Limit-point instability of a magnetoelastic membrane in a stationary magnetic field

In a previous study (Barham et al 2007 Acta Mech. 191 1-19), the finite deformation of a circular magnetoelastic membrane in an axisymmetric dipole field was calculated by specializing the equations of three-dimensional magnetoelastic equilibrium. The predicted response was found to be similar to the classical limit-point instability occurring in analogous purely mechanical problems. A limit-point instability occurs under conditions corresponding to the incipient non-existence of equilibria. Under such conditions the body is necessarily on the verge of a dynamical state. In the present setting, this corresponds to the occurrence of a maximum in the equilibrium deflection of the membrane with respect to applied field strength and proximity of the field source. The earlier conjecture of a limit-point instability, advanced in Barham et al (2007 Acta Mech. 191 1-19), is confirmed in the present work by using a variational method based on an adaptation of the energy criterion of elastic stability to the magnetoelastic setting.