A novel singular integral operator technique for analyzing semipermeable four collinear cracks in a magneto-electro-elastic solid based on the EMPS model

Abstract

This manuscript introduces a novel analytical technique using singular integral operators (SIOs) to study the strip electric-magnetic polarization saturation (EMPS) model with four collinear cracks, including two pairs of symmetrical unequal cracks, embedded in an infinite magneto-electro-elastic (MEE) solid. By applying the EMPS model and the distributed dislocation technique (DDT) while leveraging symmetry, we formulate singular integral equations for the coupled mechanical, electrical, and magnetic cracks based on their dislocation density functions. These equations are then transformed into Cauchy-type singular integral equations using nonlinear transformation. Since the reduced system of singular integral equations is defined over two disjoint intervals, an SIO-based technique is developed for the problem where Cauchy-type SIOs and inverse operators for each dislocation density function are defined, and then using their identities, the analytical solutions for the unknown dislocation density functions are derived after simplification. Iterative methods are employed to determine semipermeable crack-face conditions and the lengths of saturated zones. Numerical analysis shows that the distances between cracks and the crack length ratio significantly affect the lengths of saturated zones, crack tip opening potential (CTOP), crack tip opening induction (CTOI), crack opening displacement (COD), and the local stress intensity factor (LSIF), while inter-crack distance variations have a negligible impact on crack-face conditions. Validation of the results for the specific cases of the stated problem confirms the accuracy and efficacy of the developed SIO-based technique and the analytical solutions.

Keywords

Cauchy singular integral distributed dislocation technique EMPS model magneto-electro-elastic singular integral operator

1. Introduction

Magneto-electro-elastic (MEE) materials, which respond to magnetic and electric fields, possess the unique property of electro-magneto-mechanical coupling effects, making them valuable in various high-tech applications such as sensors, actuators, and transducers. Smart embedded structures and devices extensively utilize these materials and are frequently susceptible to fracturing when subjected to loads. Therefore, it is essential to understand the behavior of cracks in these materials, particularly when dealing with multiple or collinear cracks. Collinear cracks, which align along the same line, present significant engineering and materials science challenges due to their potential to propagate and coalesce, ultimately leading to catastrophic failure. When these cracks occur in advanced smart materials like piezoelectric and MEE materials, they can significantly impact their mechanical/electrical/magnetic properties, durability, and overall performance. Additionally, the study of the collinear cracks becomes more complex when investigated based on non-linear fracture models or the approximating nonlinear fracture models such as the Dugdale model in elastic-plastic materials, the polarization saturation (PS) model in piezoelectric materials, and the EMPS model in MEE materials. Therefore, the study and findings of collinear cracks in MEE materials based on the EMPS model can help researchers and engineers devise strategies to enhance the durability and robustness of these materials, thereby improving the safety and longevity of the smart embedded structures and devices in which they are utilised.

In elastic-plastic materials, Theocaris [1] applied the Dugdale model and complex variable approach for studying the two unequal collinear cracks problem. Collins and Cartwright [2] derived the closed-form solutions for two equal collinear cracks based on the strip yield model. Nishimura [3] implemented the integral equation method to study the strip yield model for two unequal collinear cracks problem in an infinite elastic-plastic sheet. Using the weight functions method, Xu et al. [4] validated the Collins and Cartwright solution [2] for two equal collinear cracks with a strip yield method. Bhargava and Hasan [5] extended the strip yield model for two collinear cracks problem to four symmetrical cracks with coalesced interior zones case in an infinite elastic-plastic solid. Hasan [6] proposed and studied the linear and quadratically varying modified Dugdale model for four collinear cracks with interior zones as coalesced in an infinite elastic-plastic domain using a complex variable approach. Akhtar and Hasan [7] presented the closed-form solutions for a linearly varying modified strip-yield model having three collinear unequal straight cracks. The weight functions method was also successfully implemented by Liu et al. [8] for a damage tolerance study of two unequal collinear cracked infinite elastic-plastic sheet. Using the weight function method, Zhang et al. [9] demonstrated the studies for finding the SIFs and yielding zones in a stiffened panel with multiple collinear cracks. A new analytical solution for SIFs in two unequal collinear cracked elastic-plastic domain was derived by Zhang et al. [10] using complex variable and elliptic integral function methods.

Bhargava and Jangid [11 –13] extended the PS model to two collinear cracks problems in an infinite transversely isotropic piezoelectric domain using the R-H approach. Singh et al. [14 –16] modified the PS model by proposing the polynomial varying saturation zone models and developed the solutions for two equal and unequal collinear cracks problems in an infinite two-dimensional (2D) piezoelectric domain. Liu et al. [17] employed the Fourier transform technique to find the dynamic SIFs for multiple permeable cracks in piezoelectric materials. Sharma et al. [18] developed the numerical algorithms for finding the saturated zones in a generalized strip saturated model with two equal collinear cracks in the finite and infinite 2D piezoelectric domain. Recently, Hasan et al. [19] extended their complex variable solutions for multiple collinear cracks with a strip yield–based model to study the three equal collinear cracks in the piezoelectric domain with a strip electro-mechanical yielding model.

In MEE materials, Gao et al. [20] presented the generalized solution for permeable collinear cracks subjected to arbitrary loading conditions using an extended Stroh formalism approach. Tian and Gabbert [21] studied the crack interaction problem under the multiple arbitrarily oriented and distributed cracks in MEE solids. Zhou et al. [22] implemented the generalized Almansi’s theorem to investigate the two semipermeable collinear cracks in piezoelectric and piezomagnetic solids. Zhou and his co-workers [23] also presented the analytical solutions for four parallel non-symmetric anti-plane cracks in a permeable piezoelectric and piezomagnetic composite domain. Using the integral transform method, the analytical solutions were derived by Singh et al. [24] for two mode-III collinear cracks in the MEE layer. Li [25] developed a new closed-form fundamental solution and demonstrated the numerical studies for two unequal collinear cracks in MEE solids subjected to mode-I electro-magnetic-mechanical loadings. Employing the Fourier transforms and singular integral method, Zhong [26] obtained the explicit form of solutions for two collinear dielectric cracks in MEE material. Verma and Verma [27] studied the semipermeable case of mode-III two collinear cracks in MEE strip using Fourier series and integral equation techniques. Jangid [28] investigated the effects of electric and magnetic poling direction in mode-I two unequal collinear cracks in MEE material. Jangid [29] also extended the EMPS model [30] to two semipermeable collinear cracks problem in MEE material considering only the magnetic saturation zone. Verma [31] also extended the EMPS model to mode-III two semipermeable collinear cracked MEE strip with only a magnetic saturation zone. Exploiting the DDT and the closed-form singular integral solutions, Kumar et al. [32] developed the solution for two equal collinear cracks problem based on the EMPS model in MEE material. They [33] further extended these techniques to study the moving EMPS problem with two equal collinear interface cracks in dissimilar MEE materials. The applications of the DDM and singular integral method were also emphasized in the work of Monfared and his co-researchers [34 –40]. Based on these techniques, they investigated multiple crack problems in various advanced materials, including electro-elastic, piezoelectric, MEE, and functionally graded materials (FGMs). Their studies encompass a wide range of scenarios, such as dynamic stress intensity factor evaluation under impact loading, transient crack response in nonhomogeneous layers, and the interaction of multiple defects in complex material systems.

Most of the works available in the literature are mainly on two collinear cracks problems. However, Hasan and his co-workers [5 –7] have extended the strip-yielding model from two collinear cracks to three and four collinear cracks problems in an infinite elastic-plastic domain. However, as per the authors’ knowledge, no research is available on four collinear cracks and four collinear cracks with the PS and the EMPS models in piezoelectric and MEE materials, respectively. Therefore, in this paper, authors have investigated the study of the EMPS model with four collinear cracks, including two pairs of symmetrical unequal cracks in an infinite transversely isotropic MEE solid. It is a generalized case of four equal collinear cracks problem.

In fracture mechanics, the studies of multiple/collinear cracks are mainly based on analytical and numerical mathematical techniques. However, finding analytical or closed-form solutions for a problem is always more advantageous than finding a numerical solution, as one can find the solution at any point in its domain directly or without any significant computational cost. Therefore, the researchers always try to develop a closed-form solution to their research problem rather than a numerical solution. Most of the works on collinear cracks stated above presented their studies based on the closed-form solutions developed using different mathematical techniques such as complex variable, R-H approach, integral transforms, integral equations, and Fourier series. However, limited research is available on the closed-form solutions of multiple/collinear cracks problems using the singular integral method.

Moreover, in the mathematical modeling of more than two collinear cracks and unsymmetrical collinear cracks cases, these problems are converted into simultaneous singular integral equations defined over the union of disjoint intervals, which cannot be simplified into standard singular integral equations to find their analytical solutions. Therefore, a particular mathematical approach is required to address these equations. Using the Cauchy singular operator and its inverse, Chakrabarti and George [41] proposed and employed the singular integral operator (SIO)-based method to solve a singular integral equation involving two intervals with the kernel having logarithmic and Cauchy-type singularities, which mathematically formulated with respect to (w.r.t) the linear theory of water waves problem. Chakrabarti and Sahoo [42] also explained the SIO-based method and derived the explicit solution for singular integral equations with logarithmic and Cauchy kernels subjected to two typical cases of defined intervals, one in a single finite interval and, in the other, a union of disjoint finite intervals. Further details on the SIO and associated identities can be found in the literature [43,44].

As far as the authors know, no research is available on the study of cracks and collinear cracks using the SIO theory-based method. Hence, in this paper, authors have extended the SIO theory to develop the new closed-form solutions for fracture parameters of the EMPS model with four semipermeable collinear cracks, including two pairs of symmetrical unequal cracks, embedded in an infinite MEE solid. Based on these explicit solutions, the analytical study of the problem is presented. The structure of this paper is organized as follows: section 2 outlines the fundamental equations governing MEE materials. In section 3, we present the mathematical modeling for the stated problem of four collinear cracks. Section 4 introduces the generalized methodology and analytical solutions for the derived singular integral equations and the fracture parameters employing the SIO method. Section 5 discusses the results, while section 6 concludes the study.

2. Basic equations for MEE materials

The basic equations that relate the mechanical, electrical, and magnetic components in MEE materials are given:

2.1. Constitutive equations

\begin{array}{l} σ_{ij} & = C_{ijkl} ϵ_{kl} - e_{kli} E_{k} - h_{kli} H_{k}, \end{array}

(1)

\begin{array}{l} D_{i} & = e_{ikl} ϵ_{kl} + \in_{ik} E_{k} + β_{ik} H_{k}, \end{array}

(2)

\begin{array}{l} B_{i} & = h_{ikl} ϵ_{kl} + β_{ik} E_{k} + γ_{ik} H_{k}, \end{array}

(3)

where $i, j, k, l = 1, 2, 3$ (or $x, y, z$ ).

2.2. Equilibrium equations

σ_{ij, j} = 0, D_{i, i} = 0, B_{i, i} = 0,

(4)

2.3. Kinematic equations

ϵ_{ij} = \frac{1}{2} (u_{i, j} + u_{j, i}), E_{i} = - ϕ_{, i}, H_{i} = - φ_{, i}

(5)

2.4. Boundary conditions

\begin{array}{l} σ_{ij} n_{j} & = t_{i}^{0}, on S_{σ} \end{array}

(6)

\begin{array}{l} D_{i} n_{i} & = t_{ω}^{0}, on S_{D} \end{array}

(7)

\begin{array}{l} B_{i} n_{i} & = t_{ω}^{0}, on S_{B} \end{array}

(8)

2.5. Crack-face boundary conditions

i. Impermeable boundary conditions

\begin{array}{l} σ_{ij} n_{j} & = 0, D_{y}^{+} = D_{y}^{-} = D_{y}^{c} = 0, B_{y}^{+} = B_{y}^{-} = B_{y}^{c} = 0, \end{array}

(9)

\begin{array}{l} σ_{ij} n_{j} & = 0, D_{y}^{+} = D_{y}^{-} = D_{y}^{c} \neq 0, B_{y}^{+} = B_{y}^{-} = B_{y}^{c} \neq 0, \end{array}

(10)

ii. Semi-permeable boundary conditions

\begin{array}{l} σ_{ij} n_{j} & = 0, \end{array}

(11)

\begin{array}{l} D_{y}^{+} & = D_{y}^{-} = D_{y}^{c} = - κ_{c} \frac{| | ϕ (x) | |}{| | u_{y} (x) | |}, \end{array}

(12)

\begin{array}{l} B_{y}^{+} & = B_{y}^{-} = B_{y}^{c} = - μ_{c} \frac{| | φ (x) | |}{| | u_{y} (x) | |} . \end{array}

(13)

The standard mechanical, electrical, and magnetic parameters used in the equations above are thoroughly defined in the works by Kumar et al. [32,45]. In the above equations, $σ_{ij}$ , $ϵ_{ij}$ , $D_{i}$ , $B_{i}$ , $E_{i}$ , and $H_{i}$ denote the components of the stress, strain, electric displacement, magnetic induction, electric field, and magnetic field, respectively. $C_{ijkl}, \in_{ik}$ , and $γ_{ik}$ denote the elastic, dielectric permittivities, and magnetic permeabilities constants. $e_{ikl}$ , $h_{kli}$ , and $β_{ik}$ denote the piezoelectric, piezomagnetic, and electromagnetic coupling coefficients. $κ_{c}$ and $μ_{c}$ are the permittivity and permeability of the medium between crack faces. $| | ϕ (x) | |$ and $| | φ (x) | |$ are the electrical and magnetic potential jump, and $| | u_{y} (x) | |$ is the crack opening displacement (COD).

3. Formulation of the problem

Let us consider four collinear cracks with two pairs of symmetrical unequal cracks on x-axis vary from $(- l_{0}, 0)$ to $(- j_{0}, 0)$ , $(- e_{0}, 0)$ to $(- c_{0}, 0)$ , $(c_{0}, 0)$ to $(e_{0}, 0)$ and $(j_{0}, 0)$ to $(l_{0}, 0)$ in a two-dimensional $(x, y)$ Cartesian coordinate system. The half-crack length of cracks are $a_{1}$ and $a_{2}$ and located at distances $d_{1}$ and $d_{2}$ between the centers of cracks in the MEE materials ( $BaTi O_{3} - CoFe O_{2}$ ) as shown in Figure 1. The applied loading at infinity is defined as follows:

\sum_{y}^{\infty} = {(σ_{yx}^{\infty}, σ_{yy}^{\infty}, σ_{yz}^{\infty}, D_{y}^{\infty}, B_{y}^{\infty})}^{T} .

(14)

Figure 1.

EMPS model for four collinear cracks with two pairs of symmetrical unequal cracks in an infinite MEE solid.

Considering the EMPS model, the electric yielding zone spans in $(- n_{0}, - l_{0})$ , $(- j_{0}, - h_{0})$ , $(- g_{0}, - e_{0})$ , $(- c_{0}, - a_{0})$ , $(a_{0}, c_{0})$ , $(e_{0}, g_{0})$ , $(h_{0}, j_{0})$ , and $(l_{0}, n_{0})$ intervals, and the magnetic yielding zone spans in $(- m_{0}, - l_{0})$ , $(- j_{0}, - i_{0})$ , $(- f_{0}, - e_{0})$ , $(- c_{0}, - b_{0})$ , $(b_{0}, c_{0})$ , $(e_{0}, f_{0})$ , $(i_{0}, j_{0})$ , and $(l_{0}, m_{0})$ intervals, and is depicted in Figure 1. Semipermeable crack face, saturated electric displacement, and saturated magnetic induction conditions are defined for the given problem as

\begin{array}{l} \sum_{y}^{l} & = {(σ_{yx}, σ_{yy}, σ_{yz}, D_{y}, B_{y})}^{T} = {(0, 0, 0, {(D_{y}^{c})}_{l}, {(B_{y}^{c})}_{l})}^{T}, x \in (- e_{0}, - c_{0}) \cup (c_{0}, e_{0}), \end{array}

(15)

\begin{array}{l} \sum_{y}^{r} & = {(σ_{yx}, σ_{yy}, σ_{yz}, D_{y}, B_{y})}^{T} = {(0, 0, 0, {(D_{y}^{c})}_{r}, {(B_{y}^{c})}_{r})}^{T}, x \in (- l_{0}, - j_{0}) \cup (j_{0}, l_{0}), \end{array}

(16)

\begin{array}{l} D_{y} (x, 0) & = D_{s}, x \in (- n_{0}, - l_{0}) \cup (- j_{0}, - h_{0}) \cup (- g_{0}, - e_{0}) \cup \\ (- c_{0}, - a_{0}) \cup (a_{0}, c_{0}) \cup (e_{0}, g_{0}) \cup (h_{0}, j_{0}) \cup (l_{0}, n_{0}), \end{array}

(17)

\begin{array}{l} B_{y} (x, 0) & = B_{s}, x \in (- m_{0}, - l_{0}) \cup (- j_{0}, - i_{0}) \cup (- f_{0}, - e_{0}) \cup \\ (- c_{0}, - b_{0}) \cup (b_{0}, c_{0}) \cup (e_{0}, f_{0}) \cup (i_{0}, j_{0}) \cup (l_{0}, m_{0}), \end{array}

(18)

where $D_{s}$ and $B_{s}$ are the electric and magnetic saturated loading, respectively. ${(D_{y}^{c})}_{l}$ and ${(D_{y}^{c})}_{r}$ are electric displacements at the left and right crack surfaces, respectively. ${(B_{y}^{c})}_{l}$ and ${(B_{y}^{c})}_{r}$ are magnetic induction at surfaces of left and right cracks, respectively.

Complex potential function in the infinite domain without any loading for a single edge dislocation in MEE materials is defined as [30] follows:

Φ = \frac{1}{π} Im {B < \ln (z_{k} - z_{k_{0}}) > B^{T}} b .

(19)

The relations between complex potential function and developed stresses are given as follows:

Σ_{2 j} = - Φ_{j, 1}, Σ_{1 j} = - Φ_{j, 2} .

(20)

From equations (19) and (20), stresses at point $(η, 0)$ due to a single edge dislocation located at $(ξ, 0)$ point are given as follows:

\sum_{2 j} = \frac{1}{π} Im {B < \frac{1}{η - ξ} > B^{T}} b .

(21)

Furthermore, implementing DDT [18,46] and equations (15)–(18) and equation (21), the four collinear cracks problem (without considering the EMPS model) defined from $- l_{0}$ to $- j_{0}$ , $- e_{0}$ to $- c_{0}$ , $c_{0}$ to $e_{0}$ , and $j_{0}$ to $l_{0}$ using the symmetry of the problem can be mathematically formulated as follows:

\begin{matrix} {\begin{matrix} σ_{y x}^{\infty} \\ σ_{y y}^{\infty} \\ σ_{y z}^{\infty} \\ D_{y}^{\infty} \\ B_{y}^{\infty} \end{matrix}} + \int_{c_{0}}^{e_{0}} H < \frac{2 ξ}{π (η^{2} - ξ^{2})} > b_{l} (ξ) d ξ + \int_{j_{0}}^{l_{0}} H < \frac{2 ξ}{π (η^{2} - ξ^{2})} > b_{r} (ξ) d ξ = {\begin{matrix} 0 \\ 0 \\ 0 \\ {(D_{y}^{c})}_{l} \\ {(B_{y}^{c})}_{l} \end{matrix}}, \\ c_{0} < η < e_{0}, \end{matrix}

(22)

\begin{matrix} {\begin{matrix} σ_{y x}^{\infty} \\ σ_{y y}^{\infty} \\ σ_{y z}^{\infty} \\ D_{y}^{\infty} \\ B_{y}^{\infty} \end{matrix}} + \int_{c_{0}}^{e_{0}} H < \frac{2 ξ}{π (η^{2} - ξ^{2})} > b_{l} (ξ) d ξ + \int_{j_{0}}^{l_{0}} H < \frac{2 ξ}{π (η^{2} - ξ^{2})} > b_{r} (ξ) d ξ = {\begin{matrix} 0 \\ 0 \\ 0 \\ {(D_{y}^{c})}_{r} \\ {(B_{y}^{c})}_{r} \end{matrix}}, \\ j_{0} < η < l_{0}, \end{matrix}

(23)

where $H = [\begin{array}{l} H_{1} & H_{2}^{T} & H_{3}^{T} \\ H_{2} & H_{44} & H_{45} \\ H_{3} & H_{54} & H_{55} \end{array}] = Im (B B^{T})$ , $b_{l} = {(b_{1}, b_{2}, b_{3}, b_{4}, b_{5})}^{T}$ , and $b_{r} = {(b_{6}, b_{7}, b_{8}, b_{9}, b_{10})}^{T}$ .

According to the EMPS model, the stated problem is equivalent to four mechanical cracks from $- l_{0}$ to $- j_{0}$ , $- e_{0}$ to $- c_{0}$ , $c_{0}$ to $e_{0}$ , $j_{0}$ to $l_{0}$ ; four extended electric cracks defined from $- n_{0}$ to $- h_{0}$ , $- g_{0}$ to $- a_{0}$ , $a_{0}$ to $g_{0}$ , $h_{0}$ to $n_{0}$ ; and four extended magnetic cracks from $- m_{0}$ to $- i_{0}$ and $- f_{0}$ to $- b_{0}$ , $b_{0}$ to $f_{0}$ , and $i_{0}$ to $m_{0}$ . Therefore, equations (22) and (23) can be expressed as follows:

\begin{array}{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{1} b_{l}^{*} dξ + \int_{a_{0}}^{g_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2} b_{4} dξ + \int_{b_{0}}^{f_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3} b_{5} dξ \\ + \int_{j_{0}}^{l_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{1} b_{r}^{*} dξ + \int_{h_{0}}^{n_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2} b_{9} dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3} b_{10} dξ \\ + t^{*} = 0, c_{0} \leq η \leq e_{0}, j_{0} \leq η \leq l_{0}, \end{array}

(24)

\begin{array}{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2}^{T} b_{l}^{*} dξ + \int_{a_{0}}^{g_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{44} b_{4} dξ + \int_{b_{0}}^{f_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{45} b_{5} dξ \\ + \int_{j_{0}}^{l_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2}^{T} b_{r}^{*} dξ + \int_{h_{0}}^{n_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{44} b_{9} dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{45} b_{10} dξ \\ + (D_{y}^{\infty} - {(D_{y}^{c})}_{l}) = 0, c_{0} \leq η \leq e_{0}, \end{array}

(25)

\begin{array}{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3}^{T} b_{l}^{*} dξ + \int_{a_{0}}^{g_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{54} b_{4} dξ + \int_{b_{0}}^{f_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{55} b_{5} dξ \\ + \int_{j_{0}}^{l_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3}^{T} b_{r}^{*} dξ + \int_{h_{0}}^{n_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{54} b_{9} dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{55} b_{10} dξ \\ + (B_{y}^{\infty} - {(B_{y}^{c})}_{l}) = 0, c_{0} \leq η \leq e_{0}, \end{array}

(26)

\begin{array}{l} \int_{j_{0}}^{l_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2}^{T} b_{l}^{*} dξ + \int_{h_{0}}^{n_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{44} b_{4} dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{45} b_{5} dξ \\ + \int_{c_{0}}^{e_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2}^{T} b_{r}^{*} dξ + \int_{a_{0}}^{g_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{44} b_{9} dξ + \int_{b_{0}}^{f_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{45} b_{10} dξ \\ + (D_{y}^{\infty} - {(D_{y}^{c})}_{r}) = 0, j_{0} \leq η \leq l_{0}, \end{array}

(27)

\begin{array}{l} \int_{j_{0}}^{l_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3}^{T} b_{l}^{*} dξ + \int_{h_{0}}^{n_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{54} b_{4} dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{55} b_{5} dξ \\ + \int_{c_{0}}^{e_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3}^{T} b_{r}^{*} dξ + \int_{a_{0}}^{g_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{54} b_{9} dξ + \int_{b_{0}}^{f_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{55} b_{10} dξ \\ + (B_{y}^{\infty} - {(B_{y}^{c})}_{r}) = 0, j_{0} \leq η \leq l_{0}, \end{array}

(28)

\begin{array}{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2}^{T} b_{l}^{*} dξ + \int_{a_{0}}^{g_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{44} b_{4} dξ + \int_{b_{0}}^{f_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{45} b_{5} dξ \\ + \int_{j_{0}}^{l_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{2}^{T} b_{r}^{*} dξ + \int_{h_{0}}^{n_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{44} b_{9} dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{45} b_{10} dξ \\ + (D_{y}^{\infty} - D_{s}) = 0, a_{0} \leq η \leq c_{0}, e_{0} \leq η \leq g_{0}, h_{0} \leq η \leq j_{0}, l_{0} \leq η \leq n_{0} \end{array}

(29)

\begin{array}{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3}^{T} b_{l}^{*} dξ + \int_{a_{0}}^{g_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{54} b_{4} dξ + \int_{b_{0}}^{f_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{55} b_{5} dξ \\ + \int_{j_{0}}^{l_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{3}^{T} b_{r}^{*} dξ + \int_{h_{0}}^{n_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{54} b_{9} dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{π (η^{2} - ξ^{2})} H_{55} b_{10} dξ \\ + (B_{y}^{\infty} - B_{s}) = 0, b_{0} \leq η \leq c_{0}, e_{0} \leq η \leq f_{0}, i_{0} \leq η \leq j_{0}, l_{0} \leq η \leq m_{0} \end{array}

(30)

where $t^{*} = {(σ_{yx}^{\infty}, σ_{yy}^{\infty}, σ_{yz}^{\infty})}^{T}$ , $b_{l}^{*} = {(b_{1}, b_{2}, b_{3})}^{T}$ and $b_{r}^{*} = {(b_{6}, b_{7}, b_{8})}^{T}$ .

4. Methodology for analytical solutions of dislocation densities and electric and magnetic zone lengths

4.1. Analytical solution for dislocation density, i.e., $b_{i}$ , $i = 1 t o 10$

After solving equations (24)–(26) and equations (24), (27), and (28) for mechanical dislocation density vectors $b_{l}^{*}$ and $b_{r}^{*}$ , we get

\frac{M}{π} \int_{c_{0}}^{e_{0}} \frac{2 ξ}{(η^{2} - ξ^{2})} b_{l}^{*} (ξ) dξ + \frac{M}{π} \int_{j_{0}}^{l_{0}} \frac{2 ξ}{(η^{2} - ξ^{2})} b_{r}^{*} (ξ) dξ = g_{1}^{*}, c_{0} \leq η \leq e_{0},

(31)

\frac{M}{π} \int_{j_{0}}^{l_{0}} \frac{2 ξ}{(η^{2} - ξ^{2})} b_{r}^{*} (ξ) dξ + \frac{M}{π} \int_{c_{0}}^{e_{0}} \frac{2 ξ}{(η^{2} - ξ^{2})} b_{l}^{*} (ξ) dξ = g_{2}^{*}, j_{0} \leq η \leq l_{0},

(32)

where $G_{1} = H_{1} H_{44} - H_{2} H_{2}^{T}$ , $G_{2} = H_{44} H_{3} - H_{45} H_{2}$ , $G_{3} = H_{54} H_{2}^{T} - H_{3}^{T} H_{44}$ , $G_{4} = H_{54} H_{45} - H_{44} H_{55}$ , $Q_{1} = H_{2} (D_{y}^{\infty} - D_{y}^{c}) - H_{44} t^{*}$ , $Q_{2} = H_{44} (B_{y}^{\infty} - B_{y}^{c}) - H_{54} (D_{y}^{\infty} - D_{y}^{c})$ , $M = G_{4} G_{1} - G_{2} G_{3}$ , $g_{1}^{*} = G_{4} Q_{1} - G_{2} Q_{2}$ , $g_{2}^{*} = G_{4} Q_{1} - G_{2} Q_{2}$ .

Assuming $ξ^{2} = ξ^{'}$ , $η^{2} = η^{'}$ , equations (31) and (32) reduced as follows:

\frac{M}{π} \int_{c_{0}^{2}}^{e_{0}^{2}} \frac{1}{(η^{'} - ξ^{'})} b_{l}^{*} (\sqrt{ξ^{'}}) d ξ^{'} + \frac{M}{π} \int_{j_{0}^{2}}^{l_{0}^{2}} \frac{1}{(η^{'} - ξ^{'})} b_{r}^{*} (\sqrt{ξ^{'}}) d ξ^{'} = g_{1}^{*}, c_{0}^{2} \leq η^{'} \leq e_{0}^{2},

(33)

\frac{M}{π} \int_{c_{0}^{2}}^{e_{0}^{2}} \frac{1}{(η^{'} - ξ^{'})} b_{l}^{*} (\sqrt{ξ^{'}}) d ξ^{'} + \frac{M}{π} \int_{j_{0}^{2}}^{l_{0}^{2}} \frac{1}{(η^{'} - ξ^{'})} b_{r}^{*} (\sqrt{ξ^{'}}) d ξ^{'} = g_{2}^{*}, j_{0}^{2} \leq η^{'} \leq l_{0}^{2} .

(34)

Now, defining the operators $T_{1}, T_{2}, {\tilde{T}}_{1},$ and ${\tilde{T}}_{2}$ as follows:

\begin{array}{l} T_{1} (b_{l}^{*} (\sqrt{η^{'}})) = \int_{c_{0}^{2}}^{e_{0}^{2}} \frac{b_{l}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \in (c_{0}^{2}, e_{0}^{2}); {\tilde{T}}_{1} (b_{l}^{*} (\sqrt{η^{'}})) = \int_{c_{0}^{2}}^{e_{0}^{2}} \frac{b_{l}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \notin (c_{0}^{2}, e_{0}^{2}) \end{array}

(35)

\begin{array}{l} T_{2} (b_{r}^{*} (\sqrt{η^{'}})) = \int_{j_{0}^{2}}^{l_{0}^{2}} \frac{b_{r}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \in (j_{0}^{2}, l_{0}^{2}); {\tilde{T}}_{2} (b_{r}^{*} (\sqrt{η^{'}})) = \int_{j_{0}^{2}}^{l_{0}^{2}} \frac{b_{r}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \notin (j_{0}^{2}, l_{0}^{2}) . \end{array}

(36)

Thus, equations (33) and (34) can be written in operator form as follows:

\frac{M}{π} T_{1} (b_{l}^{*} (\sqrt{η^{'}})) + \frac{M}{π} \tilde{T_{2}} (b_{r}^{*} (\sqrt{η^{'}})) = g_{1}^{*}, c_{0}^{2} \leq η^{'} \leq e_{0}^{2},

(37)

\frac{M}{π} \tilde{T_{1}} (b_{l}^{*} (\sqrt{η^{'}})) + \frac{M}{π} T_{2} (b_{r}^{*} (\sqrt{η^{'}})) = g_{2}^{*}, j_{0}^{2} \leq η^{'} \leq l_{0}^{2}

(38)

Now, as we know that the solution of Cauchy’s singular integral equation

\frac{1}{π} \int_{α}^{β} \frac{1}{(t - s)} F (s) ds = h (t), α \leq t \leq β

(39)

with unbounded singularities at both ends can be given as follows:

F (t) = \frac{C}{\sqrt{(β - t) (t - α)}} - \frac{1}{\sqrt{(β - t) (t - α)}} \frac{1}{π} \int_{α}^{β} \frac{1}{(t - s)} \sqrt{(β - t) (t - α)} h (s) ds, α \leq t \leq β .

(40)

If we write equation (39) in the operator form, then it can be expressed as

T (F (s)) = h (t), α \leq t \leq β,

(41)

where $T (F (s)) = \frac{1}{π} \int_{α}^{β} \frac{1}{(t - s)} F (s) ds$ .

Thus, equation (40) can also be written in the inverse operator form ( $T^{- 1}$ ) as per the following relation, which almost satisfies the inverse operator properties defined in equations (43,44)

T^{- 1} (h (t)) = \frac{C}{\sqrt{(β - t) (t - α)}} - \frac{1}{\sqrt{(β - t) (t - α)}} \frac{1}{π} \int_{α}^{β} \frac{1}{(t - s)} \sqrt{(β - t) (t - α)} h (s) ds, α \leq t \leq β

(42)

where

T (T^{- 1} (ϕ (t)) = ϕ (t), α \leq t \leq β

(43)

T^{- 1} (T (ϕ (t))) = \frac{C}{\sqrt{(β - t) (t - α)}} + ϕ (t), α \leq t \leq β

(44)

Hence, considering the operators defined in equations (35) and (36) and using the inverse operator defined in equation (42), we then define the inverse operators $T_{1}^{- 1}$ and $T_{2}^{- 1}$ as

\begin{array}{l} T_{1}^{- 1} (b_{l}^{*} (\sqrt{η^{'}})) = \frac{D_{1}}{△_{3} (\sqrt{η^{'}})} - \frac{T_{1} (△_{3} (η) b_{l}^{*} (\sqrt{η^{'}}))}{π^{2} △_{3} (η)}, c_{0}^{2} \leq η^{'} \leq e_{0}^{2} and \end{array}

(45)

\begin{array}{l} T_{2}^{- 1} (b_{r}^{*} (\sqrt{η^{'}})) = \frac{D_{2}}{△_{4} (\sqrt{η^{'}})} - \frac{T_{2} (△_{4} (η) b_{r}^{*} (\sqrt{η^{'}}))}{π^{2} △_{4} (η)}, j_{0}^{2} \leq η^{'} \leq l_{0}^{2}, \end{array}

(46)

where $D_{1}$ and $D_{2}$ are arbitrary constants.

Moreover, using these operators $T_{1}, T_{2}, T_{1}^{- 1}$ and $T_{2}^{- 1}$ , one can obtain the identities $I_{1}$ to $I_{5}$ [41,42] defined in Appendix 1.

Now, applying the operator $T_{1}^{- 1}$ to equation (37) and using the identity $I_{2}$ , we obtain the following:

b_{l}^{*} (\sqrt{η^{'}}) = \frac{π A_{1}}{M △_{3} (\sqrt{η^{'}})} - \frac{π^{2} \tilde{T_{2}} (△_{3} (\sqrt{η^{'}}) b_{r}^{*} (\sqrt{η^{'}})) + π T_{1} (△_{3} (\sqrt{η^{'}}) g_{1}^{*})}{M π^{2} △_{3} (\sqrt{η^{'}})}

(47)

Using equation (47) along with the identities $I_{3}$ and $I_{4}$ , one can rewrite equation (38) in the following form:

T_{2} ({\hat{△}}_{3} (\sqrt{η^{'}}) b_{r}^{*} (\sqrt{η^{'}})) = A_{1} π - \frac{1}{π} {\tilde{T}}_{1} (△_{3} (\sqrt{η^{'}}) g_{1}^{*}) + {\hat{△}}_{3} (\sqrt{η^{'}}) g_{2}^{*}, j_{0}^{2} \leq η^{'} \leq l_{0}^{2} .

(48)

Applying the operator $T_{2}^{- 1}$ to equation (48) and using the identities $I_{1}$ and $I_{5}$ , we obtain the following:

\begin{array}{l} b_{r}^{*} (\sqrt{η^{'}}) & = \frac{π (A_{1} η^{'} + B_{1})}{M △_{4} (\sqrt{η^{'}}) {\hat{△}}_{3} (\sqrt{η^{'}})} - \frac{T_{2} (\begin{array}{l} △_{4} (\sqrt{η^{'}}) {\hat{△}}_{3} (\sqrt{η^{'}}) g_{2}^{*} \end{array}) - {\tilde{T}}_{1} (\begin{array}{l} △_{3} (\sqrt{η^{'}}) {\hat{△}}_{4} (\sqrt{η^{'}}) g_{1}^{*} \end{array})}{Mπ {\hat{△}}_{3} (\sqrt{η^{'}}) △_{4} (\sqrt{η^{'}})}, \\ j_{0}^{2} < η^{'} < l_{0}^{2} \end{array}

(49)

Furthermore, using the identities $I_{5}$ to $I_{8}$ and equations (47), (49), we get $b_{l}^{*} (η)$ as

\begin{array}{l} b_{l}^{*} (\sqrt{η^{'}}) & = \frac{π (- A_{1} η^{'} - B_{1})}{M △_{3} (\sqrt{η^{'}}) {\hat{△}}_{4} (\sqrt{η^{'}})} - \frac{T_{1} (△_{3} (\sqrt{η^{'}}) {\hat{△}}_{4} (\sqrt{η^{'}}) g_{1}^{*}) - {\tilde{T}}_{2} (△_{4} (\sqrt{η^{'}}) {\hat{△}}_{3} (\sqrt{η^{'}}) g_{2}^{*})}{Mπ {\hat{△}}_{4} (\sqrt{η^{'}}) △_{3} (\sqrt{η^{'}})}, \\ c_{0}^{2} < η^{'} < e_{0}^{2} \end{array}

(50)

where $A_{1}$ and $B_{1}$ are arbitrary constants, and

\begin{array}{l} △_{3} (\sqrt{η^{'}}) = \sqrt{(η^{'} - c_{0}^{2}) (e_{0}^{2} - η^{'})}, △_{4} (\sqrt{η^{'}}) = \sqrt{(η^{'} - j_{0}^{2}) (l_{0}^{2} - η^{'})}, \\ {\hat{△}}_{3} (\sqrt{η^{'}}) = \sqrt{(η^{'} - c_{0}^{2}) (η^{'} - e_{0}^{2})}, {\hat{△}}_{4} (\sqrt{η^{'}}) = \sqrt{(j_{0}^{2} - η^{'}) (l_{0}^{2} - η^{'})} . \end{array}

Substituting $ξ^{'} = ξ^{2}$ and $η^{'} = η^{2}$ in equations (49) and (50)

\begin{array}{l} b_{l}^{*} (η) = - \frac{π (A_{1} η^{2} + B_{1})}{M Δ_{3} (η) {\hat{Δ}}_{4} (η)} & + \frac{1}{πM Δ_{3} (η) {\hat{Δ}}_{4} (η)} \\ [\int_{c}^{e} \frac{2 ξ Δ_{3} (ξ) {\hat{Δ}}_{4} (ξ) T_{1} (ξ)}{ξ^{2} - η^{2}} dξ - \int_{j}^{l} \frac{2 ξ {\hat{Δ}}_{3} (ξ) Δ_{4} (ξ) T_{2} (ξ)}{ξ^{2} - η^{2}} dξ], c_{0} \leq η \leq e_{0} \end{array}

(51)

\begin{array}{l} b_{r}^{*} (η) = \frac{π (A_{1} η^{2} + B_{1})}{M {\hat{Δ}}_{3} (η) Δ_{4} (η)} + & \frac{1}{πM {\hat{Δ}}_{3} (η) Δ_{4} (η)} \\ [\int_{j}^{l} \frac{2 ξ {\hat{Δ}}_{3} (ξ) Δ_{4} (ξ) T_{2} (ξ)}{ξ^{2} - η^{2}} dξ - \int_{c}^{e} \frac{2 ξ Δ_{3} (ξ) {\hat{Δ}}_{4} (ξ) T_{1} (ξ)}{ξ^{2} - η^{2}} dξ], j_{0} \leq η \leq l_{0} \end{array}

(52)

\begin{array}{l} △_{3} (η) & = \sqrt{(η^{2} - c_{0}^{2}) (e_{0}^{2} - η^{2})}, △_{4} (η) = \sqrt{(η^{2} - j_{0}^{2}) (l_{0}^{2} - η^{2})}, \\ {\hat{△}}_{3} (η) & = \sqrt{(η^{2} - c_{0}^{2}) (η^{2} - e_{0}^{2})}, {\hat{△}}_{4} (η) = \sqrt{(j_{0}^{2} - η^{2}) (l_{0}^{2} - η^{2})} . \end{array}

For magnetic dislocation density $b_{5} (η)$ , solving equations (24)–(26), (29), and (30), we have

\begin{array}{l} \int_{b_{0}}^{f_{0}} \frac{2 ξ}{ξ^{2} - η^{2}} (Q_{0} b_{l}^{*} (ξ) + P_{0} b_{5} (ξ)) dξ + \int_{i_{0}}^{m_{0}} \frac{2 ξ}{ξ^{2} - η^{2}} (Q_{0} b_{r}^{*} (ξ) + P_{0} b_{10} (ξ)) dξ = g^{m} (η) \\ = {\begin{array}{l} R^{l}, c_{0} \leq η \leq e_{0}, \\ R^{r}, j_{0} \leq η \leq l_{0}, \\ S, b_{0} \leq η \leq c_{0}, e_{0} \leq η \leq f_{0}, \\ i_{0} \leq η \leq j_{0}, l_{0} \leq η \leq m_{0} \end{array} \end{array}

(53)

where $R^{l} = - H_{44} (B_{y}^{\infty} - {(B_{y}^{c})}_{l}) + H_{54} (D_{y}^{\infty} - {(D_{y}^{c})}_{l})$ , $R^{r} = - H_{44} (B_{y}^{\infty} - {(B_{y}^{c})}_{r}) + H_{54} (D_{y}^{\infty} - {(D_{y}^{c})}_{r})$ , $S =$ $- H_{44} (B_{y}^{\infty} - B_{s}) + H_{54} (D_{y}^{\infty} - D_{s})$ , $P_{0} = (H_{45} H_{54} - H_{55} H_{44}) ∕ π$ , $Q_{0} = (H_{2}^{T} H_{54} - H_{3}^{T} H_{44}) ∕ π$ .

Similar to equations (31) and (32), the solutions of equation (53) are given by

\begin{array}{l} Q_{0} b_{l}^{*} (\sqrt{η^{'}}) + P_{0} b_{5} (\sqrt{η^{'}}) & = \frac{- A_{1}^{m} η^{'} - B_{1}^{m}}{△_{3}^{m} (\sqrt{η^{'}}) {\hat{△}}_{4}^{m} (\sqrt{η^{'}})} - \\ \frac{T_{1}^{m} (\begin{array}{l} △_{3}^{m} (\sqrt{η^{'}}) {\hat{△}}_{4}^{m} (η) g^{m} (\sqrt{η^{'}}) \end{array}) - {\tilde{T}}_{2}^{m} (\begin{array}{l} △_{4}^{m} (\sqrt{η^{'}}) {\hat{△}}_{3}^{m} (\sqrt{η^{'}}) g^{m} (\sqrt{η^{'}}) \end{array})}{π^{2} {\hat{△}}_{4}^{m} (\sqrt{η^{'}}) △_{3}^{m} (\sqrt{η^{'}})}, b_{0}^{2} \leq η^{'} \leq f_{0}^{2} \end{array}

\begin{array}{l} Q_{0} b_{r}^{*} (\sqrt{η^{'}}) + P_{0} b_{10} (\sqrt{η^{'}}) & = \frac{A_{1}^{m} η^{'} + B_{1}^{m}}{△_{3}^{m} (\sqrt{η^{'}}) {\hat{△}}_{4}^{m} (\sqrt{η^{'}})} - \\ \frac{T_{2}^{m} (\begin{array}{l} △_{3}^{m} (\sqrt{η^{'}}) {\hat{△}}_{4}^{m} (\sqrt{η^{'}}) g^{m} (\sqrt{η^{'}}) \end{array}) - {\tilde{T}}_{1}^{m} (\begin{array}{l} △_{4}^{m} (\sqrt{η^{'}}) {\hat{△}}_{3}^{m} (\sqrt{η^{'}}) g^{m} (\sqrt{η^{'}}) \end{array})}{π^{2} {\hat{△}}_{4}^{m} (\sqrt{η^{'}}) △_{3}^{m} (\sqrt{η^{'}})}, i_{0}^{2} \leq η^{'} \leq m_{0}^{2} \end{array}

which are obtained by employing the operators $T_{1}^{m}, T_{2}^{m}, {\tilde{T}}_{1}^{m}, {\tilde{T}}_{2}^{m}$ defined as

\begin{array}{l} T_{1}^{m} (λ_{1}^{*} (\sqrt{η^{'}})) & = \int_{b_{0}^{2}}^{f_{0}^{2}} \frac{λ_{1}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \in (b_{0}^{2}, f_{0}^{2}); {\tilde{T}}_{1}^{m} (λ_{1}^{*} (\sqrt{η^{'}})) = \int_{b_{0}^{2}}^{f_{0}^{2}} \frac{λ_{1}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \notin (b_{0}^{2}, f_{0}^{2}) \\ T_{2}^{m} (λ_{2}^{*} (\sqrt{η^{'}})) & = \int_{i_{0}^{2}}^{m_{0}^{2}} \frac{λ_{2}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \in (i_{0}^{2}, m_{0}^{2}); {\tilde{T}}_{2}^{m} (λ_{2}^{*} (\sqrt{η^{'}})) = \int_{i_{0}^{2}}^{m_{0}^{2}} \frac{λ_{2}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \notin (i_{0}^{2}, m_{0}^{2}) \end{array}

where $A_{1}^{m}$ and $B_{1}^{m}$ are arbitrary constants, and

\begin{array}{l} △_{3}^{m} (η) & = \sqrt{(η^{2} - b_{0}^{2}) (f_{0}^{2} - η^{2})}, △_{4}^{m} (η) = \sqrt{(η^{2} - i_{0}^{2}) (m_{0}^{2} - η^{2})}, \\ {\hat{△}}_{3}^{m} (η) & = \sqrt{(η^{2} - b_{0}^{2}) (η^{2} - f_{0}^{2})}, {\hat{△}}_{4}^{m} (η) = \sqrt{(i_{0}^{2} - η^{2}) (m_{0}^{2} - η^{2})} . \end{array}

Magnetic dislocation density functions $b_{5}$ and $b_{10}$ are interval wise derived as for $b_{0} \leq η \leq c_{0}$ , $e_{0} \leq η \leq f_{0}$

\begin{array}{l} b_{5} (η) & = \frac{- A_{1}^{m} η^{2} - B_{1}^{m}}{P_{0} △_{3}^{m} (η) {\hat{△}}_{4}^{m} (η)} - \frac{1}{π^{2} P_{0} {\hat{△}}_{4}^{m} (η) △_{3}^{m} (η)} {S \int_{b_{0}}^{c_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \\ + R^{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ + S \int_{e_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{i_{0}}^{j_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \\ - R^{r} \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{l_{0}}^{m_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ} \end{array}

for $c_{0} \leq η \leq e_{0}$

\begin{array}{l} b_{5} (η) & = \frac{- A_{1}^{m} η^{2} - B_{1}^{m}}{P_{0} △_{3}^{m} (η) {\hat{△}}_{4}^{m} (η)} - \frac{1}{π^{2} P_{0} {\hat{△}}_{4}^{m} (η) △_{3}^{m} (η)} {S \int_{b_{0}}^{c_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \\ + R^{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ + S \int_{e_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{i_{0}}^{j_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \end{array}

\begin{array}{l} - R^{r} \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{l_{0}}^{m_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ} - \frac{Q_{0}}{P_{0}} b_{l}^{*} \end{array}

for $i_{0} \leq η \leq j_{0}$ , $l_{0} \leq η \leq m_{0}$

\begin{array}{l} b_{10} (η) & = \frac{A_{1}^{m} η^{2} + B_{1}^{m}}{P_{0} △_{4}^{m} (η) {\hat{△}}_{3}^{m} (η)} - \frac{1}{π^{2} P_{0} {\hat{△}}_{3}^{m} (η) △_{4}^{m} (η)} {S \int_{i_{0}}^{j_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \\ + R^{r} \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ + S \int_{l_{0}}^{m_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{b_{0}}^{c_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \\ - R^{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{e_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ} \end{array}

for $j_{0} \leq η \leq l_{0}$

\begin{array}{l} b_{10} (η) & = \frac{A_{1}^{m} η^{2} + B_{1}^{m}}{P_{0} △_{4}^{m} (η) {\hat{△}}_{3}^{m} (η)} - \frac{1}{π^{2} P_{0} {\hat{△}}_{3}^{m} (η) △_{4}^{m} (η)} {S \int_{i_{0}}^{j_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \\ + R^{r} \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ + S \int_{l_{0}}^{m_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{b_{0}}^{c_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ \\ - R^{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ - S \int_{e_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ)}{ξ^{2} - η^{2}} dξ} - \frac{Q_{0}}{P_{0}} b_{r}^{*} . \end{array}

From equations (25), (27), and (29),

\begin{array}{l} \frac{1}{π} \int_{a_{0}}^{g_{0}} \frac{2 ξ (H_{2}^{T} b_{l} (ξ) + H_{44} b_{4} (ξ) + H_{45} b_{5} (ξ))}{ξ^{2} - η^{2}} dξ + & \frac{1}{π} \int_{h_{0}}^{n_{0}} \frac{2 ξ (H_{2}^{T} b_{2} (ξ) + H_{44} b_{9} (ξ) + H_{45} b_{10} (ξ))}{ξ^{2} - η^{2}} dξ \\ = g^{e} (η) = {\begin{array}{l} D_{y}^{\infty} - {(D_{y}^{c})}_{l}, c_{0} < η < e_{0} \\ D_{y}^{\infty} - {(D_{y}^{c})}_{r}, j_{0} \leq η \leq l_{0} \\ D_{y}^{\infty} - D_{s}, else \end{array} \end{array}

(54)

Similar to equations (31) and (32), assuming $ξ^{2} = ξ^{'}$ and $η^{2} = η^{'}$ , the solutions of equation (54) are given by

\begin{array}{l} \frac{H_{2}^{T} b_{1} (\sqrt{η^{'}}) + H_{44} b_{4} (\sqrt{η^{'}}) + H_{45} b_{5} (\sqrt{η^{'}})}{π} = \frac{- A_{1}^{e} η^{'} - B_{1}^{e}}{△_{3}^{e} (\sqrt{η^{'}}) {\hat{△}}_{4}^{e} (\sqrt{η^{'}})} \\ - \frac{T_{1}^{e} (\begin{array}{l} △_{3}^{e} (\sqrt{η^{'}}) {\hat{△}}_{4}^{e} (\sqrt{η^{'}}) g^{e} (\sqrt{η^{'}}) \end{array}) - {\tilde{T}}_{2}^{e} (\begin{array}{l} △_{4}^{e} (\sqrt{η^{'}}) {\hat{△}}_{3}^{e} (\sqrt{η^{'}}) g^{e} (\sqrt{η^{'}}) \end{array})}{π^{2} {\hat{△}}_{4}^{e} (\sqrt{η^{'}}) △_{3}^{e} (\sqrt{η^{'}})}, a_{0}^{2} \leq η^{'} \leq g_{0}^{2} \end{array}

(55)

\begin{array}{l} \frac{H_{2}^{T} b_{2} (\sqrt{η^{'}}) + H_{44} b_{9} (\sqrt{η^{'}}) + H_{45} b_{10} (\sqrt{η^{'}})}{π} = \frac{A_{1}^{e} η^{'} + B_{1}^{e}}{△_{3}^{e} (\sqrt{η^{'}}) {\hat{△}}_{4}^{e} (\sqrt{η^{'}})} \\ - \frac{T_{2}^{e} (\begin{array}{l} △_{3}^{e} (\sqrt{η^{'}}) {\hat{△}}_{4}^{e} (\sqrt{η^{'}}) g^{e} (\sqrt{η^{'}}) \end{array}) - {\tilde{T}}_{1}^{e} (\begin{array}{l} △_{4}^{e} (\sqrt{η^{'}}) {\hat{△}}_{3}^{e} (\sqrt{η^{'}}) g^{e} (\sqrt{η^{'}}) \end{array})}{π^{2} {\hat{△}}_{4}^{e} (\sqrt{η^{'}}) △_{3}^{e} (\sqrt{η^{'}})}, h_{0}^{2} \leq η^{'} \leq n_{0}^{2} \end{array}

(56)

which are obtained by employing the operators $T_{1}^{e}, T_{2}^{e}, {\tilde{T}}_{1}^{e}, {\tilde{T}}_{2}^{e}$ defined as

\begin{array}{l} T_{1}^{e} (λ_{1}^{*} (\sqrt{η^{'}})) & = \int_{a_{0}^{2}}^{g_{0}^{2}} \frac{λ_{1}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \in (a_{0}^{2}, g_{0}^{2}); {\tilde{T}}_{1}^{e} (λ_{1}^{*} (\sqrt{η^{'}})) = \int_{a_{0}^{2}}^{g_{0}^{2}} \frac{λ_{1}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \notin (a_{0}^{2}, g_{0}^{2}) \\ T_{2}^{e} (λ_{2}^{*} (\sqrt{η^{'}})) & = \int_{h_{0}^{2}}^{n_{0}^{2}} \frac{λ_{2}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \in (h_{0}^{2}, n_{0}^{2}); {\tilde{T}}_{2}^{e} (λ_{2}^{*} (\sqrt{η^{'}})) = \int_{h_{0}^{2}}^{n_{0}^{2}} \frac{λ_{2}^{*} (\sqrt{ξ^{'}}) d ξ^{'}}{ξ^{'} - η^{'}}, η^{'} \notin (h_{0}^{2}, n_{0}^{2}) \end{array}

where $A_{1}^{e}$ and $B_{1}^{e}$ are arbitrary constants, and

\begin{array}{l} △_{3}^{e} (η) = \sqrt{(η^{2} - a_{0}^{2}) (g_{0}^{2} - η^{2})}, △_{4}^{e} (η) = \sqrt{(η^{2} - h_{0}^{2}) (n_{0}^{2} - η^{2})}, \\ {\hat{△}}_{3}^{e} (η) = \sqrt{(η^{2} - a_{0}^{2}) (η^{2} - g_{0}^{2})}, {\hat{△}}_{4}^{e} (η) = \sqrt{(h_{0}^{2} - η^{2}) (n_{0}^{2} - η^{2})} . \end{array}

Furthermore, equations (55) and, (56) can be written as follows:

\begin{array}{l} \frac{H_{2}^{T} b_{l} (η) + H_{44} b_{4} (η) + H_{45} b_{5} (η)}{π} = \frac{- A_{1}^{e} (η^{2}) - B_{1}^{e}}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} - \frac{1}{π^{2} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} [\int_{a_{0}}^{g_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) g^{e} (ξ)}{ξ^{2} - η^{2}} dξ \\ + \int_{h_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) g^{e} (ξ)}{ξ^{2} - η^{2}} dξ], a_{0} \leq η \leq g_{0} \end{array}

\begin{array}{l} \frac{H_{2}^{T} b_{r} (η) + H_{44} b_{9} (η) + H_{45} b_{10} (η)}{π} & = \frac{A_{1}^{e} (η^{2}) + B_{1}^{e}}{△_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} - \frac{1}{π^{2} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} [\int_{h_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) g^{e} (ξ)}{ξ^{2} - η^{2}} dξ \\ + \int_{a_{0}}^{g_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) g^{e} (ξ)}{ξ^{2} - η^{2}} dξ], h_{0} \leq η \leq n_{0} . \end{array}

Electric dislocation density functions $b_{4}$ and $b_{9}$ are interval wise derived as

for $c_{0} \leq η \leq e_{0}$

\begin{array}{l} b_{4} (η) & = - \frac{π (A_{1}^{e} η^{2} + B_{1}^{e})}{H_{44} △_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} - \frac{1}{π G_{44} △_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} [\int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ + \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{ξ^{2} - η^{2}} dξ + \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ - \int_{h_{0}}^{j_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ - \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{ξ^{2} - η^{2}} dξ \\ - \int_{l_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ] - \frac{H_{45}}{H_{44}} b_{5} (η) - \frac{H_{2}^{T}}{H_{44}} b_{l}^{*}, \end{array}

for $b_{0} \leq η \leq c_{0}$ , $e_{0} \leq η \leq f_{0}$

\begin{array}{l} b_{4} (η) & = - \frac{π (A_{1}^{e} η^{2} + B_{1}^{e})}{H_{44} △_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} - \frac{1}{π H_{44} △_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} [\int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ + \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{ξ^{2} - η^{2}} dξ + \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ - \int_{h_{0}}^{j_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ - \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{ξ^{2} - η^{2}} dξ \\ - \int_{l_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ] - \frac{H_{45}}{H_{44}} b_{5} (η), \end{array}

for $a_{0} \leq η \leq b_{0}$ , $f_{0} \leq η \leq g_{0}$

\begin{array}{l} b_{4} (η) & = - \frac{π (A_{1}^{e} η^{2} + B_{1}^{e})}{H_{44} △_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} - \frac{1}{π H_{44} △_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} [\int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ + \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{ξ^{2} - η^{2}} dξ + \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \end{array}

\begin{array}{l} - \int_{h_{0}}^{j_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ - \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{ξ^{2} - η^{2}} dξ \\ - \int_{l_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ], \end{array}

for $j_{0} \leq η \leq l_{0}$

\begin{array}{l} b_{9} (η) & = \frac{π (A_{1}^{e} η^{2} + B_{1}^{e})}{H_{44} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} - \frac{1}{π H_{44} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} [\int_{h_{0}}^{j_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ + \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{ξ^{2} - η^{2}} dξ + \int_{l_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ - \int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ - \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{ξ^{2} - η^{2}} dξ \\ - \int_{c_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ] - \frac{H_{45}}{H_{44}} b_{10} (η) - \frac{H_{2}^{T}}{H_{44}} b_{r}^{*} (η), \end{array}

for $i_{0} \leq η \leq j_{0}$ , $l_{0} \leq η \leq m_{0}$

\begin{array}{l} b_{9} (η) & = \frac{π (A_{1}^{e} η^{2} + B_{1}^{e})}{H_{44} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} - \frac{1}{π H_{44} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} [\int_{h_{0}}^{j_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ + \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{ξ^{2} - η^{2}} dξ + \int_{l_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ - \int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ - \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{ξ^{2} - η^{2}} dξ \\ - \int_{c_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ] - \frac{H_{45}}{H_{44}} b_{10} (η), \end{array}

for $h_{0} \leq η \leq i_{0}$ , $m_{0} \leq η \leq n_{0}$

\begin{array}{l} b_{9} (η) & = \frac{π (A_{1}^{e} η^{2} + B_{1}^{e})}{H_{44} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} - \frac{1}{π H_{44} △_{4}^{e} (η) {\hat{△}}_{3}^{e} (η)} [\int_{h_{0}}^{j_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \end{array}

\begin{array}{l} + \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{ξ^{2} - η^{2}} dξ + \int_{l_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ \\ - \int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ - \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{ξ^{2} - η^{2}} dξ \\ - \int_{c_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{ξ^{2} - η^{2}} dξ] . \end{array}

4.2. Zone lengths

4.2.1. For magnetic zone lengths

Arbitrary constants $A_{1}^{m}$ and $B_{1}^{m}$ can be obtained after solving the uniqueness condition for $b_{5} (η)$ and $b_{10} (η)$ ), i.e.,

\begin{array}{l} \int_{b_{0}}^{f_{0}} b_{5} (η) dη = 0, \int_{i_{0}}^{m_{0}} b_{10} (η) dη = 0 . \end{array}

(57)

Substituting the value of $b_{5} (η)$ and $b_{10} (η)$ ) in equation (57), we get

\begin{array}{l} 0 = \int_{b_{0}}^{f_{0}} \frac{η^{2}}{△_{3}^{m} (η) {\hat{△}}_{4}^{m} (η)} dη A_{1}^{m} + \int_{b_{0}}^{f_{0}} \frac{1}{△_{3}^{m} (η) {\hat{△}}_{4}^{m} (η)} dη B_{1}^{m} + \frac{1}{π^{2}} \int_{b_{0}}^{f_{0}} \int_{b_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ) g^{m} (ξ)}{△_{3}^{m} (η) {\hat{△}}_{4}^{m} (η) (η^{2} - ξ^{2})} dξdη \\ - \frac{1}{π^{2}} \int_{b_{0}}^{f_{0}} \int_{i_{0}}^{m_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ) g^{m} (ξ)}{△_{3}^{m} (η) {\hat{△}}_{4}^{m} (η) (η^{2} - ξ^{2})} dξdη \end{array}

(58)

\begin{array}{l} 0 = \int_{i_{0}}^{m_{0}} \frac{η^{2}}{△_{4}^{m} (η) {\hat{△}}_{3}^{m} (η)} dη A_{1}^{m} + \int_{i_{0}}^{m_{0}} \frac{1}{△_{4}^{m} (η) {\hat{△}}_{3}^{m} (η)} dη B_{1}^{m} - \frac{1}{π^{2}} \int_{i_{0}}^{m_{0}} \int_{i_{0}}^{m_{0}} \frac{2 ξ △_{4}^{m} (ξ) {\hat{△}}_{3}^{m} (ξ) g^{m} (ξ)}{△_{4}^{m} (η) {\hat{△}}_{3}^{m} (η) (η^{2} - ξ^{2})} dξdη \\ + \frac{1}{π^{2}} \int_{i_{0}}^{m_{0}} \int_{b_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (ξ) g^{m} (ξ)}{△_{4}^{m} (η) {\hat{△}}_{3}^{m} (η) (η^{2} - ξ^{2})} dξdη . \end{array}

(59)

Solving equations (58) and (59), we get the value of arbitrary constants $A_{1}^{m}$ and $B_{1}^{m}$ . Then, imposing finite-valued condition at the extended crack tip points in the dislocation density expressions, we get four equations in unknowns $b_{0}$ , $f_{0}$ , $i_{0}$ , and $m_{0}$ , as follows: for tip $b_{0}$ ;

\begin{array}{l} - A_{1}^{m} b_{0}^{2} - B_{1}^{m} - \frac{1}{π^{2}} {S \int_{b_{0}}^{c_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - b_{0}^{2}} dξ + R^{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - b_{0}^{2}} dξ + S \int_{e_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - b_{0}^{2}} dξ \end{array}

\begin{array}{l} - S \int_{i_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - b_{0}^{2}} dξ - R^{r} \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - b_{0}^{2}} dξ - S \int_{l_{0}}^{m_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - b_{0}^{2}} dξ} = 0 \end{array}

(60)

for tip $f_{0}$ ;

\begin{array}{l} - A_{1}^{m} f_{0}^{2} - B_{1}^{m} - \frac{1}{π^{2}} {S \int_{b_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - f_{0}^{2}} dξ + R^{l} \int_{b_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - f_{0}^{2}} dξ + S \int_{b_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - f_{0}^{2}} dξ \\ - S \int_{i_{0}}^{m_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - f_{0}^{2}} dξ - R^{r} \int_{i_{0}}^{m_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - f_{0}^{2}} dξ - S \int_{i_{0}}^{m_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - f_{0}^{2}} dξ} = 0 \end{array}

(61)

for tip $i_{0}$ ;

\begin{array}{l} A_{1}^{m} i_{0}^{2} + B_{1}^{m} - \frac{1}{π^{2}} {S \int_{i_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - i_{0}^{2}} dξ + R^{r} \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - i_{0}^{2}} dξ + S \int_{l_{0}}^{m_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - i_{0}^{2}} dξ \\ - S \int_{b_{0}}^{c_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - i_{0}^{2}} dξ - R^{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - i_{0}^{2}} dξ - S \int_{e_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - i_{0}^{2}} dξ} = 0 \end{array}

(62)

for tip $m_{0}$ ;

\begin{array}{l} A_{1}^{m} m_{0}^{2} - B_{1}^{m} - \frac{1}{π^{2}} {S \int_{i_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - m_{0}^{2}} dξ + R^{r} \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - m_{0}^{2}} dξ + S \int_{l_{0}}^{m_{0}} \frac{2 ξ {\hat{△}}_{3}^{m} (ξ) △_{4}^{m} (η)}{ξ^{2} - m_{0}^{2}} dξ \\ - S \int_{b_{0}}^{c_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - m_{0}^{2}} dξ - R^{l} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - m_{0}^{2}} dξ - S \int_{e_{0}}^{f_{0}} \frac{2 ξ △_{3}^{m} (ξ) {\hat{△}}_{4}^{m} (η)}{ξ^{2} - m_{0}^{2}} dξ} = 0 . \end{array}

(63)

Solving equations (60)–(63) for unknowns $b_{0}$ , $f_{0}$ , $i_{0}$ , and $m_{0}$ , we get the magnetic zone lengths (MZLs) at all the tips.

4.2.2. For electric zone lengths

Similar to $A_{1}^{m}$ and $B_{1}^{m}$ , the arbitrary constants $A_{1}^{e}$ and $B_{1}^{e}$ can also be obtained after solving uniqueness condition for $b_{4} (η)$ and $b_{9} (η)$ , i.e.,

\begin{array}{l} \int_{a_{0}}^{g_{0}} b_{4} (η) dη = 0, \int_{h_{0}}^{n_{0}} b_{9} (η) dη = 0 . \end{array}

(64)

Substituting the value of $b_{4} (η)$ and $b_{9} (η)$ ) in equation (64), we get the following:

\begin{array}{l} \int_{a_{0}}^{g_{0}} \frac{η^{2}}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} dη A_{1}^{e} + \int_{a_{0}}^{g_{0}} \frac{1}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η)} dη B_{1}^{e} + \frac{1}{π} [\int_{a_{0}}^{g_{0}} \int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη \\ + \int_{a_{0}}^{g_{0}} \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη + \int_{a_{0}}^{g_{0}} \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη \\ - \int_{a_{0}}^{g_{0}} \int_{h_{0}}^{j_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη - \int_{a_{0}}^{g_{0}} \int_{j_{0}}^{l_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη \\ - \int_{a_{0}}^{g_{0}} \int_{l_{0}}^{n_{0}} \frac{2 ξ △_{4}^{e} (ξ) {\hat{△}}_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{△_{3}^{e} (η) {\hat{△}}_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη = 0 \end{array}

(65)

\begin{array}{l} \int_{h_{0}}^{n_{0}} \frac{η^{2}}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η)} dη A_{1}^{e} + \int_{h_{0}}^{n_{0}} \frac{1}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η)} dη B_{1}^{e} - \frac{1}{π} [\int_{h_{0}}^{n_{0}} \int_{h_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη \\ + \int_{h_{0}}^{n_{0}} \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{r})}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη + \int_{h_{0}}^{n_{0}} \int_{l_{0}}^{n_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη \\ - \int_{h_{0}}^{n_{0}} \int_{a_{0}}^{c_{0}} \frac{2 ξ {\hat{△}}_{4}^{e} (ξ) △_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη - \int_{h_{0}}^{n_{0}} \int_{c_{0}}^{e_{0}} \frac{2 ξ {\hat{△}}_{4}^{e} (ξ) △_{3}^{e} (ξ) (D_{y}^{\infty} - {(D_{y}^{c})}_{l})}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη \\ - \int_{h_{0}}^{n_{0}} \int_{e_{0}}^{g_{0}} \frac{2 ξ {\hat{△}}_{4}^{e} (ξ) △_{3}^{e} (ξ) (D_{y}^{\infty} - D_{S})}{{\hat{△}}_{3}^{e} (η) △_{4}^{e} (η) (ξ^{2} - η^{2})} dξdη = 0 . \end{array}

(66)

Solving equations (65) and (66), we get the value of arbitrary constants $A_{1}^{e}$ and $B_{1}^{e}$ . Then, imposing finite-valued condition at the extended crack tip points in dislocation density expressions, we get four equations in unknowns $a_{0}$ , $g_{0}$ , $h_{0}$ , and $n_{0}$ , as follows: for tip $a_{0}$ ;

\begin{array}{l} A_{1}^{e} a_{0}^{2} + B_{1}^{e} + \frac{1}{π^{2}} [(D_{y}^{\infty} - D_{S}) [\int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - a_{0}^{2}} dξ + (D_{y}^{\infty} - {(D_{y}^{c})}_{l}) \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - a_{0}^{2}} dξ \\ + (D_{y}^{\infty} - D_{S}) \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - a_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{h_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - a_{0}^{2}} dξ - (D_{y}^{\infty} - {(D_{y}^{c})}_{r}) \\ \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - a_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{l_{0}}^{n_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - a_{0}^{2}} dξ] = 0 \end{array}

(67)

for tip $g_{0}$ ;

\begin{array}{l} A_{1}^{e} g_{0}^{2} + B_{1}^{e} + \frac{1}{π^{2}} [(D_{y}^{\infty} - D_{S}) \int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - g_{0}^{2}} dξ + (D_{y}^{\infty} - {(D_{y}^{c})}_{l}) \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - g_{0}^{2}} dξ \\ + (D_{y}^{\infty} - D_{S}) \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - g_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{h_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - g_{0}^{2}} dξ - (D_{y}^{\infty} - {(D_{y}^{c})}_{r}) \\ \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - g_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{l_{0}}^{n_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - g_{0}^{2}} dξ] = 0 \end{array}

(68)

for tip $h_{0}$ ;

\begin{array}{l} A_{1}^{e} h_{0}^{2} + B_{1}^{e} + \frac{1}{π^{2}} [(D_{y}^{\infty} - D_{S}) \int_{h_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - h_{0}^{2}} dξ + (D_{y}^{\infty} - {(D_{y}^{c})}_{r}) \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - h_{0}^{2}} dξ \\ + (D_{y}^{\infty} - D_{S}) \int_{l_{0}}^{n_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - h_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - h_{0}^{2}} dξ - (D_{y}^{\infty} - {(D_{y}^{c})}_{l}) \\ \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - h_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - h_{0}^{2}} dξ] = 0 \end{array}

(69)

for tip $n_{0}$ ;

\begin{array}{l} A_{1}^{e} n_{0}^{2} + B_{1}^{e} + \frac{1}{π^{2}} [(D_{y}^{\infty} - D_{S}) \int_{h_{0}}^{j_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - n_{0}^{2}} dξ + (D_{y}^{\infty} - {(D_{y}^{c})}_{r}) \int_{j_{0}}^{l_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - n_{0}^{2}} dξ \\ + (D_{y}^{\infty} - D_{S}) \int_{l_{0}}^{n_{0}} \frac{2 ξ {\hat{△}}_{3}^{e} (ξ) △_{4}^{e} (η)}{ξ^{2} - n_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{a_{0}}^{c_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - n_{0}^{2}} dξ - (D_{y}^{\infty} - {(D_{y}^{c})}_{l}) \\ \int_{c_{0}}^{e_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - n_{0}^{2}} dξ - (D_{y}^{\infty} - D_{S}) \int_{e_{0}}^{g_{0}} \frac{2 ξ △_{3}^{e} (ξ) {\hat{△}}_{4}^{e} (η)}{ξ^{2} - n_{0}^{2}} dξ] = 0 . \end{array}

(70)

After solving equations (67)–(70) for unknowns $a_{0}$ , $g_{0}$ , $h_{0}$ , and $n_{0}$ , we get the electric zone lengths (EZLs).

4.3. For COD, COP, COI

The following equations can be used to calculate fracture parameters like COD, COI, and COP for both cracks based on the dislocation densities determined in the earlier subsection.

\begin{array}{l} | | u_{i} (x) | |_{l} = \int_{x}^{e_{0}} b_{n} (t) dt, i = x, y, z, n = 1, 2, 3, c_{0} \leq x \leq e_{0} \end{array}

(71)

\begin{array}{l} | | u_{i} (x) | |_{r} = \int_{x}^{l_{0}} b_{n} (t) dt, i = x, y, z, n = 6, 7, 8, j_{0} \leq x \leq l_{0} \end{array}

(72)

\begin{array}{l} | | φ (x) | |_{l} = \int_{x}^{f_{0}} b_{5} (t) dt, b_{0} \leq x \leq f_{0} \end{array}

(73)

\begin{array}{l} | | φ (x) | |_{r} = \int_{x}^{m_{0}} b_{10} (t) dt, i_{0} \leq x \leq m_{0} \end{array}

(74)

\begin{array}{l} | | ϕ (x) | |_{l} = \int_{x}^{g_{0}} b_{4} (t) dt, a_{0} \leq x \leq g_{0} . \end{array}

(75)

\begin{array}{l} | | ϕ (x) | |_{r} = \int_{x}^{n_{0}} b_{9} (t) dt, h_{0} \leq x \leq n_{0} . \end{array}

(76)

5. Results and discussion

To understand the effects of the crack interaction between four collinear cracks on the unknown saturated zone lengths, crack tip opening potential (CTOP) and crack tip opening induction (CTOI), the numerical studies are presented in this section with variations in distances $d_{1}, d_{2}$ and the crack-length ratio $\frac{a_{1}}{a_{2}}$ . Furthermore, because of the symmetry of the problem about the y-axis, the results are analyzed at the tips of cracks on the positive x-axis only, i.e., $c_{0}$ , $e_{0}$ , $j_{0}$ , and $l_{0}$ . The parameters considered for the analysis have the following values, if not explicitly stated.

Applied electro-magnetic-mechanical loading: $σ_{yy}^{\infty} = 100 MPa$ , $D_{y}^{\infty} = 0.05 {C/m}^{2}$ and $B_{y}^{\infty} = 4 N/(Am)$ .

Saturated electric displacement $(D_{s}) = 0.1 {C/m}^{2}$ ; saturated magnetic induction $(B_{s}) = 10 N/(Am)$ .

Normalized electric loading ( $D_{y}^{*}) = \frac{D_{y}^{\infty}}{D_{s}}$ , and normalized magnetic loading ( $B_{y}^{*}) = \frac{B_{y}^{\infty}}{B_{s}}$ .

Normalized saturated zone length = saturated zone length/half crack length ( $a_{2}$ ).

The material constants of MEE material $(BaTi O_{3} and CoFe O_{2})$ taken from Zhao et al. [30].

$X_{1} = (x - c_{0}) ∕ (e_{0} - c_{0})$ and $X_{2} = (x - j_{0}) ∕ (l_{0} - j_{0})$

In literature [30,45], it has been established that in the EMPS model, the semipermeable crack-face conditions, i.e., $D_{y}^{c}$ and $B_{y}^{c}$ values, are almost constant throughout the crack surface except for the points very close to the crack tips. Therefore, considering the same, in this study, these values are evaluated at the centers of the cracks using the explicit expressions of saturated zone lengths, COD, COP, and COI derived in section 4 and have been applied the same values uniformly across the crack faces.

5.1. Effects of distance $d_{1}$

To examine the impact of $d_{1}$ on fracture parameters, various ratios of $2 a_{1} ∕ d_{1}$ are considered while keeping other geometric parameters constant, such as $a_{1} = 2 m$ , $a_{2} = 1 m$ , and $2 a_{1} ∕ d_{2} = 0.8$ .

Figure 2 depicts the changes in electric displacement ( $D_{2}^{c}$ ) and magnetic induction ( $B_{2}^{c}$ ) at the centers of the crack surfaces in relation to normalized electric loading ( $D_{y}^{*}$ ) for different $2 a_{1} ∕ d_{1}$ ratios. The results show that electric displacement rises as the normalized electric loading increases while magnetic induction falls. Although $D_{2}^{c}$ and $B_{2}^{c}$ exhibit minor variations with changes in $2 a_{1} ∕ d_{1}$ , they differ significantly from the values observed in a center crack problem with high applied electrical loading. Thus, using center crack semipermeable crack-face conditions for analyzing multiple collinear cracks may not be accurate. Figure 3 shows the changes in normalized EZL at the tips $c_{0}$ , $e_{0}$ , $j_{0}$ , and $l_{0}$ w.r.t normalized electric loading ( $D_{y}^{*}$ ) for various $2 a_{1} ∕ d_{1}$ ratios. The findings indicate that EZL at all crack tips increases with normalized electric loading, consistent with previous studies [29,45]. Additionally, for a given electrical loading, the EZLs at all tips grow as $2 a_{1} ∕ d_{1}$ increases, suggesting that a reduction in the distance between two center cracks leads to larger EZLs at all tips. When comparing EZL at the left and right tips of cracks to the right of the origin, it is observed that for the crack closer to the origin, EZL depends on $2 a_{1} ∕ d_{1}$ values, whereas for the other crack, it is independent and shows higher values at the left tip ( $j_{0}$ ) compared to the right tip ( $l_{0}$ ). For the first crack, EZL is higher at the left tip for $2 a_{1} ∕ d_{1} = 0.6$ and $0.8$ , but for $2 a_{1} ∕ d_{1} = 0.2$ (larger $d_{1}$ ), the right tip ( $e_{0}$ ) shows higher EZL values compared to the left tip ( $c_{0}$ ). This behavior is influenced by the interaction between the inner tips of the two cracks along the positive x-axis. The crack closer to the origin exhibits higher EZL values compared to the one further away. This underscores the impact of crack geometry on the distribution of electric fields near the crack tips, which is crucial for understanding fracture mechanics in MEE materials. For $2 a_{1} ∕ d_{1} = 0.2$ and $2 a_{1} ∕ d_{2} = 0.8$ , the normalized EZLs converge with results from a problem with two unequal cracks of the same configurations, validating the accuracy of the analytical solutions. A tip-wise comparison of EZL w.r.t electrical loading and $2 a_{1} ∕ d_{1}$ ratio is shown in Figure 4.

Figure 2.

Variations in crack-face conditions w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{1}$ ).

Figure 3.

Variations in normalized electric zone lengths at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{1}$ ).

Figure 4.

Comparison of normalized electric zone length at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{1}$ ).

Figure 5 illustrates the variations in normalized MZL at tips $c_{0}$ , $e_{0}$ , $j_{0}$ , and $l_{0}$ relative to normalized electric loading ( $D_{y}^{*}$ ) for different $2 a_{1} ∕ d_{1}$ ratios. The MZLs follow a trend similar to the EZLs, increasing with normalized electric loading at a nearly constant rate. The rate of increase is also consistent with variations in the $2 a_{1} ∕ d_{1}$ ratio. Like the EZL, MZL values are higher at the tips of the crack nearer to the origin compared to those of the crack further away. Furthermore, the distance $d_{1}$ impacts MZL at all tips, increasing with a higher $2 a_{1} ∕ d_{1}$ ratio. This analysis highlights the effects of crack interaction and the coupled nature of electric and magnetic fields in these materials, emphasizing the need for thorough analysis in designing MEE devices. Results for MZLs with $2 a_{1} ∕ d_{1} = 0.2$ and $2 a_{1} ∕ d_{2} = 0.8$ have been validated against results from a problem with two unequal cracks of the same configurations, demonstrating the accuracy of the analytical solutions. A tip-wise comparison of MZL is provided in Figure 6.

Figure 5.

Variations in normalized magnetic zone length at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{1}$ ).

Figure 6.

Comparison of magnetic zone length at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{1}$ ).

5.2. Effects of distance $d_{2}$

In this section, we investigate the effects of $d_{2}$ on fracture parameters while keeping $a_{1} = 2 m$ , $a_{2} = 1 m$ , and $2 a_{1} ∕ d_{1} = 0.2$ constant. Figure 7 displays how electric displacement ( $D_{2}^{c}$ ) and magnetic induction ( $B_{2}^{c}$ ) vary with normalized electric loading ( $D_{y}^{*}$ ) for different $2 a_{1} ∕ d_{2}$ ratios. The observed trends in $D_{2}^{c}$ and $B_{2}^{c}$ w.r.t electric loading and $2 a_{1} ∕ d_{2}$ ratio are similar to those described in Figure 2.

Figure 7.

Variations in crack-face conditions w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{2}$ ).

Figure 8 illustrates the variations in normalized EZLs at the tips $c_{0}$ , $e_{0}$ , $j_{0}$ , and $l_{0}$ in relation to normalized electric loading ( $D_{y}^{*}$ ) for different $2 a_{1} ∕ d_{2}$ ratios. Given that $2 a_{1} ∕ d_{1} = 0.2$ , the EZLs at the right tip of the first crack (near the origin on the positive x-axis) and the left tip of the second crack (away from the origin on the positive x-axis) are higher than those at the left and right tips of their respective cracks as the $2 a_{1} ∕ d_{2}$ ratio increases (indicating a reduction in the distance between the two right cracks). For $2 a_{1} ∕ d_{2} = 0.2$ , where the distance between the two right cracks is large and matches the distance between two center cracks, the EZL values for the left and right tips converge. In this case, their zone length values are consistent with those obtained for a center crack problem [30] with the same crack lengths. Figure 9 provides similar insights for MZLs at all crack tips, showing variations with $2 a_{1} ∕ d_{2}$ ratios and electrical loading. The MZLs exhibit similar behaviour to EZLs w.r.t electrical loading, with the primary difference being a constant rate of increase.

Figure 8.

Variations in normalized electric zone length at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{2}$ ).

Figure 9.

Variations in normalized magnetic zone length at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t normalized electric loading ( $D_{y}^{*}$ ) for different ratios of ( $2 a_{1} ∕ d_{2}$ ).

5.3. Effects of crack length ratio $a_{1} ∕ a_{2}$

This section explores the effects of the crack length ratio, $a_{1} ∕ a_{2}$ , on the normalized zone lengths and other key fracture parameters of the EMPS model, including CTOP and CTOI. For this analysis, the geometric parameters are set to $a_{2} = 1 m$ , $d_{2} = 4 m$ , and $d_{1} ∕ d_{2} = 0.75, 1.0$ , and $1.8$ . Figure 10 illustrates how the normalized electric zone lengths at tips $c_{0}$ , $e_{0}$ , $j_{0}$ , and $l_{0}$ vary with the crack length ratio ( $a_{1} ∕ a_{2}$ ) for different distance ratios ( $d_{1} ∕ d_{2}$ ). The results indicate that EZL increases with an increase in the crack length ratio $a_{1} ∕ a_{2}$ at all crack tips. For the crack positioned away from the origin on the positive x-axis, it is observed that due to crack interaction, the EZL at the left tip ( $j_{0}$ ) is consistently greater than at the right tip ( $l_{0}$ ), regardless of the crack length ratio $a_{1} ∕ a_{2}$ and distance ratio $d_{1} ∕ d_{2}$ . Conversely, for the crack closer to the origin on the positive x-axis, the EZL values at the left and right tips are influenced by both the crack length ratio and the distance ratio $d_{1} ∕ d_{2}$ . Specifically, for $a_{1} ∕ a_{2} \geq 0.5$ and for all defined $d_{1} ∕ d_{2} \leq 1$ values, the EZL at the left tip is higher than at the right tip. However, for $a_{1} ∕ a_{2} < 0.5$ , the opposite behavior is observed regardless of the $d_{1} ∕ d_{2}$ value. This variation can be attributed to the reduced interaction between two center cracks, as their lengths are half those of the cracks further from the origin. Additionally, when the distance between the two center cracks is greater than the distance between the other two cracks on the positive x-axis (i.e., $d_{1} ∕ d_{2} = 1.8$ ), the EZL at the tip $e_{0}$ exceeds that at $c_{0}$ . This trend holds for all crack length ratios. Furthermore, for a fixed $a_{1} ∕ a_{2}$ value, a decrease in the $d_{1} ∕ d_{2}$ value leads to higher EZL values, highlighting the effects of crack interaction between the two symmetrical pairs of collinear cracks. Figure 11 shows a similar pattern for MZL at all crack tips w.r.t the increase in crack length ratio $a_{1} ∕ a_{2}$ and distance ratio $d_{1} ∕ d_{2}$ .

Figure 10.

Variations in normalized electric zone length at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t crack length ratio ( $a_{1} ∕ a_{2}$ ) for different distance ratios ( $d_{1} ∕ d_{2}$ ).

Figure 11.

Variations in normalized magnetic zone length at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t crack length ratio ( $a_{1} ∕ a_{2}$ ) for different distance ratios ( $d_{1} ∕ d_{2}$ ).

Figure 12 illustrates how the CTOP varies at the different crack tips $c_{0}$ , $e_{0}$ , $j_{0}$ , and $l_{0}$ with increasing crack length ratio ( $a_{1} ∕ a_{2}$ ) and for different distance ratios ( $d_{1} ∕ d_{2}$ ). The results show that CTOP values at all crack tips are negative and grow in magnitude with increasing $a_{1} ∕ a_{2}$ . Additionally, the magnitude of CTOP increases at all tips with a decrease in the $d_{1} ∕ d_{2}$ value. This trend mirrors the behavior observed for EZL and MZL, where the magnitudes at the left and right tips of cracks on the positive x-axis also change with varying $a_{1} ∕ a_{2}$ . These observations further validate the impact of crack interaction on fracture parameters in the presence of four collinear cracks. Figure 13 reveals a similar pattern in CTOI at all crack tips, showing consistent behavior w.r.t both crack length ratio and distance ratio. Figure 14 presents the variation of normalized local stress intensity factor (LSIF) at different crack tips ( $c_{0}$ , $e_{0}$ , $j_{0}$ , and $l_{0}$ ) with respect to the crack length ratio ( $a_{1} ∕ a_{2}$ ) for various distance ratios ( $d_{1} ∕ d_{2}$ ). The computed LSIF values are normalized using $σ_{yy}^{\infty} \sqrt{π a_{2}}$ . The left subplot illustrates LSIF variations at tips $c_{0}$ and $e_{0}$ , while the right subplot corresponds to tips $l_{0}$ and $j_{0}$ . Solid and dotted lines distinguish the LSIF values at different crack tips. The results reveal that LSIF increases with the crack length ratio, indicating that stress intensity becomes more pronounced as the cracks grow. Additionally, a higher distance ratio ( $d_{1} ∕ d_{2}$ ) results in lower LSIF values, suggesting that increased crack spacing reduces stress concentration effects. The differences observed between crack tips are attributed to geometric asymmetry and interaction effects. Overall, these findings emphasize the strong dependence of LSIF on crack size and spacing, which is critical for fracture mechanics and structural integrity assessments. Figure 15 depicts COD variation for both cracks at different distance ratios ( $d_{1} ∕ d_{2}$ ). The results indicate that the left crack exhibits higher numerical COD values than the right crack due to its greater length. Additionally, the COD variations for the left crack with changing $d_{1} ∕ d_{2}$ suggest that it undergoes greater deformation due to its relative positioning and interaction effects. Conversely, the COD of the right crack remains relatively stable with varying distance ratios, indicating minimal influence from the adjacent crack. The left crack, however, experiences a more pronounced interaction effect, particularly at $d_{1} ∕ d_{2} = 0.5$ . As the distance ratio increases to $d_{1} ∕ d_{2} = 1.8$ , the interaction effect diminishes, resulting in a lower COD compared to $d_{1} ∕ d_{2} = 0.5$ . This trend highlights that closer crack spacing intensifies interaction, while greater spacing reduces its impact.

Figure 12.

Variations in CTOP at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t crack length ratio ( $a_{1} ∕ a_{2}$ ) for different distance ratios ( $d_{1} ∕ d_{2}$ ).

Figure 13.

Variations in CTOI at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t crack length ratio ( $a_{1} ∕ a_{2}$ ) for different distance ratios ( $d_{1} ∕ d_{2}$ ).

Figure 14.

Variations in normalized LSIF at $c_{0}, e_{0}, j_{0}$ , and $l_{0}$ tips w.r.t crack length ratio ( $a_{1} ∕ a_{2}$ ) for different distance ratios ( $d_{1} ∕ d_{2}$ ).

Figure 15.

Variations in COD for different distance ratios ( $d_{1} ∕ d_{2}$ ).

6. Conclusion

Based on the solutions and numerical studies conducted on four collinear cracks, including two pairs of symmetrical unequal cracks in a MEE solid, as described by the EMPS model, the following conclusions can be drawn:

The validation of results for specific cases, including two unequal collinear cracks and a center crack problem, demonstrates the accuracy and effectiveness of the developed SIO technique and the analytical solutions. This confirms the reliability of the solution approach for collinear cracks in MEE solids and can be extended to collinear cracks problems in other materials.

Variations in inter-crack space distances $d_{1}$ and $d_{2}$ do not significantly affect the crack-face conditions. However, their values do have a considerable impact when compared to the crack-face conditions observed in the center crack problem, and their difference increases with the increase in electrical loading conditions.

The effects of inter-crack space distances on electrical and magnetic saturated zone lengths have been observed. Specifically, as the distance $d_{1}$ (the distance between the two centrally located cracks) decreases, the lengths of the saturated zones at all crack tips increase. Moreover, as the distance $d_{2}$ increases (with $d_{1}$ held constant), the saturated zone lengths at the tips of the cracks further from the origin decrease and approach those of the center crack problem with equivalent crack length.

The lengths of the saturated zones at the left and right tips of the crack nearest to the origin are influenced by both the crack length ratio $a_{1} ∕ a_{2}$ and the distance ratio $d_{1} ∕ d_{2}$ . For $d_{1} ∕ d_{2} = 1.8$ , the saturated zone length at the right crack tip exceeds that at the left crack tip for all defined $a_{1} ∕ a_{2}$ values. A similar pattern is observed for $a_{1} ∕ a_{2} < 0.5$ across all distance ratio ( $d_{1} ∕ d_{2}$ ) values.

The magnitudes of the saturated zone lengths, as well as the CTOP and CTOI, are greater for larger cracks compared to smaller ones. Additionally, these magnitudes increase with an increasing $a_{1} ∕ a_{2}$ ratio.

The crack length ratio and spacing impact have been observed on LSIF and COD. The LSIF and COD increase with crack length while decreasing with an increase in inter-crack space distances.

Footnotes

Appendix 1

\begin{array}{l} I_{1} - & T_{1} (△_{3} (η)) = π (- η + \frac{c_{0} + e_{0}}{2}), η \in (c_{0}, e_{0}), \\ I_{2} - & T_{1} (△_{3} (η) \tilde{T_{2}} (b_{r}^{*} (η))) = π \int_{j_{0}}^{l_{0}} b_{r}^{*} (η) dη - π \tilde{T_{2}} ({\hat{△}}_{3} (η) b_{r}^{*} (η)), η \in (c_{0}, e_{0}), \\ I_{3} - & {\tilde{T}}_{1} (\frac{1}{△_{3} (η)} \tilde{T_{2}} ({\hat{△}}_{3} (η) b_{r}^{*} (η))) = π T_{2} (b_{r}^{*} (η)) - \frac{π}{{\hat{△}}_{3} (η)} T_{2} ({\hat{△}}_{3} (η) b_{r}^{*} (η)), η \in (j_{0}, l_{0}), \\ I_{4} - & {\tilde{T}}_{1} (\frac{1}{△_{3} (η)} T_{1} (△_{1} (η) g_{1}^{*})) = - \frac{π}{{\hat{△}}_{3} (η)} \tilde{T_{1}} (△_{3} (η) g_{1}^{*}), η \in (j_{0}, l_{0}), \\ I_{5} - & T_{2} (△_{4} (η) {\tilde{T}}_{1} (△_{3} (η) g_{1}^{*})) = π \int_{c_{0}}^{e_{0}} △_{3} (η) g_{1}^{*} dη + π {\tilde{T}}_{1} (△_{3} (η) {\hat{△}}_{4} (η) g_{1}^{*}), η \in (j_{0}, l_{0}), \\ I_{6} - & {\tilde{T}}_{2} (\frac{1}{△_{4} (η)} {\tilde{T}}_{1} (△_{3} (η) {\hat{△}}_{4} (η) g_{1}^{*})) = \frac{π}{{\hat{△}}_{4} (η)} T_{1} (△_{3} (η) {\hat{△}}_{4} (η) g_{1}^{*}) - π T_{1} (△_{3} (η) g_{1}^{*}), η \in (c_{0}, e_{0}), \\ I_{7} - & {\tilde{T}}_{2} (\frac{1}{△_{4} (η)} T_{2} ({\hat{△}}_{3} (η) △_{4} (η) g_{2}^{*})) = \frac{π}{{\hat{△}}_{4} (η)} {\tilde{T}}_{2} (△_{4} (η) {\hat{△}}_{3} (η) g_{2}^{*}), η \in (c_{0}, e_{0}), \\ I_{8} - & {\tilde{T}}_{2} (\frac{η}{{\hat{△}}_{4} (η)}) = π (1 + \frac{η}{{\hat{△}}_{4} (η)}), η \in (c_{0}, e_{0}) . \end{array}

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial assistance from the CSIR-UGC, Government of India, New Delhi, is greatly appreciated by the first author.

ORCID iD

Kuldeep Sharma

References

Theocaris

. Dugdale models for two collinear unequal cracks. Engng Fract Mech 1983; 18(3): 545–559.

Collins

Cartwright

. An analytical solution for two equal-length collinear strip yield cracks. Engng Fract Mech 2001; 68(7): 915–924.

Nishimura

. Strip yield analysis of two collinear unequal cracks in an infinite sheet. Engng Fract Mech 2002; 69(11): 1173–1191.

Wang

. Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet. Engng Fract Mech 2011; 78(11): 2356–2368.

Hasan

. Application of modified Dugdale model to two pairs of collinear cracks with coalesced yield zones. Appl Math Model 2016; 40(4): 3381–3399.

Hasan

. Modified Dugdale model for four collinear straight cracks with coalesced yield zones. Theor Appl Fract Mech 2016; 85: 227–235.

Akhtar

Hasan

. Assessment of the interaction between three collinear unequal straight cracks with unified yield zones. AIMS Mater Sci 2017; 4(2): 302–316.

Liu

, et al. Weight functions and stress intensity factors for two unequal-length collinear cracks in an infinite sheet. Engng Fract Mech 2019; 209: 173–186.

Zhang

, et al. Stress intensity factors and plastic zones of stiffened panels with multiple collinear cracks. Theor Appl Fract Mech 2020; 110: 102816.

10.

Zhang

Long

Zheng

, et al. Theoretical analysis of stress intensity factor for two unequal collinear cracks subjected to internal pressure and compressive stress. J Mech 2021; 37: 584–596.

11.

Bhargava

Jangid

. Strip electro-mechanical yielding model for piezoelectric plate cut along two equal collinear cracks. Appl Math Model 2013; 37(22): 9101–9116.

12.

Bhargava

Jangid

. Strip-saturation model for piezoelectric plane weakened by two collinear cracks with coalesced interior zones. Appl Math Model 2013; 37(6): 4093–4102.

13.

Bhargava

Jangid

. Strip-coalesced interior zone model for two unequal collinear cracks weakening piezoelectric media. Appl Math Mech 2014; 35: 1249–1260.

14.

Singh

Sharma

Bhargava

. Analytical solution for two equal collinear modified strip saturated cracks in 2-D semipermeable piezoelectric media. ZAMM 2019; 99(9): e201800244.

15.

Singh

Sharma

Bhargava

. Modified strip saturated models for two equal collinear cracks with coalesced zones in piezoelectric media. Appl Math Mech 2019; 40: 1097–1118.

16.

Singh

Sharma

Bui

. Closed-form solutions for modified polarization saturated models in two unequal collinear cracked piezoelectric media considering coalesced interior zones. Eur J Mech A Solids 2021; 89: 104313.

17.

Liu

. Dynamic intensity factors for multiple permeable cracks in piezoelectric materials. ZAMM 2020; 100(9): e201900344.

18.

Sharma

Singh

Bui

. Simplified numerical algorithms for generalized strip saturated two equal collinear cracks in piezoelectric media using distributed dislocation technique. ZAMM 2023; 103(11): e202300004.

19.

Hasan

Akhtar

. Strip electro-mechanical yield model for three equal straight collinear cracks in piezoelectric materials. Eur J Mech A Solids 2024; 106: 105335.

20.

Gao

Kessler

Balke

. Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks. Int J Eng Sci 2003; 41(9): 983–994.

21.

Tian

Gabbert

. Multiple crack interaction problem in magnetoelectroelastic solids. Eur J Mech A Solids 2004; 23(4): 599–614.

22.

Zhou

Zhang

. Solutions to a limited-permeable crack or two limited-permeable collinear cracks in piezoelectric/piezomagnetic materials. Arch Appl Mech 2007; 77: 861–882.

23.

Zhou

Zhang

. Basic solution of four parallel non-symmetric permeable mode-iii cracks in a piezoelectric/piezomagnetic composite plane. Philos Mag 2008; 88(8): 1153–1186.

24.

Singh

Rokne

Dhaliwal

. Closed-form solutions for two anti-plane collinear cracks in a magnetoelectroelastic layer. Eur J Mech A Solids 2009; 28(3): 599–609.

25.

Lee

. Collinear unequal crack series in magnetoelectroelastic materials: mode I case solved via new real fundamental solutions. Engng Fract Mech 2010; 77(14): 2772–2790.

26.

Zhong

. Closed-form solutions for two collinear dielectric cracks in a magnetoelectroelastic solid. Appl Math Model 2011; 35(6): 2930–2944.

27.

Verma

. Magneto-electro-elastic analysis for two mode-III semi-permeable collinear cracks in piezo-electro-magnetic strip. Int J Math Model Numer Optim 2020; 10(2): 187–213.

28.

Jangid

. A study of electric and magnetic polarizations on a piezoelectromagnetic media weakened by two unequal collinear semi-permeable cracks. Int J Comput Method Eng Sci Mech 2020; 21(6): 292–311.

29.

Jangid

. Magnetic-saturation zone model for two semipermeable cracks in magneto-electro-elastic medium. Int J Comput Method Eng Sci Mech 2018; 19(2): 129–137.

30.

Zhao

Zhang

Fan

, et al. Semi-permeable crack analysis in a 2D magnetoelectroelastic medium based on the EMPS model. Mech Res Commun 2014; 55: 30–39.

31.

Verma

. Magnetic-yielding zone model for assessment of two mode-III semi-permeable collinear cracks in piezo-electro-magnetic strip. Mech Adv Mater Struct 2022; 29(11): 1529–1542.

32.

Kumar

Sharma

Bui

. Singular integral method based closed form solution for moving EMPS model in semipermeable two-equal-collinear cracked MEE material. ZAMM J Appl Math Mech 2024; 104: e202300731.

33.

Kumar

Sharma

Bui

. Moving EMPS model for semipermeable two equal collinear interface cracks in dissimilar magneto-electro-elastic materials. Mech Adv Mater Struct 2024; 32: 873–892.

34.

Monfared

Ayatollahi

. Elastodynamic analysis of a cracked orthotropic half-plane. Appl Math Modell 2012; 36(6): 2350–2359.

35.

Hejazi

Ayatollahi

Bagheri

, et al. Dislocation technique to obtain the dynamic stress intensity factors for multiple cracks in a half-plane under impact load. Arch Appl Mech 2014; 84: 95–107.

36.

Monfared

Bagheri

. Multiple interacting arbitrary shaped cracks in an FGM plane. Theor Appl Fract Mech 2016; 86: 161–170.

37.

Sourki

Ilyaei

Bastanfar

, et al. Multiple cracks analysis in a FG orthotropic layer with FGPM coating under anti-plane loading. J Braz Soc Mech Sci Eng 2018; 40: 1–12.

38.

Bagheri

Ayatollahi

Mirzaei

, et al. Multiple defects in a piezoelectric half-plane under electro-elastic in-plane loadings. Theor Appl Fract Mech 2019; 103: 102316.

39.

Hosseini

Bagheri

Monfared

. Transient response of several cracks in a nonhomogeneous half-layer bonded to a magneto-electro-elastic coating. Theor Appl Fract Mech 2020; 110: 102821.

40.

Monfared

Bagheri

. In-plane stress analysis in a cracked functionally graded piezoelectric plane. Mech Based Design Struct Machines 2023; 51(8): 4508–4534.

41.

Chakrabarti

George

. Solution of a singular integral equation involving two intervals arising in the theory of water waves. Appl Math Lett 1994; 7: 43–47.

42.

Chakrabarti

Sahoo

. Solution of singular integral equations with logarithmic and Cauchy kernels. Proc Indian Acad Sci Math Sci 1996; 106: 181–196.

43.

Tricomi

. Integral equations. Chelmsford, MA: Courier Corporation, 1985.

44.

Estrada

Kanwal

. Singular integral equations. Cham: Springer, 2012.

45.

Kumar

Sharma

Bui

. Singular integral based closed-form solutions for modified EMPS models in semipermeable magneto-electro-elastic materials. Appl Math Modell 2024; 129: 673–695.

46.

Hills

Kelly

Dai

, et al. Solution of crack problems. Cham: Springer, 1996.

A novel singular integral operator technique for analyzing semipermeable four collinear cracks in a magneto-electro-elastic solid based on the EMPS model

Abstract

Keywords

1. Introduction

2. Basic equations for MEE materials

2.1. Constitutive equations

2.2. Equilibrium equations

2.3. Kinematic equations

2.4. Boundary conditions

2.5. Crack-face boundary conditions

3. Formulation of the problem

4. Methodology for analytical solutions of dislocation densities and electric and magnetic zone lengths

4.1. Analytical solution for dislocation density, i.e., b i , i = 1 t o 10

4.2. Zone lengths

4.2.1. For magnetic zone lengths

4.2.2. For electric zone lengths

4.3. For COD, COP, COI

5. Results and discussion

5.1. Effects of distance d 1

5.2. Effects of distance d 2

5.3. Effects of crack length ratio a 1 ∕ a 2

6. Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References

4.1. Analytical solution for dislocation density, i.e., $b_{i}$ , $i = 1 t o 10$

5.1. Effects of distance $d_{1}$

5.2. Effects of distance $d_{2}$

5.3. Effects of crack length ratio $a_{1} ∕ a_{2}$