ArmstrongFrederick1D

1D Armstrong-Frederick Steel Model

This model is a uni-axial version of the ArmstrongFrederick steel model. Readers can also refer to the corresponding section in Constitutive Modelling Cookbook for details on the theory.

Theory

A von Mises type yield function is used. The associated plasticity is assumed. Both isotropic and kinematic hardening rules are employed.

Isotropic Hardening

An exponential function is added to the linear hardening law.

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

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, \(\mathrm{d}p=\Big|\mathrm{d}\varepsilon^p\Big|\) is the rate of accumulated plastic strain \(p\).

Kinematic Hardening

The Armstrong-Frederick type rule is used. Multiple back stresses are defined,

\[ \beta=\sum\beta^i \]

in which

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

where \(a^i\) and \(b^i\) are material constants.

Syntax

material ArmstrongFrederick1D (1) (2) (3) (4) (5) (6) [(7) (8)...] [9]
# (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, a
# (8) double, b
# [9] double, density, default: 0.0

Example

Kinematic Hardening Only With No Elastic Range

material ArmstrongFrederick1D 1 2E2 0. 0. 0. 0. 50 500.

The maximum stress can be computed as

\[ \sigma_{max}=\sigma_y+\sum\dfrac{a^i}{b^i}=100~\mathrm{MPa}. \]