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Elastic material in large strain analyses

Submitted by musto on

Hi there,

 I came across the following question on another website, which I found intriguing:

<...> what happens if a linear material is used within a non-linear, large-strain analysis in commercial codes such as ABAQUS, MARC?

I can think about two options:

1) linear strain measure is used in the material law for the linear materials

2) the linear material is "interpreted" as a non-linear material: by replacing linear strain with Green strain and insiting on a linear stress-strain relationship one would end up with a Saint-Venant-Kirchoff material

The problem is that both are unsatisfactory: the first would not accout for rotation, while the second will give unphysical behaviour incompression!

What happens them say in ABAQUS?The problem must be well known in practice, as similar analyses must be quite common, thinking of steel-reinforced rubber bellows, etc. hence I trust a solution has been found.

In principle one could easily derive hyperelastic constants for a Neo-Hookean material to match in the small strain regime a linear material, would this not be the most consistent approach?

<...>

 Then option of using a corotational frame is not mentioned and equally plausible, can anybody answer this doubt?

 Thank you very much


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musto

Hi All,

 one researcher, particularly shy I assume, contacted me on my mail pointing out the Updated Lagrangian approach was not listed on the options, and might be the correct one.

I very much doubt, as I can not see how the use of an Updated Lagrangian coupled with a linear material could cope with anything else than infinitesimlarotations, but I am eager to hear the comments of more expert workers in the field.

 Thanks

 

Marco

Mon, 06/10/2013 - 12:35 Permalink
M. Jahanshahi

When the deformation is large we have to take account of the geometric nonlinearity although the material behavior is elastic. In order to do so we have to use an updated/total lagrangian finite element formulation to solve the problem at hand. As a good reference on the subject one can consult the following book:

Nonlinear Continuum Mechanics for Finite Element Analysis, J. Bonet & R.D. Wood

Mohsen

Tue, 06/11/2013 - 16:41 Permalink
M. Jahanshahi

Marco,

You should make clear what you mean by linear behavior. Your problem probably stems from the fact that you are not using the correct material model. If you want to use large deformation with reversible material behavior you should consider hyperelastic models. For these models the stress always derives from a potentian and this potential is a proper function of F (deformation gradient) or C/b (right Cauchy-Green tensor/left Cauchy-Green tensor). In these types of models the material behavior is independent of the path of loading.

Therefore, I can itemize the followings for you:

1. You should decide whether your deformations are intrinsically large or infinitesimal depending on the nature of your problem.

2. In the case of infinitesimal deformations, it is not necessary to use large deformations.

3. If you have large deformations but elastic behavior you should consider hyperelastic materials. In this case the book in my previous post can be very helpful.

Mohsen

Thu, 06/13/2013 - 18:22 Permalink
musto

Mohsen,

 what you say is understandable, but I did ask what the best way to proceed is, I asked, what happens if a linear material (in the sense, a material for which the Cauchy stress is a linear transformation of the infinitesimal strain tensor) is used in an analysis involving large strains, i.e. how the code handles it from the algorithmic point of view.

 I agree hyperelastic materials are path-independent (if we are happy to consider tautologies as valid logical  statements).

Thanks

 

Fri, 06/14/2013 - 10:19 Permalink
M. Jahanshahi

Marco,

I vever encountered the incidence of doing so but I can say the results can be unpredictable. Many contradictions may arise that can not be resolved so easily. For example because the large deformation capability is enabled the software tries to compute the strain from the lagragian version E = 1/2 (C - I) or eulerian version e = 1/2 (I - b^-1) while this might not be possible probably because it does not compute F at all due to the linearity of material. These are what I think might happen besides other sort of things that can only be explained by the actual developers.

Mohsen

Sat, 06/15/2013 - 06:11 Permalink
Akumar

In abaqus, the linear elastic material model is based on small strain theory. Therefore, in their manual it's clearly written that hyperelastic material model should be used in case of analysis involving large strain. I don't think anyone can do any better job then letting be aware of when to use something and when notto. Abaqus has done an excellent job in documenting limitations. If a user chooses to use linear elastic material in problems involving large strain, it's entirely his fault (wheather he knows about it or not, it doesn't matter). Sorry for being harsh. :)

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Thu, 06/13/2013 - 15:19 Permalink