Magnetoelastic problem for a bodywith periodic elastic inclusions

Abstract

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

Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

“…The unknown constants a j ( j = 1, 2, 3) in the electroelastic problem with conditions (14) can be found from the following systems of equations: …”

Analyzing the viscoelastic state of a plate with elliptic or linear elastic inclusions

“…Note that problems for bodies with circular and elliptic stress concentrators were addressed in [13][14][15][16][17].…”

Elastic state of a transversely isotropic piezoelectric body with an arbitrarily oriented elliptic crack

“…However, solving three-dimensional problems of electroelasticity involves certain mathematical difficulties because the original equations of electrostressed state constitute a complicated coupled system of differential equations [1,4]. Plane problems of electroelasticity are addressed in [11,13,14,22,26], which analyze the two-dimensional electroelastic state near single cavities, inclusions, cracks and the interaction of stress concentrators in electric and mechanical fields. Similar approaches are proposed in [5,23] to construct general solutions to the coupled equations of electroelasticity for transversely isotropic bodies and to find the exact solutions to some problems of electroelasticity for a special orientation of the polarization axis of the ceramic body.…”