Biomedical Engineering Reference
In-Depth Information
Fig. 25.2
An anisotropic
representative volume
element for a network of
semi-flexible chains (Bischoff
et al.,
2002b
,
2002c
)
25.2.1 3D Hyperelastic Constitutive Models
Collagen fibers and elastin are long-chain molecules. Their mechanical responses
can be categorized into three regimes: flexible, semi-flexible, and stiff (Palmer et al.,
2010
). Many biopolymer inspired constitutive models have been proposed, such as
the freely jointed chain and semi-flexible chain models. MacKintosh et al. (
1995
)
proposed a model that describes the nonlinear behavior of semi-flexible biopolymers
at small strains well (MacKintosh et al.,
1995
). This model has been implemented in
an eight-chain framework (Arruda and Boyce,
1993
) to capture the response of an F-
actin network (Palmer and Boyce,
2008
). The Cauchy stress tensor
T
A
is expressed
as
L
c
/l
p
−
6
(
1
−
λ
c
r
0
/L
c
)
nkΘ
A
3
l
p
r
0
λ
c
1
T
A
=
−
λ
c
r
0
/L
c
)
B
p
I
,
(25.5)
λ
c
λ
0
/L
c
)
2
4
(
1
−
L
c
/l
p
−
2
(
1
−
tr
(
B
)/
3
.
λ
c
=
(25.6)
In this equation,
n
is the chain density of the network,
k
is Boltzmann's constant,
Θ
A
is the temperature, and
p
is the hydrostatic pressure. For the MacKintosh chain
network
l
p
represents the persistence length,
L
c
represents the contour length,
r
0
is
the initial vector chain length and
λ
c
is the chain stretch. Since these are all physical
parameters, potentially, they are measurable experimentally. The MacKintosh chain
network of the micromechanical model is embedded within an initially isotropic
or anisotropic 8-chain framework (Arruda and Boyce,
1993
; Bischoff et al.,
2002b
,
2002c
), as in Fig.
25.2
to mathematically model the mechanical behavior of a struc-
tural protein network or inorganic elastomer network.
