Willam–Warnke yield criterion

The Willam–Warnke yield criterion ^[1] 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

${\displaystyle f(I_{1},J_{2},J_{3})=0\,}$

Three-parameter Willam-Warnke yield surface.

where ${\displaystyle I_{1}}$ is the first invariant of the Cauchy stress tensor, and ${\displaystyle J_{2},J_{3}}$ are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters (${\displaystyle \sigma _{c}}$ - the uniaxial compressive strength, ${\displaystyle \sigma _{t}}$ – the uniaxial tensile strength, ${\displaystyle \sigma _{b}}$ - the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure.

In terms of ${\displaystyle I_{1},J_{2},J_{3}}$, the Willam-Warnke yield criterion can be expressed as

${\displaystyle f:={\sqrt {J_{2}}}+\lambda (J_{2},J_{3})~({\tfrac {I_{1}}{3}}-B)=0}$

where ${\displaystyle \lambda }$ is a function that depends on ${\displaystyle J_{2},J_{3}}$ and the three material parameters and ${\displaystyle B}$ depends only on the material parameters. The function ${\displaystyle \lambda }$ can be interpreted as the friction angle which depends on the Lode angle (${\displaystyle \theta }$). The quantity ${\displaystyle B}$ 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

View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$

Trace of the three-parameter Willam-Warnke yield surface in the ${\displaystyle \sigma _{1}-\sigma _{2}}$-plane for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$

In the original paper, the three-parameter Willam-Warnke yield function was expressed as

${\displaystyle f={\cfrac {1}{3z}}~{\cfrac {I_{1}}{\sigma _{c}}}+{\sqrt {\cfrac {2}{5}}}~{\cfrac {1}{r(\theta )}}{\cfrac {\sqrt {J_{2}}}{\sigma _{c}}}-1\leq 0}$

where ${\displaystyle I_{1}}$ is the first invariant of the stress tensor, ${\displaystyle J_{2}}$ is the second invariant of the deviatoric part of the stress tensor, ${\displaystyle \sigma _{c}}$ is the yield stress in uniaxial compression, and ${\displaystyle \theta }$ is the Lode angle given by

${\displaystyle \theta ={\tfrac {1}{3}}\cos ^{-1}\left({\cfrac {3{\sqrt {3}}}{2}}~{\cfrac {J_{3}}{J_{2}^{3/2}}}\right)~.}$

The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity ${\displaystyle r(\theta )}$ which is given by

${\displaystyle r(\theta ):={\cfrac {u(\theta )+v(\theta )}{w(\theta )}}}$

where

${\displaystyle {\begin{aligned}u(\theta ):=&2~r_{c}~(r_{c}^{2}-r_{t}^{2})~\cos \theta \\v(\theta ):=&r_{c}~(2~r_{t}-r_{c}){\sqrt {4~(r_{c}^{2}-r_{t}^{2})~\cos ^{2}\theta +5~r_{t}^{2}-4~r_{t}~r_{c}}}\\w(\theta ):=&4(r_{c}^{2}-r_{t}^{2})\cos ^{2}\theta +(r_{c}-2~r_{t})^{2}\end{aligned}}}$

The quantities ${\displaystyle r_{t}}$ and ${\displaystyle r_{c}}$ describe the position vectors at the locations ${\displaystyle \theta =0^{\circ },60^{\circ }}$ and can be expressed in terms of ${\displaystyle \sigma _{c},\sigma _{b},\sigma _{t}}$ as (here ${\displaystyle \sigma _{b}}$ is the failure stress under equi-biaxial compression and ${\displaystyle \sigma _{t}}$ is the failure stress under uniaxial tension)

${\displaystyle r_{c}:={\sqrt {\cfrac {6}{5}}}\left[{\cfrac {\sigma _{b}\sigma _{t}}{3\sigma _{b}\sigma _{t}+\sigma _{c}(\sigma _{b}-\sigma _{t})}}\right]~;~~r_{t}:={\sqrt {\cfrac {6}{5}}}\left[{\cfrac {\sigma _{b}\sigma _{t}}{\sigma _{c}(2\sigma _{b}+\sigma _{t})}}\right]}$

The parameter ${\displaystyle z}$ in the model is given by

${\displaystyle z:={\cfrac {\sigma _{b}\sigma _{t}}{\sigma _{c}(\sigma _{b}-\sigma _{t})}}~.}$

The Haigh-Westergaard representation of the Willam-Warnke yield condition can be written as

${\displaystyle f(\xi ,\rho ,\theta )=0\,\quad \equiv \quad f:={\bar {\lambda }}(\theta )~\rho +{\bar {B}}~\xi -\sigma _{c}\leq 0}$

where

${\displaystyle {\bar {B}}:={\cfrac {1}{{\sqrt {3}}~z}}~;~~{\bar {\lambda }}:={\cfrac {1}{{\sqrt {5}}~r(\theta )}}~.}$

Modified forms of the Willam-Warnke yield criterion

Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the ${\displaystyle \pi }$-plane for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$

An alternative form of the Willam-Warnke yield criterion in Haigh-Westergaard coordinates is the Ulm-Coussy-Bazant form:^[2]

${\displaystyle f(\xi ,\rho ,\theta )=0\,\quad {\text{or}}\quad f:=\rho +{\bar {\lambda }}(\theta )~\left(\xi -{\bar {B}}\right)=0}$

where

${\displaystyle {\bar {\lambda }}:={\sqrt {\tfrac {2}{3}}}~{\cfrac {u(\theta )+v(\theta )}{w(\theta )}}~;~~{\bar {B}}:={\tfrac {1}{\sqrt {3}}}~\left[{\cfrac {\sigma _{b}\sigma _{t}}{\sigma _{b}-\sigma _{t}}}\right]}$

and

${\displaystyle {\begin{aligned}r_{t}:=&{\cfrac {{\sqrt {3}}~(\sigma _{b}-\sigma _{t})}{2\sigma _{b}-\sigma _{t}}}\\r_{c}:=&{\cfrac {{\sqrt {3}}~\sigma _{c}~(\sigma _{b}-\sigma _{t})}{(\sigma _{c}+\sigma _{t})\sigma _{b}-\sigma _{c}\sigma _{t}}}\end{aligned}}}$

The quantities ${\displaystyle r_{c},r_{t}}$ are interpreted as friction coefficients. For the yield surface to be convex, the Willam-Warnke yield criterion requires that ${\displaystyle 2~r_{t}\geq r_{c}\geq r_{t}/2}$ and ${\displaystyle 0\leq \theta \leq {\cfrac {\pi }{3}}}$.

View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$	Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the ${\displaystyle \sigma _{1}-\sigma _{2}}$-plane for ${\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}$

See also

• Yield (engineering)
• Yield surface
• Plasticity (physics)