A gradient-enriched continuum model for magneto-elastic coupling: formulation, finite element implementation and in-plane problems

A general piezo-magnetic continuum model with gradients of strain, magnetic field and piezo-magnetic coupling terms is proposed in this work. An energy variational principle with strain, strain gradient, magnetic field and magnetic field gradient as independent variables is presented to develop the constitutive equations and governing equations. Three internal length parameters are introduced to represent the underlying microstructure. Finite element implementations are obtained by extending the Ru-Aifantis ‘operator split’ method from gradient elasticity to gradient magneto-elasticity. Numerical results and discussion of two-dimensional in-plane problems show the effects of gradients on static piezo-magnetic analysis, in particular (1) removal of singularities from magnetic fields as well as mechanical fields, and (2) capturing size-dependent piezo-magnetic response. The individual effects of the mechanical length scale, magnetic length scale and coupling length scale on the removal of singularities from magnetic field and mechanical field and the prediction of size-dependent piezo-magnetic response are discussed in detail.