Biomedical Engineering Reference
In-Depth Information
with
ν
ε
0
=
ν
l
δ
0
D
K
=
k
B
T
ρ
and
2
0
D
ε
10
−
23
JK
−
1
Here
k
B
=
1
.
38
×
is the Boltzmann constant,
T
is the absolute temper-
ature,
is the number of molecules per unit surface in the equatorial plane,
D
is
the length of the axial collagen period after the removal of all kinks, and
ρ
ν
is the
δ
total number of kinks of a molecule
per D
-period. Moreover,
0
denotes the average
<δ>
value
, obtained in the absence of external stress. Thus, the entropic regime
may account for the initial regime of stress-strain behavior for fibril strains up to
8%, as is shown in figure 6 in the Misof
et al
. paper, cf. also [244].
Hence, when
ε
ε
is small compared to
0
, the prediction is a linear elastic behavior
ε
ε
with the elastic modulus
K
.When
approaches
0
(that is, when almost all of the
σ
kinks are removed), the tension
required to get a further extension by the removal
of the remaining kinks tends to infinity, which means that other mechanisms must
become dominant for the sample behavior. The stress-strain curve corresponding
to the Misof relation is bent upward. When tension
σ
becomes too large, other
mechanisms of collagen elasticity, like a direct stretching or a side-by-side gliding
of the molecules, come into play and lead to a linear stress-strain relation instead
of the Misof relation.
When all the kinks are straightened, another mechanism of deformation must
prevail and explain the linear dependence of stress and strain in this region of
the force-elongation curve. The most probable processes are thought to be the
stretching of the collagen triple helices and the cross-links between the helices,
implying a side-by-side gliding of neighboring molecules, leading to structural
changes at the level of the collagen fibrils.
A hierarchical structure of a collagen tendon for the model of strain-rate-
dependent effects was proposed by Puxkandl
et al
. [259]. The tendon was considered
as a composite material with collagen fibrils embedded in a proteoglycan-rich
matrix. This matrix is mostly loaded under shear. Because the spacing between
fibrils is much smaller than their length, the shear stress
τ
effectively applied to
the matrix is much smaller than the tensile stress
σ
on the tendon. In reality,
τ
≈
σ
H
/
L
,where
L
is the length of the fibrils and
H
is their spacing, [266]. The
aspect ratio
L
/
H
is of the order of 100-1000. One may suppose that the elastic
response of the matrix is mostly due to the entanglement of molecules attached
to the collagen fibrils, such as proteoglycans. In addition, there is a considerable
viscosity in the matrix due to the many hydrogen bonds that can form in this glassy
structure.
Gupta
et al
. in 2004 carried out tensile measurements by using synchrotron X-ray
diffraction [260]. A load cell was mounted on one grip while the other was moved
with a motorized translation. To maintain the tendon in a native state, the sample
was partially immersed in a physiological solution of phosphate-buffered saline
during the test.
The most striking result was a peak splitting of the diffraction spectrum for large
macroscopic tendon strains,
≥
2
−
3%. The splitting implies an inhomogeneous
