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# FEM modeling of incompressibility in rubber hyperelasticity

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(@kk473)
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Joined: 13 years ago

Hi Jorgen:

This is my first post, and thanks for starting and maintaining this site. I have been studying FEM implementations of incompressible hyperelasticity models for rubber-like materials, and have a few questions that I was hoping to get some clarification on. To guide the discussion, my goal is to solve generalized 3D boundary value problems using FEM of rubber-like materials (i.e. Gent/Arruda-Boyce free energy functions) with the ideal starting point of having a strain energy function that incorporates the effects of incompressibility such that it can be directly differentiated to get stress and stiffness. However, if other approaches are better, it would be great to get your point of view.

One approach I tried was following equation (6.57) in Holzapfels book, where the modified strain energy function is defined as: Psi = Psi(F) - p(J-1), where p is the pressure/lagrange multiplier to be solved for depending on the boundary value problem. For example, I took Psi(F) to be say the Gent strain energy function, used known solutions for p (i.e.: [url] http://en.wikipedia.org/wiki/Gent_%28hyperelastic_model%29#Stress-deformation_relations [/url]), and took derivatives of Psi with respect to F (or C) to get stress and stiffness. My stress values are reasonable and are similar those published on the wikipedia page. However, my stiffness tensor has some negative values and the acoustical tensor is 0, which suggests that something isnt quite right - note that I derived both the stress and stiffness symbolically/analytically from the modified strain energy. Is this the correct procedure for the incompressible implementation, or am I missing a trick someplace?

Ive also seen some researchers in the literature use modified strain energy functions to enforce incompressibility by doing something like (see equation (6) of Zhao and Suo, APL 2008, 93:251902 ([url] http://apl.aip.org/resource/1/applab/v93/i25/p251902_s1?isAuthorized=no [/url])) W = mu/2 * (F*F - 2Log(DetF) - 3) + K(DetF-1)^2, where the ratio of K/mu is supposed to be large to enforce incompressibility. However, I have been unable to find a general theoretical framework for doing this for arbitrary strain energy functions in the literature - do you know how stable and general this sort of approach is?

Finally, I am aware of quasi-incompressible approaches based on Simo that use a volumetric/deviatoric split of the deformation gradient, though I havent tried this approach yet.

If in your experience for FEM implementation there is a better way of enforcing incompressibility for hyperelastic constitutive models, I would be grateful to hear your thoughts.

Thanks, and regards,

Harold

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(@jorgen)
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Interesting work. I am aware of the Lagrange multiplier approach which should allow you to get the exact results for an incompressible material. There is no particular trick needed. If your results are not quite right, then I suspect that your derivation may have some problem. Note that you have to be careful when you take partial derivatives of Psi when the energy function contain an unknown multiplier since the multiplier may have a functional dependence.

These days I almost exclusively work with compressible materials, since it is more physical and equally easy to solve in FE applications.

-Jorgen

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(@kk473)
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Dear Jorgen - thanks for your response. I found a very helpful reference by Bonet and Wood on nonlinear continuum mechanics that discusses FEM implementation of incompressible materials, and resolved my issues above.

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Hello all,

Im working on large viscoelastic material behavior of rubbers. Implementing my developed model into a UMAT, I could get correct results for single elements in tension or in shear loadings. The problem is that I dont know how to incorporate the incompressibility constraint in a UMAT. In practice, I increase the bulk modulus but this decreases the time increment sizes terribly specially in structures with a large number of elements (not in single elements). Should I use a UHYPER instead of a UMAT? Moreover, I have a tensorial and a scalar internal variable in my code. Does the UHYPER can solve my problem?

Best regards

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(@vahtuan)
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Joined: 13 years ago

For a large deformation viscoelastic model, you have to implement that in UMAT.

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Hi mark98306
Why do you say that? Does it mean that VUMAT does not work for large strains? or there is a problem with viscoelastic behavior!
If you mention about the reason I will be so grateful.

Regards

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