Biomedical Engineering Reference
In-Depth Information
u
)
T
+
2
E
u
), the three equations of motion (3.37), the one fluid
content-stress-pressure relation (
8.17
) (or the one fluid content-strain-pressure relation
(
8.18
)) and the one mass conservation equation (
8.56
) and a relation between the fluid
pressure and the density
p
¼
((
D
D
r
f
) which is not specified here. The parameters of a
poroelasticity problem are the drained effective elastic constants of porous matrix
material
S
d
, the Biot effective stress coefficients
¼
p
(
A
,
C
eff
, the fluid viscosity
m
,the
intrinsic permeability tensor
K
, and the action-at-a-distance force
d
,whichareall
assumed to be known. If the displacement vector
u
is taken as the independent
variable, no further equations are necessary. However, if it is not, use of the compati-
bility equations (2.54) is necessary to insure that the displacements are consistent.
There are many methods of approach to the solution of poroelastic problems for
compressible media. The method selected depends upon the information that is
provided and the fields that are to be calculated. One approach that has been
effective is to solve for the variation in fluid content
if the stress or the strain
field is known or may be calculated without reference to the variation in fluid
content
z
. The diffusion equation for the variation in volume fraction is obtained by
first substituting Darcy's law (
8.17
) into the expression (
8.56
) for the conservation
of mass and subsequently eliminating the pore pressure by use of (
8.43
), thus
z
@z
@
1
mL
1
mL
K
O
K
O
½ A
E
t
z ¼
:
(8.58)
is due either to fluid
flux or to volume changes caused by the strain field. It is possible to replace the
strain on the right hand side of (
8.58
) by stress in which case (
8.58
) becomes
This shows that the time rate of change of the fluid content
z
@z
@
1
1
S
d
C
eff
K
O
C
eff
K
O
½ A
T
t
z ¼
:
(8.59)
m
m
Diffusion equations for the pressure field are also employed in the solution of
poroelastic problems. The first diffusion equation for the pore pressure field is
obtained by substituting Darcy's law (
8.17
) into the expression (
8.56
) for the
conservation of mass and subsequently eliminating the variation in fluid content
z
by use of (
8.17
), thus
@
E
@
@
p
1
mL
1
L
K
Op
A
t
¼
t
:
(8.60)
@
The alternative diffusion equation for the pore pressure field is obtained by
replacing the strain on the right hand side of (
8.60
) by stress, thus
!
:
Þ
@
T
@
@
p
1
1
C
eff
ðS
m
S
d
C
eff
K
Op
U
t
¼
(8.61)
@
t
m
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