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  1. Material
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ArmstrongFrederick

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Last updated 8 months ago

Armstrong-Frederick Steel Model

References

Theory

Implementation can be found in a separate .

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

Although the plastic flow is associative, the hardening rules are not. As the result, the consistent tangent modulus is not symmetric.

Isotropic Hardening

An exponential function is added to the linear hardening law.

k=σy+ks(1−e−mp)+klp,k=\sigma_y+k_s(1-e^{-mp})+k_lp,k=σy​+ks​(1−e−mp)+kl​p,

where σy\sigma_yσy​ is the initial elastic limit (yielding stress), ksk_sks​ is the saturation stress, klk_lkl​ is the linear hardening modulus, mmm is a constant that controls the speed of hardening, dp=23dεp:dεp\mathrm{d}p=\sqrt{\dfrac{2}{3}\mathrm{d}\varepsilon^p:\mathrm{d}\varepsilon^p}dp=32​dεp:dεp​ is the rate of accumulated plastic strain ppp.

Kinematic Hardening

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

β=∑βi\beta=\sum\beta^iβ=∑βi

in which

dβi=23ai dεp−biβ dp,\mathrm{d}\beta^i=\sqrt{\dfrac{2}{3}}a^i~\mathrm{d}\varepsilon^p-b^i\beta~\mathrm{d}p,dβi=32​​ai dεp−biβ dp,

where aia^iai and bib^ibi are material constants. Note here a slightly different definition is adopted as in the original literature 23\dfrac{2}{3}32​ is used instead of 23\sqrt{\dfrac{2}{3}}32​​. This is purely for a slightly more tidy derivation and does not affect anything.

Syntax

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

material ArmstrongFrederick (1) (2) (3) (4) (5) (6) (7) [(8) (9)...] [10]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, poissons ratio
# (4) double, yield stress
# (5) double, linear hardening modulus
# (6) double, saturation stress
# (7) double, m, saturation rate
# (8) double, a, kinematic hardening parameter
# (9) double, b, kinematic hardening parameter
# [10] double, density, default: 0.0

History Layout

location
parameter

initial_history(0)

accumulated plastic strain

initial_history(1-6)

initial_history(7-12)

...

more back stresses

Example

Here a few examples are shown.

Isotropic Hardening Only

There is no difference between the classic J2 plasticity model and this AF steel model if only isotropic hardening is defined.

material ArmstrongFrederick 1 2E2 .2 .1 1. .05 1000.

Kinematic Hardening Only

If a≠0a\neq0a=0 and b=0b=0b=0, there is no difference between the classic J2 plasticity model and this AF steel model. A linear kinematic hardening rule is implied. Normally at least one set of a1a^1a1 and b1b^1b1 is defined.

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

By definition, if one set of a1a^1a1 and b1b^1b1 is defined, then the maximum stress can be computed as

σmax=σy+32a1b1.\sigma_{max}=\sigma_y+\sqrt{\dfrac{3}{2}}\dfrac{a^1}{b^1}.σmax​=σy​+23​​b1a1​.

In this case, it is

σmax=0.1 GPa+3250500 GPa=222.47 MPa.\sigma_{max}=0.1~\mathrm{GPa}+\sqrt{\dfrac{3}{2}}\dfrac{50}{500}~\mathrm{GPa}=222.47~\mathrm{MPa}.σmax​=0.1 GPa+23​​50050​ GPa=222.47 MPa.

The cyclic response is shown as follows.

Of course, multiple sets of aia^iai and bib^ibi can be defined.

material ArmstrongFrederick 1 2E2 .2 .1 0. 0. 0. 50. 500. 100. 600.

Accordingly, the maximum stress is

σmax=σy+32∑aibi=0.1+32(50500+100600)=426.60 MPa.\sigma_{max}=\sigma_y+\sqrt{\dfrac{3}{2}}\sum\dfrac{a^i}{b^i}=0.1+\sqrt{\dfrac{3}{2}}\left( \dfrac{50}{500}+\dfrac{100}{600}\right)=426.60~\mathrm{MPa}.σmax​=σy​+23​​∑biai​=0.1+23​​(50050​+600100​)=426.60 MPa.

The cyclic response is shown as follows.

Zero Elastic Range

It is possible to define a zero plastic range response, although the initial stiffness cannot be explicitly assigned.

material ArmstrongFrederick 1 2E2 .2 0. 0. 0. 0. 40.82482305 500.

The corresponding maximum stress is 100 MPa100~\mathrm{MPa}100 MPa.

With some linear isotropic hardening,

material ArmstrongFrederick 1 2E2 .2 0. .1 0. 0. 40.82482305 500.

back stress for the first pair of and

back stress for the second pair of and if defined

a1a^1a1
b1b^1b1
a2a^2a2
b2b^2b2
10.1016/S0749-6419(01)00049-3
10.1115/1.3264257
10.1179/096034007X207589
document
Example 1
Example 2
Example 3
Example 3
Example 5
Example 6
Example 7