Magnetoelectroelastic (MEE) material is a new type of intelligent material which can achieve the conversion of magnetic energy, electric energy, and mechanical energy, and has wide applications in lots of multi-functional equipment. So far, great progress has been made regarding the interaction among multi-defects in magnetoelectric composites.^[1–11] Hao et al.^[12] studied the interaction of a screw dislocation with a semi-infinite interfacial crack in an MEE bi-material. Fang et al.^[13,14] derived the solutions of the interaction between a screw dislocation and other defects such as rigid lines, interfacial cracks, and inhomogeneity in MEE material. Hu and Li^[15] presented the general solutions of the singular stress, electric and magnetic field in a piezoelectromagnetic strip containing a Griffith crack. Xiao et al.^[16] discussed the generalized screw dislocation interacting with a wedge-shaped MEE bi-material interface. Liu and Guo^[17] dealt with the problem of the interaction between a screw dislocation and an oblique edge crack in a half-infinite MEE solid. As mentioned above, few studies on the interaction between many dislocations and cracks have been done up to now. In this paper, we will study the interaction between many parallel dislocations and a semi-infinite crack in an MEE solid. The analytic solutions of the dislocations and the crack are obtained according to the method in Ref. [18]. This study expands the research scope of fracture mechanics of the MEE solid, and provides the theoretical basis for the application of this material in the fields of engineering and technology.

For an MEE solid, supposing that its polarized directions of the electric and magnetic fields are along the x₃ axis in the three-dimensional (3D) space coordinate system with an isotropic x₁ – x₂ plane, we consider the mechanical–electric–magnetic coupling anti-plane deformation problem. It means that all the quantities can be determined by anti-plane displacement u₃ (x₁, x₂), in-plane electric potential φ(x₁, x₂), and magnetic potential ψ(x₁, x₂). The basic equations can be written as follows.^[19]

Constitutive equations are given by 1

Generalized strain-displacement relations are given by 2

Equilibrium equations are given by 3 where none of the body force, body volume charge density, and body volume current are considered; σ_3k, ε_3k, and u₃ are the stress, the strain, and the displacement, respectively; D_k, E_k, and φ are the electric displacement, the electric field, and the electric potential, respectively, with k = 1,2; B_k, H_k, and ψ are the magnetic induction, the magnetic field, and the magnetic potential, respectively; C₄₄, κ₁₁, and μ₁₁ are the elastic coefficient, the dielectric coefficient, and the magnetic permeability coefficient, respectively; e₁₅, q₁₅, and d₁₁ are piezoelectric, piezomagnetic, and magnetoelectric coupling coefficient, respectively.

Substituting Eqs. (1) and (2) into Eq. (3), we obtain 4 in which 5 6 7 . Because C is nonsingular, equation (4) is equivalent to 8

The physical model considered in this paper is shown in Fig. 1. There are n screw dislocations in the vicinity of a semi-infinite crack in an MEE solid that is infinitely large.

Fig. 1.

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	Fig. 1. Many parallel dislocations around a semi-infinite crack in MEE solid.

We first consider the analytic solution of a dislocation. Let one of the dislocations be located at the z₀ point in the x₁ − x₂ plane and its Burgers vector is (0,0,b₃,b_φ, b_ψ) in the problem under consideration. The dislocation conditions are given by 9 where λ denotes the Burgers contour surrounding the dislocation z; b₃ is the displacement jump value along x₃; b_φ is the potential jump value; b_ψ is the magnetic potential jump value. Note that the three integrals are conducted in physical space.

By means of properties of the analytic function, from Eq. (8), we define 10 where f(z) represents the analytic function vector and Re denotes its real part.

We introduce the following representations of symbols: 11 12 According to the force conditions of the dislocation, the form of analytic function f(z) can be taken as 13 where z = x₁ + ix₂.

Moreover, 14 From Eq. (14), we have 15 From Eqs. (10), (13), and (15), we can obtain the complex representation of the generalized displacement.

Based on Eq. (1), equation (11) can be written as 16 Equation (16) is the generalized stress in the form of the matrix.

According to Eqs. (2) and (12), the generalized strain can be obtained as 17 From Eq. (16), the generalized stress induced by dislocation z_i acting on z₁ is 18 Supposing that there are n parallel screw dislocations located at points z₁, z₂,…,z_n, respectively, and their Burgers vectors are . Hence, the form of the corresponding matrix is taken as A_(i). Using the superposition principle, the field force of the dislocation z_i, acting on an arbitrary point z in the x₁–x₂ plane is 19 Equation (19) is the analytic solution of n parallel dislocations in the coupling field.

In this section, we consider the interactions among n parallel dislocations. According to Ref. [20], using the superposition principle, the force induced by dislocations z₂, z₃,…,z_n, acting on z₁, is given by 20 where Res denotes the residue on variable z, and refers to the complex conjugate of variable in the bracket. According to residue theory, we have 21 Equation (21) is the Peach–Koehler formula of n parallel dislocations in an MEE solid. It is indicated that the force induced by dislocations z₂, z₃,…,z_n, acting on z₁ is inversely proportional to the distance from it to the dislocations, and it is proportional to the Burgers vectors in an MEE solid.

The interaction between n parallel dislocations and a semi-infinite crack has received more attention.

From Eq. (19), the generalized distribution force in the crack surface, produced by n parallel dislocations is 22 Because the crack is free of traction on its surface, the generalized stress produced by the dislocations in the crack surface is counteracted by the equivalent and reverse additional distribution force produced by the crack surface. The boundary conditions of the crack are expressed as follows: 23 The three sets of equations for the MEE solid together with Eq. (23) constitute the Riemann–Hilbert boundary problem.^[21,22]

According to Muskhelishvili’s method,^[21] we obtain the generalized stress field induced by additional distribution force, namely, 24 Submitting Eq. (19) to Eq. (24), the generalized stress field of the interaction between n parallel dislocations and a semi-infinite crack is derived and given by 25

To discuss the interaction between many parallel dislocations and a semi-infinite crack in an MEE solid, the material constants we choose are listed in Table 1.^[23]

Table 1.

Table 1.

Table 1. Material constants in MEE solid. . C44/N · m−2 e15/C · m−2 κ11/C2· N−1· m−2 q15/N·(A · m)−1 μ11/N · s2· C−2 d11/N · s · V−1· C−1 4.4 × 1010 5.8 56.4 × 10−10 275 297 × 10−6 5.2 × 10−12

Table 1. Material constants in MEE solid. .

All the Burgers vectors of the dislocations are taken to be 10⁻⁹ in magnitude. In addition, F_σ represents the stress of the elastic field, F_D the stress of the electric field, and F_B the stress of the magnetic field in Fig. 2.

Fig. 2.

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	Fig. 2. (color online) (a) Stress produced by elastic field, Fσ, (b) stress produced by electric field, FD, and (c) stress of magnetic field, FB, varying with the position of point z.

It is found from Fig. 2 that the generalized stress increases with the decrease of the distance from the points on the crack to the dislocation. The singularity of the stress in the dislocation core is of the first order in MEE materials.

Letting n = 5, and z₁ = 1, z₂ = 2, z₃ = 3, z₄ = 4, and z₅ = 5, figure 3 shows that the stress concentrations occur, respectively, at the dislocation cores and the tip of the crack. The stress field, electric field, and magnetic field from the crack tip to the dislocation z₁ first decrease and then increase. From Figs. 3(a), 3(b), and 3(c), we can see that the interaction between dislocations and the crack has a greater influence on the elastic field than the electric field and the magnetic field. This is because the material constants relating to the elastic field are far greater than those relating to the electric field and magnetic field.

Fig. 3.

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	Fig. 3. (color online) Total stresses of (a) elastic field, (b) electric field, and (c) magnetic field varying position of point z.

Taking the stress of the elastic field for example, figure 4 illustrates that the crack weakens the stress field of n parallel dislocations. The resultant interaction makes the system stay in a lower energy state. Besides, the interaction among dislocations plays a major role in the generalized stress field as the position of point z goes away from the crack tip.

Fig. 4.

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	Fig. 4. (color online) Comparison between the total stress and stress produced by dislocations of elastic field.

In this paper, a mathematical model of the interaction between n parallel screw dislocations and a semi-infinite crack in an MEE solid is established, and the analytic solutions of the stress field, electric field, and magnetic field are obtained. Finally, the interaction among n parallel screw dislocations and the interaction between n parallel dislocations and a semi-infinite crack are studied further by numerical examples.

(i) Equation (21) gives the Peach–Koehler formula of n parallel screw dislocations in an MEE solid. The results indicate that the Burgers vectors and the distance between dislocations affect the interaction among n parallel screw dislocations.

(ii) From Eq. (25), we can see that the generalized stress in an MEE solid is related to the Burgers vectors and the material constants. In addition, the interaction in the elastic field is more obvious than that in the electric field or magnetic field.

(iii) The generalized stress first decreases and then increases from the dislocation z₁ and the crack tip. At the position far away from the crack, the generalized stress is mainly affected by the dislocations.

(iv) The stress accumulates at the dislocation core and the tip of the crack. The resultant interaction makes the system stay in a lower energy state.