Determination of the constitutive relation and critical condition for the shock compression of cellular solids

This study aims at understanding the constitutive relation and critical condition for the shock compression of cellular solids. A 2D virtual foam is constructed from the cross-section of a closed-cell aluminium foam imaged by micro X-ray computed tomography, which enables the realistic consideration of meso-scale structural effect in numerical modelling. Quasi-static and shock compressions of the 2D foam are simulated. A series of Hugoniot relations between shock speed (and other mechanical quantities) and impact speed are determined from the finite element (FE) simulations. It is found that the shock speed increases approximately linearly with impact speed, similar to that observed for condensed solids, but the related material constants for cellular solids have different physical implications, whereas the shock strain, stress and energy increase with impact speed nonlinearly, due to shock-enhanced cell compaction and cell-wall plastic deformation. Based on conservation laws in continuum mechanics, other Hugoniot relations are derived from the basic linear one, which agree well with those obtained from the FE simulations. It is thus demonstrated that the unique linear Hugoniot relation can be used to characterise the shock constitutive behaviour which is distinct from the quasi-static one. Furthermore, a new analytical method based on the linear Hugoniot relation is proposed to estimate the critical impact speed for shock initiation, which has reasonable agreement with the present FE simulations and previous experimental and numerical results, and outperforms the existing methods.