Drucker–Prager yield criterion

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

${\displaystyle {\sqrt {J_{2}}}=A+B~I_{1}}$

where ${\displaystyle I_{1}}$ is the first invariant of the Cauchy stress and ${\displaystyle J_{2}}$ is the second invariant of the deviatoric part of the Cauchy stress. The constants ${\displaystyle A,B}$ 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

${\displaystyle \sigma _{e}=a+b~\sigma _{m}}$

where ${\displaystyle \sigma _{e}}$ is the equivalent stress, ${\displaystyle \sigma _{m}}$ is the hydrostatic stress, and ${\displaystyle a,b}$ are material constants. The Drucker–Prager yield criterion expressed in Haigh–Westergaard coordinates is

${\displaystyle {\tfrac {1}{\sqrt {2}}}\rho -{\sqrt {3}}~B\xi =A}$

The Drucker–Prager yield surface is a smooth version of the Mohr–Coulomb yield surface.