Biomedical Engineering Reference
In-Depth Information
Fig. 3.9 Change of length
and strain of a tensile bar
Dl
¼
l
l
0
:
ð
3
:
37
Þ
Strain is expressed by the ratio of a change in length Dl per unit of the original
or initial length l
0
. As long as the bar is exposed to homogenous deformation (i.e.
identical deformation at each point of the specimen), the (normal) strain e reads
e :
¼
Dl
l
0
¼
l
l
0
l
0
¼
l
l
0
1
:
ð
3
:
38
Þ
If the deformation is not homogenous, in general, a different deformation at
every point X of the material is apparent. Considering an (undeformed) line ele-
ment with length dX at the position X in the ICFG and the (deformed) line element
with length dx at the position x in the CCFG, the change in length is obtained using
the displacement u
ð
x
Þ
of the element at position X (cf. Fig.
3.9
)by
du
¼
dx
dX
with
u
ð
X
Þ¼
x
X
:
ð
3
:
39
Þ
In addition, the local strain or ''engineering strain'' (note that again, according
to definition (
3.38
) the ratio of change in length du and initial length dX is
expressed) thus derives to
e
¼
du
dX
¼
dx
dX
d
dX
ð
x
X
Þ¼
dx
dX
1
:
ð
3
:
40
Þ
dX
Relations (
3.37
)-(
3.40
) are elementary mechanics relations. In continuum
mechanics the strain e, the (current) coordinate x in the CCFG and the displace-
ment coordinate u generally depend on the position X in the ICFG as well as the
time t. More precisely, equation (
3.40
) thus reads
e
ð
X
;
t
Þ¼
o
u
ð
X
;
t
Þ
oX
¼
o
x
ð
X
;
t
Þ
oX
1
ð
3
:
41
Þ
