Where is the problem? Generate a FE model of your representative volume element (RVE), load it by uniaxial stress and calculate the strains in stress direction and in transversal direction. Apply the usual formulae of Hooke's law to calculate Young's modulus and the Poisson's ratio. It is the same procedure for any FEM software.

A second thought shows that some care is needed. First, you want the stress of the RVE without voids. Therefore, you need to convert your stress loading to the effective area. Second, the displacement boundary conditions need to eliminate rigid body motions without restricting the contractions, which are caused by the Poisson effect.

Dear Santosh Bagewadi,

A) Below is the step by step process for calculating effective modulus of elasticity.

1. After loading the geometry in FE environment, apply constraint on all the nodes located on the back surface of Representative Volume Element (RVE) in such that prevents axial movement along the displacement boundary condition explained below. Either apply a weak spring to your model that prevents your geometry from rigid body motion (preferred method) or consider a zero displacement (for all directions) on one of the corners located on the back surface (this method creates a very small deviation from true result but it is still OK).

2. Apply a very small axial displacement on the front side of the RVE which enables creating a uniform strain through out your model.

3. Calculate the strain by using the following equation.

Strain = Ln ((RVELength + SmallDisplacementApplied)/RVELength)

4.Extract normal stress results (along the direction which displacement is applied) from all the nodes located on the front surface of the RVE.

5. Use the following equation for calculating the modulus of elasticity for each node located on the front surface of the RVE. Remember that Strain presented in this equation is the one that is calculated above and it is going to be the same for all the nodes.

E(i) = ExtractedStress(i)/ Strain

6. The average of results for the modulus of elasticity calculated for each node is going to give you the effective modulus of elasticity.

B) Below is the step by step process for calculating effective Poisson's ratio.

1. Extract Normal and transverse strain from the front surface of the RVE and used the following equation for calculating Poisson's ratio.

Nu(i) = TransversStrain(i) / NormalStrain(i)

2. The average value of these results is going to give the effective Poisson's Ratio.

**Remember that you can do the above process on multiple cross section of the RVE and check how much deviation you will get and may need to go with the average of these results.

Hope this answer helps!

Dr. Moghaddam procedure is better than mine because a prescribed displacement keeps the loaded plane plane. This is not the case with stress loading because the stiffness of the RVE is not constant.

Dear Dr. Manfred Staat,

Thank you so much! I tried different methods for calculating effective modulus of elasticity, Poisson's ratio and coefficient of thermal expansion of filler modified polymers and found that the procedure above provides the most reliable results.

Thanks,

Thank you Hamidreza Ahmadi Moghaddam and Manfred Staat .

Hamidreza Ahmadi Moghaddam sir can you give reference for your solution?