Biomedical Engineering Reference
In-Depth Information
Fig. 20.4
Biphasic macromodel
20.2.1.1 Mixture Theory, Concept of Volume Fraction and Kinematics
The microscopic structure is represented within a statistical distribution of the
constituents (solid and fluid) over a representative elementary volume (REV), see
Fig. 20.4 . The constituents ϕ α will be represented by an averaging volume frac-
tion n α . Thereby, the volume fractions n α refer to the volume elements dv α of the
individual constituents ϕ α from the bulk volume element dv with
κ
κ
dv α
dv ,
ρ α
ρ α R =
n α ( x , t )
n α ( x , t )
=
=
1
∈{
S , F
}
, (20.1)
α =
1
α =
1
where x is the position vector of the spatial point x in the actual placement and t is
the time; we proceeded to a homogenized model with superimposed continua. The
now separated partial volumes dv α will again be interconnected constitutively by so-
called interaction quantities. In general, these quantities are a mass exchange
ρ α ,an
ˆ
e α . The volume fractions n α in ( 20.1 ) 1
satisfy the volume fraction condition ( 20.1 ) 2 for κ constituents ϕ α . Moreover, the
partial density ρ α
p α , and an energy exchange
interaction force
ˆ
ˆ
n α ρ α R of the constituent ϕ α is related to the real density of the
materials ρ α R involved via the volume fractions n α ;see( 20.1 ) 2 . Due to the volume
fraction concept, all geometric and physical quantities, such as motion, deformation,
and stress, are defined in the total control space. Thus, they can be interpreted as the
statistical average values of the real quantities.
The saturated porous solid will be treated as an immiscible mixture of all con-
stituents ϕ α with particles X α , where at any time t each spatial point X S of the
current solid placement is simultaneously occupied by fluid particles X F . These
particles proceed from different reference positions X α at time t
=
=
t 0 .
det F α , where F α = (∂ x α )/(∂ X α ) =
Grad α χ α is the deformation gradient of the constituent ϕ α . During the deformation
process F α is restricted to satisfy det F α > 0.
For scalar fields depending on x and t, the material time derivatives are defined
as (...) α = ∂(...)/∂ t
Furthermore, the Jacobian is defined as J α =
x α , with grad (...) = ∂(...)/∂ x . In order to use a
+[
grad (...)
Search WWH ::




Custom Search