Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 269

The derivation of ( 10.29 ) involves the expressions for the time derivatives
( 10.11 ) and ( 10.12 ), the constituent-specific mass balance ( 10.20 ) and the definition
of the density-weighted sum r ϖ in terms of the constituent-specific quantity per

unit mass denoted by ϖ ( a ) ,( 10.28 ). When the result ( 10.30 ) is incorporated in

( 10.29 ) it takes the form

X

N
a¼ 1 r ðaÞ

D a

X

N s

b¼ 1 rð ˆ ðbÞ r ðbÞ n ðb=sÞ Þ ˆ rrð

D s ˆ ðaÞ

Dt ¼ r

Dt þ

v v ðsÞ Þ

( )

X

N
a¼ 1 ˆ ðaÞ ^

^ þ ˆ ð t Þ

ðaÞ ð

t Þ (10.31)

Note that ( 10.30 ) reduces to

X X

N
a¼ 1 r ðaÞ

D a

Ns

D s

ˆ ðaÞ
Dt

Dt þ

¼ r

1 rð ˆ ðbÞ r ðbÞ n ðb=sÞ Þ

b ¼

( )

X

N

^

1 ˆ ðaÞ ^

þ ˆ ð t Þ

Þ ð

t Þ (10.32)

ð a

a ¼

when v ðsÞ ¼ v and to

X

N
a¼ 1 r ðaÞ

D a

D

ˆ ðaÞ
Dt ¼ r

Dt

(10.33)

when

ˆ ðaÞ is assumed not to have a dependence upon the index a . Application of the
formula ( 10.31 ) relating the sum of the density-weighted, constituent-specific, time
derivatives to the time derivative following the selected component to the special
case of the velocity v ðaÞ yields the following representation:

X X

N

D a v ð a Þ
Dt ¼r

N

D s v
Dt þ

1 r ðaÞ 1 r½r ðaÞ v ða=sÞ

v ða=sÞ

a ¼ a ¼

( ) ( )

X X X

N N N

1
r

v ^ v ðaÞ ^

1 rðr ðaÞ v ða=sÞ Þ 1 r ðaÞ v ða=sÞ

þ ð t Þ

Þ ð

t Þ

ð a

a ¼ a ¼ a ¼ 1

(10.34)

which reduces to

( )

X X

N

D a v ð a Þ
Dt ¼ r

N

D s v
Dt þ

v ^ v ðaÞ ^

1 r ðaÞ

ð t Þ

Þ ð

t Þ (10.35)

ð a

a ¼ a ¼ 1