Biomedical Engineering Reference
In-Depth Information
where the partial derivative o
=
oX is now used since the quantities e, u and x
depend on X and t. Generally, the motion of a continuum body is expressed by the
mapping function v through
x
¼
v
ð
X
;
t
Þ
ð
3
:
42
Þ
where v is referred to as motion. The motion links both configurations (ICFG and
CCFG), thus establishing a definite relation between x and X at each point in time t
in terms of deformation processes (tension, pressure, shear etc.) acting on the
body. In case of a tensile bar, the relation is given as follows (cf. Fig.
3.9
)
x
¼
v
ð
X
;
t
Þ¼
X
þ
u
ð
X
;
t
Þ:
ð
3
:
43
Þ
It is common use to denote X the material or L
AGRANGIAN
coordinate and x the
spatial or E
ULERIAN
coordinate. Both differential quotients in (
3.41
)
F
ð
X
;
t
Þ
:
¼
ox
oX
o
v
ð
X
;
t
Þ
H
ð
X
;
t
Þ
:
¼
o
u
ð
X
;
t
Þ
oX
ð
3
:
44
Þ
and
oX
are denoted as deformation gradient F
and displacement gradient H. Equa-
tions (
3.41
) may thus be expressed by
e
¼
H
¼
F
1
F
¼
1
þ
H
ð
3
:
45
Þ
and
In continuum mechanics, F plays a central role and constitutes a building block
to generate appropriate deformation measures.
Note: According to (
3.45
), the ''engineering strain'' e is equal to the displace-
ment gradient H and thus is a linear function of H
:
Hence, it is referred to as linear
strain.
Some important terms of one dimensional continuum mechanics have hence
been introduced, using the example of a tensile bar. Based upon on this intro-
duction, a transfer can be made to three dimensional continuum mechanics. It is to
note that using the knowledge of (
3.42
)to(
3.44
) the result (
3.40
) deduced from
Fig.
3.9
could also have been derived by substituting (
3.43
)in(
3.44
)
1
i.e. without
visual help which however, represents a more comprehensible approach. To
establish more complex relations, the latter approach however, is more advisable
due to lack of visual perception. In addition, the three dimensional formulations
are most comfortably established using vector and tensor analysis.
3.2.3.2 Continuum and Body
In continuum mechanics a (biological) body is perceived as a continuous mass
rather than as a conglomeration of atomic particles. Such a body is also called a
point continuum (see Fig.
3.10
), where X stands for any material point. One
envisions a material point as an infinitesimally large volume element dV with an
