Multiscale stability analysis of periodic magnetorheological elastomers

Abstract We analyze instability phenomena of periodic magnetorheological elastomers in the framework of computational homogenization. Our focus is on two kinds of instabilities given by macroscopic material and microscopic structural instabilities. While the first are related to the rank-one convexity of an associated homogenized energy density, the latter are related to the coercivity of an associated microscopic boundary value problem. At both scales we detect instabilities of equilibrium states by superimposing wave-like perturbations. At macroscopic scale we consider classical plane waves giving rise to the definition of a generalized acoustic tensor. At microscopic scale we exploit Bloch–Floquet wave analysis, which allows to determine critical buckling modes of microstructures based on computations at unit-cell level. The microscopic material response is governed by a four-field variational principle of magneto-elasticity that is embedded in a framework of first-order computational homogenization. A series of numerical simulations reveals a spectrum of complex pattern transformations that could be triggered by appropriate microstructure design and coupled magneto-mechanical loading.