Static deformations and dislocation theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 501

Substituting for the normal tractions from equation (9.53), the first integral in (9.54)
is transformed to

ρ 0 ( b − ) V 1 u s V 1 u s r = b + d g 0 ρ 0 ( b − ) u s u s u s u s r = b + d − − S+ − S

S S

ρ 0 ( b − ) V 1 u s − V 1 u s r = b + d =− S . (9.55)

S

The physical equivalent of equation (9.44) is

1
g 0 V 1 .

u s ( b − )

=

(9.56)

This allows the second integral in (9.54) to be written as

ρ 0 ( b − ) V 1 1

g 0 V 1 d

ρ 0 ( b − ) V 1 u s − V 1 u s r = b + d V 1 1

g 0 V 1 −

S− S

S S

ρ 0 ( b − ) V 1 u s − V 1 u s r = b + d =− S . (9.57)

S

Adding expressions (9.55) and (9.57), we find that the extra surface integrals over
the core-mantle boundary (9.54) cancel each other. Similar arguments may be con-
structed for the inner core boundary. Thus, the form (9.49) of Betti's reciprocal
theorem is unaltered for discontinuities in the variable y 6 given by equation (9.47).
Applied to the mantle and crust (the shell), the surface integrals in Betti's recip-
rocal theorem remain to be considered for the surface of the Earth and the faces
of the dislocation. By arguments analogous to those used at the core-mantle and
inner core boundaries, the surface integrals can be shown to vanish over the surface
of the Earth. The surface integrals

V 1 ∂ V 1

4π G ρ 0 u i V 1 ∂ V 1 4π G ρ 0 u i

1
4π G ∂ x i −

−

∂ x i −

ν i d S (9.58)

S

over the faces of the dislocation cancel each other for slip faults, since for slip faults
there is no displacement component normal to the faces of the dislocation and the
normal vectors on the two sides of the dislocation are in opposite directions. The
surface integrals

t i − ρ 0 g 0 u s ν i u i d

− ρ 0 g 0 u s ν i ) u i d ( t i S and S (9.59)

S S

over the faces of the dislocation cancel each other since, for slip faults, there
is no displacement component normal to the faces of the dislocation, and the
tractions required to maintain the dislocation are equal and opposite on the two
faces. The remaining steps in the derivation of Volterra's formula using Betti's
reciprocal theorem follow those in Section 9.1.1. The radius vector to the field