Principal planes are the planes on which maximum normal stresses act with no shear stresses and these stresses are known as “principal stresses.” These are desig­nated as CTj, o2, and o3 implying no shear stresses or

Sij — Sij dj with dij — 0 for i — j. (A.6)

Such principal stresses can be found for the 2D case using Mohr’s circle; however, for the general 3D case, one can determine them as the three solutions to the deter­minant of the stress tensor:

where the indices 1, 2, and 3 are used in place of x, y, and z. The three solutions for the determinant are the principal stresses and it is common practice to designate them to be s1 > o2 > o3 taking into account the sign as well.

Expansion of the determinant gives

0 — 03 — (on + 022 + a33)o2 + (О11О22 + S22S33 + О33О11 — o22 — o23 — o2Jo

— (o11022033 + 2o12023031 — s11 o23 — s22o21 — s33o22)

(A.7a)

or

o3 — I1o2 — I2o — I3 — 0, (A.7b)

where I’s are invariants of the stress tensor:

11 — (011 + 022 + 033),

12 — -(011022 + 022033 + 033011 — of2 — o:;3 — o|1),

13 — o11o22o33 + 2o12o23o31 — o11o23 — o22°21 — o33o22.

Note that I1 is the trace of the determinant (sum of diagonal terms).

A.3