ABSTRACT
THERMODYNAMICS OF DEFORMATION
4.1. Energy Equation
A deforming body, or a given portion of it, can be considered to be a thermo-
dynamic system in continuum mechanics. The rst law of thermodynamics
relates the mechanical work done on the system and the heat transferred
into the system to the change in total energy of the system. The rate at
which external surface and body forces are doing work on a body currently
occupying the volume V bounded by the surface S is given by Eq. (3.5.6),
i.e.,
P =
d
dt
Z
V
v v dV +
Z
V
: D dV: (4.1.1)
Let q be a vector whose magnitude gives the rate of heat ow by conduction
across a unit area normal to q. The direction of q is the direction of heat
ow, so that in time dt the heat amount q dt would ow through a unit
area normal to q. If the area dS is oriented so that its normal n is not
in the direction of q, the rate of outward heat ow through dS is q n dS
(Fig. 4.1). Let a scalar r be the rate of heat input per unit mass due to
distributed internal heat sources. The total heat input rate into the system
is then
Q =
Z
S
q n dS +
Z
V
r dV =
Z
V
(r q+ r) dV: (4.1.2)
According to the rst law of thermodynamics there exists a state function
of a thermodynamic system, called the total energy of the system E
tot
, such
that its rate of change is
_
E
tot
= P +Q: (4.1.3)
Neither P nor Q is in general the rate of any state function, but their sum
is. The total energy of the system consists of the macroscopic kinetic energy
Figure 4.1. The heat ow vector q through the surface
element dS with a unit normal n.
