Magnetoelectroelastic Bimaterials Research Articles

Propagation of conductive crack along interface of piezoelectric/piezomagnetic bimaterials

This paper investigates the fracture characteristics of a Yoffe conductive crack moving along the interface of piezoelectric (PE)/piezomagnetic (PM) bimaterials. By assuming that the tangential electric- and magnetic-fields along the crack surface are zero and that the speed of the moving crack is lower than the minimum shear wave speed of the bimaterial system, the considered problem can be transformed into a Riemann–Hilbert boundary value problem of vector form. Then, the singularity parameters are exactly solved for different speed regions. In contrast to the anti-plane moving crack model including impermeable and permeable crack-face assumptions along the interface of magnetoelectroelastic (MEE) bimaterials studied before, which shows inverse square-root singularity, three novel kinds of singularities are found as the speed of the moving crack is varied for the present PE/PM interface model, which can be defined as δ1,2 = − 1/2 ± ie (Case 1), δ1,2 = − 1 ± ie (Case 2) and δ1,2 = − 1/2 ± κ (Case 3), and the third parameter δ3 = − 1/2 always holds true for all three cases. Two bimaterial combinations, i.e., BaTiO3/CoFe2O4 and BaTiO3/Terfenol-D, are numerically examined. Different from the piezoelectric case, Case 3 does not appear for BaTiO3/CoFe2O4 bimaterial combination. Above all, the singularity parameters significantly depend on the speed of the moving crack and the material properties of bimaterial systems.

An isogeometric-symplectic coupling approach for fracture analysis of magnetoelectroelastic bimaterials with crack terminating at the interface

A new and novel approach that capitalizes the advantages of isogeometric analysis (IGA) and symplectic method is proposed for the mode III fracture analysis of magnetoelectroelastic bimaterials with crack terminating at the interface. In this new approach, IGA is adopted for discretizing the overall physical domain and the analytical symplectic eigenfunctions are applied in the singular field near the crack tip. New explicit expressions for singular physical field variables are derived and new intensity coefficients are introduced and determined. The physical consequences and engineering fracture implications of this innovative coupled isogeometric-symplectic approach are fully described in a few practical, numerical examples.

Singularity of subsonic and transonic crack propagations along interfaces of magnetoelectroelastic bimaterials

The unified method for addressing the propagation of subsonic and transonic cracks along the interfaces of anisotropic bimaterials in Shen and Nishioka (2000) is applied in this paper for the plain strain problem of subsonic and transonic crack propagation along the interfaces of anisotropic magnetoelectroelastic (MEE) bimaterials. Using a modified Eshelby–Stroh formalism and analytic continuation, the problem here leads to a Riemann–Hilbert problem. After solving the Riemann–Hilbert equation, near-tip asymptotic fields are obtained, which shows the properties of the subsonic and transonic crack propagation in an MEE bimaterial system. Finally, the numerical results related to different singularity powers of the crack-tip are presented and discussed for barium titanate-cobalt ferrite (BaTiO3CoFe2O4) composites. The effects of piezoelectricity and piezomagnetics are also examined.

Fracture analysis of magnetoelectroelastic bimaterials with imperfect interfaces by symplectic expansion

A Hamiltonian-based analytical method is used to study the mode III interface cracks in magnetoelectroelastic bimaterials with an imperfect interface. By introducing an unknown vector, the governing equations are reformulated in sets of first-order ordinary differential equations. Using separation of variables, eigensolutions in the symplectic space are obtained. An exact solution of the unknown vector is obtained and expressed in terms of symplectic eigensolutions. Singularities of mechanical, electric, and magnetic fields are evaluated with the generalized intensity factors. Comparisons are made to verify accuracy and stability of the proposed method. Numerical examples including mixed boundary conditions are given.

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 1: Theoretical solution

ABSTRACTThe extended displacement discontinuity boundary integral–differential equation method is developed for the analysis of an interface crack of arbitrary shape in a three-dimensional (3D), transversely isotropic magnetoelectrothermoelastic bimaterial. The extended displacement discontinuities (EDDs) include conventional displacement discontinuities, electric potential discontinuity, magnetic potential discontinuity as well as temperature discontinuity across the interface crack faces, correspondingly, while the extended stresses represent conventional mechanical stresses, electric displacement, magnetic induction, heat flux, etc. By virtue of the potential functions and Hankel transformation technique, the fundamental solutions for unit-point EDDs on the interface in a 3D transversely isotropic magnetoelectrothermoelastic bimaterial are derived, then the extended displacements and stresses are all obtained in terms of EDDs. An analysis method is proposed based on the analogy with the solution in an isotropic thermoelastic bimaterial. The singular indices and the singular behaviors of the near crack-tip fields are studied. The combined extended stress intensity factors for three new fracture modes are derived in terms of the EDDs and are compared with those in magnetoelectroelastic bimaterials.

Integral Equations of Plane Magnetoelectroelasticity for a Cracked Bimaterial With Thin Inclusions

Based on the combined application of the Stroh formalism and the theory of functions of complex variable, we deduce dual integral equations for a magnetoelectroelastic bimaterial. For the first time, we construct the integral representations of the Stroh complex potentials and the explicit expressions for all kernels in terms of the parameters and matrices of the applied formalism. This noticeably reduces the amount of computations required to get the governing equations of the boundary-element methods. The explicit formulas are obtained for the principal parts of the complex potentials. These formulas enable us to consider homogeneous magnetoelectromechanical loads applied at infinity. The obtained equations, together with previously developed models of thin deformable inclusions, are introduced in the computational algorithm of the boundary-element method with jump functions. The solution of test problems reveals high accuracy and efficiency of the proposed approach. Some solutions are presented for new problems posed for a magnetoelectroelastic bimaterial with thin inclusion.

Crack tip enrichment functions for extended finite element analysis of two-dimensional interface cracks in anisotropic magnetoelectroelastic bimaterials

In this paper, the extended finite element method (X-FEM) is employed to present a static fracture analysis of two-dimensional interfacial crack problems in linear magnetoelectroelastic (MEE) bimaterials. Magnetoelectrically impermeable crack-face boundary conditions are adopted and the multi-field coupled effect in MEE bodies is considered. In order to capture the oscillating singularity of the extended stresses near the interfacial crack tip, suitable crack tip enrichment functions for anisotropic and transversely isotropic MEE bimaterials are newly derived and further applied to perform X-FEM analysis. As the fracture parameter, the J-integral is evaluated using the domain form of the contour integral. By comparing yielded results with the analytical and numerical solutions of the corresponding interfacial crack problems, the validity of the proposed formulation is verified. Moreover, it is shown that the results obtained by way of the new enrichment functions are superior to those obtained by the fourfold enrichment functions and twelvefold enrichment functions, especially in the case of topological enrichment. If there is no special requirement for precision, the fourfold enrichment functions with less computational cost can also be used to conduct X-FEM analysis for the present problem.

Integral identities based on symmetric and skew-symmetric weight functions for a semi-infinite interfacial crack in anisotropic magnetoelectroelastic bimaterials

In this paper, we address a semi-infinite interfacial crack problem in an anisotropic magnetoelectroelastic (MEE) bimaterial system subjected to a magnetoelectromechanical asymmetric load on the crack surface. First, the symmetric and skew-symmetric weight functions are derived for a two-dimensional (2-D) deformation problem. Using these weight functions and extending the Betti formula to MEE materials, the integral identities are further obtained and the present crack problem is formulated in terms of singular integral equations, which establish the relationship between the applied external load and the generalized displacement jump across the crack faces. The illustrative examples in relation to Mode III, and Mode I and Mode II problems show that the method developed in this study avoids the use of Green's function and is very convenient for the fracture analysis of MEE solids, in which a multi-field coupled effect is observed.

2D fracture analysis of magnetoelectroelastic composites by the SBFEM

This paper extends the scaled boundary finite element method (SBFEM) to 2D fracture analysis of magnetoelectroelastic composites. The generalized extended stress intensity factors, i.e., the stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor, for different crack configurations and material combinations are evaluated. The governing equations of the SBFEM for magnetoelectroelastic materials are derived based on the virtual work principle. The generalized extended stress intensity factors are determined from the solution of the singular stress, electric displacement and magnetic induction fields by extending the definition of the generalized stress intensity factors. The electrically and magnetically impermeable and permeable crack face boundary conditions are implemented and discussed. Numerical examples including the cracks in magnetoelectroelastic materials, the interface cracks between magnetoelectroelastic bimaterials and magnetoelectroelastic bimaterial wedge are presented to validate the proposed technique. The accuracy of the proposed technique is verified by comparison with the reference results.

Green’s functions and extended displacement discontinuity method for interfacial cracks in three-dimensional transversely isotropic magneto-electro-elastic bi-materials

In this paper, we derive the Green’s functions of constant extended interfacial displacement discontinuities within a rectangular element and of point extended interfacial displacement discontinuities in three-dimensional transversely isotropic magneto-electro-elastic (MEE) bi-materials. The derived Green’s functions along with the extended displacement discontinuity method are applied to analyze the electrically and magnetically impermeable interfacial cracks in the three-dimensional MEE bi-materials. To deal with the oscillatory singularities at the crack front, the Dirac delta function in the Green’s functions is replaced by the Gaussian distribution function, and correspondingly, the unit Heaviside function is approximated by the Error function. Numerical study illustrates the effect of the ε parameter in the Gaussian distribution function on the J-integral. The stress intensity factors, electric displacement intensity factor, and magnetic induction intensity factor are expressed in terms of the extended displacement discontinuities. The influence of different MEE material mismatches as well as different extended loadings (uniformly or non-uniformly distributed on the crack face) on the fracture parameters is investigated. Different rectangular crack sizes are also considered in the numerical simulation.

Analytical solutions of 3D anisotropic magneto-electro-elastic bi-materials under extended non-uniform dislocations and tractions over a circular area

In this paper, we derive the analytical solutions in a three-dimensional magneto-electro-elastic (MEE) bi-material under extended dislocations and tractions generally distributed over a horizontal circular area by virtue of the extended Stroh formalism and Fourier transformation. Explicit analytic results of the Hankel transform integrals of different orders and three kernel integrals are obtained so that the final solutions in the physical domain can be expressed in simple line-integral form. Some existing solutions in MEE materials can be reduced directly from the general solutions presented in this paper. The effect of different non-uniform loadings (dislocations and tractions) on the induced fields in MEE bi-material is investigated. Our analytical solutions can serve as benchmarks for future numerical analysis of various problems in MEE bi-materials.

Evaluation of mode III interface cracks in magnetoelectroelastic bimaterials by symplectic expansion

The mode III interface crack between two dissimilar magnetoelectroelastic materials is studied by means of symplectic expansion. Four crack surface electromagnetic assumptions, that is, electrically and magnetically permeable, electrically permeable and magnetically impermeable, electrically impermeable and magnetically permeable, and electrically and magnetically impermeable, are taken into account. In the symplectic space, stress, electric displacement, and magnetic induction are found to be the dual variables of the displacement, electric potential, and magnetic potential. The Hamiltonian governing equations that allow the use of the method of separation of variables are formulated by employing the variational principle of mixed energy. Expressions for the mechanical, electric, and magnetic fields are derived explicitly in the series of symplectic eigensolutions with coefficients to be determined by the boundary conditions, and so are the fracture parameters, such as field intensity factors and energy release rates for dissimilar bimaterials. Numerical results are provided to verify the validity of the present method and reveal the influences of the material constants and boundary conditions.

Transient dynamic interface crack analysis in magnetoelectroelastic bi-materials by a time-domain BEM

Abstract This paper aims to gain a deeper insight into the effects of the poling directions, the volume fractions and the combined loading parameters on the dynamic field intensity factors at the interfacial crack-tips in two dimensional, piecewise homogeneous and magnetoelectroelastic composite bi-materials under the combined magnetoelectromechanical impact loadings. For this purpose, a time-domain boundary element method (BEM) together with the sub-domain technique is applied. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the interfacial crack-tips. An explicit displacement extrapolating formula is proposed to determine the dynamic field intensity factors of stresses, electrical displacements and magnetic inductions at the interfacial crack-tips. Numerical examples for different poling directions, different volume fractions and different combined loading parameters are presented and discussed. Based on these numerical results, some useful conclusions are drawn on the effects of various material and loading combinations on the dynamic field intensity factors.

Dislocation and traction loads over an elliptical region in anisotropic magnetoelectroelastic bimaterials

In indentation tests of material properties and in the analysis of the interaction between a structure and its foundation, the solution for a uniform loading over an elliptical area in a half-space is essential. Thus, in this paper, we derive the analytical solution for a general magnetoelectroelastic bimaterial system under the action of extended traction and dislocation uniformly distributed over a horizontal ellipse. The solution is obtained by making use of two-dimensional Fourier transformation combined with the Stroh formalism. To deal with the elliptical shape, a simple scale transformation technique is applied. As such, our solution is very general and contains various decoupled material systems and reduced material domains (infinite and half-space) as special cases. As numerical examples, a bimaterial system made of BaTiO3/CoFe2O4 is studied under both traction and dislocation loads within the elliptical area with various semi-axes ratios. It is shown that the induced field due to traction is smoother than that due to dislocation and that both the elliptical semi-axes ratios and material orientation can significantly influence the induced elastic, electric and magnetic fields.

Fields induced by three-dimensional dislocation loops in anisotropic magneto-electro-elastic bimaterials

The coupled elastic, electric and magnetic fields produced by an arbitrarily shaped three-dimensional dislocation loop in general anisotropic magneto-electro-elastic (MEE) bimaterials are derived. First, we develop line-integral expressions for the fields induced by a general dislocation loop. Then, we obtain analytical solutions for the fields, including the extended Peach–Koehler force, due to some useful dislocation segments such as straight line and elliptic arc. The present solutions contain the piezoelectric, piezomagnetic and purely elastic solutions as special cases. As numerical examples, the fields induced by a square and an elliptic dislocation loop in MEE bimaterials are studied. Our numerical results show the coupling effects among different fields, along with various interesting features associated with the dislocation and interface.

Analytical solutions of uniform extended dislocations and tractions over a circular area in anisotropic magnetoelectroelastic bimaterials

In this paper, we derive the analytical solutions in a three-dimensional anisotropic magnetoelectroelastic bimaterial space subject to uniform extended dislocations and tractions within a horizontal circular area. By virtue of the Stroh formalism and Fourier transformation, the final expression of solutions in the physical domain contains only line integrals over [0, 2π] rather than infinite integrals. As the reduced cases, the half-space and homogeneous full-space solutions can be directly derived from the present solutions. Also, in terms of material domains, the present solutions can be reduced to the piezoelectric, piezomagnetic, purely elastic materials with different symmetries of material property. To carry out numerical calculations, Gauss quadrature is adopted. In the numerical examples, the effect of different loading locations on the response at the interface is analyzed. It is shown that, when the magnetic traction or electric dislocation is applied, the physical quantities on the interface may not decrease monotonically as the loading area moves away from the interface. The distributions of different in-plane physical quantities on the upper and lower interfaces under various extended horizontal loadings are compared and the differences are discussed. The work presented in this paper can serve as benchmarks for future numerical studies in related research fields.

An electrically impermeable and magnetically permeable interface crack with a contact zone in magnetoelectroelastic bimaterials under a thermal flux and magnetoelectromechanical loads

An interface crack with a frictionless contact zone at the right crack-tip between two dissimilar magnetoelectroelastic materials under the action of a thermal flux and remote magnetoelectromechanical loads is considered. The open part of the crack is assumed to be electrically impermeable and magnetically permeable, and the crack faces are assumed to be heat insulted. The inhomogeneous combined Dirichlet–Riemann and Hilbert boundary value problems are, respectively, formulated and solved analytically. Stress, electrical displacement intensity factors as well as energy release rate are found in analytical forms, and analytical expressions for the contact zone length have been obtained for both the general case and the case of small contact zone length. Some numerical results are presented, which show clearly the effects of thermal and magnetoelectromechanical loads on the contact zone length, stress intensity factor and energy release rate. Results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic structures and devices.

A mode III crack crossing the magnetoelectroelastic bimaterial interface under concentrated magnetoelectromechanical loads

A mode III crack cutting perpendicularly across the interface between two dissimilar semi-infinite magnetoelectroelastic solid is studied under the combined loads of a line force, a line electric charge and a line magnetic charge at an arbitrary location. The impermeable conditions are implied on the crack faces. The technique developed in literature for the elastic bimaterial with a crack cutting interface is exploited to treat the magnetoelectroelastic bimaterial. The Riemann-Hilbert problem can be formulated and solved based on complex variable method. Analytical solutions can be obtained for the entire plane. The intensity factors around crack tips can be defined for the elastic, electric and magnetic fields. It shows that, no matter where the load position is, the electric displacement intensity factors (EDIFs), as well as the magnetic induction intensity factors (MIIFs), are identical in magnitude but opposite in sign for both crack tips, on condition that a line force is solely applied. Alternatively, if only a line electric charge is considered, then the stress intensity factors (SIFs) and the MIIFs exhibit the behavior. Likewise, if only a line magnetic charge is applied, it turns to the SIFs and the EDIFs instead. In addition, the dependence of the intensity factors is graphically shown with respect to the location of a line force. It is found that the SIF for a crack tip tends to be infinite if the applied force is approaching the tip itself, but the EDIF, with the complete opposite trend, tends to be vanishing. Finally, focusing on the more practical case of piezoelectric/piezomagnetic bimaterial, variation of the SIF along with the moduli as well as the piezo constitutive coefficients is explored. These analyses may provide some guidance for material selection by minimizing the SIF. It is also believed that the results obtained in this paper can serve as the Green’s function for the dissimilar magnetoelectroelastic semi-infinite bimaterial with a crack cutting the interface under general magnetoelectromechanical loads.

Image singularities of planar magnetoelectroelastic bimaterials for generalized line forces and edge dislocations

Abstract This study presents two-dimensional explicit full-field solutions of transversely isotropic magnetoelectroelastic bimaterials subjected to generalized line forces and edge dislocations using the Fourier-transform technique. One of the major objectives of this study is to analyze the physical meaning and the structure of the solution. Complete solutions for this problem consist only of the simplest solutions for an infinite medium. The solutions include Green's function of originally applied singularities in an infinite medium and thirty-two image singularities which are induced to satisfy interface continuity conditions. It is shown that the physical meaning of the solution is the image method. The mathematical method used in this study provides an automatic determination for the locations and magnitudes of image singularities. The locations and magnitudes of image singularities are dependent on the roots of the characteristic equation for bimaterials. The number and distribution for image singularities are discussed according to characteristic roots features. With the aid of the generalized Peach–Koehler formula, the explicit expressions of image forces acting on generalized edge dislocations are easily derived from the full-field solutions of the generalized stresses. Numerical results for the full-field distributions of stresses, electric fields, and magnetic fields in bimaterials are presented. The image forces and equilibrium positions of one dislocation, two dislocations, and an array of dislocations are presented by numerical calculations and are discussed in detail.

Theoretical analysis of in-plane problem in functionally graded nonhomogeneous magnetoelectroelastic bimaterials

By using the Fourier-transform technique, the two-dimensional full-field solutions of transversely isotropic functionally graded magnetoelectroelastic bimaterials subjected to generalized line forces and edge dislocations are presented in this study. For the nonhomogeneous problem, the mathematical derivation is much complicated than the homogeneous case since the material properties vary with coordinate. From the framework of the generalized stress function approach, only the generalized strain compatibility equations needed to be satisfied. A powerful analytical method is developed to solve the functionally graded magnetoelectroelastic planar problem. For the special case of the nonhomogeneous bimaterial with continuous material properties at the interface, it is shown in this study that all magnetoelectroelastic fields are continuous at the interface. Furthermore, this functionally graded bimaterial has the identical contour slopes for the generalized stresses (except for σ xx ( j ) ) across the interface. For the computational results, the full-field distributions of generalized stresses and strains in the nonhomogeneous bimaterial subjected to line forces or edge dislocations are presented with different functionally graded parameters.