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 Keywords: anisotropic body, complex potentials, crack, hole, inclusion, magnetic field intensity, magnetic flux density, magnetoelasticity, plane problemIntroduction. In recent years, interest in piezomagnetic materials has heightened. Development of analytic and numerical methods for solving specific classes of problems is still one of the urgent tasks in the theory of magnetoelasticity of anisotropic piezomagnetic bodies. The governing equations of magnetoelasticity were derived in [6], internal and external two-dimensional problems of magnetostatics were solved in [1], and the interaction of mechanical strains in solids with an electromagnetic field was investigated in [12]. The great prospects for piezomagnetic materials in modern electronics and engineering generate interest in their effective properties [3-5], interaction of magnetic and mechanical fields [9], and magnetoelastic problems for piezomagnetic plates [10,11], bodies with inclusions [7], holes, and cracks [2].In the present paper, we extend the general approaches to the solution of two-dimensional electroelastic problems for multiply connected bodies [8] to magnetoelastic problems for piezomagnetic bodies with holes and cracks. We will introduce complex potentials for a two-dimensional magnetoelastic problem, derive formulas for the basic magnetoelastic characteristics, formulate boundary conditions for the potentials, obtain their general representations for multiply connected domains, find a magnetoelastic solution for a body with an elliptic (circular) cavity or a crack, and present numerical results.1. Problem Formulation. Let us consider a multiply connected cylindrical anisotropic piezomagnetic body weakened by L longitudinal cavities with generatrices parallel to the cylinder axis. We will use a rectangular coordinate frame Oxyz with the z-axis directed along the cavity generatrices. The cross section of the body by the plane Oxy is a multiply connected domain S bounded by the external boundary L 0 and the outlines L l ( , ) l L = 1 of the holes. As a special case where the outside surface is at

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 Introduction. The interaction of mechanical, thermal, and electromagnetic fields is of much interest for solid mechanics [2,3,16]. This is first because of the prospective use of magnetic materials in modern electronics, engineering, and instrumentation [10] and ample opportunities for predicting and modeling the effective properties of available materials and creating new materials with prescribed magnetoelastic properties for specific structures [9,11,12,[14][15][16][17][18][19]. In studying issues of magnetoelasticity, special attention is given to the magnetoelastic state of multiply connected piezomagnetic materials. The papers [6, 13] offer a method to solve two-dimensional problems of magnetoelasticity for piezomagnetic bodies with holes and cracks and for a body with an elliptic piezomagnetic inclusion made of a different material [1], which can go over into a plane inclusion in a specific case (a rectilinear inclusion in a plate). This method is used here to solve a two-dimensional periodic problem of magnetoelasticity for a body (plate) with inclusions made of a different piezomagnetic (magnetostrictive) material.Problem Formulation. Consider a piezomagnetic matrix body with a periodic row of identical elliptic cavities with parallel generating surfaces. Inclusions of a different piezomagnetic material are soldered-in without interference into the cavities to provide perfect contact conditions. The body is subjected at infinity to constant external forces and magnetic field of constant intensity such that the matrix and inclusions are in a two-dimensional magnetoelastic state that does not vary along the generatrices of the cylindrical cavities (inclusions). Body forces, initial magnetization, and rigid-body rotations of the body as a whole and of each inclusion as a whole are absent.We choose a rectangular coordinate system Oxyz with the Oz-axis directed along the generatrices of the cylindrical cavities (inclusions). The cross-section of the piecewise-homogeneous body by the plane Oxy is a multiply connected plane S bounded by the identical and equally spaced boundaries L l ( , , , ) l = ± ± 0 1 2 K of the elliptic holes with semiaxes a and b and centers aligned along the Ox-axis (Fig. 1) and finite domains S l bounded by the boundaries L l . Denote the center-to-center