## Abstract

**Residual Stresses are inherent** in most materials and structures which have been exposed to a machining or manufacturing process. They are known to have both a beneficial as well as detrimental influence on the performance of manufactured components and yet often go by undetected.

**This paper presents the results of a research project** aimed at determining the use of Digital Shearography as a suitable method to identify inherent material and structural properties. In order to apply the technique, 3 samples were prepared, one in its fully annealed state and the other 2 with different levels of residual stresses introduced into one of the sample surfaces. This was repeated for three different materials. Investigations were then conducted to determine the samples deflection curvatures in response to an applied load. From these deflections the investigation attempted to determine the Young’s Modulus and magnitude of residual stresses present. The results are presented and compared with tensile specimen results for accuracy. From the results obtained it is apparent that Digital Shearography cannot necessarily be used to detect the presence of residual stresses, but can determine the material’s Young’s modulus.

## Introduction

**Residual stresses are present in virtually all manufactured components.** They are introduced into materials and parts as a result of forming and machining processes applied, are contained within the components surface region and can vary in magnitude from part to part. The presence of residual stresses is often undesirable and in such cases heat treating procedures can be applied to remove them. In other instances the presence of residual stresses is desired1 – compressive residual stresses are known to counteract the onset and propagation of fatigue and stress corrosion cracking.

In the challenge to reduce the weight of components without scarifying their performance, there is an increasing need to have a better knowledge of the presence and magnitude of residual stresses2. This can be achieved through non destructive as well as destructive testing techniques. As destructive techniques rely on some form of material removal, non destructive techniques are preferred due to the part still being intact after the test.

**Optical NDT techniques such as** Electronic Speckle Pattern Interferometry and Digital Shearography are non contacting inspection techniques suitable for the inspection of objects for both surface and subsurface defects and have also been used to detect the presence of residual stresses. Digital Shearography, on the face of it, appears to be particularly suited for residual stress investigations, as the technique records the rate of surface deformation in response to an applied stress. This possibility was highlighted in a pilot study, the results of which were presented at the 2010 BINDT annual conference3. The work focussed on using Digital Shearography to investigate the deflection characteristics of a set of mild steel cantilever beams, some with induced residual stresses, and concluded that the technique showed promise in determining both a material’s Young’s Modulus as well as the presence of residual stresses.

This paper extends this work by investigating the ability to detect residual stresses in three different materials using Digital Shearography and compares the calculated Young’s Modulus with experimentally determined values.

### Theory

**Digital Shearography is a laser based non-contacting** interferometric technique4. The technique relies on an expanded monochromatic laser to illuminate the object to be inspected. The light reflected off the surface of the object is viewed through a CCD camera which in turn is connected to a PC for image processing purposes. In front of the camera a purpose built shearing device is placed. The function of the shearing device is to split the image of the object into 2 distinct images which overlap each other. In a Michelson interferometer setup this is achieved by using a 45° beamsplitter to split an incoming image into two images. For each of these images a mirror is used to reflect the images back onto the beamsplitter where they recombine and then are captured by the camera. If one of the mirrors has the ability to be manipulated in the x and y direction (the shearing mirror), the magnitude as well as position of image overlap, or shear can be controlled. A typical optical set-up is shown in figure 1 below.

**Figure 1. Typical Shearography set-up **

**The overlap of the two images forms a unique** speckle pattern which is captured and stored in a PC. If the object is deformed due to an applied stress, and there is relative movement between the two overlapped images, a change in the speckle pattern occurs. In addition a controlled phase shift is introduced into the beam path length. Comparing the speckle images before and after for areas of correlation and decorrelation produces a saw tooth fringe pattern, where the direction of the fringe intensity gradient provides information on the direction of the displacement gradient. The mathematical formula for this process is outlined below5.

where i = 1,2,3,4

= phase distribution after stressing

In order to calculate the magnitude of the displacement gradient, the following equation can be used:

where

= displacement gradient in the x (or y) direction, = wavelength of the laser light N = number of fringes counted, S = shear magnitude

**A cantilever beam is a beam which is** securely mounted at its one end and free to move at the other end. If a force is applied at the free end, perpendicular to the face of the cantilever, the force will cause the beam to bend, placing one face of the beam into tension and the other into compression, as shown in figure 2 below.

**Figure 2. Schematic of a cantilever and the applied load. With this controlled loading environment, the cantilever deflection can easily be modelled according to equation 5 below.**

Differentiating this equation yields the slope of the deflection.

**The resultant stress produced in the surface of the cantilever is defined as:**

where:

E = Young’s modulus, L = Length of the beam t = beam thickness

P = Load,

I = second moment of area for a rectangular beam

x = position along the beam

y = beam deflection.

**When a force is applied perpendicular to** the cantilever, it deflects accordingly in the direction of the applied force. With the aid of Digital Shearography the rate of deflection can be determined. Using equation 4, the magnitude of the rate of deflection, can be determined without any knowledge of the material properties. The theoretical rate of displacement is defined in equation 6. All constants and dimensions, with the exception of the Young’s Modulus, can be determined from the dimensions of the cantilever sample used. By selecting a suitable value for the Young’s Modulus and determining the rate of displacement curve, equation 6 can be manipulated to fit the rate of displacement curve obtained experimentally in equation 4, thus determining the best suited Young’s Modulus to fit the experimental data.

**When considering the cantilever**, residual stresses, if present, will manifest themselves in the surface layer of the cantilever. It has been suggested that these locked-in surface stresses have the ability to enhance or resist the expected deflection curve, depending on whether they are compressive or tensile stresses, when a transverse load is applied. By comparing these deflection curves with those of "stress free" cantilever beams, initial results indicate that it is possible to detect and quantify the magnitude of the residual stresses. By recording the applied force and rate of displacement in a residual stress sample, this is achieved by establishing the equivalent force required in a "stress free" sample using equation 4, which would produce the same displacement gradient as that recorded in the experiment. Equation 7 would then have to be applied to compute the stress distributions in the surfaces of the cantilevers for both the "stress free" case and the equivalent load case. The difference in surface stress levels could then be directly determined and attributed to the presence the residual stresses.