VAFCRP1D

Viscous J2 Steel Model

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

References

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

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

So the plastic flow is

$$\dot{\varepsilon}^p=\gamma\dfrac{\partial{}F}{\partial{}\sigma}=\gamma{}n,$$

where $n=\dfrac{\eta}{\Big|\eta\Big|}=\dfrac{\sigma-\beta}{\Big|\sigma-\beta\Big|}=\mathrm{sign}~\left( \sigma-\beta\right)$.

V

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

$$k=\sigma_y+k_s(1-e^{-mp})+k_lp,$$

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, $\mathrm{d}p=\Big|\mathrm{d}\varepsilon^p\Big|=\gamma$ 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

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

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.

$$\dfrac{\gamma}{\Delta{}t}=\dot{\gamma}=\dfrac{1}{\mu}\left(\left(\dfrac{\Big|\eta\Big|}{k}\right) ^{\dfrac{1}{\epsilon}}-1\right),$$

where $\mu$ and $\epsilon$ are two material constants that controls viscosity. Note either $\mu$ or $\epsilon$ 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