S11 S12 S13

13 = det S12 S22 S23

S13 S23 S33

The three roots of Equation 1.12, herein denoted as ffP1, ffP2, and cP3, are the principal stresses acting on the three orthogonal planes. The components of a unit vector that defines the principal plane (i.e., n1Pi, n2Pi, n3Pi) corresponding to a specific principal stress sPi (with i = 1, 2, 3) can be evaluated using any two of the following equations:

«1P> (S11 — Spi ) + «2P; S12 + «3P; S13 = 0 «1P> S12 + «2Pi (s22 — Spi ) + «3pi S23 = 0 «1Pi S13 + «2Pi S23 + «3Pi (s33 — SPi ) = 0

The unit vector calculated for each value of sPi represents the direction of a principal axis. Thus, three principal axes that correspond to the three principal planes can be identified.

Note that the three stress invariants in Equations 1.13 can also be written in terms of the principal stresses:

13 = Sp1 Sp2Sp3

0 0

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