CONTINUUM MECHANICS for ENGINEERS

θ =
η
∂u
∂
(5.8-2)
Furthermore, partial differentiation of the internal energy with respect to the
thermodynamic substate variables results in thermodynamic tensions
(5.8-3)
The preceding equations can be used to write a differential form of the
internal energy as follows:
(5.8-4)
From Eqs 5.8-1 and 5.8-2 we see that both temperature and thermodynamic
tensions are functions of entropy and the substate parameters.
Assuming that all the functions defined in the section are continuously
differentiable as many times as necessary, it is possible to solve for entropy
in terms of temperature
(5.8-5)
This result may be substituted into the caloric equation of state to yield
internal energy as a function of temperature and substate parameters
(5.8-6)
Using this result in Eq 5.8-3 allows the definition of the thermal equations
of state
which inverts to give the substate parameters
τ a
u
= ∂
∂v
du = θdη+∑τ dv
(5.8-7)
(5.8-8)
The principles of thermodynamics are often posed in terms of thermodynamic
potentials which may be defined as follows:
a
a
η= η( θ,va)
u= u( θ,va)
τ τ θ,v
= ( )
a a a
v v
= ( θτ ,
)
a a a
a a