Stress-Deformation Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 718

Stress-strain models for saturated
and unsaturated soils

Elastic
models

Elastoplastic
models

Linear models

Perfect plastic
models

Hardening
models

Nonlinear models

E - μ models

Associated flow rules

Associated flow rules

K - G models

Nonassociated flow rules Nonassociated flow rules

Isotropic models

Isotropic hardening

Tresca, Von Mises,
Mohr-Coulomb,
Druker-Prager,
spatially mobilized plane,
etc.

Anisotropic models

Kinematic hardening

Generalised Hooke's law,
State surface,
Stress-induced anisotropy,
Hyperbolic model,
etc.

Cam-Clay,
modified Cam-Clay,
Barcelona basic model,
etc.

Figure 13.66 Stress-strain constitutive models for saturated-unsaturated soils.

h T

1
G dτ xz

= [ h 1 h 2 h 3 h 4 h 5 h 6 ]

dγ xz = (13.147)

T

σ = [ σ x σ y σ z τ xy τ yz τ zx ]

1
G dτ yz

T

δ = [11100 ] dγ yz = (13.148)

where:

The bold characters indicate matrices and vectors. The
superscript T designates a transposed matrix.

E = Young's modulus,

μ = Poisson's ratio,

13.7.5 Elastic Models
Elastic models for unsaturated soils (left branch in
Fig. 13.66) are usually based on extensions of Hooke's
law (Fredlund and Morgenstern, 1976). These strain-stress
equations were presented earlier in this chapter using the
σ − u a and u a − u w stress state variables:

H = elastic modulus for the soil structure with respect to

changes in suction, and

G = shear modulus, G = E/ 2 ( 1 − μ) .

The constitutive matrices D, H , and h can be substituted
into Eqs. 13.141 and 13.142 (Fredlund and Gitirana, 2005).
Some nonlinear strain characteristics can be accounted for
by using incremental analysis and the concept of a state
surface (Matyas and Radhakrishna, 1968). The values of E
and H can be obtained for each incremental step through
use of coefficients of compressibility obtained from the void
ratio state constitutive surface. For example,

1
E d(σ x − u a ) −

μ
E d(σ y + σ z −

dε x = 2 u a )

1
H d(u a − u w )

+ (13.143)

1
E d(σ y − u a ) −

μ
E d(σ x + σ z −

dε y = 2 u a )

3 ( 1 2 μ)

m 1

−

E = (13.149)

1
H d(u a − u w )

+ (13.144)

3
m 2

1
E d(σ z − u a ) −

μ
E d(σ x + σ y −

H = (13.150)

dε z = 2 u a )

1
H d(u a − u w )

where:

+ (13.145)

dε v
d(σ mean − u a ) =

1 de

d(σ mean − u a )

1
G dτ xy

m 1 =

dγ xy = (13.146)

1 + e 0