VAFCRP

Viscous J2 Steel Model

Before I can find a proper name for it, I would call it VAFCRP model. Although the name is a bit weird, it contains all the initials of researchers. Similar models are available as: ArmstrongFrederick , ExpJ2 and NonlinearPeric.

Reference

1. https://doi.org/10.1017/S0368393100118759
2. https://doi.org/10.1179/096034007X207589
3. https://doi.org/10.1016/0749-6419(89)90015-6
4. https://doi.org/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

\[ F=\sqrt{\dfrac{3}{2}(s-\beta):(s-\beta)}-k=q-k. \]

So the plastic flow is

\[ \dot{\varepsilon}^p=\gamma\dfrac{\partial{}F}{\partial{}\sigma}=\sqrt{\dfrac{3}{2}}\gamma{}n, \]

where \(n=\dfrac{\eta}{|\eta|}=\dfrac{s-\beta}{|s-\beta|}\).

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, \(\mathrm{d}p=\sqrt{\dfrac{2}{3}\mathrm{d}\varepsilon^p:\mathrm{d}\varepsilon^p}\) 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

\[ \mathrm{d}\beta^i=\sqrt{\dfrac{2}{3}}a^i~\mathrm{d}\varepsilon^p-b^i\beta~\mathrm{d}p, \]

where \(a^i\) and \(b^i\) are material constants. Note here a slightly different definition is adopted as in the original literature \(\dfrac{2}{3}\) is used instead of \(\sqrt{\dfrac{2}{3}}\). This is purely for a slightly more tidy derivation and does not affect anything.

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.

\[ \dfrac{\gamma}{\Delta{}t}=\dot{\gamma}=\dfrac{1}{\mu}\left(\left(\dfrac{q}{k}\right)^{\dfrac{1}{\epsilon}}-1\right), \]

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.

Also note the Perzyna type definition, which is defined as

\[ \dfrac{\gamma}{\Delta{}t}=\dot{\gamma}=\dfrac{1}{\mu}\left(\dfrac{q}{k}-1\right)^{\dfrac{1}{\epsilon}}, \]

is not used. It shall in fact be avoided as it is less numerically stable than the Peric definition since it is not known whether \(\dfrac{q}{k}-1\) is greater or smaller than \(1\).

Syntax

material VAFCRP (1) (2) (3) (4) (5) (6) (7) (8) (9) [10 11...] [12]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, poissons ratio
# (4) double, yield stress
# (5) double, saturated stress
# (6) double, linear hardening modulus
# (7) double, m
# (8) double, mu
# (9) double, epsilon
# (10) double, a
# (11) double, b
# [12] double, density, default: 0.0

Example

This model is essentially a viscous extension of the ArmstrongFrederick model. Only some different behavior is shown here.

Viscosity

For static analysis with viscosity material, the step time is not analytical time anymore, it represents real time as it is used in the computation of viscous response. The step time shall be properly set to be consistent with the material parameters used in the model.

material VAFCRP 1 2E2 .2 .1 0. 0. 0. 1. 0. 50. 500. 100. 600.
material VAFCRP 2 2E2 .2 .1 0. 0. 0. 1. 10. 50. 500. 100. 600.
material VAFCRP 3 2E2 .2 .1 0. 0. 0. 1. 20. 50. 500. 100. 600.
material VAFCRP 4 2E2 .2 .1 0. 0. 0. 1. 50. 50. 500. 100. 600.

Relaxation

material VAFCRP 1 2E2 .2 .1 0. 0. 0. 1. 10.