Finite element implementation of a multi-scale dynamic piezomagnetic continuum model

A gradient-enriched dynamic piezomagnetic model is presented. The gradient enrichment introduces a number of microstructural terms in the model that allow the description of dispersive wave propagation. A novel derivation based on homogenisation principles is shown to lead to a multi-scale formulation in which the micro-scale displacements and magnetic potential are included alongside the macro-scale displacements and magnetic potential. The multi-scale formulation of the model has the significant advantage that all higher-order terms are rewritten as second-order spatial derivatives. As a consequence, a standard C⁰-continuous finite element discretisation can be used. Details of the finite element implementation are given. A series of one and two-dimensional examples shows the effectiveness of the model to describe dispersive wave propagation and remove singularities in a coupled elasto-magnetic context.