CONTINUUM MECHANICS for ENGINEERS

5.18 Show that one way to express the rate of change of kinetic energy of
the material currently occupying the volume V is by the equation
∫ i i
ij i,j i i
V ∫V
∫S
˙
ˆ
K = ρbvdV − σ v dV + v t
( n)
dS
and give an interpretation of each of the above integrals.
5.19 Consider a contiuum for which the stress is σij = –p0δij and which
obeys the heat conduction law qi = –κθ ,i. Show that for this medium
the energy equation takes the form
ρ ˙u
= – p 0v i,i – ρr + κ θ ,ii
5.20 If mechanical energy only is considered, the energy balance can be
derived from the equations of motion. Thus, by forming the scalar
product of each term of Eq 5.4-4 with the velocity v i and integrating
the resulting equation term-by-term over the volume V, we obtain the
energy equation. Verify that one form of the result is
1
ρ v v D ρb
v v
2 ⋅
•
( ) + tr( ⋅ )− ⋅ + div ⋅
5.21 If a continuum has the constitutive equation
σ ij = –pδ ij + αD ij + βD ikD kj
where p, α and β are constants, and if the material is incompressible
(D ii = 0), show that
σ ii = – 3p – 2β II D
where IID is the second invariant of the rate of deformation tensor.
5.22 Starting with Eq 5.12-2 for isotropic elastic behavior, show that
σ ii = (3λ + 2µ)ε ii
and, using this result, deduce that
ε
ij
µ σ
λ
λ µ δσ
⎛
1 ⎜ = ⎜ −
2 ⎜ 3 + 2
⎝
ij ij
⎞
⎟
kk
⎟
⎠
( )
5.23 For a Newtonian fluid, the constitutive equation is given by
= 0