Geoscience Reference
In-Depth Information
lithosphere. In this case, Eq. (10.13) becomes
D
d
4
w
d
x
4
+
H
d
2
w
d
x
2
=−
(
ρ
m
−
ρ
w
)
gw
(10.18)
The sine function
w
=
w
0
sin(2
π
x
/λ
)isasolution to this equation for values
of
λ
given by
D
2
π
λ
4
−
H
2
π
λ
2
=−
(
ρ
m
−
ρ
w
)
g
(10.19)
(To check this, differentiate the expression for
w
and substitute into Eq. (10.18).)
Since Eq. (10.19)isaquadratic equation in (2
π/λ
)
2
, the solution is
±
H
2
2
2
π
λ
H
−
4
D
(
ρ
m
−
ρ
w
)
g
=
(10.20)
2
D
)
2
must be real, not imaginary, the term under the square-root sign
must not be negative:
Because (2
π/λ
H
2
≥
4
D
(
ρ
m
−
ρ
w
)
g
(10.21)
The smallest value of
H
for which there is a real solution is given by
H
=
4
D
(
ρ
m
−
ρ
w
)
g
(10.22)
Forvalues of the horizontal force less than this value, there is no deformation;
but, at this critical value, the plate deforms into a sine curve. The wavelength of
the deformation for this critical force is then obtained by substituting the critical
value for
H
from Eq. (10.22) into Eq. (10.20), which gives
2
π
λ
4
D
(
ρ
m
−
ρ
w
)
g
2
D
2
=
(10.23)
Upon reorganization, Eq. (10.23) yields
1
/
4
D
(
ρ
m
−
ρ
w
)
g
λ
=
2
π
(10.24)
Foranelastic plate with flexural rigidity of 10
25
Nm,avalue that may be appro-
priate for the lithosphere, the critical compressive force as given by Eq. (10.22)
is therefore 10
15
Nm
−
1
,which corresponds to a critical horizontal compressive
stress of 10
10
Nm
−
2
(10 GPa). Even a flexural rigidity of 10
24
Nmcorresponds to
a critical compressive stress of more than 6 GPa. Such values of the compressive
stress are much greater than reasonable failure limits of the lithosphere. Buckling
would not occur in reality: failure by the formation of faults would take place
first.
However, if the lithosphere is modelled as a viscoelastic plate subjected to
some irregular normal load, it can be shown that, under constant compression,
the initial deflections due to this load are magnified and increase with time.
Such deformations occur for compressive forces an order of magnitude less than
the critical buckling forces. With time, failure of the crust by thrust faulting
presumably also occurs. Figure 10.49(d) shows the cross section of the crust
