Biomedical Engineering Reference
In-Depth Information
Two cases associated with the effects of fluid pressure are considered for the
purpose of measuring the elastic constants in a poroelastic material, the drained and
the undrained. In the drained case the fluid pressure in the pores is zero; draining all
the pores before the test or executing the test very slowly achieves this because the
fluid in the pores will drain from the negligible fluid pressure associated with slow
fluid movement. In a porous medium the pores are assumed to be connected; there
are no unconnected pores that prevent the flow of fluid through them. In the
undrained case the pores that permit fluid to exit the test specimen are sealed; thus
pressure will build in the specimen when other forces load it, but the pressure cannot
cause the fluid to move out of the specimen. Paraphrasing the opening quote of this
chapter, the representative volume element (RVE) for a porous medium (Fig. 4.2 )is
considered as a small cube (Fig. 4.1 ) that is large enough, compared to the size of the
largest pores, that it may be treated as homogeneous, and at the same time small
enough, compared to the scale of the macroscopic phenomena which are of interest,
that it may be considered as infinitesimal in the mathematical treatment. The creator
of the poroelastic theory advanced this description of the RVE for a saturated porous
medium before the terminology RVE came into wide usage. Maurice Anthony
Biot (1905-1985), a Belgian-American engineer who made many theoretical
contributions to mechanics, developed poroelastic theory. It is his 1941 paper
(Biot 1941 ) that is the basis of the isotropic form of the theory described here.
There are three sets of elastic constants employed in this poroelasticity theory,
the drained, S d , the undrained, S u , and those of the matrix material, S m . The RVE
associated with the large box in Fig. 4.2 is used for the determination of the drained
and the undrained elastic constants while the RVE associated with the smallest box
is used for the characterization of the matrix elastic constants. Unlike the large box
RVE, the small box RVE contains no pores. The elastic compliance matrices S u , S d ,
and S m for these materials have a similar representation:
2
3
S 11
S 12
S 13
S 14
S 15
S 16
4
5
S 12
S 22
S 23
S 24
S 25
S 26
S 13
S 23
S 33
S 34
S 35
S 36
S x
¼
;
S 14
S 24
S 34
S 44
S 45
S 46
S 15
S 25
S 35
S 45
S 55
S 56
S 16
S 26
S 36
S 46
S 56
S 66
where x
d , u ,or m stands for drained, undrained, and matrix, respectively. The
special forms of
¼
S x
associated with particular elastic symmetries are listed in
Tables 4.4 and 4.5.
There are seven scalar stress variables and seven scalar strain variables in
poroelasticity. The seven scalar stress variables are the six components of the stress
tensor T and fluid pressure p in the pores. The seven scalar strain variables are the
components of the strain tensor E and the variation in fluid content
, a dimension-
less measure of the fluid mass per unit volume of the porous material. The variation
z
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