Plastic strain
In plastic analysis, the materi_strain_elasti rate follows by subtracting from the materi_strain_total rate the materi_strain_plasti rate
where the materi_strain_total rate is
The materi_strain_plasti rate follows from the
condition that the stress cannot exceed the yield surface.
This condition is specified by a yield function
.
The direction of the plastic strain rate is specified by the stress gradient
of a flow function
.
If the yield function and flow function are chosen to be the same,
the plasticity is called associative, else it is non-associative.
Von-Mises is typically used for metal plasticity. Mohr-Coulomb and Drucker-Prager are typically used for soils and other frictional materials. The plasticity models can freely be combined; the combination of the plasticity surfaces defines the total plasticity surface.
Von-Mises stress, mean stress and deviatoric stress
First some stress quantities which are used in most of the plasticity models are listed.
Equivalent Von-Mises stress:
Compression limiting plasticity model
This group_materi_plasti_compression model uses a special definition for the equivalent stress
where ,
and
are the first, second and third
principal stress respectively.
Each of these is only incorporated if it is a compression stress.
The model now reads
This plasticity surface limits the allowed compression stresses.
Drucker-Prager plasticity model
The group_materi_plasti_druckprag model reads
Here c is the cohesion, which needs to be specified both for the yield function and the flow rule; by choosing different values non-associative plasticity is obtained.
Gurson plasticity model
The group_materi_plasti_gurson model reads
Here f* is the volume fraction of voids.
The rate equation
Tension limiting plasticity model
This group_materi_plasti_tension model uses a special definition for the equivalent stress
where ,
and
are the first, second and third
principal stress respectively.
Each of these is only incorporated if it is a tension stress.
The model now reads
This plasticity surface limits the allowed tension stresses.
A simple model for concrete can be obtained as follows. Ue group_materi_plasti_tension to limit the tension strength ft. Use group_materi_plasti_compression to limit the compressive strength fc. The tension strength could be softened to zero over a plastic strain of, say, 1 percent. The compressive strength could be softened to zero over a plastic strain of, say, 10 percent.
Von-Mises plasticity model
The group_materi_plasti_vonmis model reads
Mohr-Coulomb plasticity model
The group_materi_plasti_mohrcoul model reads
Here c is the cohesion,
is the maximal principal stress
and
is the minimal principal stress.
The angle
needs to be specified for
both the yield condition and the flow rule; by choosing different
values, non-associative plasticity is obtained.
Isotropic Hardening
The size of the plastic strains rate is measured
by the materi_plasti_kappa parameter
This parameter can be used for isotropic hardening. Use the dependency_diagram for this.
Kinematic Hardening
The materi_plasti_rho matrix ,
governs the kinematic
hardening in the plasticity models.
It is used in the yield rule and flow rule to get a new origin by using the
argument
:
Plastic heat generation
The plastic energy loss can be partially turned into heat rate per unit volume q:
where
is a user specified parameter (between 0 and 1)
specifying which part of the plastic energy loss is turned into heat
(see group_materi_plasti_heat_generation).