Environmental Engineering Reference
In-Depth Information
'
()
zh
<
σ
z
=
ρ
gz
cos
α
w
z
'
()
(
)(
)
⎡
⎤
[3.1]
zh
>
σ
z
=
ρ
h
+
ρ ρ
−
zh gz
−
cos
α
⎣
⎦
w
z
w
w
w
where
z
is the current depth measured normally at the surface,
h
w
the depth of the
phreatic surface, α
the local angle of the slope, ρ
the density and
g
the gravity.
For the case of secondary creep and for an incompressible mass, the material law
is reduced to [VUL 00]:
(
)
Df
=
Jt
[3.2]
ij
2
D
ij
with
D
ij
the stretching tensor,
t
ij
the deviatoric part of the stress tensor,
J
2D
its second
invariant and
f
(.) a scalar function of its argument. This equation is reduced, in the
case of an infinite slope, to a case of simple shear:
1
()()
&
γ
()
z
=
f
ττ
z
[3.3]
2
&
where τ(
z
)
is the shear stress at depth
z
and
( )
z
is the shear strain rate.
Knowing an estimate of τ(
z
)
from equation [3.1] and the value of
γ
(z)
by means
of inclinometric measurements, we can thus determine the function
f
(τ), whose
nature is that of the inverse of viscosity. In the case of a Newtonian material
f
(τ) will
be a non-linear function of the stress (see [VUL 00]). Figure 3.4 gives an illustration
of this principle.
In the event of a slide occurring in a limited zone (shear band or sliding base),
the constitutive relationship between slide velocity on the interface
v
B
and stress
condition can be written on the interface by [VUL 88c]:
(
)
v
=
F
τστ
,'
[3.4]
Bi
BBB
i
where τ
Bi
is the vector of tangential stress on the base, τ
B
its norm, σ'
B
the norm of
the normal stress on the base and
F
B
(.) the sliding law. Figure 3.4 (right) illustrates
the case of the Sallèdes landslide [POU 85], where we note the strong non-linearity
between the sliding velocity and the normalized shear stress.
