Description
This criterion is a non-associated criterion (flow direction is not the
same as the normal to the yield surface \(\pder{f}{\ten \sigma}\). It is
commonly used for soils and plastics.
\(\beta\) friction angle (general coef)
\(\phi\) dilatation angle (general coef)
\(K\) ratio of triaxial yield in tension to triaxial yield in compression; \(0.778\leq K \leq 1\)
(355)\[\begin{split}\begin{aligned}
\ten S &= \tenf U:\ten \sigma = \ten \sigma + p\ten 1 \\
p &= -\frac{1}{3}\ten 1:\ten \sigma \\
q &= \left[\dfrac{3}{2}\ten S:\ten S\right]^{1/2} \\
r^3 &= \left[\dfrac{9}{2}\ten S\cdot\ten S:\ten S\right] \\
\end{aligned}\end{split}\]
(356)\[\begin{split}\begin{aligned}
t& = \dfrac{q}{2}\left[1+\frac{1}{K}-\left(1-\frac{1}{K}\right)\dfrac{r^3}{q^3}\right]\\
f& = t - p~\hbox{tan}\beta - d
\end{aligned}\end{split}\]
(357)\[\begin{split}\begin{aligned}
\hskip1cm&d=\left(1-\dfrac{1}{3}\hbox{tan}\beta\right)\sigma_c \qquad
\hbox{ $\sigma_c$ is the compressive yield} \\
\hskip1cm&d=\left(1+\dfrac{1}{3}\hbox{tan}\beta\right)\sigma_t \qquad
\hbox{ $\sigma_t$ is the tensile yield} \\
\hskip1cm&d=d\equiv \frac{\sqrt{3}}{2}\tau(1+K^{-1}) \qquad
\hbox{ $\tau$ = shear yield (cohesion)} \\
\end{aligned}\end{split}\]
(358)\[\begin{split}\begin{aligned}
g& = t - p\hbox{tan}\psi \\
\vect n &= \frac{1}{c}\pder{g}{\ten \sigma}
\end{aligned}\end{split}\]
(359)\[\begin{split}\begin{aligned}
\hskip1cm&c=1-\dfrac{1}{3}\hbox{tan}\psi \qquad
\hbox{defined in compression yield} \\
\hskip1cm&c=\frac{1}{K}\left(1+\dfrac{1}{3}\hbox{tan}\psi\right) \qquad
\hbox{defined in tension yield} \\
\hskip1cm&c=\left[\frac{5}{2}+\frac{5}{2K^2} -\frac{4}{K}\right]^{1/2}\qquad
\hbox{defined in pure shear (cohesion)} \\
\end{aligned}\end{split}\]
Note
Since this criterion is not associated, it should only be used with
behaviors and criterion which accept a different criterion function from
normal definition.
Syntax
The criterion takes a few options and parameters. The following assumes
that the criterion is selected with a **criterion command as in
gen_evp behaviors.
**criterion linear_drucker_prager
\(~\,~\,\) friction_angle COEFFICIENT
\(~\,~\,\) dilatation_angle COEFFICIENT
\(~\,~\,\) K COEFFICIENT
\(~\,\) [ use_sigma_c ]
\(~\,\) [ use_sigma_t ]
\(~\,\) [ use_cohesion ]
