Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 37

dx 1 /dt

dx 1

Fig. 2.5 An illustration for the geometric interpretation of the D 11 component of the rate-of-
deformation tensor D . A vector of infinitesimal length representing the present position of an
infinitesimal material filament coinciding with the x 1 at time t is denoted by dx 1 . The instantaneous
time rate of change of the material filament instantaneously coincident with dx 1 is dv 1 ¼ D 11 dx 1 .

The expression dv 1 D 11 dx 1 shows dv 1 as a linear function of dx 1 at any point x and time t . Thus

the geometric interpretation of D 11 ¼ð

¼

d x 1 = _ dx 1 Þ is that it is the instantaneous time rate of change of

dx 1 at time t relative to dx 1 at time t

filaments instantaneously situate upon the 2 and 3 axes, the 1 and 3 axes, and the 1
and 2 axes, respectively.
The rate-of-deformation tensor D represents instantaneous rates of change, that
is to say how much a quantity is changing compared to its present size. Let dx 1 be a
vector of infinitesimal length representing the present position of an infinitesimal
material filament coinciding with the x 1 at time t , Fig. 2.5 . The instantaneous time
rate of change of the material filament instantaneously coincident with dx 1 is
dv 1 ¼

D 11 dx 1 , a result that follows from the entry in the first column and first
row of ( 2.33 ). The expression dv 1 ¼ D 11 dx 1 shows dv 1 as a linear function of dx 1 at

any point x and time t . Thus the geometric interpretation of D 11 ¼ x 1 / dx 1 ) is that it

is the instantaneous time rate of change of dx 1 at time t relative to dx 1 at time t .
Similar geometric interpretations exist for D 22 and D 33 .
The geometric interpretation of the normal rate of shearing components D 11 ,
D 22 , and D 33 is easily extended to obtain a geometric interpretation of the trace of D
which is also the divergence of the velocity, tr D

( d _

¼ ∇ v .If dv represents an

element of volume in the spatial coordinate system, dv

dx 1 dx 2 dx 3 (Fig. 2.6 ),
the material time rate of change of dv can be computed using the type of formula
developed in the previous paragraph; d

¼

x 1 ¼ _ D 11 dx 1 , d x 2 ¼ _ D 22 dx 2 , and d x 3 ¼ _

D 33 dx 3 , thus

D
Dt ð

d v _ ¼ dx 1 dx 2 dx 3 Þ¼ð D 11 þ D 22 þ D 33 Þ dx 1 dx 2 dx 3 ¼ð tr D Þ dv (2.34)

or, noting from the definition of D that

tr D ¼ D 11 þ D 22 þ D 33 ¼rv ¼ div v (2.35)

it follows that

d v

dv :

_

tr D ¼r v ¼ (2.36)