Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 276

and the derivatives of the free energy with respect to temperature, strain, and

volume fraction yield the entropy, stress, and electrochemical (or chemical) poten-
tial, respectively:

C

!

E;r ðaÞ ; y;r ðaÞ ; m ðaÞ ¼

¼ @C
@y

@C
@

@C
@r ðaÞ

T ¼ :

(10.58)

E

E;y

The time derivative of the free energy C may then be expressed as follows:

X

D s

D s D s

N
a¼ 1 m ðaÞ

r ðaÞ
D t

D t ¼ y

D t þ

T : D þ : (10.59)

It is assumed that each constituent of the mixture has the regular properties of a
thermodynamic substance, thus the Helmholtz free energy of each constituent

C ( a )

y e ( a ) and

is related to the temperature and constituent-specific internal energy

entropy ( a ) by the component-specific form of ( 10.57 )

C ðaÞ ¼ e ðaÞ y ðaÞ ; (10.60)

where

X

N
a¼ 1 r ðaÞ C ðaÞ :

1
r

C ¼ (10.61)

D s N ð i Þ
D t

D s N ð e Þ
D t

D s N

0 or the integral ( 10.49 ) or the field
equation ( 10.50 ) are called the Clausius Duhem inequality for internal entropy
production. They are equivalent statements of the second law of thermodynamics.
In order to generalize the inequality ( 10.50 ) to a mixture, three substitutions into
( 10.50 ) are made. First the

Either the condition ¼

D t

in ( 10.50 ) is replaced by the density-weighted average
of the constituent-specific internal entropy

( a ) , thus

X

N

1
r

¼

¼ 1 r ðaÞ ðaÞ ;

(10.62)

a

and, second, a similar replacement, the second of ( 10.37 ) is made for r . Third, the

formula ( 10.26 ) for the density-weighted sum of all the time derivatives of

r

ˆ ðaÞ
following all the constituents is related to the time derivative following the selected
constituent is applied to the density-weighted average of the constituent-specific
internal entropy

( a ) ,

n o

:

X

D a

X

N
a¼ 1 r ðaÞ

ðaÞ
D t

D s

Ns
b¼ 1 r½ ðbÞ r ðbÞ

D t þ

¼r u ðbÞ r½r ðbÞ u ðbÞ (10.63)