Finite deformation with mixed 2D element

[K.Li]

Dear Prof. Taylor and FEAPpv users,

A few years ago, I have tried to use the Neo-Hookean material Model for incompressible material under finite deformation in the FEAPpv, but it did not work out. You can find my old post here,

http://feap.berkeley.edu/forum/index.php?topic=1300

Now it seems the 2D mixed element has been added to the new FEAPpv 4.1. So I have prepared a user material subroutine (umati5.f and umatl5.f, see attached) based on the modified Neo-Hookean subroutine came with FEAPpv. The reason why I am doing this with FEAPpv is to provide an example user material subroutine for my students to learn.

I have tested this subroutine with a rectangular model under uniaxial tension test, see the attached input file "iuni2d.txt". When I set the bulk modulus K (parameter "bk" in my input file) to 2.0d4, it worked well when compared with the analytical solution, see the green and orange curves in the plot below. But if I change the bulk modulus to 1d5, the stress becomes very large, see the blue curve in the plot. I have experienced similar behavior with FEAP too. So can someone please explain what has caused this strange behavior? How should I choose a proper bulk modulus for incompressible materials?

[Prof. S. Govindjee]

What happens when you use the built in neoh model instead of your user model?

[Prof. R.L. Taylor]

First, what problem are you trying to solve?  A fully incompressible one or a compressible one?  If the former than you should carefully check that the augmenting really worked.  The Uzawa algorithm will converge very slowly if parameters are not proper.  I upped the bulk modulus a few orders and it converged much better (on feap).  So check carefully the output file and the log file if you run with feap.  Also output an element stress state and check what you get.  You may want to modify feappv to give extra things (like feap does) to help.

So the moral is: set parameters to replicate near incompressibility (Poisson ratio above 0.49 if possible) then use Uzawa augmenting.

[Prof. R.L. Taylor]

One other observation is that the modified Neohook works much better than standard Neohook model.

[K.Li]

Hi, thank you for all your help and information!

I have tried the built-in neo-Hookean and modified neo-Hookean models in FEAPpv first, I was not able to get anything close to the analytical solution no matter how I change the Poisson's ratio. That was why I decided to try the user material subroutine in FEAPpv since the 2D mixed element is now available.

Yes, I am trying to solve a fully incompressible problem. But it seems this is difficult with FEAPpv. Although I got a good agreement with the analytical solution when I set the bulk modulus to 2.0d4 last time, I checked the volume change of my model, I found that the volume is not consistent meaning it becomes compressible already.

I checked the user subroutine with FEAP 8.4. With a higher bulk modulus, I was able to get a more incompressible behavior. The volume of the model is almost consistent. But the stress result is a little bit off. I will check the subroutine with FEAP 8.5 again to see if it is better.   

[Prof. R.L. Taylor]

1. There is a bug in the mixed finite deformation element in FEAPpv - it will be updated in vers 5.1 soon.

2. FEAPpv does not have the augmenting fully implemented.

3. What is meant by "volume is almost consistent"?  How does the solution differ depending on using NEOH or MNEOhook model?

[K.Li]

Dear Prof. Taylor:

Thank you again for your help! It would be great if we can get the mixed finite deformation element working in FEAPpv. That will definitely help our students learning and testing the mixed finite element formulation discussed in your FEM book.

I tested the same subroutine in FEAP 8.5 too. With a higher bulk modulus such as 1d8, the model becomes incompressible meaning the total area (volume if 3D) of the model stays the same as it was in the initial configuration. 

With the built-in MNEOhook model in FEAP 8.5, I was able to get an incompressible behavior with a Poisson's ratio of 0.49999. But the stress result is much smaller than the analytical solution.  With the NEOH model, I got the same result. 



With the built-in MNEOhook model in FEAPpv 4.1i, I was able to get a result with a Poisson's ratio of 0.49999. But the stress result is way too high, and the deformed shape does not look correct. With the NEOH model, it blows up. 



 

[K.Li]

I have now tested the user material subroutine with FEAP 3D elements, it turns out to be perfect.




I think this shows that the subroutine is correct, but somehow it does not work well with the 2D mixed element formulation in FEAP.

[Prof. R.L. Taylor]

The mixed 2-d finite deformation element (fld2d2.f) may be corrected by replacing 'xl' by 'xu' in the call to 'interp2d'.   This does not correct the missing 'augment' option, which will be added in ver5.1 soon.

[K.Li]

Thank you for the note! It does improve the solution with a higher bulk modulus.

I look forward to the new release with support for augmented iteration