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