The Drucker–Prager yield criterion is a pressure-dependent model for determining whether a material has failed or undergone plastic yielding. The criterion was introduced to deal with the plastic deformation of soils. It and its many variants have been applied to rock, concrete, polymers, foams, and other pressure-dependent materials.
The Drucker–Prager yield criterion has the form
formula_1
where formula_2 is the first invariant of the Cauchy stress and formula_3 is the second invariant of the deviatoric part of the Cauchy stress. The constants formula_4 are determined from experiments.
In terms of the equivalent stress (or von Mises stress) and the hydrostatic (or mean) stress, the Drucker–Prager criterion can be expressed as
formula_5
where formula_6 is the equivalent stress, formula_7 is the hydrostatic stress, and
formula_8 are material constants. The Drucker–Prager yield criterion expressed in Haigh–Westergaard coordinates is
formula_9
The Drucker–Prager yield surface is a smooth version of the Mohr–Coulomb yield surface.
Expressions for A and B.
The Drucker–Prager model can be written in terms of the principal stresses as
formula_10
If formula_11 is the yield stress in uniaxial tension, the Drucker–Prager criterion implies
formula_12
If formula_13 is the yield stress in uniaxial compression, the Drucker–Prager criterion implies
formula_14
Solving these two equations gives
formula_15
Uniaxial asymmetry ratio.
Different uniaxial yield stresses in tension and in compression are predicted by the Drucker–Prager model. The uniaxial asymmetry ratio for the Drucker–Prager model is 
formula_16
Expressions in terms of cohesion and friction angle.
Since the Drucker–Prager yield surface is a smooth version of the Mohr–Coulomb yield surface, it is often expressed in terms of the cohesion (formula_17) and the angle of internal friction (formula_18) that are used to describe the Mohr–Coulomb yield surface. If we assume that the Drucker–Prager yield surface circumscribes the Mohr–Coulomb yield surface then the expressions for formula_19 and formula_20 are
formula_21
If the Drucker–Prager yield surface middle circumscribes the Mohr–Coulomb yield surface then
formula_22
If the Drucker–Prager yield surface inscribes the Mohr–Coulomb yield surface then
formula_23
Drucker–Prager model for polymers.
The Drucker–Prager model has been used to model polymers such as polyoxymethylene and polypropylene. For polyoxymethylene the yield stress is a linear function of the pressure. However, polypropylene shows a quadratic pressure-dependence of the yield stress.
Drucker–Prager model for foams.
For foams, the GAZT model uses
formula_24
where formula_25 is a critical stress for failure in tension or compression, formula_26 is the density of the foam, and formula_27 is the density of the base material.
Extensions of the isotropic Drucker–Prager model.
The Drucker–Prager criterion can also be expressed in the alternative form
formula_28
Deshpande–Fleck yield criterion or isotropic foam yield criterion.
The Deshpande–Fleck yield criterion for foams has the form given in above equation. The parameters formula_29 for the Deshpande–Fleck criterion are
formula_30
where formula_31 is a parameter that determines the shape of the yield surface, and formula_32 is the yield stress in tension or compression.
Anisotropic Drucker–Prager yield criterion.
An anisotropic form of the Drucker–Prager yield criterion is the Liu–Huang–Stout yield criterion. This yield criterion is an extension of the generalized Hill yield criterion and has the form
formula_33
The coefficients formula_34 are 
formula_35
where
formula_36
and formula_37 are the uniaxial yield stresses in compression in the three principal directions of anisotropy, formula_38 are the uniaxial yield stresses in tension, and formula_39 are the yield stresses in pure shear. It has been assumed in the above that the quantities formula_40 are positive and formula_41 are negative.
The Drucker yield criterion.
The Drucker–Prager criterion should not be confused with the earlier Drucker criterion which is independent of the pressure (formula_2). The Drucker yield criterion has the form
formula_43
where formula_3 is the second invariant of the deviatoric stress, formula_45 is the third invariant of the deviatoric stress, formula_46 is a constant that lies between -27/8 and 9/4 (for the yield surface to be convex), formula_47 is a constant that varies with the value of formula_46. For formula_49, formula_50 where formula_32 is the yield stress in uniaxial tension.
Anisotropic Drucker Criterion.
An anisotropic version of the Drucker yield criterion is the Cazacu–Barlat (CZ) yield criterion which has the form
formula_52
where formula_53 are generalized forms of the deviatoric stress and are defined as
formula_54
Cazacu–Barlat yield criterion for plane stress.
For thin sheet metals, the state of stress can be approximated as plane stress. In that case the Cazacu–Barlat yield criterion reduces to its two-dimensional version with
formula_55
For thin sheets of metals and alloys, the parameters of the Cazacu–Barlat yield criterion are