Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 36

1.4.2 Conditions of equilibrium
Let the body force per unit mass be f i . The total force acting on the volume

V ,

bounded by the surface S , vanishes in static equilibrium. Thus,

τ ji ν j d S+ ρ f i d V= 0, (1.215)

S V

with ρ representing the mass density.
In equilibrium, the moment about any point must also vanish. For the moment
about the origin, we have

ξ ijk x j τ lk ν l d S+ ξ ijk x j ρ f k d V= 0. (1.216)

S V

where ξ ijk is the alternating tensor.
The divergence theorem of Gauss allows us to convert the surface integrals to
volume integrals, giving, for force equilibrium,

∂τ ji

∂ x j + ρ f i d V=

0, (1.217)

V

and, for moment equilibrium,

ξ ijk ∂ + ρ x j f k d

∂ x l x j τ lk

V

V

ξ ijk x j ∂τ lk

∂ x l + ρ f k d

(1.218)

= τ jk + V= 0.

V

Since V is an arbitrary volume,

∂τ ji
∂ x j =− ρ f i

(1.219)

and

ξ ijk x j ∂τ lk

∂ x l + ρ f k

τ jk + = ξ ijk τ jk = 0. (1.220)

The latter condition for moment equilibrium expands to three equations:
τ 23 − τ 32 = 0, − τ 13 + τ 31 = 0,τ 12 − τ 21 = 0. Hence, moment equilibrium requires that

τ ij = τ ji ,orthat the stress tensor is symmetric . The condition for force equilibrium

can then alternatively be written as

∂τ ij
∂ x j =− ρ f i ,

(1.221)

referred to as the equation of equilibrium .