Biomedical Engineering Reference
In-Depth Information
which is to say that the current apparent mass density depends on its original value
ρ α ( 0 ) and the kinetic loss of the original material via Q α (s)
∈[
]
,aswellas
both the subsequent true production m α (τ) and associated loss q α (s,τ)
0 , 1
∈[
0 , 1
]
of
material after s
0 (the time at which a perturbation initiates G&R). Because the
constituent mass densities are apparent, not true, densities, the total mass density is
computed easily via
=
ρ α (s)
φ α (s),
ρ(s)
=
1
=
(9.9)
where φ α (s)
ρ α (s)/ρ(s) are usual mass fractions. Of course, we must recover
=
ρ α ( 0 ) at s
0, which reveals that Q α ( 0 )
1inEq.( 9.8 ).
Because we employ a rule-of-mixtures relation for the stress, linear momentum
for quasi-static G&R is simply the same as that in classical continuum mechan-
ics, namely div t
=
=
0 , where t is the Cauchy stress. As in most of biomechanics,
therefore, the significant challenges lie first in formulating appropriate constitutive
relations and second in solving initial-boundary value problems of interest.
Although it is natural to seek constitutive relations for stress directly (Humphrey
and Rajagopal, 2002 ), it proves useful to follow advances in nonlinear elasticity
and alternatively seek constitutive relations for the stored energy W α (s) , whereby a
rule-of-mixtures approach can be written conceptually as
=
n
φ α (s) W α (s),
W(s) =
(9.10)
α
=
1
noting of course that the stored energy depends on the (finite) deformation experi-
enced by the material, which is to say each of its load-bearing constituents. Prior
studies have suggested, however, that such an approach is limited in its ability to
capture contributions of individual constituents that may turnover continuously at
different rates and to different extents. Hence, following Baek et al. ( 2006 ), we let
n
W α (s),
W(s)
=
(9.11)
α
=
1
where we postulated, constituent-specific, forms motivated by Eq. ( 9.8 )(whichwas
derived directly), namely
ρ α ( 0 )Q α (s)
ρ(s)
W α C n( 0 ) (s)
W α (s)
=
s
(9.12)
m α (τ)q α (s
W α C n(τ) (s) d τ,
τ)
+
ρ(s)
0
where the energy stored in individual constituents is assumed to depend on de-
formations experienced by those constituents, which by the principle of material
frame indifference requires dependence on the deformation gradient through the
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