Biomedical Engineering Reference
In-Depth Information
development
the inverse of (
5.7H
),
the strain-stress relation rather than the
stress-strain relation, is employed
¼ C
1
E
¼ S
T
S
;
;
(5.12H)
where
S
is the compliance tensor of elastic material coefficients. The form and
symmetry of
C
and
S
are identical for any material, and it is easy to show that the
symmetry of one implies the symmetry of the other. The symmetry of
C
and
S
follows from the requirement that the work done on an elastic material in a closed
cycle vanish. This requirement stems from the argument that if work can be done on
the material in some closed cycle, then the cycle can be reversed and the material
can do work in the reversed closed cycle. This would imply that work could be
extracted from the material in a closed loading cycle. Thus one would be able to
take an inert elastic material and extract work from it. This situation is not logical
and therefore it is required that the work done on an elastic material in a closed
loading cycle vanish. We express the work done on the material between the strain
E
ð
1
Þ
and the strain
E
ð
2
Þ
by
ð
2
T
dE
W
12
¼
;
(5.13H)
1
and for a closed loading cycle it is required that
þ
T
dE
¼
0
:
(5.14H)
Consider the work done in a closed loading cycle applied to a unit cube of a
linear anisotropic elastic material. The loading cycle begins from an unstressed
state and contains the following four loading sequences (Fig.
5.2(a)
): O
A, the
stress is increased slowly from 0 to
T
i
;A
!
B, holding the stress state
T
i
constant
the second stress is increased slowly from
T
i
!
to
T
i
+
T
i
,
T
i
6¼ T
i
;B
C, holding
the second stress state
T
i
constant the first stress is decreased slowly from
T
i
!
+
T
i
to
T
i
O, the stress is decreased slowly from
T
i
to 0. At the end of this
loading cycle the object is again in an unstressed state. The work done in (
5.13H
)on
each of these loading sequences is expressed as an integral in stress:
; and C
!
ð
ð
2
2
T
d
E
E
dT
W
12
¼
¼
:
(5.15H)
1
1
The integral over the first loading sequence of the cycle, from 0 to
T
i
, is given by
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