# Effective thermal expansion coefficient

Hi
I am trying to calculate the effective thermal expansion coefficient (CTE) of a composite with numerical homogenization. I start from the homogenization tutorial shown in here.
However, the literature I have read so far shows how to calculate CTE, the formulation is different from the tutorial so I am not sure how to extend it to CTE.
Can anyone suggest how can i extend the tutorial to calculate CTE ?
Any help is appreciated.

Thanks

Hi,
it would be best if you started with a first implementation or the formulation you would like to use, do you want periodic BCs or uniform temperature gradient BCs ?
Homogenization of thermal expansion coefficient is just the scalar version of elasticity homogenization. In the tutorial you referenced, just replace elastic stiffness with expansion coefficient, displacement with temperature, strain with temperature gradient, stress with heat flux and macroscopic strain with macroscopic temperature gradient. Periodic BCs are treated in the same way, and you should also fix temperature at a given point or enforce zero temperature average.

I want periodic BC.
Thanks for the suggestions. I will try to implement it as suggested.

I just had a few questions.

1. if I assume unit thermal conductivity then stress (heat flux) and strain (temperature gradient) are going to be the same.
2. In that case, would elastic stiffness (expansion coefficient) remain unused?

Thanks

1. Yes
2. No actually, it is a good test case. The code returns the homogenized expansion coefficient in the domain, so in the uniform case it should return the input thermal conductivity, which is the identity matrix in your case