Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 72

ð
O rð

K

v ¼ v Þ d v :

The formula ( 3.51 ) shows that the total mechanical power supplied to the object
is equal to the time rate of change of kinetic energy plus an integral representing
power involved in deforming the object.
The point form statement of the principle of energy conservation will now be
obtained by placing the integral representations for U , Q , and P , equations ( 3.44 ),
( 3.46 ), and ( 3.51 ), respectively, into the global statement ( 3.43 ), thus
ð
O f

T : D þ r r r q reg d v ¼ 0 : (3.52)

Now, for the last time in this chapter, the argument employed in the transition
from ( 3.4 )to( 3.5 ) is applied here again (see the discussion following ( 3.5 )), thus the
integrand of ( 3.52 ) must be zero everywhere in the object O ;

re ¼ T : D þ r r r q : (3.53)

This is the desired point form of the principle of energy conservation. It states
that the time rate of change of the specific internal energy

e multiplied by the

density is equal to the sum of the stress power, the negative of the divergence of

the heat flux, and the internal heat supply term.
Before leaving these considerations of energy, a formula for the quasistatic work
done during a loading of an object will be obtained. The mechanical power P
delivered to the object, ( 3.47 ), is the rate of work. The desired new formula relates
to the work done rather that to the rate of doing work or power. A formula identical
to formula ( 3.47 ) for the rate of work, in every regard except that the velocity is
replaced by the displacement, is employed, thus

r

ð ð

O rd u d v

W ¼ t u d a þ : (3.54)

@ O

W is the mechanical work delivered to the object in a quasistatic loading, t is the
surface traction acting on the surface of the object O , d is the action-at-a-distance
force and u is the displacement vector. The terms t

u d a and r d u d v both represent

the mechanical work done on the object, t u d a is the work of surface forces and

r T · n ,

into ( 3.54 ) and subsequent application of the divergence theorem (A184) to the
surface integral in the resulting expression yields

d u d v is the work of action-at-a-distance forces. Substitution of ( 3.16 ), t ¼

ð
O fr ð

ð
O fððr

W ¼ T u Þþr d u g d v ¼ T Þ u Þþ T : ðr u Þþr d u g d v ;