CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.10B
Table displaying direction cosines of principal axes Px 1*x 2*x 3* relative to axes Px 1x 2x 3.
q
for (q = 1,2,3), which upon introducing the identity aqi ≡ nibecomes
σijaqj = σ(
. Now, multiplying each side of this equation by ami and using
q) aqi
the symmetry property of the stress tensor, we have
The left-hand side of this expression is simply , from Eq 3.6-13. Since,
by orthogonality, a a = δ on the right-hand side, the final result is
(3.6-14)
which demonstrates that when referred to principal axes, the stress tensor
is a diagonal tensor with principal stress values on the main diagonal. In
matrix form, therefore,
⎡σ
I 0 0 ⎤
* ⎢
⎥
or [ σ σ
(3.6-15)
ij]=
⎢
0 II 0
⎥
⎣
⎢ 0 0
σ ⎦
⎥
III
where in the second equation the notation serves to indicate that the principal
stresses are ordered, σ I ≥ σ II ≥ σ III, with positive stresses considered greater
than negative stresses regardless of numerical values. In terms of the principal
stresses, the stress invariants may be written
I = σ
+ (1) σ + (2) σ = (3) σ + Iσ
+ IIσ
III (3.6-16a)
II = σ (1)σ (2) + σ (2)σ (3) + σ (3)σ (1) = σ Iσ II + σ IIσ III + σ IIIσ I
III
qi mi qm
⎡σ
⎤
( 1)
0 0
* ⎢
⎥
[ σ ij]=
⎢ 0 σ ( 2)
0 ⎥
⎢
⎣ 0 0 σ ⎥
( 3)
⎦
σ a a σ a a
ji qj mi = ( q) qi mi
*
σ δ σ
qm = qm ( q)
*
σ qm
(3.6-16b)
= σ σ σ (1) = (2) σ (3) σ σ I II III (3.6-16c)
( )