Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 284

p is a Lagrange multiplier representing the fluid pressure. We now employ the
Coleman and Noll ( 1963 ) argument described in the example above, namely that
the assumed constitutive dependencies ( 10.84 ) must be restricted so that the
inequality ( 10.86 ) is true for any value of the final set of independent variables.
In view of this set of independent variables, the last summand of ( 10.86 ) is linear in
the time derivative of the diffusion velocity

D s v ð b = s Þ
D t

. This time derivative is not

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

contained in the set of independent variables

y; f ðbÞ ; 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. It follows
that the coefficient @C
@v ðb=sÞ

v ðb=sÞ Þ

of D s v ð b = s Þ
D t

must vanish, thus the functional dependence of C

¼ Cð E ; y; ry; f ðbÞ ; v ðb=sÞ Þ is reduced to C ¼ Cð E ; y; ry; f ðbÞ Þ . Repeating the same

of D s

argument, the coefficient @C
@ry

ry
D t

must vanish, thus the functional dependence of

C ¼ Cð E ; y; ry; f ðbÞ Þ is reduced to C ¼ Cð E ; y; f ðbÞ Þ . In view of the reduced

dependence of , a repetition of the previous argument for this

reduced set of independent variables is applied to the coefficient of the first
summand of ( 10.86 ) involving the time derivative of the temperature

C ¼ Cð E ; y; f ðbÞ Þ

D s y

D t ,andso
the coefficient of the third summand is linear in the solid rate of deformation tensor
D ðsÞ ,thus

¼ @C
@y

¼ r @C
@

and T eff

E :

(10.88)

This result shows that the entropy and effective stress of a porous medium
can be derived from a regular strain energy function ,

which physically has the same meaning as in single phase or multiphasic media,
but which can depend on both strain and solute concentrations in the medium.
Concerning the summands within the summation signs in ( 10.86 ) there is one
summand linear in the gradient of the diffusion velocity

C ¼ Cð E ; y; f ðbÞ Þ

v ðb=sÞ , and this term
appears nowhere else in the reduced inequality. In order to satisfy this restriction the
following equation for the partial stress is implied:

r

T ðbÞ ¼ f ðbÞ ðg ðbÞ C ðbÞ m ðbÞ Þ 1 ; (10.89)

thus the partial stress of the fluid and the solutes are all seen to be scalars.
When the reduced functional dependence of the free energy to ; y; f ðbÞ Þ

and the three restrictions ( 10.88 ) and ( 10.89 ) obtained on the inequality ( 10.87 ) are
substituted back into ( 10.87 ), it reduces to the following:

C ¼ Cð E

)

X

Ns
b¼ 1 f ðbÞ

1
y

q ry v ðb=sÞ rm ðbÞ 0 : (10.90)