non_associated#

Description#

The non_associated behavior is a ZebFront behavior used as an example for non-associated deformation with kinematic hardening variables, which can have deviatoric and spherical components 1there is only a Runge-Kutta implementation in Z8.0 – this model will be implemented in gen_evp in Z8.1. The source for this model can be found in the developer manual.

The model uses elasticity, criterion, flow, and isotropic classes. An interesting use of this model is with a non-associated criterion such as linear_drucker_prager. There is however no such requirement, so the model could be used with von Mises as well.

Any number of kinematic hardening variables are possible. For kinematic \(i\), the back stress \(\ten X\) is calculated as follows:

(295)#\[\begin{split}\begin{aligned} X_{s_i} &= C_{s_i} \alpha_{s_i} \\ \ten{X}_{d_i}& = C_{d_i} \ten{\alpha}_{d_i} \\ \ten{X}_i & = \ten{X}_{d_i} + \delta X_{s_i} \end{aligned}\end{split}\]

Noting that different back stress moduli are available for the two components.

The evolution of back stress is also separated in two components:

(296)#\[\dot{\alpha_{s_i}} = \dot{\lambda}\left[ \mbox{Tr}(\ten n) - \frac{D_{s_i}}{C_{s_i}} X_{s_i} \right]\]
(297)#\[\dot{\ten \alpha_{d_i}} = \dot{\lambda}\left[ \ten n' - \frac{D_{d_i}}{C_{d_i}} \ten X_{d_i} \right]\]

Note here that \(\ten n\) is the criterions normal and not \(\pder{f}{\ten \sigma}\).

Syntax#

The syntax is in standard ZebFront format, with a number of standard sub-classes.

***behavior non_associated \(~\,\) **elasticity <ELASTICITY> \(~\,\) **flow <FLOW> \(~\,\) **isotropic <ISOTROPIC_HARDENING> \(~\,\) **criterion <CRITERION> \(~\,\) **model_coef \(~\,~\,~\,\) Cd1 COEFFICIENT \(~\,~\,~\,\) Cs1 COEFFICIENT \(~\,~\,~\,\) Dd1 COEFFICIENT \(~\,~\,~\,\) Ds1 COEFFICIENT \(~\,~\,~\,\) ...

Stored Variables

prefix

size

description

default

F

UT-2

deformation gradient

yes

sig

T-2

total Cauchy stress

yes

Fp

UT-2

plastic deformation gradient

yes

gamma#

V

resolved shear strains

yes

alpha#

V

back strains on slip system

yes

crss#

V

current resolved shear stress

yes

Example#

***behavior drucker_prager
 **elasticity isotropic
      young   2.25
      poisson 0.125
 **isotropic constant
      R0     0.0011547
 **flow norton
      K 1.e-6
      n 2.
 **criterion linear_drucker_prager
  friction_angle 20.0
  dilatation_angle 20.0
  K        .9
 **model_coef
     Cd1 1.e-1
     Dd1 0.0
     Cs1 1.e-1
     Ds1 0.0
***return