Biomedical Engineering Reference
In-Depth Information
Because the origin is a relative position, it can be considered that γ|
x
=0
= 0.
Finally, the solution is
G
wE
2
(
γ =−
τ+τ
−
h
x
0
1
x
1
−
e
m
(9.28)
2
G
The solution (9.28) indicates that the sign of shear strain γ depends
entirely on the difference between τ
0
and τ
1
(τ
0
> τ
1
according to traditional
beam theory). The conclusion can be drawn that the shear stress deter-
mined by traditional beam theory is responsible for the sign of piezovolt-
age in bone. Because the exponent term in Equation (9.28) includes elastic
modulus
E,
normal stresses still contribute to the shear strain. However,
the normal stress can change only the amplitudes, not the sign of the shear
strain. That is, no matter how small τ
0
and τ
1
are, or how large the normal
stresses σ
0
and σ
1
are, the signs of shear strain in collagen fibrils depend
exclusively on the shear stresses τ
0
and τ
1
. This conclusion explains why the
signs of piezovoltages remain the same with shear stress under three-point
bending.
Fu et al. [23] demonstrated experimentally and theoretically that the signs
of piezovoltages of bone under bending deformation depend only on shear
stress. It seems doubtful that normal stress contributes only to the amplitude
of piezovoltage and not to changing its sign. If normal stress itself can gen-
erate a piezovoltage in bone, it must dominate the sign of the voltage, as a
voltage is a physical quantity with both magnitude and sign.
A possible reason is the coupling effect between normal stress and shear
stress. Macroscopically, only normal stress operates on a cross section of
a sample under four-point bending. However, because cortical bone has
a hierarchical structure and collagen fibrils are distributed in the mineral
matrix in a very complex and random manner, the coupling effect between
normal stress and shear stress becomes microscopically stronger. Perhaps
the piezovoltages in the pure bending zone of the sample under four-point
bending arise from the coupled shear stresses. If so, the contribution of
normal stress to piezovoltages in bone still comes substantially from shear
stresses.
Other possible contributors to bone piezoelectricity are the cross links,
which are covalent bonds between two adjacent collagen molecules
(Figure 9.19). Pollack et al. [45] found experimentally that the piezoelectricity
of bone increased with an increasing number of cross links. Minary-Jolandan
and Yu [38] formed the opinion that cross links enable collagen molecules
to transmit mechanical forces to neighboring collagen molecules. In terms
of their transmission function, the cross links are affected by shear stress.
The contribution of piezosignals from the cross links to the piezovoltages
from the bending experiment is an interesting research issue.
