Microstructure with PCB + fixed point

[sicheng6699]

Dear Prof. Taylor,

I have a (short) question about the boundary condition in FE2.
First of all, the micro structure (or RVE) is computed alone, not in parallel.
The large deformation is used, and only the first term G11 is defined in the input file for the uniaxial tension case.
 
Below in the animation you can see that the microstructure has additional rigid body motion.
The displacement curves (ux, uy, uz) of the node at left bottom corner, w.r.t. the load factor, are not smooth.

This is a problem for our special investigation. The solution is to fix the middle point of the RVE. I checked the homogenized stress and tangent moduli. It seems that there is no problem.

My question is whether it is correct to do so in feap, and in FE2 version, with a middle node fixed?
Thank you for your reply in advance!

Ps. FEAP version 8.4, linux system.

[sicheng6699]

Two more files are additionally to run the input file, for the plot purpose.
The last one is the displacement results of the left bottom node.

[JStorm]

Within quasi-static homogenisation theory using RVEs you are allowed to fix the six global degrees of freedom (3 translations, 3 rotations) in any way you like. This is not influencing the homogenised stress and tangent. Thus, to avoid instabilities with the statically indeterminate system just add appropriate displacement BCs.

[sicheng6699]

Hello JStorm,

thanks a lot for your very quick reply! First of all, I am glad to hear that the translation dofs could be fixed.

It is hard to explain what I really need, but generally I need all the deformation vectors.
To fixed the translation of RVE is more important than the fixed rotation. In may case the small rotation is not a problem, even it exist.

May I ask further whether it is necessary to fixe the rotational ones?
If I calculate the six deformations G11, G22, G33, G12+G21, G23+G32, G13+G31, the last three have rigid body rotations.
So maybe it should not be fixed, for the situation if only one model is used.

My question now is, maybe there is no free rotation at all, because of PBC? Can you please give some hints? Thank you so much!!

[Prof. R.L. Taylor]

In the current implementation of RVE in feap you must fix all the vertex nodes with fixed boundary conditions.  These nodes will be displaced by the specified G_ij so will deform.   It is not sufficient to just fix the center node.

[JStorm]

PBCs relates the relative displacements between the nodes of opposite surfaces. In general, these do not fix the 6 global DOFs.

But the often used implementation of Kouznetsova (phd thesis 2002) applies PBCs and fixes the global DOFs in one step because all surface DOFs are related to the corner nodes and these are explicitly given in her method.

I am  not familiar with the FE2-feature in FEAP and do not know whether it follows the method of Kouznetsova or a more general way.

[JStorm]

Prof. Taylor was faster 

By the way, which implementation of periodic BCs is used in FEAP?

[sicheng6699]

Many thanks again.

Ich have tried to fix the 8 corner nodes, instead of center node. They seem to be the same in my case..... as you can see one mark for the fixed node at the center. Otherwise the stress and displacement are identical.

For this plot, I calculate in the FE2 framework. The macro structure is just one brick element under uniaxial tension for small deformation. The RVE consist of two different matrials. (The above model by using two different E and v), also with small deformation.

[sicheng6699]

see inputfiles in attachments  with center node fixe. You can also comment this setting and active another setting which fixes the corner nodes.

[sicheng6699]

continued...

and hopefully the input files are complete. 

[Prof. R.L. Taylor]

I had not tried this before but you are getting the same result with either form.  I assume this is true no matter which component of G_ij you specify, since there are no warnings on solving the equations.

In answer to the other  question the FE2 in feap is based on Kouznetsova's work.  The main difference was to combine the transformation to Kirchhoff stress to reduce the number of constrains from 9 to 6.  We did not study her treatment of corner constraints since imposing the vertex displacements was not difficult (and the original algorithm to impose the periodic boundary behavior did not work at the vertices for multiple directions).

We till only treat symmetric nodal meshing for periodic behavior - whereas imposing all the boundary displacements can solve the general case but probably with some error due to mismatches.

[JStorm]

In answer to the other  question the FE2 in feap is based on Kouznetsova's work.  The main difference was to combine the transformation to Kirchhoff stress to reduce the number of constrains from 9 to 6.  We did not study her treatment of corner constraints since imposing the vertex displacements was not difficult (and the original algorithm to impose the periodic boundary behavior did not work at the vertices for multiple directions).

We till only treat symmetric nodal meshing for periodic behavior - whereas imposing all the boundary displacements can solve the general case but probably with some error due to mismatches.

Thank you Prof. Taylor for that clarification.

[sicheng6699]

I had not tried this before but you are getting the same result with either form.  I assume this is true no matter which component of G_ij you specify, since there are no warnings on solving the equations.

In answer to the other  question the FE2 in feap is based on Kouznetsova's work.  The main difference was to combine the transformation to Kirchhoff stress to reduce the number of constrains from 9 to 6.  We did not study her treatment of corner constraints since imposing the vertex displacements was not difficult (and the original algorithm to impose the periodic boundary behavior did not work at the vertices for multiple directions).

We till only treat symmetric nodal meshing for periodic behavior - whereas imposing all the boundary displacements can solve the general case but probably with some error due to mismatches.

Dear Prof. Taylor,

I am reading the dissertation of Kouznetsova, and am trying to understand the algorithm. Based on that we want to develop some new idea.
The main problem is the implementation is not so easy to understand.

Is there any FEAP document what would be helpful to understand in more details, especially how is the Kirchoff stress averaged, and how to compute and push the material elasticity tensor (or matrix)? For example is there any FEAP FE2 theory manual? Many thanks!

One more question, is there any publication which uses FEAP FE2?

[Prof. R.L. Taylor]

There is not FE2 theory manual other than that at the web site.  The theory implemented is summarized in the 6th and 7th edition of the FE books by Zienkiewicz and Taylor (the second volume on Solids and Structures).