Structural Instabilities in Soft Electro-Magneto-Elastic Cylindrical Membranes

Abstract

In the present work, a nonlinear coupled electro-magneto-elastic membrane formulation is developed for soft functional materials starting from the variational form of 3D governing equations. The resulting 2D model is applied to an internally pressurized cylindrical membrane placed in an azimuthal magnetic field and radial electric field. The results of our model are verified with existing literature for some special cases. The model is subsequently used to analyze mechanical and electrical limit-point instabilities, and the effect of external fields on the onset of these instabilities. It is observed that the onset of mechanical limit-point instability, defined by a loss of monotonicity in the pressure versus deformation plots, is dependent on the material properties, geometrical parameters and applied electromagnetic fields. Magnetic field induced instability results in an initial dip in the pressure versus stretch plots, subsequently converging to the purely hyperelastic membrane behavior at larger stretches. Application of electric field, on the other hand, results in an early onset of limit-point instability (i.e., at smaller stretches) compared to the hyperelastic case. Additionally, electrical limit-point instability, characterized by a loss of monotonicity in voltage versus stretch plots, is observed and its dependence on magnetic field and force inputs is studied. Finally, we study the effect of Maxwell stress due to electromagnetic fields. It is observed that ignoring the Maxwell stress related traction boundary term results in an error of up to 10\%, depending on the force and magnetic fields inputs. In summary, our membrane model describes the interactions between the electromagnetic fields and deformation in a soft functional material, and can be applied towards design of soft actuator and sensor devices.