1 Introduction

The magneto-electro-elastic (MEE) or multiferroic composite materials,which exhibit the multi-field couplings among the elastic,electric,and magnetic fields,have recently stimulated a great deal of scientific research for their potential applications to the multifunctional devices such as memories,harvesters,sensors,transducers,and actuators^{[1, 2]}. MEE materials are usually composites made of multi-phases or multi-layers,such as particulate composites,rodarray composites,and laminate composites^[1]. The desirable magneto-electric (ME) coupling usually arises from the strain-mediated elastic interaction across piezomagnetic-piezoelectric interfaces^{[1, 2]}. Therefore,the interfaces can have a great influence on the property of MEE materials and the performance of MEE devices. At the same time,as common defects in crystalline solids,dislocations inevitably interact with the microstructures and interfaces of the MEE materials and therefore affect the mechanical,electrical,and magnetic properties of MEE materials. Due to the above facts,it is necessary to investigate the interaction of dislocations with interfaces in MEE composite materials.

For the problem of dislocation-interface interactions,most related work in the literature is confined to purely elastic materials^{[3, 4, 5, 6, 7, 8]}. Little work has been done to treat the interaction of an arbitrary three-dimensional (3D) dislocation loop with the interface/surface of MEE composite materials. Recently,Han et al.^[9] derived analytical expressions of the extended displacements and extended stresses induced by a planar dislocation loop of arbitrary shape in a generally anisotropic MEE bimaterial. As an extension to our recent work^[3],by utilizing the potential theory,in this paper we mainly solve the extended displacements and extended stresses due to a 3D dislocation loop of arbitrary shape,and the interaction energy between two arbitrarily shaped 3D dislocation loops in transversely isotropic MEE bimaterials. The MEE bimaterial considered here is composed of two dissimilar semi-infinite transversely isotropic MEE solids, which are perfectly bonded together at a planar interface. We assume that the bimaterial interface is parallel to the isotropic plane of both MEE solids (i.e.,perpendicular to the poling direction of both MEE solids).

The present paper is organized as follows. In Section 2,we express the point-force Green’s function for the non-degenerate transversely isotropicMEE bimaterials in a new way for the sake of later derivations of the dislocation solutions. In Section 3,using Green’s function method, we derive line-integral expressions for the extended displacements and extended stresses of an arbitrary 3D dislocation loop,and the interaction energy between two arbitrary 3D dislocation loops in transversely isotropic MEE bimaterials. In Section 4,we provide several numerical examples to show the multi-field coupling effect and the influence of material interface/surface on the extended displacements,extended stresses,and interaction energy of dislocation loops. Concluding remarks are drawn in Section 5. 2 Point-force Green’s function for transversely isotropic MEE bimaterials

Before the discussion,it would be pointed out that throughout the paper we follow the conventions below: (i) The range of Roman indices is from 1 to 3 for lowercase letters (i,j,k, etc.) and from 1 to 5 for uppercase letters (I,J,K,etc.),and the range of Greek indices (α,β, γ,etc.) is from 1 to 2,unless otherwise specified. (ii) When dealing with bimaterials,the Greek index λ or μ in the square bracket (i.e.,[λ] or [μ]) indicates the corresponding relationship with material λ or material μ.

In the Cartesian coordinates (x₁,x₂,x₃),we consider a bimaterial which is composed of two joined half-spaces,as shown in Fig. 1(a). In this bimaterial,one half-space (x₃ >0) is occupied by transversely isotropic MEE material 1,and the other half-space (x₃ <0) is occupied by transversely isotropic MEE material 2. The two half-spaces are perfectly bonded together at the planar interface (x₃=0),which means that both the extended displacements u_I and the extended stresses σ_3J are continuous at x₃=0 (the extended displacement and extended stress are defined in (A2)). We further assume that the isotropic plane of each material is parallel to the bimaterial interface (x₃=0).

We denote the point-force Green’s function for transversely isotropic MEE bimaterials as (y; x),which means the Ith component of the extended displacement at point y (y₁,y₂, y₃) in material λ due to the Jth component (of unit magnitude) of the extended body force applied at point x (x₁,x₂,x₃) in material μ (the extended body force is defined in (A2)). By virtue of the image method,the point-force Green’s function for bimaterials can be divided into two parts as follows:

where u_IJ ^[μ] (y; x) is the point-force Green’s function for the transversely isotropic MEE full space occupied by material μ,while U_IJ^[λ][μ](y; x) is the complementary part from the image sources due to the bimaterial interface; δ_ij is the 3×3 Kronecker delta defined as According to the reciprocity theorem in linear magneto-electro-elasticity^[10],we can find that For the sake of later derivations,we now express u_IJ ^[μ] (y; x) and U_IJ^[λ][μ] (y; x) in terms of second derivatives of certain potential functions as follows: where and In (4) and (5b),the material constants c₄₄ ^[μ] and γ₅ ^[μ] are defined in (A3b) and (A6),respectively. In (4),the extended partial derivative is defined as follows:  and the extended Kronecker delta is defined as Similarly,for easy use in this paper,we further define the extended differential as and the extended permutation symbol as where ε_ijk is the classic permutation symbol defined as Based upon the general solutions of the coupled equations for transversely isotropic MEE media summarized in Appendix A[11−12],the unknown functions in (5a) and (5b) for the nondegenerate bimaterials are found to be and In (8b),the material constant m_nj ^[μ] (n = 1,2,3,4) is defined in (A11). In (8a) and (8b), the symbol “” means that,without confusion,ψ_* ^[μ] (y; x) and ψ_* ^[λ][μ] (y; x) can be written as ψ_* ^[μ] and ψ_* ^[λ][μ] for short,respectively. From (8b),we can further obtain the following useful relations: For the bimaterial with a perfectly-bonded interface,the unknown coefficients in (8a) and (8b) are determined by^[11] and in which the material constants s_J ^[μ] ,α_nj ^[μ] ,ω_nj ^[μ],and θ_nj ^[μ] (n = 1,2,3,4) are defined in (A6) and (A11),respectively.

Also in (8a) and (8b),the unknown functions are defined as^{[13, 14]}

or where The functions defined in (12) satisfy the following useful relations: and 3 Line-integral forms of extended displacements,stresses,and interac- tion energy of arbitrarily shaped 3D dislocation loops in transversely isotropic MEE bimaterials 3.1 Extended displacements of arbitrarily shaped 3D dislocation loop

Now,we consider an arbitrarily shaped 3D dislocation loop C bounding a curved surface A in transversely isotropic MEE bimaterials (see Fig. 1(b)). Based upon the classical theory of dislocations,the extended displacements induced by this dislocation loop can be expressed as^[15]

where u_N ^[μ][λ] (x) is the Nth component of the extended displacement at point x (x₁,x₂,x₃) in material μ due to the dislocation loop C located in material λ,dA_i at point y (y₁,y₂, y₃) is the ith component of the vector area element dA,and the positive normal of dA is associated with the positive direction of the closed dislocation line according to the right-hand rule (Fig. 1(b)),the extended elastic coefficient tensor c_iJKl ^[λ] is defined by (A3a) and (A3b),b_J is the Jth component of the extended Burgers vector b defined as in which u_J ^- and u_J ⁺ denote the Jth component of the extended displacement at the same point on the lower and upper surface of the cut face,respectively. The positive normal of dA (see Fig. 1(b)) should point from the lower surface to the upper surface of the cut face. In this paper, we assume that the extended Burgers vector b is constant over the dislocation surface.

Substituting (1) into (17),we can obtain that

where The above u_N ^[λ] (x) denotes the Nth component of the extended displacement at point x (x₁, x₂,x₃) induced by the dislocation loop C in a transversely isotropic MEE full space occupied by material λ,and U_N ^[μ][λ] (x) is the complementary part from the image sources due to the bimaterial interface.

Substituting (A3a),(A3b),(4),(5a),and (5b) into (20),and then making use of (15),(16), (A6),(A13),and the following Stokes’ theorem^[16]

or we can transform the area integrals in (20) into the line-integral form as follows: where ℓ_[μ] ^[λ] = c₄₄ ^[λ] /c₄₄ ^[μ] , in which the material constants C_ij ^[λ] and C_J ^[λ] are defined in (A12a) and (A12b),respectively. It can be observed from (23c) and (23d) that Note that the area integral Ω ₅ ^[λ] (x) in (23a) denotes the quasi solid angle subtended by the cut face A of the dislocation loop C in material λ at point x,which can also be transformed into a line integral^[17]. 3.2 Extended stresses of arbitrarily shaped 3D dislocation loop

Using the constitutive relation for transversely isotropic MEE materials as shown in (A1b), we can derive from (19) the extended stresses as follows:

in which where σ_iJ ^[μ][λ] (x) denotes the extended stress at point x (x₁,x₂,x₃) in transversely isotropic MEE material μ due to an arbitrarily shaped 3D dislocation loop C located in transversely isotropic MEE material λ,σ_iJ ^[λ] (x) is the extended stresses at point x (x₁,x₂,x₃) induced by the same dislocation loop C in a transversely isotropic MEE full space occupied by material λ, S_iJ ^[μ][λ] (x) is the complementary part from the image sources due to the bimaterial interface.

Substituting (22) into (26),we can express σ_iJ ^[λ] (x) and S_iJ ^[μ][λ] (x) in terms of line integrals

where

During the derivation of (27),we have made use of (15) and (16),along with the following relations:

and (30) can be verified by Stokes’ theorem in (21).

By virtue of (9),we can obtain from (28b) and (28c) that

and 3.3 Interaction energy between two arbitrarily shaped 3D dislocation loops

Suppose that there are two arbitrarily shaped 3D dislocation loops C and in transversely isotropic MEE bimaterials. The dislocation loop C is located in material λ and bounds a curved surface A with an extended Burgers vector b,the dislocation loop is located in material μ and bounds a curved surface with an extended Burgers vector . Based upon the classical theory of dislocations^[15],the interaction energy w_I ^[μ][λ] between the two dislocation loops can be expressed as

in which where σ_pQ ^[λ](x) and S_pQ ^[μ][λ] (x) are the two parts of the extended stress induced by the dislocation loop C,as given in (27),d _p is the pth component of the vector area element d defined at point x (x₁,x₂,x₃) on the dislocation surface . The above w_I ^[λ] is the interaction energy between the two dislocation loops and C in a transversely isotropic MEE full space occupied by material λ,and W_I ^[μ][λ] is the complementary part from the image sources due to the bimaterial interface.

Substituting (27) into (33),and then making use of (16),the Stokes theorem in (21) and the following relations:

we can thus express the interaction energy in terms of double line integrals as in which and As an immediate application of (36),we now consider one dislocation loop C with an extended Burgers vector (b1,b2,b3,0,0) and another dislocation loop ˜ C with an extended Burgers vector (₁,₂,₃,0,0),both of which are located in the transversely isotropic MEE full space occupied by material λ. In this special case,the interaction energy between the two dislocation loops can be expressed in an elegant way as in which During the derivation of (38) from (36),we have made use of (16) and (34),along with the following relations: and (40) can be verified by Stokes’ theorem in (21).

Now,we consider a single dislocation loop C with an extended Burgers vector b,which bounds a curved surface A in material λ of the transversely isotropicMEE bimaterial. According to Hirth and Lothe^[15],we can express the self-energy w_S ^[λ] of the dislocation loop C as

where in which σ_pQ ^[λ](x) and S_pQ^[μ][λ](x) are the two parts of the extended stress induced by the dislocation loop C,as given in (27); dA_p is the pth component of the vector area element dA defined at point x (x₁,x₂,x₃) on the dislocation surface A. The above w_FS^[λ] is the self-energy of the dislocation loop C in a transversely isotropic MEE full space occupied by material λ,and W_IS^[λ] is the image self-energy from the image sources due to the bimaterial interface.

Following the derivation of (36),similarly,we can obtain from (43) that

and In summary,we have presented line-integral expressions for the extended displacements, extended stresses,interaction energy,and self-energy of arbitrarily shaped 3D dislocation loops in transversely isotropic MEE bimaterials,as shown in (22),(27),(36),(44a),and (44b). These line-integral expressions are the main results of this paper. 4 Numerical examples and discussion

In this section,we utilize our dislocation solutions in Section 3 to investigate the multi-field coupling effect and the interface/surface effect in transversely isotropic MEE materials. Before presenting the numerical examples,we point out that our formulations have been verified to be correct by comparing our numerical results with those in Han and Pan^[18].

Example 1 The extended displacements and extended stresses of a cardioid dislocation loop in transversely isotropic MEE bimaterials.

In the Cartesian coordinates,we consider a transversely isotropic bimaterial which is composed of the MEE BaTiO₃-CoFe₂O₄ (material 1,x₃ > 0) and the piezoelectric BaTiO₃ (material 2,x₃ < 0),with the isotropic plane of both materials being parallel to the perfectly bonded interface (i.e.,x₃=0). The MEE BaTiO₃-CoFe₂O₄ composite is based on the 25% BaTiO₃ and 75% CoFe₂O₄. The material coefficients used here are listed as follows^[9]:

We consider a cardioid dislocation loop C₁ whose parametric equation is described by

with h being a real constant and a being the shape parameter of the cardioid (see Fig. 2). In this example,the cardioid dislocation loop is assumed to be located on the plane x₃ = h = 0.5a in material 1 (i.e.,MEE BaTiO₃-CoFe₂O₄).

Shown in Figs. 3 and 4 are the electric potential φ,magnetic potential ψ,electric displacement component D₃,and magnetic induction component B₃ on a vertical line (i.e.,x₁ = 0,x₂ = 0.5a,−4a 6 x₃ 6 5a) due to the loop C₁ with an extended Burgers vector (0,0,0,Δφ₁, 0) or (0,0,0,0,Δψ₁). Three different cases are investigated,in which the loop C₁ is located in the perfectly-bonded MEE bimaterial as mentioned above,an MEE BaTiO₃-CoFe₂O₄ half space (x₃ ≥ 0) with a free surface at x₃ = 0 (see Appendix B),and MEE BaTiO₃-CoFe₂O₄ full space. It can be observed from Figs. 3 and 4 that the curves are very sensitive to different boundary conditions except for those in Figs. 3(d) and 4(d). This is mainly caused by the fact that BaTiO₃ (i.e.,material 2) has relatively large dielectric and piezoelectric coefficients while but extremely small piezomagnetic and magnetic-permeability coefficients. In addition, Figs. 3(b),3(c),4(b),and (4c) show an obvious ME coupling effect in MEE materials.

Example 2 The interaction energy between two cardioid dislocation loops in transversely isotropic MEE bimaterials

In this example,we investigate the influence of the bimaterial interface and the half-space surface on the interaction energy between two cardioid dislocation loops C₁ and C₂ in MEE materials. The models of the MEE bimaterial,half space (x₃ ≥ 0) and full space are exactly the same as those in Example 1. The first loop C₁ with the extended Burgers vector (0,0,0, Δφ₁,0) or (0,0,0,0,Δψ₁) is also described by (45) with h = 0.5a. The second loop C₂ with the extended Burgers vector (0,0,0,Δφ₂,0) or (0,0,0,0,Δψ₂) is described by the following parametric equation:

with X₃ being independent of the parameter t. It is noted that (46) is just a simple translation relative to (45).

The numerical results for the interaction energy between loops C₁ and C₂ are shown in Fig. 5. Similarly to Figs. 3 and 4 in Example 1,the interaction energy in Fig. 5 is also sensitive to different boundary conditions,and an obvious ME coupling effect is also observed.

Example 3 The image self-energy of a cardioid dislocation loop in transversely isotropic MEE bimaterials.

In this example,the model of the MEE bimaterial is also exactly the same as in Example 1. An MEE BaTiO₃-CoFe₂O₄ half space (x₃ ≥ 0) and a piezoelectric BaTiO₃ half space (x₃≤6 0), both with a free surface at x₃ = 0,are also considered here.

We first investigate the influence of the distance between the interface/surface and the cardioid dislocation loop C₂ on its image self-energy. Here,the cardioid loop C₂ is also described by (46),with its extended Burgers vector being (0,0,0,Δφ₂,0) or (0,0,0,0,Δψ₂). The numerical results are shown in Fig. 6. It can be observed from Fig. 6 that (i) for the loop C₂ with an extended Burgers vector (0,0,0,Δφ₂,0),the perfect bimaterial interface imposes an attractive force upon this loop if it is located within MEE BaTiO₃-CoFe₂O₄ which exhibits a relatively weaker piezoelectric effect,and imposes a repulsive force upon this loop if it is located within BaTiO₃ which exhibits a relatively stronger piezoelectric effect; (ii) for the loop C₂ with an extended Burgers vector (0,0,0,0,Δψ₂),the perfect bimaterial interface imposes a repulsive force upon this loop if it is located within MEE BaTiO₃-CoFe₂O₄ which exhibits a relatively stronger piezomagnetic effect,and imposes an attractive force upon this loop if it is located within BaTiO₃ which exhibits a relatively weaker piezomagnetic effect; (iii) for the loop C₂ with an extended Burgers vector either (0,0,0,Δφ₂,0) or (0,0,0,0,Δψ₂),the free surface always imposes a repulsive force upon this loop. In other words,if the interface or surface is present,the dislocation loop with an electric-potential (or a magnetic-potential) discontinuity always has the tendency to move towards the material with a relatively stronger piezoelectric (or piezomagnetic) effect. This phenomenon in MEE materials is quite different from that for dislocation loops in purely elastic materials^[3].

Then,we investigate the influence of the size of the cardioid dislocation loop C₁ on its image self-energy. Here,the cardioid loop C₁ with an extended Burgers vector (0,0,0,Δφ₁, 0) or (0,0,0,0,Δψ₁) is also described by (45) with h = 0.5_a0 or −0.5_a0 (a₀ = const. > 0), but we change the size of this cardioid loop by simply changing its shape parameter a. The numerical results are shown in Fig. 7. It can be observed from Fig. 7 that (i) for the loop C₁ with an extended Burgers vector (0,0,0,Δφ₁,0),the perfect bimaterial interface imposes an expanding force upon this loop if it is located within MEE BaTiO₃-CoFe₂O₄ which exhibits a relatively weaker piezoelectric effect,and imposes a shrinking force upon this loop if it is located within BaTiO₃ which exhibits a relatively stronger piezoelectric effect; (ii) for the loop C₁ with an extended Burgers vector (0,0,0,0,Δψ₁),the perfect bimaterial interface imposes a shrinking force upon this loop if it is located within MEE BaTiO₃-CoFe₂O₄ which exhibits a relatively stronger piezomagnetic effect,and imposes an expanding force upon this loop if it is located within BaTiO₃ which exhibits a relatively weaker piezomagnetic effect; (iii) for the loop C₁ with an extended Burgers vector either (0,0,0,Δφ₁,0) or (0,0,0,0,Δψ₁),the free surface always imposes a shrinking force upon this loop.

5 Conclusions

In this paper,we derive simple and elegant line-integral expressions for the extended displacements, extended stresses,interaction energy,and self-energy of arbitrarily shaped 3D dislocation loops with constant extended Burgers vector in transversely isotropic MEE bimaterials. These expressions are very similar to their elastic isotropic full-space counterparts,such as Burgers’ formula for displacements^[19],Peach-Koehler’s formula for stresses^[20],and Blin’s formula for the interaction energy^[21]. Moreover,it is straightforward to reduce our line-integral expressions for MEE materials to those for piezoelectric,piezomagnetic,or purely elastic materials.

Our line-integral expressions for transversely isotropic MEE bimaterials are also suitable for a transversely isotropic MEE half space,provided that we slightly modify some coefficients of the point-force Green’s function for bimaterials,as shown in Appendix B.

All possible degenerate cases of these expressions can be treated systematically via proper limiting processes^[3]. In numerical calculations,we can also deal with these degenerate cases by means of a slight perturbation to material coefficients.

Our numerical examples show clearly the multi-field coupling and interface/surface effects on the extended displacements,extended stresses,and interaction energy of dislocation loops in the MEE materials. It is also observed from Example 3 that,for a dislocation loop with an electric-potential (or a magnetic-potential) discontinuity,the bimaterial interface would impose repulsive and shrinking forces to the dislocation loop embedded in the MEE material with a relatively stronger piezoelectric (or piezomagnetic) effect,while it would impose attractive and expanding forces to the dislocation loop embedded in the MEE material with a relatively weaker piezoelectric (or piezomagnetic) effect; however,the half-space surface would always impose repulsive and shrinking forces to the dislocation loop.

Acknowledgements The first author would like to dedicate this paper to the late Professor Y. H. PAO from College of Civil Engineering and Architecture in Zhejiang University for his supervision. Appendix A General solutions of coupled equations for transversely isotropic MEE media

Using the extended Barnett and Lothe notation^{[22, 23]},in Cartesian coordinates (x₁,x₂,x₃),the governing equations and constitutive relations for transversely isotropic MEE media can be expressed as

in which the extended body force,extended displacement,and extended stress are defined as where f_i,f_e,and f_m are the body force,electric charge,and electric current (or called magnetic charge), respectively; u_i,φ,and Ψ are the elastic displacement,electric potential,and magnetic potential, respectively; σ_ij ,D_i,and B_i are the stress,electric displacement,and magnetic induction,respectively.

Also in (A1a) and (A1b),the extended elastic coefficient tensor for transversely isotropic MEE media can be written as

where c_ijkl,ε_ij ,and μ_ij are the elastic,dielectric,and magnetic permeability coefficients,respectively; e_ijk,q_ijk,and d_ij are the piezoelectric,piezomagnetic,and magnetoelectric coefficients,respectively. If the isotropic plane of the transversely isotropic MEE medium is parallel to the x₁x₂-plane,then the above coefficients can be expressed explicitly as where c_mn,e_in,and q_in (m,n=1,2,3,4,5,6) are the contracted elastic,piezoelectric,and piezomagnetic coefficients; δ_ij is the 3×3 Kronecker delta.

According to Hou et al.^[11] and Chen et al.^[12] ,the general solution of (A1a) and (A1b) can be expressed compactly as

and where i = satisfies the quasi-harmonic equations as follows: with In (A4b),(A5),and (A6), Υ_n (n=1,2,3,4) satisfies the following characteristic equation: where In other words,Υ_n (n=1,2,3,4) are the four roots (with positive real part) of the following algebraic equation: where

Other coefficients in (A4a) and (A4b) are defined as

where Λ_n;pq (n,p,q = 1,2,3,4) are the cofactors of the pq element of the matrix Π_n defined by (A8); C_ij is the element of a symmetric matrix defined by and C_J is defined by Obviously,we have C_1j = C_j1 = C_j+2.

Notice that m_nj (n=1,2,3,4) satisfy the following relation:

in which “T” denotes the transpose of a matrix or a vector.

Moreover,we have the following relations:

which can be verified by direct substitutions. In (A14),H_n (n = 1,2,3,4) are given by (10). Appendix B Dislocation loops in transversely isotropic MEE half space with free surface

For arbitrarily shaped 3D dislocation loops in a transversely isotropic MEE half space with a free surface,the line-integral expressions given in (22),(27),(36),(44a),and (44b) are still applicable,pro- vided that we make modifications to (11a) and (11b) according to the free-surface boundary condition, and then set λ = μ=1 for the half space x₃ ≥0 while λ = μ = 2 for the half space x₃ ≤0. Here,we also assume that the isotropic plane of the transversely isotropic material is parallel to the free surface of the half space (i.e.,the x₁x₂-plane).

The free-surface boundary condition means the vanishing of the extended stresses σ_3J at x₃ = 0, which gives,corresponding to (11a) and (11b),that

and