Static deformations and dislocation theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 484

at r , observed at r . Applying Betti's reciprocal theorem to the two point-force
systems of displacement fields, the surface integrals vanish, since the surface of
the half-space is force free, and the volume integrals produce the identity
u i ( r , r ) u k ( r , r ). = (9.6)

With this identity, Volterra's formula can be modified to give the displacement
at the field point as an integral over the dislocation surface of derivatives of the
displacements at the point forces:

u i ( r ) ⎣ λ ∂ u k ( r , r )

⎝ ⎠ ⎦ ν j dS . (9.7)

∂ u k ( r , r )
∂ x i

∂ u k ( r , r )

i
j

u k ( r ) =

S Δ

δ + μ

∂ x j + ∂ x

With this formulation, the point forces are on the surface of dislocation and the
partial derivatives may be interpreted as opposing double forces. The term pro-
portional to the Kronecker delta δ

i

j arises when the discontinuity over the surface
of dislocation is normal to the surface. The opposing double forces then represent
points of dilatation. When the discontinuity over the surface of dislocation is paral-
lel to the surface, paired double forces without moment occur. These double force
combinations are often called nuclei of strain .

9.1.1 The displacement fields of inclined faults
We begin by adopting a right-hand system of Cartesian co-ordinates ( x 1 , x 2 , x 3 ),
with ( x 1 , x 2 ) being the co-ordinates on the surface of the force-free elastic half-
space, with x 3 measured vertically downward into the half-space. u i then repres-
ents the i th component of displacement observed at ( x 1 , x 2 , x 3 ) due to a point force
of unit magnitude acting in the j -direction at (ξ 1 ,ξ 2 ,ξ 3 ). The dislocation surface is
then taken to be the rectangular fault surface shown in Figure 9.1. In this config-
uration, with a fault inclined at angle θ to the horizontal, and with ξ the down-dip
co-ordinate, the fault surface extends over

− L ≤ ξ 1 ≤ L and d ≤ ξ ≤ D , with the

total fault length given by 2 L .
In applications of dislocation theory to earthquake displacement fields, the dis-
continuity over the surface of dislocation is parallel to that surface. The faults
are then described as slip faults . The term proportional to δ

i j is then missing in
Volterra's formula (9.7). The outward unit normal vector to the fault surface is
ν

cosθ). If the slip is down the fault surface it is called dip-slip .Fora
dip-slip of magnitude U , the slip vector is

= (0,sinθ, −

(0, U cosθ, U sinθ). Carrying out
the implied summations over i and j , Volterra's formula for a dip-slip fault gives
u k = μ U

Δ u =

⎣ ⎝ ⎠ ⎝ ⎠ ⎦

∂ u k ∂ u k

∂ξ 3

∂ u k ∂ u k

∂ξ 2

dS .

∂ξ 2 −

sin 2θ −

∂ξ 3 +

cos 2θ (9.8)

S