Periodic boundary conditions -> Shouldn't supports hinder all motion?

[NewStuff]

Hello everyone,

I am currently trying to understand periodic boundary conditions for the mechanical investigation of mechanical properties of a RVE. I found a good video explaining the theory behind it:

But something is unclear to me: At the above linked time step, the individual conical equations are shown (basically saying, that nodes on opposite faces should have the same displacement and thereby connecting the different node pairs). So far this is logical.

But once I look at the corner nodes (1&2 in the video) it becomes a little unclear: If I use a fixed support at Node 1 to prevent rigid body motion (which equals a 0/0 displacement) shouldn't that also restrain Node 2 to a 0/0 displacement (according to the equation that is shown)?

Now in the video this issue does not arise, because the equations for the corner nodes are connected to the equations of the internal nodes: InternalNodeA - InternalNodeB = CornerNode1 - CornerNode2

In this connected form, it is not a problem any more, because if the displacement of CornerNode1 = 0 than there is still this equation remaining:
InternalNodeA - InternalNodeB = CornerNode2

And now I can apply my displacement load at Corner Node 2 and everything is fine. But looking at the original equation (CornerNode1-CornerNode2 = 0) this wouldn't work.

So in short:
(1) InternalNodeA - InternalNodeB = 0
(2) CornerNode1 - CornerNode2 = 0
(3) InternalNodeA - InternalNodeB = CornerNode1 - CornerNode2

Equation 2 by it self does not make sense to me as CornerNode1 is a fixed support and CornerNode2 is used to apply a load. Once (1)&(2) are connected they work.

It is most likely just a simple thinking error, but I would really like to understand the reason behind it.

Kind regards
Mike

 

[berkeman]

Welcome to PF.

I am currently trying to understand periodic boundary conditions for the mechanical investigation of mechanical properties of a RVE.

Do you mean REV? https://en.wikipedia.org/wiki/Representative_elementary_volume

I found a good video explaining the theory behind it:

A video alone is generally not a great thread starter. Could you please summarize your question using your own screenshots of your simulations? Thanks.

 

[NewStuff]

Thanks Berkeman :)

Do you mean REV? https://en.wikipedia.org/wiki/Representative_elementary_volume

Yes, but I think REV and RVE can be used interchangeably.

A video alone is generally not a great thread starter. Could you please summarize your question using your own screenshots of your simulations? Thanks.

The simulation currently poses no problem, what I am wondering about is the theoretical background.

But I can try it again. So the underlying equations of periodic boundary conditions are as follows:

Source: https://www.physicsforums.com/threa...-workbench-modal-analysis.985108/post-6317474

So if look at these equations isolated: If u¹ = 0 than u³ should be 0 as well. This would be the case if Node 1 was a fixed support.

But, if I combine the equations (e.g. the bottom two on the left) something like this results:

u¹-u³=u⁷-u⁸.

Now you set u¹ = 0 and apply a displacement constraint to Node3 (which represents the applied load). If these combined equations are implemente in Abaqus it does result in periodic deformations, so it works. As an example (just for demonstration purposes) a combined shear/tensile load (the bottom left node is a fixed support):

But the underlying equations now don't seem to valid anymore (to my mind). And I can't figure out why. What these combined equations seem to do is couple the corner nodes to the internal nodes. But for some reason, I can't wrap my head around the logic behind that (or how it works)

A more graphical illustration (from the video linked above):

Kind regards
Mike

 

[FEAnalyst]

In practice, RVE is more common than REV. The former stands for Representative Volume Element.

The goal of this equation constraint in Abaqus is to equalize displacements in a selected DOF for two nodes/node sets. And if you want to apply prescribed displacement then you could do it as it’s described in that older thread you cited.

Check the output files generated during this analysis, Abaqus may warn you about some conflicting comstraints and tell you how ot handled them.

 

[NewStuff]

The goal of this equation constraint in Abaqus is to equalize displacements in a selected DOF for two nodes/node sets. And if you want to apply prescribed displacement then you could do it as it’s described in that older thread you cited.

I basically want to apply combined shear and tensile loads (not necessary via displacement). If I only couple opposing nodes, how do I then introduce the load? Via forces? And how do I support the RVE?

Check the output files generated during this analysis, Abaqus may warn you about some conflicting comstraints and tell you how ot handled them.

Which specific file should I look at? The .log files of the Jobs do not show any error messages.

 

[FEAnalyst]

I basically want to apply combined shear and tensile loads (not necessary via displacement). If I only couple opposing nodes, how do I then introduce the load? Via forces? And how do I support the RVE?

Check the Micromechanics plugin for Abaqus, it automates the process of RVE definition and allows you to apply various driving fields, including strain.

Which specific file should I look at? The .log files of the Jobs do not show any error messages.

Warning messages can be found in .dat and .msg files.