Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
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 .
The specimen is subjected to a monotonically increasing hydrostatic stress p, and the volumetric strain is measured.
Calculate a relationship between the pressure and volumetric strain, in terms
of the initial strength of the soil 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
while its plastic properties are characterized by M and c. The fluid has a bulk modulus .
At time the soil has a cavity volume fraction and strength ,
The specimen is subjected to a monotonically increasing hydrostatic pressure p, and is then unloaded. The volumetric
strain is measured. Assume that both elastic and
plastic strains are small. Show that the
relationship between the normalized pressure and normalized volumetric strain is a function of only three dimensionless
material properties: ,
and . Plot the dimensionless pressure-volume curves
(showing both the elastic and plastic parts of the loading cycle for a few
representative values of ,
3.11.3. A drained specimen of Cam-clay is first subjected to a
monotonically increasing confining pressure p,
with maximum value . The confining pressure is then held
constant, and the specimen is subjected to a monotonically increasing shear
stress q. Calculate the volumetric strain and the shear strain 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.