Dear Moussa Mirehei,

imagine a porous solid consisting of a solid phase and a hollow phase for which

you want to compute the homogenized elasticity tensor. Imagine the case that the solid phase is originally connected but then disconnected by -for simplicity- a planar cut in 3d or a linear cut in 2d thus creating a gap of finite width in the porous solid. As a consequence of the discarded connectivity there is no mechanical restistance in terms of non-zero stiffness against pulling apart the

porous specimen in the direction perpendicular to the cut. Instead, there is a relative translation as a rigid body motion of the separated parts. This rigid body motion indicating zero stiffness manifests in the homogenized elasticity tensor in terms of a zero eigenvalue thus making the elasticity tensor singular.

Mathematical homogenization of linear elastic, microheterogeneous solids underpins the above statement more precisely, e.g. in the appendix of our paper [1] which you can find here on RG.

I hope that the physical/mechanical notion of disconnection enabling a rigid body motion and its mathematical consequence for homogenizing linear elastic solids is helpful.

Best

Bernhard

[1] "The Heterogeneous Multiscale Finite Element Method for the Homogenization of Linear Elastic Solids and a Comparison with the FE$^2$ Method, Computer Methods in Applied Mechanics and Engineering 329,

DOI: 10.1016/j.cma.2017.10.001"

Dear Prof. Bernhard Eidel!

Very accurate and correct comments and explanation.

I am ready to subscribe to every word.

Dear Mr. S Moussa Mirehei !

You can not find a better answer than the comments of Prof. Bernhard Eidel.

Thank you all very much. Best, -Moussa

It is not clear that this "connectivity of a component" is a mathematical requirement of the homogenization analysis, but otherwise one is likely to be facing a much more difficult problem for which the mathematical tools are not yet available. Consider, for example, the seepage of water through a pile of rocks supported by sand so no one of the 3 phases would be connected.

I do not understand: how could water, as a fluid, be flowing through, not in isolated regions, and be disconnected? In addition, the physical support of rocks by the sand requires connectivity in the solid phase which in this case comprises of rocks and sand. It is true, though, that the solid constituents would be connected and discrete. Perhaps, the scenario would be a good example to show that connectivity, not being discrete, is the requirement. It is my understanding that, since both phases are connected in the mentioned case, upscaling would produce valid results.