Geology Reference
In-Depth Information
In symbolic notation this is
2
u
λ
∇
(
∇·
u
)
+
μ
∇
+
μ
∇
(
∇·
u
)
=−
F
.
(1.269)
Using the vector identity (A.14),
2
u
∇
=−∇×
(
∇×
u
)
+∇
(
∇·
u
),
(1.270)
we obtain the
Navier equation
for the displacement field,
(λ
+
2μ)
∇
(
∇·
u
)
−
μ
∇×
(
∇×
u
)
=−
F
.
(1.271)
1.4.5 Solutions of the Navier equation
By the Helmholtz separation considered in Section 1.2, the vector displacement
field
u
can be broken into lamellar and solenoidal parts written as
u
=∇
L
+∇×
A
,
(1.272)
where
L
is the scalar lamellar potential, and
A
is the vector potential, giving rise
to the solenoidal part of the vector displacement field, with the gauge
∇·
A
=
0.
Similarly, the vector body force per unit volume can be separated as
F
=∇
L
F
+∇×
A
F
,
(1.273)
with the gauge
0. A further separation derives from the Lamb-Backus
decomposition (1.174) of a vector field into lamellar, poloidal and toroidal or tor-
sional parts. Applied to both
u
and
F
,wehave
∇·
A
F
=
u
=
L
+
P
+
T
=∇
L
+∇×
(
∇×
r
P
)
+∇×
r
T
,
(1.274)
and
F
=
L
F
+
P
F
+
T
F
=∇
L
F
+∇×
(
∇×
r
P
F
)
+∇×
r
T
F
,
(1.275)
where
L
,
P
,
T
are the scalars for
u
,and
L
F
,
P
F
,
T
F
are the scalars for
F
. Applying
the Helmholtz separation to the Navier equation of equilibrium (1.271) itself yields
2
L
(λ
+
2μ)
∇
∇
=−∇
L
F
,
(1.276)
−
μ
∇×
∇×
A
)
=−∇×
(
∇×
A
F
.
(1.277)
Taking the divergence of (1.276) gives
1
λ
+
4
L
2
L
F
,
∇
=−
2μ
∇
(1.278)
Search WWH ::
Custom Search