Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 71

ð ð

O r

P ¼ t v d a þ d v d v ; (3.47)

@O

where t is the surface traction acting on the surface of the object O , d is the action-
at-a-distance force and v is the velocity vector. The terms t v d a and r d v d v both

represent the rate at which mechanical work is done on the object, t v d a is the rate of

work of surface forces and rdv d v is the rate of work of action-at-a-distance forces.
Substitution of ( 3.16 ), t ¼ Tn , into ( 3.47 ) and subsequent application of the
divergence theorem (A184) to the surface integral in the resulting expression yields

ð
O fr ð

ð
O fððr

P ¼ T v Þþr d v g d v ¼ T Þ v Þþ T : ðr v Þþr d v g d v ;

or

ð
O fðr

P ¼ T þ r d Þ v þ T : L g d v ; (3.48)

where L is tensor of velocity gradients defined by (2.31). This result may be
further reduced by using the stress equations of motion ( 3.38 ) to replace

∇

T þ r d

r _ by v , thus

ð
O fr _

P ¼ v v þ T : L g d v : (3.49)

Two more manipulations of this expression for P will be performed. First, recall
from (2.31) that L ¼rv

T and from (2.32) that L is decomposed into a
symmetric part D and a skew-symmetric part W by L ¼ D þ W . It follows
then that

½

T:L ¼ T:D þ T:W ; (3.50)

but T : W is zero because T is symmetric by ( 3.37 ) and W is skew-symmetric, hence
T : L

T : D . The second manipulation of ( 3.49 ) is to observe that the first integral in
( 3.49 ) is the material time rate of change of the kinetic energy K defined by ( 3.41 ).
To see that the first term in the integral of ( 3.49 )is K , apply ( 3.8 )to( 3.41 ). With
these two changes, the integral expression for P ( 3.49 ) now has the form

¼

ð

¼ K P þ T:D d v ; (3.51)

O

since, from ( 3.41 ),