Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 65

This solenoidal displacement field has the associated normal stresses

Q 3 1

− ξ 1 ) 2
Q 2

A

3 ( x 1

τ 11 = − , (1.452)

Q 3 1

3 ( x 2 − ξ 2 ) 2
Q 2

A

τ 22 = − , (1.453)

Q 3 1

+ ξ 3 ) 2
Q 2

A

3 ( x 3 τ 33 = − , (1.454)

and the associated shear stresses

3 A ( x 1

− ξ 1 )( x 3 + ξ 3 )

τ 13 =− , (1.455)

Q 5

3 A ( x 2

− ξ 2 )( x 3 + ξ 3 )

τ 23 =− , (1.456)

Q 5

3 A ( x 1 − ξ 1 )( x 2 − ξ 2 )
Q 5

τ 12 =− . (1.457)

This stress field generated by the Galerkin vector G 1 produces a traction per unit
area, on a small sphere of radius a surrounding the image point, with components
F 1 = τ 11 ( x 1

− ξ 1 )

a + τ 12 ( x 2

− ξ 2 )

a + τ 13 ( x 3

+ ξ 3 )
a =−

2 A ( x 1

− ξ 1 )
a 4

, (1.458)

= τ 12 ( x 1

− ξ 1 )

a + τ 22 ( x 2

− ξ 2 )

a + τ 23 ( x 3

+ ξ 3 )
a =−

2 A ( x 2

− ξ 2 )
a 4

F 2 , (1.459)

= τ 13 ( x 1 − ξ 1 )
a

+ τ 23 ( x 2 − ξ 2 )
a

+ τ 33 ( x 3 + ξ 3 )
a

2 A ( x 3 + ξ 3 )
a 4

F 3 =− . (1.460)

The components of the total traction on the small sphere are then

a 2 2π
0

π

T i = F i sinθ d θ d φ. (1.461)

0

Thus, T 1 = 0.

For the second Galerkin vector G 2 ,wehave

T 2 = T 3 =

− ξ 1 ) 1

x 3 + ξ 3

Q 3

∇ ( ∇· G 2 ) = B e 1 ( x 1

Q ( Q + x 3 + ξ 3 ) −

B e 2 ( x 2 − ξ 2 )

1 x 3 + ξ 3

Q 3

+

+ ξ 3 ) − Q ( Q + x 3

B e 3 2

( x 3 + ξ 3 ) 2
Q 3

+

Q −

, (1.462)

and

B e 3 2

2 G 2 =

∇

Q .

(1.463)