Biomedical Engineering Reference
In-Depth Information
in fluid content
may be changed in two ways, either the density of the fluid may
change, or the porosity
z
may change. This set of variables may be viewed as the
conjugate pairs of stress measures (
T
,
p
) and strain measures (
E
,
f
) appearing in
the following form in an expression for work done on the poroelastic medium:
d
W
z
¼
T
:d
E +p
d
z
. The linear stress-strain-pore pressure constitutive relation
E
E
(
T
,
p
), consisting of six scalar equations, is described in Sect.
8.2
and the
single-scalar-linear equation fluid content-stress-pore pressure constitutive relation
z ¼ z
¼
(
T
,
p
) is described in Sect.
8.3
. Thus the seven scalar stress variables will be
linearly related to the seven scalar strain variables. The poroelastic theory consid-
ered here is fully saturated, which means that the volume fraction of fluid is equal to
the porosity,
, of the solid matrix. Darcy's law, which relates the gradient of the
pore pressure to the mass flow rate, is the subject of Sect.
8.4
. The special form of
poroelasticity theory in the case when both the matrix material and the pore fluid are
considered to be incompressible is developed in Sect.
8.5
. Formulas for the
undrained elastic coefficients
S
u
f
, the bulk modulus of the pore
fluid,
K
f
, and drained and matrix elastic constants,
S
d
and
S
m
, respectively,
S
u
as functions of
f
¼
S
u
ðS
d
; S
m
are developed in Sect.
8.6
. The objective of Sects.
8.2
,
8.3
,
8.4
,
8.5
,
8.6
, and
8.7
is to develop the background for the basic system of equations for
poroelasticity recorded in Sects.
8.8
and
8.9
. If the reader would like to have a
preview of where the Sects.
8.2
,
8.3
,
8.4
,
8.5
,
8.6
,
8.7
, and
8.8
are leading, they can
peruse Sect.
8.9
and observe how the various elements in Sects.
8.2
,
8.3
,
8.4
,
8.5
,
8.6
,
8.7
, and
8.8
are combined with the conservation of mass (3.6) and the
conservation of momentum (3.37) to form poroelasticity theory. Examples of
the solutions to poroelastic problems are given in Sect.
8.10
and a summary of
the literature, with references, appears in Sect.
8.11
.
; f;
K
f
Þ
8.2 The Stress-Strain-Pore Pressure Constitutive Relation
The basic hypothesis is that the average strain
E
in the RVE of the saturated porous
medium is related, not only to the average stress
T
in the RVE, but also to the fluid
pressure
p
in the fluid-filled pores. Thus the stress-strain-pore pressure constitutive
relation for a saturated porous medium linearly relates the strain
E
in the saturated
porous medium not only to the stress
T
, but also to the fluid pressure
p
in the fluid-
filled pores; this is expressed as the strain-stress-pore pressure relation
E
¼ S
d
T
þ S
d
Ap
(8.1)
or the stress-pore pressure strain relation
¼ C
d
T
þ Ap
E
;
(8.2)
where
S
d
represents the drained anisotropic compliance elastic constants of the
saturated porous medium and
C
d
is its reciprocal, the drained anisotropic elasticity
tensor. This constitutive equation is a modification of the elastic strain-stress relation
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