Static deformations and dislocation theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 483

Now let the unprimed quantities refer to the problem of a dislocation in an elastic
half-space and the primed quantities refer to the deformation caused by a point
force F i at the point x i in the elastic half-space. The surface integral on the left-
hand side of the reciprocal theorem vanishes, because t i is zero on the free surface
of the half-space and because it changes sign across the dislocation while u i is con-
tinuous there. The volume integral on the left-hand side of the reciprocal theorem
vanishes because there are no body forces in the dislocation problem. To evaluate
the surface integral on the right-hand side of the reciprocal theorem, let ν i be the
unit outward normal vector from the side of the dislocation with displacement u i .
The traction on this side of the dislocation is then t i = τ ij ν j . On the other side of the
dislocation, the normal vector is reversed and the displacement is u i

. The traction

on that side is then t i =− τ ij ν j . If the dislocation is u i , the contribution

to the surface integral on the right-hand side of the reciprocal theorem from the
dislocation is

u i −

Δ u i =

S t i Δ

u i dS . Since t i vanishes on the free surface of the half-space
there is no contribution to the surface integral from there. If the point force F i is of
unit strength in the k -direction, generating a displacement field u i , the right-hand
side of the reciprocal theorem gives

−

u k ( x i ) t i Δ = u i dS , (9.2)

S

where the integral is over the surface of dislocation. By Hooke's law,

= ν j ⎣ λ ∂ u k

⎝ ⎠ ⎦ .

∂ u j
∂ x i

∂ u k
i

t i = τ ij ν j

i
j

∂ x δ

+ μ

∂ x j +

(9.3)

We are thus led to Volterra's formula,

u i ⎣ λ ∂ u k

⎝ ⎠ ⎦ ν j dS .

∂ u j
∂ x i

∂ u k

∂ x δ

i

u k ( x i )

i
j

=

S Δ

+ μ

∂ x j +

(9.4)

In this form of Volterra's formula, the displacement field is found by integrat-
ing derivatives of the point force solution over the fault surface, with the point
force located at the field point. If we denote the radius to the field point as r ,and
the radius to a point on the dislocation surface by r , then a point force in the k -
direction at r generates a displacement field u i ( r , r ) observed at r . With these
conventions, Volterra's formula takes the form

u i ( r ) ⎣ λ ∂ u k

⎝ ⎠ ⎦ ν j dS . (9.5)

∂ u j ( r , r )
∂ x i

( r , r )
∂ x

∂ u k
i

( r , r )
∂ x j +

i
j

u k ( r ) =

S Δ

δ + μ

Now introduce a second displacement field u k ( r , r ), which by our conventions is
the k th component of displacement produced by a unit point force in the i -direction