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 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 dε_p is the incremental plastic strain tensor in one time step.