Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 283

requires only that k be positive, since the temperature is positive as a consequence of
its definition.
Finally, note that the argument involving D s ry= D t was performed first, then the

argument involving D s D t . This ordering of the arguments is required because D s

y=

, D s

D t ¼ @yð x ; t Þ

y

y= D t depends on ry þ v ðsÞ r½yð x ; t Þ , a result that is a special case of

@ t

( 10.11 ).
The method of this example will next be applied to the mixture of interest.
Constitutive assumptions will now be made for the free energy C , the entropy , the

effective stress T eff , each constituent-specific free energy

C ( b ) , each constituent-
specific partial stress T ðbÞ and the heat flux q . The independent variables included in
the constitutive assumptions are the localized small strain tensor E , the temperature
y r ( b ) and the diffusion

velocities v ðb=sÞ . These functional dependencies are expressed as equations in the
following forms:

, the temperature gradient ry , the constituent densities

C ¼ Cð E ; y; ry; f ðbÞ ; v ðb=sÞ Þ;¼ ð E ; y; ry; f ðbÞ ; v ðb=sÞ Þ;

T ef f T eff ¼ ð E ; y; ry; f ðbÞ ; v ðb=sÞ Þ; C ðcÞ ¼ C ðcÞ ð E ; y; ry; f ðbÞ ; v ðb=sÞ Þ

T ðcÞ ¼ T ðcÞ ð E ; y; ry; f ðbÞ ; v ðb=sÞ Þ; q ¼ q ð E ; y; ry; f ðbÞ ; v ðb=sÞ Þ

(10.84)

These constitutive assumptions are now substituted into the entropy inequality
( 10.82 ); in the case of the free energy, C , the chain rule is applied, thus

D s D s v ð a = s Þ

D s D s D s E

f ðaÞ
D t þ

D s D t ¼ @C

D t þ @C

y

D t þ @C

@C

D t þ @C

ry
D t E : @ry

;

@y @ @f ðaÞ @ v ðb=sÞ

(10.85)

thus

D s

D s

r þ @C
@y

y
Dt

r @C
@

D ðsÞ r @C

ry
Dt

1
y

T eff

q ry þ :

@ry E

!

X

N
a¼ 1 r

X

N s

b¼ 1 f½

D s v ð b = s Þ
Dt

@C

þ

v ðb=sÞ

þ T ðbÞ þ f ðbÞ m ðbÞ 1 r ðbÞ C ðbÞ 1 : ½r v ðb=sÞ g

@

X

Ns
b¼ 1 f½r

þ T ðbÞ þ m ðbÞ rf ðbÞ rr ðbÞ C ðbÞ v ðb=sÞ g 0 ;

(10.86)

where are the electrochemical potentials of the constituents other than the

porous solid,

m ðbÞ

@C
@f ðbÞ þ g ðbÞ e þ

m ðbÞ ¼ r p ; (10.87)