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 \le 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{align} 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{align} $$

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 \le 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{align} 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{align} $$ The quantities $$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 \ge r_c \ge r_t/2$$ and $$0 \le \theta \le \cfrac{\pi}{3}$$.