Nonlinear multi-scale homogenization with different structural models at different scales

Abstract

We present an extension of the computational homogenization theory to cases where different structural models are used at different scales and no energy potential can be defined at the small scale. We observe that volumetric averaging, which is not applicable in such cases unless similarities exist in the macro-scale and micro-scale models, is not a necessary prerequisite to carry out computational homogenization. At each material point of the macro-model, we replace the conventional representative volume element with a representative domain element (RDE). To link the large-scale and small-scale problems, we then introduce a linear operator, mapping the smooth part of the small-scale displacement field of each RDE to the large-scale strain field and a trace operator to impose boundary conditions in the RDE. The latter is defined on the basis of engineering judgement, analogously to the conventional theory. A generalized Hill’s condition, rather than being invoked, is derived from duality principles and is used to recover the stress measures at the large scale. For the implementation in a nonlinear finite-element analysis, ‘control nodes’ and constraint equationsare used. The effectiveness of the procedure is demonstrated for three beam-to-truss example problems, for which multi-scale convergence is numerically analysed.