Magnetoelastic deformation of a circular membrane: Wrinkling and limit point instabilities

Abstract

We study the inflation of a weakly magnetizable isotropic incompressible circular membrane in the presence of magnetic field generated by a magnetic dipole. Following the approach in recent papers by (

Coupled electro-elastic deformation and instabilities of a toroidal membrane

“…This theory has been applied to study wrinkles in axisymmetric hyperelastic membranes [37,38] and to model wrinkles in skin during wound closure [69], to name a few applications. A generalisation of the tension field theory has been attempted (although without a rigorous mathematical proof) for the case of electroelasticity by De Tommasi et al [14], De Tommasi et al [15], Greaney et al [26], and for the case of magnetoelasticity by Reddy and Saxena [59,60], Saxena et al [63]. Wong and Pellegrino [79] developed an analytical method to quantify the location, amplitude, and wavelengths of linear elastic membranes.…”

“…This typically requires checking the sign of the second variation of the total potential energy to determine the stability state. Chaudhuri and Dasgupta [11] studied the perturbed deformations of inflated hyperelastic circular membranes, Venkata and Saxena [77] analysed buckling of hyperelastic toroidal membranes, Xie et al [80] analysed the shape bifurcations of a dielectric elastomeric sphere through a direct perturbation approach, and Reddy and Saxena [59,60], Saxena et al [63] analysed shape bifurcations of magnetoelastic membranes.…”

Modeling and analysis of a magnetoelastic annular membrane placed in an azimuthal magnetic field

“…Reddy and Saxena [46, 47] studied deformation of toroidal and cylindrical magnetoelastic membranes in the presence of external magnetic field. This approach was extended by Saxena et al [48] to study the limit point and wrinkling instabilities occurring in a circular MRE membrane.…”

Modelling instabilities in electroelastic and magnetoelastic membranes: a very brief review

“…Barham et al (2007Barham et al ( , 2008Barham et al ( , 2010Barham et al ( , 2012 modelled large deformation and limit point instabilities in magnetoelastic membranes using both, analytical and finite element, methods. Saxena (2017, 2018); Saxena et al (2019) performed comprehensive analysis of all the instabilities listed above in circular, cylindrical, and toroidal magnetoelastic membranes using a variational approach. Choice of symmetric geometries allowed them to convert the resulting partial differential equations to ordinary differential equations.…”