122
3 Continuum Mechanics and Nonlinear Elasticity
Fig. 3.22 Sources of
mechanical and thermal
energy in a 3D body (current
configuration)
n
T
b
V
dV
dA
x 2
x 1
x 3
o
q
r
and the total internal energy is given by
U =
V
u ρ dV ,
(3.175)
where u represents the internal energy per unit mass. The total mechanical power
input is
P in =
A
T · v dA +
V
b · v dV ,
(3.176)
where T and b are the surface traction vector (or stress vector) and body force,
respectively, relative to the current geometry (Fig. 3.22).
Heat enters the body through its surface, as well as from internal sources. We
introduce the heat flux vector q such that q ·n represents the rate at which heat flows
outward through the surface area element n dA, with n being the unit normal to the
surface. In addition, the heat production rate r is defined per unit deformed volume
(Fig. 3.22). In terms of these quantities, the rate at which heat is added to the body
is given by
Q in = −
A
q · n dA +
V
r dV,
(3.177)
where the minus sign gives inward heat flow when q · n is positive.
Next, we prepare the various integrals for substitution into Eq. (3.163). First,
since the mass ρ dV is constant for each particle, the time derivatives of Eqs. (3.174)
and (3.175) yield