Finite deformation modeling of wrinkling in thin magnetoelastic sheets

Abstract

This paper presents an asymptotic two-dimensional (2D) finite-deformation theory for a thin magnetoelastic sheet starting from a three-dimensional (3D) variational form. This theory is used to model wrinkling phenomena under applied traction and external magnetic field. The resulting 2D potential energy is specialized to an isotropic, in compressible material, and the expressions for material constants, i.e., elasticity tensor, magnetoelastic coupling constant, and permeability, are obtained as a function of the applied magnetic field. The formulation is further simplified assuming reflection symmetry about the mid-surface of the sheet. Strong form of the resulting 2D governing equations are obtained by setting the first variation of potential energy functional to zero. The resulting governing equations are used to analyze wrinkling in a stretched rectangular sheet placed in a uniform magnetic field. Approximate analytical solutions are obtained by linearizing the 2D plate equation for small slopes. The effect of magnetic field and applied stretch on onset of wrinkling and amplitude of wrinkles is studied.