Homogenization of linear elastic materials

[valvikby]

Hi, sckit-fem team)
Сould you present a quick example considering homogenization of linear elastic materials, like the one shown
https://comet-fenics.readthedocs.io/en/latest/demo/periodic_homog_elas/periodic_homog_elas.html?

Thanks in advance for your reply

[valvikby]

Hi, kinnala) I'm trying to solve a similar problem as indicated in the link above. My problem is based on MeshHex1 and I use ElementHex1 wrapped in ElementVector.
I need to add another strain tensor to the tensor T in the stress-strain function.

def linear_stress(Lambda=1., Mu=1.):

    def C(T):
        """Linear-elastic stress-strain relationship."""
        return 2. * Mu * T + Lambda * eye(trace(T), T.shape[0])

    return C

For instance, another tensor looks like this:

$$\begin{pmatrix} 1 & 0 & 0 \\\ 0 & 0 & 0 \\\ 0 & 0 & 0 \end{pmatrix}$$

In this case, will the stress-strain function look like below or not?

def linear_stress(Lambda=1., Mu=1.):

    def C(T):
        T[0, 0] += 1.
        return 2. * Mu * T + Lambda * eye(trace(T), T.shape[0])

    return C

[kinnala]

Yes.

Sorry for not answering the original question. I believe we have all the required functionality, but it is somewhat new topic for me so I would need to read it carefully in order to come up with an example.

[valvikby]

Hi, everyone) Here's a little question. What does the strain-stress relationship look like in the scikit-fem?

[kinnala]

The library does not enforce any specific stress-strain relationship, the only one we have implemented as an example for others to use is here:

scikit-fem/skfem/models/elasticity.py

Line 39 in 02a80c3

return 2. * Mu * T + Lambda * eye(trace(T), T.shape[0])