Mixture Theory-Based Poroelasticity and the Second Law of Thermodynamics - Continuum Mechanics of Anisotropic Materials - page 264

X ðaÞ fixed ;

D a x ð a Þ

D t ¼ @ x ð a Þ ð

X ðaÞ ; t Þ

v ðaÞ ¼ x ðaÞ ¼ (10.7)

@ t

X ðaÞ fixed ;

2

D a v ð a Þ
D t

¼ @

w ðaÞ ð X ðaÞ ; t Þ

a ðaÞ ¼ x ðaÞ ¼ (10.8)

@ t 2

where X ( a ) is held fixed because it is the velocity or acceleration X ( a ) that is being
determined. The spatial description of motion of the particle X ( a ) (as opposed to the
material description of motion of the particle X ( a ) represented by ( 10.1 )) is obtained
by substituting ( 10.2 ) into the expressions ( 10.7 ) for the velocity; thus v ðaÞ ¼

D a

D t w ðaÞ

ð X ðaÞ ; t Þ

becomes

D a
D t w ðaÞ w 1

v ðaÞ ¼

Þ ð

x ; t Þ; t ¼ v ðaÞ ð x ; t Þ; (10.9)

ð a

which is a generalization of (2.26). The material time derivative of the a th constitu-
ent is the time derivative following the material particle X ( a ) ; it is denoted by D a

D t
and is defined as the partial derivative with respect to time with X ( a ) held constant. If
G

=

( x , t ) represents a function of x and t , the material time derivative of the a th
constituent is given by:

X ðaÞ fixed :

@G w ðaÞ ð X ðaÞ ; t Þ; t

D a
D t ¼

(10.10)

@ t

This definition is simply a generalization of ( 10.7 ) to an arbitrary function G ( x , t ).

It then follows from (2.29) that

D a

D t ¼ @Gð

G x , t Þ

t þ

v ðaÞ r½Gð x ; t Þ: (10.11)

@

The modeler may select one component of the mixture as special because, from
the viewpoint of the modeler, that constituent serves as a key reference relative to
which the movement of all the other constituents may be referred. This constituent
of the mixture is denoted by s . The motion of the selected constituent is, from
( 10.1 ), given by x

O s (0). The material time derivative
following the selected constituent is given by ( 10.11 ) with the label a replaced by
the label s . A relationship between the time derivative following the selected
constituent s and the time derivative following the generic constituent a is obtained
by subtracting the two formulas for the time derivatives:

¼ w

( X s , t ) for all X s

D a D s

G
D t ¼

G

t þ

v ða=sÞ r½Gð x ; t Þ;

(10.12)

@