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Residual stresses have been associated with humans ever since civilization began. The making of intricate clay components using fire in early days was actually an art that maintained the balance between reducing the residual stress gradient and achieving the desired shape of products. A stronger sword was often the result of a thin layer of compressive residual stress induced by repeated hammering at a controlled elevated temperature. Even today, the presence of residual stresses still dictates the design of many components, whether in a spacecraft or a tiny integrated circuit.

All mechanical methods of residual stress measurement are based on the principles of elasticity and linear superposition. In particular, the superposition for the slitting method as shown in Fig. 2.1, is extended from Bueckner’s principle for crack propagation [9]. When a cut of depth a is introduced to a part with residual stress (case A), the stress on the site of cut is released (case B). This process is the same as imposing a stress field of the same magnitude of the stress in (case A) with a different sign on the site of the cut (case C), which leads to a stress-free slit face in case B. To compute the deformation or the compliance functions due to introduction of the cut in case B, we make use of case C because there is no change in deformation in case A. Note the superposition shown in Fig. 2.1 remains valid when external loads are present. For a body with prescribed displacement boundary conditions, however, the boundary condition should be properly maintained, as shown in Fig. 2.2. Note that the displacement at the boundary for case C should be set to zero. The stress estimated from the deformation measured from case B and the compliance functions computed from case C is due to both the residual stress and the prescribed boundary condition in case A.

When a crack or a cut of finite width is introduced to a surface, the release of the residual stresses on the plane of the cut leads to deformation that can be measured and used to predict the residual stress that existed on the plane before the cut was made. Since a cut of finite width is much easier to introduce than a crack, and less likely to experience face closure associated with releasing compressive stresses, it is of practical importance to obtain the compliance functions for a cut of finite width. For this reason, the approach based on a cut of finite width introduced in this chapter for near surface stress measurement is often referred to as the slitting method.

Crack compliances, as discussed throughout this book, are referred to as the unit elastic response of a cracked body to surface tractions acting on the crack faces. The surface tractions may be those actually applied on the crack faces or the residual stresses released by introducing a thin cut. Two approaches have been used to obtain crack compliances as a function of crack sizes for an arbitrary surface traction. One uses the stress intensity factor solutions from linear elastic fracture mechanics (LEFM). Another uses numerical computations based on the finite element method, which will be introduced in the next chapter.

Many configurations which require residual stress measurement differ from those we discussed in the previous chapter and do not have analytical solutions available from fracture mechanics. In this case we can use the finite element method (FEM) [133] to obtain numerically the crack compliance functions for a specimen of a complex geometry. Since this approach allows the compliance functions to be obtained for a crack as well as a slit of finite width,in what follows,we will use the term “compliance functions” interchangeably for a crack and a slot.

Before we introduce the methods for estimating residual stresses from measured strains, it is helpful to examine strain responses to some typical residual stresses released by a cut of progressively increasing depth. Two stress distributions are considered, one with the peak stress at one surface (Fig. 6.1.a), another, though unsymmetrical, with the same peak stress on both surfaces (Fig. 6.2.a). Different strain responses are obtained depending upon whether the strain gage is located on the right or left face, as shown in Figs. 6.1.b and 6.1.c, or Figs. 6.2.b and 6.2.c respectively. In general, when a tensile stress is released near the surface, the measured strain is initially negative, and vice versa. This response is expected from the superposition principle demonstrated in Section 2.1 that the measured deformation is the same as that produced by applying the released stress to the faces of the cut with an opposite sign. Alternatively, a more intuitive explanation is that the release of a tensile stress opens the cut and leads to a compressive bending stress on the face opposite the cut. As the depth of cut increases, the measured strain variations will take very different forms. Note the more than tenfold difference in the magnitude of measured peak strains shown in Figs. 6.1.b and 6.1.c. In the first case the high stress near the surface is quickly balanced by a compressive stress below the surface, leading to a very gradual strain variation.

Measurement of residual stresses through thickness using the layer-removal or sectioning method is traditionally a very time-consuming process, which often makes obtaining a large number of data impractical. For localized residual stresses such as due to welding, the measurement may become more tedious and prone to error in both measurement and analysis. Fortunately, the slitting method, when combined with wire EDM, makes the experimental procedure straightforward, requiring only a single strain gage installed on the surface opposite the cut. A large amount of data can be recorded as the depth of cut is being extended incrementally with high precision wire EDM. The basic assumptions of the method are that the stress does not vary in the transverse direction, and the deformation due to cutting is linear elastic. In this chapter we will present the procedures to be used in measurement of the through- thickness residual stresses.

The traditional approach for measurement of axisymmetric residual stresses in a hollow cylinder in plane strain involves removing annular layers from the inside surface while measuring axial and hoop strains at the outside surface. Although normally attributed to Sachs (1927)[113], the equations required for this approach were first developed by Mesnager (1919)[82]. He also pointed out that deformation could be measured at the inside surface while annular layers were removed from the outside. Work prior to Mesnager by Heyn and Bauer (1910)[64] only considered axial residual stresses. For a solid rod, the Mesnager-Sachs procedure requires that a central core be removed before layer removal is carried out. Strain measurements on the outside diameter allow the effect of core removal on the stresses in the cylinder to be computed. However, only average values of the stresses in the core region can be obtained.

Residual stresses in many long structural parts can be idealized as distributed uniformly in the axial direction with little out-of-plane shear stresses. Examples are rods, beams, rails, and long butt or fillet welds between two plates. In many cases the residual stresses arise under conditions which are not easily simulated by numerical computation. Figure 9.1-A shows schematically a section of rod and beam while Fig. 9.1-B shows a section of butt weld be- tween two plates. In these examples the constraint is often such that residual stress in the -direction would be expected to be the most severe. Therefore, an essential aspect in assessing the integrity of these parts is to measure the distribution of axial (-axis) stress.

The measurement of residual stresses in a part and the influence of residual stresses on fracture are related topics which have been studied extensively in the literature. Generally, compressive residual stresses are found to be beneficial in fracture calculations while tensile residual stresses degrade the strength of a part. However, in the first part of this chapter we will point out that the local compressive residual stresses, and sub-surface cracks produced by scratching glass at very low loads are responsible for the low tensile strength of conventional glass specimens. Also, if parts containing surface cracks are exposed to processes, such as shot peening, which induce high, near surface compressive stresses, the internal end of the crack will experience tensile loading. What is often ignored is that the compressive stresses close the crack at the surface and make it more difficult to detect by dye penetrant techniques [83]. An attempt is made here to quantify these observations using procedures based on linear elastic fracture mechanics (LEFM).