Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 265

where

v ða=sÞ ¼ v ðaÞ v ðsÞ ; (10.13)

represents the diffusion velocity of the a th constituent relative to the s constituent.
The tensor of velocity gradients for the a th constituent L ( a ) is formed by taking
the spatial gradient of the velocity field for the a th constituent v ðaÞ ¼

v ðaÞ ð x ; t Þ

, thus

L ðaÞ ¼r v ðaÞ :

(10.14)

Please note that this definition is completely analogous to the definition of the
tensor of velocity gradients for a single constituent material, L , given by (2.31).
Using the chain rule it is easy to show that L ( a ) also has the representation

D a F ð a Þ
D t

F 1 L ðaÞ ¼

ðaÞ :

(10.15)

r ( a ) denotes the density of the a th constituent, then the density of the mixture
may be defined by

If

X

N

rð x ; t Þ¼

1 r ðaÞ ð

x ; t Þ: (10.16)

a ¼

r ( a ) represents the mass of the a th constituent per unit volume of the
mixture. The true material density for the a th constituent is denoted by

Physically,

g ( a ) and
represents the mass of the a th constituent per unit volume of the a th constituent.
The quantity

r ( a ) is sometimes called the bulk density as opposed to the true
material density, f ( a ) , that is to

say the volume of the a th constituent per unit volume of the mixture, is defined by

g ( a ) . The volume fraction of the a th constituent,

Þ¼ r ð a Þ ð

x ; t Þ

f ðaÞ ð x ; t

Þ ;

(10.17)

g ðaÞ ð x ; t

which may be viewed a factoring the bulk density into two components.

r ðaÞ ð x ; t Þ¼f ðaÞ ð x ; t Þg ðaÞ ð x ; t Þ: (10.18)

It is assumed that the sum of all volume fractions divided by the total volume is
equal to one,

X

N
a¼ 1 f ðaÞ ð

x ; t Þ¼ 1 : (10.19)

The porosity of the a th constituent is 1- f ( a ) .Ifthe a th constituent is incompress-

ible, then g ( a ) is a constant. Observe from ( 10.18 ) that the bulk density r ( a ) need not be