Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 71

Force equilibrium for the infinite elastic medium then requires that

4π λ + 4π λ +

2μ
λ + μ

4π μ
λ + μ

2μ
λ + μ ,

− D − B =− (1.512)

or

B μ
λ +

2μ + D =

1. (1.513)

On the surface x 3 = 0, we have Q = R . The condition that the normal stress van-

ishes on this surface is then

⎣ ⎦ − ⎣ ⎦ + ⎣ ⎦

2
3
R 2

2
3
R 2

2
3
R 2

3 ξ 3 ξ 3 ξ

ξ 3
R 3

μ
λ + μ +

D ξ 3
R 3

μ
λ + μ +

A 1
R 3

1 −

⎣ ⎦ − ⎣ ⎦ =

3
R 2

3
R 2

3 ξ 3 ξ

B ξ 3
R 3

λ
λ + μ −

C μ
λ + μ

1
R 3

+ 1 − 0. (1.514)

=

The condition that the shear stresses vanish on the surface x 3 0is

⎣ ⎦ − ⎣ ⎦ −

2
3
R 2

2
3
R 2

3 ξ 3 ξ

1
R 3

μ
λ + μ +

D 1
R 3

μ
λ + μ +

3 A ξ 3
R 5

−

⎣ ⎦ −

2
3
R 2

3 ξ B 1

R 3

λ
λ + μ −

3 C λ
λ + μ

ξ 3
R 5 =

+ 0. (1.515)

cients of terms in 1/ R 3 and of terms in 1/ R 5 must vanish.
For the normal stress this yields

In both cases, the coe

B ξ 3 λ C μ D ξ 3 μ

λ + μ =− ξ 3 μ

A +

λ + μ − λ + μ − λ + μ ,

(1.516)

C μ A + B ξ 3 −

λ + μ +

D ξ 3 = ξ 3 , (1.517)

while for the shear stresses we have

B λ − D μ = μ, (1.518)

C λ A + B ξ 3 +

λ + μ +

D ξ 3 =− ξ 3 . (1.519)

Equations (1.513) and (1.518) can be solved to give

B = 2μ λ +

2μ
( λ + μ ) 2 = 4 ( 1 − σ )( 1 − 2σ ) ,

(1.520)

2λ λ +

2μ
(λ + μ) 2 −

D = 1 = 8σ(1 − σ) − 1. (1.521)