1 Introduction

Due to the intrinsic magneto-electro-mechanical coupling behavior, dissimilar magnetoelectroelastic (MEE) bimaterials or layered MEE composites which are composed of piezoelectric and magnetostrictive materials have been widely used in smart devices and structures such as transducers, magnetic field probes, medical ultrasonic imaging, and actuators^[1]. However, it is well known that MEE materials are very brittle and susceptible to fracture.Debondings and cracks occur at the interface and lead to failures of the devices either during fabrication or in service. Therefore, it is necessary to find a way to enhance the smart structures.

A large number of publications on the fracture analysis of MEE materials and structures have appeared in the recent literature^[2-22]. However, most of them focused on the infinite cracked body which cannot develop an effective way to enhance the finite-size devices or structures in practice. In the study of brittle ceramics, Lawn et al.^[23] found out that the imperfect interface (weak bonding interface) could increase the fracture resistance of bimaterials or layered structures. This phenomenon provided a way of designing fracture-resistant MEE materials. Thereafter, piezoelectric or MEE materials with imperfect interface have attracted attention from researchers^[24-30]. Comparing with the large number of studies on perfect bonding bimaterials, only a few studies have been carried out on the crack at a weak interface in MEE materials.Li et al.^[31-34] and Li and Lee^[35-36]proposed four models of the bimaterial interface and analyzed the fracture behaviors of functionally gradient (FG) MEE materials with imperfect interfaces. Wang^[37] studied the dynamic electromechanical behavior of a triple-layer piezoelectric composite cylinder with imperfect interface using a linear spring model to describe the weakness of imperfect interface. Li et al.^[38]presented a new model with six coefficients to formulate three kinds of imperfections in the anti-plane deformation case of multiferroic composites based on a classical spring model.

In view of the existing works outlined above, it is found that the studies on the cracked MEE structures with imperfect interface were very limited. It is necessary to develop new analytical methods to reveal the fracture mechanism and to guide the design. Motivated by this reason, we present an analytical symplectic approach for the mode Ⅲ fracture analysis of MEE bimaterials with imperfect interfaces. The symplectic method was first proposed by Yao et al.^[39] and Lim and Xu^[40], which was developed for some basic problems in solid mechanics and elasticity which have long been bottlenecks. In recent years, the research works on the symplectic method have been extended to many new areas, e.g., bending of plates^[41-43], free vibration of plates^[44-46], buckling of shells^[47-49], linear fracture mechanics^[50-59], and other elastic problems^[60-62]. Unlike the conventional semi-inverse methods, the new symplectic approach serves as a completely rational and accurate model in analyzing the fracture problems. The high-order partial differential governing equations are reduced to a set of ordinary differential equations in the symplectic space by introducing an unknown vector. The singular variables are represented by a series of symplectic eigenfunctions, and fracture parameters are obtained simultaneously.

This paper is organized as follows. The basic equations are presented in Section 2. The Hamiltonian dual governing equation is established in Section3. The symplectic eigensolutions are derived in Section 4. The complete solution is given in Section 5. Numerical examples are performed in Section6.

2 Fundamental equations

An MEE bimaterial with an interface crack is considered in Fig. 1.A cylindrical coordinate system (r, θ, z) with origin at the crack tip is selected. The crack faces are at θ = ± π, and the imperfect interface is at θ =0. The upper and lower material elements are denoted by Material 1 (Ω⁽¹⁾) and Material 2 (Ω⁽²⁾), respectively.

The constitutive equations for the MEE bimaterial in the Cartesian coordinate system are

(1)

(2)

(3)

where the subscript n=1, 2 stands for the nth material element, σ_ij⁽ⁿ⁾, D_i⁽ⁿ⁾, B_i⁽ⁿ⁾, ε_kl⁽ⁿ⁾, E_l⁽ⁿ⁾, and H_l⁽ⁿ⁾ are the stress, electric displacement, magnetic induction, strain, electric field, and magnetic field, respectively, C_ijkl⁽ⁿ⁾, e_lij⁽ⁿ⁾, h_lij⁽ⁿ⁾, χ_il⁽ⁿ⁾, κ_il⁽ⁿ⁾, and g_il⁽ⁿ⁾ denote the shear modulus, piezoelectric constant, piezomagnetic constant, magnetic permeability, dielectric constant, and electromagnetic constant, respectively, u_i, Φ, and ψ are the elastic displacements, electric potential, and magnetic potential, respectively, and f_b⁽ⁿ⁾, f_e⁽ⁿ⁾, and f_c⁽ⁿ⁾ are the body force, electric charge density, and electric current density, respectively.

For the anti-plane problem, only the out-of-plane displacement, in-plane electric field, and in-plane magnetic field are taken into account. Then, (1) -(3) can be rewritten as

(4)

(5)

(6)

where k=r, θ, and M⁽ⁿ⁾ is the matrix of material constants (Appendix A (A1)).

For mathematical modeling, the imperfect interface is modeled as a set of linear springs^[63], and the boundary conditions at θ =0 are described as

(7)

where k(r) denotes the interfacial parameter characterizing the interfacial imperfection. It should be pointed out that k(r)=∞ and k(r)=0 represent the perfect bonding and debonding, respectively. Assume that the crack faces at θ = ±π are traction free, i.e.,

(8)

Four types of ideal electromagnetic assumptions are considered as follows.

Case 1   Electrically impermeable and magnetically impermeable crack

(9)

Case 2   Electrically permeable and magnetically impermeable crack

(10)

Case 3   Electrically impermeable and magnetically permeable crack

(11)

Case 4   Electrically permeable and magnetically permeable crack

(12)

3 Hamiltonian system and governing equations

To transform the Lagrangian into Hamiltonian, a generalized coordinate ξ = ln r is introduced in analogy to the time coordinate in Hamiltonian mechanics. Define the original variable q and its dual variable p as

(13)

The governing equation in the Hamiltonian system can be expressed as^[51]

(14)

where ψ⁽ⁱ⁾=[(q⁽ⁱ⁾)^T, (p⁽ⁱ⁾)^T]^T is the total unknown vector, and H⁽ⁿ⁾ and f⁽ⁿ⁾ are given by

(15)

Making use of the original variable q, the imperfect interface condition (7) can be rewritten as

(16)

where g_w⁽ⁿ⁾(n=1, 2) are expressed in Appendix A (A2).The associated crack-face conditions (9) -(12) can also be rewritten in the Hamiltonian form, i.e.,

(17)

where g_L⁽ⁿ⁾(n=1, 2; L=A, B, C, D) are listed in Appendix A ((A3) -(A6)).

Here, it should be pointed out that there are two important properties of the Hamiltonian operator matrix.

(ⅰ) The symplectic eigenfunctions can be divided into two groups of α and β,

(18)

(19)

(ⅱ) The symplectic eigenfunctions satisfy the adjoint symplectic orthonormal relations,

(20)

where is a unit symplectic matrix,

is an inner product operation, I is a unit identity matrix, and δ_jk is the Kronecker delta.

The solution to (14) can be represented by the superposition of the complementary solution ψ_c to the homogeneous part and the particular solution ψ_p which satisfy the non-homogeneous part. The particular solution has been solved in the authors' previous work^[55-56].The remaining complementary solution will be derived in the following section.

4 Eigenvalues and eigensolutions

In the symplectic space, the eigenfunctions consist of zero eigensolutions and non-zero eigensolutions and can be solved similar to Refs. [51] and [55].

The zero eigensolutions (μ =0) are

(21)

(22)

∆_ij⁽ⁿ⁾ can be found in Appendix A (A7). Here, (21) and (22) are the Saint-Venant type solutions. It is obvious that(21) is the rigid translation along the z-coordinate, constant electric potential, and constant magnetic potential, and (22) is the uniform shear stress, uniform electric displacement, and uniform magnetic induction.

The non-zero eigensolutions (μ ≠ 0) are

(23)

where are undetermined coefficients for Ω⁽ⁿ⁾. The relations between Λ⁽¹⁾ and Λ⁽²⁾ can be obtained by (16), i.e.,

(24)

where are listed in Appendix A (A8). Consequently, only Λ_ij⁽ⁿ⁾ are independent undetermined coefficients which can be determined by the crack conditions. Substituting (23) into (17) yields

(25)

where are listed in Appendix A (A9) -(A12). Making use of the condition of non-trivial-solutions to (25), the determinant of Z_L vanishes, that is,

(26)

where k=3, l=2 for Case 1, k=4, l=1 for Cases 2 and 3, k=5, l=0 for Case 4, and

(27)

The corresponding eigenvalues are

(28)

(29)

(30)

(31)

where n=1, 2, …, μ₁ < 0.5 and can be determined by(27). From (28) -(31), it can be concluded that μ=μ₁ corresponds to the singularity of the mechanical field which is raised by the imperfect interface, while μ =0.5 corresponds to the singularity of electric and magnetic fields.

The eigensolution ψ_j⁽ⁿ⁾ for each material element Ω⁽ⁱ⁾ is obtained by (25) so that the eigensolution for the overall MEE bimaterial can be expressed as . These non-zero eigensolutions are usually neglected by the Saint-Venant principle as they represent self-equilibrating and localized configurations. In particular, the eigensolutions which satisfy 0 < μ < 1 represent the singular fields near the crack-tip.

Therefore, the complementary solution is obtained as follows:

(32)

where a_{0, j}, b_{0, j}, b_j^(α), and b_j^(β) are undetermined coefficients, and N is the number of non-zero eigensolutions taken into account.

5 Outer boundary conditions

The remaining coefficients in (32) will be solved by considering a circular domain. The non-circular domain can be obtained by a least-square method^[64]. It is clear that ψ_{0, j}^(β) in (22) and ψ_j^(β) in(23) lead to singularities of the generalized displacements at r=0so that they should be neglected in this study. The complementary solution (32) can be simplified in the form of

(33)

where ψ_j is composed of ψ_{0, j}^(α) and ψ_j^(α), and c_j is composed of a_{0, j}, b_{0, j}, b_j^(α), and b_j^(β).

Both the given generalized displacement boundary ∂ Ω_D and the given generalized stress boundary ∂ Ω_S at r=r₀ are taken into consideration, i.e.,

(34)

By using (33), (34) can be rewritten as

(35)

where . According to the symplectic adjoint orthogonal relations (20), the unknown coefficients c_j can be determined by a set of linear algebraic equations,

(36)

where

Finally, the complementary solution (33) is achieved. In the numerical examples, we usually take all zero eigensolutions and the first N terms of non-zero eigensolutions. Therefore, there are N+3 unknown c_j and N+3 algebraic equations. In this way, the approximate solution can be obtained.

6 Numerical examples

Three examples are presented for the fracture behaviors of MEE bimaterial with an imperfect interface. The singularities of mechanical, electric, and magnetic fields are evaluated by the generalized field intensity factors which are defined as^[51]

(37)

(38)

where 0 < μ < 1, K_S, K_E, and K_H are the generalized strain, electric field, and magnetic field intensity factors, respectively, and K₃, K_D, and K_B are the generalized stress intensity factors (SIFs), electric displacement intensity factors (EDIFs), and magnetic induction intensity factors (MIIFs), respectively. The material constants are selected in Table 1.

6.1 Validation and convergence study

A bimaterial MEE circular disk with an edge crack is considered in Fig. 2. The disk is subjected to concentrated loads on its circumference (a pair of out of-plane concentrated forces P_r, in-plane charges Q_r, and in-plane magnetic inductions B_r).The crack length, load angle, and interface parameter are taken as r₀, θ₀ and k, respectively.

To verify accuracy and convergence of the symplectic method, the non-dimensional generalized intensity factors are introduced as K_S =K_S /r₀^(1-μ), K_E =K_E / r₀^(1-μ), K_H=K_H/r₀^(1-μ), K₃ =K₃ /(C₄₄⁽¹⁾ r₀^(1-μ)), K_D =K_D/(C₄₄⁽¹⁾ r₀^(1-μ) × 10^-10), and K_B =K_B /(C₄₄⁽¹⁾r₀^(1-μ) × 10^-10). Assuming P_r =1, Q_r =1, B_r =0, θ₀ = ± π, and Materials 1 and 2 as MEE-1, the MEE bimaterial is simplified into a purely piezoelectric medium in which the magnetic quantities are made to vanish. The variations of SIFs and EDIFs versus the number of symplectic eigensolutions N for the electromagnetic permeable crack (Cases 2 and 4) are plotted in Fig. 3. The results of the perfect interface (k=∞) are compared with those reported by Liu and Chue^[17], and excellent agreement is observed. It implies that the symplectic method is sufficiently accurate. Furthermore, Fig. 3 shows that N does not have a significant effect on the intensity factors when N≥6.For more accurate results, 10 terms of symplectic eigensolutions or more are taken to compute the intensity factors.

6.2 Effects of interfacial parameter

After validating accuracy of the symplectic method, the influence of concentrated loads on the generalized SIFs is investigated. The model and calculating parameters are the same as those in Subsection6.1. From (28) -(31), the eigenvalue μ can be divided into two groups, μ < 0.5 and μ =0.5. Therefore, there are also two groups of generalized intensity factors. Three combinations of concentrated loads are considered as follows:

(ⅰ) P_r =1, Q_r =0, B_r =0; (ⅱ) P_r =0, Q_r =1, B_r =0;(ⅲ) P_r =0, Q_r =0, B_r =1.

Let Materials 1 and 2 be MEE-1 and MEE-2, respectively. The characteristics of generalized SIFs, EDIFs, and MIIFs for θ₀= ± π are tabulated in Table 2. It is noted that the generalized SIFs of μ =0.5 are absent for all types of crack-face assumptions. It can be deduced that the weak bonding interface changes the singularity of the mechanical field. It is also worth noting that the concentrated mechanical load is the major influencing factor on the singularities of the MEE bimaterial, because it leads to the singularities of the mechanical, electric, and magnetic fields simultaneously, while the electromagnetic loads only produce the singular electromagnetic fields. For a better illustration, the influence of the interface parameter k on the generalized SIFs under load combination (ⅰ) is shown in Fig. 4. It is interesting to find that only the generalized intensity factors of μ < 0.5 increase with the increase of k, while those with μ =0.5 remain unchanged. The observations are in accord with the expressions of eigenvalues ((28) -(31)). It indicates that the eigenvalue μ < 0.5 is caused by the imperfect interface.Moreover, it is noted that, compared with the results of the perfect interface (k=∞), the singularity of stress for the imperfect interface crack is improved by the interface parameter k. An appropriate k may prevent crack initiation or crack propagation.

6.3 Mixed boundary

To illustrate the applicability of the mixed boundary conditions, the mixed boundary including both given generalized displacement and stress boundaries are considered here. Figure 5 depicts an edge-cracked bimaterial MEE circular disk subject to a pair of concentrated loads P_r =1 and magnetic field H=1. The interface parameter is taken as k=1. Materials 1 and 2 are selected as MEE-1 and MEE-2, respectively.

The complex mixed loads are computed by the symplectic method, and variations of generalized SIFs, EDIFs, and MIIFs are plotted in Fig. 6. It is seen that the generalized intensity factors of μ < 0.5 have similar variation trends but different values at specific Ω₀ (see Figs. 6(a), (c), (e)), while those of μ =0.5 are quite different from each other. It indicates that the mixed boundary has more significant influence on the generalized intensity factors of μ =0.5 which is raised by the interface crack. Unlike the continuous boundary in Subsection6.2, the mixed boundary may lead to electromagnetic failure.

7 Conclusions

In this paper, the edge-cracked MEE bimaterials with imperfect interfaces are studied by an analytical symplectic approach. The spring model is taken to describe the imperfect interface. Closed form solutions of mechanical, electric, and magnetic fields and generalized intensity factors for four types of crack surface assumptions are derived by means of symplectic eigensolutions.Several conclusions can be made based on the results obtained in numerical examples. (ⅰ) The eigenvalues which are related to the singular orders can be divided into two groups, i.e., μ < 0.5 which is raised by the imperfect interface and μ =0.5 corresponding to the singularities of electric and magnetic fields.(ⅱ) The generalized SIFs, EDIFs, and MIIFs with μ < 0.5 are proportional to the interface parameter, while those with μ =0.5 are independent of the interface parameter. (ⅲ) The present method is suitable for both the continuous boundary and the mixed boundary, and the singularities of the MEE bimaterial can be improved by adjusting the applied loads, which can be used as a guideline for design of such fracture-resistant MEE structures.

Appendix A

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

where ∆₁₁⁽ⁱ⁾ =-∆₁⁽ⁱ⁾/(2π ∆⁽ⁱ⁾), ∆₁₂⁽ⁱ⁾ =-∆₄⁽ⁱ⁾/(2π ∆⁽ⁱ⁾), ∆₁₃⁽ⁱ⁾ =-∆₅⁽ⁱ⁾/(2π ∆⁽ⁱ⁾), ∆₂₂⁽ⁱ⁾ =-∆₂⁽ⁱ⁾ /(2π ∆⁽ⁱ⁾), ∆₂₃⁽ⁱ⁾ =-∆₆⁽ⁱ⁾ /(2π ∆⁽ⁱ⁾), and ∆₃₃⁽ⁱ⁾ =-∆₃⁽ⁱ⁾ /(2π ∆⁽ⁱ⁾), in which ∆₁⁽ⁱ⁾ =-κ₁₁⁽ⁱ⁾ χ₁₁⁽ⁱ⁾ +(g₁₁⁽ⁱ⁾)², ∆₂⁽ⁱ⁾ =C₄₄⁽ⁱ⁾χ₁₁⁽ⁱ⁾ +(h₁₅⁽ⁱ⁾)², ∆₃⁽ⁱ⁾ =C₄₄⁽ⁱ⁾ κ₁₁⁽ⁱ⁾ +(e₁₅⁽ⁱ⁾)², ∆₄⁽ⁱ⁾=-e₁₅⁽ⁱ⁾ χ₁₁⁽ⁱ⁾ +h₁₅⁽ⁱ⁾ g₁₁⁽ⁱ⁾, ∆₅⁽ⁱ⁾ =e₁₅⁽ⁱ⁾ g₁₁⁽ⁱ⁾ -h₁₅⁽ⁱ⁾κ₁₁⁽ⁱ⁾, ∆₆⁽ⁱ⁾ =-C₄₄⁽ⁱ⁾ g₁₁⁽ⁱ⁾ -e₁₅⁽ⁱ⁾ h₁₅⁽ⁱ⁾, and ∆⁽ⁱ⁾=det(Mⁱ).

(A8)

(A9)

(A10)

(A11)

(A12)