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£zz = - 2v)om + — (au - om) E 2(j with Yxy = W2G; Yyz = W2G; y2X - rw/2G.

The terms (<?„ — om), (ervv — om) and (oa — om) are the deviatoric components of stress. The volumetric strain em associated with the hydrostatic or mean stress om is then:

£m = ^ — £xx + £yy + Szz where K is the bulk modulus.

8.21. Deviatoric strains

As for the deviatoric stresses the deviatoric strains are also defined with reference to some selected "false zero" or datum value,

£ = 5(«i +£2 + 63) = mean of the three principal strain values.

Thus, referred to the new datum, the principal strain values become

and these are the so-called deviatoric strains. It may now be observed that the following relationship applies: ^ ^

It can also be shown that the deviatoric strains are related to the principal strains as follows: (e',)2 + (s'2)2 + (e3)2 = ![(£, - s2f + (e2 - s3)2 + (e3 - e,)2] (8.66)

822. Plane stress and plane strain

If a body consists of two parallel planes a constant thickness apart and bounded by any closed surface as shown in Fig. 8.28, it is said to be a plane body. Associated with this type of body there is a particular class of problems within the general theory of elasticity which are termed plane elastic problems, and these allow a number of simplifying assumptions in their treatment.

Fig. 8.28. A plane element.

In order to qualify for these simplifications, however, there are a number of restrictions which must be placed on the applied load system:

(1) no loads may be applied to the top and bottom plane surfaces (in practice there is often a uniform stress in the Z direction on the planes but this can always be reduced to zero by superimposing a suitable stress au of opposite sign);

(2) the loads on the lateral boundaries (and the surface shears) must be in the plane of the body and must be uniformly distributed across the thickness;

(3) similarly, body forces in the X and Y directions directions must be uniform across the thickness and the bodv force in the Z direction must be zero.

There is no limitation on the thickness of the plane body and, indeed, the thickness serves as a means of classification within the general type of problem. Normally a plane stress approach is applied to members which are relatively thin in relation to their other dimensions, whereas plane strain methods are employed for relatively thick members. The terms plane stress and plane strain are defined in detail below.

The plane elastic type of problem may thus be defined as one in which stresses and strains do not vary in the Z direction. Additionally, lines parallel to the Z axis remain straight and parallel to the axis throughout loading.

(The problem of torsion provides an exception to this rule.)

8.22.1. Plane stress

A plane stress problem is taken to be one in which azz is zero. As stated above, cases where a uniform stress is applied to the plane surfaces can easily be reduced to this condition by application of a suitable ozz stress of opposite sign. Shear components in the Z direction must also be zero.

Under these conditions the stress equations of equilibrium in cartesian coordinates reduce to dx 3y dT*> dCTyy y. _ Q

dx dy

The following stress and strain relationships then apply:

Plane stress systems are often referred to as two-dimensional or bi-axial stress systems, a typical example of which is the case of thin plates loaded at their edges with forces applied in the plane of the plate.

8.22.2. Plane strain

Plane strain problems are normally defined as those in which the strains in the Z direction are zero. Again, problems with a uniform strain in the Z direction at all points on the plane surface can be reduced to the above case by the addition of a suitable uniform stress azz, the additional lateral strains and displacements so introduced being easily calculated. Thus

Also, from the basic assumptions of plane elastic problems,

The equations of stress equilibrium in this case reduce to

derxx

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