Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 54

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

⎣ ⎦ ,

x 2

x 2 2

x 1

x 1 2 x 3

− +

−

2μ B
R 0 ( R 0

τ 11 = − (1.353)

R 0 R 0 + x 3

+ x 3 )

⎣ ⎦ ,

x 1

x 1 2

x 2

x 2 2 x 3

− +

−

2μ B
R 0 ( R 0

τ 22 = − (1.354)

R 0 R 0 + x 3

+ x 3 )

2μ B x 3

τ 33 =−

R 0 ,

(1.355)

with shear stresses,

x 1

2μ B x 1

−

τ 13 =−

R 0 ,

(1.356)

x 2

2μ B x 2

−

τ 23 =−

R 0 ,

(1.357)

2μ B x 1

x 1 x 2 x 2 2 R 0 x 3 − − +

τ 12 =− . (1.358)

R 0 ( R 0 x 3 ) 2 +

The normal stress (1.355) on the surface x 3 0 again vanishes, and the stresses

τ 13 , expressed by (1.356), and τ 23 , expressed by (1.357), have the same form as
their counterparts, expressed by (1.349) and (1.350). This suggests that a linear
combination of the displacement fields (1.345) and (1.352) would leave the surface
x 3

=

0 stress free.
The problem of a concentrated force acting normal to the free surface of an
elastic half-space was first solved by J. Boussinesq in 1878 and is known as the
Boussinesq problem . As suggested, we form a linear combination of the displace-
ment fields (1.345) and (1.352) to get

=

x 1

x 1

x 1

x 1 −

A
2μ

x 3
R 0

u 1 = − + B

x 3 ) ,

(1.359)

R 0 ( R 0 +

x 2 −

x 2

x 2 − x 2
R 0 ( R 0

A
2μ

x 3
R 0

u 2 = + B

x 3 ) ,

(1.360)

+

⎣ ⎦ +

x 3
R 0

A
2μ

λ + 3μ

λ + μ

1
R 0 +

B
R 0 .

u 3 = (1.361)

For determination of the constants, A and B , two conditions are required. One is
provided by the condition that the surface x 3 0 is stress free, except at the point of

application of the normal force. A second is provided by the condition that the total
traction on a hemisphere in the medium, centred on the point of application of the

=