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or
1
2
(
ik
=
u
i,k
+
u
k,i
)
(1.19)
which is called the
Cauchy
6
strain tensor.
In matrix form, these become
x
y
z
γ
xy
γ
xz
γ
yz
∂
x
00
0
∂
y
0
u
x
u
y
u
z
00
∂
z
=
(1.20)
∂
y
∂
x
0
∂
z
0
∂
x
0
∂
z
∂
y
or
=
D
u
(1.21)
with the differential operator matrix
∂
x
00
0
y
0
00
∂
∂
z
D
=
(1.22)
∂
y
∂
x
0
∂
z
0
∂
x
0
∂
z
∂
y
Two-Dimensional Problems
For the special case of two-dimensional behavior in the
xy
plane,
u
y
]
T
,
xy
]
T
u
=
[
u
x
=
[
γ
(1.23)
x
y
and Eq. (1.20) becomes
=
u
x
u
y
∂
x
0
x
∂
0
y
y
γ
∂
∂
x
xy
y
(1.24)
=
Du
EXAMPLE 1.2 Physical Interpretation of the Strain Components
The strain-displacement relations will be rederived here in a manner that may provide more
physical insight into the strain components.
Consider the projection of a deformed differential element onto the
xy
plane as shown
in Fig. 1.8. Points 0,
A
, and
B
in the unstrained element move to points 0
,A
,
and
B
of the
6
Augustin Louis Cauchy (1789-1857) was a French mathematician and engineer. His father fled Paris during the
French revolution, going to a nearby village where many outstanding scientists often met at Laplace's house.
There the exceptional mathematical prowess of the young Cauchy was noticed by Lagrange. Beginning in 1815,
he held positions at the Ecole Polytechnique, Faculte des Sciences and the College de France. In 1830 his refusal
to sign an oath of allegiance cost him his positions and he left for the University of Turin. He returned to Paris
in 1838. In the theory of elasticity he introduced the concept of stress and derived the conditions of equilibrium,
including the surface conditions called Cauchy's formula.
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