Hyper elasticity is used to model rubbers. It should be combined with a total lagrange formulation for the memory of the material (so use -total or -total_piola for group_materi_memory).
The stresses follow from a strain energy function (with components of the matrix , and where is the deformation tensor and is the stretch tensor following from the polar decomposition of the deformation tensor)
To obtain a purely deviatoric function, the following strain measures are used (with , and the first, second and third invariant of the elastic strain matrix respectively)
The group_materi_hyper_besseling function reads ( with , and user defined constants)
The group_materi_hyper_mooney_blatz_ko function reads (with and user defined constants)
This Blatz-Ko hyperelastic material hardens in compression, and softens slightly in tension; it models a foamlike rubber.
The group_materi_hyper_mooney_rivlin function reads (with and user defined constants)
The group_materi_hyper_neohookean function reads (with a user defined constant)
The group_materi_hyper_reducedpolynomial function reads (with user defined constants)
where a summation over is applied.
We define . Now a volumetric part can be added to the strain energy.
The group_materi_hyper_volumetric_linear contribution reads:
The group_materi_hyper_volumetric_murnaghan contribution reads:
The group_materi_hyper_volumetric_polynomial contribution reads:
The group_materi_hyper_volumetric_simotaylor contribution reads:
The group_materi_hyper_volumetric_ogden contribution reads:
As an example, you can combine the group_materi_hyper_mooney_rivlin energy function with the group_materi_hyper_volumetric_linear so that the total strain energy function becomes:
Here the initial shear modulus and bulk modulus are included as: