Computational analysis of limit point instability in circular magnetoelastic membranes

Membranes made of soft materials are prone to limit point instability, characterized by a loss of monotonicity in pressure-deflection relationship. In soft active materials that respond to external fields (e.g. magneto-/electro-elastic materials), onset of this instability can be controlled using the external field. In this work, limit point instability in magnetoelastic circular membranes is analyzed in the presence of magnetic field and transverse pressure. Forces and deformation in the membrane are studied for a weakly magnetizable material medium under axisymmetric loading and transverse magnetic field while incorporating material nonlinearity, Maxwell stress, and pre-stretch effects. An $h$-order membrane theory is presented and the resulting nonlinear system of ordinary differential equations are solved using a boundary value problem (BVP) solver in MATLAB. BVP solvers are prone to convergence issues for nonlinear problems and exhibit a high sensitivity to the initial guesses, particularly in the unstable regime. An iterative computational scheme is proposed here to alleviate this issue by improving the initial guesses provided to the solver. The results are validated with existing literature for special cases and several parametric studies are performed to understand the response of a magnetoelastic membrane actuator under combined magnetomechanical loading. Improved convergence for a wide range of input values is observed, allowing a more comprehensive study of soft magnetoelastic membrane actuators. The computational framework presented in this work can be applied towards device design in soft robotics.