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Definition of the First Piola-Kirchoff tensor

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Dear Sirs,

I am currently using Comsol 4.2a. I am performing numerical tests of structural problems both using the Structural Mechanics module and the General PDE module. I am modelling the material as Neo-Hookean in the nearly-incompressible form.

My problem is that, using the two modules, performing the same test on the same geometry and using the same BCS, the results are clearly different. It's evident I am making some errors.

Since in the General PDE - Weak form (that is what I am using) I have to specify the weak formulation of the problem, I define the first Piola-Kirchhoff tensor and then use. I am wondering where, when the Structural Mechanics module is used, I can see the definition of the first Piola Kirchhoff tensor.

Please note: As you know, in the Structural Mechanics module, when I insert the strain energy function and I activate the Equation view, I can already find a definition of the First Piola-Kirchhoff tensor. That one is useless to check how the tensor P is defined. In fact, it is formulated as the product of F*S where F is the deformation gradient and S is the second Piola Kirchhoff tensor. Recursively, the definition of the tensor S is given in terms of general derivatives of the strain energy functions ( \partial W / \partial C) where W is the strain energy and C is the right Cauchy-Green tensor.

Is there a way to check the tensorial expression of P and not its symbolic one?

Thanks a lot,

I wish you a nice day.

Paolo Tricerri

5 Replies Last Post 17 janv. 2015, 14:16 UTC−5
Nagi Elabbasi Facebook Reality Labs

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Posted: 1 decade ago 30 juil. 2012, 09:32 UTC−4
Hi Paolo,

Based on the equations provided in COMSOL it seems that they formulate the hyperelastic materials derivatives primarily in terms of the Second Piola-Kirchhoff stress tensor (as the derivative of the strain energy density function with respect to the right Cauchy Green tensor). In that case the calculation of the First Piola-Kirchhoff stress is the secondary expression that you mentioned. You are trying to formulate your PDE implementation it seems using the First Piola-Kirchhoff stress so you cannot directly compare expressions. The final solutions (displacements, stresses, etc.) should be the same though.

Nagi Elabbasi
Veryst Engineering
Hi Paolo, Based on the equations provided in COMSOL it seems that they formulate the hyperelastic materials derivatives primarily in terms of the Second Piola-Kirchhoff stress tensor (as the derivative of the strain energy density function with respect to the right Cauchy Green tensor). In that case the calculation of the First Piola-Kirchhoff stress is the secondary expression that you mentioned. You are trying to formulate your PDE implementation it seems using the First Piola-Kirchhoff stress so you cannot directly compare expressions. The final solutions (displacements, stresses, etc.) should be the same though. Nagi Elabbasi Veryst Engineering

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Posted: 1 decade ago 30 juil. 2012, 10:55 UTC−4
Dear Mr. Elabbasi,

I thank you for your answer. You are perfectly right when you say that the results have to be same no matters which form I use. By the way, I had a look at the Documentation of Comsol and I have seen that even in the nearly incompressible case, Comsol uses a mixed formulation computing also the pressure field. I am totally missing that in my General PDE setting. That is probably why I am getting so different results.

I will try to add that part in my formulation.

Many thanks again,

Paolo
Dear Mr. Elabbasi, I thank you for your answer. You are perfectly right when you say that the results have to be same no matters which form I use. By the way, I had a look at the Documentation of Comsol and I have seen that even in the nearly incompressible case, Comsol uses a mixed formulation computing also the pressure field. I am totally missing that in my General PDE setting. That is probably why I am getting so different results. I will try to add that part in my formulation. Many thanks again, Paolo

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Posted: 1 decade ago 30 juil. 2012, 11:28 UTC−4
Good afternoon,

I small update to close the thread.

I have moved to the incompressible case using also the pressure. The results are much closer now to my "reference" solution than before. I guess the problem was clearly the mixed formulation I was missing.

Thanks again for the support.

Paolo
Good afternoon, I small update to close the thread. I have moved to the incompressible case using also the pressure. The results are much closer now to my "reference" solution than before. I guess the problem was clearly the mixed formulation I was missing. Thanks again for the support. Paolo

Nagi Elabbasi Facebook Reality Labs

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Posted: 1 decade ago 30 juil. 2012, 11:31 UTC−4
You’re welcome Paolo, and thanks for sharing the latest update about the mixed formulation.
You’re welcome Paolo, and thanks for sharing the latest update about the mixed formulation.

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Posted: 10 years ago 17 janv. 2015, 14:16 UTC−5
can u tell me how u have defined piols kirchoff tensor?
can u tell me how u have defined piols kirchoff tensor?

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