CONTINUUM MECHANICS for ENGINEERS

5.12 Determine the form which the equations of motion take if the stress
components are given by σij = –pδij where p = p(x,t).
Answer: ρai = –p ,i + ρbi 5.13 Let a material contiuum have the constitutive equation
σ ij = αδ ijD kk + 2βD ij
where α and β are constants. Determine the form which the equations
of motion take in terms of the velocity gradients for this material.
Answer: = ρb i + (α + β)v j,ij + β v i,jj
ρ ˙v i
5.14 Assume that distributed body moments mi act throughout a
continuum in motion. Show that the equations of motion are still valid
in the form of Eq 5.4-4, but that the angular momentum principle now
requires
ε σ
m + =0
ijk jk i
implying that the stress tensor can no longer be taken as symmetric.
5.15 For a rigid body rotation about the origin, the velocity field may be
expressed by vi = εijkωjxk where ωj is the angular velocity vector. Show
that for this situation the angular momentum principle is given by
M ω I
= ( ) •
i j ij
where M i is the total moment about the origin of all surface and body
forces, and I ij is the moment of inertia of the body defined by the tensor
( )
∫ ρδij k k i j
V
Iij = xx −xxdV
5.16 Determine expressions for the stress power σijDij in terms of
(a) the first Piola-Kirchoff stress tensor
(b) the second Piola-Kirchoff stress tensor.
Answers: (a) σij ij ρ ρ (b)
5.17 Show that, for a rigid body rotation about the origin, the kinetic
energy integral Eq 5.7-1 reduces to the form given in rigid body
dynamics, that is,
D = ˙ o
FP iA iA / 0 σijD ij = ρs C˙
AB AB /2ρ0 K = 1
2 ωω i jIij where I ij is the inertia tensor defined in Problem 5.15.