Page 286 - Intro to Tensor Calculus

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i
I 44. Consider a material with body forces per unit volume ρF ,i =1, 2, 3 and surface tractions denoted by
r
rj
σ = σ n j , where n j is a unit surface normal. Further, let δu i denote a small displacement vector associated
with a small variation in the strain δe ij .
Z
i
(a) Show the work done during a small variation in strain is δW = δW B +δW S where δW B = ρF δu i dτ
V
Z
r
is a volume integral representing the work done by the body forces and δW S = σ δu r dS is a surface
S
integral representing the work done by the surface forces.
(b) Using the Gauss divergence theorem show that the work done can be represented as
1 Z ijmn 1 Z ij
δW = c δ[e mn e ij ] dτ or W = σ e ij dτ.
2 V 2 V
ij
1
The scalar quantity σ e ij is called the strain energy density or strain energy per unit volume.
2
R ij
Hint: Interchange subscripts, add terms and calculate 2W = σ [δu i,j + δu j,i ] dτ.
V
I 45. Consider a spherical shell subjected to an internal pressure p i and external pressure p o . Let a denote
the inner radius and b the outer radius of the spherical shell. Find the displacement and stress fields in
spherical coordinates (ρ, θ, φ).
Hint: Assume symmetry in the θ and φ directions and let the physical components of displacements satisfy
the relations u ρ = u ρ (ρ), u θ = u φ =0.

I 46. (a) Verify the average normal stress is proportional to the dilatation, where the proportionality
1
i
e = ke where k is the bulk modulus
constant is the bulk modulus of elasticity. i.e. Show that σ = E 1 i i
3 i 1−2ν 3 i i
of elasticity.
(b) Define the quantities of strain deviation and stress deviation in terms of the average normal stress
i
1
1 i
s = σ and average cubic dilatation e = e as follows
3 i 3 i
i
i
strain deviator ε = e − eδ i
j j j
i
i
stress deviator s = σ − sδ j i
j
j
Show that zero results when a contraction is performed on the stress and strain deviators. (The above
definitions are used to split the strain tensor into two parts. One part represents pure dilatation and
the other part represents pure distortion.)
(c) Show that (1 − 2ν)s = Ee or s =(3λ +2µ)e
(d) Express Hooke’s law in terms of the strain and stress deviator and show
i
i
i
E(ε + eδ )= (1 + ν)s +(1 − 2ν)sδ i
j j j j
i
i
which simplifies to s =2µε .
j j
I 47. Show the strain energy density (problem 44) can be written in terms of the stress and strain deviators
(problem 46) and
1 Z ij 1 Z ij
W = σ e ij dτ = (3se + s ε ij ) dτ
2 V 2 V
and from Hooke’s law
3 Z 2 2µ ij
W = ((3λ +2µ)e + ε ε ij ) dτ.
2 V 3