Following the same procedure for sP2 or, more conveniently, by realizing that the two principal planes are orthogonal to each other, we have

(Note that the planes on which the maximum shear stress acts make an angle of ±45° with the principal planes, that is, 0s1 = 0P1 — (p/4), 0s2 = 0P2 — (p/4) = 0P1 + (p/4).)

1.1.5 Octahedral, Mean, and Deviatoric Stresses

Octahedral normal and shear stresses are stresses that act on planes with direction indices satisfying the condition «1 = «2 = «2 = 3 with respect to the three principal axes of an infinitesimal volume element. Since there are eight such planes, which together form an octahedron, the stresses acting on these planes are referred to as octahedral stresses. The equations for the octahedral normal and shear stresses are given by soct — 1

where I1 and I2 are the first and second stress invariants defined in Equations 1.13 or in Equations 1.16. Octahedral stresses are used to define certain failure criteria (e.g., von Mises) for ductile materials.

Mean stress is obtained as the arithmetic average of three normal stresses (or the three principal stresses):

Sm — 5(^11 + S22 + S33 ) — |(ffpi + Sp2 + ffp3) — 1

Deviatoric stress is defined by the stress tensor

0 0

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