Size effects in two-dimensional Voronoi foams: A comparison between generalized continua and discrete models

In view of size effects in cellular solids, we critically compare the analytical results of generalized continuum theories with the computational results of discrete models. Representatives are studied from two classes of generalized continuum theories: the strain divergence theory from the class of higher-grade continua and the micropolar theory from the class of higher-order continua. In the former, the divergence of strain is proposed as an additional deformation measure, while in the latter the microrotation gradients act as the source for extra internal energy. We analytically solve a range of basic boundary value problems (simple shear, pure bending and the strain concentration around a hole) and compare the results with discrete, numerical calculations that are based on a Voronoi representation of the cellular microstructure. By comparing both the local deformation fields and the overall elastic response, we critically assess the capabilities of both theories in capturing size effects in cellular solids.