Closed-form solutions for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix

This paper presents two different analytical methods to investigate the magneto-mechanical coupling effect for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix. First, the magnetoelastic solution is expressed in terms of magnetoelastic Green's function that can be decoupled into elastic Green's function and magnetic Green's function. Second, the problem is analyzed by the equivalent inclusion method, and then, the formulation of the inhomogeneity problem can be decoupled into an elastic problem and a magnetic inhomogeneity problem connected by some eigenstrain and eigenmagnetic fields. For the piezomagnetic composites with a non-piezomagnetic matrix, these two solutions are completely equivalent each other though they are obtained by means of two different methods. Moreover, based upon the unified energy method, the effective magnetoelastic moduli of the composites are expressed explicitly in terms of phase properties and volume fractions. Then the dilute and Mori–Tanaka schemes are discussed, respectively. Finally, the calculations are made to predict the effective magnetoelastic moduli and illustrate the performance of each model.     

The Stress State of a Ferromagnetic with a Spheroidal Inclusion in a Homogeneous Magnetic Field

The stress–strain state of an infinite isotropic magnetically soft ferromagnetic body with a spheroidal inclusion is analyzed. It is assumed that the body is in an external magnetic field. The basic stress–strain characteristics and the induced magnetic field near and inside the inclusion are analyzed. The plots and the table presented show how the total magnetoelastic and Maxwell stresses near and inside the inclusion depend on the ratio of the spheroid axes, the latitude angle, and the magnetic induction when the medium and the inclusion are dissimilar materials.     

On multiple circular inclusions in plane magnetoelasticity

A general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper. By the use of complex variable theory and Laurent series expansion method, the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived. Expanding the definition of the Airy’s stress function of pure elastic field into the magnetoelastic field and applying the superposition method, the general expression then can be reduced to a set of linear algebraic equations and solved in a series form. An approximate closed form solution for the case of two arbitrarily located inclusions is also provided. For illustrating the effect of the pertinent parameters, the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form.     

Stress Concentration in a Soft Ferromagnetic with an Elliptic Inclusion in a Homogeneous Magnetic Field

The paper addresses a stress–strain problem for an infinite soft ferromagnetic body with an elliptic inclusion. The body is in a homogeneous magnetic field B ₀₁. The basic stress–strain characteristics and induced magnetic field in the body and inclusion are determined and their features in the neighborhood of the inclusion are studied. The magnetoelastic and Maxwell stresses are plotted against the ratio of ellipse axes and the latitude angle. Maximum stresses versus magnetic induction and mechanical and magnetic properties of the material are tabulated     

A Magnetoelastic Field in a Ferromagnetic with an Elliptic Inclusion

A stress–strain problem is solved for an infinite isotropic magnetically soft body containing an elliptic inclusion. It is assumed that the body is in an external magnetic field. The basic characteristics of the stress–strain state and the induced magnetic field are determined and their features at the inclusion are analyzed. Graphs are drawn for the total magnetoelastic and Maxwell stresses versus the ratio of the ellipse axes and the angle of dip, and tabular maximum stresses versus the magnetic induction and the magnetic properties of the material.     

Exact solutions for functionally graded pressure vessels in a uniform magnetic field

Analytical studies for magnetoelastic behavior of functionally graded material (FGM) cylindrical and spherical vessels placed in a uniform magnetic field, subjected to internal pressure are presented. Exact solutions for displacement, stress and perturbation of magnetic field vector in FGM cylindrical and spherical vessels are determined by using the infinitesimal theory of magnetoelasticity. The material stiffness and magnetic permeability obeying a simple power law are assumed to vary through the wall thickness and Poisson’s ratio is assumed constant. Stresses and perturbation of magnetic field vector distributions depending on an inhomogeneous constant are compared with those of the homogeneous case and presented in the form of graphs. The inhomogeneous constant, which includes continuously varying volume fraction of the constituents, is empirically determined. The values used in this study are arbitrary chosen to demonstrate the effect of inhomogeneity on stresses and perturbation of magnetic field vector distributions.     

Two-Dimensional Magnetoelastic Problem for a Multiply Connected Piezomagnetic Body

A general method based on complex variable theory is proposed to determine the magnetic and elastic fields of a piezomagnetic body. This method is used to derive the basic relations for complex potentials in the two-dimensional problem of magnetoelasticity, their general representations for a multiply connected domain, expressions for stresses, displacements, vectors of magnetic field intensity and magnetic flux density, and magnetic field potential. A closed-form solution is obtained for a body with an elliptic (circular) hole or crack subjected at infinity to the action of a constant magnetoelastic field. Numerical results for a piezomagnetic plate with a circular hole are presented __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 64–74, October 2005.     

Local elastodynamic stresses in the unit cell of a woven fabric composite

Summary A theoretical study of the local elastodynamic stresses of woven fabric composites under dynamic loadings is presented in this article. The analysis focuses on the unit cell of an orthogonal woven fabric composite, which is composed of two sets of mutually orthogonal yarns of either the same fiber (nonhybrid fabric) or different fibers (hybrid fabric) in a matrix material. Using the mosaic model for simplifying woven fabric composites and a shear lag approach to account for the inter-yarn deformation, a one-dimensional analysis has been developed to predict the local elastodynamic and elastostatic behavior. The initial and boundary value problems are formulated and then solved using Laplace transforms. Closed form solutions of the dynamic displacements and stresses in each yarn and the bond shearing stresses at the interfaces between adjacent yarns are obtained in the time domain for any type of in-plane impact loadings. When time tends to infinity, the dynamic solutions approach to their corresponding static solutions, which are also developed in this article. Solutions of certain special cases are identical to those reported in the literature. Lastly, the dynamic stresses and bond shearing stresses of plain weave composites subjected to step uniform impacts are presented and discussed as an example of the general analytical model. Received 3 May 1999; accepted for publication 22 September 1999     

Interaction of an edge dislocation with a coated elliptic inclusion

This paper presents an analytical solution for plane elasticity problems of an elliptically cylindrical layered media subject to an arbitrary edge dislocation. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the displacements and stresses, where an edge dislocation is located in matrix, coating layer and inclusion are obtained. The numerical results of image forces exerted on a generalized edge dislocation are carried out by using the generalized Peach–Koehler equation. As a numerical illustration, both the image forces and equilibrium positions are presented for different material combinations and relative thickness of a coating layer. The result shows that the thickness and the shear modulus of the coating layer have a strong influence on the stability of dislocation.     

Une transformation du problème d'élasticité linéaire en vue d'application au problème de l'inclusion et aux fonctions de Green

A simple transformation of the problem of the linear elastic structure is presented. The transformed problem corresponds to a new problem of linear elastic structure with different behaviour, geometry and prescribed forces and displacements. The transformed problem can be easier to study, or can correspond to cases with well-known solutions. By means of this transformation, the problem of ellipsoidal inclusion is transformed into a problem of spherical inclusion, the analytical results known for the Eshelby tensor for an isotropic or transversely isotropic matrix are extended to more general cases of matrix behaviour, and finally, close form expressions of the Green function for an infinite medium are derived for some cases of elastic behaviour without transversal isotropy or orthotropy.     

A generalized screw dislocation interacting with interfacial cracks along a circular inhomogeneity in magnetoelectroelastic solids

The interaction of a generalized screw dislocation with circular arc interfacial cracks under remote antiplane shear stresses, in-plane electric and magnetic loads in transversely isotropic magnetoelectroelastic solids is dealt with. By using the complex variable method, the general solutions to the problem are presented. The closed-form expressions of complex potentials in both the inhomogeneity and the matrix are derived for a single circular-arc interfacial crack. The intensity factors of stress, electric displacement and magnetic induction are provided explicitly. The image forces acting on the dislocation are also calculated by using the generalized Peach–Koehler formula. For the case of piezoelectric matrix and piezomagnetic inclusion, the shielding and anti-shielding effect of the dislocation upon the stress intensity factors is evaluated in detail. The results indicate that if the distance between the dislocation and the crack tip remains constant, the dislocation in the interface will have a largest shielding effect which retards the crack propagation. In addition, the influence of the interfacial crack geometry and materials magnetoelectroelastic mismatch upon the image force is discussed. Numerical computations show that the perturbation effect of the above parameters upon the image force is significant. The main result shows that a stable or unstable equilibrium point may be found when a screw dislocation approaches the surface of the crack from infinity which differs from the perfect bonded case under the same conditions. The present solutions contain a number of previously known results which can be shown to be special cases.     

Exact solution for elliptical inclusion in magnetoelectroelastic materials

This paper considers the magneto-electro-mechanical coupling between an inclusion and matrix, which are both of magnetoelectroelastic materials. The general cases including the mode I, mode II and mode III are studied. Analytical solutions for an elliptical cylinder inclusion inside an infinite magnetoelectroelastic medium under combined mechanical–electrical–magnetic loadings are formulated via the Stroh formalism. Crack problem is also investigated and the stress, electric and magnetic fields in the vicinity of the crack tip are determined by a complex vector of intensity factors. Various special cases, including an impermeable inclusion, a permeable crack, a rigid and permeable inclusion, a rigid and permeable line inclusion, a permeable cavity, and an impermeable cavity are discussed.     

Magnetoelastic interactions in a soft ferromagnetic body with a nonlinear law of magnetization: Some applications

Linearized equations and boundary conditions of a magnetoelastic ferromagnetic body are obtained with the nonlinear law of magnetization. Magnetoelastic interactions in a multi-domain ferromagnetic materials are considered for magneto soft materials, i.e. the case when the magnetic field intensity vector and magnetization vector are parallel. As a special case, the following two problems are considered: (1) the magnetoelastic stability of a ferromagnetic plate-strip in a homogeneous transverse magnetic field; (2) the stress–strain state of a ferromagnetic plane with a moving crack in a transverse magnetic field. It is shown that the modeling of magnetoelastic equations with a nonlinear law of magnetization provides qualitative and quantitative predictions on physical quantities including critical loads and stresses. In particular, it is shown that the critical magnetic field in plate stability problems found with the nonlinear law of magnetization is in better agreement with the experimental finding than the one found with a linear law. Furthermore, it is also shown that the stress concentration factor around a crack predicted with the nonlinear law of magnetization is more accurate than the one obtained with a linear counterpart. Numerical results are presented for above mentioned two problems and for various forms of nonlinear laws of magnetization.     

A crack along the interface of a circular inclusion embedded in an infinite solid

The two-dimensional problem of an arc shaped crack lying along the interface of a circular elastic inclusion embedded in an infinite matrix with different elastic constants is considered. Based on the complex variable method of Muskhelishvili, closed-form solutions for the stresses and the displacements around the crack are obtained when general biaxial loads are applied at infinity. These solutions are then combined with A.A. Griffith's virtual work argument to give a criterion of crack extension, namely the de-bonding of the interface. The critical applied loads are expressed explicitly in terms of a function of the inclusion radius and the central angle subtended by the crack arc. In the case of simple tension the critical load is inversely proportional to the square-root of the inclusion radius. By analyzing the variation of the cleavage stress near the crack tip, the deviation of the crack into the matrix is discussed. The case of uniaxial tension is worked out in detail.     

Electromechanical response of (3–0, 3–1) particulate,fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.     

Magnetoelastic problem for a bodywith periodic elastic inclusions

A general approach based on complex variable theory is proposed to determine the magnetoelastic state of a body with an infinite row of elliptic inclusions under the action of magnetic and elastic fields. Numerical solutions to a two-dimensional problem for a body made of Terfenol-D magnetostrictive material and piezomagnetic ceramic material and having circular, elliptic, and rectilinear inclusions made of a different material are presented depending on the geometry of the inclusions, their material characteristics, the spacing between them, and the type of applied load __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 32–40, September 2006.     

Thermal stresses in an isotropic trimaterial interacted with a pair of point heat source and heat sink

A general analytical solution for an isotropic trimaterial interacted with a point heat source is provided in this paper. Based on the method of analytical continuation in conjunction with the alternating technique, the solutions to heat conduction and thermoelasticity problems for three dissimilar media are first derived. A rapidly convergent series solution for both the temperature and stress functions, which is expressed in terms of an explicit general term of the complex potential of the corresponding homogeneous problem, is obtained in an elegant form. As a numerical illustration, the distributions of thermal stresses along the interface are presented for various material combinations and for different positions of the applied heat source and heat sink.     

Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.