Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 35

¼ w 1 X ð x ; t Þ for all X O ð 0 Þ (2.25)

and substituting the result into the first of the expressions ( 2.24 ) for the velocity;
thus v

¼ _ ¼ wð

x X ; t Þ becomes

¼ wðw 1 v ¼ _ x ð x ; t Þ; t Þ¼ v ð x ; t Þ (2.26a)

or

ðw 1 1 Þ; w 2 1 Þ; w 3 1

v ð x ; t Þ¼ v ð x ; t ð x ; t ð x ; t Þ; t Þ; (2.26b)

which emphasizes that the time dependence of the spatial representation of velocity
is both explicit and implicit. This representation of the velocity with the places x as
independent variables is called the spatial representation of motion . A quantity is
said to be in the spatial representation if its independent variables are the places x
and not the particles X . In the material representation the independent variables
are the particles X ; compare the material description of motion, ( 2.2 ), with (2.26).
The material time derivative is the time derivative following the material particle
X ; it is denoted by a superposed dot or D / Dt and it is defined as the partial derivative
with respect to time with X held constant. The material time derivative is easy to
calculate in the material representation. It is more complicated to calculate in
the spatial representation. To determine the acceleration in the spatial representa-
tion we must calculate the material time rate of the spatial representation of velocity
(2.26). The notation D / Dt introduced above is illustrated using the definitions
of ( 2.24 ):

X fixed ¼ @ X fixed

2

¼ @

w v Dv

Dt :

a (2.27)

@ t 2 @ t

A formula for Dv / Dt is obtained by observing the explicit and implicit time
dependence of the spatial representation of velocity ( 2.26b ) and noting that the time
derivative associated with the implicit dependencies may be obtained using the
chain rule, thus

x fixed þ @ X fixed ¼ @ x fixed þ @

Dv

Dt ¼ @

v

x i @

v x i

@

v v

x i v i ;

(2.28)

@ t @ t @ t @

a result that may be written more simply as

x fixed þ

Dv

Dt ¼ @

v
@

v r v : (2.29)

t