The influence of dispersity in geometric structure on the stability of cellular solids

Geometric structures in cellular solids span the spectrum from perfectly periodic to strictly random. Depending on the degree of disorder at the cellular-scale, the corresponding continuum-scale mechanical response can admit instabilities or can remain stable for the entire range of compressive deformations. In the present work, the response of cellular materials to quasi-static uni-axial compression is investigated. The underlying geometric structures in these materials are allowed to range from highly ordered to highly disordered, and the corresponding transition from unstable to stable mechanical response is explored. A stochastic constitutive model is developed and used for this purpose. Model development begins with an established cellular-scale mechanical response description, but this cellular-scale model is generalized to accommodate finite strain. A continuum-scale constitutive model is established by averaging the cellular-scale model over an ensemble of foam cells, and stochastic variation in cellular-scale geometric structure and material properties is considered through the use of probability density functions for the associated model parameters. Results show that dispersity in geometric structure has little to no effect on the initial elastic properties of the cellular materials under investigation. For deformations occurring prior to any occurrence of instability, however, increasing dispersity is accompanied by decreasing stiffness, an increase in critical strain, and a decrease in the extent of localized deformation. Most notably, materials with the highest degrees of dispersity in their cellular structures exhibit mechanical response that remains stable for the entire range of compressive deformations, demonstrating a general stabilizing effect of dispersity in geometric structure on the continuum-scale mechanical response of cellular solids.