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, E_p 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 yielding material points cross an entire cross section of a material under load, but is usually stable otherwise. It other words, is can be used to model localized to yield zones. To prevent non-physical negative values at 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 converted 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 dε_p is the incremental plastic strain tensor in one time step.