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

Abstract

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.

Limit points in the free inflation of a magnetoelastic toroidal membrane

“…Note that the formulation here is based on the assumption that the self-generated magnetic field is negligible compared to the applied field. Now, we have the following relations (Barham et al, 2008)…”

“…Note that we use the same form of energy density function here as used in other theoretical works (Dorfmann and Ogden, 2005;Barham et al, 2008) for simplicity. Similar analysis can be applied to energy density functions that are more grounded in experiments, such as those by, (Bustamante, 2010;Danas et al, 2012b;Itskov and Khim, 2016).…”

“…Only non-dimensional quantities are used henceforth. We use the same values of the material parameter and magnetic susceptibility in our calculations that were used by Tamadapu and DasGupta (2014a) and Barham et al (2008) and are listed in Table 1.…”

“…They claim that magnetization per unit mass can be a better choice for an independent variable along with deformation gradient because, unlike magnetic flux or intensity, it vanishes outside the material. Their work (with magnetization as an independent variable in free energy density function) has later been utilised by Barham et al (2007Barham et al ( , 2008 for studying thin magnetoelastic membranes. We follow the same formulation in the current study.…”

“…However, none of these studies considered wrinkling or limit point instabilities in membranes. Barham et al (2008) evaluated the solution for inflation of a circular magnetoelastic membrane under the influence of a stationary magnetic dipole. The arrangement is such that upon inflation, the membrane gets closer to the dipole.…”