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Use of the binomial expansion
n
(
n
−
1
)
n
a
2
for
a
2
(
1
+
a
)
=
1
+
na
+
+···
<
1
2!
permits the radical in (2) to be rewritten
2
∂
2
∂
2
∂v
∂
1
2
u
u
x
=
1
+
x
+
+
+···−
1
∂
∂
x
x
∂
2
2
∂v
∂
≈
∂
u
1
2
u
x
+
+
(3)
∂
∂
x
x
where powers of the derivatives higher than the second are neglected. If the displacement
in the
z
direction were also taken into account, (3) would be replaced by
∂
2
2
∂v
∂
2
∂w
∂
=
∂
u
1
2
u
x
+
+
+
(4)
x
∂
∂
x
x
x
Similarly, the other normal strains are found to be the expressions of Eqs. (1.15b) and (1.15c).
The shear strain at a point is defined as the change due to deformation in the value
of the cosine of an angle that in the unstrained state was a right angle. In Fig. 1.8, let
γ
xy
be the shear strain referred to in the
xy
coordinate system. Consider the cosine of
angle
A
0
B
.
cos
2
−
γ
xy
cos
A
0
B
=
=
sin
γ
(5)
xy
For small angles, the sine of an angle may be replaced by the angle. Thus, cos
A
0
B
=
γ
xy
.
From analytical geometry, the cosine of the angle between two lines can be replaced by the
sum of the products of their direction cosines, i.e.,
cos
A
0
B
=
γ
=
+
a
x
1
a
y
1
a
x
2
a
y
2
(6)
xy
where
a
x
1
,a
x
2
are the direction cosines of 0
A
,
and
a
y
1
,a
y
2
are the direction cosines of 0
B
.
Using Fig. 1.8, these direction cosines are found to be
a
x
1
=
(
1
+
∂
u
/∂
x
)
dx
0
A
=
(∂v/∂
x
)
dx
a
x
2
0
A
(7)
=
(∂
u
/∂
y
)
dy
a
y
1
0
B
a
y
2
=
(
1
+
∂v/∂
y
)
dy
0
B
Then
1
∂
1
+
∂
u
u
y
+
∂v
+
∂v
∂
dx dy
γ
xy
=
(8)
∂
x
∂
∂
x
y
(
0
A
)(
0
B
)
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