Computational first-order homogenization in chemo-mechanics

In the present contribution, we consider species diffusion coupled to finite deformations in strongly heterogeneous microstructures. A semi-dual energy formulation parameterized in terms of the chemical potential is obtained by Legendre transformation of the free energy. Doing so avoids the presence of higher gradients of the deformation field. The constitutive response at the macroscopic level is obtained using variationally consistent homogenization (Larson et al. in Int J Numer Method Eng 81(13):1659–1686, 2010. doi: 10.1002/nme.2747). This approach allows to treat transient microscale problems on a representative volume element which has a finite size, i.e., the scales are not clearly separated. Full details of the implementation are provided. A series of numerical examples compares the homogenization results to single-scale formulations which fully resolve all microstructural features