CONTINUUM MECHANICS for ENGINEERS

One of the uses for the Clausius-Duhem form of the second law is to infer
restrictions on the constitutive responses. Taking Eq 5.8-20 as the form of the
Clausius-Duhem equation we see that functions for stress, free energy,
entropy, and heat flux must be specified. The starting point for a constitutive
response for a particular material is the principle of equipresence (Coleman
and Mizel, 1963):
An independent variable present in one constitutive equation of a material
should be so present in all, unless its presence is in direct contradiction
with the assumed symmetry of the material, the principle of material
objectivity, or the laws of thermodynamics.
For an elastic material, it is assumed that the response functions will depend
on the deformation gradient, the temperature, and the temperature gradient.
Thus, we assume
σ σ˜ F , θ, g ; ψ ψ˜ F , θ,
g ;
= ( ) = ( )
ij ij iA i iA i
η ˜ η F , θ, g ; q q˜ F , θ,
g
= ( ) = ( )
iA i i i iA i
(5.9-2)
These response functions are written to distinguish between the functions
and their value. A superposed tilde is used to designate the response function
rather than the response value. If an independent variable of one of the
response functions is shown to contradict material symmetry, material frame
indifference, or the Clausius-Duhem inequality, it is removed from that function’s
list.
In using Eq 5.8-20, the derivative of ψ must be formed in terms of its
independent variables
ψ˜ ˙ ˙ ψ˜
˙
˜
ψ
˙
θ θ
ψ
= ∂
+
∂
∂
+
∂
∂
F ∂
F
g g
iA
i
iA
(5.9-3)
This equation is simplified by using Eq 4.10-7 to replace the time derivative
of the deformation gradient in terms of the velocity gradient and deformation
gradient
ψ˜ ψ˜
˙
˙
˜
ψ
˙
θ θ
ψ
= ∂
+
∂
∂
+
∂
∂
F ∂
LF
g g
ij jA
i
iA
(5.9-4)
Substitution of Eq 5.9-3 into Eq 5.8-20 and factoring common terms results in
∂
−
∂ +
⎛ ⎞ ⎛ ∂ ⎞
⎜ ⎟ + ⎜ −
⎝ ⎠ ⎝ ∂
⎟ −
⎠
∂
ρ − ≥
∂
ψ
ψ
ηθ σ ρ ρ
θ ψ
˜
˜ ˙
˜ ˜ 1
˙ ˜
ij FjA Lij
gi qigi 0
F g θ
iA
i
i
i
(5.9-5)