In the above derivation the cartesian stresses axx, oyy and a^ could have been used in place of the principal stresses a\, oj and (73 to yield more general expressions but of identicial form. It therefore follows that the stress and associated strain in any given direction within a complex three-dimensional stress system is given by eqns. (8.30) and (8.31) which must satisfy the three-dimensional Mohr's circle construction.

Comparison of eqns. (8.30) and (8.31) indicates that

Thus, having constructed the three-dimensional Mohr's stress circle representations, the equivalent strain values may be obtained simply by reference to a new axis displaced a distance (3u/(l + v))<r as shown in Fig. 8.12 bringing the new axis origin to O'.

Fig. 8.12. The "combined Mohr diagram" for three-dimensional stress and strain systems.

Distances from the new axis to any principal stress value, e.g. o\, will then be 2G times the corresponding £i principal strain value, i.e. 0'<j\ -+-2G = S]

Thus the same circle construction will apply for both stresses and strains provided that:

(a) the shear strain axis is offset a distance-a to the right of the shear stress axis;

(b) a scale factor of 2G, [= E/{ 1 + v)], is applied to measurements from the new axis.

8.8. Application of the combined circle to two-dimensional stress systems

The procedure of §14. 13t uses a common set of axes and a common centre for Mohr's stress and strain circles, each having an appropriate radius and scale factor. An alternative procedure utilises the combined circle approach introduced above where a single circle can be used in association with two different origins to obtain both stress and strain values. As in the above section the relationship between the stress and strain scales is stress scale E

strain scale (1+d)

This is in fact the condition for both the stress and strain circles to have the same radius^ and should not be confused with the condition required in §14.13^ of the alternative approach for the two circles to be concentric, when the ratio of scales is E/( 1 — v).

î E.J. Hearn, Mechanics of Materials I, Butterworth-Heinemann, 1997. i For equal radii of both the stress and strain circles

(gj - <72) _ (g| -£2) 2 x stress scale 2 x strain scale stress scale (o\ —02) (<ti — 02) E E

strain scale (ei — s2) (<xi — 02) (1 + v) (1-1- v)

0 0

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