Yield Stress.

A Bingham plastic can absorb some stress without deforming the material. Deformation (strain) only occurs after stress exceeds a minimum threshold. That minimum threshold required before stress causes strain, is the yield stress ( $\begin{array}{l}\tau_y\end{array}$ ), which is the intercept of the stress-strain relationship. HEC-RAS provides three methods for yield stress:

User-specified constant
Exponential formulation
Mohr-Coulomb formula

Exponential

A widely used formula to estimate the yield stress is the exponential formulation (Chien and Ma, 1958; Dai et al., 2014; O'Brien and Julien, 1988)

$\begin{array}{l}\displaystyle \tau_y = a e^{ (b C_v )}\end{array}$

where $\begin{array}{l}a\end{array}$ and $\begin{array}{l}b\end{array}$ are calibration coefficients, and $\begin{array}{l}C_v\end{array}$ is the volumetric concentration between 0 and 1.

Table: Yield stress parameters for the Exponential equation from Julian (1995)

Material	a (Pa)	b
"Typical soil"	0.005	7.5
Kaolinite	0.05	9
Sensitive Clays	0.03	10
Bentonite	0.002	100

The exponential equation works relatively well for hyperconcentrated flows with concentrations between 5% and 30%. However, for high concentrations or very low concentrations the formulation does not work as well. For example a zero concentration produces a yield stress equal to the coefficient a and does not go to zero as it should theoretically.

Mohr-Coulomb

The Mohr-Coulomb yield stress model is given by

$\begin{array}{l}\displaystyle \tau_y = c + \mu \sigma\end{array}$

$\begin{array}{l}\displaystyle \sigma = ( \rho_m - \rho_w ) g h \cos^2 \theta\end{array}$

$\begin{array}{l}\displaystyle \mu = \tan \phi\end{array}$

where $\begin{array}{l}c\end{array}$ is the cohesion or cohesive strength, $\begin{array}{l}\mu\end{array}$ is the Coulomb friction coefficient, $\begin{array}{l}\sigma\end{array}$ is the normal stress at the bottom of the mixture, $\begin{array}{l}\theta\end{array}$ is the bed slope angle, h is the vertical flow depth, and $\begin{array}{l}\phi\end{array}$ is the internal friction angle. The normal stress is computed assuming the flow is parallel to the bed as in Hergarten and Robl (2015). In addition the mixture is assumed to be fully saturated. The internal friction angle depends on mixture but its values are typically between 2.5º and 15º.