Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 268

where e ðaÞ is the partial internal energy density, q ðaÞ is the partial heat flux vector, r ðaÞ

is the heat supply density, (1/2) ( L ( a ) +( L ( a ) ) T )

by the extension of (2.32) to each constituent. The energy supply

^

e is the energy supply and D ( a ) ¼

ðaÞ

^

is the only
term that is not directly associated with a term in the single constituent continuum
form of energy conservation (3.52); it represents the transfer of energy from the
other constituents to constituent a .
If the sum of all mass supplies to a constituent from other constituents, denoted
by ^ and defined by ( 10.22 ) is zero, then the summation of the forms of the balance
of mass ( 10.20 ), the balance of momentum ( 10.26 ) and the balance of energy
( 10.27 ) for each constituent over all the constituents is required to produce again
the single constituent continuum forms of the balance of mass (3.6), the balance of
momentum (3.29) and the balance of energy (3.52), respectively. In the case when
the summation is over the density-weighted time derivatives of specific quantities
following the generic constituent as, for example, on the left hand side of ( 10.27 ),
the result is difficult to interpret. Thus a formula relating the time derivative of the
selected component to the sum of the density-weighted time derivatives has been
developed. Let the constituent-specific quantity per unit mass be denoted by

e

ð a Þ

ϖ ( a )

and its density-weighted sum by r ϖ , thus

X

N

1
r

ˆ ¼

1 r ðaÞ ˆ ðaÞ :

(10.28)

a ¼

The desired formula relating the sum of the density-weighted, constituent-
specific, time derivatives to the time derivative following the selected component

X

N
a¼ 1 r ðaÞ

X

Ns
b¼ 1 rð ˆ ðbÞ r ðbÞ n ðb=sÞ Þ ˆ r

X

Ns
b¼ 1 r ðbÞ

D a D s

ˆ ðaÞ
Dt ¼ r

Dt þ

v ðb=sÞ

( )

X

N
a¼ 1 ˆ ðaÞ ^

^

þ ˆ ð t Þ

ðaÞ ð

t Þ ;

(10.29)

where v ða=sÞ is the diffusion velocity relative to the selected component defined by

( 10.13 ).
The derivation of ( 10.29 ) is given in the Appendix to this Chapter. The deriva-
tion involves the following relationship that follows from ( 10.25 ) and ( 10.16 ) with
the use of ( 10.13 ) to note that v ðs=sÞ must be zero:

X X

N
a¼ 1 r ðaÞ v ða=sÞ ¼

N s

b¼ 1 r ðbÞ v ðb=sÞ ;

rð v v ðsÞ Þ¼ (10.30)