finite_strain_crystal#

Description#

This behavior is a simple implementation of a finite strain formulation of a single crystal 1Only Runge-Kutta integration is implemented. This behavior is included as an example of ZebFront programming, and the source can be found in the developers manual. This model only allows one crystal orientation to be input, and works for Runge-Kutta integration only.

This model works with the assumption of multiplicative plasticity, where the deformation gradient \(\ten F\) is broken into an elastic part \(\ten F_e\) and a plastic part \(\ten F_p\).

(281)#\[\ten F_e = \ten F \ten F_p^{-1};\]

The elastic deformation gradient is separated into a rotation component and a stress component.

(282)#\[\ten F_e = \ten R_e \ten U_e\]

and the stress is calculated using the logarithmic strain measure from the elastic stretch.

(283)#\[\ten \sigma = \tenf D_{el}:\log(\ten F_e)\]

There is a yield criterion for each slip system.

(284)#\[f_i = \ten m_i:\ten \sigma - C \alpha_i - R(\gamma_i)\]

which is used to calculate the evolution.

(285)#\[\dot{\ten F}_p = \sum_i\dot{v}(f_i)\ten m_i\ten F_p\]

\(\gamma_i\) is the integration of \(\dot{v}(f_i)\).

(286)#\[\dot{\alpha}_i = \dot{\gamma}_i\text{sign}(\ten m_i:\ten \sigma-C\alpha_i) - D\alpha_i\dot{\gamma}_i\]

Syntax#

***behavior finite_strain_crystal \(~\,\) **elasticity <ELASTICITY> \(~\,\) **flow <FLOW> \(~\,\) **isotropic <ISOTROPIC_HARDENING> \(~\,\) **orientation <CRYSTAL_ORIENTATION> \(~\,\) **model_coef \(~\,~\,~\,\) C COEFFICIENT \(~\,~\,~\,\) D 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