Willam-Warnke Yield Criterion - Modified Forms of The Willam-Warnke Yield Criterion | Modified Forms Willam-Warnke Yield Criterion

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

$\bar{\lambda} := \sqrt{\tfrac{2}{3}}~\cfrac{u(\theta)+v(\theta)}{w(\theta)} ~;~~ \bar{B} := \tfrac{1}{\sqrt{3}}~\left$

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 are interpreted as friction coefficients. For the yield surface to be convex, the Willam-Warnke yield criterion requires that and .

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