Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 273

P E ¼ T : D ; Q E ¼ r r r h ; (10.44)

where h is the heat flux vector. The internal mechanical power is represented by the
power generated by the time rate of change of the internal parameters { E ,

r ( a ) }
contracted with the corresponding internal force systems to form scalar quantities
with the dimension of energy. The internal nonmechanical power is represented by
the product of the time rate of change of the specific entropy with the density and
the temperature y , thus

D s

D t

Q I ¼ ry : (10.45)

From ( 10.43 ) and ( 10.45 ) it follows that

D s

Dt ¼

ry P E P I þ Q E ; (10.46)

thus the production of specific entropy is the difference between the external and
internal mechanical work plus the external nonmechanical power.
Let N denote the entropy of an entire object and let denote the entropy per unit

mass or specific entropy at the particle X of the object, thus

ð
o r

N ¼ d v : (10.47)

The time rate of change of the N and are related by

ð
o r

D s N
Dt ¼

D s

Dt d v

(10.48)

which follows from (3.8). The total entropy production D s D t is written as the sum of

the external production D s N ð e Þ
D t

and the internal production D s N ð i Þ
D t

, thus

D s N
D t ¼

D s N ð i Þ
D t þ

D s N ð e Þ
Dt

: (10.49)

The external entropy production D s N ð e Þ

D t is associated with the total flux of
nonmechanical power, divided by the absolute temperature

y , thus

þ ð

D s N ð e Þ
D t

h n r r

y

¼

d a

þ

d v

;

(10.50)

y

@ O O