CONTINUUM MECHANICS for ENGINEERS

One final form of the Clausius-Duhem equation is obtained by using
Eq 5.8-9b to obtain the local dissipation inequality
ψ˙ ηθ˙ 1
+ −Dijσij − qigi≥0 θ
5.9 Restrictions on Elastic Materials by the
Second Law of Thermodynamics
(5.8-20)
In general, the thermomechanical continuum body must be specified by
response functions that involve mechanical and thermodynamic quantities.
To completely specify the continuum, a thermodynamic process must be
defined. For a continuum body B having material points X a thermodynamic
process is described by eight functions of the material point and time. These
functions would be as follows:
1. Spatial position xi χ i X, t
2. Stress tensor σij σij
3. Body force per unit mass bi = bi(X,t) 4. Specific internal energy u = u(X,t)
5. Heat flux vector qi = qi(X,t) 6. Heat supply per unit mass r = r(X,t)
7. Specific entropy η = η(X,t)
8. Temperature (always positive) θ = θ(X,t)
Xt = ( ) ,
= ( )
A set of these eight functions which are compatible with the balance of linear
momentum and the conservation of energy makes up a thermodynamic
process. These two balance laws are given in their local form in Eqs 5.4-4
and 5.7-13 and are repeated below in a slightly different form:
σ − ρv˙= −ρb
ji, j i i
ρu˙− σ D + q =
ρr
ij ij i, i
(5.9-1)
In writing the balance laws this way, the external influences on the body,
heat supply, and body force have been placed on the right-hand side of the
equal signs. From this it is noted that it is sufficient to specify x i, σ ij, ε, q i, η,
and θ and the remaining two process functions r and b i are determined from
Eqs 5.9-1.