ArmstrongFrederick
Last updated
Last updated
Armstrong-Frederick Steel Model
Implementation can be found in a separate document.
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.
An exponential function is added to the linear hardening law.
The Armstrong-Frederick type rule is used. Multiple back stresses are defined,
in which
The following applies to v3.6
and later. Check the older syntax in the older version of the documentation.
initial_history(0)
accumulated plastic strain
initial_history(1-6)
initial_history(7-12)
...
more back stresses
Here a few examples are shown.
There is no difference between the classic J2 plasticity model and this AF steel model if only isotropic hardening is defined.
In this case, it is
The cyclic response is shown as follows.
Accordingly, the maximum stress is
The cyclic response is shown as follows.
It is possible to define a zero plastic range response, although the initial stiffness cannot be explicitly assigned.
With some linear isotropic hardening,
where is the initial elastic limit (yielding stress), is the saturation stress, is the linear hardening modulus, is a constant that controls the speed of hardening, is the rate of accumulated plastic strain .
where and are material constants. Note here a slightly different definition is adopted as in the original literature is used instead of . This is purely for a slightly more tidy derivation and does not affect anything.
back stress for the first pair of and
back stress for the second pair of and if defined
If and , 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 and is defined.
By definition, if one set of and is defined, then the maximum stress can be computed as
Of course, multiple sets of and can be defined.
The corresponding maximum stress is .