Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 278

X

N
a¼ 1 yr½r ðbÞ

D s

y
Dt

1
y

r q ry þ u ðbÞ

X

N

D s

y
Dt

1
y

þr q ry þ

¼ 1 yr½r ðbÞ u ðbÞ a

X

N

D a

C ðaÞ
Dt

^

þ T ðaÞ : D ðaÞ v ðaÞ

ðaÞ r ðaÞ

0 : (10.68)

a ¼ 1

The expression relating the terms in ( 10.68 ) containing the time derivatives of
the specific free energy density for the mixture C ðaÞ is replaced by

n o

X X

N
a¼ 1 r ðaÞ

D a

D s

Ns
b¼ 1 r½C ðbÞ r ðbÞ

C ðaÞ
D t ¼ r

D t þ

u ðbÞ Cr½r ðbÞ u ðbÞ ; (10.69)

a result that was obtained by substituting C ðaÞ for ˆ ðaÞ in ( 10.23 ); thus ( 10.68 )

becomes

X

N
a¼ 1 er½r ðbÞ

D s D s

D t r y

D t

1
y

r q ry þ u ðbÞ

h i

X

N

^

þ T ðaÞ : D ðaÞ v ðaÞ

ðaÞ r½C ðbÞ r ðbÞ u ðbÞ

0 ; (10.70)

a ¼ 1

where use of been made of ( 10.57 ) in setting .

The entropy inequality ( 10.70 ) will now be restricted to the case of accelerationless
processes. Neglecting both the acceleration and the action-at-a-distance forces, the
balance ofmomentumfor the continuum(3.29) reduces to

C þ y ¼ e

0and the balance of
momentum for each constituent of the continuum ( 10.20 ) reduces to ^

r T ¼

T ðaÞ .
Thus we now have the following representations for the divergence of the total
stress and the divergence of the constituent-specific partial stress

ðaÞ ¼r

^

r T ¼ 0 ; r T ðaÞ ¼

ðaÞ :

(10.71)

An algebraic development will now be used to obtain an alternate representation
for the first two terms of the sum in ( 10.70 ). This manipulation begins with the
identity that follows easily from ( 10.71 ) and the separation of the selected constitu-
ent stress-related components from the other stress-related components;

X

N
a¼ 1 ½

X

N
a¼ 1 ½

^

T ðaÞ : D ðaÞ v ðaÞ

Þ ¼

T ðaÞ : D ðaÞ þ v ðaÞ r T ðaÞ

ð a

X

N s

b¼ 1 ½

¼ T ðsÞ : D ðsÞ þ v ðsÞ r T ðsÞ þ T ðbÞ : D ðbÞ þ v ðbÞ r T ðbÞ : (10.72)