Microbuckling-field theory: a new model for material instabilities in cellular solids

Abstract

We introduce a general theory of the microbuckling field for modelling microbuckling
and other material instabilities in cellular solids such as foams, lattices, and mechanical metamaterials. The model’s main novelty lies in the assignment of a scalar-valued microbuckling field
indicating the degree to which a material point is locally microbuckled. Two independent strain
energies are prescribed: one for locally unbuckled and another for locally microbuckled material
points. The former energy corresponds to linear elasticity of the underlying cells, while the latter
to their microbuckling and eventual densification. These energies are constructed using standard
Gibson-Ashby formulae for effective cellular solid constitutive parameters in terms the microstructural topology and composition. The microbuckling field unifies the pair of strain energies into
a single overall energy via a first-order regularization of the microbuckling field, i.e., including a
penalty based on its normed gradient. Unlike existing continuum models for material instabilities,
none of the strain energies need violate polyconvexity. In this manner, localized material behavior
is described not by the absence of convexity but instead by the extended kinematic setting. The
length scales conferred by the first-order regularization are precisely those relevant to the localized
material behavior. We demonstrate that the model is amenable to both analytical and computational treatments and recovers phenomena distinctive of cellular solids such as localized compression
bands and global softening mechanisms.