DuncanSelig

Plane Strain Duncan-Selig Soil Model

References

1. 10.3141/2511-07

Syntax

material DuncanSelig (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) [11]
# (1) int, unique material tag
# (2) double, reference atmospheric pressure, postive, p_a
# (3) double, reference elatic modulus, E_r
# (4) double, elastic modulus exponent, n
# (5) double, reference bulk modulus, B_r
# (6) double, bulk modulus exponent, m
# (7) double, initial friction angle, \phi_i
# (8) double, friction angle slope, \Delta\phi
# (9) double, ratio of actual failure to ultimate failure, r_f
# (10) double, cohesion, c
# [11] double, density, default: 0.0

Theory

The constitutive relationship can be expressed as

$$\dot{\sigma }={\displaystyle \frac{3B}{9B-E}}\left[\begin{array}{ccc}3B+E& 3B-E& 0\\ 3B-E& 3B+E& 0\\ 0& 0& E\end{array}\right]\dot{\epsilon }.$$

Note it is an incremental form of the constitutive relationship. Symbols $B$ and $E$ denote bulk and elastic modulus, respectively.

The elastic modulus $E$ is a function of stress.

$$E=E_i{(1-{\displaystyle \frac{{\sigma }_d}{{\sigma }_{d,max}}})}^2,$$

with

$$E_i=E_r{\left({\displaystyle \frac{{\sigma }_3}{p_a}}\right)}^n,{\sigma }_{d,max}={\displaystyle \frac{2}{r_f}}{\displaystyle \frac{c\cos \varphi +{\sigma }_3\sin \varphi }{1-\sin \varphi }}.$$

The friction angle $\varphi$ decreases with increasing ${\sigma }_3$.

$$\varphi ={\varphi }_i-\mathrm{\Delta }\psi {\log }_{10}\left({\displaystyle \frac{{\sigma }_3}{p_a}}\right).$$

The deviatoric stress ${\sigma }_d$ is the difference between the major and minor principal stresses.

$${\sigma }_d={\sigma }_1-{\sigma }_3.$$

The bulk modulus $B$ is a function of ${\sigma }_3$.

$$B=B_r{\left({\displaystyle \frac{{\sigma }_3}{p_a}}\right)}^m.$$