Geology Reference
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Then, using (1.326), the vector displacement field is found to be represented by
B
Φ
1
u
=
Φ
−∇
+
σ) R
·
.
(1.333)
4 (1
This representation of the displacement field was developed by Papkovich (1932)
and Neuber (1934) and is referred to as the Papkovich-Neuber solution of the
Navier equation. The vector potential Φ obeys the vector Poisson equation (1.325),
while the scalar potential B obeys the scalar Poisson equation (1.331). Outside
regions containing body forces, the displacement field is represented by the four
scalar, harmonic functions
Φ 3 and B . There has been much debate as to
whether or not all four are independent, or whether one can be set to zero without
loss of generality (see Sokolniko
Φ 1 ,
Φ 2 ,
(1956), p. 331). From equations (1.279), (1.284)
and (1.285), the displacement field appears to depend on the lamellar, poloidal and
toroidal or torsional scalars. In the absence of body forces, the lamellar scalar is
determined to within a constant as a harmonic function, and the poloidal and tor-
oidal or torsional scalars are harmonic. Indeed, in the case of Kelvin's problem of a
force F 0 concentrated at (ξ 1 2 3 ) in an infinite medium, only the vector potential,
ff
1
F 0
R ,
Φ
=
(1.334)
with R 2
ξ 1 ) 2
ξ 2 ) 2
ξ 3 ) 2 , is required, and B may be set to zero.
=
( x 1
+
( x 2
+
( x 3
1.4.8 The Galerkin vector
Another representation of the displacement field related to the Papkovich-Neuber
solution of the Navier equation is that given by the Galerkin vector G ,
2 G
u
=
2 (1
σ)
−∇
(
∇·
G ).
(1.335)
If we set
1
2 G
=
σ) Φ
(1.336)
2 (1
and
∇·
G
=
L ,
(1.337)
1
the Galerkin vector representation reduces to equation (1.326) defining the vector
potential Φ ,
u
=
Φ
L .
(1.338)
1
 
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