Biomedical Engineering Reference
In-Depth Information
6.3.2 Kinematics of the anisotropic mechanism
In general, the collagen fibres will display a distribution of fibre crimp. Namely,
there will not be a
single
discrete strain associated with the initiation of load bearing
of all collagen fibres. For the moment, we consider an arbitrary fibre family at an
arbitrary material point in the body. We denote the configuration where this fibre
is recruited to load bearing (activated) as
κ
a
, Fig. 6.2. For simplicity of the ensuing
discussion, we refer to the time at which this configuration is reached as
t
a
. The
length and orientation of an infinitesimal material segment of fibre in
a
will be
denoted as
dS
a
and
a
a
, respectively. Under an affine transformation, this material
element will be mapped back to a material element in
κ
κ
0
which we denote as
dS
0
a
0
,
so that,
dS
a
a
a
=
F
0
(
t
a
)
dS
0
a
0
.
(6.9)
The stretch of a material element in
κ
(
t
)
relative to its length is
κ
0
is denoted by
λ
0
so that,
2
0
λ
0
=
ds
/
dS
0
,
and
λ
=
C
0
(
t
)
:
(
a
0
⊗
a
0
)
.
(6.10)
For definiteness, we take
a
0
and
a
a
to be unit vectors. Based on histological infor-
mation, it is clear the fibre is in a buckled state in
κ
0
. Therefore, the direction
a
0
defined in (6.9) is not the local orientation of the fibre, Fig. 6.2 and
dS
0
is not the
fibre length in
κ
0
. We denote the stretch of the underlying material element during
the deformation from
κ
0
and
κ
a
as
λ
a
,
2
a
λ
a
=
dS
a
/
dS
0
,
and
λ
=
C
0
(
t
a
)
:
(
a
0
⊗
a
0
)
.
(6.11)
The value of
λ
a
will be treated as a material property of the fibre reinforced mate-
rial defining the necessary material stretch for the fibre to become uncrimped. In
Sect. 6.3.5, we will discuss means of acquiring an estimate of
λ
a
from imaging
data. In principal, a theory could be developed to relate the geometry of the fibre
in
λ
a
. We do not address this issue here. In fact, one of the advantages of
the current approach arises from the fact that the constitutive contribution of col-
lagen is independent of this uncrimping process. Rather, the response of the col-
lagen fibres depends only on the deformation of collagen relative to the reference
κ
0
to
Fig. 6.2.
Schematic of configurations used for the fibre recruitment constitutive model
