# Prof. Yu's Research Group in the Cloud

## Forum

### Periodic Boundary Conditions

1. I am trying to implement periodic boundary in a 3D RVE using dummy_nodes. My goal is found the coefficients of the elastic tensor, I am successfully getting the values for the coefficients C11, C22, C33, C44, C55 and C66. I perform three independent simulations 3 axial loadings and 3 shear loadings. For example, for the axial loading in x-direction, I impose a displacement in the direction x in the dummy_node_x that represents the average strain divided to the RVE length. After find the average stress with the strain energy relation (U=0.5*V*average_strain(x)*average_stress(x)) I obtain the C11 value.
However, I am not seen how can I found the value of the coefficient C12 that should be equal to average_stress_x divided to the averge_strain_y. But i am not able to find the value of the average_strain_y using the dummy_node_x degrees of freedom.

Replying to pedro carneiro

1. Did you get your problem resolved? Is there a reason you want to find the properties by using a strain energy relation? You should be able to find the effective stiffness as described in the attachment from Dr. Yu’s book. Let me know what you think.

Replying to Ernesto Camarena

1. Can you let me know the title of Dr. Yu's book ? I would like to buy it for my group.

2. Hi,
Thanks for the answer.
Sorry for the delay in the response but I had been occupied with other things.
I am now back to this specific work.
I will try to explain better what is my questions.
I am trying to implement periodic boundary conditions in a Finite Element software through the use of dummy_nodes.
For example, in the case of the e11=1 and all ohter strain equal to zero, these strain is applied in the RVE as a displacement in the dummy_node referent to x_axis, which corresponds to:
UX(fx+)-UX(fx-)=UX(DNx)
i.e. the difference in the displacement of the two opposite faces in the x direction is equal to the displacement of the dummy_node_x.
So, the average strain in x direction is equal to the displacement in the dummy_node_x divided to the edge length of the RVE.
My question is how can I calculate the average strain in the remaining directions in order to find the first column of the compliance matrix?
What are the definition of average stress and average strain in the paper that you sent to me?
I really appreciate your help.
Thanks for your attention and time.

Replying to pedro carneiro