Civil Engineering Reference
In-Depth Information
For materials that are isotropic and elastic and perfectly plastic (see Sec. 3.9),
J
1
=
J
2
,
and the stiffness matrix is symmetric, while for materials that are isotropic and elastic,
J
1
J
2
0 (see Sec. 3.8) so that shearing and volumetric effects are decoupled.
Alternatively, a constitutive equation can be written as
=
=
δε
C
11
C
12
C
21
C
22
δ
q
s
δε
=
(3.26)
p
δ
v
where [
C
] is a compliance matrix containing compliance parameters. Comparing
Eqs. (3.21) and (3.26), [
C
] is the inverse of [
S
] and, in general, there are no simple
relationships between the stiffness parameters in [
S
] and the compliance parameters
in [
C
]. However, for materials that are isotropic and elastic, shear and volumetric
effects are decoupled so that
C
12
=
1/3
G
and
C
21
=
0 and in this case
C
11
=
1/
S
11
=
1/
K
.
Since the stress-strain behaviour of soil is largely non-linear stiffness and compliance
parameters will not be constants, but will vary with strain. They also depend on the
current stresses and on the history of loading and unloading.
C
22
=
1/
S
22
=
3.8 Elasticity
Materials that are elastic are conservative so that all of the work done by the external
stresses during an increment of deformation is stored and is recovered on unloading:
this means that all the strains that occur during an increment of loading are recovered
if the increment is removed. An important feature of isotropic and elastic materials is
that shear and volumetric effects are decoupled so that the stiffness parameters
J
1
and
J
2
are both zero and Eq. (3.21) becomes
δ
3
G
δε
q
s
0
=
(3.27)
p
K
v
δ
0
δε
(where the superscripts e denote elastic strains) and the complete behaviour is as shown
in Fig. 3.8. For materials that are elastic but anisotropic the coupling moduli
J
1
and
J
2
are equal, so that the matrix in Eq. (3.21) is symmetric. Elastic materials can be
non-linear, in which case all the elastic moduli vary with changing stress or strain.
(Stretching and relaxing a rubber band is an example of non-linear and recoverable
elastic behaviour.)
The more usual elastic parameters are Young's modulus
E
and Poisson's ratio
ν
.
These are obtained directly from the results of uniaxial compression (or extension)
tests with the radial stress held constant (or zero), and are given by
σ
a
d
E
=
(3.28)
ε
a
d
r
d
ε
ν
=−
(3.29)
a
d
ε
