:
The interpretation of the hardening input can be seen from the equation that defines the state of plastic material.
This function is the yield function:
Where,
is the deviatoric stress
is the yield stress
is the back stress
If 
, the material is in an elastic
state. If 
, the material is in a plastic
state.
And then if 
and the deviatoric strain rate
is not zero (
), then the plastic flow is
occurring.
Note that it is not possible for 
. The deviatoric stress is given
by
Where,
is the shear modulus
is the total deviatoric
strain
is the elastic deviatoric
strain
is the plastic deviatoric
strain
The deviatoric strains satisfy the equation:
The relationship between the total deviatoric strain and the true strain 
is:
Where is the true strain,
measured using small strain theory, and 
is the 3x3 identity
matrix.
The yield stress implements the isotropic
hardening. If the isotropic hardening is present, will increase as plastic flow
occurs. If no isotropic hardening is present, then is a constant and does not
change with plastic flow. The exact form of depends on the type of
isotropic hardening that is chosen. In all 3 types of isotropic hardening, is a function of 
, where is defined by the differential
equation:
Where, 
and for any arbitrary 3x3 matrix 
:
The back stress implements the kinematic
hardening. It permits the yield surface defined by 
to shift without changing its
size. This is easiest to understand with a 1-dimensional example. If uni-axial
specimen is stretched, then the kinematic hardening would allow the yield point
in tension to rise and simultaneously for the yield point in compression to
decrease in magnitude.
In 3-dimensional plastic deformation, the back stress allows for the center of the
plastic yield surface to shift in the direction of plastic flow.