DruckerPrager model  DruckerPrager
The DruckerPrager plasticity model^{1} is an isotropic elastoplastic model based
on a yield function

(19) 
with the pressuredependent equivalent stress

(20) 
As usual,
is the stress tensor,
is the yield stress
under pure shear, and and are the first invariant and second
deviatoric invariant of the stress tensor.
The friction coefficient is a positive parameter that
controls the influence of the pressure on the yield limit, important for
cohesivefrictional materials such as concrete, soils or other
geomaterials. Regarding MohrCoulomb plasticity, relation to cohesion, , and
the angle of friction, , exists for the DruckerPrager model



(21) 



(22) 
The flow rule is derived from the plastic potential

(23) 
where
is the dilatancy coefficient. An associated
model with
would overestimate the dilatancy of
concrete, so the dilatancy coefficient is usually chosen smaller than the
friction coefficient.
The model is described by the equations

(28) 
which represent the linear elastic law, hardening law, evolution laws
for plastic strain and hardening variable, and the
loadingunloading conditions.
In the above,
is the elastic stiffness
tensor,
is the strain tensor,
is the plastic strain tensor,
is the plastic multiplier,
is the unit
secondorder tensor,
is the
deviatoric stress tensor, is the hardening variable, and a
superior dot marks the derivative with respect to time.
The flow rule has the form given in Eq. (26) at all
points of the conical yield surface with the exception of its vertex,
located on the hydrostatic axis.
For the present model, the evolution
of the hardening variable can be explicitly linked to the plastic
multiplier.
Substituting the flow rule
(26) into Eq. (27) and computing the norm
leads to

(29) 
with a constant parameter
, so the
hardening variable is proportional to the plastic multiplier.
For
, the associated plasticity model
is recovered as a special case.
In the simplest case of linear hardening, the hardening function is a
linear function of , given by

(30) 
where is the initial yield stress, and is the
hardening modulus normalized with the elastic modulus.
Alternatively, an exponential hardening function

(31) 
can be used for a more realistic description of hardening.
The stressreturn algorithm is based on the Newtoniteration.
In plasticity, this is commonly called ClosestPointProjection (CPP),
and it generally leads to quadratic convergence.
The implemented algorithm is convergent in any stress case, but
in the vicinity of the vertex region, quadratic convergence might be
lost because of insufficient regularity of the yield function.
The algorithmic tangent stiffness matrix is implemented for both the
regular case and the vertex region.
Generally, the error decreases quadratically (of course only asymptotically).
Again, in the vicinity of the vertex region, quadratic convergence
might be lost due to insufficient regularity.
Furthermore, the tangent stiffness matrix does not always exist for
the vertex case. In these cases, the elastic stiffness is used
instead.
It is generally safer (but slower) to use the elastic stiffness if you
encounter any convergence problems, especially if your problem is
tensiondominated.
Table 8:
DP material  summary.
Description 
DP material 
Record Format 
DruckerPrager num(in) #
d(rn) # tAlpha(rn) # E(rn) # n(rn) #
alpha(rn) # alphaPsi(rn) # ht(in) #
iys(rn) # lys(rn) # hm(rn) # kc(rn) # [ yieldtol(rn) #] 
Parameters 
 num material model number 

 d material density 

 tAlpha thermal dilatation coefficient 

 E Young modulus 

 n Poisson ratio 

 alpha friction coefficient 

 alphaPsi dilatancy coefficient 

 ht hardening type, 1: linear hardening, 2: exponential
hardening 

 iys initial yield stress in shear, 

 lys limit yield stress for exponential hardening,


 hm hardening modulus normalized with Emodulus (!) 

 kc for the exponential softening law 

 yieldtol tolerance of the error in the yield criterion, default value
1.e14 

 newtonIter maximum number of iterations in search, default value 30 
Supported modes 
3dMat, PlaneStrain, 3dRotContinuum 

Borek Patzak
20180102