Problems for Chapter 3

 

Constitutive Models: Relations between Stress and Strain

 

 

 

3.11.    Critical State Soils

 

 

3.11.1.  A drained specimen of a soil can be idealized as Cam-clay, using the constitutive equations listed in Section 3.11.  At time t=0 the soil has a strength $a_0$. The specimen is subjected to a monotonically increasing hydrostatic stress p, and the volumetric strain $\Delta V/V={\epsilon }_{kk}$ is measured.  Calculate a relationship between the pressure and volumetric strain, in terms of the initial strength of the soil $a_0$ and the hardening rate c.

 

 

 

3.11.2.  An undrained specimen of a soil can be idealized as Cam-clay, using the constitutive equations listed in Section 3.11.  The elastic constants of the soil are characterized by its bulk modulus K and Poisson’s ratio $\nu$, while its plastic properties are characterized by M and c.  The fluid has a bulk modulus $K_w$. At time $t=0$ the soil has a cavity volume fraction $n_0$ and strength $a_0$, and $p_s=p_w=0$. The specimen is subjected to a monotonically increasing hydrostatic pressure p, and is then unloaded. The volumetric strain $\Delta V/V={\epsilon }_{kk}$ is measured. Assume that both elastic and plastic strains are small.  Show that the relationship between the normalized pressure $p/a_0$ and normalized volumetric strain $K{\epsilon }_{kk}/a_0$ is a function of only three dimensionless material properties: $\alpha =K_w/a_0$, $\beta =K/a_{0}c$ and $\gamma =n_{0}c$.  Plot the dimensionless pressure-volume curves (showing both the elastic and plastic parts of the loading cycle for a few representative values of $\alpha$, $\beta$ and $\gamma$.

 

 

 

3.11.3.  A drained specimen of Cam-clay is first subjected to a monotonically increasing confining pressure p, with maximum value $p>a_0$.   The confining pressure is then held constant, and the specimen is subjected to a monotonically increasing shear stress q.   Calculate the volumetric strain ${\epsilon }_{kk}^p$ and the shear strain ${\epsilon }_{12}^p$ during the shear loading as a function of q and appropriate material properties, and plot the resulting shear stress-shear strain and volumetric strain-shear strain curves as indicated in the figure.