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 Camclay, 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 Camclay, 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 pressurevolume 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 Camclay 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 stressshear strain and volumetric strainshear strain curves as indicated in the figure.