Residual Stress

In 200 B.C., Chinese artisans manufactured thick bronze discs that were flat and polished on one side and had a relief cast on the other face. These were heated and quenched. When such a “magic mirror” was illuminated on the flat face, the reflection showed the pattern of the relief on the other side of the disc! Due to the different cooling rates of the sections with various thicknesses, distortions occurred on the flat side, which mimicked the pattern of the relief. To our knowledge, this is the first deliberate use of residual stresses and strains. Today we know these play a key role in the behavior of welded structures (and hence in ship construction, pipelines and oil rigs), in the response of heat treated or finished parts (ground gears, shot-peened or sand blasted pieces, or material subjected to laser heat treatments, or quenched after a heat treatment). These stresses are also a key factor in the fatigue response of solids, and in the phenomenon known as stress corrosion.

In this chapter the basic theorems of linear elasticity are reviewed, and some fundamental definitions in plasticity theory are discussed. These concepts form the basis of all types of stress analysis, and will be utilized in subsequent chapters for the analysis of the deformation distributions that cause residual stress fields.

In one of the most comprehensive books on mieroplasticity, Mura[l] defines residual stresses as the “self-equilibrating internal stresses existing in a free body which has no external forces or constraints acting on its boundary”. These stresses arise from the elastic response of the material to an inhomogeneous distribution of nonelastic strains such as plastic strains, precipitation, phase transformation, misfit, thermal expansion strains, etc. For example, mechanical deformation processes that cause plastic deformation in the surface layers of the material, such as shot-peening, grinding, machining, etc., cause residual stresses in these layers because of the constraining effect of the bulk, where plastic deformation is minimal. These stresses are called macrostresses. Since the surface layers will also constrain the bulk in return, the bulk material will also have residual stresses even though it may not have suffered deformation. It is also possible to have residual stresses when a multi-phase body, where the phases have different yield points, is pulled in uniaxial tension. The macrostress field in the deformed material will be negligible since the material will have the same plastic strains at all depths. The inhomogeneous distribution of yield points in the material volume, however, causes an inhomogeneous partitioning of the plastic strains between the phases, which, due to the constraining effect of the stronger phases on the weaker ones, causes a residual stress field to form. Residual stresses of this type are called microstresses.

Diffraction methods of residual stress determination basically measure the angles at which the maximum diffracted intensity occur when a crystalline sample is irradiated with x-rays or neutrons. From these angles one then obtains the spacing of the diffracting lattice planes by using Bragg’s law. If the material is under load, these values will be different than the unstressed plane spacing, and the difference will be proportional to the stress acting on the planes. At this point one can use elasticity theory, Is discussed in Chap. 2 and 3, to determine the stress (residual or applied) acting on these planes. Thus, no matter how sophisticated the elasticity analysis, the final stress results are only as good as the data supplied by the diffraction methods.

Up to this point the mechanical and micromechanical behavior of solids and basic concepts of x-ray and neutron scattering from crystalline solids have been presented. In this chapter these concepts are combined in the derivation of the basic equations of residual stress determination with diffraction. The fundamental assumptions inherent in these derivations and the limits they impose on the applicability of the stress measurement will also be discussed. Various problems in an actual stress measurement, such as the effect of stress gradients, the separation of micro and macrostresses, determination of stresses in thin films and single crystals, etc., are also considered, with special emphasis on the interpretation of the data within the limitations of the theory.

The basic methods of residual stress measurements with x-rays and neutrons, as well as the theory underlying the measurements, have been covered in the previous chapters. In any scientific experiment, however, the error associated with the measured quantity is just as important as the measured value itself and must be known for the correct interpretation of the results. Evaluation of errors by theoretical fomiulae is also of primary importance in automating a measurement

In this chapter, we consider the various ways to make measurements of residual stresses on real parts, either in an ordinary x-ray laboratory (Sect. 7.2) to which such samples may be brought occasionally, in a factory where frequent inspection is required, or in the field, for example at large construction sites, on oil-rigs or pipelines or power stations, or for parts which are too large for a commercial diffractometer. There are several units now available for such situations, but this equipment is new and can be expected to change frequently. Therefore we prefer here to provide a list of ideal requirements for the software and hardware (Sect. 7.3), against which the reader can examine what is available commercially, emphasizing his own priorities in these lists. (Also, it is always advisable to contact several users of such equipment, as well as the supplier.) We will then briefly describe the available equipment at this writing (Sect. 7.4), and conclude with a detailed examination of several examples of the use of stress measurements in such situations (Sect. 7.5).

During processing, the shape of a diffraction peak can change (as well as shift) and this is called “X-ray Line Broadening”. This broadening can be an important signature. For example, as shown in Fig. 8.1 (from [1]), for peened soft steels this can be even more important in controlling fatigue life than the induced stress. For a well-annealed sample, the peak shape depends on the size of the source, finite divergences of the slits at the x-ray tube and the receiving slits, the range of wavelengths in the beam, and the fact that in a diffractometer a flat specimen is only tangent to the focusing circle at one point.¹ As a soft sample is deformed, there are local strains due to the strain fields from dislocations and dislocation arrays. The material is broken into regions with slight tilts with respect to one another due to these arrays, subgrain boundaries, etc. Constructive interference occurs only within each such region, and the peak (from such a region) is larger in angular extent the smaller is such a region. This is so because when the region is small there are not as many planes to cause destructive interference away from the exact Bragg angle. To see this more clearly we adopt a treatment by Cullity [2].