CONTINUUM MECHANICS for ENGINEERS

internal energy u (5.8-9a)
free energy ψ = −ηθ
(5.8-9b)
enthalpy (5.8-9c)
u
χ = u −∑τa
a v
free enthalpy ζ = χ − ηθ = u −ηθ −∑τ
(5.8-9d)
a a v
These potentials are related through the relationship
(5.8-10)
All of the energy potentials may be written in terms of any one of the
following independent variable sets
η, v a; θ, v a; η, τ a; θ, τ a (5.8.11)
In order to describe the motion of a purely mechanical continuum the
function x i = x i(X A, t) is needed. Adding the thermodynamic response
requires the addition of temperature, θ, or, equivalently, entropy, η, both
being a function of position and time
θ = θ(X A, t) or η = η(X A, t) (5.8-12)
When considered for a portion P of the body, the total entropy is given as
and the entropy production in the portion P is given by
a
u − ψ + ζ − χ = 0
Η= ∫ ρηdV
P
Γ= ∫ ργ dV
P
(5.8-13)
(5.8-14)
where the scalar γ is the specific entropy production. The second law can be
stated as follows: the time rate-of-change in the entropy equals the change
in entropy due to heat supply, heat flux entering the portion, plus the internal
entropy production. For a portion P of the body, this is written as
d
ρr
qi⋅ni ρηdV
= dV − dS + ργ dV
dt∫
∫ θ ∫ θ ∫
P P
a
∂P
P
(5.8-15)