CONTINUUM MECHANICS for ENGINEERS

in Section 5.9 had response functions ψ, σ ij, η, and θ i postulated. However, it
could have just as easily been postulated as a function of u, θ, σ ij, and q i.
As an example of how Eqs 5.11-3 could restrict the independent variables,
consider a fluid whose stress response function is assumed to be a function
of density, ρ, velocity, v i, and velocity gradient, L ij. With these assumptions,
the restrictions of Eq 5.11-3c would be
+ + + + +
σ ( ρ , v , D , W )= Q σ ( ρ,
v , D , W ) Q
ij k mn mn ip pl k mn mn lj
(5.11-4)
where the velocity gradient has been decomposed into its symmetric and
skew-symmetric parts, D mn and W mn, respectively. Using the results of Problem
5.31, Eqs 5.10-21c, 5.10-24, and 5.10-25, we find
˜ σ ( ρ, v + c , Q D Q , Q W Q + Ω )= Q σ˜ ( ρ,
v , D , W ) Q
ij k k mp pq qn mp pq qn mn ip pl k mn mn lj
(5.11-5)
Since Eq 5.11-5 must hold for all motions, a specific rigid body rotation may
be chosen to reduce the constitutive assumption of Eq 5.11-4. For this pur-
˙Q ij = 0
pose, suppose that Q ij = δ ij and thus . Using this motion and Eq 5.11-5
implies
( )= ( )
˜ σ ρ, v + c , D , W σ˜ ρ,
v , D , W
ij k k mn mn ij k mn mn
(5.11-6)
where it is noted that Eq 5.10-15 has been used. In this case, vector ck is
simply equal to the time derivative of the superposed rigid body motion
integration factor (see Eq 5.10-8). This arbitrary nature would allow for
Eq 5.11-6 to be violated if the stress function has velocity as an independent
variable. Thus, velocity must be removed from the independent variable list
for stress, leaving
˙a k
σ σ˜ ρ,
D , W
= ( )
ij ij mn mn
(5.11-7)
Again, with the modified response function Eq 5.11-7, the invariance condition
under superposed rigid body motion may be written as
˜ σ ( ρ, Q D Q , Q W Q + Ω )= Q σ˜ ( ρ,
D , W ) Q
ij mp pq qn mp pq qn mn ip pl mn mn lj
(5.11-8)
which must hold for all motions. Select a motion such that Qij = δij as before,
but now require . Substitution of this into Eq 5.11-8 leaves
Ωij ≠ 0
( )= ( )
˜ σ ρ, D , W + Ω
σ˜ ρ,
D , W
ij mn mn mn ij mn mn
(5.11-9)