KINEMATIC#
Description#
This object defines the model of kinematic hardening (tensorial back
stresses) applicable in behaviors or potential objects in the
gen_evp
behavior. The models generally store an additional tensorial
internal variable, and therefore may support static recovery and state
coupling without modification. The name of this internal variable will
be determined normally by the object to which it belongs (a
POTENTIAL
for example). Kinematic hardening acts as a translation of
the yield surface in stress space (often deviatoric).
Syntax#
The syntax consists of specifying the type of kinematic object,
supplying an optional name on the same line, and giving the appropriate
coefficient definitions on the following lines. This is summarized below
where the keyword specifying a KINEMATIC
object is assumed to be
*kinematic
:
*kinematic
type [ name ]
where the possible substitutions for type are:
CODE |
CRITERION |
---|---|
|
linear hardening |
|
nonlinear Armstrong-Fredrick hardening with static recovery |
|
same as the |
|
nonlinear hardening based on the Euclidean norm |
|
nonlinear with cumulative influence |
|
nonlinear with reduced ratcheting |
|
linear-nonlinear with criterion |
|
nonlinear model using coefficient matrices for the coefficients |
Kinematic models are all formulated in state variable form, where the stored (integrated) variable is analogous to a strain translation in tensorial strain space. The “back stress” component is then calculated using an appropriate modulus. Most of the kinematic models use the convention for calculating the back stress:
with C
should always be positive.
Evolution of the internal variable is written in the form:
where
linear
The linear kinematic hardening has its internal variable evolve with the inelastic strain:
The only coefficient for this model is the
C
modulus (units of stress). The slope in a uniaxial test will be equal to theC
value.
nonlinear
The nonlinear model evolves much as the isotropic
nonlinear
model does (with a slight change in coefficient meaning). The evolution is the following:The coefficients are
C
andD
for the strain evolution, andM
andm
for the static recovery term. In the absence of these latter two, there will be no static recovery calculation. The saturation of this model occurs at in a uniaxial test at a rate determined byD
. Increases inD
yield faster saturating (more nonlinear) behavior.Static recovery is seen to follow a Norton type formulation in the back stress tensorial space. This kinematic model supports RK, TM, or Reduced TM integrations.
nonlinear_phi
This model installs a kinematic variable with the following evolution:
where the function
is:where the material coefficients
and are namedphim
andomega
. Note that defines the rapidity of saturation in the non-linear coefficient, and is a factor between zero and one interpolating theD
coefficient between initial and saturated values. This kinematic model works in RK, TM, or Reduced TM.
nonlinear_evrad
This model limits the ratcheting effect in biaxial conditions by comparing the flow direction with the back stress. Evolution is calculated as follows:
The coefficients are defined such that uniaxial response is equivalent to the
nonlinear
model given equivalent coefficients. The coefficient names areC
,D
, andeta
. This model works in RK, TM, or Reduced TM, but has no static recovery.
ziegler
This model describes a back stress with hydrostatic and deviatoric components:
which takes the coefficients
C
,D
,M
andm
. In the absence of the latter two, there will be no static recovery. The Ziegler model may not be used with thereduced_plastic
integration.
nonlinear_with_crit
Example#
*kinematic nonlinear_evrad
C 40000.
D 500.
eta 0.4