Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 73

or

ð
O fðr

W ¼ T þ r d Þ u þ T : ðr u Þg d v : (3.55)

This result may be further reduced by using the stress equations of motion ( 3.38 )
in the case when v

¼

∇

T þ r d by 0, thus

0 to replace

ð

W ¼ T : ðr u Þ

d v

:

(3.56)

O

u ) T may be decomposed into a symmetric part E
and a skew-symmetric part Y by (

Recall from (2.49) that (

∇

u ) T

∇

¼ E þ Y . It follows then that

T: ðr u Þ¼ T: E þ T: Y ; (3.57)

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

∇

u ) ¼ T : E . The work done on the object is then given by

ð

W ¼ T : E d v : (3.58)

O

This means that the local work done is T : E . This result will be of interest in the
consideration of elastic objects.

Problem

3.5.1. In terms of the concepts introduced in this section how would one specify a
system that was functioning adiabatically globally but not locally? How
would one specify a system that was adiabatic locally or point wise? If a
system is adiabatic locally what type of energy is the mechanical work done
on the object transferred into? If the system is not adiabatic and if the
system's internal energy does not change as mechanical work is done on
the system, into what type of energy is the mechanical work then converted?

References

Callen HB (1960) Thermodynamics. Wiley, New York
Truesdell CA, Toupin RA (1960) The classical field theories. In: Flugge S (ed) Handbuch der
Physik, vol III/1. Springer, Berlin