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\ge0\lt \math\gt and \lt math\gt K\ge0 }[/math]. The law can also be used to model linear softening by entering a negative modulus. Entry of a negative [math]\displaystyle{ E_p\lt \math\gt , however, has to be done by enterin a negative \lt math\gt K }[/math]. This entry will result in [math]\displaystyle{ E_p=K \sigma_{Y0} \lt \math\gt . 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 Laws|hardening law]] are defined with the following properties: {| class="wikitable" |- |- ! Property !! Description !! Units !! Default |- | yield || The initial yield stress (enter in [[ConsistentUnits Command#Legacy and Consistent Units|pressure units]]). This stress corresponds to the axial stress at yield during uniaxial, 3D loading. || [[ConsistentUnits Command#Legacy and Consistent Units|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 \lt tt\gt Khard\lt /tt\gt . || [[ConsistentUnits Command#Legacy and Consistent Units|pressure units]] || 0.0 |- | Khard || Alternatively, you can enter this dimensionless parameter for hardening. It is only used if E\lt sub\gt p\lt /sub\gt is not entered and when entered, it is convert to E\lt sub\gt p\lt /sub\gt using E\lt sub\gt p\lt /sub\gt = \lt math\gt \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.