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  1. Material
  2. Material1D
  3. vonMises

ArmstrongFrederick1D

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Last updated 8 months ago

1D Armstrong-Frederick Steel Model

This model is a uni-axial version of the steel model. Readers can also refer to the corresponding section in 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.

σy=σi+Kq+σs(1−e−msq),\sigma^y=\sigma^i+Kq+\sigma^s(1-e^{-m^sq}),σy=σi+Kq+σs(1−e−msq),

where σi\sigma^iσi is the initial elastic limit (yielding stress), σs\sigma^sσs is the saturation stress, KKK is the linear hardening modulus, msm^sms is a constant that controls the speed of hardening, dq=∣dεp∣\mathrm{d}q=|\mathrm{d}\varepsilon^p|dq=∣dεp∣ is the rate of accumulated plastic strain qqq.

Kinematic Hardening

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

β=∑βi\beta=\sum\beta_iβ=∑βi​

in which

Syntax

The following applies to v3.6 and later. Check the older syntax in the older version of the documentation.

material ArmstrongFrederick1D (1) (2) (3) (4) (5) (6) [(7) (8)...] [9]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, yield stress, \sigma^i
# (4) double, linear hardening modulus, K
# (5) double, saturation stress, \sigma^s
# (6) double, m^s, saturation rate
# (7) double, a_i, kinematic hardening
# (8) double, b_i, kinematic hardening
# [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

dβi=ai dεp−biβ dq,\mathrm{d}\beta_i=a_i~\mathrm{d}\varepsilon^p-b_i\beta~\mathrm{d}q,dβi​=ai​ dεp−bi​β dq,

where aia_iai​ and bib_ibi​ are material constants.

σmax=σi+∑aibi=50500=0.1.\sigma_{\text{max}}=\sigma^i+\sum\dfrac{a_i}{b_i}=\dfrac{50}{500}=0.1.σmax​=σi+∑bi​ai​​=50050​=0.1.
ArmstrongFrederick
Constitutive Modelling Cookbook
Example 1