Dynamic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 237

It is interesting to note that Biot's elasticity tensor Z d differs from the drained
elasticity tensor C d by the term M

ðA A

, which is M times the open product of
Biot effective stress coefficient vector A with itself. Equations ( 9.6 ) and ( 9.7 ) take
the following forms when the strain-displacement relations ( 2.49 ) and ( 9.3 ) are
employed:

Þ

T ij ¼ Z ijkm u k;m þ M ij w k;k (9.9)

and

p ¼ M km u k;m Mw k;k : (9.10)

The balance of momentum in the form of the dynamical stress equations of
motion

T T

rx

¼r T þ r d ; T ¼ ð 3 : 37 repeated Þ

will now be applied twice, once to the solid phase and once to the fluid phase. In
both cases of its application the action-at-a-distance force d is neglected. The
application to the solid phase involves the mass times acceleration terms for the
fluid saturated solid phase fr f U

þð 1 fÞr s € u and may be reduced to r€ u þ r f € w

when ( 9.2 ) is used as well as the definition of r ,

r ¼ð 1 fÞr s þ fr f ; (9.11)

where r s represents the density of the solid matrix material, thus

¼ r € þ fr f €

r T u w : (9.12)

In the application of the balance of momentum to the fluid phase the mass
times acceleration term may first be written as

r f U and may be rendered in the form

r f ð€ by use of ( 9.2 ). However Biot ( 1962a , b ) extends this formulation of

this mass times acceleration term of include J , the micro-macro velocity average
tensor, thus

u þð€ w =fÞ

r f ð€ u þð€ w =fÞÞ becomes r f ð€ u þ J € w Þ where the newly defined J

f 1 . The micro-macro velocity average tensor J functions
like a density distribution function that relates the relative micro-solid-fluid velo-
city to its bulk volume average w . In introducing this concept Biot was clearly
viewing and modeling the poroelastic medium as hierarchical. The use of

incorporates the factor

r f ðu þ J

wÞ yields a balance of linear momentum for the pore fluid phase in the form

r p ¼ r f ð€ u þ J € w Þ: (9.13)