CONTINUUM MECHANICS for ENGINEERS

or, in expanded form, using σ = σ ,
ij ji
(σ 11 – σ)n 1 + σ 12 n 2 + σ 13 n 3 = 0 (3.6-3a)
σ 12n 1 + (σ 22 – σ)n 2 + σ 23n 3 = 0 (3.6-3b)
σ 13n 1 + σ 23 n 2 + (σ 33 – σ)n 3 = 0 (3.6-3c)
In the three linear homogeneous equations expressed by Eq 3.6-3, the tensor
components σ ij are assumed known; the unknowns are the three components
of the principal normal n i, and the corresponding principal stress σ. To
complete the system of equations for these four unknowns, we use the
normalizing condition on the direction cosines,
n i n i = 1 (3.6-4)
For non-trivial solutions of Eq 3.6.2 (the solution n j = 0 is not compatible
with Eq 3.6-4), the determinant of coefficients on n j must vanish. That is,
σij − δijσ =0
(3.6-5)
which upon expansion yields a cubic in σ (called the characteristic equation
of the stress tensor),
3 2
σ − I σ + II σ − III = 0
(3.6-6)
whose roots σ (1), σ (2), σ (3) are the principal stress values of σij. The coefficients
, , and are known as the first, second, and third invariants, respec-
I II III tively, of σij and may be expressed in terms of its components by
Iσ ii
= σ = tr σ
[ ]
II 1
1 2 2
σ = ( σσ ii jj −σσ
ij ji)=
( tr σ) − tr(
σ )
2
2
III ijk
= ε σ σ σ = det
1i 2j 3k
(3.6-7a)
(3.6-7b)
(3.6-7c)
Because the stress tensor σij is a symmetric tensor having real components,
the three stress invariants are real, and likewise, the principal stresses being
roots of Eq 3.6-6 are also real. To show this, we recall from the theory of equations
that for a cubic with real coefficients at least one root is real, call it σ (1),
and let the associated principal direction be designated by . Introduce a
second set of Cartesian axes so that is in the direction of . In
n ( 1)
i
Px1′ x2′ x3
′ x1 ′
n ( 1)
i