Effective elasticity tensor

Hi Community,

I have a question about the numerical homogenization of the effective stiffness tensor. This is done in this excellent example of legacy fenics.

I apply this procedure in 3D. As a result, I get an unsymmetric elasticity tensor. The reason for that is also explained by @bleyerj in the example mentioned above:

" Note: The macroscopic stiffness is not exactly symmetric because we computed it from the average stress which is not stricly verifying local equilibrium on the unit cell due to the FE discretization. A truly symmetric version can be obtained from the computation of the bilinear form for a pair of solutions to the elementary load cases. "

However, I don’t really understand how to get the truly symmetric stiffness tensor. Which pair of solutions are meant? Could anyone describe the procedure in more detail or refer to the literature?

Say you have computed two solutions u^1 and u^2 corresponding to two loading cases E^1 and E^2, then evaluating a(u^1,u^2) would give |\Omega|E^1.C.E^2 which provides the different components of C when considering different pairs of load cases. Since a is a symmetric bilinear form by construction, you will necessarily end up with a symmetric stiffness tensor C.

1 Like

Got it. Thank you very much!