A constitutive model for the compressive response of metallic closed-cell foams including micro-inertia effects

Metallic foams have known a keen interest in the last decades. Their ability to undergo very large deformations while transmitting low stress levels make them capable of performing functions of protective layers against intense loadings and of energy absorbers, for instance. The behaviour of metal foams varies considerably between quasi-static and dynamic regimes. Those differences can be linked to the strain-rate sensitivity of the skeleton material and to micro-inertial effects (induced by the crushing of the foam cells). In the present work, a micromechanical model has been developed to take into account micro-inertia effects on the macroscopic behaviour of closed-cell foams under dynamic loading conditions. The proposed modelling is based on the dynamic homogenisation procedure introduced by Molinari and Mercier (J. Mech. Phys. Solids 49 (2001) 1497–1516). Within this framework, the macrostress is the sum of two terms. The first one is a static stress, that can be described with any existing model of metal foam. The second contribution is a dynamic stress related to micro-inertia effects. Considering an initially spherical shell as a Representative Volume Element (RVE) of the foam material, a closed-form expression of the dynamic stress was obtained. The proposed modelling was applied to shock propagation in aluminium foams (it should however be noted that the present theory is not restricted to uniaxial deformation but can be applied to arbitrary loadings). From experimental data of the literature, it is observed that incorporating micro-inertia effects allows one to achieve a better description of the foam shock response. This indicates that micro-inertia may have a significant influence on the dynamic behaviour of metallic foams