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 normally 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 will be unstable if is crosses entire cross section of a material under applied load, but is usually stable otherwise or when localized to a yield zone. To prevent non-physical negative values are large plastic strain, the softened yield stress must be limited to a minimum yield stress value (the default minimum is zero).
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. 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 |
| yieldMin | The minimum yield stress. This minimum only matters when using negative Khard or when modeling softening plasticity | pressure units | 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.