Hyper elasticity is used to model rubbers. The stresses follow from a strain function (with Cij components of the matrix C, and where F is the deformation tensor and U is the stretch tensor following from the polar decomposition of the deformation tensor)
Strictly speaking W is not a strain energy function, because
the Cauchy stresses
are not conjugate to the strain
matrix Cij; the approach obeys the restriction of objectivity however.
The stress rates follow from the time derivative of this law. Typically,
this law is chosen such that it gives only a deviatoric stress contribution.
The hydrostatic stress is obtained by including group_materi_elasti_compressibility.
To obtain a purely deviatoric function, the following strain measures
are used (with I1, I2 and I3 the first, second and third
invariant of the elastic strain matrix respectively)
The group_materi_hyper_besseling function reads ( with K1, K2
and
user defined constants)
The group_materi_hyper_mooney_rivlin function reads (with K1 and K2 user defined constants)