Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 49

The vector displacement field is found from (1.272) using expressions (1.311)
and (1.315) for L and A , respectively. The vector identities,

F 0 ·

R F 0

R −

( F 0 · R )

∇ = R , (1.317)

R R 3

and

F 0

× R F 0

R +

( F 0 · R )

∇× = R ,

(1.318)

R R 3

allow simplification of the final expression for u to

3μ
8πμ(λ +

λ + F 0

R +

λ + μ
8πμ(λ +

( F 0 · R )

u = R . (1.319)

2μ) 2μ) R 3

This result, for the displacement field arising from a concentrated point force in
an infinite, uniform medium, was first obtained by Lord Kelvin in 1848, and is
generally known as the solution to Kelvin's problem .
With the constant

λ + μ
8πμ(λ +

H =

2μ) ,

(1.320)

the associated normal stresses are

R 3 3 ( x 1

R

2 F 01 ( x 1 − ξ 1 )

− ξ 1 ) 2 F 0

H · R μ

λ + μ

τ 11 =− 2μ + − F 0 · ,

R 2

R 3 3 ( x 2

R

2 F 02 ( x 2 − ξ 2 )

− ξ 2 ) 2 F 0 H · R μ

λ + μ

τ 22 =− 2μ + − F 0 · , (1.321)

R 2

R 3 3 ( x 3

R

2 F 03 ( x 3 − ξ 3 )

− ξ 3 ) 2 F 0

H · R μ

λ + μ

τ 33 =− 2μ + − F 0 · ,

R 2

and the associated shear stresses are

R 3 3 ( x 1

H − ξ 1 )( x 3 − ξ 3 ) F 0 · R

τ 13 =− 2μ

R 2

− ξ 1 )

F 01 ( x 3

μ
λ + μ

+ − ξ 3 ) + F 03 ( x 1 ,

R 3 3 ( x 2

H − ξ 2 )( x 3 − ξ 3 ) F 0 · R

τ 23 =− 2μ

R 2

− ξ 2 )

F 02 ( x 3

μ
λ + μ

+ − ξ 3 ) +

F 03 ( x 2

, (1.322)

R 3 3 ( x 1

− ξ 1 )( x 2 − ξ 2 ) F 0 · R

H

τ 12 =− 2μ

R 2

F 02 ( x 1 − ξ 1 )

F 01 ( x 2 − ξ 2 )

μ
λ + μ

+ + .