Experimental Mechanics (2010) 50:245–253 DOI 10.1007/s11340-009-9285-y

Hole-Drilling Residual Stress Measurements at 75: Origins, Advances, Opportunities G.S. Schajer

Received: 22 June 2009 / Accepted: 17 August 2009 / Published online: 3 September 2009 # Society for Experimental Mechanics 2009

Abstract Since its inception by Mathar in the 1930s, the hole-drilling method has grown to be the most widely used general-purpose technique for measuring residual stresses in materials. During its history, the method has progressed greatly in both sophistication and scope, with substantial advances in all three of its main aspects. Drilling procedures have developed from the early use of a conventional low-speed drill to the modern use of highspeed orbiting endmills and, for very hard materials, abrasive machining. Deformation measurements have advanced from the use of a mechanical extensometer to strain gauges and full-field optical measurements such as Moiré, ESPI and Digital Image Correlation. Computation techniques have progressed from empirical calibrations for discrete measurements within uniform stress fields to finite-element inverse solutions for multiple measurements within non-uniform stress fields. This paper gives an overview of the history and progress of all three aspects of the hole-drilling method, and indicates some promising directions for future developments.

in materials. It has the advantages of good accuracy and reliability, standardized test procedures, and convenient practical implementation. The damage caused to the specimen is localized to the small, drilled hole, and is often tolerable or repairable. The modern hole-drilling method derives from the pioneering work of Mathar in the 1930s [1]. Since that time, the method has grown and developed remarkably, with contributions from numerous researchers. The hole-drilling method is now well-established, with an ASTM Standard Test Procedure [2] and extensive instructional literature [3– 5]. It is a tribute to the fertility of Mathar’s original concept that, even after 75 years, interest in hole drilling continues to grow, with frequent new developments. The hole-drilling method involves three major aspects:

Keywords Hole-drilling . Residual stress measurement

Introduction

All three aspects of the hole-drilling method have developed greatly since the time of Mathar. This paper describes the advances in each of the three aspects and suggests some promising directions for future developments.

The hole-drilling method is one of the most widely used general-purpose technique for measuring residual stresses

Mathar’s Foundational Work

Invited paper, presented at the SEM XII International Congress & Exposition on Experimental and Applied Mechanics, Albuquerque, NM, June 1–4, 2009. G.S. Schajer (*, SEM member) Department Mechanical Engineering, University of British Columbia, Vancouver V6T 1Z4, Canada e-mail: [email protected]

1. drilling a small hole in the specimen in the area of interest, 2. measuring the resulting deformations around the hole, and 3. computing the corresponding residual stresses.

Josef Mathar was a Privat-dozent (Assistant Professor) working in the laboratory of von Kármán at the Technical University of Aachen, Germany, in the 1920s and early 1930s. Figure 1 illustrates the hole-drilling apparatus he developed for measuring residual stresses in metal plates [1]. Mathar used a low-speed drill to cut a 12 mm diameter through-hole in a metal plate specimen, a mechanical

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Fig. 1 Mathar’s hole-drilling apparatus [1]

extensometer to characterize the material deformation, and experimental calibrations to evaluate the axial stresses. Thus, all three major aspects of the hole-drilling method were present from the start. Mathar’s work was initially published in German [6] in 1932. Tragically, Josef Mathar died in his early 30s in 1933. His seminal hole-drilling paper in the Transactions of the American Society of Mechanical Engineers [1] was published posthumously in 1934. The following sections describe the advances in the hole-drilling method since the time of Mathar. The discussion focuses on the three major aspects of the method listed in the Introduction. For convenience of presentation, the order of the first two aspects is interchanged.

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is cut in the center and deformation measurements are made on the surrounding material. In the ring-core method, the deformation measurements are made in the center area while an annular groove is cut in the surrounding material. Figure 2 schematically illustrates the methods. The two methods are identical mathematically, and differ only in the numerical constants used for the residual stress evaluations. The ring-core method has the advantage of producing larger relieved strains. However, the hole-drilling method remains the more commonly used procedure because of its ease of use and lesser specimen damage. The modern application of the strain gauge hole-drilling method dates from the work of Rendler and Vigness [13] in 1966. They established a standardized strain gauge geometry for residual stress measurements and developed the hole-drilling method into a systematic and repeatable procedure. Their work provided the basis for the establishment of ASTM Standard Test Method E837 in 1981, updated several times since then [2]. A large literature on strain gauge hole-drilling measurements has developed, with descriptive information [4], good practice guide [5], and measurement accuracy analysis [14]. Variant strain gauge rosette geometries have been proposed for specialized applications, for example an 8-gauge design [15] to improve measurement accuracy, 12-gauge [16] and 6-gauge [17] designs to provide thermal compensation and increased sensitivity, and 4-gauge [18] and 9-gauge [19] designs to allow consideration of plastic deformations. Thus, strain gauge hole-drilling is both a prominent standard test method and a fertile field for further development.

Deformation Measurements—Strain Gauges From an early stage, the mechanical extensometer used by Mathar was recognized as a major factor limiting the accuracy and reliability of hole-drilling residual stress measurements. The development of strain gauges in the 1940s provided an opportunity for substantial improvements in deformation measurement quality. In 1950, Soete [7] introduced the use of strain gauges for hole-drilling measurements, greatly improving measurement accuracy and reliability, and allowing smaller holes to be used. The use of strain gauges was further investigated by Riparbelli [8] and Boiten and Ten Cate [9]. In the same period, Milbradt [10] introduced the ring-core method, with subsequent developments by Gunnert [11] and Hast [12]. The ring-core method is an “inside-out” adaptation of the hole-drilling method. In conventional hole-drilling, the hole

Fig. 2 Residual stress measurement methods, (a) hole-drilling, (b) ring-core [3]

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A variant measurement approach is to replace the traditional resistance-type strain gages with optical strain gages based on interferometric techniques [20]. These interferometric strain gages are very small and can make very localized measurements. Strain gage rosettes have been developed and successfully applied to hole-drilling and ring-core residual stress measurements [21]. A further variant approach is the deep-hole method. It is useful for determining the residual stresses within the interior of material. The method was initially developed as a means of measuring geological stresses within large rock masses [22, 23], and was later extended to the measurement of residual stresses in large metal components such as castings [24, 25]. The method involves drilling a deep hole into the test material and then measuring the change in diameter as the surrounding material is overcored. The method involves some mechanical elements of the holedrilling and ring-core methods, but it differs significantly in that the measurements are made in the interior of the hole rather than at the surface. This is an important point because the location of the measurements controls the location of the measured stresses. Conventional holedrilling and ring coring involve measurements at the surface, and so they are mostly sensitive to the residual stresses at the surface, with some diminishing sensitivity to stresses within a depth approximately equal to the hole radius. For deep-hole measurements, the diameter change can be monitored along the entire length of the hole, creating a detailed stress vs. depth profile.

Deformation Measurements—Full-Field Optical Techniques Starting in the 1980s and 1990s, several optical techniques have been introduced for evaluating residual stresses by the hole-drilling method. These techniques have the advantage of providing full-field data, which are useful for data averaging, error checking and extraction of detailed information. Figure 3 compares the localized information provided by strain gauges with the much richer information available from full-field displacement measurements. Three main optical techniques have been applied so far to hole-drilling residual stress measurements: Moiré interferometry, Holographic Interferometry, and Digital Image Correlation. When using Moiré interferometry [26–30], a diffraction grating consisting of finely ruled lines, typically 600–1200 lines/mm, is attached or made directly on the specimen surface. This area is illuminated by two symmetric light beams that are derived from a single coherent laser source. Diffraction of the light beams creates a “virtual grating”, giving interference fringes consisting of light and dark lines that are imaged by a video camera. The fringe

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Fig. 3 Comparison of strain gauge and full-field data [36]

lines represent contours of in-plane surface displacement at intervals typically of about 0.5 μm. Holographic interferometry [31–34] provides a further important method for measuring the surface displacements around a drilled hole. A modern variant, Electronic Speckle Pattern Interferometry (ESPI) has become popular because its use of a video camera allows “live” fringe patterns to be produced by image subtraction [35–37]. Figure 4 shows an example ESPI fringe pattern created by hole-drilling. Inplane, out-of-plane or surface slope ESPI measurements [38] are possible, depending on the optical configuration used. A significant feature of ESPI is that it can work with a plain specimen surface, without attachment of the diffraction grating needed for Moiré measurements. This makes it possible to do ESPI measurements rapidly, and potentially to use the method as an industrial quality control tool. Digital Image Correlation (DIC) is another optical technique that can be used for hole-drilling residual stress measurements [39, 40]. The technique involves painting a textured pattern on the specimen surface and imaging the region of interest using a high-resolution digital camera. The camera, set perpendicular to the surface, records images of the textured surface before and after deformation. The local details within the two images are then mathematically correlated and their relative displacements determined. The algorithms used for doing this have become quite sophisticated, and with a well-calibrated optical system displacements of ±0.02 pixel can be resolved. When using a single camera, DIC can identify displacements in two in-plane directions. 3-D displacement measurements can also be made by stereoscopic imaging using two cameras. The additional outof-plane deformation data created could potentially improve

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Hole-Drilling Technique

Fig. 4 ESPI fringe pattern

the accuracy of residual stress evaluations from hole-drilling measurements. However, the effect is likely to be modest because the out-of-plane displacements are much smaller and therefore less influential than the in-plane displacements. The full-field optical techniques are complementary to the strain gauge technique, each approach having generally opposite advantages and disadvantages. Table 1 lists some of their features. The optical techniques have the advantage that they can provide dramatically larger data sets. The availability of “excess” data creates the possibility to improve stress evaluation accuracy and reliability by data averaging, and to be able to identify errors, outliers or additional features. However, the optical methods generally require fairly controlled conditions, while strain gauges are much better suited to field use. The two measurement approaches also have opposite cost characteristics, with strain gauges having relatively low equipment cost but high per-measurement cost. Optical apparatus has high equipment cost but relatively low per-measurement cost.

The drilling of the hole for residual stress measurements needs to be done with significant care to avoid introduction of errors. There are three main error sources: introduction of machining stresses (adding to the residual stresses to be measured), non-cylindrical hole shape, and eccentricity of the hole relative to the strain gauge rosette. Hole-drilling technique has been of concern since the time of Mathar. Initially, the drills used for cutting the hole had specific shapes aimed at improving hole quality. For example, early researchers used a twist-drill with a small pilot cutter ahead of the main cutter to direct the cutting action [1, 9]. Subsequent researchers [13, 41] started using small milling cutters instead because they give squarebottom holes of well-defined depth, thereby simplifying the associated stress/strain calibrations. Drilling speed is another procedural aspect that evolved over time. Formerly, it was common to use conventional cutting speeds for hole-drilling, for example, the 1966 work of Rendler and Vigness [13] describes the use of a handdrill. In the 1980s it became common to use high-speed dental burs in place of conventional endmills. Such burs are available with carbide tips, and thus are able to maintain their sharpness during the hole drilling process. In addition, the manufacturing accuracy of such burs is relatively high because the size suitable for residual stress work is at the large end of the available range, while for industrial cutters the size is typically at the small end of the available range. The air turbines commonly used to drive dental burs provided a further advance in drilling technique. The very high rotation speeds of air turbines, 40 000 to 400 000 rpm, were found to reduce machining stresses significantly, and thereby decrease induced artifacts in the computed residual stresses [42]. In addition, it was found advantageous to orbit the cutter in a circular path, as shown in Fig. 5, thereby creating a hole larger than the cutter diameter [43]. Such orbiting motion shifts much of the cutting action from the end-surface cutting edges to the side-surface cutting

Table 1 Features of strain gauge and optical measurements Strain gauge measurements

Optical measurements

• Moderate equipment cost, high per-measurement cost • Significant preparation and measurement time • Small number of very accurate and reliable measurements

• High equipment cost, moderate per-measurement cost • Preparation and measurement time can be short • Large number of moderately accurate measurements available for averaging • Stress calculations often quite large • Extensive capabilities for data averaging and self-consistency checking

• Stress calculations are relatively compact • Modest capabilities for data averaging and self-consistency checking • Relatively rugged, suitable for field use • Sensitive to hole-eccentricity errors

• Less rugged, more suited to lab use • Hole center can be identified accurately

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Fig. 5 Orbit drilling technique [43]

edges. This change reduces induced machining stresses because the chips produced from the side-surface cutting edges have a shorter and less constrained exit path. The orbiting motion can also help improve hole circularity and concentricity. Orbiting motion with a much larger orbit radius is used when doing measurements using the ringcore method. Measurements on very hard materials such as highstrength steels and ceramics present serious challenges because conventional cutting tools rapidly become dull and then induce substantial machining stresses. Use of a jet of high-speed abrasive particles has been demonstrated as an effective hole-cutting technique for such “difficult” materials [44–46]. The hole produced by an abrasive jet has less well-defined boundaries than by a hole produced by a milling cutter, so careful alignment of the jet and orbiting motion are required to maintain hole circularity. Some rounding occurs at both surface and sub-surface corners, and these features must be accounted for in the associated stress/strain calibrations. Since the design of Mathar’s device shown in Fig. 1, the importance of using a purpose-built device for aligning and guiding the drill has been clearly recognized. In the early years, such devices were individually made, but with the increasing popularity of the hole-drilling method, sophisticated hole-drilling and ring-coring devices have become available commercially, now also coupled to computerized control and data acquisition systems.

Computation of Uniform Stresses Residual stress computation is the third procedural step required when making hole-drilling measurements. A defining characteristic of the hole-drilling method and almost all other destructive methods for measuring residual

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stresses is that they involve removal of stressed material in one area of the specimen and the measurement of deformations in a different nearby area. This difference in the locations of the target stresses and the measured deformations creates a substantial computational challenge. The resulting mathematical form of the deformation/ stress relationship is as an “inverse” equation [47]. Such equations are known for the sensitivity of their solutions to small errors in the data. Two types of stress calculation are possible, the first where the in-plane stresses can be assumed not to vary with depth from the specimen surface, “uniform stresses”, and the second where they do vary with depth, “stress profiling”. The first case is the simpler one because there are only three unknown in-plane residual stresses to be determined from the measured data. The uniform stress assumption greatly reduces the mathematical sensitivity to data errors. The simplest experimental procedure involves measuring the deformations as the hole is directly drilled from zero to “full” depth, approximately equal to the hole diameter. When making measurements in a thin plate specimen, the hole goes all the way through the material. For the uniform stress case when making strain gauge measurements, three strain measurements are made to evaluate the three unknown residual stresses. The relationship between the measured strain and the in-plane residual stresses is: s x þ s y að 1 þ vÞ s x
s y b "¼ þ cos 2q ð1Þ E E 2 2 þt xy

b sin 2q E

where σx, σy and τxy are the in-plane Cartesian stresses and θ is the angle between the strain gauge axis and the xdirection. a and b are calibration constants that define the strain/stress sensitivity of the measurement. Hole drilling only partially relieves the strains at the strain gauge location, typically just a third of the residual strain. Thus, the measured strains tend to be small, causing the relative effect of noise to be large. The Ring Core method relieves all the residual stress, thus giving larger measured strains and a smaller relative effect of measurement noise. When making optical measurements with hole-drilling, surface displacements are measured rather than surface strains. Equation (1) still applies, with surface displacement replacing surface strain on the left side. The numerical values of the corresponding calibration constants are different, but the trigonometric form of the equation is preserved. A practical way to improve residual stress evaluation accuracy when hole drilling is to make strain measurements at a series of small depth increments as the hole is drilled from zero to full depth. All measured data can be

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considered, outliers identified and removed, and an averaging method used to minimize the effect of measurement noise [48]. The use of eight hole depth increments is specified in ASTM E837 [2], and is an effective procedure for improving residual stress evaluation quality [14]. In early strain gauge work, calibration constants a and b were evaluated experimentally using measurements made on specimens with known applied stresses [1, 7, 41]. This method is practical for uniform stress measurements, although time consuming. In later years, finite element calculations were introduced to provide the needed calibrations [44, 49–51]. Such calibrations are much more consistent than experimental values. Detailed modeling of the strain gauges is necessary to achieve accurate results [52]. The notation for the calibration constants using a superscript “bar” acknowledges the averaging over the strain gauge area. Previously, point strain values were sometimes used. The trigonometric relationship in equation (1) is identical in form to the equations for stress and strain axis transformation using Mohr’s Circle. The first term
 s x þ s y 2 represents the
 isotropic stress, and the second and third terms s x þ s y 2 and τxy the shear (deviatoric) stresses. This feature lead some early authors to define of strain/stress relationship using alternative calibration constants that parallel Hooke’s Law expressed in terms of Young’s modulus E and Poisson’s ratio ν. However, this analogy is not valid because the locations of the measured strains and the calculated stresses are different. Even so, the trigonometric relationship in equation (1) is sufficient to allow the use of 2-D axi-symmetric “harmonic” finite element models to compute a and b instead of a much larger 3-D calculation, without introducing any approximation [49]. An interesting example of a case where equation (1) does not apply is for hole-drilling residual stress measurements in an orthotropic material. The strain/stress relationship is not trigonometric, requiring the residual stresses to be evaluated in a more complex matrix format using seven calibration constants [53, 54]. A significant limitation of the hole-drilling method is that the hole creates a stress concentration that can cause localized plastic deformations if the nearby residual stresses are high. Typical residual stress computation methods, such as described by equation (1) rely on material linearity. The localized yielding near the hole boundary caused by stress concentrations starts to cause noticeable deviations from linearity for residual stresses greater than 60% of the material yield stress. Fortunately, in most cases, the effect is to overestimate the size of the residual stress, often to values significantly above the material yield stress. Thus, the existence of problematic results is readily apparent. For strain gage measurements, correction procedures have been

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developed to allow accurate measurement of residual stresses up to 90% of the material yield stress [18, 55, 56].

Computation of Stress vs. Depth Profiles The more general case of residual stress evaluation occurs when the stresses vary with depth from the specimen surface. This process is called “stress profiling”. The stress calculation is done using the deformation data measured at a sequence of hole depth increments. Early strain gauge methods for evaluating the stress profiles [41, 57] relied on experimental calibrations of the strain vs. stress relationships. Of necessity, these methods were approximate because the experimental calibrations could not provide all the detailed calibration data needed. The subsequent development of finite element calculations provided the needed detailed calibrations [51]. They enabled the introduction of more accurate and reliable stress computation methods, notably the Integral and Power Series methods [50, 51]. The Integral method is a direct generalization of the uniform stress calculation method, with equation (1) rewritten in vector-matrix format. The strain ε becomes a vector of strains measured after a series of small increments in hole depth. The various stresses σ and τ become vectors of the stresses contained within the hole depth increments. The calibration constants a and b become matrix quantities relating the various stresses and strains. Figure 6 shows a physical interpretation of matrix a [3]. Coefficient a32 represents the strain caused by a unit stress within increment 2 of a hole 3 increments deep. The matrix is lower triangular because only stresses that exist within the hole contribute to the measured strains.

Fig. 6 Physical interpretation of matrix coefficients a for the holedrilling method

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Figure 6 illustrates the significance of the finite element calibrations because the various coefficients shown cannot be determined using experimental calibrations. The matrix equations to determine the residual stresses from the measured strain data are “inverse” equations, whose solution is very sensitive to any modeling or data errors [14, 58, 59]. Thus, the accuracy and consistency of the finite element results are very important, as well as strict attention to meticulous experimental technique [5]. In mathematical terms, the computational error sensitivity derives from the numerical ill-conditioning of matrices a and b. The small size of their diagonal elements reduces the matrix determinants almost to zero. A computational technique called “regularization” can be used to stabilize the matrices and reduce the sensitivity to measurement noise [59, 60]. Although much more complex and error sensitive than uniform stress evaluations, stress profiling hole-drilling measurements are now widely used. The ASTM Standard Test Method E837 [2] has recently been revised to include a standardized procedure to evaluate residual stress vs. depth profiles.

Stress Computation from Optical Data Most of the early development of residual stress computation methods from hole-drilling data related to strain gauge measurements. These calculations keep to a moderate size because the typical three-element strain gauge rosettes provide the minimum data set required to determine the three in-plane residual stresses. Optical measurements introduce both a challenge and an opportunity through their massive quantity of full-field data from hundreds of thousands, even millions of pixels. The data quantity greatly exceeds the number of discrete measurements available from strain gauges. A big challenge is to extract the information content from a very large and noisy data set in an effective and efficient way. Initial optical measurements for hole-drilling used calculation methods parallel to those used for strain gauges [32, 34, 61]. Typically, they involved visually picking a small number of opportune points within the measured image, interpreting their fringe orders, and then doing a strain gauge style calculation. Further computational opportunities exist by making larger use of the available data. Desirable features of full-field residual stress computations using optical data are that they: & &

take advantage of the wealth of data available within an optical image extract the data from the image with a minimum of human interaction, preferably none

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&

use the available data in a compact and stable computation, preferably a linear one.

Full-field optical data tend to be highly over-determined; each measurement is mathematically connected with the others. This feature enables extensive averaging of measurements to be done, thereby substantially ameliorating the effect of the large measurement noise typically contained within individual measurements. In addition, the excess data enable data consistency checking to be done, and this ability provides a valuable tool for identification and size estimation of experimental errors. Linear computation methods, for example using a least-squares approach [62– 64] have been applied successfully. Non-linear procedures can also be used, but only when essential because they are much more computationally intensive and potentially less stable. With careful data management and choice of computation technique, the required handling of large quantities of data is not excessively burdensome, especially with modern computer equipment. The data richness of optical measurements provides opportunities for more detailed analysis of the underlying residual stresses, in particular the evaluation of stresses that vary in-plane and stresses whose size approaches the yield stress. The latter case is of particular importance because hole-drilling creates substantial stress concentrations at the hole boundary, causing localized yielding to occur even when the far-field stress is significantly below the material yield stress. The localized yielding causes deviations from linear-elastic response that can be observed within the optical data, thereby providing opportunities for evaluation procedures for residual stresses approaching the material yield stress. For strain gage measurements, computational procedures have been developed to allow accurate measurement of residual stresses up to 90% of the material yield stress [18, 55, 56]. Development of similar procedures for optical measurements on both isotropic and orthotropic materials is an important area for further exploration.

Concluding Remarks The hole-drilling method of residual stress measurement has grown and developed substantially since the pioneering work of Mathar. The technique has become well established, with its own ASTM Standard Test Procedure. The hole-drilling method continues to advance actively in all three of its main aspects: hole-drilling, deformation measurement and computational methods. Recent work has concentrated on the use of full-field optical techniques to measure the deformations around a drilled hole. These developments have greatly expanded the scope of hole-drilling residual stress measurements, notably by providing a very rich source of available

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data. These additional data can provide detailed information about residual stress distributions, and can enable issues such as non-linear material behavior and non-uniform stresses to be taken into account. Mathar’s idea to use the deformations around a drilled hole to evaluate material residual stresses has proven to be a very fertile one. Seventy-five years of active development has followed his pioneering work. Based on the accelerating interest in the technique, it seems a safe bet to suggest that the next seventy-five years will be even more active and that the range and quality of the applications with continue to grow. Acknowledgments This work was financially supported by the Natural Science and Engineering Research Council of Canada.

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