Geoscience Reference
In-Depth Information
Introducing scaling factors
L
for length,
0
for pressure,
g
0
for gravity, and
T
for
time provides
:
7
2
%
g
L
F
c
v
T
F
F
:
7
8
9
5
6
0
G
and
(9.5)
9
6
2
2
X
X
L
X
Y
0
'
u
g
x
t
with
%
,
F
,
G
,
X
,
Y
g
L
T
0
0
0
0
and a
consolidation number
c
v
T/L
2
. A convenient requirement is similarity of stress,
Two dimensionless numbers occur: an acceleration number
g
0
L/
0m =
0p
and
u
m
=
u
p
. Similarity of mass, internal friction, cohesion, and compressibility
by using the same materials provides identical deformation and failure states in
model and prototype. The gravity is different:
g
0m
/
g
0p
=
N
, which may vary up to
100 and more, depending on the rotation speed of the centrifuge. Similarity of
equilibrium is then satisfied by adopting
L
m
/
L
p
= 1/
N
. Physical similarity of
consolidation involves beside length also time and the consolidation coefficient
c
v
.
Since
c
v
= k/
g
=
/
and one may adopt similarity of viscosity
and
40
, and the fact that the intrinsic permeability
compressibility
is proportional to
D
2
(
D
is the representative grain size), it shows that the consolidation number
c
v
T/L
2
satisfies similarity, if also the grain size is scaled according to
D
m
/
D
p
= 1/
N
.
Then, time is equal in model and prototype. However, then the same material
cannot be used in the model and in the prototype, and similarity of intrinsic
parameters friction, cohesion, and compressibility may not hold. Usually the grain
size is not scaled in centrifuge tests, and as a consequence permeability is
N
2
larger
and consolidation proceeds equally faster, which implies that transient boundary
conditions must be faster by
N
2
as well.
Implying scaling principles in modelling physical phenomena requires special
expertise.
B
CONSTITUTIVE MODELS
Soil is a complicated material that behaves non-linearly, anisotropic and time-
dependently, when subjected to loading. It behaves differently for primary loading,
unloading and reloading and it shows plastic and brittle failure behaviour. Since it
is not practically possible to describe the true mechanical behaviour of saturated
porous granular materials, approximate constitutive models are adopted at larger
scale where soils are considered as continuous media. Though such models are
commonly based on fundamental physics (continuum mechanics), the parameters
involved are empirical in nature and should be verified by lab and field-testing, and
in some cases monitored during construction and use. Observed phenomena, e.g.
deformations, stresses, pressures and rates offer indirect information about these
40
Note, that in reality
=
/
'
and therefore
consolidation in (9.4) is a non-linear partial
differential equation.
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