Modification and verification of the Deshpande-Fleck foam model: A variable ellipticity

The ellipticity involved in the Deshpande-Fleck foam model describing the constitutive behaviour of cellular materials was usually considered to be constant, but some very different values were suggested in the literature. A cell-based finite element model of closed-cell foam under uni-/multi-axial compression is employed to verify the Deshpande-Fleck foam model. The ellipticity is determined by applying uniaxial and hydrostatic compression tests and it is found to vary with the equivalent plastic strain, i.e., the ellipticity decreases with the equivalent plastic strain and then increases sustainably before full densification. According to the understandings from the numerical results, a fitting relation between the ellipticity and the equivalent plastic strain is suggested. The ellipticities of an open-cell foam and a closed-cell foam studied experimentally in the literature are fitted well with this relation. A modification to the Deshpande-Fleck foam model with a variable ellipticity is thus proposed. The modified Deshpande-Fleck foam model brings much accurate predictions with using a rigid-plastic hardening (RPH) idealisation model, which describes the stress-strain relation of cellular material under uniaxial compression. Good agreement is also observed between the experimentally measured stress-strain responses and the predictions of the modified Deshpande-Fleck foam model, especially when considering the effect of the plastic Poisson's ratio with a non-associated flow rule. The findings herein are helpful to improve the prediction accuracy of the Deshpande-Fleck foam model.