# Willam–Warnke yield criterion

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
    $f(I_{1},J_{2},J_{3})=0\,$
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 $I_{1},J_{2},J_{3}$, the Willam-Warnke yield criterion can be expressed as
    $f:={\sqrt {J_{2}}}+\lambda (J_{2},J_{3})~({\tfrac {I_{1}}{3}}-B)=0$
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
    $f={\cfrac {1}{3z}}~{\cfrac {I_{1}}{\sigma _{c}}}+{\sqrt {\cfrac {2}{5}}}~{\cfrac {1}{r(\theta )}}{\cfrac {\sqrt {J_{2}}}{\sigma _{c}}}-1\leq 0$
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
    $\theta ={\tfrac {1}{3}}\cos ^{-1}\left({\cfrac {3{\sqrt {3}}}{2}}~{\cfrac {J_{3}}{J_{2}^{3/2}}}\right)~.$
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
    $r(\theta ):={\cfrac {u(\theta )+v(\theta )}{w(\theta )}}$
where
    ${\begin{aligned}u(\theta ):=&2~r_{c}~(r_{c}^{2}-r_{t}^{2})~\cos \theta \\v(\theta ):=&r_{c}~(2~r_{t}-r_{c}){\sqrt {4~(r_{c}^{2}-r_{t}^{2})~\cos ^{2}\theta +5~r_{t}^{2}-4~r_{t}~r_{c}}}\\w(\theta ):=&4(r_{c}^{2}-r_{t}^{2})\cos ^{2}\theta +(r_{c}-2~r_{t})^{2}\end{aligned}}$
The quantities $r_{t}$ and $r_{c}$ describe the position vectors at the locations $\theta =0^{\circ },60^{\circ }$ and can be expressed in terms of $\sigma _{c},\sigma _{b},\sigma _{t}$ as (here $\sigma _{b}$ is the failure stress under equi-biaxial compression and $\sigma _{t}$ is the failure stress under uniaxial tension)
    $r_{c}:={\sqrt {\cfrac {6}{5}}}\left[{\cfrac {\sigma _{b}\sigma _{t}}{3\sigma _{b}\sigma _{t}+\sigma _{c}(\sigma _{b}-\sigma _{t})}}\right]~;~~r_{t}:={\sqrt {\cfrac {6}{5}}}\left[{\cfrac {\sigma _{b}\sigma _{t}}{\sigma _{c}(2\sigma _{b}+\sigma _{t})}}\right]$
The parameter $z$ in the model is given by
    $z:={\cfrac {\sigma _{b}\sigma _{t}}{\sigma _{c}(\sigma _{b}-\sigma _{t})}}~.$
The Haigh-Westergaard representation of the Willam-Warnke yield condition can be written as
    $f(\xi ,\rho ,\theta )=0\,\quad \equiv \quad f:={\bar {\lambda }}(\theta )~\rho +{\bar {B}}~\xi -\sigma _{c}\leq 0$
where
    ${\bar {B}}:={\cfrac {1}{{\sqrt {3}}~z}}~;~~{\bar {\lambda }}:={\cfrac {1}{{\sqrt {5}}~r(\theta )}}~.$

## 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:
    $f(\xi ,\rho ,\theta )=0\,\quad {\text{or}}\quad f:=\rho +{\bar {\lambda }}(\theta )~\left(\xi -{\bar {B}}\right)=0$
where
    ${\bar {\lambda }}:={\sqrt {\tfrac {2}{3}}}~{\cfrac {u(\theta )+v(\theta )}{w(\theta )}}~;~~{\bar {B}}:={\tfrac {1}{\sqrt {3}}}~\left[{\cfrac {\sigma _{b}\sigma _{t}}{\sigma _{b}-\sigma _{t}}}\right]$
and
    ${\begin{aligned}r_{t}:=&{\cfrac {{\sqrt {3}}~(\sigma _{b}-\sigma _{t})}{2\sigma _{b}-\sigma _{t}}}\\r_{c}:=&{\cfrac {{\sqrt {3}}~\sigma _{c}~(\sigma _{b}-\sigma _{t})}{(\sigma _{c}+\sigma _{t})\sigma _{b}-\sigma _{c}\sigma _{t}}}\end{aligned}}$
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 $2~r_{t}\geq r_{c}\geq r_{t}/2$ and $0\leq \theta \leq {\cfrac {\pi }{3}}$.
| View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for $\sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7$ | Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the $\sigma _{1}-\sigma _{2}$-plane for $\sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7$ |