Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 215

The i th component of the combination − μ ∇× ( ∇× u ) +∇× ( u ×∇ μ) +∇ ( u ·∇ μ)

is then

∂ u i
∂ x k +

2 u i 2 u k

∂ x k ∂ x i −

2

∂μ
∂ x k

∂ u k
∂ x i

+ μ ∂

∂ x k − μ ∂

∂ x i ∂ u k

∂μ

u i ∂

μ
∂ x k .

∂ x k +

(3.18)

2

If we now add μ to this combina-

tion, we obtain the left side of the equilibrium equation (3.15). In symbolic vector
notation it takes the form

∇ λ ∇· u + λ ∇ ( ∇· u ) + 2μ ∇ ( ∇· u ) +∇ μ ∇· u − u ∇

(λ + 2μ) ∇ ( ∇· u ) − μ ∇× ( ∇× u )

2

+ ( ∇ λ +∇ μ) ∇· u +∇× ( u ×∇ μ) +∇ ( u ·∇ μ) − u ∇ μ

=−∇ (ρ 0 u · g 0 ) − g 0 ρ 1 − g 1 ρ 0 − F . (3.19)

The first two terms on the left side and the last term on the right side appear in
the classical Navier equation of equilibrium (1.271) for uniform elastic solids, free
of self-gravitation. The extra terms on the left side reflect the spatial variation of
elastic properties. The extra terms on the right side reflect, respectively, the e ects

of the extra pressure arising from transport through the hydrostatic pressure field,
the equilibrium gravity acting on the perturbed density and the perturbed grav-
ity acting on the equilibrium density. To complete the description of the elasto-
gravitational system, (3.19) must be augmented by the gravitational equation (3.11)
expressed in symbolic vector notation,

ff

2 V 1

∇ = 4π G ∇· (ρ 0 u ). (3.20)

3.2 The reciprocal theorem of Betti
The reciprocal theorem of Betti (1.266) is one of the basic theorems of classical
elasticity theory. It can be regarded as the basis for generating formulae of the
Green's theorem type, where the solution at the field point is an integral over point
sources in the source region. In this section, we generalise the reciprocal theorem of
Betti to elasto-gravitational systems whose elastic properties are spatially varying,
and which are subject to hydrostatic pre-stress in the equilibrium reference state,
as described in the previous section.
Consider two systems of surface tractions and body forces (primed and unprimed),
t i , F i ,and t i , F i , producing, respectively, displacement fields u i and u i . Let these
force systems act on material contained in a volume

by a surface S , which is
such as to allow the divergence theorem of Gauss to be applied. The work done by
the unprimed tractions and forces, acting through the primed displacements, is

V

t i u i dS F i u i d

S τ ji ν j u i dS

F i u i d + V= + V , (3.21)

S V V