Incompressible Mooney-Rivlin/Neo-Hookean material

[navneet roshan]

Dear Members, 

I changed step-44 and reduced the code to 2-field, results for 2-field and 3-field formulation were matching closely. Then I changed the material to incompressible Neo-Hookean, I choose to use Lagrange Multiplier method formulations i.e. energy functional is as following

with 

 

, (fourth order tensor)

Boundary Condition: I am applying pressure from top, bottom face is fixed( ). 

After solving for   and . I found even if the force is increasing linearly with the time steps(quasi-static), the geometry is not deforming after first time step. Any suggestions will be highly appreciated. 

Attachments:

1) c++ code, runs with the, parameters.prm file in step-44.

2) Plot for deformations after first time step, note that the material does not deform further even if force is increasing.  

Regards,

Navneet R

[Wolfgang Bangerth]

Navneet,
I think I'm not clear what precisely your question is. Are you hoping that we
take a look at your code and debug it for you?

Best
WB

On 5/13/20 11:38 PM, navneet roshan wrote:
>
> Dear Members,
>
> I changed step-44 and reduced the code to 2-field, results for 2-field and
> 3-field formulation were matching closely. Then I changed the material to
> incompressible Neo-Hookean, I choose to use Lagrange Multiplier method
> formulations i.e. energy functional is as following
>

> \Psi = c_1\left(I_1 - 3\right) + p\left( J - 1\right)
> with

> \sigma =  -p \bar{\bar{I}} + 2c_1b
>

> J\mathfrak{c} = 0, (fourth order tensor)
>
> *Boundary Condition:* I am applying pressure from top, bottom face is fixed(
> U_x = U_y= U_z = 0).
> After solving for U  and p. I found even if the force is increasing linearly

> with the time steps(quasi-static), the geometry is not deforming after first
> time step. Any suggestions will be highly appreciated.
>

> *Attachments*:

> 1) c++ code, runs with the, parameters.prm file in step-44.
> 2) Plot for deformations after first time step, note that the material does
> not deform further even if force is increasing.
>
> Regards,
> Navneet R
>

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--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[navneet roshan]

Dear Prof. 

Thank you for the response, Sorry, I did't put the precise question.

My geometry is not deforming after first loading step,

Is it locking behaviour? Do I need to give some special treatment to the incompressibility constraint? as

some references suggests using augmented lagrangian and using selectively reduced integration.

  

Regards,

Navneet R 

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[Wolfgang Bangerth]

On 5/14/20 10:28 AM, navneet roshan wrote:
>
> Thank you for the response, Sorry, I did't put the precise question.
> My geometry is not deforming after first loading step,
> Is it locking behaviour? Do I need to give some special treatment to the
> incompressibility constraint? as
> some references suggests using augmented lagrangian and using selectively
> reduced integration.

It may be that others can speak with more knowledge about the question.

But what have you done already to debug the problem? Have you simplified the
material law by setting certain parameters to zero? Have you considered
uniaxial tests for which you can derive an exact solution, and compared your
code to that exact solution?

From your question, I infer that you have derived a complicated code whose
results you don't understand. That may be because you don't understand the
underlying mathematics (is it locking? do I need a different formulation?) or
because the code has a bug. You need to develop the mental tools to deal with
this kind of question -- you will be in this situation many more times in the
future. My approach is generally to make the problem simpler (like in the
examples above) so that I can derive what I expect theoretically and compare
with what I get practically.

Best
W.

[Ester Comellas]

Hi Navneet,

I agree with Wolfgang's advice. Specifically, I suggest you start by double-checking the formulation you implemented in deal.ii. Surely, your Jc tangent tensor can't be zero! Also, shouldn't the volumetric part of your strain energy density be a function of the bulk modulus kappa, not of p? Something like:

\Psi = c_1\left(I_1 - 3\right) + \frac{1}{2} \kappa \left( J - 1\right)^2

and then the pressure appears in the stress tensor because

 p = - \frac{\partial \Psi_{vol}}{\partial J}

I implemented a mixed u/p quasi-incompressible hyperelastic formulation in a Fortran code during my PhD. I think it is similar to what you're trying to implement. You'll find all the analytical derivations for tangent tensors and the outline of the algorithm I used in my dissertation on ResearchGate. I really recommend the following references, if you haven't checked them already:

Crisfield (1991), Non-linear finite element analysis of solids and structures. John Wiley & Sons.

Bonet and Wood (2008), Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press.

Hope this helps!

Ester

El dijous, 14 maig de 2020 12:38:56 UTC-4, Wolfgang Bangerth va escriure: