Stress-Deformation Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 666

CHAPTER 13

Stress-Deformation Analysis for Unsaturated Soils

13.1 INTRODUCTION for the formulation of constitutive relations. One set of

constitutive relationships was proposed for the soil structure
and another set of constitutive relations was proposed for
the water phase. Volumetric deformations of unsaturated
soil specimens were studied under triaxial

The constitutive relationships for volume change relate defor-
mation state variables to stress state variables. Constitutive
relationships extend beyond those required for overall volume
change because it is necessary to predict the relative quantity
(or volume) of each phase comprising the unsaturated soil
(i.e., solid phase, water phase, and air phase).
Several forms of the volume change constitutive equations
can be formulated (e.g., soil mechanics formulation, com-
pressibility formulation, and elasticity formulation). The pri-
mary emphasis is on the application of incremental elasticity;
however, some reference is made to elasto-plastic models.
The geotechnical engineer must understand how one set of
constitutive parameters relate to another set of parameters
defined when using another system. For example, soil param-
eters are sometimes measured in the laboratory and defined
in one system but will need to be converted to another system
of variables when solving practical engineering problems.
Experimental results used for the verification of the con-
stitutive equations are described. Soil properties used in the
volume change constitutive equations come under the gen-
eral term of “volumetric deformation coefficients.”

test loading

conditions.
Matyas and Radhakrishna (1968) introduced the concept
of state parameters for an unsaturated soil. The state param-
eters consisted of several stress variables [i.e., σ m = (σ 1 +
2 σ 3 )/ 3

− u a , σ 1 − σ 3 , and u a − u w for triaxial compression]
along with the initial void ratio and degree of saturation
(i.e., e 0 and S 0 ). Tests were performed on essentially “iden-
tical” soil specimens compacted at the same water content
and dry density. The stress parameters reduced to σ 3 − u a
and u a − u w for isotropic compression. The void ratio and
degree of saturation were used to represent the deformation
state of the unsaturated soil.
Three-dimensional state surfaces were formed with void
ratio and degree of saturation plotted versus the independent
state parameters σ − u a and u a − u w . The state surfaces are,
in essence, constitutive surfaces. Matyas and Radhakrishna
(1968) experimentally tested the uniqueness of the consti-
tutive surfaces while following a series of selected stress
paths. Isotropic and K 0 compression tests were performed
on mixtures of 80% flint powder and 20% kaolin. The total,
pore-air, and pore-water pressures were controlled during
the tests. The constitutive surfaces of void ratio and degree
of saturation versus the σ − u a and u a − u w stress variables
were experimentally measured and used to test the surfaces
for uniqueness.
The void ratio results (Matyas and Radhakrishna, 1968)
produced a single warped surface with the soil structure
always decreasing in volume as the u a − u w stress was
decreased or as the σ − u a stress was increased (Fig. 13.1a).
The results showed that the soil had a metastable soil
structure which collapsed as a result of a gradual reduction

13.1.1 Background of Stress and Deformation Studies
for Unsaturated Soils
Biot (1941) presented a three-dimensional consolidation the-
ory based on the assumption that the soil was isotropic and
behaved in a linear elastic manner. The soil was assumed to be
unsaturated in that the pore-water contained occluded air bub-
bles. Two constitutive relationships were proposed in order to
completely describe the deformation state of the unsaturated
soil. One constitutive relationship was formulated for the soil
structure, and the other constitutive relationship was formu-
lated for the water phase. Two independent stress variables
were used in the formulations. In total, four volumetric defor-
mation coefficients were required to provide linkage between
the stress and deformation states.
Coleman (1962) assumed that the stress components for
an unsaturated soil could be used in an independent manner

in matric suction u a − u w . A soil with a stable structure
would have swelled when the matric suction was decreased.
The results showed a unique constitutive surface for the
soil structure even though the soil structure experienced