The decomposed form of magnetoelastic beams with free faces

Abstract

The decomposed form of an isotropic elastic plate is extended to a bending magnetoelastic beam with free faces, and the corresponding decomposition theorem is presented. It is shown that the stress state of the magnetoelastic beam without transverse surface loadings can be decomposed into three parts: the interior state, the Papkovich-Fadle state (in shortened form the P-F state) and the magnetic state. Based on the three new lemmata, a rigorous mathematical proof of the decomposition theorem is given concisely and directly, which is independent of the Papkovich-Fadle eigenfunction expansion of biharmonic functions. In the proof course, some basic mathematical methods are used only, so the proof is more convenient for being understood. More importantly, these three states correspond to the three equations derived in the refined theory of magnetoelastic beams: the fourth-order equation, the transcendental equation and the magnetic equation, respectively. Therefore, this work virtually proves the consistency between the decomposition theorem and the refined theory of bending magnetoelastic beams.