Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 286

importance in the development of the mixture theories is the application by Bowen
( 1967 , 1976 , 1980 , 1982 ) of a thermodynamically based analytical approach devel-
oped by Coleman and Noll ( 1963 ) to restrict the form of constitutive equations.
At that time (1966-1967) the thermodynamically based Coleman and Noll approach
was very new. In his history of the development of theories for porous media, de
Boer ( 2000 ) credits Goodman and Cowin ( 1972 ) with the first use of the decompo-
sition of bulk density into a volume fraction and a true or material density ( 10.16 ).
The introductory material in this chapter is taken from Bowen ( 1976 , 1980 , 1982 ).

Appendix

The purpose of this appendix is to record the derivation of (30) and some related
auxiliary results. Recall that

ϖ ( a ) denotes a generic component-specific property

e ðaÞ and we seek a simple formula for P

N
a¼ 1 r ðaÞ

D a ˆ ðaÞ

D t

such as v ðaÞ or to be used in

determining the continuum level form of the conservation laws by summing over
the single constituent continuum forms of the conservation laws. A formula relating
the density-weighted sum of the time derivatives of the selected components to the
sum of the density-weighted time derivatives is desired. Recall that the sum of
generic constituent-specific quantity per unit mass

ϖ ( a ) is related to its density-

weighted sum by ( 10.29 ). The time derivative of ( 10.29 ) with respect to the

selected component is given by

ϖ

X X

N
a¼ 1 r ðaÞ

D s

N
a¼ 1 ˆ ðaÞ

D s

D s D s

r ðaÞ
D t

r
D t ¼

ˆ ðaÞ
D t þ

D t þ ˆ

r (10.95)

which may be solved for P

N

D s

ˆ ðaÞ
D t

1 r ðaÞ

, thus

a ¼

X X

D s

N
a¼ 1 r ðaÞ

D s

N
a¼ 1 ˆ ðaÞ

r ðaÞ
D t

D s D s

ˆ ðaÞ
D t

r
D t

D t þ ˆ

¼ r :

(10.96)

The relationship between the time derivatives with respect to the selected
component and with respect to the “ a ” component is obtained using ( 10.12 )

X

N
a¼ 1 r ðaÞ

D s

X

N
a¼ 1 r ðaÞ

D a

X

N
a¼ 1 r ðaÞ

X

N
a¼ 1 r ðaÞ

ˆ ðaÞ
D t ¼

ˆ ðaÞ
D t þ

v ðsÞ r ˆ ðaÞ v ðaÞ r ˆ ðaÞ ;

(10.97)