VAFCRP1D

Viscous J2 Steel Model

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

References

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

Text Only

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