Question

# Why is there no body force considered in the equilibrium equations of the RVE in the homogenization technique?

The overall response of the RVE is obtained by solving the RVE boundary value problem under prescribed boundary conditions (either traction or displacement BC) without considering the body force.

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• You're right, it is possible to add it. But it is more complicated taking into account the density of each phases. Moreover, there are a few problems for which these kind of field are relevant. Regularly, the body force is negligible in comparison with the given forces taking as boundary conditions. Good point!
• Ahmad Akbari R. · Cardiff University
Thanks. I agree that in most of the problems the body forces are negligible in comparison with boundary conditions. But here is an extract from Sia Nemat-Nasser & Muneo Hori book that is not clear for me. "The balance equations contain the body forces representing the effect of the materials not in contact with the considered continuum and the inertia forces due to the motion of the continuum itself, as well as the associated force and displacement boundary data which represent the effect of the other continua in contact with the considered continuum. Therefore, in formulating boundary-value problems associated with an RVE, it is not necessary to include the body forces." ( microstructure: overall properties of heterogeneous materials ,2nd Edition, pp 19 )
Could you please explain to me the above in your own words.
• Your question needs an accurate answer. I will try to be clear!
1- It is normal that the balance equations contain the body forces. Without, it is wrong.
2- The homogenization theory permits to replace the elastic properties of an heterogeneous materials by an homogeneous one, with isotropic or anisotropic properties. Therefore, it is possible to calculate the Hooke tensor of the equivalent homogeneous material. For this last step, it is not necessary to take into account the body force. If you take it, that means that you want to take into account the difference of density of each phase and their influence on the homogenization result. The calculation is possible but it depends on the orientation of the RVE with the body force direction. It is complicated and really not very useful?
3- If you calculate the RVE properties (without body force) by homogenization, you can use the result and put it into the previous balance equations, which include the body force.
Regards
• Ahmad Akbari R. · Cardiff University