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AFCO1D

This is an extension of ArmstrongFrederick1D model.

The name represents the following.

  • AF --- Armstrong-Frederick type kinematic hardening rule
  • CO --- Chaboche-Ohno type isotropic hardening reduction rule

References

  1. 1840.20/26854
  2. 10.1016/0013-7944(85)90095-5
  3. 10.1115/1.3162603

Theory

Compared to ArmstrongFrederick1D, this AFCO1D model further adopts the concept of a non-hardening plastic strain region. It is described by the following surface.

\[ h=|\varepsilon^p-\theta|-r \]

In which, \(\theta\) points to the centre of the non-hardening plastic strain region. It may be called the back plastic strain. The field \(r\) describes the size of this surface. It is further used to introduce a reduction term in the isotropic hardening.

\[ \sigma^y=\sigma^i+Kq+\sigma^s\left(1-e^{-m^sq}\right)-\sigma^r\left(1-e^{-m^rr}\right). \]

Further details can be found in the corresponding section in Constitutive Modelling Cookbook.

Syntax

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material AFCO1D (1) (2) (3) (4) (5) (6) (7) (8) (9) [(10) (11)...] [12]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, yield stress, \sigma^i
# (4) double, linear isotropic hardening modulus, K
# (5) double, saturation stress, \sigma^s
# (6) double, m^s, saturation rate
# (7) double, c, between 0 and 1
# (8) double, reduction, \sigma^r
# (9) double, m^r, reduction rate
# (10) double, a, kinematic hardening
# (11) double, b, kinematic hardening
# [12] double, density, default: 0.0

Examples

With a perfectly plastic response, the ArmstrongFrederick1D presents a non-degrading envelop.

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material ArmstrongFrederick1D 1 2E5 4E2 0 0 0
materialTest1D 1 1E-4 100 200 200 200 200 200 200 200 200 200 200 200 200 200 200 300 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400
exit

The cyclic response is the following.

no degradation

By introducing the reduction, one can obtain the following.

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material AFCO1D 1 2E5 4E2 0 0 0 .2 20. 300.
materialTest1D 1 1E-4 100 200 200 200 200 200 200 200 200 200 200 200 200 200 200 300 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400
exit

with degradation