Drucker–Prager yield criterion

The Drucker–Prager yield criterion[1] 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 where

is the first invariant of the Cauchy stress and

is the second invariant of the deviatoric part of the Cauchy stress.

The Drucker–Prager model can be written in terms of the principal stresses as If

is the yield stress in uniaxial tension, the Drucker–Prager criterion implies If

is the yield stress in uniaxial compression, the Drucker–Prager criterion implies Solving these two equations gives 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 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 (

) that are used to describe the Mohr–Coulomb yield surface.

[2] If we assume that the Drucker–Prager yield surface circumscribes the Mohr–Coulomb yield surface then the expressions for

, then at those points the Mohr–Coulomb yield surface can be expressed as or, The Drucker–Prager yield criterion expressed in Haigh–Westergaard coordinates is Comparing equations (1.1) and (1.2), we have These are the expressions for

[3] For polyoxymethylene the yield stress is a linear function of the pressure.

However, polypropylene shows a quadratic pressure-dependence of the yield stress.

For foams, the GAZT model[4] uses where

is a critical stress for failure in tension or compression,

The Drucker–Prager criterion can also be expressed in the alternative form The Deshpande–Fleck yield criterion[5] for foams has the form given in above equation.

is a parameter[6] that determines the shape of the yield surface, and

is the yield stress in tension or compression.

[7] This yield criterion is an extension of the generalized Hill yield criterion and has the form The coefficients

are the uniaxial yield stresses in compression in the three principal directions of anisotropy,

are the uniaxial yield stresses in tension, and

are the yield stresses in pure shear.

The Drucker yield criterion has the form where

is a constant that lies between -27/8 and 9/4 (for the yield surface to be convex),

is the yield stress in uniaxial tension.

An anisotropic version of the Drucker yield criterion is the Cazacu–Barlat (CZ) yield criterion [9] which has the form where

are generalized forms of the deviatoric stress and are defined as 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 For thin sheets of metals and alloys, the parameters of the Cazacu–Barlat yield criterion are

Figure 1: View of Drucker–Prager yield surface in 3D space of principal stresses for
Comparison of Drucker–Prager and Mohr–Coulomb (inscribed) yield surfaces in the -plane for
Comparison of Drucker–Prager and Mohr–Coulomb (circumscribed) yield surfaces in the -plane for
Figure 2: Drucker–Prager yield surface in the -plane for
Figure 3: Trace of the Drucker–Prager and Mohr–Coulomb yield surfaces in the -plane for . Yellow = Mohr–Coulomb, Cyan = Drucker–Prager.