Biomedical Engineering Reference
In-Depth Information
u
(
x
)
u
(
x
+
D
x
)
x
x
+
D
x
Figure 6.5
Displacements of a thin slice within a continuous bar.
x
, as depicted in Fig.
6.5
. The linear
Consider a slice of the bar having length
strain
ε
is expressed in the stretch
λ
of the slice by
ε
=
λ
−
1,
(6.9)
where the stretch is the ratio of the deformed length of the slice and the initial
length. At position
x
the displacement of the cross section of the bar equals
u
(
x
),
while at
x
+
x
the displacement equals
u
(
x
+
x
). The initial length of the slice
equals:
0
=
x
,
(6.10)
while the current length is given by
=
x
+
u
(
x
+
x
)
−
u
(
x
) .
(6.11)
Therefore, the stretch, that is the ratio of the deformed length over the initial length
is given by:
λ
=
x
+
u
(
x
+
x
)
−
u
(
x
)
.
(6.12)
x
Consequently, if the width of the slice
x
approaches zero, the strain is computed
from
x
→
0
x
+
u
(
x
+
x
)
−
u
(
x
)
ε
=
lim
−
1
x
u
(
x
+
x
)
−
u
(
x
)
=
lim
x
→
0
.
(6.13)
x
Using the definition of the derivative this yields
du
dx
.
ε
=
(6.14)
In conclusion, the strain is defined as the derivative of the displacement field
u
with respect to the coordinate
x
.
