Fig. 3.35 Space diagonal and octahedral plane.


<rr = radial, or cell, pressure cra = axial stress on sample due to applied loading (71 — <7a = for compression tests and o\ — or for extension tests.

Consider again the point P and consider the octahedral plane passing through it. This plane will cut the space diagonal at the point P' such that P' represents the stress system (op, op, where

It is seen therefore that it is often convenient to divide a general stress system into two components:

(i) the hydrostatic component, trp = (cri + 02 + o3)/3 (point P' in Fig. 3.35b);

(ii) a deviator stress component accounting for the remainder (the distance PP' in Fig. 3.35b).

The magnitude of the distance PP' can be found as follows:

OP, the stress tensor of P (i.e. the length of OP) is:

As the space diagonal is at right angles to the octahedral plane, OPP' is a right-angled triangle and PP' = VOP2 OP'2. Hence

PP'2 = -L vWi - a2)2 + (<72 - CT3)2 + (a3 - 0-1)2]

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