Biomedical Engineering Reference
In-Depth Information
analogous to Eq. 5-12 and where D is a damping coefficient. The real
and imaginary moduli can be calculated directly from
π
P 0
h 0
2
E S
)
cosφ
(5-48)
= (1 − ν
2
A C
π
P 0
h 0
2
E L
)
sin
φ
(5-49)
=
(1
− ν
2
A C
The derivations of these expressions and caveats concerning their use can
be found in other sources. 38-40 In practice, the typical experiment would
include a number of different measurements for different frequencies
(“frequency sweep”) and the evaluation of the real and imaginary parts
of the complex modulus as a function of frequency and potentially also
as a function of static ( P max ) and dynamic ( P 0 ) load levels.
6.2. Poroelastic contact
Poroelasticity concerns the flow of fluid through a saturated porous
elastic solid. 41 Although the components (fluid, solid) themselves are
taken to be time-independent, the flow problem creates time-dependence
in the response. The basic elastic problem is augmented in poroelasticity
with two additional scalar variables, the pore pressure, p , and an
increment of fluid content,
, the change in fluid volume relative to a
control volume. The poroelastic framework considered here is for an
isotropic elastic porous material with elastic material properties shear
modulus, G , and Poisson's ratio,
. There are two other parameters to
characterize the fluid and the fluid-solid interactions, and B , which
are bounded between zero and unity. With the additional variables ( p , )
and material constants (, B ), the stress-strain ( ij ij ) relationship
for elasticity is modified by the addition of a term including the pore
pressure (
p ij ) to become:
2 G
ν
ε kk δ ij
+ α
p
δ ij
+
(5-50)
σ ij
=
2 G
ε ij
1
2
ν
ij is the Kronecker delta. 26 A single additional scalar expression is
added to the six elastic equations to total seven constitutive equations:
where
 
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