Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 280

relative to which the movement of all the other constituents may be referenced.
It is therefore singled out for special consideration in models of the porous medium.
Since the true density of each constituent, g ( a ) , is assumed to be constant, the

local statement of mass conservation for a single constituent continuum, ( 10.20 ), is
written for each constituent a

¼ ...

, N in terms of the local volume fractions,

1, 2,

f ( a ) ,as

@f ð a Þ
@ t þrðf ðaÞ

v ðaÞ Þ¼ 0 : (10.76)

Since the solid constituent is treated as special, the restriction ( 10.19 ) is rewritten as

X

N s

b¼ 1 f ðbÞ ð

f s ð x ; t Þþ x ; t Þ¼ 1 ; (10.77)

where the summation index b runs over all the constituents except s . Summing all
the constituent mass conservation equations ( 10.76 ), and employing ( 10.19 ), it
follows that

!

X

N
a¼ 1 f ðaÞ

r v ðaÞ ¼ 0 : (10.78)

Multiplying ( 10.19 )by v s and subsequently taking the divergence of the result, it
follows that

!

X

N

r v s r

1 f ðaÞ v ðsÞ

¼ 0 ; (10.79)

a ¼

the sum of ( 10.78 ) and ( 10.79 ) yields

!

X

N
a¼ 1 f ðaÞ

r v s þr u ðaÞ ¼ 0 ; (10.80a)

where the definition of the diffusion velocity, ( 10.13 )or v ða=sÞ ¼ v ðaÞ v ðsÞ , has been

employed. However, since v ðs=sÞ ¼ 0, it follows that ( 10.80a ) is equivalent to

!

X

N s

b¼ 1 f ðbÞ u ðbÞ

r v s þr ¼ 0 : (10.80b)

It is required that this entropy inequality ( 10.75 ) hold for all states of the mixture
complying with the balance laws, the incompressibility condition and that it hold