During plastic deformation, a stress state remains on the boundary of the elastic region, that is, in a plastic state. In other words, loading from a plastic state leads to another plastic state. This can be mathematically represented using the consistency condition,
where 
is the
gradient of the yield function, f, in stress space and a comma represents
the derivative with respect to the corresponding variable. The consistency
condition allows us to define the scalar factor, L, which in turn
represents the magnitude of the plastic strain increment. Using Eqn. 10 and the elastic relations, Eqn. 4, together with
the consistency condition, Eqn. 17, we get
where 
is related to the plastic modulus and is defined as
Using Eqn. 18, we can solve for L,
Substituting Eqn. 12 into Eqn. 20, we get
where 
denotes a tensor product in the sense 
for any second order tensors a and b. Plugging
Eqn. 21 into Eqn. 12, the
elasto-plastic tangent stiffness, 
, during plastic loading,
becomes
