Dyv

Therefore the criterion of failure becomes

~Oy = 1 [(en - ai)1 + {a2 - <r3)2 + - or,)2]1'2 i.e. 2a2 = (<n - a2)2 + (<t2 - <r3)2 + (<r3 - atf (8.59)

This is clearly the same criterion as that referred to earlier as the Maxwell/von Mises distortion or shear strain energy theory.

8.20. Deviatoric stresses

It is sometimes convenient to consider stresses with reference to some false zero, i.e. to measure their values above or below some selected datum stress value, and not their absolute values. This is particularly useful in advanced analysis using the theory of plasticity.

The selected datum stress a or "false zero" is taken to be that stress which produces only a change in volume. This is the stress which acts equally in all directions and is referred to earlier (page 251) as the hydrostatic or dilatational stress. This is defined in terms of the principal stresses or the cartesian stresses as follows:

O = 5(<7i + <72 + <73) = |(cr„ + ayy + a(8.60) i.e. a = mean of the three principal stress values.

The principal stresses in any three-dimensional complex stress system may now be written in the form o\ — mean stress + deviation from the mean

= hydrostatic stress + deviatoric stress

Thus the additional terms required to make up any stress value from the datum to the absolute value are termed the deviatoric stresses and written with a prime superscript, i.e. cti = a + cr', etc.

Cartesian stresses axx, oyy and azz can now be referred to the new datum as follows:

All the above values then represent deviatoric stresses.

It may be observed that the system used for representing stresses in terms of the datum stress and the deviation from the datum is, in effect, a consideration of the normal and shear stresses respectively, on the octahedral planes, since the octahedral and deviatoric planes are equally inclined to all three axes (/ = m — n — ±1/V3) and the selected datum stress o = 5(0-1 +o2 + 03)

is also the octahedral normal stress value.

As stated earlier when discussing octahedral stresses, this has a particular relevance to the yield behaviour of materials.

Whilst any detailed study of the theory of plasticity is beyond the scope of this text, the fundamental requirements of the theory should be understood. These are:

(a) the volume of material remains constant under plastic deformation;

(b) the hydrostatic stress component a does not cause yielding of the material;

(c) the hydrostatic stress component a does not influence the point at which yielding occurs.

From these points it is clear that it is therefore the deviatoric or octahedral shear stresses which must govern the yield behaviour of materials. This is supported by the accuracy of the octahedral shear stress (distortion energy) theory and, to a lesser extent, the maximum shear stress theory, in predicting the elastic failure of ductile materials. Both theories involve stress differences, i.e. shear stresses, and are therefore independent of the hydrostatic stress as indicated by (b) above.

The representation of a principal stress system in terms of the octahedral and deviatoric stresses may thus be shown as in Fig. 8.27.

It should now be clear that the terms hydrostatic, volumetric, mean, dilational and octahedral normal stresses all indicate the same quantity.

The standard elastic stress-strain relationships of eqn. (8.71)

Principal stresses

dilatationol or nyarosianc stresses

Octohedrol sheor or

Octohedrol, meon, dilatationol or hydrostatic stresses deviatoric stresses

Principal stresses dilatationol or nyarosianc stresses for failure ]

Fig. 8.27. Representation of a principal stress system in terms of octahedral and deviatoric stresses.

may be re-written in a form which distinguishes between those parts which contribute only to a change in volume and those producing a change of shape.

Thus, for a hydrostatic or mean stress om = | (oxx + oyy + cr„) and remembering the relationship between the elastic constants E = 2G(1 + v)

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