Hyper elasticity is used to model rubbers. The stresses follow from a
strain 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)
Strictly speaking
is not a strain energy function, because
the Cauchy stresses
are not conjugate to the strain
matrix
;
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
,
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_rivlin function reads (with
and
user
defined constants)