A Geometrically Exact Micromorphic Model for Elastic Metallic Foams Accounting for Affine Microstructure. Modelling, Existence of Minimizers, Identification of Moduli and Computational Results

We investigate a geometrically exact generalized continua of micromorphic type in the sense of Eringen for the phenomenological description of metallic foams. The two-field problem for the macrodeformation ϕ and the “affine microdeformation” $\overline{P} \in {\text{GL}}^{{\text{ + }}} {\left( 3 \right)}$ in the quasistatic, conservative elastic case is investigated in a variational form. The elastic stress-strain relation is taken for simplicity as physically linear. Depending on material constants different mathematical existence theorems in Sobolev-spaces are recalled for the resulting nonlinear boundary value problems. These results include existence results obtained by the first author for the micro-incompressible case $\overline{P} \in {\text{SL}}{\left( 3 \right)}$ and the micropolar case $\overline{P} \in {\text{SO}}{\left( 3 \right)}$. In order to mathematically treat external loads for large deformations a new condition, called bounded external work, has to be included, overcoming the conditional coercivity of the formulation. The observed possible lack of coercivity is related to fracture of the substructure of the metallic foam. We identify the relevant effective material parameters by comparison with the linear micromorphic model and its classical response for large scale samples. We corroborate the performance of the micromorphic model by presenting numerical calculations based on a linearized version of the finite-strain model and comparing the predictions to experimental results showing a marked size effect.