The refined theory for a magnetoelastic body–I. plate problems

Abstract

The problem of deducing two-dimensional theory from three-dimensional theory for a soft ferromagnetic elastic isotropic body is investigated. By using the general solution for the soft ferromagnetic elastic solids and Lur'e method, it is shown that the general deformation can be decomposed into two independent parts: the asymmetric deformations (plate problems) and the symmetric part (plane problems). A refined plate theory which takes into account the transverse shear deformation can now be explicitly established without employing assumptions. The exact equations and solutions for the plate with homogeneous boundary conditions are derived, and the equations can be decomposed into four governing differential equations. Moreover, the approximate equations and solutions for the plate under transverse loadings and magnetic field perturbations are derived directly from the refined plate theory. By omitting higher order terms and coupling effect, the refined plate theory can be degenerated into other well-known elastic and magnetoelastic theoretical models.

The refined theory for a magnetoelastic body – II. Plane problems

“…As the continuation of the previous work [1], the present paper deals with the plane problems of magnetoelasticity under the framework of the 3D refined theory for a magnetoelastic plate of rectangular cross-section. In this case, the magnetoelastic plate is subjected only to symmetrical loadings and edge conditions.…”

“…where B, M and H are magnetic induction vector, magnetization vector and magnetic intensity vector, respectively, t, σ and u denote the magnetomechanical stress tensor, elastic stress tensor and displacement vector, respectively, λ and µ represent the Lame constants, µ 0 and χ are the magnetic permeability and the magnetic susceptibility, respectively, χ 1 and χ 2 are the parameters determined by some theoretical models for magnetoelastic interaction [1,[5][6][7], ∇ 0 are the 3D gradient operator. The barred quantities are the magnetic fields in the rigid-body state with no mechanical singularities; the quantities in lower case represent singularities and are assumed much smaller than the undisturbed state.…”

“…That is, the plate is subjected only to a set of symmetrical loadings, the perturbation of magnetic intensity and edge conditions with respect to the x − y plane, thus only even functions of z are required for u x and u y , and odd functions of z for u z and ϕ. Using the same manipulation in the paper [1], we treat Eq. (6) as an ordinary differential equation in z with constant coefficients, and obtain the following symbolic solutions of Eq.…”

“…The two-dimensional (2D) problems associated with the two types of deformation are called the plate problems and the plane problems, respectively. The refined theory for plate bending problems of magnetoelasticity was discussed in our previous work [1]. It was shown that in the magnetoelastic plate problems the approximate governing equations of refined theory have the same form as those of other well-know theoretical models of magnetoelastic plates, such as the Pao-Yeh model [2] from the axiomatic system, the Eringen-Maugin model [3,4] from the rational mechanical theorem, the magnetic couple model [5] and the magnetic force model [6] from the magnetic dipole model, and the variational principle model [7,8].…”

“…Also with no a priori assumption, Piltner [10] studied the elastic thick and thin plates, making use of his representations of complex variables and complex functions for 3D elasticity problems [11]. For piezoelectric and soft ferromagnetic elastic materials, the same problems were considered by Ding and Chen [12] and Gao and Zhao [1], respectively.…”