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uniaxial loading
(σ
)
along the
x
direction, the strain in the
x
direction is simply
σ
/
E,
x
x
where
E
is called
Young's
10
modulus
or the modulus of elasticity. The lateral strains due to
the
σ
x
are the contractions
=
=−
νσ
/
E,
where
ν
is a material constant, which is
y
z
x
referred to as
Poisson's
11
ratio
. In turn, imposed normal stresses
σ
y
and
σ
z
would produce
strains in the
x
direction of magnitudes
E
. As the normal strains
are not influenced by shear deformation, the normal strains in a direction can be obtained
by superimposing these effects. For
=−
νσ
/
E,
=−
νσ
/
x
y
x
z
x
we find
1
E
[
x
=
σ
x
−
ν(σ
y
+
σ
z
)
]
The shear strain corresponding to
τ
xy
is simply
γ
xy
=
τ
xy
/
G
, where
G
is called the
shear
modulus of elasticity
.
The modulus of elasticity and Poisson's ratio are material constants that can be deter-
mined experimentally. Typical values of
E
are 70 GN
m
2
10
6
psi
/
(
10
.
1
×
)
for aluminum and
m
2
10
6
psi
210 GN
for steel. Poisson's ratio is usually between 0.25 and 0.35 for met-
als. As demonstrated in elementary textbooks, the theoretical lower bound for
/
(
29
×
)
ν
is 0, which
corresponds to a very compressible material such as cork. Certain concretes can have a
ν
as low as 0.1. The upper bound of 0.5 applies for very incompressible materials, e.g., water,
such as is often assumed to be the case for materials in the plastic range. Rubber, which
is quite incompressible, is sometimes incorrectly considered to be compressible because a
change in shape may be mistaken for a change in volume.
It can be shown that for an isotropic material there are only two independent mate-
rial constants. Thus,
G
,
ν
, and
E
are interrelated. More specifically (Problem 1.33),
G
=
E
/
.
If all of the equations are placed together in matrix notation, we have
2
(
1
+
ν)
.
x
y
z
σ
x
1
−
ν
−
ν
.
σ
y
−
ν
1
−
ν
0
.
σ
z
...
−
ν
−
ν
1
1
E
...
=
...
...
...
.
...
...
...
(1.32a)
.
γ
xy
γ
xz
γ
yz
2
(
1
+
ν)
0
0
τ
xy
.
τ
0
0
2
(
1
+
ν)
0
xz
.
τ
0
0
2
(
1
+
ν)
yz
E
−
1
=
σ
(1.32b)
Gresham College. In 1679, in his brief collection of his studies,
Lectiones Cutleriane
, he enunciated Hooke's law. It
is believed that his correspondence with Isaac Newton formed the basis of the theory of gravitation. He was an
important geologist and even more important instrumentation inventor.
10
Thomas Young (1773-1829) was an English physicist who started early as somewhat of a prodigy. He was
largely self-educated in modern languages and literature and was well-read in physics and mathematics. In 1819,
he proposed the key to decipher Egyptian hieroglyphics. He became a physician, but continued his work in
physics, including considerable efforts in the mechanics of solids. He discovered the principle of interference of
light and employed what is now known as Young's modulus.
11
Simeon-Denis Poisson (1781-1840) was a French mathematician who solved many mechanics problems of prac-
tical importance. After an unsuccessful attempt to learn surgery, he entered the Ecole Polytechnique, where his
work attracted the attention of Lagrange and Laplace. In terms of fundamental ideas, he introduced the concept
of tension of a bar being accompanied by lateral contraction. His efforts in mathematical physics were diverse
and significant.
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