Magnetoelastic surface Green’s function and the stability of a soft magnetoactive half-space

Surface Green’s functions for a pre-deformed compressible soft magnetoactive (SMA) half-space are derived in this study. Two typical magnetic conditions are considered, one accounting for the effect of the external magnetic field, while the other does not. These Green’s functions are obtained for the first time using the principles of magneto-elastic continuum mechanics. By employing a generalized neo-Hookean ideal magnetic model, fundamental solutions are explicitly presented for a half-space subject to an in-plane biaxial stretch and a transverse magnetic biasing field, which are then utilized to analyze related boundary-value problems. We consider the flat-ended cylindrical indentation of a pre-deformed SMA half-space, with either prescribed magnetic induction or magnetic field intensity within the contact region. The critical criteria for vanishing indentation force in the case of equi-biaxial pre-stretch and the generalizations in the case of unequal biaxial pre-stretch are obtained. For the sake of illustration, the specific analysis is conducted for a compressible neo-Hookean model. We find that the above criteria are essentially equivalent to those acquired from bifurcation analysis of a half-space, where the instability always first occurs along the direction of principal stretch. The outcomes demonstrate that the increase in the magnetic biasing field exerts a consistent adverse effect on the evolution of surface instability, whether considering the external field or not. The present study provides a general tool for investigating diverse mechanical behaviors of soft smart materials via the powerful Green’s functions.