E$4\mathrm{.}5.$ A sample of dense dry sand is set up in a plane strain biaxial apparatus with initial stresses ${}_x^{\text{'}}={}_y^{\text{'}}={}_z^{\text{'}}=100$kPa. A test is carried out in which ${}_y^{\text{'}}$ is kept constant at $100$ kPa and ${}_x^{\text{'}}$ is steadily increased, with no strain$-$in the $z$ direction. It is assumed that ${}_x^{\text{'}}$ and ${}_{,}^{\text{'}},$ remain principal stresses throughout the test. Appropriate stress and strain variables for presentation of results of plane strain tests were introduced in Section $1\mathrm{.}5.$

The experimental results in Fig. $4.$E$1$ suggest an idealisation using a perfectly elastic, perfectly plastic model. Yielding of the ideal material in these plane$-$strain tests is governed by the function

$f(s^{\text{'}},t)=t-s^{\text{'}}sin{}^{\text{'}}=0$

and plastic flow is governed by the plastic potential

$g(s^{\text{'}},t)=t-s^{\text{'}}sin=0$

where ${}^{\text{'}}$ and $$ are angles of friction and dilation, respectively. The elastic behaviour is represented in matrix form by

$\left[\begin{array}{c}{s}^{\text{'}}\\ t\end{array}\right]=\left[\begin{array}{cc}A& B\\ B& C\end{array}\right]\left[\begin{array}{c}{}_s\\ {}_t\end{array}\right]$

Show that $A=K^{\text{'}}+\frac{G^{\text{'}}}{3},B=0,$ and $C=G^{\text{'}},$ where $K^{\text{'}}$ is the bulk

Fig. $4.$E$1$ Biaxial test on dense sand.

$1$

Exercises

$111$

modulus, and $G^{\text{'}}$ is the shear modulus. Estimate values of $K^{\text{'}},G^{\text{'}},$ and Poisson's ratio $v^{\text{'}},$ and the friction angle ${}^{\text{'}}$ for the sand.

Show that after yield the ideal material experiences no elastic strains, and hence derive a value for the dilation angle $.$