Homogenization and Geometric Linearization for Multi‐Well Energies

Abstract

AbstractThe concept of linear elasticity is to assume that the stored energy has one‐well structure and to consider small displacements. The energy function is then expanded around the equilibrium state and higher‐order terms are neglected. It was proved in [3] that this concept really gives results that provide a good approximation for the actual problem. On the other hand, for materials with fine periodic structure we accomplish a simplification of the problem in an analogous manner by homogenizing the energy. It was recently shown by Müller and Neukamm in [8] that both processes, homogenization and linearization, are interchangeable for such elastic energies. If we consider an energy with multiple wells then, under some reasonable conditions, it is also possible to (geometrically) linearize the problem. The second author proved that quasiconvexification of the limit function yields the desired result. We present that in this case both processes still commute. This is not a priori clear since the proof by Müller and Neukamm significantly rests upon the one‐well structure and properties of quadratic forms. They also provide an example when the statement does not hold. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)