Geology Reference
In-Depth Information
the geopotential in equilibrium, as demonstrated by equation (5.107). If s is the
co-ordinate in the orthometric direction (opposite to gravity), then
g 0 j ∂ρ 0
x i u j u i = g 0 ∂ρ 0
s u s u s .
(3.28)
The last term in the integrand can be replaced by the identity
x i
V 1 V 1
G ρ 0 u i
x i V 1
1
G
V 1
ρ 0 g 1 i u i =
x i
.
(3.29)
x i
Finally, with the aid of Gauss's theorem, we are able to write that
t i u i dS
F i u i d
+
V
S
V
λ u i
u i
u j
x j + μ
x j u i
x i u i
u j
+ g 0 ∂ρ 0
s u s u s
=
x j +
x i
x j
V
d
x j u i g 0 i
x i V 1
ρ 0 u j
ρ 0 u j
G V 1
1
( u i g 0 i )
V
x j
x i
V 1 V 1
G ρ 0 u i
1
G
S ρ 0 g 0 u i u s ν i dS
+
x i
ν i dS .
(3.30)
S
With the use of the second form of the right side of equation (3.27), on exchange
of the primed and unprimed systems of forces and displacements, it is seen that
of all the terms on the right side of expression (3.30), only the surface integrals
change. We then have the required generalisation of the reciprocal theorem of Betti,
expressed as
ρg 0 u s ν i ) u i dS
F i u i d
( t i
+
V
S
V
t i ρ 0 g 0 u s ν i u i dS
F i u i d
=
+
V
S
V
V 1 V 1
G ρ 0 u i
V 1 V 1
G ρ 0 u i
1
G
+
x i
x i
ν i dS .
(3.31)
S
The surface tractions are modified by the contributions from transport through the
initial hydrostatic stress field, and new terms arise to account for the e
ff
ect of self-
gravitation.
3.3 Radial equations: spheroidal and torsional
The Earth departs by only about one part in 300 from spherical shape. The internal
surfaces of equal density, geopotential and pressure are close to spherical, and
in close approximation the orthometric co-ordinate becomes the radius. In this
 
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