Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 59

This displacement field has associated normal stresses

⎣ ⎦ ,

3 x 1 − x 1 2 C x 1 − x 1

R 0

2λ + 3μ

λ + μ

τ 11 =−

R 0 −

(1.403)

⎣ ⎦ ,

3 x 2 − x 2 2 C x 1 − x 1

R 0

μ
λ + μ

τ 22 =−

R 0 −

(1.404)

⎣ ⎦ ,

x 1

3 x 3 C x 1 −

2λ + 3μ

λ + μ

τ 33 =−

R 0 +

(1.405)

R 0

with shear stresses

x 3 3 x 1 −

x 1 2

1

C
R 0

τ 13 =−

R 0 −

⎦ , 1

R 0
R 0 +

λ + 2μ

λ + μ

− x 1 x 1 2 2 R 0 + x 3 − − (1.406)

x 3

R 0 ( R 0 + x 3 )

⎣ ⎦ ,

C x 1 − x 1 x 2 − x 2
R 0

3 x 3

λ + 2μ

λ + μ

2 R 0 + x 3

τ 23 =−

R 0 +

(1.407)

x 3 ) 2

( R 0 +

3 x 1 − x 1 2

1 C x 2 − x 2

R 0

τ 12 =−

R 0 −

. (1.408)

In order for the two shear stresses, τ 13 andτ 23 ,tovanishontheplane x 3 = 0, the

two free constants of the extra displacement fields must be related by

λ + 2μ

λ + μ

2μ B = C . (1.409)

Combining the extra displacement fields, they are found to have the associated
normal stresses

⎣

x 3 ) 2 3

C x 1 x 1

λ + 2μ

λ + μ

1
R 0 ( R 0

− x 1 x 1 2 3 R 0

+ x 3

τ 11 =− − −

R 0 ( R 0 + + x 3 )

3 x 1 −

x 1 2

,

1
R 0

2λ + 3μ

λ + μ

+

R 0 −

(1.410)

x 1 λ +

x 3 ) 2 1

C x 1 −

2μ
λ + μ

1
R 0 ( R 0 +

− x 2 − x 2 2 3 R 0

+ x 3

τ 22 =−

R 0 ( R 0

+ x 3 )

3 x 2 − x 2 2

,

1
R 0 R 0 − μ

+ (1.411)

λ + μ

⎣

1 ⎦ ,

x 1

3 x 3

C x 1 −

τ 33 =−

R 0 +

(1.412)

R 0