Effective thermal expansion coefficient

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 ?
or point to some readings that can be helpful.
Any help is appreciated.


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.

Thanks for your suggestion.
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?


  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

Thanks for your prompt reply.
My query is shouldn’t it return the thermal expansion instead of thermal conductivity for the uniform case?

Sorry, I did not realize that you were interested in the coefficient of expansion and not conductivity. My first answer was in fact about the conductivity properties. For the coefficient of thermal expansion, you should consider a variant of the homogenization problem where you impose zero macroscopic strain E but add in the constitutive equation thermal strains for an imposed temperature variation \Delta T.
Indeed, we define the homogenized CTE \boldsymbol{\alpha}_\text{hom} such that the total strain partitions into an elastic and a thermal strain as follows:
\boldsymbol{E}=\boldsymbol{E}_{el}+\boldsymbol{E}_{th} = \boldsymbol{C}_{hom}^{-1}\boldsymbol{\Sigma} + \boldsymbol{\alpha}_{hom}\Delta T

If you consider a problem with \boldsymbol{E}=0, then taking the average of the resulting stress will give you:
\boldsymbol{\Sigma} = \langle \boldsymbol{\sigma}\rangle = - \boldsymbol{\kappa}_{hom}\Delta T = -\boldsymbol{C}_{hom}\boldsymbol{\alpha}_{hom}\Delta T