Multi-scale computational homogenization analysis of foams with micro-buckling

When studying the behavior of foams by multi-scale computational homogenization procedure,
the micro-buckling may occur at the cell walls and edges and reduces the effective stiffness of the
structures at macro-scale. This instability can be enhanced by plastic deformation at micro-scale.
At sufficiently large value of macro-strain, even if the micro-tangent moduli of micro-material is
still elliptic, the homogenized tangent moduli at macro-scale can lose its ellipticity that implies
the localization occurs at macro-scale. When localization occurs, the characteristic size of macro-
scopic deformation is the same order of the microscopic size. The assumption of material action in
standard multi-scale computational homogenization approach where the stress only depends on the
strain at this point is no-longer suitable. And the material behavior at given point depends also on
the neighborhood of this point. To cover this problem, the second-order multi-scale computational
homogenization is suitably used. At macroscopic problem, the high-order stress and the high-order
strain are enhanced to the standard formulation by using the Discontinuous-Galerkin formulation
while at the micro-scale, the standard continuum formulation is still used. By this procedure, the
influence of micro-buckling of foams on structural behaviour is studied.