The main elasticity-based principle for design of components is that the working-stress must remain within the elastic range. The efficient use of materials makes the designer to keep the working-stress quite close to the yield stress but definitely below it. The stress commonly calculated (total load divided by the total cross-sectional area) is called the nominal-stress.
But the actual working-stress may be much different than this calculated nominal-stress because of many reasons, two of which are:
1. The presence of residual-stresses in a component can increase the actual stress to be much different than the nominal-stress. Residual stresses are induced in a component basically due to non-uniform plastic deformation during casting, heat treatment, cold working, or welding of the component.
2. Any geometrical discontinuity in the component such as a hole, a notch, or a sudden change in cross-sectional area, Fig. 6.39 & 6.40 results in non-uniform stress distribution near the discontinuity or change in the cross-section. Such a discontinuity or change in shape is a stress-raiser.
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For example, a circular hole in a plate, Fig. 6.39 under a tensile load makes the axial stress to reach a high tensile value of three times the uniform nominal stress (σmax = 3 σn) at the edges of the hole, and a compressive stress of equal magnitude exists at the edge of the hole, perpendicular to the axis of loading in the plane of the plate.
This high tensile stress drops rapidly with the distance away from this edge of the hole. The development of high tensile stress at the top as well as at the bottom of the hole can be imagined to be produced by the inability of the stress to pass across the hole, i.e., the fibres of the plate around the hole must bear the hole-part of the stress in addition to its nominal share.
The increase in stress is described by a stress-concentration factor, K, which is defined as-
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K = σmax / σnominal = 3 (for Fig. 6.39 a) …(6.50)
where, σnominal is the stress based on net cross-section.
The effect of change in shape is shown in Fig. 6.40 (a) & (b). The sudden change in cross-section near x – x’ results in stress-raising effect to raise the value twice of nominal stress at x and x’ but drops rapidly to even become less than the nominal stress in the central part. Fig. 6.39 (b) illustrates an elliptical hole in a plate. The maximum stress at the ends of the hole is given by,
and for a circular hole, a = b, and thus,
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σmax = 3 σn … (6.52)
The equation (6.51) proves that maximum stress increases as a/b ratio increases. Thus, a very thin hole (like a crack) normal to the tensile axis causes very high stress- concentration to become equal to the theoretical cohesive strength of the material to propagate the crack.
Stress-raisers are not very dangerous in a ductile material, but are in a brittle material. In the ductile material at the point of maximum stress, plastic deformation occurs as soon as yield stress is exceeded, to partially relieve the stress. At this critically stressed region, local increase of strain occurs with little increase of stress.
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Because of strain-hardening, the stress increases in regions adjacent to the stress-raiser, and thus, the stress-distribution becomes essentially uniform. In brittle material, the absence of plastic deformation is Unable to cause redistribution of the stress, and thus, the stress-concentration becomes equal to the theoretical cohesive value. In gray cast iron, the graphite flakes make the iron brittle, and also act like internal stress-raisers. The impact is much more severe than the small notches made on its surface.
Residual-Stresses:
The mechanical behaviour of materials in service is dependent not only on the stresses produced by the application of external load, but also on the residual-stresses present in them. The latter type of stresses is induced in materials during casting, heat treatment, cold working or welding.
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The magnitude of the residual- stresses in a freshly-quenched rolling mill-roll can be visualised, as in some reported cases, that cracked pieces of the roll flew quite a few meters-while it was lying unstressed. Each heat-treatment shop has one or more such incidents to relate in its history.
The severe internal-stresses set up by quenching are the residual-stresses in such cases. Another quoted example is of welded steel-bridge breaking into two parts with a loud noise to collapse when there was no traffic on it.
All types of stresses can be classified into two main categories:
1. Body-Stress:
This is the type of stress which varies with macroscopic position in a component.
2. Textural-Stress:
This is the type of stress which varies on microscopical scale. These stresses are present due to microscopic heterogeneity in a metal. For example, the stress-field around a dislocation. A screw dislocation has shear-stress field around and along the length of dislocation line.
Textural-stresses are developed in adjacent grains across the grain boundary during plastic deformation. The difference in orientation of atoms in them results in different deformation behaviour. Even the coherency strains obtained around coherent-precipitates during age-hardening are textural-stresses.
Body stresses can be broadly classified into two main categories:
1. Contingent-Body Stresses:
These stresses remain in metals as long as the source of the stress is present, but disappear when the source is removed. The stresses produced by an external load within the elastic range are contingent-body stresses. Nitriding or carburising the steel-surface results in development of compressive type of stresses on the surface, and remain in metal until the source is removed.
2. Residual-Stresses:
These are the body-stresses, which remain in the metal even when the source has been removed. The basic cause of residual-stresses is the occurrence of non-uniform plastic deformation.
A few examples of development of stress-patterns of residual-stresses are:
(a) On an average about 10% of the cold-working energy is stored in cold-worked metal as residual stresses, but the type of stress-pattern developed depends on the type of cold-working done. If the cold working is done so that plastic deformation occurs only in surface layers of the metal, then Fig 6.41 (a) shows the type of stress-pattern developed, i.e., the surface layers are under compression because the undeformed central part tries to restrain the elongating surface layers, while the surface layers put the central part under tension. Such a pattern is obtained by skin-rolling, shot-peening light cold-drawing. Such a pattern improves fatigue-properties.
(b) A cold-drawn rod which has been severely worked so that deformation occurs throughout the cross-section of the rod. Surface layers, due to friction, elongate to a lesser extent than the central part, and thus, the surface layers are in tension (Fig. 6.41 b). Such a stress-pattern reduces the resistance to cracking specially during fatigue test, and thus,
(c) A skin (light) rolling of the cold-drawn rod as above can be made to have compressive stresses in the surface as illustrated by Fig. 6.41 (c).
(d) If a hot, solid metal plate is quenched in water, the surface cools faster than the centre, and thus contracts faster. This non-uniform contraction compresses plastically the hotter and softer interior of the plate. When the surface has cooled down, the interior is still contracting as its temperature is still falling.
This contraction of the interior puts compressive stresses in surface layers and ultimately the residual stress-pattern resembles as in Fig. 6.41 (a). A slower cooling rate producing-coolant such as oil reduces the residual-stresses. Water-quenching can induce residual-stresses of value -20 x 107 Nm-2 at the surface.
(e) To a metallurgist, the expansion occurring due to phase transformation of austenite to martensite during hardening of steel is of importance in heat treatment of steel. When the central layers are transforming to martensite, i.e., expansion is occurring there, the brittle, hard martensite in surface layers is put under high tensile stresses (≈ 55 x 107 Nm-2) which could cause distortion, or even cracks in the part. The distortion or cracking of steel is still more likely to occur when the component has sharp corners, re-entrant angles and abrupt changes in section, even holes in sections.
(f) Shrinkage occurs during the solidification of a casting. The stresses, thus produced, if become large, can produce cracks when the casting is still solidifying. This phenomenon is called hot-tearing. The basic cause is that after 80% of the solidification has been completed, the solidifying casting becomes ‘connected solid’ through interconnection of dendrites. Cracking can occur only after attaining this connection stage. If the remaining 20% solidification occurs at a constant temperature (by choosing such a composition that 20% is the eutectic liquid), development of hot-tearing can be avoided.
Residual stresses can be removed or reduced to a safe level by a heat treatment, called stress-relieving heat treatment. In this treatment, the alloy is heated to a predetermined temperature (depending on the alloy) at which local plastic deformation occurs under the action of residual-stresses present in it. Some temperatures for stress relieving annealing in one hour are-
