The Willam–Warnke yield criterion is a function that is used to predict when failure will occur in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. This yield criterion has the functional form where I_1 is the first invariant of the Cauchy stress tensor, and J_2, J_3 are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters (\sigma_c - the uniaxial compressive strength, \sigma_t – the uniaxial tensile strength, \sigma_b - the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure. In terms of, the Willam-Warnke yield criterion can be expressed as where \lambda is a function that depends on J_2,J_3 and the three material parameters and B depends only on the material parameters. The function \lambda can be interpreted as the friction angle which depends on the Lode angle (\theta). The quantity B is interpreted as a cohesion pressure. The Willam-Warnke yield criterion may therefore be viewed as a combination of the Mohr–Coulomb and the Drucker–Prager yield criteria.

Willam-Warnke yield function

In the original paper, the three-parameter Willam-Warnke yield function was expressed as where I_1 is the first invariant of the stress tensor, J_2 is the second invariant of the deviatoric part of the stress tensor, \sigma_c is the yield stress in uniaxial compression, and \theta is the Lode angle given by The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity r(\theta) which is given by where The quantities r_t and r_c describe the position vectors at the locations and can be expressed in terms of as (here \sigma_b is the failure stress under equi-biaxial compression and \sigma_t is the failure stress under uniaxial tension) The parameter z in the model is given by The Haigh-Westergaard representation of the Willam-Warnke yield condition can be written as where

Modified forms of the Willam-Warnke yield criterion

An alternative form of the Willam-Warnke yield criterion in Haigh-Westergaard coordinates is the Ulm-Coussy-Bazant form: where and The quantities r_c, r_t are interpreted as friction coefficients. For the yield surface to be convex, the Willam-Warnke yield criterion requires that and.