Biomedical Engineering Reference
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Fig. 12 Viscoelastic responses of bone with respect to the applied strain: a Total stress
relaxation of bone in both tension and compression. b Time constants for both tension and
compression obtained from the stress-time curves in the progressive tests
unimportant at low frequency [
94
]. In addition, temperature-scan tests of bone
specimens with different concentration of denatured collagens indicate that the
hydration condition, not the collagen phase, is significantly related to the behavior
of tand [
92
]. Moreover, damping increases with accumulation of mechanical
damage, in terms of increases in tand and frequency sensitivity of storage modulus
(E') of bone [
95
,
96
].
Viscoelastic behavior of bone changes with applied strain. As bone is contin-
uously deformed, stress relaxation increases up to an equilibrium state after
yielding (Fig.
12
). Total stress relaxation (Dr, a measure of the viscous contri-
bution to total load bearing) is much greater in compression than tension, and the
time constant (s, inversely related to damping capability) is slightly greater in
compression. Since hysteresis energy dissipation is viscoelastic in nature, it is most
likely dependent on the magnitude of both Dr and s. Thus, a positive effect on the
hysteresis energy dissipation due to an increase in Dr could be cancelled out by a
negative effect induced by a decrease in damping property (s)[
84
]. This could
explain the fact that only a slight difference in hysteresis energy dissipation
between compression and tension is observed (Fig.
13
).
In general, an exponential relaxation function (e
-t/s
, the Debye model) is often
used to describe stress relaxation behavior of viscoelastic materials. However, this
model is not suitable for bone. A linear combination of the Kohlrausch-Williams-
Watts (KWW) function and the Debye function can be employed to predict the
viscous response of bone [
97
,
98
]:
Þ
b
r
¼
A
1
e
t
=
s
1
ð
þ
A
2
e
t
=
s
2
þ
C
ð
1
Þ
where, A
1
and A
2
represent the contribution to the total stress relaxation, and s
1
and
s
2
are the time constants for the KWW and Debye processes, respectively, b is a
material constant that determines the acuteness of the KWW stress relaxation, and
C is a constant representing the stress component by the pure elastic deformation.
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