Biomedical Engineering Reference
In-Depth Information
r
fo
@fr
f
1
1
r
fo
rðfr
f
v
t
þ
Þ¼
0
:
(8.54)
@
The form of (
8.54
) is changed by use of the time derivative of (
8.16
),
@z
@
r
fo
@fr
f
1
t
¼
(8.55)
@
t
and (
8.26
), thus
@z=@t þr
q
¼
:
0
(8.56)
In the case of incompressibility,
r
f
¼ r
fo
, and (
8.54
) becomes
@f
@
t
þrðf
v
Þ¼
0
:
(8.57)
The stress equations of motion in three dimensions,
T
T
r€
u
¼r
T
þ r
d
;
T
¼
;
ð
3
:
37
Þ
repeated
have no simple representation in 6D vector notation, and the conventional notation
is employed;
€
u
represents the acceleration and
d
the action-at-a-distance force.
Problems
8.7.1. Evaluate each of the following formulas in the limit as
f !
0 (note that
0): (a) (
8.8
), (b) (
8.13
), (c) (
8.17
), (d) (
8.22
), (e) (
8.27
),
(f) (
8.49
), (g) (
8.50
), (h) (
8.53
), (i) (
8.54
).
8.7.2. Evaluate each of the following formulas in the limit as
f !
0 implies
f
o
!
f !
1 (note that
f !
1): (a) (
8.3
), (b) (
8.2
), (c) (
8.17
), (d) (
8.18
). The last
two results requires the easily justified restriction that 1
1 implies
f
o
!
K
eff
!
=
0as
f !
1.
8.8 The Basic Equations of Poroelasticity
An overview of the theory of poroelastic materials can be obtained by considering it as
a system of 18 equations in 18 scalar unknowns. This system of equations and
unknowns, a combination of conservation principles and constitutive equations, is
described in this section. The 18 scalar unknowns are the six components of the stress
tensor
T
, the fluid pressure
p
, the fluid density
, the six
components of the strain tensor
E
and the three components of the displacement vector
u
. The 18 scalar equations of the theory of poroelastic solids are the six equations of
the strain-stress-pressure relation (
8.1
), the six strain displacement relations (2.49),
r
f
, the variation in fluid content
z
Search WWH ::
Custom Search