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The desired solution of Eq. (1.63) would be a set of single-valued, continuous displace-
ments that satisfy all boundary conditions. The existence of such a solution is due to the
uniqueness principle
[Timoshenko and Goodier, 1970, p. 269].
In scalar form, the governing equations of Eq. (1.63) for an isotropic solid with constant
E,
ν
are
∂
1
∂
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
∂
p
Vx
G
2
u
x
+
∇
+
=
0
1
−
2
ν
x
z
∂
1
∂
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
∂
p
Vy
G
2
u
y
+
∇
+
=
0
(1.64)
1
−
2
ν
y
z
∂
1
∂
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
∂
p
Vz
G
2
u
z
+
∇
+
=
0
1
−
2
ν
z
z
2
x
y
z
is the
Laplacian
15
or
harmonic
operator. Frequently, these equations
where
∇
=
∂
+
∂
+
∂
are written in the form
2
u
i
+
(λ
+
G
∇
G
)
u
k,ki
+
p
Vi
=
0
(1.65)
with the Lame constants
ν
E
E
λ
=
=
G
(1.66)
(
1
+
ν)(
1
−
2
ν)
2
(
1
+
ν)
Equation (1.
65
) is called the
Navier
16
or
Lame-Navier
equations of elasticity. In the absence
of body forces
p
V
i
,
Eq. (1.65) can be written as the
biharmonic (differential) equation
2
2
u
i
=
∇
∇
0
(1.67)
which is a frequently occurring equation in mathematical physics and, in particular, in the
theory of elasticity.
For this displacement formulation, in which the governing differential equations for
the displacements are the conditions of equilibrium, the compatibility requirements are
often satisfied trivially, i.e., by inspection or by dealing only with single-valued, contin-
uous displacements. Equation (1.67) is sometimes simplified by choosing displacement
15
Pierre Simon Laplace (1749-1827) was a French astronomer and mathematician. He was born into an upper
middle-class family. At sixteen he entered the University of Caen and shortly thereafter he became a professor of
mathematics at Ecole Militaire in Paris. Among his numerous remarkable achievements was his work on the idea
of a potential. He showed that the potential satisfied what is now known as Laplace's equation. In addition to his
efforts in astronomy, he made important contributions in the mathematical theory of probability. He is considered
to be one of the most influential scientists of all time. Although he was referred to as “the Newton of France,”
questions have often been raised as to his sense of “honor.” As evidenced by his dedications to the different
volumes of his five volume work
Mecanique celeste
, published between 1799 and 1825, Laplace was politically
flexible during the French revolution and the times of Napoleon. More “honorable” scientists such as Lavoisier,
who collaborated with Laplace in the study of specific heats, met their fate on the guillotine.
16
Claude-Louis-Marie-Henri Navier (1785-1836) was a French engineer who made wide-ranging contributions
to mechanics. Some of his efforts were guided by the somewhat earlier works of Coulomb. Navier worked on
torsional and bending stress formulations; however, both were based on erroneous suppositions. He designed
bridges in France and Italy and became a professor of calculus and mechanics. His published material enjoyed
considerable popularity among French engineers for many years.
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