Biomedical Engineering Reference
In-Depth Information
E
C
E
zero strains
E
>
0
for all non
(5.34H)
¼ C
T
C
is a positive definite symmetric tensor,
C
Thus it follows that
and
E
C
E
0 for all non-zero strains
E
.
>
5.12 Summary of Results
In this chapter a progressive development of four constitutive relations has been
presented. Beginning with the constitutive idea, restrictions associated with the
notions of localization, invariance under rigid object motions, determinism, coor-
dinate invariance, and material symmetry were imposed. In the development the
constitutive equations were linearized and the definition of homogeneous versus
inhomogeneous constitutive models was reviewed. Restrictions due to material
symmetry, the symmetry of the material coefficient tensors, and restrictions on
the coefficients representing material properties were developed. The results of
these considerations are the following constitutive equations
H
T
q
¼ fr
f
v
=r
o
¼
H
ð
p
Þr
p
ð
x
;
t
Þ;
H
ð
p
Þ¼
ð
p
Þ;
(5.36D)
¼ C
T
T
¼ C
E
where
C
;
(5.36H)
where
H
(
p
) and
C
are positive definite, and
T
¼
p
1
þ l
ðÞ
tr
D
1
þ
2
m
D
;
(5.36N)
m
>
m
>
where
p
is the fluid pressure and
l
and
m
are viscosity coefficients (3
2
2
),
l
and
1
T
G
ÞD
¼
ð
s
ð
x
;
t
s
Þ
d
s
;
(5.36V)
s
¼
0
G
where there are no symmetry restrictions on
. All the constitutive equations
developed in this chapter, including Darcy's law, can be developed from many
different arguments. Darcy's law can also be developed from experimental or
empirical results for seepage flow in non-deformable porous media; all of the
other constitutive equations in this chapter have experimental or empirical basis.
Analytical arguments for these constitutive equations are presented so that it is
understood by the reader that they also have an analytical basis for their existence.
Darcy's law is a form of the balance of linear momentum and could include a body
force term; however, such a body force would normally be a constant, and since it is
ð
s
Þ
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