Second-order homogenization with body-forces for non-linear cellular- and meta-materials

Among the different existing computational homogenisation formulations, first order computational homogenisation considers a classical continuum at the macro-scale, while second-order computational homogenisation considers a higher order continuum at the macro-scale. When considering lattices or meta-material local instabilities, corresponding to a change of the micro-structure morphology, the former homogenisation method cannot capture localisation bands while the latter one 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. As a result, a non-uniform body force is introduced at the micro-scale and 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 is expressed in terms of the micro-scale strain localisation tensor, i.e. the relation between the micro-scale and macro-scale deformation gradients [1]. This is in contrast to the uniform body force introduced in [2] which results from an asymptotic homogenisation formulation.

The consistency and accuracy of the approach are illustrated by considering non-linear elastic meta-materials and elasto-plastic cellular materials. In particular it is shown that this approach reduces the RVE size dependency on the homogenised response.

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

REFERENCES
[1] 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.
[2] 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.