Linear Hardening
In the linear hardening law, the yield stress is given by
[math]\displaystyle{ \sigma_y = \sigma_{Y0} + E_p\alpha = \sigma_{Y0}(1+K\alpha) }[/math]
where [math]\displaystyle{ \sigma_{Y0} }[/math] is initial yield stress, Ep is the plastic modulus, α is cumulative equivalent plastic strain, and K is a hardening coefficient.
Hardening and Softening
As implied by the name "Linear Hardening", the law normal models a response where the yield strength increases with plastic strain [math]\displaystyle{ \alpha }[/math]. In other words, [math]\displaystyle{ E_p=\ge 0 }[/math] and [math]\displaystyle{ K\ge0 }[/math]. This law can also be used to model linear softening by entering a negative modulus. Entry of a negative [math]\displaystyle{ E_p }[/math], however, has to be done by entering a negative [math]\displaystyle{ K }[/math]. This entry will result in [math]\displaystyle{ E_p=K \sigma_{Y0} }[/math]. Softening behavior tends to lead to unstable simulations, but it can be used when yielding is localized or maybe with addition of effecting damping.
Hardening Law Properties
The material parameters in this hardening law are defined with the following properties:
| Property | Description | Units | Default |
|---|---|---|---|
| yield | The initial yield stress (enter in pressure units). This stress corresponds to the axial stress at yield during uniaxial, 3D loading. | pressure units | Very Large |
| Ep | The plastic modulus. This modulus is the slope of total stress as a function of plastic strain during uniaxial, 3D loading. The default of 0.0 results in an elastic-perfectly plastic material or a material with no hardening. It must be non-negative, but you can enter softening by using negative Khard. | pressure units | 0.0 |
| Khard | Alternatively, you can enter this dimensionless parameter for hardening. It is only used if Ep is not entered and when entered, it is convert to Ep using Ep = [math]\displaystyle{ \sigma_{Y0}K }[/math]. K can be positive (hardening) or negative (softening) | pressure units | 0.0 |
History Data
This hardening law defines one history variable, which is stored as history variable #1. It stores the the cumulative equivalent plastic strain (absolute) defined as
[math]\displaystyle{ \alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p|| }[/math]
where dεp is the incremental plastic strain tensor in one time step.