The elastic stress rate is

where is the elastic modulus tensor (which is a doubly symmetric tensor: , and ), and is the elastic strain rate. See the plasticity section for a definition of the elastic strain rate.

For an __isotropic__ material

with **group_materi_elasti_young** modulus and **group_materi_elasti_poisson** ratio
(the remaining non-zero moduli follow from the double symmetry conditions).

For a __transverse isotropic__ material the material has one unique
direction (think of an material with fibers in one direction).
Here we take 'a' as the unique direction; 'b' and 'c' are
the transverse directions. The material is fully defined by ,
, , and and the unique direction
in space (see **group_materi_elasti_transverse_isotropy**).
The other non-zero moduli follow from
,
,
and from the
double symmetry conditions.

The __nonlinear elasticity polynomials__ is a strain dependent model.
In this model, the 'young's stiffness' modulus is made dependend of the size
of the strains via a series of polynomials

(1) |

where

(2) |

with the components of the strain matrix.
The parameters etc. need to be specified in the
**group_materi_elasti_young_polynomial** record.

The __power law nonlinear elasticity__ is a stress dependent model which typically is used
to model the elastic behavior of granular materials.
It can be combined with plastic models, by example with the di Prisco plasticity model for soils,
and with a poisson ratio.

In this model, the 'young's stiffness' modulus is made a function of the average stress state:

(3) |

where is the pressure.
Furthermore, is the reference stiffness at reference pressure , and is a soil dependent
power coefficient.
The parameters , , and need to be specified in the **group_materi_elasti_young_power** record.

The __Lade nonlinear elasticity__ is a stress dependent model which typically is used
to model the elastic behavior of granular materials.
It can be combined with plastic models, by example with the di Prisco plasticity model for soils.

The stress rates are linked to the strain rates by the equation:

(4) |

where the function is

where

with pressure and deviatoric stresses .

The model contains three user specified constants , ,
which need to be specified in the **group_materi_elasti_lade** record.
and are defined by means of an isotropic unloading test, and by means of
an unloading-standard-triaxial-compression test.
For example for a loose sand , , .
See [7] for the details.

The model cannot be used in combination with a poisson ratio.