Dynamic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 235

passing through this poroelastic medium to obtain an appropriate average of
properties in the poroelastic RVE, what condition must be satisfied between
the wavelength of a wave and the characteristic length L RVE associated

with the volume element employed in the poroelastic model?
9.1.2. What parameter of an experiment may the experimentalist control to insure
that the criterion that is the answer to question 9.1.1 above is satisfied?

l

9.2 Historical Backgrounds and the Relationship
to the Quasistatic Theory

The formulation of the theory of wave motions in the context of poroelastic theory
presented here is consistent with the presentations of Biot ( 1941 , 1955 , 1956a , b ,
1962a , b ), Plona and Johnson ( 1983 ), Sharma ( 2005 , 2008 ) and many others. The
origins of this analysis appear in the work of Frenkel ( 1944 ). Unchanged by the
addition of anisotropy is the fact that the total elastic volumetric response in
poroelasticity described in the previous chapter is due to a combination of the
elastic volumetric response of the matrix material of the porous solid, the volumet-
ric elastic response of the pore fluid, and the pore volume changes in the porous
medium. The poroelastic constitutive equations are described in this and the
following section follow Biot's ( 1956a , b , 1962a , b ) formulation of the appropriate
two coupled wave equations. In the following section the coupled wave equations
(( 9.16 ) and ( 9.17 )) for the propagation of plane waves in an anisotropic, saturated
porous medium are developed and, in the section after that, the relationships
between the material coefficients and the fabric are recorded. The algebra
associated with the representation of plane waves is developed in Sect. 9.4 , and
the fabric dependence of the coefficients is recorded in Sect. 9.5 . The propagation of
plane waves in a principal direction material symmetry and the direction that is not
a principal direction of material symmetry, are recorded in Sects. 9.6 and 9.7 ,
respectively.
In his 1956 papers on wave propagation Biot ( 1956a , b ) let u represent the
displacement vector of the solid matrix phase as has been done in this chapter, and
U represent the displacement vector of the fluid phase, which is not done in this
chapter. These were the two basic kinematic quantities employed in Biot ( 1956a , b ).
In Biot ( 1962a ) the displacement vector of the fluid phase U was replaced by the
displacement vector w of the fluid relative to the solid, thus

w ¼ fð U u Þ: (9.1)

The present development follows Biot ( 1962a , b ) and the two basic kinematic
fields are considered to be the displacement vectors u and w . The relative velocity
of the fluid and solid components is, from ( 9.1 ),