## Introduction To Advanced Elasticity Theory

8.1. Type of stress

Any element of material may be subjected to three independent types of stress. Two of these have been considered in detai previously, namely direct stresses and shear stresses, and need not be considered further here. The third type, however, has not been specifically mentioned previously although it has in fact been present in some of the loading cases considered in earlier chapters; these are the so-called body-force stresses. These body forces arise by virtue of the bulk of the material, typical examples being:

(a) gravitational force due to a component's own weight: this has particular significance in civil engineering applications, e.g. dam and chimney design;

(b) centrifugal force, depending on radius and speed of rotation, with particular significance in high-speed engine or turbine design;

(c) magnetic field forces.

In many practical engineering applications the only body force present is the gravitational one, and in the majority of cases its effect is minimal compared with the other applied forces due to mechanical loading. In such cases it is therefore normally neglected. In highspeed dynamic loading situations such as the instances quoted in (b) above, however, the centrifugal forces far exceed any other form of loading and are therefore the primary factor for consideration.

Unlike direct and shear stresses, body force stresses are defined as force per unit volume, and particular note must be taken of this definition in relation to the proofs of formulae which follow.

8.2. The cartesian stress components: notation and sign convention

Consider an element of material subjected to a complex stress system in three dimensions. Whatever the type of applied loading the resulting stresses can always be reduced to the nine components, i.e. three direct and six shear, shown in Fig. 8.1.

It will be observed that in this case a modified notation is used for the stresses. This is termed the double-suffix notation and it is particularly useful in the detailed study of stress problems since it indicates both the direction of the stress and the plane on which it acts. The first suffix gives the direction of the stress.

The second suffix gives the direction of the normal of the plane on which the stress acts. Thus, for example, z o.

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