Body-force enhanced second-order computational homogenisation for non-linear cellular materials and metamaterials

When considering lattices or metamaterial local instabilities, corresponding to a change of the micro-structure morphology, classical computational homogenisation methods fail: first order computational homogenisation, which considers a classical continuum at the macro-scale, cannot capture localisation bands while 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. The second-order computational homogenisation was thus reformulated using the idea of
an equivalent homogenised volume, from which arises at the micro-scale a non-uniform
body force that acts as a supplementary volume term over the RVE. In the presented
method, this non-uniform body-force expression arises from the Hill-Mandel condition
and depends mainly on the relation between the micro-scale and macro-scale deformation
gradients [1]. We show by considering elastic and elasto-plastic metamaterials and cellular
materials that this approach reduces the RVE size dependency on the homogenised
response. [1] Wu, L. and Mustafa, S. M. and Segurado, J. and Noels, L.. Second-order computational homogenisation enhanced with non-uniform body forces for non-linear cellularmaterials and metamaterials. Comput. Meth. in Appl. Mech. Eng. (2023) 407: 115931