For elasto-plastic materials undergoing infinitesimal deformation, the total
strain increments, 
are decomposed additively
into an elastic, 
, and a plastic, 
component. That is,
The elastic component can be represented using the generalized Hooke's law,
that is,
where 
and 
are the tensors of
elastic compliance and elastic moduli respectively, and 
. For isotropic and Green
elastic materials, 
can be represented in terms of two Lame
constants, 
and 
. That is,
A yield stress is a stress state which marks the onset of plastic deformation. A yield surface specify the locus of yield stress states. The existence of a convex yield surface allows to separate the regions of purely elastic and plastic response in stress space. In general, the yield function is assumed to be represented by a function of the form:
where 
is the current state of stress and 
are the
internal variables that define the hardening or softening behavior.
Plastically admissible stress states, , are restrained to
a convex set, such that
The plastic constitutive relations require the direction of plastic loading 
and the loading function L, which in turn determine the
plastic deformation as
It is often assumed that the direction of plastic loading, , may
be derived from a scalar function 
, called a plastic
potential, by means of
Eqn. 7 defines the flow rule. For the simplest case, when the
yield function and the plastic potential function coincide, f=G, the flow
rule is associative. In this case:
however, if 
, the flow rule is non-associative. If the internal
variables are made function of the plastic strains, the evolution rule for can be given by
where h is a scalar function of the plastic strain incremental tensor, 
, and it will be called the hardening
function.
Combining Eqn. 2, Eqn. 3, and Eqn. 7 , we get,
At a plastic state for which 
, and considering
that the scalar factor L is nonzero only when plastic deformations occur,
one may establish the criterion of loading-unloading as :
Or via the Kuhn-Tucker conditions,
This means that, in the case of loading, the stress point tries to move outward of the current yield surface and further plastic deformations occur. Instead, in the case of neutral loading or unloading, the stress point tries to move on or inward from the current yield surface and no further plastic deformations occur.
