Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 282

Example 10.8.1
Restrict the functional dependence of the free energy and the heat

flux vector q in a rigid isotropic heat conductor using the entropy inequality and
arguments concerning the functional dependence of these three functions.
Solution : Democratic but elementary constitutive assumptions are made for the free
energy

C , the entropy

, and the heat flux q . The independent variables for these
three functions are assumed to be same for all three quantities; they are the tempera-
ture y and the temperature gradient ry , thus C ¼ Cðy; ryÞ , ¼ ðy; ryÞ , q ¼ qð
y; ryÞ

C

, the entropy

. The time rate of change of the free energy C ¼ Cðy; ryÞ , determined using

the chain rule,

D s D s D s D t ¼ @C

D t þ @C

y ry

D t @ry

;

@y

and the constitutive assumptions are

then substituted into the entropy inequality ( 10.82 ) and one obtains the inequality

C ¼ Cðy; ryÞ , ¼ ðy; ryÞ , q ¼ q ðy; ryÞ

D s

D s

r þ @C
@y

y
D t

1
y

ry r @C
@ry

ry
D t

q 0 :

The argument originated by Coleman and Noll ( 1963 ) is that this inequality must
be true for all possible physical processes and the functional dependence upon the
independent variables, in this case , must be restricted so that is the case.

In view of this set of independent variables, the last summand of the inequality
above is linear in the time derivative of the temperature gradient D s

fy; ryg

ry=

D t .This

fy; ryg

time derivative is not contained in the set of independent variables and it

does not appear elsewhere in the inequality and thus it may be varied when the set
of independent variables at a point is held fixed. In order that it not be varied in way
that inequality be violated it is necessary that the coefficient

D s @C

@ry

of

ry
D t

must

vanish. However, if @C

@ry ¼

0, it follows that the function dependence of C ¼ Cðy

; ryÞ is reduced to C ¼ CðyÞ . When the reduced dependence, C ¼ CðyÞ , the

inequality above reduces to

D s

r þ @C
@y

y
D t

1
y

q ry :

0

The argument made above is now repeated to show that the term involving the time
derivative of the temperature, D s y= D t , must vanish, thus one concludes that

¼ð@C=@yÞ . When the restrictions @C=@ry ¼ 0and ¼ð@C=@yÞ are

substituted back into the form of the entropy inequality above, it reduces to ð 1 =yÞ

q 0. In the case when the isotropic form of the Fourier law of heat conduction

gives the heat flux, q

ry

¼ k ry , the inequality reduces to ð k =yÞry ry 0andit