Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 58

with B a free constant, in the Helmholtz separation (1.272), and is

⎣

1 ⎦ ,

x 1

x 1

2

−

B
R 0 +

u 1 =−

x 3 ) −

(1.390)

x 3 R 0 ( R 0 +

B x 1 − x 1 x 2 − x 2
R 0 ( R 0

u 2 =−

x 3 ) 2 ,

(1.391)

+

x 1 x 1 −

u 3 =− B

x 3 ) .

(1.392)

R 0 ( R 0 +

This displacement field is both lamellar and solenoidal, and it has associated nor-
mal stresses

⎣ ⎦ ,

x 1

x 1

x 1 2

x 1 −

3 R 0 + x 3
R 0 ( R 0 +

τ 11 =− 2μ B 3 − − (1.393)

x 3 ) 2

x 3 )

R 0 ( R 0 +

⎣ ⎦ ,

x 1 − x 1
R 0 ( R 0 +

x 2 −

x 2 2

3 R 0 + x 3

τ 22 =− 2μ B 1 − (1.394)

x 3 ) 2 R 0 ( R 0

+ x 3 )

2μ B x 1 − x 1 τ 33 =

R 0 ,

(1.395)

with shear stresses

⎣ ⎦ ,

x 1

x 1 2

2μ B
R 0 ( R 0

2 R 0 + x 3

τ 13 =− 1 − − (1.396)

+

x 3 )

R 0 ( R 0

x 3 )

+

2μ B x 1

x 1 x 2 x 2

2 R 0 +

x 3 ) 2

− −

x 3

τ 23 = , (1.397)

R 0

( R 0 +

⎣ ⎦ .

x 2 − x 2
R 0 ( R 0 +

x 1 −

x 1 2

3 R 0 + x 3

τ 12 =− 2μ B 1 − (1.398)

x 3 ) 2 R 0 ( R 0

+ x 3 )

The second extra displacement field derives from the Galerkin vector,
C x 1 x 1 log( R 0 − + x 3 ) e 3 , (1.399)

with C a free constant. From (1.335), the corresponding displacement field is found
to be

⎣

1 ⎦ ,

x 1 −

x 1 2

C
2μ

1
R 0

u 1 =

R 0 −

(1.400)

x 1

x 1 x 2 x 2 − −

C
2μ

u 2 = , (1.401)

R 0

⎣ ⎦ .

x 1 − x 1
R 0

C
2μ

x 3
R 0 +

λ + 2μ

λ + μ

2

u 3 = (1.402)

R 0 + x 3