EN0175
10 / 10 / 06
Strain in a solid
x
v
y
v
u
v
1
d
y
v
2
d
y
v
1
d
x
v
2
d
x
v
1
3
2
Consider an arbitrary fiber within the elastic body,
In the undeformed configuration, we can represent the fiber as a small vetor:
where
is the length and
is the unit vector along the fiber direction (orientation of the fiber).
0
d
d
l
m
x
v
v
=
0
d
l
m
v
In the deformed configuration, the same fiber is represented as
l
n
y
d
d
v
v
=
. Write the deformed
position of a particle as
)
,
,
(
)
,
,
(
3
2
1
3
2
1
x
x
x
u
x
x
x
x
y
y
r
r
v
v
+
=
=
where
is clearly the displacement vector. We can write a differential segment
)
,
,
(
3
2
1
x
x
x
u
r
y
v
d
as
x
F
y
v
v
d
d
=
where
j
i
ij
x
y
F
∂
∂
=
is called the deform gradient tensor. This suggests that
0
d
d
l
m
F
l
n
v
v
=
The ratio between the deformed length to undeformed length:
ε
λ
+
=
=
1
d
d
0
l
l
(
ε
: strain) is
defined as stretch. Therefore,
n
m
F
v
v
λ
=
1

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*