## You are here

# Correct Average of Stress/Strain Microfields

This is a general micromechanic question.

Suppose we have a microstructured 2-phases material with random inclusions (or a porous material with random pores) and we make a real tensile test (monoaxial loading in the vertical direction of a specimen with nominal length L). We want to model such tensile test with FEM (for instance with Abaqus) like in this pic: We can model the microstructure of the material, meshing a 2D subregion of the real specimen (like in the attached pic), so that the FEM solver can output a "strain field" across the 2D region at each time step of the simulation. The experiment measures the strain as dL/L, where dL is the total elongation of the sample at the initial stress-free reference configuration.

The simulation makes the Bottom nodes of the model fixed in y and assigns to the TOP nodes a prescribed maximal strain (for instance 1%). Left and Right nodes can move (poisson effect) but parallel on the vertical axis.

I am now going to compare the experimental stress/strain with the calculated ones. How to calculate correctly the "homogenized"(average) stress/strain to get a 1:1 comparison with the experiment?

I see two options:

A. AVERAGING STRESS/STRAIN OVER TOP LINE. We take the simulated strain at the TOP boundary nodes during all the steps of the sim (the final one is the prescribed strain), because this seems to be what is actually measured by the experiment. The same for the stress, since the force is transmitted trought the TOP line of the model.

B. AVERAGING STRESS/STRAIN OVER THE WHOLE REGION. This is what states the theory of micromechanics: we should average both local stress and strain fields over the surface (volume in 3D). (Of course we have to average over the area of the single elements to get mesh-indipendent results)

Question 1): What is the right way to calculate the average strain in order to compare it with experimental curves? A or B?1b) in the ideal case, are both ways equivalent? Question 2) IN case B is appropriate and the second phase is not meshed (pores, like in the pic above), should we average over the whole region including pores (where both stress and strain are zero) or just over the solid region (green one in the pic) ? Thanks, Alex PS: I calculated strain both ways (A and B). The mismatch seems not negligible, but maybe this is due to a numerical artifacts. Still unclear to me.

## It should be A

You should use A to compare with the experimental results. Theoretically, the average stress over the whole region does not equal the average stress over the top line because the stress on the boundary is not constant. The volume average stress equal the stress on the boundary only when the stress on the boundary is constant. To make an estimation, the value of B stress is smaller than that of A. I hope your results can verify this. Many thanks.