Magnetoelectroelastic Medium Research Articles

Exact solutions of axisymmetric free vibration of transversely isotropic magnetoelectroelastic laminated circular plates

The axisymmetric free vibrations of transversely isotropic magnetoelectroelastic laminated circular plates are studied. Based on the three-dimensional governing equations of magnetoelectroelastic medium, the state space equations of laminated circular plates are obtained. By using the finite Hankel transform and rendering the free terms left by the transform in terms of the boundary quantities, the solutions of the state space equations are given for two kinds of boundary conditions. The frequency equations of the free vibration are derived using the propagator matrix method and the boundary conditions at top and bottom surfaces. By virtue of the inverse Hankel transform, the mode shapes are also determined. Since the solutions strictly satisfy the governing equations in the region and the boundary conditions at the edges, they are the three-dimensionally exact. Finally, the natural frequencies of such plates are tabulated and compared with those of the piezoelectric and elastic plates in the numerical example.

A magnetoelectroelastic medium with an elliptical cavity under combined mechanical–electric–magnetic loading

The solution for an elliptical cavity in an infinite two-dimensional magnetoelectroelastic medium subject to remotely uniformly applied combined mechanical–electric–magnetic loadings is obtained by using the Stroh formalism and the exact boundary conditions along the surface of the cavity. By letting the minor-axis of the cavity to zero the solution for a crack is deduced. A self-consistent method is proposed to calculate the real crack opening under the combined mechanical–electric–magnetic loadings. The method requires that the crack opening is the minor-axis of the elliptical opening profile. Beside the real crack solution, four different extreme models, i.e., the impermeable crack, permeable crack, electrically impermeable and magnetically permeable crack and electrically permeable and magnetically impermeable crack, are discussed. An expression of the strain energy density factor is derived. Numerical results of the strain energy density at the crack tip are given for a BaTiO 3–CoFe 2O 4 composite with the piezoelectric BaTiO 3 material being the inclusion and the magnetostrictive CoFe 2O 4 material being the matrix. The effects of the proportion of the two phases, permeability of the crack to electric and magnetic fields, the electric and magnetic loadings on the strain energy density factor are discussed.

Moving crack at the interface between two dissimilar magnetoelectroelastic materials

Analytical solutions for an anti-plane Griffith crack moving at the interface between two dissimilar magnetoelectroelastic media under the conditions of permeable crack faces are formulated using the integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic materials. Expressions for stresses, electric displacements, and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities, and it is found that the dynamic stress intensity factor (DSIF), the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF) are independent of the remote electromagnetic loads. The moving speed of the crack has influence on the DEDIF and the DMIIF. When the crack is moving at lower speeds 0 ≤ M≤ Mc1 or higher speeds Mc2 < M < 1, the crack will propagate along its original plane, while in the range of Mc1 < M < Mc2 , the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.

A moving dislocation in a magneto–electro-elastic solid

This work considers the generalized plane problem of a moving dislocation in an anisotropic elastic medium with piezoelectric, piezomagnetic and magnetoelectric effects. The closed-form expressions for the elastic, electric and magnetic fields are obtained using the extended Stroh formalism for steady-state motion. The radial components, E r and H r , of the electric and magnetic fields as well as the hoop components, D θ and B θ , of electric displacement and magnetic flux density are found to be independent of θ in a polar coordinate system. This interesting phenomenon is proven to be is a consequence of the electric and magnetic fields, electric displacement and magnetic flux density that exhibit the singularity r −1 near the dislocation core. As an illustrative example, the more explicit results for a moving dislocation in a transversely isotropic magneto–electro-elastic medium are provided and the behavior of the coupled fields is analyzed in detail.

Green's functions of magnetoelectroelastic solids with a half-plane boundary or bimaterial interface

Green's functions for magnetoelectroelastic medium with an arbitrarily oriented half-plane or bimaterial interface are presented in this paper. The derivation is based on an extended Stroh's formalism and coordinate-transform technique. In particular, a new coordinate variable is introduced to handle vertical or other boundary problems. These Green's functions satisfy related boundary or interface conditions. The Green's functions obtained can be used to establish boundary-element formulation and to analyse fracture behaviour involving half-plane boundaries or bimaterial interfaces.

Dynamic analysis of a cracked magnetoelectroelastic medium under antiplane mechanical and inplane electric and magnetic impacts

The transient analysis of a magnetoelectroelastic medium containing a crack is made under antiplane mechanical and inplane electric and magnetic impacts. The crack is assumed to penetrate through the solid along the poling direction. By using the Fourier and Laplace transforms, the associated mixed boundary value problem is reduced to a Fredholm integral equation of the second kind, which is solved numerically. By means of a numerical inversion of the Laplace transform, dynamic field intensity factors are obtained in the time domain. Numerical results are presented graphically to show the effects of the material properties and applied electric and magnetic impacts on the dynamic intensity factors of COD and stress, and dynamic energy density factors. The results indicate that except for the intensity factors of electric displacement and magnetic induction, other field intensity factors exhibit apparent transient feature. Moreover, they depend strongly on mechanical input as well as electric and magnetic impacts.

Stress concentration around a hyperboloidal notch under tension in a magnetoelectroelastic material

AbstractA transversely isotropic magnetoelectroelastic medium made of piezoelectric BaTiO3 and magnetostrictive CoFe2O4 is bounded by a single sheeted hyperboloid of revolution which is traction, electric charges, and magnetic charges free. This solid is subjected to a finite tensile force and electric and magnetic charges at infinity. A closed‐form solution based upon the potential functions approach is obtained. The present solutions include all the solutions, such as piezoelectric, piezomagnetic, magnetostrictive, and purely elastic solutions, as special cases. Numerical examples of the tensile stress at the narrowest section are presented and show that the stress concentration is significantly affected by material property and applied electric charges, and not appreciably influenced by magnetic charges.

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M c1 < M < M c2 , the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.

Macrocrack–microcrack interaction problem in magnetoelectroelastic solids

Aiming at investigating the macrocrack–microcrack interactions in magnetoelectroelastic solid, the elementary solutions for a semi-infinite macrocrack and a microcrack subjected to various loading conditions are derived first. Then by using the pseudo-traction–electric-displacement–magnetic-induction method that is motivated by the well-known pseudo-traction method [J. Mech. Phys. Solids 35 (1987) 601], the present interaction problem is reduced to a system of integral equations. After solving these equations numerically by using the Chebyshev numerical integration technique, the crack tip stress intensity factors (SIFs), the electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF) as well as the mechanical strain energy release rate (MSERR) are evaluated subsequently. The J-integral for magnetoelectroelastic medium is further deduced, and it is employed to examine the correctness of the calculation results derived. Finally, take both the stress intensity factors and the mechanical strain energy release rate as measures to investigate the influences of various loading conditions, the crack location and orientation as well as the number of microcracks on the macrocrack–microcrack interactions. It is found that both the electric and magnetic loadings have a significant influence on the fracture toughness of magnetoelectroelastic solid.

Analytical solutions of simply supported magnetoelectroelastic circular plate under uniform loads.

In this paper, the axisymmetric general solutions of transversely isotropic magnetoelectroelastic media are expressed with four harmonic displacement functions at first. Then, based on the solutions, the analytical three-dimensional solutions are provided for a simply supported magnetoelectroelastic circular plate subjected to uniform loads. Finally, the example of circular plate is presented.

Crack tip field in piezoelectric/piezomagnetic media

This paper considers the magnetoelectroelastic problem of a crack in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects. Based on the extended Stroh formalism, the general two-dimensional solutions to the magnetoelectroelastic problem are obtained, involving five analytic functions of different variables. The magnetoelectroelastic field around the crack tip is given. It contains five modes of square root singularities. Expressions of the stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived and the field intensity factors are provided. The path-independent conservative integral is derived. The energy release rate is written in terms of those field intensity factors. The explicit algebraic results are given for a special case of an anti-plane crack in a magnetoelectroelastic medium.

Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation

While the method of solution for crack problems in anisotropic elasticity and magnetoelectroelasticity is nearly identical, the prediction of crack initiation and/or growth behavior based on the asymptotic stress fields can differ widely depending on the chosen failure criterion. The latter has no haven to hide. Avoidance of contraction and satisfaction of the first law of thermodynamics are the rules that need to be observed. A negative crack tip energy release rate would infer the creation of energy and a negative strain energy density would imply the non-uniqueness of solution in classical linear continuum mechanics theories. Under these conditions, the crack initiation and growth behavior in a magnetoelectroelastic material will be examined. For the sake of notation and continuity, this work gives a brief account of the well-known solution for a line crack in a magnetoelectroelastic medium. Not to be underestimated is the derivation of the asymptotic form of the strain energy density function that was first given for the piezoelectroelastic crack problem. The equivalent expression for the magnetoelectroelastic case is given here. This sets the stage for a physical interpretation of how a crack would behave in a composite possessing the mixed properties of piezoelectric and piezomagnetic materials. The individual terms in the strain energy density function shows how the applied electric and magnetic field would affect the critical applied mechanical normal and shear stress to trigger crack initiation, respectively. When both normal and shear action prevail, the direction of crack initiation is no longer obvious and it needs to be determined since the energy stored ahead of the crack depends on the direction of prospective crack initiation. Numerical results are given for a BaTiO 3–CoFe 2O 4 composite. The inclusion is the piezoelectric BaTiO 3 material and the matrix is the magnetostrictive CoFe 2O 4 material. The proportion of the two phases can be varied by adjusting the volume fraction of the inclusions upon which the fluctuation of the energy density field near the crack depends and the material properties of the constituents. Directions of the applied electric and magnetic field can also be changed; their effects on crack initiation are discussed using the strain energy density criterion. Indeed, the additional magnetostrictive effect can have an influence on crack initiation as the applied field directions are altered. Possible behavior of crack initiation can be made for design-specific magnetoelectroelastic composites.

Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks

In this paper, we study mechanically traction-free and electromagnetically permeable crack problems in infinite magnetoelectroelastic solids with linear coupling between the elastic and electromagnetic fields. Using the Stroh-formalism, we first obtain the general solution for collinear cracks in a magnetoelectroelastic medium subjected to arbitrary loads. Then, we give specific solutions for several examples: finite or infinite number of collinear crack subjected to arbitrary remote loads, and a single crack subjected to a line load at an arbitrary point. It is found that in the most general cases, the singularity of electric-magnetic field is always dependent on that of stress. Especially when the medium is only loaded by the remote uniform field, the intensity factor of stress is the same as that of isotropic materials, and the electric-magnetic field inside any crack is uniform.

Crack problems in magnetoelectroelastic solids. Part I: exact solution of a crack

This paper presents an exact treatment on the problem of an elliptic hole or a crack in a magnetoelectroelastic solid subject to the farfield loadings. First, based on the extended version of Eshelby-Stroh’s formulation, the general solution of an elliptical hole is obtained according to exact boundary conditions at the rim of the hole. Then, when the hole degenerates into a crack, explicit solutions are given for the field intensity factors and electric–magnetic fields inside the crack. It is shown that all the singularities of fields are dependent on the applied mechanical loads, not on the applied electric–magnetic loads. Due to its explicitness, the present solution for a crack can also serve as a benchmark to test the validity of various analysis approaches or assumptions to more complicated crack problems in magnetoelectroelastic media.

Green's functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack

Based on the extended Stroh formalism combined with the technique of conformal mapping and the Laurent series expansion, Green's functions for an infinite two-dimensional anisotropic magnetoelectroelastic medium containing an elliptical cavity are obtained. The exact electromagnetic boundary conditions on the cavity surface are adopted in this analysis. When the elliptic cavity degenerates into a slit crack, the explicit, closed-form expressions for the coupled fields and the intensity factors are also provided. When the piezomagnetic and electromagnetic constants vanish, our results reduce to existing solutions of the piezoelectric solids with an elliptical cavity, which shows that our results are correct and universal.

Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials

We have studied the average magnetoelectroelastic field in a multi-inclusion or inhomogeneity embedded in an infinite matrix. The magnetoelectroelastic inclusion and inhomogeneity problems are discussed [1], and a numerical algorithm to evaluate the magnetoelectroelastic Eshelby’s tensors for the general material symmetry and ellipsoidal inclusion shape is developed. The solutions for the magnetoelectroelastic inclusion and inhomogeneity problems are applied to study the multi-inclusion and inhomogeneity problems. It is shown that the average field in an annulus surrounding an inclusion embedded in an infinite magnetoelectroelastic medium only depends on the shapes and orientations of two ellipsoids, which generalizes Tanaka and Mori's observation in elasticity [2]. The average field in a multi-inclusion is then determined exactly, from which the average field in a multi-inhomogeneity is obtained, using the equivalent-inclusion concept [3]. The solutions of multi-inclusion and inhomogeneity problems serve as basis for an averaging scheme to model the effective magnetoelectroelastic moduli of heterogeneous materials, which generalizes Nemat-Nasser and Hori's multi-inclusion model in elasticity [4]. The model is further extended to predict the effective thermal moduli of the heterogeneous magnetoelectroelastic solids, generalizing the recent work of Li on the thermal expansion coefficients of elastic composites [5]. The proposed model recovers Mori–Tanaka and self-consistent approaches as special cases. Finally, some numerical results are given to demonstrate the applicability of the model. The potential techniques to enhance the magnetoelectric effect in practical composites are also discussed.