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Definition of the First Piola-Kirchoff tensor
Posted 30 juil. 2012, 08:04 UTC−4 Structural Mechanics Version 4.2a 5 Replies
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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
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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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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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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
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