Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 287

this is used to rewrite ( 10.96 )as

X

N
a¼ 1 r ðaÞ

X

N
a¼ 1 ˆ ðaÞ

D s

X

N
a¼ 1 r ðaÞ

D a D s D s

r ðaÞ
D t

r
D t

ˆ ðaÞ
D t ¼ r

D t þ ˆ

v ðsÞ r ˆ ðaÞ

X

N

þ

¼ 1 r ðaÞ v ðaÞ r ˆ ðaÞ : a

(10.98)

The following relationships, the first obtained from the conservation of mass for
the mixture ( 10.25 ),

D s

D t ¼ @r

r

^

t þ

v ðsÞ rr ¼ ð t Þrðr v Þþ v ðsÞ rr (10.99)

@

and the second obtained from the conservation of mass for the constituent ( 10.23 )

D s

D t ¼ @r ð a Þ

r ðaÞ

^

t þ

v ðsÞ rr ðaÞ ¼

Þ ð

t Þrðr ðaÞ v ðaÞ Þþ v ðsÞ rr ðaÞ (10.100)

ð a

@

will now be used in ( 10.98 ). However, before using ( 10.100 ) it is multiplied by and
summed over all values of “ a ,” thus

X

N
a¼ 1 r ðaÞ

D a

D s

ˆ ðaÞ
Dt

Dt þ ˆ ð

¼ r ^ ð t Þrðr v Þþ v ðsÞ rrÞ

X X

N
a¼ 1 ˆ ðaÞ ^

N
a¼ 1 ˆ ðaÞ rðr ðaÞ

ðaÞ ð

t Þþ v ðaÞ Þ

X

N
a¼ 1 ˆ ðaÞ v ðsÞ rr ðaÞ

X

N
a¼ 1 r ðaÞ

v ðsÞ r ˆ ðaÞ

X

N

þ

1 r ðaÞ v ðaÞ r ˆ ðaÞ :

(10.101)

a ¼

The second line of the result above is condensed

X X

N
a¼ 1 r ðaÞ

D a

D s

N
a¼ 1 ˆ ðaÞ ^

ˆ ðaÞ
D t

D t þ ˆ ð

^ ¼ r ð t Þrðr v Þþ v ðsÞ rrÞ

ðaÞ ð

t Þ

X

N
a¼ 1 rð ˆ ðaÞ r ðaÞ

X

N

þ v ðaÞ Þ v ðsÞ r ˆ ðaÞ r ðaÞ

a¼ 1

(10.102)

and then the entire equation is algebraically reduced to ( 10.30 ).