eqnarray}\displaystyle \text{} & \text{}K_{V}(c)={\displaystyle \frac{\sqrt{2{\pi}}\left({\Lambda}_{55}{\Delta}_{3}+{\Lambda}_{54}{\Delta}_{2}-{\Lambda}_{52}{\Delta}_{1}\right)}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}c^{2}+C_{1}c+C_{2}}{\sqrt{(a-c)(b-c)(c-d)}}}\right\}, & \displaystyle\end{eqnarray}$ (50) $\begin{eqnarray}\displaystyle \text{} & \text{}K_{V}(b)={\displaystyle \frac{\sqrt{2{\pi}}({\Lambda}_{55}{\Delta}_{3}+{\Lambda}_{54}{\Delta}_{2}-{\Lambda}_{52}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}b^{2}+C_{1}b+C_{2}}{\sqrt{(a-b)(b-c)(b-d)}}}\right\}, & \displaystyle\end{eqnarray}$ (51) $\begin{eqnarray}\displaystyle \text{} & \text{}K_{V}(a)={\displaystyle \frac{-\sqrt{2{\pi}}({\Lambda}_{55}{\Delta}_{3}+{\Lambda}_{54}{\Delta}_{2}-{\Lambda}_{52}{\Delta}_{1})}{{\Delta}(a_{11}a_{22}-a_{12}a_{21})}}\left\{{\displaystyle \frac{C_{0}a^{2}+C_{1}a+C_{2}}{\sqrt{(a-b)(a-c)(a-d)}}}\right\}. & \displaystyle\end{eqnarray}$ (52)

8. Numerical case study

In this section, the effect of inter-crack distance, volume fraction, and electric and magnetic loads are shown on various fracture parameters (discussed in the previous section).

Piezo-electromagnetic composite BaTiO₃-CoFe₂O₄ is selected for the numerical case study considering BaTiO₃ as inclusion and CoFe₂O₄ as matrix. The volume fraction of the inclusion is denoted by V _f. The proportion of the two phases can be varied by adjusting the volume fraction of inclusion and the matrix. The elastic constants, dielectric permittivities and magnetic permeabilities, as well as piezo-electric and piezo-magnetic constants, are obtained by fraction rule (taken from Wang and Mai [17]) $\begin{eqnarray}\displaystyle {\kappa}_{is}^{c}=V_{f}\cdot {\kappa}_{is}^{i}+(1-V_{f}).{\kappa}_{is}^{m}, & & \displaystyle\end{eqnarray}$ (53) where the superscripts c, i and m represent composite, inclusion and matrix, respectively. 𝜅_is denote the dielectric permittivities.

This rule cannot be applied to the determination of electromagnetic constants, since no electromagnetic coupling is present in any of the single phases. We assume the crack faces are semi-permeable (𝜅_r = 𝛾_r =1). The length of the bigger crack, L ₁, smaller crack, L ₂, prescribed mechanical load, electric displacement and magnetic induction are 2a ₀₁(= 5mm), 2a ₀₂(= 4mm), ${\sigma}_{22}^{\infty }=5$ MPa, $D_{2}^{\infty }=2(e_{33}/c_{33}){\sigma}_{22}^{\infty }$ and $B_{2}^{\infty }=2(h_{33}/c_{33}){\sigma}_{22}^{\infty }$ , respectively, unless otherwise specified. Material constants for BaTiO₃-CoFe₂O₄ for different volume fractions are listed in Table 1 (taken from Zhong [22]).

Table 1

Material constants for BaTiO₃-CoFe₂O₄for different volume fractions

Material constants	V _f (0.25)	V _f (0.50)	V _f (0.75)
c ₁₁ (10⁹ N/m²)	245	215	186
c ₁₂ (10⁹ N/m²)	145	125	115
c ₁₃ (10⁹ N/m²)	144	112	90
c ₃₃ (10⁹ N/m²)	235	210	181
c ₄₄ (10⁹ N/m²)	46	50	51
e ₃₁ (C/m²)	−1.5	−2.8	−3.8
e ₃₃ (C/m²)	4.2	8.	13.2
e ₁₅ (C/m²)	0.0	0.2	0.3
h ₃₁ (N/Am)	380	220	90
h ₃₃ (N/Am)	475	290	135
h ₁₅ (N/Am)	335	180	75
𝜅₁₁ (10⁻⁹ C²/Nm²)	0.1	0.25	0.5
𝜅₃₃ (10⁻⁹ C²/Nm²)	3.2	6.3	9.4
𝛾₁₁ (10⁻⁶ Ns²/C²)	−3.55	−2.00	−0.90
𝛾₃₃ (10⁻⁶ Ns²/C²)	1.2	0.8	0.45
𝛽₁₁ (10⁻⁹ Ns/VC)	3.1	5.3	6.8
𝛽₃₃ (10⁻⁹ Ns/VC)	2350	2750	1800

8.1. Effect of inter-crack distance

Figure 2 depicts the COD profile over the crack surfaces for different inter-crack distances. It may be seen that the COD profile is increased as the inter-crack distance decreases; this is because of the mutual interactions between the two cracks. It may also be seen that the two cracks may get coalesced if the inter-crack distance decreases. Similarly, Figs 3 and 4 depict the COPD profile and COID profile over the crack surfaces for different inter-crack distances.

Fig. 2.

COD profile over the crack surfaces for different inter-crack distances.

Fig. 3.

COPD profile over the crack surfaces for different inter-crack distances.

Fig. 4.

COID profile over the crack surfaces for different inter-crack distances.

Figure 5 shows the variation of stress intensity factors (SIFs) versus normalized inter-crack distance for different volume fractions. It may be seen that due to the mutual interactions of two cracks, the SIFs at the crack tips are increased as the inter-crack distance decreases. It may also be seen that SIF at the inner crack tips (at x ₁ = c and x ₁ = b) is higher compared to that at the outer crack tips (at x ₁ = d and x ₁ = a), which implies that the cracks will open more at the inner tips as compared to outer tips. Moreover, K _I stabilizes for d ₀∕a ₀₂ ≥ 3. Also, SIF is decreased as the volume fraction increases. Similarly, Figs 6 and 7 show the variations of EDIF and MIIF versus inter-crack distance for different volume fractions.

Fig. 5.

Effect of normalized inter-crack distance on SIF for different volume fractions.

Fig. 6.

Effect of normalized inter-crack distance on EDIF for different volume fractions.

Fig. 7.

Effect of normalized inter-crack distance on MIIF for different volume fractions.

8.2. Effect of crack length

The effect of crack length a ₀₂ on the stress intensity factor (SIF), K _I, for different volume fractions is shown in Fig. 8. It may be seen from the figure that at the interior and exterior tips of the longer crack, K _I increases at both tips as the crack length is increased. an increase in K _I at the interior tip is more steep compared to the exterior tip. A similar variation is observed at the interior and exterior tips of the shorter crack. It should also be noted that for the half length of the crack equal to 2.5 mm (i.e., the length of both cracks is equal), the curves for K _I at the interior tips of both cracks and exterior tips of the cracks become equal. Figures 9 and 10 show the same variations for EDIF and MIIF.

Fig. 8.

Effect of crack length a ₀₂ on SIF for different volume fractions.

Fig. 9.

Effect of crack length a ₀₂ on EDIF for different volume fractions.

Fig. 10.

Effect of crack length a ₀₂ on MIIF for different volume fractions.

9. Conclusions

Considering the aforementioned analytical and numerical studies done on the proposed model, the following conclusions can be drawn.

A complex variable and Stroh’s formalism technique are successfully applied to study the two unequal collinear semi-permeable cracks in a piezo-electromagnetic media.

The closed form analytic expressions are derived for the various fracture parameters such as COD, COPD, COID, SIF, EDIF and MIIF for the proposed model.

To obtain the electric displacement and magnetic induction inside the crack gap media, two non-linear equations are derived.

The effect of volume fraction is observed on the fracture parameters. All parameters are decreased when the volume fraction is increased.

The effect of the inter-crack distance is observed on the fracture parameters. All parameters are increased when the inter-crack distance is decreased.

The effect of crack length is observed on the fracture parameters. All parameters are increased when the crack length is increased.

Footnotes

Acknowledgements

The author would like to show his gratitude to the editor of this esteemed Journal. He is also immensely grateful to the reviewers for their comments on an earlier version of the manuscript.

Appendix A.

$\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle X_{1}(z)=\sqrt{(z-a)(z-b)(z-c)(z-d)},\quad P_{1}(z)=C_{0}z^{2}+C_{1}(z)+C_{2};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad C_{0}=a_{11}a_{22}-a_{12}a_{21},\quad C_{1}=a_{20}a_{12}-a_{10}a_{22},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\Delta}={\Lambda}_{22}({\Lambda}_{44}{\Lambda}_{55}-{\Lambda}_{45}{\Lambda}_{54})-{\Lambda}_{24}({\Lambda}_{42}{\Lambda}_{55}-{\Lambda}_{45}{\Lambda}_{52})+{\Lambda}_{25}({\Lambda}_{42}{\Lambda}_{54}-{\Lambda}_{44}{\Lambda}_{52});\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad C_{2}=a_{21}a_{10}-a_{11}a_{20},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\Delta}_{1}=-{\sigma}_{22}^{\infty }({\Lambda}_{44}{\Lambda}_{55}-{\Lambda}_{45}{\Lambda}_{54})-(D^{c}-D_{2}^{\infty })({\Lambda}_{24}{\Lambda}_{55}-{\Lambda}_{25}{\Lambda}_{54})+(B^{c}-B_{2}^{\infty })({\Lambda}_{25}{\Lambda}_{44}-{\Lambda}_{24}{\Lambda}_{45});\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\Delta}_{2}={\sigma}_{22}^{\infty }({\Lambda}_{42}{\Lambda}_{55}-{\Lambda}_{45}{\Lambda}_{52})+(D^{c}-D_{2}^{\infty })({\Lambda}_{22}{\Lambda}_{55}-{\Lambda}_{25}{\Lambda}_{52})+(B^{c}-B_{2}^{\infty })({\Lambda}_{25}{\Lambda}_{42}-{\Lambda}_{22}{\Lambda}_{45});\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\Delta}_{3}={\sigma}_{22}^{\infty }({\Lambda}_{44}{\Lambda}_{52}-{\Lambda}_{42}{\Lambda}_{54})+(D^{c}-D_{2}^{\infty })({\Lambda}_{24}{\Lambda}_{52}-{\Lambda}_{22}{\Lambda}_{54})+(B^{c}-B_{2}^{\infty })({\Lambda}_{22}{\Lambda}_{44}-{\Lambda}_{24}{\Lambda}_{42});\nonumber\end{eqnarray}$ $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle k^{2}={\displaystyle \frac{(a-b)(c-d)}{(a-c)(b-d)}};\quad g={\displaystyle \frac{2}{\sqrt{(a-c)(b-d)}}},\quad {\alpha}^{2}={\displaystyle \frac{d-c}{a-c}},\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad {\beta}^{2}={\displaystyle \frac{a-b}{a-c}},\quad a_{11}=g[aF(k)+(d-a){\Pi}({\alpha}^{2},k)];\quad a_{12}=gF(k),\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle a_{21}=g[cF(k)+(b-c){\Pi}({\beta}^{2},k)],\quad a_{22}=gF(k);\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle a_{10}=g[a^{2}F(k)+2a(d-a){\Pi}({\alpha}^{2},k)+(d-a)^{2}V_{2}];\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle a_{20}=g[c^{2}F(k)+2c(b-c){\Pi}({\beta}^{2},k)+(b-c)^{2}V_{3}];\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle V_{2}={\displaystyle \frac{1}{2({\alpha}^{2}-1)(k^{2}-{\alpha}^{2})}}\{{\alpha}^{2}E(k)+(k^{2}-{\alpha}^{2})F(k)+(2{\alpha}^{2}k^{2}+2{\alpha}^{2}-{\alpha}^{4}-3k^{2}){\Pi}({\alpha}^{2},k)\};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle V_{3}={\displaystyle \frac{1}{2({\beta}^{2}-1)(k^{2}-{\beta}^{2})}}\{{\beta}^{2}E(k)+(k^{2}-{\beta}^{2})F(k)+(2{\beta}^{2}k^{2}+2{\beta}^{2}-{\beta}^{4}-3k^{2}){\Pi}({\beta}^{2},k)\};\nonumber\end{eqnarray}$ where F (k), E (k) and 𝛱(𝛼², k) are the complete elliptic integrals of the first, second and third kinds, respectively.

Appendix B.

$\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{1}^{2}={\displaystyle \frac{a}{d}}{\alpha}^{2},\quad {\beta}_{1}^{2}={\displaystyle \frac{c}{b}}{\beta}^{2},\quad {\nu}=\sin ^{-1}\sqrt{\frac{(a-c)(y-d)}{(d-c)(a-y)}},\quad {\psi}=\sin ^{-1}\sqrt{\frac{(a-c)(y-b)}{(a-b)(y-c)}};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle S_{1}={\alpha}^{2}E\left({\nu},k\right)+\left(k^{2}-{\alpha}^{2}\right)F\left({\nu},k\right)+\left(2{\alpha}^{2}k^{2}+2{\alpha}^{2}-{\alpha}^{4}-3k^{2}\right){\Pi}\left({\nu},{\alpha}^{2},k\right)-\frac{{\alpha}^{4}\text{sn}u\text{cn}u\text{dn}u}{1-{\alpha}^{2}\text{sn}^{2}u};\nonumber\end{eqnarray}$ where snu, cnu and dnu are the Jacobian elliptic functions. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle S_{2}={\beta}^{2}E({\psi},k)+(k^{2}-{\beta}^{2})F({\psi},k)+(2{\beta}^{2}k^{2}+2{\beta}^{2}-{\beta}^{4}-3k^{2}){\Pi}({\psi},{\beta}^{2},k)-{\displaystyle \frac{{\beta}^{4}\text{sn}u\text{cn}u\text{dn}u}{1-{\beta}^{2}\text{sn}^{2}u}};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle S_{3}=d^{2}g{\displaystyle \frac{{\alpha}_{1}^{4}}{{\alpha}^{4}}}\left\{F({\nu},k)+{\displaystyle \frac{2({\alpha}^{2}-{\alpha}_{1}^{2})}{{\alpha}_{1}^{2}}}{\Pi}({\nu},{\alpha}^{2},k)+{\displaystyle \frac{({\alpha}^{2}-{\alpha}_{1}^{2})^{2}}{2{\alpha}_{1}^{4}({\alpha}^{2}-1)(k^{2}-{\alpha}^{2})}}S_{1}\right\};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle S_{4}=dg{\displaystyle \frac{{\alpha}_{1}^{2}}{{\alpha}^{2}}}\left\{F({\nu},k)+{\displaystyle \frac{{\alpha}^{2}-{\alpha}_{1}^{2}}{{\alpha}_{1}^{2}}}{\Pi}({\nu},{\alpha}^{2},k)\right\},\quad S_{5}=gF({\nu},k);\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle S_{6}=b^{2}g{\displaystyle \frac{{\beta}_{1}^{4}}{{\beta}^{4}}}\left\{F({\psi},k)+{\displaystyle \frac{2({\beta}^{2}-{\beta}_{1}^{2})}{{\beta}_{1}^{2}}}{\Pi}({\psi},{\beta}^{2},k)+{\displaystyle \frac{({\beta}^{2}-{\beta}_{1}^{2})^{2}}{2{\beta}_{1}^{4}({\beta}^{2}-1)(k^{2}-{\beta}^{2})}}S_{2}\right\};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle S_{7}=bg{\displaystyle \frac{{\beta}_{1}^{2}}{{\beta}^{2}}}\left\{F({\psi},k)+{\displaystyle \frac{{\beta}^{2}-{\beta}_{1}^{2}}{{\beta}_{1}^{2}}}{\Pi}({\psi},{\beta}^{2},k)\right\},\quad S_{8}=gF({\psi},k);\nonumber\end{eqnarray}$ where F (, k), E (, k) and 𝛱(, k) are the incomplete elliptic integrals of the first, second and third kinds, respectively.

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