Multi-scale simulation of non-linear cellular- and meta-materials with body-force-enhanced second-order homogenisation

Multi-scale simulation of lattices, cellular materials and meta-materials faces the difficulty of handling the local instabilities which correspond to a change of the micro-structure morphology. On the one hand, first order computational homogenisation, which considers a classical continuum at the macro-scale, cannot capture localisation bands. On the other hand, second-order computational homogenisation, which considers a higher order continuum at the macro-scale, introduces a size effect with respect to the Representative Volume Element (RVE) size.

By reformulating second-order computational homogenisation as an equivalent homogenised volume, non-uniform body forces arise at the micro-scale and act as a supplementary volume term over the RVE. Contrarily to the original uniform body forces resulting from an asymptotic homogenization [1], the devised non-uniform body forces arise from the Hill-Mandel condition and are expressed in terms of the micro-scale strain localization tensor, i.e. the relation between the micro-scale and macro-scale deformation gradients [1].

The consistency and accuracy of the approach are illustrated by simulating non-linear elastic meta-materials and elasto-plastic cellular materials under compressive loading.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 862015.

REFERENCES
[1] V. Monchiet, N. Auffray, J. Yvonnet, Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods, Mechanics of Materials 143 (2020) 103309.
[2] L. Wu, S. M. Mustafa, J. Segurado and L. Noels. Second-order computational homogenisation enhanced with non-uniform body forces for non-linear cellular materials and metamaterials. Computer Methods in Applied Mechanics Engineering, 407: 115931, 2023.