VAFCRP1D

Viscous J2 Steel Model

The VAFCRP1D model is the uniaxial version of the VAFCRP model.

Reference

1. 10.1017/S0368393100118759
2. 10.1179/096034007X207589
3. 10.1016/0749-6419(89)90015-6
4. 10.1002/nme.1620360807

Theory

The VAFCRP model is a von Mises J2 yield criterion based model and uses an associative plasticity flow. The yield function is defined as

$$F=|\sigma -\beta |-k.$$

So the plastic flow is

$${\dot{\epsilon }}^p=\gamma {\displaystyle \frac{\partial F}{\partial \sigma }}=\gamma n,$$

where $n={\displaystyle \frac{\eta }{|\eta |}}={\displaystyle \frac{\sigma -\beta }{|\sigma -\beta |}}=\mathrm{sign}(\sigma -\beta )$.

V

The Voce (1955) type isotropic hardening equation is used.

$$k={\sigma }_y+k_s(1-e^{-mp})+k_{l}p,$$

where ${\sigma }_y$ is the initial elastic limit (yielding stress), $k_s$ is the saturated stress, $k_l$ is the linear hardening modulus, $m$ is a constant that controls the speed of hardening, $\mathrm{d}p=|\mathrm{d}{\epsilon }^p|=\gamma$ is the rate of accumulated plastic strain $p$.

AF

The Armstrong-Frederick (1966) kinematic hardening rule is used. The rate form of back stress ${\beta }^i$ is

$$\mathrm{d}{\beta }^i=a^i\mathrm{d}{\epsilon }^p-b^i\beta \mathrm{d}p,$$

where $a^i$ and $b^i$ are material constants.

CR

A multiplicative formulation (Chaboche and Rousselier, 1983) is used for the total back stress.

$$\beta =\sum {\beta }^i.$$

P

The Peric (1993) type definition is used for viscosity.

$${\displaystyle \frac{\gamma }{\mathrm{\Delta }t}}=\dot{\gamma }={\displaystyle \frac{1}{\mu }}({\left({\displaystyle \frac{|\eta |}{k}}\right)}^{{\displaystyle \frac{1}{ϵ}}}-1),$$

where $\mu$ and $ϵ$ are two material constants that controls viscosity. Note either ($\mu$ or ($ϵ$ can be set to zero to disable rate-dependent response, in that case this model is identical to the Armstrong-Frederick model.

Syntax

material VAFCRP1D (1) (2) (3) (4) (5) (6) (7) (8) [(9) (10)...] [11]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, yield stress
# (4) double, saturated stress
# (5) double, linear hardening modulus
# (6) double, m
# (7) double, mu
# (8) double, epsilon
# (9) double, a
# (10) double, b
# [11] double, density, default: 0.0