Biomedical Engineering Reference
In-Depth Information
associated with Stokes theorem at the end of the section before last, in three
dimensions, a differential d
q
is an exact differential in a simply-connected
3D region
R
of the
x
1
,
x
2
,
x
3
space if between the functions
v
1
,
v
2
,
and v
3
there exist
the relations (A.190) or (A.192). If d
q
¼ v
d
x
is an exact differential in a simply-
connected 3D space, then from (A.189) the integral about any closed path in the
space is zero, and furthermore it follows that when the vector field
v
satisfies the
condition (A.189) all closed paths, it may be represented as the gradient of a
potential,
v
¼ v
d
x
.
Having now built the idea of an exact differential from one to three dimensions,
we now extend it to six dimensions and the consideration a 6D work differential
¼rf
¼ T
d
E
d
W
:
(A.193)
In Chap.
6
it is shown that, in order that no work can be extracted from an elastic
material in a closed path, it is necessary for the work done in all closed paths to be
zero,
Þ
T
d
E
(6.14H repeated)
Thus the 6D work differential (A.193) is an exact differential for an elastic
material. The conditions parallel to those in 3D, namely (A.190) and (A.191) are
that
¼
0
:
r
E
T
be symmetric,
@
T
i
@ E
j
¼
@ T
j
@ E
i
;
(A.194)
or
T
r
E
T
¼ðr
E
T
Þ
;
(A.195)
respectively. The parallel to the existence of a potential in 3D is the strain energy
U
defined by (6.25H) and related to the stress and strain by (6.26H).
Problem
14.1 The conditions for an exact differential are (A.192) in 2D, (A.190) or (A.191)
in 3D, and (A.194) or (A.195) in 6D. What are the conditions for an
exact differential
in 4D if
the differential
is denoted by d
q
¼ a
d
x
,
d
q
¼
a
1
d
x
1
þ
a
2
d
x
2
þ
a
3
d
x
3
þ
a
4
d
x
4
?
A.15 Tensor Components in Cylindrical Coordinates
In several places use is made of cylindrical coordinates in the solutions to
problems in this text. The base vectors in curvilinear coordinates are not generally
unit vectors nor do they have the same dimensions. However local Cartesian
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