Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 68

From relation (1.335) the displacement field is found to have the components

1

C
2μ

x 1 − ξ 1

Q 3

3 x 3 ( x 3

+ ξ 3 )
Q 2

= −

u 1 , (1.484)

1

− ξ 2
Q 3

3 x 3 ( x 3

+ ξ 3 )
Q 2

C
2μ

x 2

u 2 = − , (1.485)

Q 3 1

+ ξ 3 ) 2
Q 2

C
2μ

x 3

3 ( x 3

C
λ + μ

x 3 + ξ 3

Q 3 ,

u 3 = − − (1.486)

with derivatives

2μ Q 3 1

1

− ξ 1 ) 2
Q 2

− ξ 1 ) 2
Q 2

∂ u 1
∂ x 1 =

C

3 ( x 1

3 C
2μ

x 3 ( x 3 + ξ 3 )

Q 5

5 ( x 1

− − −

, (1.487)

2μ Q 3 1

1

− ξ 2 ) 2
Q 2

− ξ 2 ) 2
Q 2

∂ u 2 C

3 ( x 2

3 C
2μ

x 3 ( x 3 + ξ 3 )

Q 5

5 ( x 2

∂ x 2 =

− − −

, (1.488)

2μ Q 3 1

1

3 ( x 3 + ξ 3 ) 2
Q 2

5 ( x 3 + ξ 3 ) 2
Q 2

∂ u 3
∂ x 3 =

C 3 C

2μ

x 3 ( x 3 + ξ 3 )
Q 5

− − −

Q 3 1

3 ( x 3 + ξ 3 ) 2
Q 2

6 C
2μ

x 3 ( x 3 + ξ 3 )
Q 5

C
λ + μ

1

− − − . (1.489)

Summing the derivatives, the cubical dilatation is found to be

Q 3 1

+ ξ 3 ) 2
Q 2

C
λ + μ

1

3 ( x 3 ∇· u =− − . (1.490)

This displacement field has the associated normal stresses

+ ξ 3 ) 2
Q 5

C μ
λ + μ

1
Q 3 +

3 C λ
λ + μ

( x 3

3 C x 3 ( x 3

+ ξ 3 )
Q 5

τ 11 = −

1

3 C ( x 1 − ξ 1 ) 2
Q 5

5 x 3 ( x 3 + ξ 3 )
Q 2

− − , (1.491)

( x 3 + ξ 3 ) 2
Q 5

C μ
λ + μ

1
Q 3 +

3 C λ
λ + μ

3 C x 3 ( x 3 + ξ 3 )
Q 5

τ 22 = −

1

− ξ 2 ) 2
Q 5

3 C ( x 2 5 x 3 ( x 3

+ ξ 3 )
Q 2

− − , (1.492)

Q 3 1

3

+ ξ 3 ) 2
Q 2

+ ξ 3 ) 2
Q 2

τ 33 =− μ
λ + μ

1

3 ( x 3

x 3 ( x 3 + ξ 3 )

Q 5

5 ( x 3

− − − , (1.493)