CONTINUUM MECHANICS for ENGINEERS

6.8 Show that the distortion energy density W for a linear elastic
( 2)
medium may be expressed in terms of (a) the principal stresses,
σ , σ , σ , and (b) the principal strains, ε , ε , ε , in the form
( 1) ( 2) ( 3)
2
2
( ) + ( − ) + −
( )
(a)
(b)
6.9 Beginning with the definition for W (Eq 6.1-21a), show for a linear elastic
material represented by Eq 6.2-2 or by Eq 6.2-7 that and
W(
2)
=
σ( 1) − σ( 2)
σ( 2) σ( 3)
12G
σ( 3) σ(
1)
1
2
2
2
⎡
⎤
W( G
2) = ( ε 1 2 2 3 3 1
3 ( ) −ε(
) ) + ( ε( ) −ε(
) ) + ( ε( ) −ε(
) )
⎣⎢
⎦⎥
∂ ∂ε = σ
∂ ∂σ = ε . (Note that ∂ε / ∂ε = δ δ .)
W / ij ij
ij mn im jn
( 1) ( 2) ( 3)
6.10 For an isotropic, linear elastic solid, the principal axes of stress and
strain coincide, as was shown in Example 6.2-1. Show that, in terms
of engineering constants E and v this result is given by
ε
( q)
[ ]
( ) − + +
1+
v σ v σ σ σ
=
E
( q) ( 1) ( 2) ( 3)
, (q = 1, 2, 3).
Thus, let E = 10 6 psi and v = 0.25, and determine the principal strains
for a body subjected to the stress field (in ksi)
[ σ ij]=
⎡12
0 4⎤
⎢ ⎥
⎢
0 0 0
⎥
⎣
⎢ 4 0 6⎦
⎥
Answer: ε (1) = –4.5 × 10 –6 , ε (2) = 0.5 × 10 –6 , ε (3) = 13 × 10 –6
6.11 Show for an isotropic elastic medium that
1 2
(a) , (b) , (c) ,
1+
3 2
(d) 2µ(1+ν) = 3K(1–2ν)
6.12 Let the x1x3 plane be a plane of elastic symmetry such that the transformation
matrix between Ox1x2x3 and is
=
( λ + µ ) v
v λ + µ 1− v 2
=
λ 2µν 3Kν
=
λ + µ 1−2ν1+ ν
O xxx ′ ′ ′
⎡1
0 0⎤
⎥
⎢
0 1 0
⎥
⎣
⎢0
0 1⎦
⎥
⎢ [ aij]= −
1 2 3
2
W / ij ij