Working Group on Residual Stresses

Transcription

1 Working Group on Residual Stresses A.Ajovalasit, M.Scafidi, B.Zuccarello, University of Palermo M.Beghini, L.Bertini, C.Santus - University of Pisa E.Valentini, A.Benincasa, L.Bertelli SINT Technology s.r.l. AIAS TR0:00 The hole-drilling strain gauge method for the measurement of uniform or non-uniform residual stresses Revision: AIAS TR-0:00 Page of 70

2 PREFACE This test method is the result of work by the AIAS Working Group on Residual Stresses over the period from 006 to 00. The objective was to draw up a draft set of recommendations for the measurement of residual stresses by the incremental hole-drilling technique, also known as the hole-drilling strain-gauge method. Both terms are used without distinction in this document. The hole-drilling strain-gauge method is the test method which is the most widely used in industry to determine near-surface residual stresses. The technical standard on the subject (ASTM E ), which is an indispensable reference, has a restricted field of application as it does not consider: cases in which stresses exceed 50% of the yield stress. corrections where the drilled hole is eccentric to the centre of the rosette; the effects of plasticity within the hole boundary. the effects of any fillet radius at the bottom of the hole. All these effects, nevertheless, influence the quality and accuracy of measurement. The latest revision of the standard, ASTM E837-08, introduced computation of non-uniform stresses, however, the static nature of the method means that it is impossible to evaluate residual stresses in many practical cases. While acknowledging the progress that has been achieved thanks to the ASTM E standard, the purpose of this guide is to go a step further, integrating new methods of correcting and calculating residual stress values with the considerations set out in the ASTM standard. This method presents detailed instructions for the test reports and provides considerations regarding uncertainty analysis in residual stress measurement. The contributions presented herein reflect the results of the work carried out on these subjects by Italian researchers both in the theoretical-experimental field and in design and construction of new measurement instruments. Thanks go to the researchers of the University of Palermo, the University of Pisa and the company SINT Technology srl for the invaluable contributions they have given both to the scientific works developed over these years and to the preparation of this test method guide. Emilio Valentini Coordinator of the A.I.A.S. Residual Stress Working Group Florence, July 00 AIAS TR-0:00 Page of 70

3 CONTENTS INTRODUCTION... 7 SCOPE REFERENCED DOCUMENTS SYMBOLS PRINCIPLE OF MEASUREMENT PRACTICAL ISSUES ASSOCIATED WITH THE MEASUREMENT APPLICABILITY OF THE METHOD PARAMETERS OF THE MATERIAL ACCESSIBILITY OF THE MEASUREMENT AREA EFFECT OF NON-UNIFORMITY AND PLASTICITY STRAIN GAUGE ROSETTE SELECTION ROSETTE DESIGNS ROSETTE DIMENSIONS OTHER FACTORS INFLUENCING SELECTION SURFACE PREPARATION AND INSTALLATION SURFACE PREPARATION CHOICE OF ADHESIVE STRAIN-MEASUREMENT INSTRUMENTATION ALIGNMENT PERPENDICULARITY EFFECTS OF THE FILLET RADIUS AT THE BOTTOM OF THE HOLE HOLE SPACING DISTANCE FROM GEOMETRIC DISCONTINUITIES ZERO DEPTH DETECTION ELECTRICAL CONTACT DETECTION OBLIQUE OBSERVATION OF DRILLING HOLE-PRODUCING TECHNIQUES HIGH-SPEED DRILLING MEDIUM-SPEED DRILLING LOW-SPEED DRILLING ABRASIVE JET MACHINING ELECTRO-CHEMICAL MACHINING HIGH-SPEED ORBITAL DRILLING DRILLING CUTTERS VERIFICATION OF THE DRILLING PROCESS SELECTION OF DRILL DEPTH INCREMENTS MEASUREMENT OF STRAIN EFFECT OF THE TURBINE AIR SUPPLY TEMPERATURE HEAT GENERATED DURING THE DRILLING PROCESS MEASUREMENT OF HOLE DIMENSIONS AND ECCENTRICITY FINAL HOLE DEPTH MEASUREMENT CHECK PRACTICAL EXAMPLE OF APPLICATION RESIDUAL STRESS ANALYSIS TECHNIQUES STANDARD ASTM E837-08: GENERAL STRAIN GAUGE ROSETTES STRAIN RELIEF IN PROXIMITY TO THE HOLE NUMERICAL VALUES OF a AND b SENSITIVITY OF THE METHOD STANDARD ASTM E837-08: CALCULATION OF RESIDUAL STRESSES AIAS TR-0:00 Page 3 of 70

4 7.. THIN WORKPIECE THICK WORKPIECE RESIDUAL STRESS UNIFORMITY TEST CALCULATION OF UNIFORM RESIDUAL STRESSES CALCULATION OF NON-UNIFORM RESIDUAL STRESSES INTERMEDIATE THICKNESS WORKPIECE CALCULATION OF NON-UNIFORM RESIDUAL STRESSES. OTHER METHODS INTEGRAL METHOD INCREMENTAL STRAIN METHOD (ALSO KNOWN AS THE SCHWARZ KOCHELMANN METHOD) HDM METHOD NON-UNIFORM RESIDUAL STRESSES WITH AN OFF-CENTRE HOLE CORRECTION FOR PLASTICITY (ELASTIC RELAXATION OF STRESSES) CORRECTION WITH A 3-ELEMENT ROSETTE CORRECTION WITH A SPECIAL 4-ELEMENT ROSETTE CORRECTION FOR ECCENTRICITY CORRECTION FOR ECCENTRICITY: THROUGH HOLE CORRECTION BY HDM TECHNIQUES CORRECTION USING THE SPECIAL 6-ELEMENT ROSETTE RESIDUAL STRESS ANALYSIS SOFTWARE FEATURES TEST REPORT CONTENTS OF THE TEST REPORT GENERAL PRESENTATION OF THE RESULTS UNCERTAINTY ANALYSIS SUMMARY OF THE SOURCES OF UNCERTAINTY CORRECTION OF THE MAIN ERRORS AFFECTING MEASUREMENT EVALUATION OF UNCERTAINTIES ON STRESSES REFERENCES AIAS TR-0:00 Page 4 of 70

5 INDEX OF FIGURES Figure - Symbols used in this publication. (On the left the symbols necessary for determining the state of stress, on the right the symbols used for correct definition of the geometry of the rosettes). 0 Figure - Relaxation of residual stresses after hole-drilling. Figure 3 - Diagram of the measurement chain using a high-speed air turbine. Figure 4 - Designs of strain gauge rosettes recommended by standard ASTM E Figure 5 - On the left a CW numbering scheme, on the right a rosette with CCW gauge identification. 5 Figure 6 - Hole drilling apparatus with a high speed air turbine (MTS SINT Technology) 0 Figure 7 Hole drilling device: on the left alignment, on the right rotation of the drilling head. Figure 8 - Checking the vertical perpendicularity of the hole-drilling tool. Figure 9 - Hole sections: on the left and in the centre a hole made by high speed drilling with inverted-cone tungsten carbide cutters, on the right a hole made by EDM. 3 Figure 0 - D (left) and 3-D (right) BEM models for studying the effects of the hole-bottom fillet radius. 3 Figure - Identifying the zero cutter depth by an electrical connection. 5 Figure - Types of holes that can be produced with the techniques studied by Flaman: 6 Figure 3 - High speed drilling technique 6 Figure 4 - Medium-speed drilling technique. 7 Figure 5 - High-speed orbital hole-drilling 8 Figure 6 - High-speed orbital hole-drilling technique. Detail of the cutting tool 8 Figure 7 - Cutters used for high-speed drilling 9 Figure 8 - Hardness ranges for which the three types of cutters are recommended 9 Figure 9 - Measurement of hole diameter and eccentricity 3 Figure 0 - Off-centre hole, parameters necessary for calculating hole-rosette eccentricity 3 Figure - Instrument for measuring hole depth 3 Figure - Graphical test of through-thickness stress uniformity (ASTM E837-08) 39 Figure 3 - Schwarz Kochelmann method. 48 Figure 4 - On the right, calibration functions Kx and Ky for the HBM rosette shown on the left. 49 Figure 5 -. Symbols used in the HDM method. 50 Figure 6- Assumed material constitutive law: bilinear isotropic hardening 53 Figure 7- Ratio between the measured relaxed strains versus plasticity factor 54 Figure 8.HBM 4-element Rosette 0/90/57,5/5 (Left), Angles between gauges (Right) 56 Figure 9: (a) Principal Angle (least squares minimisation); (b) Reconstruction of measured strain versus angle. 56 Figure 30 Equi-biaxial Stress Field: difference between the values of strain measured in the absence (above) and presence (bottom) of eccentricity (e=0. mm) 57 Figure 3 - Notations relating to a rosette with an off-centre hole 57 Figure 3-6-element rosette for eccentricity correction 59 Figure 33 - Hole-drilling software. Endmill Positioning Tool (left) and Drilling System Setup (right) 60 Figure 34 - Measured and interpoled strains versus depth. 60 Figure 35 - Residual stress evaluation: above analysis in accordance with ASTM E837-08, below stress analysis with the Integral Method. 6 AIAS TR-0:00 Page 5 of 70

6 INDEX OF TABLES Table - Symbols. 0 Table - Typical dimensions of type A, B and C rosettes described by standard ASTM E Table 3 - Rosettes produced by HBM and Vishay Measurement Group. 7 Table 4 - Maximum and minimum workpiece thicknesses and hole diameters, and drilling depths recommended by standard ASTM E Table 5 - Residual stress calculation methods: principal features. 34 Table 6 - Numerical values of coefficients a and b provided by standard ASTM E for type A, B and C rosettes for uniform stress evaluations with through holes and blind holes. 36 Table 7 - Convention used for placement of angle β (ASTM E837-08). 38 Table 8 - Coefficients a and b for type A rosettes for non-uniform residual stress evaluations (ASTM E837-08). 4 Table 9 - Coefficients a and b for type B rosettes for non-uniform residual stress evaluations (ASTM E837-08). 4 Table 0 - Coefficients a and b for type C rosettes for non-uniform residual stress evaluations (ASTM E837-08). 43 Table - Coefficients a and b of the integral method for type A, B and C rosettes. 47 Table - Errors due to hole-rosette eccentricity for some types of rosette considered in standard ASTM Table 3 - Contributions of uncertainty in residual stress measurement. 65 AIAS TR-0:00 Page 6 of 70

7 Introduction Residual stresses are present in almost all structures. They may be caused by manufacturing processes or may be created during the life of a mechanical component. Residual stresses are often a predominant factor contributing to structural failure, particularly of structures subject to alternating service loads or corrosive environments. The effect on properties can also be beneficial, in which case residual stresses are created purposely to improve the behavior of a material, for example, the compressive stresses produced by shot peening. In either case, it is important to determine the residual stresses in order to be able to foresee static resistance and fatigue strength. The hole-drilling method is a practical, inexpensive and widely used method for determining residual stresses near the surface of a component to be analysed. It can be applied to a wide range of materials. It involves attaching a three-element strain rosette to the surface, drilling a hole in a series of depth increments through the centre of the rosette, and measuring the strains that are produced reflecting the stress relaxation which takes place with the removal of material. Scope This test method specifies an incremental hole-drilling procedure for determining residual stress profiles near the surface of an isotropic linearly elastic homogeneous material. The test method is applicable also to plastic materials and composite materials: these materials present a different mechanical behavior from that of metal materials and also require particular attention in the choice of hole-drilling procedure. The test method may be considered semi-destructive because the damage that it causes is localized and often does not affect use of the component to which it is applied. The method, which is a development of the hole-drilling procedure specified by standard ASTM E [], may also be applied in cases where: a) residual stresses vary with depth, b) there is a small eccentricity between the axis of the hole and the centre of the strain gauge rosette. This test method is limited to cases where the maximum residual stresses do not exceed 50% of the material yield stress. A correction method is specified for stresses exceeding 50% of yield stress, which can only be applied where the stresses remain constant with depth. However, the limitation relating to the thickness of a component reported in the ASTM standard holds and if the thickness is between 0.4 D and. D the results have to be considered approximate. 3 Referenced documents Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain Gauge Method, ASTM E Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain Gauge Method, ASTM E Grant P.V., Lord J.D., Whitehead P.S., The Measurement of Residual Stresses by the Incremental Hole Drilling Technique, NPL Materials Centre, Measurement Good Practice Guide No.53, National Physical Laboratory, UK, 00. LU J., Handbook of Measurement of Residual Stresses, Society for Experimental Mechanics, Fairmont Press, Lilburn, GA, 996, Chapter. AIAS TR-0:00 Page 7 of 70

8 4 Symbols The diagrams shown in Figure are useful for understanding the majority of the symbols listed in Table. Symbol Definition Units a b Calibration constant for isotropic stresses Calibration constant for shear stresses a jk Calibration matrix for isotropic stresses b jk Calibration matrix for shear stresses D Gauge circle diameter mm G L Grid length mm G W Grid width mm R Distance from the centre of the rosette to the internal edge of the grid mm R Distance from the centre of the rosette to the external edge of the grid W Rated resistance of the strain gauge rosette Ω D0 Diameter of the drilled hole mm E Young s modulus MPa Ep Plastic modulus of proportionality MPa r j k Strain hardening ratio of the material Poisson s ratio Number of drilled hole depth steps Sequence number for hole depth steps z Depth of drilling mm P Uniform isotropic stress MPa Pk Uniform isotropic stress within hole depth step k MPa p Uniform isotropic strain m/m pk Uniform isotropic strain after hole depth step k m/m Q Uniform 45 shear stress MPa Qk 45 shear stress within hole depth step k MPa q Uniform 45 shear strain m/m qk 45 shear strain after hole depth step k m/m T Uniform shear stress in x-y direction MPa Tk x-y shear stress within hole depth step k MPa t Uniform shear strain in x-y direction m/m tk x-y shear strain after hole depth step k m/m mm αp Regularization factor for P stresses AIAS TR-0:00 Page 8 of 70

9 αq αt Regularization factor for Q stresses Regularization factor for T stresses Angle measured clockwise from rto max direction Relieved strain for uniform stress case m/m r Relieved strain measured by the gauge, in radial direction m/m,,3 Relieved strains measured by the strain gauge grids m/m j Relieved strain measured after j hole depth steps have been drilled m/m 0 Maximum relievable strain m/m θ Angle of strain gauge from the x-axis max Maximum principal stress MPa min Minimum principal stress MPa x Stress in x direction MPa x)k Stress in x direction within hole depth k MPa y Stress in y direction MPa y)k Stress in y direction within hole depth k MPa xy Shear xy-stress MPa xy)k Shear xy-stress within hole depth step k MPa Ra Surface roughness m/m S Ω C f(c) Sensitivity merit index Biaxiality ratio Plasticity corrective coefficient Dimensionless load parameter X,X Hole radiuses measured in x direction mm Y,Y Hole radiuses measured in y direction mm Dx Hole diameter measured in x direction mm Dy Hole diameter measured in y direction mm D0,m Average diameter of the measured hole mm ex Eccentric radius measured in x direction mm ey Eccentric radius measured in y direction mm e Eccentric radius mm φ Eccentric angle p(h j), q(h j), t(h j), A(H,h j), B(H,h j) K x, K y ψ j (),ψ j (33),ψ j (3) K j (),K j (33),K j (3) ξ u(x) p, q and t values calculated for the hole depth steps by the integral functions proposed by Schajer Influence functions of the integral method Numerical/experimental calibration functions Influence functions describing the state of stress (HDM) Coefficients for the calculation of strains (HDM) Objective function Uncertainty tied to factor x MPa AIAS TR-0:00 Page 9 of 70

10 ci Uc(y) k y U V Weight of uncertainty associated with parameter x Total uncertainty associated with the measurement Normal distribution of uncertainty coverage factor Quantity measured in the test Extended uncertainty associated with the measurement Result of the test Table - Symbols. Figure - Symbols used in this publication. (On the left the symbols necessary for determining the state of stress, on the right the symbols used for correct definition of the geometry of the rosettes). 5 Principle of measurement The hole-drilling method involves drilling a small hole into the surface of a component, at the centre of a special strain gauge rosette, and measuring the relieved strains. The maximum depth of hole is approximately equal to 0.4 D. The single measurements represent the average values of surface strain in the area of the grids caused by relaxation of the stresses and the value of the readings is more sensitive to relaxation of the material the closer they are taken to the surface. This sensitivity decreases as the depth increases until it reaches zero. The residual stresses originally present at the hole location are then calculated from the measured strain values. The relieved strains depend on the stresses that originally existed at the boundaries of the drilled hole (the residual stresses are assumed to act uniformly over the in-plane region around the rosette and to vary only through the thickness of the material) and are not affected by the stresses beyond the hole boundary. AIAS TR-0:00 Page 0 of 70

11 It is also assumed that the drilling technique does not introduce plastic local strains: as will be pointed out later, the drilling operation calls for techniques and specific measures to eliminate Figure shows relaxation of the stresses after drilling a hole for measurement of residual stresses. Relieved stress Modified stress due to hole Strain Gauge Hole Diameter - Do Stress before drilling Hole Depth - h Figure - Relaxation of residual stresses after hole-drilling. The relieved strains decrease rapidly with distance from the edge of the hole and the strain gauges measure only a strain corresponding to 5% to 40% of the original residual stress present in the hole area. The measurement involves the following steps, which are described in greater detail in Section 6 of this guide: Installation of a special strain gauge rosette, with a minimum of three grids, on the component to be analysed for residual stress; Connection of the rosette to suitable instrumentation for recording of strains; Alignment and setting up of the drilling fixture; Establishing zero depth, particularly important for incremental drilling; Drilling in a series of depth increments to obtain data on the variation of stresses with depth; Recording of the strains measured at each depth increment; Calculation of the residual stress state applying a series of equations to the measured values. These calculations are described in Section 7. The typical rosettes used for these measurements are shown in Section 6.: the size of the hole strictly depends on the size of the strain gauge used. The maximum depth of a hole is approximately equal to 0.4 D. Any greater depths are pointless because the surface strain gauges are not sensitive to contributions at subsequent depth increments. It is necessary to use an accurate alignment and drilling system for making these measurements. Excellent results are achieved drilling with a high speed air turbine. AIAS TR-0:00 Page of 70

12 It is always preferable to drill the hole in small increments of depth, recording the measured strains and hole depth at each increment. It is advisable that the drilling system for the incremental method is automatic and electronically controlled: for example, Figure 3 shows a typical diagram of the measurement chain using a high-speed air turbine. Figure 3 - Diagram of the measurement chain using a high-speed air turbine. (Restan MTS 3000, SINT Technology s.r.l.) Also where stresses can be considered to be uniform, incremental hole drilling allows considerations to be made on the uniformity of the stresses. The basic method described in ASTM E and presented in Section 7. is strictly valid where the stresses do not exceed approximately 50% of the yield strength. In these cases the experimentally derived strain calibration coefficients experimentally developed from test specimens with known stress fields can be used. The numerical determination (finite element solutions) of calibration data (influence coefficients) has opened new possibilities for improving the calculation of non-uniform residual stresses from incremental strain data using the so-called integral method []. With this method, the contributions to the total measured strain relaxation of the stresses at all depths are considered simultaneously. It will be examined in greater detail in Section AIAS TR-0:00 Page of 70

13 6 Practical issues associated with the measurement There are two major factors that influence uncertainty associated with the measurements obtained by the hole-drilling method, which are: the way the hole is produced, the procedure used to evaluate the residual stresses originally present, based on the strain measurements. These factors will be considered separately in the following sections. Some of the practical issues are considered below, and recommendations on the analysis methods are presented in Section 7. The practical issues addressed in the following section include: applicability of the method and planning of measurements, strain gauge rosette selection, surface preparation and installation, strain gauge instrumentation, alignment, perpendicularity, hole diameter, effects of the fillet radius at the bottom of the hole, hole spacing, distance from geometric discontinuities, zero depth detection, hole-producing technique, drilling cutters, selection of drilling steps, measurement of strain, measurement of hole dimensions and eccentricity, final hole depth measurement check. 6. Applicability of the method Hole-drilling is a semi-destructive technique with relatively low sensitivity and can analyse residual stress profiles in proximity to the surface of a material. It is the least expensive and most widely used technique for measuring residual stress. 6.. Parameters of the material A component on which the test for determining residual stress is to be carried out should be made of an isotropic material and the properties of the material should be known. If possible, values for Young s modulus (E) and Poisson s ratio () experimentally determined on a sample of the material under investigation should be used, particularly for non-standard alloys and materials where handbook data is not available. Handbook values are correct only for some well-defined, homogenous materials. Typical uncertainties in the mechanical properties of common steel and aluminium alloys are roughly considered to be in the - 4% range and can therefore contribute significantly to the overall uncertainty in the measurement. AIAS TR-0:00 Page 3 of 70

14 6.. Accessibility of the measurement area It is necessary to be able to access the areas of the component to be analysed both in order to apply the strain gauge rosette and to align and make the hole. Ideally, the sample should be flat and the hole location far from any geometric discontinuity. In practice, tests often have to be conducted on curved surfaces or at a location close to an edge, hole or some other feature. In such cases, although the results may provide sufficient information, the validity of the stress values must be considered carefully. In the most critical cases, departures from the ideal can be evaluated by using a finite element model to calculate the influence functions ( a and b coefficients) for the specific installation Effect of non-uniformity and plasticity Standard ASTM E is applicable to residual stress profile determinations where the stresses may be uniform or non-uniform through the thickness of the component under investigation. In addition, the test method provides accurate results if the stresses are less than approximately 50% of the yield stress. There are many circumstances where these requirements are not met, for example, residual stress measurements on a shot peened surface, close to a weld or a hole. This does not mean that the hole-drilling technique cannot be applied, but numerical corrections are required to take account of these effects. For example, the welding process generates high residual stress values that may reach and even exceed the yield strength of the base metal being welded, and in this case the two principal sources of error are: the assumption of uniformity in the stress field, the plasticity around the hole. The methods of evaluating non-uniform through-thickness stresses are analysed in detail in Section 7. The error in residual stress measurements due to the effect of localized yielding has been analysed in literature from both an experimental and an analytical point of view. Beghini and Bertini [3,47,49] have studied the effects of plasticity in the region around the hole: if the value of the stresses in that area exceeds the yield strength of the material, some relations have been proposed to correct the value of stresses, clearing obtained results of the effect of plasticity. The influence of plasticity is discussed in detail in Section Strain gauge rosette selection 6.. Rosette designs A number of commercial strain gauge rosette designs are available, designed specifically for the hole-drilling technique. Rosettes are available with self-temperature-compensation for some materials. All of the rosette designs incorporate centering marks for aligning the drilling tool precisely at the centre of the gauge circle. Standard ASTM E describes the three strain gauge designs which are shown in Figure 4. AIAS TR-0:00 Page 4 of 70

15 Figure 4 - Designs of strain gauge rosettes recommended by standard ASTM E Standard ASTM E distinguishes rosettes also by the arrangement of the measurement grids: the numbering scheme can follow a clockwise (CW) convention if a clockwise rotation is necessary to go from grid (or a) to grid 3 (or c); rosettes can have counter-clockwise (CCW) gauge numbering if a counter-clockwise convention is used. Whether a rosette is CW or CCW type therefore depends on the location of grids and 3: whereas the position of grid determines the type of rosette (type A, B or C). Figure 5 shows both identification schemes. Figure 5 - On the left a CW numbering scheme, on the right a rosette with CCW gauge identification. Type A (with grids in two quadrants) is recommended for general-purpose use, type B (with all grids in a single quadrant) is used for measurements near an obstacle, such as a fillet radius or weld, and type C for situations where high strain sensitivity and high thermal stability are required. The type C rosette consists of six grids forming three pairs, with radially and tangentially aligned grid axes. The opposed grids (for example, T and R in Figure 4) are to be wired in half-bridge configurations.the type C gauge has increased sensitivity (varying from +70% to +40%) in relation to type A and B designs. The disadvantages in using this type include a higher cost, limited availability, and the extra preparation time and instrumentation associated with the six strain gauges (connected to three measurement channels). Table shows the typical geometric dimensions of type A, B and C rosettes described by standard ASTM E A variety of sizes and types of strain gauge currently produced by HBM and Vishay Measurement Group are presented in Table Rosette dimensions The first factor to be considered in selecting a strain gauge is size. The size of strain gauge to use is dependent on the following factors: the size of the available area on the component (proximity of edges, weld features, etc.), AIAS TR-0:00 Page 5 of 70

16 the depth required for the residual stress analysis (larger gauges are more suitable for determining the stress profile at greater depths whereas smaller gauges are suitable for a near-surface analysis), acceptable damage (smaller holes are introduced with the smaller gauges). The most widely used gauge size is the one with an individual gauge length measuring.5.57 mm. This size of gauge is capable of providing useful residual stress data to a depth of approximately mm. It should be noted that the experimental errors associated with the measurements from small strain gauges (hole eccentricity, control of depth, etc) are higher than those associated with the corresponding measurements with larger gauges. However, the larger strain gauges should be selected with caution because of the size of drills required and the large amount of material to be removed during the drilling process. Table - Typical dimensions of type A, B and C rosettes described by standard ASTM E Other factors influencing selection Others factors to be considered in selecting the most suitable strain gauge rosette include: the time required for installation and wiring, temperature compensation, the ease of handling, availability, cost. AIAS TR-0:00 Page 6 of 70

17 Encapsulated designs are available complete with soldering tabs. These are particularly suitable for use in harsh environmental conditions where special protection for the gauge is required. Gauge Pattern Designation Manufacturer ASTM E837 Type Resistance (W) Nominal Gauge Factor Grid Lenght (mm) Grid Ctr'line Dia. (mm) Min. dia. hole - dmin (mm) Max. dia. hole - dmin (mm) Carrier Lenght (mm) Carrier Width (mm) dmin/d dmax/d d/d -RY6-,5/0S HBM Type A (CCW) RY6-,5/0K HBM Type B (CCW) RY6-,5/0R HBM Type B (CCW) K-RY6-,5/0R (with pre-attached leads) HBM Type B (CCW) VY6-,5/0S HBM NK-XX-030 RR Vishay - Measurement Group Type C (CW) EA-XX-03 RE Vishay - Measurement Group Type A (CW) EA-XX-06 RE Vishay - Measurement Group Type A (CW) CEA-XX-06 UL Vishay - Measurement Group Type A (CW) CEA-XX-06 UM Vishay - Measurement Group Type B (CW) EA-XX-5 RE Vishay - Measurement Group Type A (CW) Table 3 - Rosettes produced by HBM and Vishay Measurement Group. Open-faced strain gauges are more suitable for installation on irregular surfaces where the stiffness of encapsulating layers precludes conforming the gauge to the workpiece surface. Configurations with pre-attached leads considerably facilitate installation work avoiding soldering on the strain gauge solder tabs and reduce strain gauge installation time and errors. AIAS TR-0:00 Page 7 of 70

18 6.3 Surface preparation and installation Installation of the strain gauge rosette should be carried out by qualified personnel in accordance with the strain gauge and adhesive manufacturers instructions. [4]. The instructions provided by the UNI 0478 standards [5-9] should be followed for correct installation of strain gauges. Surface-preparation and gauge-installation procedures must be of the highest quality as they have a direct influence on the accuracy of the strain measurements. As a rule, it is also useful to refer to material manufacturers instructions for surface-preparation and gauge-installation procedures Surface preparation To ensure a high-quality bond between the strain gauge and the component, the surface must be properly prepared. This is particularly important when using the incremental hole-drilling technique as the strains measured are generally very small (typically only several μm/m in the first depth increments). The purpose of surface preparation is to develop a surface texture suitable for bonding without altering the state of the surface stresses. Nevertheless, any oxides, rust or paint should always be removed. The UNI standard suggests a surface roughness (Ra) in the μm range for gauge bonding with a cyanoacrylate-based adhesive [6]. However, it is recommended that mechanical abrading be avoided as much as possible if the incremental hole-drilling method is to be used for determining near-surface stresses [0-] Surface abrasion influences only the range of depth nearest the surface and the importance of it depends on the residual stress gradients and the measurement requirements. It should be noted that extremely rough surfaces must be avoided due to ambiguity in establishing the zero depth for incremental hole-drilling []. ASTM E also recommends restricting surface preparation to those methods that have been demonstrated to induce no significant residual stresses (particularly for workpieces that contain sharp near-surface stress gradients) Choice of adhesive. The simplest, quickest and most common method of bonding the strain gauge to the specimen is to use a conventional cyanoacrylate adhesive. These adhesives consist of a single component with a short cure time (- minutes), and are realtively easy to use. If the surface of the component is particularly rough, it is important that the chosen adhesive fills the asperities and irregularities to achieve a good bond. In such cases, a more viscous, twocomponent epoxy adhesive may be more suitable. 6.4 Strain-measurement instrumentation It is important that the instrumentation chosen for strain measurement is calibrated and suitable to be used for this application. ASTM E stipulates that the instrumentation for recording of strains should have a strain resolution of ± μm/m and that stability and repeatability should also be ± μm/m. Generally, most modern strain-measurement instrumentation has the required resolution and stability for measuring the small strains in incremental hole drilling. AIAS TR-0:00 Page 8 of 70

19 However, the following minimum requirements are believed to be advisable for incremental hole-drilling applications: strain resolution of ±0.5 μm/m, stability ±0.5 μm/m, repeatability ±0.5 μm/m. With the more conventional rosettes (types A and B) a three-wire quarter bridge circuit should be used (self-temperature-compensating for as far as regards apparent thermal strain of the leads) with conveniently short leadwires. Half-bridge circuits should be used with type C rosettes. A particularly high acquisition frequency is not necessary for these measurements. It is advised that the average of the values measured (recommended value between 0 and 50 acquisitions) be made for every measurement interval. ASTM E recommends checking the integrity of the gauge installation by applying a small load to induce strains and evaluating the mechanical hysteresis of the strain gauges forming the rosette. The standard also recommends visual inspection of the rosette installation. For the strain gauge installation, however, it is advisable to refer to the preliminary checks specified by the standard UNI [7]. 6.5 Alignment. Eccentricity between the hole and gauge centre can introduce significant errors into the measurement of residual stresses. Alignment between these centres is normally achieved with the aid of a microscope incorporating a reticle in the focus of the objective, the centre of which should coincide with the centre of the endmill for drilling the hole. After installation of the strain gauge rosette, the mechanical part of the measurement system is moved close to the point where the measurement is to be made, and is positioned so that the strain gauge centering marks are within the field of view of the microscope. Two adjustments set at 90 to each other are used for centering until the microscope reticle coincides with the strain gauge centering marks. A typical alignment and air turbine drilling system is shown in Figure 6. In this setup, the microscope is incorporated in the measurement system and is not taken off during measurements: all that is necessary is a rotation of the drilling head as it is aligned with the microscope (Figure 7). The drilling tool is fitted in front of the microscope after the alignment procedure. In other measurement systems the microscope is replaced with the drilling tool after alignment. This reduces (but does not eliminate) eccentricity as alignment of the reticle does not allow the uncertainty in positioning the tool holder (in the region of a few microns) to be taken into account. ASTM E states that the centre of the drilled hole should be aligned concentric with the strain gauge circle to within ±0.004 D. AIAS TR-0:00 Page 9 of 70

20 Stepping motor for fine positioning - Knob for slow manual feed 3 - Eyepiece 4 - Turbine release pushbutton 5 - Compressed air connection 6 - Air turbine 7 - Chuck 8 - Endmill 9 - Knob for fast vertical movement 0 - Rear cap for closing the turbine - Threaded dowels for microscope alignment - Support feet 3 - Microscope 4 - Knob for horizontal movement 5 - Eyepiece reticle 6 - Vertical height adjustment Figure 6 - Hole drilling apparatus with a high speed air turbine (MTS SINT Technology) AIAS TR-0:00 Page 0 of 70

21 Figure 7 Hole drilling device: on the left alignment, on the right rotation of the drilling head. The standard recommends using an optical system to align the axis of rotation of the cutter in relation to the centre of the strain gauge rosette. In other cases it is necessary to align the apparatus using a microscope, then remove the microscope and fit the air turbine hole-drilling system. Section 7.5 deals with the influence of eccentricity and methodologies for correcting the effect of eccentricity. 6.6 Perpendicularity It is essential that the cutter is positioned perpendicular to the surface of the component to be analysed. Figure 8 - Checking the vertical perpendicularity of the hole-drilling tool. For example, if a mm. diameter hole is to be made using a rosette with strain gauges with.5.57 mm long grids, a angle off the perpendicular will lead to a difference in depth of 7 μm between the outer edge and the centre of the cutter. AIAS TR-0:00 Page of 70

22 This corresponds to a substantial error in depth in the typical increments that are used in incremental measurements: its effect will depend on the orientation between the angle axis and the rosette configuration []. It is important that the drilling system be checked before any test to avoid any errors caused by the drill not being perpendicular: this is not always easy, particularly for in-situ measurements. It is therefore important that the drilling system incorporates a means of adjusting perpendicularity to ensure that the cutter is correctly positioned. Apparatuses usually have three magnetic feet that can be used for regulating perpendicularity. This operation can be checked with precision squares and levels (Figure 8). It is recommended that a margin of at least 0.30 mm be maintained between the hole and the strain gauge grid endloops to protect the grids. The need for this margin limits the maximum allowable diameter of the drilled hole D 0. The recommended minimum hole diameter is 60% of the maximum allowable diameter. Table 4 indicates the maximum and minimum diameters recommended for standardized, type A, B, and C rosettes. Table 4 - Maximum and minimum workpiece thicknesses and hole diameters, and drilling depths recommended by standard ASTM E As indicated in Section 7..4, it is important to note that as the ratio of D 0/D increases, the sensitivity of the method increases in approximate proportion to (D 0/D). Consequently, larger holes are recommended to achieve higher sensitivity. Drilling diameters between.6 and.0 mm are normally used for rosettes with grids from.5.57 mm long. If orbital drilling is used, the hole diamter is significantly larger than the drill diameter. 6.7 Effects of the fillet radius at the bottom of the hole. The drilling techniques that can be used with the hole-drilling method for determining residual stresses generally produce a blind hole with a significant fillet radius at the bottom of the hole. For example, if the high-speed drilling technique is used, the hole-bottom fillet radius varies between 4% and 0% of the hole diameter D 0; whereas with electrical-discharge machining (EDM) or abrasive jet machining techniques the fillet radius can reach values greater than 30%. AIAS TR-0:00 Page of 70

23 Figure 9 shows the section and hole-bottom radius of three holes made with different holedrilling techniques and endmills. r r Figure 9 - Hole sections: on the left and in the centre a hole made by high speed drilling with invertedcone tungsten carbide cutters, on the right a hole made by EDM. The hole-bottom fillet radius has an effect on residual stress values measured by the holedrilling method. It is possible to study the effect with -D and 3-D BEM models (Boundary Element Method, Beasy code) (Figure 0). D/ D 0 / r z ER Figure 0 - D (left) and 3-D (right) BEM models for studying the effects of the hole-bottom fillet radius. The study by M. Scafidi and B. Zuccarello [4] has shown that the hole-bottom fillet radius ranges from 0.04 D 0 to 0.0 D 0 using inverted-cone tungsten carbide cutters and is in the region of 0.30 D 0 with EDM techniques. The effect of the hole-bottom fillet radius on relaxed strains was evaluated by numerical simulations performed with the BEM models shown in Figure 0: it must be taken into due account particularly in the initial drilling steps. The effect increases as the hole-bottom fillet radius increases and decreases with hole depth. The hole-bottom fillet radius can significantly influence the test specified by ASTM E for evaluating the uniformity of stresses. For example, considering an equi-biaxial stress field, it is found that: a hole-bottom fillet radius equal to r=0.0 D 0 leads to a maximum deviation in relaxed strains of 5%, a hole-bottom fillet radius equal to r=0.30 D 0 leads to a deviation in relaxed strains greater than 0%. In both cases the bottom-hole fillet radius influences the stress uniformity test and therefore stress measurement by the ASTM E method: these deviations can actually influence determination of the stress field since uniformity of the field is guaranteed if the deviations in strain between the measured value and the theoretical value are lower than ±3% according to the standard. AIAS TR-0:00 Page 3 of 70

24 6.8 Hole spacing The presence of a hole in the vicinity of a new hole alters the residual stresses present in the material as the first hole-drilling process produces a relaxation of the residual stresses which extends a certain area around the hole. The extension of the relaxed area depends on the type and diameter of the hole. It is recommended that the minimum spacing between holes should be equivalent to at least six times the hole diameter. When possible, strain gauge grids should be installed well removed from adjacent holes, and not between adjacent holes. [5,6]. 6.9 Distance from geometric discontinuities ASTM E introduced a number of considerations relating to the minimum distance necessary between the centre of a hole and the closest geometric discontinuity. A geometric discontinuity means an abrupt geometric change or an abrupt change in the thickness of a component: these geometric discontinuities can locally influence the value of residual stresses present in a component. The minimum distance from the nearest discontinuity depends on the diameter and type of strain gauge used. If type A strain gauge rosettes are used, the distance between the hole centre and the discontinuity must be at least.5 D; this is reduced to 0.5 D using a type B rosette and positioning the grids diametrically opposed to the discontinuity. 6.0 Zero depth detection Accurate detection of zero depth, ie, the point at which the drill is in initial contact with the surface of the component, is particularly important in measuring residual stress variation with depth (using the incremental hole-drilling technique). Before actually drilling into the material, the drill should be lowered so that it cuts through the backing film on the strain gauge without touching the surface below. Zero depth is the point at which drilling and acquisition of the strain measurements start after cutting through the backing film. Exact identification of the zero point may be affected by the following causes of uncertainty: surface roughness causing uncertainty in identifying a single zero depth; any error in drill alignment (off perpendicular) leading to initial contact on one side of the hole; a concave profile at the end mill cutting edge resulting in an initial ring contact around the end mill circumference rather than over the whole end face; axial clearance in drill motor bearings (in particular those of air turbines) may cause some ambiguity in the absolute position of the end mill cutting edge; uncertainty relating to actual removal of strain gauge backing and encapsulation material. These causes of uncertainty can affect the accuracy of stress measurements at initial depth increments and cannot be identified or corrected by examination of the strain data. The techniques presented in sections 6.0. and 6.0. can be applied to determine the instant when the strain gauge backing is broken Electrical contact detection Considering the zero point to be the point when the strain gauge backing is removed, it is possible to use the electrical contact technique. AIAS TR-0:00 Page 4 of 70

25 This technique can be applied when analysing conductive materials and providing the air turbine conducts electricity [7]. Figure shows zero depth detection by the electrical contact technique. Figure - Identifying the zero cutter depth by an electrical connection. The advantages of this method are the simplicity in determining the initial contact, the short time required (a few seconds), the low cost (no auxiliary equipment is needed except for an electrical lead) and automation of the method (managed by the electronic control system and measurement instrumentation software) [8,9,0] The measurement system shown in Figure has an automatic procedure for determining the initial drilling point, removing the strain gauge backing and positioning the end mill cutter in contact with the workpiece metal surface Oblique observation of drilling The technique consists in carrying out oblique observation of the drilling process through a mini video camera, magnifying eyeglass or a microscope. The device should be held close to the hole location and cold light reflected from the strain gauge backing makes it possible to detect the thinning and subsequent elimination of the strain gauge backing. A cold light source does not generate significant heat, whereas use of conventional inspection lamps may introduce undesired thermal strains []. This technique for determining the zero position provides a less accurate detection of zero depth than the electrical contact method. It may be applied to all types of materials and not just materials which carry electricity. Oblique observation has the advantage of observing the drilling area in detail and consequently the errors due to bad perpendicular alignment between the endmill and workpiece can be minimized []. 6. Hole-producing techniques The two key factors to be considered in selecting the hole-producing technique are the following: introduction of additional residual stresses during the machining process; the ability of the technique to produce geometrically well-defined holes. In fact, calculation of residual stresses with one of the techniques available requires a cylindrical hole with a flat bottom. AIAS TR-0:00 Page 5 of 70

26 M.T.Flaman and J.A.Herring [] studied four different techniques which were compared quantitatively on the basis of induced stresses and hole geometry and qualitatively in terms of portability and ease of use. In addition, a fifth drilling technique, the orbital hole-drilling technique, was introduced and later studied by the aforementioned M.T.Flaman []. The main techniques are: high-speed drilling, low-speed drilling, abrasive jet machining, electro-chemical machining, high-speed orbital drilling. A diagram is provided in Figure showing the geometric characteristics of the holes that can be made by the four techniques studied by M.T.Flaman. Figure - Types of holes that can be produced with the techniques studied by Flaman: A High-speed drilling; B Conventional low-speed drilling; C Abrasive jet machining; D Electro-chemical machining. Figure 3 - High speed drilling technique These methods for residual stress measurement are described and analysed in detail in the following sections. 6.. High-speed drilling High-speed drilling was first used by M.TFlaman [] employing an air turbine drilling system rotating at speeds of up to 400,000 rpm (Figure 3). The typical cutting tool is an inverted-cone tungsten carbide cutter, which produces a circular hole with straight sides and a flat bottom. AIAS TR-0:00 Page 6 of 70

27 High-speed drilling is considered suitable for most materials as it does not introduce significant machining stresses due also to the modest torque applied to the tool during the drilling process. 6.. Medium-speed drilling The incremental hole drilling method can also be applied using a high speed electric motor (approx RPM). Recent research shows that even lower speeds (4,000 40,000 rpm) can produce reasonable results. [3]. An electric motor system is an alternative when a compressed air supply for a high speed turbine is not available and with an automatic drilling system a high speed electric motor can be operated automatically. The electric motor drilling system can be housed in a special drilling head (Figure 4) and the centering and measurement of the hole diameter can be done by changing the drilling instrument holder with a microscope with a centering reticle. The drill shank has a diameter of.35 or 3.0 mm. and the cutter diameter can be.6 to.8 mm or approx. 3 mm depending on the rosette diameter Low-speed drilling Figure 4 - Medium-speed drilling technique. The low-speed drilling was the first technique used for measuring residual stress by the holedrilling method. Rendler and Vigness [4] introduced the low-speed drilling technique in 966 with specially developed endmills. The technique produces holes that are geometrically suitable for determining residual stresses by the hole-drilling method. However, the results of Flaman s comparison of hole-producing techniques [] showed that the low-speed milling technique induces high stresses and therefore must be considered unsuitable for the hole-drilling method Abrasive jet machining Hole machining by the abrasive jet technique, proposed and developed by Beaney and Proctor, is achieved by directing a small diameter jet of cutting powder at high pressure at the surface of the workpiece. The jet of air and powder removes material and quickly produces a hole. Abrasive jet machining produces a fairly irregular hole shape that is little suited to the hole-drilling method (type C in Figure ): in fact, it allows little control of the hole diameter and shape. AIAS TR-0:00 Page 7 of 70

28 In addition, abrasive jet machining cannot be used for determining non-uniform residual stresses as it does not allow sufficient control of hole depth and diameter. It is not recommended for the less hard materials. [] 6..5 Electro-chemical machining Electro-chemical hole-producing techniques refer to electrical discharge machining (EDM) and electro-chemical machining (ECM). The hole shape they produce is acceptable for the hole-drilling method of measuring residual stress although convexities are formed on the bottom of the hole, as can be seen for type D in Figure, which can influence the measured value of residual stress. Use of these hole machining processes is limited to electrically conductive materials: the presence of high electric discharges that generate stresses on the surface layers of the material plus the presence of chemical agents can cause problems for protection of the strain gauge grids. These factors have prevented development and diffusion of these techniques in producing holes for the measurement of residual stresses. [] 6..6 High-speed orbital drilling Another technique available for measurement of residual stresses by the hole-drilling method is high-speed orbital drilling. It was first introduced by Flaman []. With this technique, the drill is deliberately offset from the centre of the strain gauge and the hole is drilled with an orbital motion. The diameter of the cutting tool is smaller than the diameter of the hole (figures 5 and 6). Figure 5 - High-speed orbital hole-drilling Figure 6 - High-speed orbital hole-drilling technique. Detail of the cutting tool The orbital drilling technique is an effective method for drilling hard, highly abrasive materials such as spring and bearing steels and cast aluminium alloys with a silicon content greater than 6% (for example AlSi9Cu3 and AlSi7Mg). With the orbital drilling technique the removal and extraction of chips is facilitated and more efficient. A further advantage are greater drilling diameters. 6. Drilling cutters For high-speed drilling the recommended drill for most materials is the inverted-cone tungsten carbide type. An inverted-cone polycrystalline diamond coated cutter can be used for harder materials. AIAS TR-0:00 Page 8 of 70

29 The inverted-cone cutting tools that may be used for high-speed drilling with an air turbine are illustrated in Figure 7. Figure 7 - Cutters used for high-speed drilling (on the left a tungsten carbide (TC) endmill, in the centre a TiAlN-coated tungsten carbide (TTC) endmill, on the right a diamond-coated (D) cutting tool) Milling cutters are available in a range of diameters (from 0.6 to. mm) and with.6 mm shanks (for an air turbine) or.3 mm shanks (for coupling to an electric motor). The end face cutting edge must be flat or slightly concave; the side relief of the inverted cone gives clearance for chip removal without affecting the cutting surface. To avoid ambiguities in hole diameter identification, ASTM E prescribes that the radial clearance angles of the cutting edges should not exceed (to avoid ambiguities in hole depth identification by ensuring that the depth is uniform within at least % of the tool diameter) and that the taper angle should not exceed 5 (to allow the diameter of the drilled hole to be identified with certainty). The cutting edge outer angle should be as sharp as possible. Excessive blunting or too high a radius can produce unacceptable errors (see 6.8) Milling cutters should be visually inspected (for example, with a magnifying lens) prior to use and on completion of the drilling of a hole. It is advisable to change the cutter for every holedrilling operation. Figure 8 shows the HV0 hardness ranges of metal materials for which tungsten carbide and diamond-coated cutters are used. Figure 8 - Hardness ranges for which the three types of cutters are recommended AIAS TR-0:00 Page 9 of 70

30 As can be noted, uncoated tungsten carbide cutters can be used on materials with a hardness ranging between 00 and 00 HV0. Tungsten carbide cutters with TiAlN coating can be used with materials with hardnesses up to 550 HV0. Inverted-cone diamond-coated cutters (type D in Figure 7) are recommended for extremely hard materials (cemented and nitrided steels, ceramics, glass, etc.). Diamond-coated cutters do not cut a hole with a sharp angle; the small radius represents a departure from the ideal case for which coefficients were evaluated, therefore the residual stress data from near-surface increments should be treated with caution as they are affected by greater uncertainty. 6.3 Verification of the drilling process Verification of the selected drilling technique is recommended, when no prior experience is available, in order to prevent any residual stresses induced by the drilling method from significantly influencing the accuracy of the results. Verification could consist in applying a strain gauge rosette identical to the rosette used in the test method on a stress-free specimen of the same material, and then drilling a hole. If the drilling method is satisfactory, the stresses produced by the drilling will be small. According to standard ASTM E the drilling method is acceptable if the measured strains are within ±8 µm/m. If necessary (or if the strains induced by the drilling process exceed ± 8 µm/m), it is possible to use coolants during the drilling process. The coolant used must be electrically non-conductive (water-based coolants are not suitable). The most common method for obtaining stress-free specimens consists of annealing heat treatment. 6.4 Selection of drill depth increments It is recommended that the measurement system for the incremental hole-drilling techique have the possibility of selecting the number and position of the depth increments to be drilled. It may be useful to increase the number of increments in the area near the workpiece surface and to reduce depth increments further from the surface. Modern automatic measurement systems make it possible to drill in increments even smaller than 0.0 mm [7]. 6.5 Measurement of strain After the zero point corresponding to the workpiece surface is identified, strain at the set drilling depth increments is measured by each strain gauge in a rosette Effect of the turbine air supply temperature It is important that an air turbine drilling system have a side air exhaust to minimize the effects of any difference between the air temperature and the temperature of the workpiece. Should the exhaust not be in a lateral position, before beginning drilling the drill (if the air turbine type) should be made to run for a period of time and then stopped while monitoring the strain gauge outputs. If any effect of the turbine exhaust on the strain gauge measurements is noted, it may be necessary to allow time for the strain gauge readings to stabilize Heat generated during the drilling process Heat is generated during the drilling process and this causes localized heating of the strain gauge area. AIAS TR-0:00 Page 30 of 70

31 It is therefore necessary to wait for undesired thermal strains to gradually reduce and for gauge output to stabilize. This is particularly important for materials with poor thermal conductivity. The delay time before acquisition of strain measurements depends on the material, the shape of the workpiece, and the ambient temperature. Standard ASTM E prescribes waiting at least 5 seconds between the end of drilling and reading strain gauge output to allow the surface to cool. The cutter need not be retracted. In practice, the signal stabilization time depends on the thermal conductivity and thickness of the material. In metal materials stabilization occurs in 3 to 0 seconds. 6.6 Measurement of hole dimensions and eccentricity. After completion of the hole-drilling process it is necessary to measure all the geometric characteristics of the drilled hole. Figure 9 - Measurement of hole diameter and eccentricity The diameter and eccentricity are measured starting with measurement of four radiuses of the hole in two directions perpendicular to each other. An optical microscope incorporating a graticule may be of aid for obtaining an enlarged image of the drilled hole boundary, utilizing visible light. The overall hole shape should be analysed to check for any irregularities. It is usual for the cutting edge of the gauge backing material to be irregular. Displacement is measured at these four positions with two dial indicators (if possible with graduations of 0.00 mm) (Figure 9). The four displacement measurements (X,X,Y,Y ) are then used to calculate the measurement of the hole diameter (average value in the two directions) and eccentricity as indicated herebelow (Figure 0) Indeed, the hole diameter in the two orthogonal directions (X and Y) and the average diameter have the following definitions: D X D Y ( X X ) ( Y Y ) ( ) ( ) D 0, M ( D X DY ) ( 3 ) AIAS TR-0:00 Page 3 of 70

32 Figure 0 - Off-centre hole, parameters necessary for calculating hole-rosette eccentricity while hole eccentricity and orientation are: e X e Y e ( X ) X ( Y ) Y e X e Y and the eccentric angle is expressed as: e arctan e Y X 80 ( 4 ) ( 5 ) ( 6 ) ( 7 ) 6.7 Final hole depth measurement check After removing the strain gauge, the final hole depth can be measured using a conventional depth gauge. A depth measuring instrument like the one shown in Figure can be used. Figure - Instrument for measuring hole depth AIAS TR-0:00 Page 3 of 70

33 Any difference from the expected hole depth (recorded during drilling by the micrometer gauge of the drilling apparatus) should be taken into consideration. Cutter wear, the grip between the tool holder and cutter shank, and inadequate stiffness between the component and the drilling apparatus can all contribute to hole depth errors. 6.8 Practical example of application Automatic residual stress measurement systems are generally used having the advantage of enabling numerous depth increments to be drilled with adequate accuracy. Briefly summarized, the measurement procedure involves the following steps: installation of the strain gauge rosette and wiring of the gauge grids, connection to the strain recording instrumentation, positioning of the measurement system, centring of the drilling tool over the centre of the rosette (aligned with the microscope), manual advancement of the cutter to the surface of the workpiece using the fast vertical advance, set-up of the test parameters. For example: o Hole depth:.0 mm, o Number of drilling increments: 40, o Hole drilling curve: linear. An automatic procedure makes it possible to: start the high-speed turbine by acting on the air supply system, determine the initial drilling point (identification of the zero reference surface) by an electrical contact that is made with removal of the strain gauge backing film and bringing the endmill into contact with the metal surface, zero-balance the strain gauge circuits by a command to the strain recording system. The automatic system drills the hole automatically in the set depth increments. On completion of each depth increment and the time interval, the system records the three strain gauge readings. Hole-drilling procedure example: tool:.6 mm. diameter, inverted cone, surface-treated, tungsten carbide endmill, speed of rotation (typical): from 350,000 to 400,000 rpm, feed rate: 0.mm/min, depth increment: 0.05 mm, delay time: 5 seconds. Typical results are presented in section 8 (Residual stress analysis software features) AIAS TR-0:00 Page 33 of 70

34 7 Residual stress analysis techniques The following analysis techniques are presented in this section: methods proposed by standard ASTM E (uniform and non-uniform stresses), other methods for analysing non-uniform stresses within the hole depth. Treatment of uniform stresses substantially follows the recommendations of ASTM E837-08, making the following distinctions: thin workpiece, thick workpiece, workpiece with a thickness between 0.4 D and. D (0.48 D and.44 D for type C rosettes). With respect to the prescriptions of ASTM E837-08, in addition to considering the case of rosettes which are not necessarily the types A, B or C contemplated by the standard, the procedures are provided here for plasticity correction and hole-rosette eccentricity correction. Although a uniform stress profile is rare in reality, it is useful for study purposes and for evaluating uncertainties in the measurement procedure. Other methods are also proposed for analysis of non-uniform stresses, for example, the integral method [, 5, 6], the Schwarz Kochelmann method [7] and lastly the HDM method [8, 9, 30, 3, 3, 33, 34, 35]. The methods are made more generally suitable considering: the possibility of varying both the depth between two successive drilling increments and the number of measured acquisition points (Integral, Schwarz-Kochelmann, HDM), the possibility of having an eccentric hole, a centred hole being a particular case (HDM), the use of improved influence functions using a more accurate database and a better representation of influence functions versus Poisson s ratio (HDM), the possibility of describing the state of stress with various functions (HDM). Feature Stress State / Methods of calculation Uniform stresses Non-uniform stresses Blind hole in a Through-hole plate of Intermediate Thickness thick plate ASTM E ASTM E Mod. Integral Schwarz - Kochelmann HDM Drilling spacing and depths No No No No No Yes Yes Yes Optimization of calculation steps No No No No No No No Yes Type of rosettes Yes (*) Yes (*) Yes (*) Yes (*) Yes (*) Yes (*) HBM rosette only Yes Eccentricity Correction Yes (Ajovalasit [38]) Yes (**) (Ajovalasit [38]) Yes (**) (Ajovalasit [38]) No Yes (HDM) Yes (HDM) No Yes Plasticity Correction Yes (***) (Beghini-Bertini [3,47]) Yes (***) (Beghini- Bertini [3,47]) Yes (***) (Beghini- Bertini [3,47]) No No No No No (*) Any type of rosette can be used: however, the new coefficients need to be calculated. (**) In the case of blind hole, the correction is indicative. (***) The accuracy of the correction depends on the stress state and on the type of rosette used in the test. Table 5 - Residual stress calculation methods: principal features. AIAS TR-0:00 Page 34 of 70

35 Table 5 summarizes the techniques and major features of the residual stress analysis methods. The main corrections that can be applied to the results are also indicated. 7. Standard ASTM E837-08: general The ASTM E standard is the procedure that can be used for measuring residual stresses in homogeneous isotropic linear-elastic materials. Application of this test method is limited to low levels of eccentricity. The standard allows residual stresses to be calculated directly when using the rosettes specified in the standard (A, B and C). Nevertheless, it is possible to extend the standard by recalculating the coefficients for other rosettes. The standard provides accurate results if: the equi-biaxial component of the residual stresses is less than 50% of the yield stress, shear stresses in any direction are less than 5% of the yield stress. However, in practice satisfactory results are achieved providing the residual stresses do not exceed 60% of the material yield stress. 7.. Strain gauge rosettes Figure 4 shows the geometry of the strain gauge rosette and the preferred numbering for the direction of the principal stresses. The centres of the three radially oriented gauges are D/ from the gauge target and the centre of the hole. Although, in theory, the angles between the strain gauges can be chosen arbitrarily, the majority of commercially available rosettes are rectangular with gauges oriented at 0 /45 /90. The types of strain gauge rosette standardized by ASTM E are presented in Table. In the ASTM type A rosette design (gauges in two quadrants, ie, at 0 /5 /90 ), gauge (or b) has been transposed to be diametrically opposite its original position to give more sampling about the hole position and a larger grid size. The type B rosette has all three gauges in a single quadrant, ie, at 0 /45 /90, to allow the gauge to be used closer to obstructions such as corners or welds. The ASTM type C rosette has a circular configuration and is formed of six diametrically opposed circumferential and radial grids. Compared to the other rosettes, this design provides greater sensitivity and accuracy. 7.. Strain relief in proximity to the hole Considering the state of uniform stress in proximity to the hole, schematically illustrated in Figure surface strain relief is tied to residual stresses x, y, xy by the following relationship: x y x y r a b cos xy E E E b sin The two calibration constants a and b are dimensionless, almost independent of the properties of the material, and vary with hole depth, as indicated in Table. In the case of a through-hole in a thin workpiece, a is independent of the Poisson s ratio. Whereas, considering the case of non-uniform stresses within depth, the surface strain relief associated with the hole depth step j ( k j ) is tied to the relieved principal stresses by the following relationship: ( 8 ) AIAS TR-0:00 Page 35 of 70

36 j j j x y x y j a jk b jk cos jk xy k E k E k E k k k b ( ) sin ( 9 ) The two calibration constants a jk and b jk indicate the strains relieved by the drilling process at the depth associated with hole step j Numerical values of a and b Table 6 shows the numerical values of a and b for a blind hole and through hole, for the type A, B and C rosettes specified by the ASTM E standard. Table indicates the dimensions provided in standard ASTM E for type A, B and C rosettes. Finally, Table 4 gives the hole diameters and depth steps recommended for each rosette. Table 6 - Numerical values of coefficients a and b provided by standard ASTM E for type A, B and C rosettes for uniform stress evaluations with through holes and blind holes Sensitivity of the method The strain relieved by hole-drilling is a fraction of the maximum relievable strain. For a uniaxial tensile stress field (ε /ε = ε x/ε y = 0) relieved strain in direction β=0 is obtained on the basis of x x r a b a b E E E E ( 0 ) AIAS TR-0:00 Page 36 of 70

37 Whereas the maximum relievable strain is: 0 ( ) E Assuming as the sensitivity merit index the equation: S r 0 it follows that: a E S E E b A 0 B 0 ( ) ( 3 ) where the values of A 0 and B 0 are to be found in literature [37] ]: A 0 D0 G R R W ( 4 ) B D 0 D0 4 sin sin sin cos GW R R R 0 D0 R sin cos where: ( 5 ) d arctan ( 6 ) R d arctan ( 7 ) R To increase strain relieving efficiency it is necessary to [37] ]: adopt high values for (D 0/)/R, that is, drill holes with the biggest diameter possible, compatibly with the need to avoid parasitic effects on the inner edge of the strain gauge (paragraph 6.7), use rosettes with a short gauge length (low R /R values), use rosettes with reduced grid width (low GW/R values), have the usual S values around 0.3. Special rosettes are also available (ASTM Type C) with six grids, three of which are radial and three circumferential, which are wired in a half-bridge configuration (using a radial grid and the diametrically opposed circumferential grid). This achieves a sensitivity equal to.3 times the sensitivity of the corresponding ASTM standard Type A or B. AIAS TR-0:00 Page 37 of 70

38 7. Standard ASTM E837-08: calculation of residual stresses 7.. Thin workpiece For a thin workpiece or a through hole (thicknesss 0.4 D plane stress state conditions) the stresses are considered uniform in the depth direction and only a single reading of strains ε, ε and ε 3 is required. The following quantities can be calculated from the measured strains: 3 3 p q 3 t where p is the hydrostatic strain component and q and t are the shear strain components. ( 8 ) The stress components P, Q and T are calculated from p, q and t with the following equations: x y Ep y P x Eq Et Q T xy a ( ) b b ( 9 ) Finally, the principal stresses are calculated using: MAX, MIN P Q T ( 0 ) The angle β, which the maximum principal stress σ max forms with the direction of strain gauge, (measured clockwise for the CW rosettes and counterclockwise for CCW rosettes), is calculated with the following equation: T arctan Q ( ) The direction of the angle is defined by Table 7, dependent on signs T and Q. Q>0 Q=0 Q<0 T<0 45 < β <90 45 T=0 90 undefined 0 T> Table 7 - Convention used for placement of angle β (ASTM E837-08). 7.. Thick workpiece To calculate the values and directions of the principal stresses for a thick workpiece (thickness. D), it is recommended that eight sets of strain readings ε, ε and ε 3 are measured as the hole depth is increased in increments of 0.05 D with type A and B rosettes and 0.06 D with type C rosettes. The sequence of the relieved strains (eight ε, ε and ε 3 strain values) and the corresponding hole depths are recorded. The combination strains p, q and t are calculated for each hole depth from the measured relieved strains ε, ε and ε 3 using the equations ( ): p 3 q 3 The procedure continues with: t 3 verification of residual stress field uniformity (section 7..3), calculating the residual stresses (section 7..4 and 7..5). ( 3 ) AIAS TR-0:00 Page 38 of 70

39 7..3 Residual stress uniformity test In the case of a thick workpiece it is necessary to verify that the residual stresses are uniform within the hole depth. This involves: identifying the set of combination strains q or t that contains larger absolute values, expressing each set of combinations strains (p and the larger of q and t) as a percentage of their values at the hole depth corresponding to 0.4 D, plotting these percent strains versus hole depth (D0/D). These graphs should yield data points very close to the curves shown in Figure 3. Data points that are separated from the curves in Figure 3 by more than ±3% indicate either substantial stress non-uniformity through the material thickness, or strain measurement errors. Figure - Graphical test of through-thickness stress uniformity (ASTM E837-08) In either case, the measured data are not acceptable for the residual stress calculations described in the ASTM E standard. This graphical test is not a sensitive indicator of stress field uniformity. Workpieces with significantly non-uniform stress fields can yield percentage relieved strain curves substantially similar to those shown in Figure. The main purpose of the test is to identify grossly non-uniform stress fields and strain measurement errors. This stress uniformity test may be applied only to thick workpieces Calculation of uniform residual stresses When working with thick workpieces, all eight sets of ε, ε, ε 3 measurements are used for calculating the magnitude and direction of the principal stresses. For each of the hole depths corresponding to the eight sets of ε, ε, ε 3, measurements, the numerical values of the calibration constants a and b, corresponding to the hole depth and diameter, and the type of rosette used, are determined using Table 5. The three combination stresses P, Q and T, corresponding to the three sets of combination strains p, q and t are calculated using the following formulas: AIAS TR-0:00 Page 39 of 70

40 P E a p a Q b q E b T b t E b where the summation is of the indicated values for the eight hole depths. Angle β is calculated with the formula: T arctan Q arctan bt bq ( 4 ) ( 5 ) The measurement direction for the angle is determined referring to Table 7, dependent on the signs of T and Q. The principal stresses are calculated using: max, min P Q T ( 6 ) The basic calculations described above are not to be used for measuring residual stresses in non-uniform stress fields Calculation of non-uniform residual stresses Should the uniformity test referred to in section 7..3 prove negative and stresses be found to be non-uniform within hole depth, ASTM E provides a procedure for calculating the residual stress profile of the workpiece whereby a residual stress value is associated with each hole depth. For each hole depth step j, it is possible to calculate: p j ( ) 3 j q j ( ) 3 j t j ( 3 ) j also estimating the standard error in the combinations strains with the equations: p q STD STD n 3 j n 3 j ( p ( q j j 3 p 3 q j j 3 p j 0( n 3) 3 q j 0( n 3) q p j3 ) j3 ) ( 7 ) ( 8 ) n 3 ( t j 3 t j 3 t j t j3) t STD j 0( n 3) where the symbol j refers to the serial numbers of the hole depth steps corresponding to the successive sets of measured strains, and n is the number of sets of strain data at the various hole depth steps. The summation is carried out over the range j n 3. The calibration matrices a jk and b jk for the three types of rosettes prescribed by the standard are given in Table 8, Table 9 and Table 0. The hole-drilling test consists in drilling 0 equal hole depth steps for type A and B rosettes; 5 for type C rosettes. AIAS TR-0:00 Page 40 of 70

41 Table 8 - Coefficients a and b for type A rosettes for non-uniform residual stress evaluations (ASTM E837-08). The tabulated numbers refer to a /6 inch (5.3 mm) nominal size rosette: if a /3 inch (.56 mm) rosette is used, all hole and stress depths in the tables should be multiplied by 0.5; if a /8 AIAS TR-0:00 Page 4 of 70

42 in. (0.6 mm) rosette is used, they should be multiplied by. Since the tabulated numbers refer to a nominal hole diameter of mm, the numbers have to be adjusted once the actual hole diameter is measured and be multiplied by the following corrective factor: (actual diameter/ nominal diameter). Table 9 - Coefficients a and b for type B rosettes for non-uniform residual stress evaluations (ASTM E837-08). AIAS TR-0:00 Page 4 of 70

43 Table 0 - Coefficients a and b for type C rosettes for non-uniform residual stress evaluations (ASTM E837-08). AIAS TR-0:00 Page 43 of 70

44 AIAS TR-0:00 Page 44 of 70 The depth increments should be 0.05 mm for.56 mm diameter rosettes; 0.05 mm for 5.3 mm diameter rosettes and, finally, 0.0 mm for 0.6 mm diameter rosettes. The residual stresses are calculated for each hole depth step j by solving the following matrix equations: p E P a q E Q b t E T b ( 9 ) in which: ] ) ( ) [( k x k y P k ] ) ( ) [( k x k y Q k k xy k T ) ( ( 30 ) When a large number of hole depth steps are used, the matrices a and b are numerically illconditioned: small errors in the input data lead to large errors in output. The results are filtered by Tikhonov regularization to reduce this effect. Using a regularization matrix of the type: c ( 3 ) and applying the Tikhonov regularization, the equations for calculating residual stresses are: p a E P c c a a T T P T ) ( q b E Q c c b b T T Q T ) ( ( 3 ) t b E T c c b b T T T T ) ( The factors α P,α Q and α T control the amount of regularization that is used: an initial value for these factors may be chosen in the range 0-4 to 0-6. On account of the regularization used, the unregularized strains that correspond to the calculated stresses (P, Q and T) do not exactly correspond to the actual strains p, q and t. The difference in terms of strain is indicated by the misfit vectors, calculated as : ap E p p misfit b Q E q q misfit b T E t t misfit ( 33 ) Parameters p rms, q rms and t rms are defined as the mean squares of the misfit vectors at the various depths: n j j rms p misfit n p ) ( n j j rms q misfit n q ) ( n j j rms t misfit n t ) ( ( 34 ) If the values of p rms, q rms and t rms differ 5% from the values of p std, q std e t std regularization values α P,α Q and α T need to be modified and recalculated through an iterative process. The new values of α P, α Q and α T will be: old P rms std new P p p ) ( ) (

45 q std ( Q ) new qrms std ( T ) new trms t ( ) T Q ( ) old old ( 35 ) If the difference is within 5% the maximum and minimum residual stress values are calculated for every hole depth step by the following equations: ( MAX ) k,( MIN ) k Pk Qk Tk Tk k arctan Qk ( 36 ) ( 37 ) 7..6 Intermediate thickness workpiece In the intermediate case of a workpiece with a thickness between 0.4 D and. D the ASTM E test method provides an approximate result. An approximate result can be obtained in such a case by using the calculation for a through hole and interpolating the blind hole and through-the-thickness hole calibration data given in Table 6. Residual stress results obtained in this way should be reported as nonstandard and approximate. 7.3 Calculation of non-uniform residual stresses. Other methods In addition to the ASTM standard test method, other methods for evaluating non-uniform stress are possible. The Integral method, the Schwarz-Kochelmann method and the Hole-Drilling method are summarized below Integral Method The integral method for analysing non-uniform residual stresses was proposed by G. S. Schajer in 988 [, 5, 6] in order to overcome the limits of the procedure given in ASTM E837 for evaluating uniform residual stress fields. In this method values p, q and t are calculated for every hole depth; the stress profile can be obtained by solving the three integral equations indicated below: p( hi ) E h i 0 Aˆ( H, h ) P( H ) dh i q( h ) i t( h ) i E E h i h i 0 0 Bˆ( H, h ) Q( H ) dh i Bˆ( H, h ) T( H ) dh i ( 38 ) where  and Bˆ are the influence functions for a hydrostatic stress state and a shear stress state, respectively, and take account of the effect of the relaxed stresses at depth H for a measurement depth h. In order to simplify the problem of residual stress evaluation, Schajer proposed that the stress field can be determined by means of functions defined in intervals with constant values in each AIAS TR-0:00 Page 45 of 70

46 depth step considered: with this procedure the integral equations seen above can easily be evaluated, provided the influence functions can be calculated for each calculation step. If this can be done, the equations shown above can be expressed in discrete form as: i i i E pi ai, j Pj E qi bi, jq j E ti bi, jt j j j j ( 39 ) where n indicates the hole depth step considered and a i, j and b i, j indicate the relieved strains due to unit stresses P, Q and T at depth j for hole depth step i. The a, coefficients are related to the functions A ˆ( H, h) as follows: i j Hi Hi ai j Aˆ, H, hi dh ( 40 ) Discrete formulation of the problem therefore implies solution of a linear system with a lower triangular matrix of coefficients. Figure 4 - Drilling depths: physical interpretation of coefficients a,. i j With the aid of a finite element calculation, coefficients calculating the following functions A H h, h Aˆ H, 0 h i dh a, have been determined by i j ( 4 ) by which coefficients a, are evaluated as: i j H h AH h a, i, j A j, i j i ( 4 ) Functions A and B have been provided for ratios D 0/D equal to 0.3, 0.4 and 0.5 for calculation depth h between 0.05 and The coefficients are obtained by interpolation for different D 0/D ratio and calculation depth h values. The values of the coefficients proposed by G. S. Schajer for calculating residual stresses by the integral method are indicated in Table. AIAS TR-0:00 Page 46 of 70

47 Table - Coefficients a and b of the integral method for type A, B and C rosettes Integral Method calculation steps To calculate the residual stresses from the relaxed strain, the following steps are necessary: The hole should be produced in many small drilling increments so that the resulting strain data can be smoothed to reduce noise. At a smaller number of calculation increments, combination strains p, q and t are calculated from the smoothed strain data. Cumulative strain relaxation functions ( A and B ) are calculated (for the measured hole diameters), by interpolation, from the values of the coefficients of the triangular matrices provided. Coefficients a i, j and b i, j are calculated directly by subtraction of adjacent elements in the cumulative strain function matrices. Stresses P, Q and T are calculated for successive increments using the relationships: i i i E pi ai, j Pj E qi bi, jq j E ti bi, jt j j j j ( 43 ) Residual stresses and residual stress orientation (σ max, σ min, β), for each calculation increment, are obtained from the corresponding combination stresses. AIAS TR-0:00 Page 47 of 70

48 7.3. Incremental strain method (also known as the Schwarz Kochelmann method) The incremental strain method, proposed by T.Schwarz and H.Kochelmann [7] in 993, is based on measurement of the strain rate during the drilling operation (Figure 3). Figure 3 - Schwarz Kochelmann method. The method involves a preliminary stage of experimental/numerical determination of the relaxation functions defined as: ( 44 ) ( 45 ) where x and y are the strains measured respectively by the strain gauge grids oriented parallel to the loading direction and perpendicular to the loading direction in the case of uniaxial loading. Figure 4 shows relaxation functions K x and K y calculated for an HBM rosette, type -Y6-.5/0S for d m/d 0=3. AIAS TR-0:00 Page 48 of 70

49 Figure 4 - On the right, calibration functions Kx and Ky for the HBM rosette shown on the left. After the relaxation functions have been defined, the stress field can be calculated applying the following formulas: ( 46 ) ( 47 ) The principal stress values can be calculated by the following equation: ( 48 ) ( 49 ) ( 50 ) This method can be applied only for HBM rosettes as the numerical/experimental values of functions K x and K y have been calculated only for this type of rosette. Although the residual stress results obtained with this method may agree with those evaluated with the integral method, it must be pointed out that the method is approximate because it does not take account of the change in hole geometry with depth (but only of the residual stress in the removed stratum of material) HDM Method The HDM Method [8, 9, 30, 3, 3, 33, 34, 35] was originally proposed by the University of Pisa in the nineties as an improvement of the integral method. It is based on three equations proposed by G. S. Schajer (43) and it has been generalized by analytical definition of the influence functions A ˆ( H, h) and B ˆ( H, h). The main advantages of the hole-drilling method over the other methods are: a parametric description of the strain gauge rosette which eliminates dependence on the model of rosette used, AIAS TR-0:00 Page 49 of 70

50 a greater accuracy in calculation of the influence coefficients in relation to the values provided by G. S. Schajer (in the case of stress evaluation by the integral method), if the hole depths do not match those provided in his study, the possibility of correcting the effect of hole-rosette eccentricity a more accurate FEM model for definition of the influence functions, the possibility of describing the state of stress within a component with several functions, in order to reconstruct the experimental measurements more accurately by using spline functions, directly inserted into the integral equations, in addition to power series and Fourier series (basic functions), availability of an optimization algorithm which takes account also of the effects of experimental measurement errors, the possibility of minimizing the effect of experimental measurement errors by solutions which are no longer direct but based on statistical instruments, such as the least squares method, introducing more experimental measurement points, within each stress calculation. The hole-drilling method makes it possible to chose from five possible functions that represent the state of stress: piecewise constant splines, linear spline, cubic spline, power series, Fourier series Non-uniform residual stresses with an off-centre hole The HDM method was subsequently extended to the off-centre hole by the same authors, with the contribution of W. Rosellini, [33, 34, 35]: if an off-centre hole is to be considered, the equations used in the ASTM method and in the integral method are no longer applicable as the symmetry of the problem is lost. A more general formulation is therefore necessary. Figure 5 -. Symbols used in the HDM method. AIAS TR-0:00 Page 50 of 70

51 Observing Figure 5 and considering an isotropic linearly elastic homogeneous material, the relationship between measured strain and related stresses can be described as follows: ( 5 ) where the influence functions A A 9 depend on the properties of the material, hole depth and eccentricity and rosette geometry. Each of these influence functions can be described with a double power series, the coefficients of which have been calculated by a finite-element regression analysis of surface displacements for every particular configuration of the problem under examination (Poisson s ratio, ratio between the hole radius and rosette mean radius, etc). Knowing the form of the functions and the relieved strains, it is possible to solve the system of equations seen above by an inverse formulation, in order to determine the state of residual stress existing in the component. Supposing that each stress component may be described with a series of functions, the following expressions can be obtained: where: J, J and J 3 are the degrees of freeedom of the stress field, ( 5 ) represent the functions used to describe the stress state, are constant coefficients determined by the least squares method for best reconstructing the experimental strain measurements. By combining the two systems of equations, it is possible to obtain the following relationship: where i =, n = number of hole-drilling steps. ( 53 ) Knowing the form of the influence functions, the integrals in the equations can easily be analytically solved, and therefore the whole relationship is reduced to a linear system of 3n equations in which J +J +J 3 are unknowns, that can be simplified to: ( 54 ) This system can be solved directly when 3n=J +J +J 3, and with the least squares method when 3n> J +J +J 3. The latter analysis technique is better because it reduces the influence of random experimental errors. AIAS TR-0:00 Page 5 of 70

52 Nevertheless, the liberty granted in selecting the ψ j functions used to describe the state of residual stress implies introduction of some new parameters, the definition of which influences the accuracy of the result (for example, in the case of the power series, the maximum superscript at which to stop expansion). To obtain the optimum solution with the hole-drilling method, such parameters can be selected by automatic methods, for example, by genetic algorithms [36]. The choice of these parameters is very important as it actually corresponds to the level of flexibility that is given to the representation of the state of stress, and therefore, to the ability to reproduce even highly complex stress functions. A low level of flexibility can lead to approximate solutions whereas too high a flexibility can cause excessive sensitivity to measurement error, which is always present in acquired strain data. The criterion of optimization follows the principle that the strains obtained from calculated residual stresses have to reconstruct the acquired state of strain with the same accuracy of the method. Any inferior accuracy leads to an approximate solution, whereas a higher accuracy results in reproducing the measurement errors that are normally associated with a high instability of the results. For this reason, the following objective function is defined: ( 55 ) where σ exp is the estimated standard deviation of the measurement error (which may be experimentally calculated from acquired strain data), and ~ is the standard deviation of the error between acquired strain and strain obtained from the calculated residual stress state. To find the minima of that function, the spline methods use a genetic algorithm, whereas serial methods employ an exhaustive algorithm: the genetic optimization algorithm makes it possible to position base points randomly in the interval (0, z max), where z max is the maximum hole depth, collapses neighbouring base points if their distance is less than a threshold (the threshold is set at 5% of z max) and then identifies the best arrangement and the best number of base points with a process of evolution of the solution typical of genetic algorithms, until the condition ξ<0.0 is verified, the exhaustive algorithm identifies the solution with the least value of ξ, within the domain of all possible solutions, compatibly with the number of measurements made (for which the condition 3n> J +J +J 3 is verified). 7.4 Correction for plasticity (elastic relaxation of stresses) The local plastic deformation which occurs around the drilled hole can introduce significant errors in the calculation of residual stresses: standard ASTM E actually specifies a limit to the maximum measureable residual stress of about 50% of the yield stress of the material. The University of Pisa has developed a procedure [3, 47, 49] for correcting this effect and HBM has produced a special 4-element rosette (Figure 7) to overcome the limitations set by the correction procedure. There are two approaches for correcting the effect of plastic deformation: To use a standard 3-element rosette. The two perpendicular elements should be oriented in the principal directions. To use a special rosette with 4 elements at 0/90/57,5/5 (for example HBM -VY6-.5/0S. Figure 8): the fourth grid allows the principal directions to be calculated approximating the stress profile with the use of a fourth-order Fourier series development. AIAS TR-0:00 Page 5 of 70

53 7.4. Correction with a 3-element rosette The hole-drilling method in accordance with standard ASTM E poses a limit related to the maximum measureable residual stress: for the relationships between strains and stresses defined by the standard to be valid, the measured stress components should not exceed about 50% of the yield stress of the material, in absolute value. The local plastic deformation which occurs around the drilled hole, must be considered in order to determine correct residual stress values. In the case of uniform stress through the thickness of a workpiece, it is possible to correct the residual stress values and to estimate the actual value of residual stresses in a component [3,47] In recent studies [49] a more accurate FE elastic plastic parametric analysis has been developed and a more accurate, and general, correcting procedure is provided which can be applied to all the rosettes available on the market. Figure 6- Assumed material constitutive law: bilinear isotropic hardening To correct stresses for the effect of local yielding it is necessary to know the yield stress and strain hardening ratio of the material (R) (Figure 6): E plastic R (56) E E plastic indicates the ratio between deformation and displacement in the plastic field. The von Mises equivalent stresses eq x y x y was assumed to quantify the effect of biaxiality and a dimensionless plasticity factor f is introduced: f eq Y eq, i eq, i (57) where σ eq,i is the equivalent residual stress producing the onset of plasticity in the D case, and σ Y is the material yield stress. The condition of f=0 represents the highest residual stress that still does not produce plasticity, while f= is related to the residual stress producing general yielding in the whole body. The plasticity factor measures the residual stress intensity with respect to the approximate onset of plasticity given by the plane Kirsch solution [5]. For the correction algorithm, it is AIAS TR-0:00 Page 53 of 70

54 necessary to consider the biaxial stress ratio Ω. The ratio between the measured relaxed strains along the principal directions x/y depends on the stress ratio Ω but it is almost unaffected by the plasticity factor, as shown in Figure 7. As a consequence, the biaxiality ratio Ω can be approximated by the ratio between the elastically calculated residual stress components σ x, el, σ y, el y W x W el y, el x, el (58) Figure 7- Ratio between the measured relaxed strains versus plasticity factor The equivalent residual stress at the plasticity onset can be expressed as a function of the biaxiality ratio Ω, according to the plane stress Kirsch solution [5]. W W eq, i (59) Y 3 W The acquired strain can be used to obtain the as elastically-evaluated residual stresses and for the elastically-evaluated equivalent stress definition: (60) eq, el x, el y, el x, el y, el The related elastically-evaluated plasticity factor is obtained by the equation: eq, el eq, i f el (6) Y eq, i If a significant plasticity is produced, the elastically-calculated plasticity factor is larger than the actual plasticity factor. As the plasticity is not expected to play a significant role for a plasticity factor near 0, it follows that f el<f, when f. The following function: f el f Wf (6) was found accurately to fit the relationship between f and f el for any considered configuration. It can be observed that the asymptotic behaviour of the function at low f values is fulfilled by the proposed expression for any values of the parameters W and μ. The parameters W and μ were found for any analysed material and geometrical configuration by means of a least-squares fitting, which was found to produce excellent results. AIAS TR-0:00 Page 54 of 70

55 The parameters W and μ depend on the biaxial ratio Ω, the hardening ratio of the material R, the hole depth Z, the hole diameter D 0 and the strain gage average diameter. The procedure is summarized in the following steps: The strain gauge rosette is applied to the surface of the body affected by residual stress with the and 3 grids aligned with the known (or at least assumed) principal residual stress directions and the signals are set to 0. The relaxed strains,, 3, are measured after the hole, the x axis is chosen parallel to the grid measuring the maximum absolute value; therefore, if > 3, then x= and y=3, otherwise x=3 and y=. The elastically-evaluated residual stresses σ x,el, σ y,el are determined. The elastically-calculated biaxiality ratio Ω el is assumed to be an accurate approximation of the actual residual stress biaxiality ratio Ω The elastically-evaluated plasticity factor f el can be obtained with the equivalent residual stress at the yield onset σ eq,ei If f el=0, no correction is required and the elastically-evaluated residual stresses are assumed as the actual residual stresses; otherwise, the correction is needed Calculation of the parameters W and μ Calculation of the actual f from f el by inverting equation (6): f f Wf el To solve this unelementary equation, the Newton Raphson algorithm is recommended which gives an accurate numerical approximation of the plasticity factor, hereafter called fˆ Estimation of the equivalent residual stress ˆ eq obtained from the plasticity factor fˆ ˆ eq eq i fˆ, Y eq, i (63) Calculation of the principal residual stress components ˆ x, ˆ, x eq W W ˆ y obtained from ˆ eq : ˆ W ˆ (64) y x 7.4. Correction with a special 4-element rosette As previously observed, a limitation in this procedure is that of requiring the relaxed strains along the principal directions x and y. This implies prior knowledge of the principal directions of the residual stress distribution, which can actually often be deduced from the technological history and background of the component. Should the principal directions not be known, the procedure could still be used managing to deduce the relaxed strains that would be read if the grids were oriented along the principal directions of stress, starting with the relaxed strains with a generic rosette orientation. This objective can be achieved in part by using a 4-element rosette developed specifically for addressing the effect of plasticity (Figure 8). Assuming the principal directions are unknown, there are four strain readings in four directions: a = ( = 0 ); b = ( = 90 ); c = ( = 80 - :5 ); d = ( = ) (65) while the four values to be obtained are,, and φ according to the formula: ' 0 ' ) cos4( ) ' ' ' ( ) 0 cos ( 4 (66) Inversion of the calculation is not linear; the principal angle is calculated by solving a least squares minimisation problem. Figure 9a shows the determination of the principal angle and Figure 9b shows the reconstruction of measured strain versus angle. ' 4 AIAS TR-0:00 Page 55 of 70

56 Figure 8.HBM 4-element Rosette 0/90/57,5/5 (Left), Angles between gauges (Right) Unknown dependance Simulated readings Re-evaluated dependence Figure 9: (a) Principal Angle (least squares minimisation); (b) Reconstruction of measured strain versus angle. 7.5 Correction for eccentricity Hole-rosette eccentricity influences strain measurement, particularly when small rosettes are used, making the formulas developed for the centre-hole method of measuring residual stresses less accurate. When dealing with small grid dimensions, it is not always possible to ensure the concentricity of the rosette and hole centres. Nevertheless, this operation should always be carried out as accurately as possible. Standard ASTM E presents limitations in the field of application related to the maximum permitted hole eccentricity. It considers a maximum acceptable eccentricity of 0.04*D (for example 0.0 mm for a 5.3 mm diam. rosette). Figure 30 shows the effect of the hole eccentricity on the measured strains. Reference may be made to the studies of A. Ajovalasit [38] for formulas which allow the determination of residual stresses even in the presence of misalignment between the hole and rosette centres in the case of a through hole and to the studies of J.P.Sandifer and G.E.Bowie for a blind hole method with an off-centre hole [39]. In both studies strains measured in the presence of eccentricity are corrected to zero eccentricity: in A. Ajovalasit s publication the correction is applied to the values of calibration constants A and B (and therefore to a and b) presented in the ASTM standard for the through hole case; whereas, in J.P.Sandifer s publication the correction is made on the measured strains and is applied to blind holes. AIAS TR-0:00 Page 56 of 70

57 Strain [µm/m] Strain [µm/m] The hole-drilling strain gauge method for the measurement of uniform or non-uniform residual stresses Grid A Grid B Grid C Depth [mm] Grid A Grid B Grid C Depth [mm] Figure 30 Equi-biaxial Stress Field: difference between the values of strain measured in the absence (above) and presence (bottom) of eccentricity (e=0. mm) 7.5. Correction for eccentricity: through hole In the case of an eccentric hole in relation to the centre O of the rosette, the relationship between strain measured by the strain gauge and residual stresses is of the type: o o o ' A B C i i i i ( ) cos sin i i ( ), ( i a, b, c) E E E (67) where the coefficients A i, B i and C i depend not only on the radius of the hole and geometry of the grid but also on the orientation of the grid and eccentricity (Figure 3). Figure 3 - Notations relating to a rosette with an off-centre hole For the through hole, there are theoretical expressions of A i, B i and C i that take account of the active grid length of the strain gauge [38]. In the presence of misalignment between the hole and rosette centres, residual stresses can be determined using the standard procedure for a rosette with a centered hole. To do so, it is sufficient to calculate the strains reduced to zero eccentricity, ie, the strains that the strain gauges would measure if the rosette were centered, by formulas of the type: AIAS TR-0:00 Page 57 of 70

58 J ' J ' J ' a aa a ab b ac c J ' J ' J ' b ba a bb b bc c J ' J ' J ' c ca a cb b cc c where the expressions of coefficient J are to be found in [38]. Residual stresses and stress orientation are calculated by introducing strains a, b, c in the formulas used for a centered rosette. If eccentricity between the hole and rosette is neglected, the principal stresses σ and σ are affected by the following errors: ' E 00 (7) ' E 00 (7) where σ' and σ' are the calculated principal stresses, that is, without taking account of eccentricity; σ and σ are the actual principal stresses, that is, calculated taking account of eccentricity. Only the error relating to the greater stress in absolute value in the most unfavourable conditions of γ (orientaton of the greater principal strain) and β (orientaton of the eccentricity) is considered. For a given relative eccentricity e'%= (e/r )%, and with other conditions being the same, 0 /45 /90 rosettes are more unfavourable than 0 /5 /90 rosettes. In the most unfavourable conditions of σ /σ (σ /σ =-) the errors relating to the type A and B rosettes considered in standard ASTM E837, calculated for the minimum hole diameter are: ' ASTM type A rosettes (0 /5 /90 ): E 5,6 % (73) e ' ASTM type B rosettes (0 /45 /90 ): E 8, % (74) e The formulas stated above therefore make it possible to determine the upper error limit, in the most unfavourable conditions of γ, β and σ /σ ASTM type of rosette Type A -/3 0 /5 /90 Type A -/6 0 /5 /90 Type B-/6 0 /45 /90 Average diam. of the rosette D=r (mm) Distance of the inner edge of the grid from the centre of the rosette r (mm) Hole-centre eccentricity e (mm) Per cent relative eccentricity e'%=(e/r )% (68) (69) (70) Error relating to maximum stress E % % % % Table - Errors due to hole-rosette eccentricity for some types of rosette considered in standard ASTM AIAS TR-0:00 Page 58 of 70

59 Table provides the error limit calculated by formulas (73) and (74) for some types of rosettes specified by standard ASTM E837. If the orientation of the cross of the principal stresses is known, the rosette can be set in a known angular position in relation to the cross. Thus it is possible to reduce the hole-rosette eccentricity error. The formulas for a centre-hole rosette are usually used, but keeping eccentricity within set limits by using centering and drilling devices. The data in the table points to the need for careful checking of hole-rosette eccentricity when using the standard centre-hole rosette formulas Correction by HDM techniques Eccentricity can be corrected by the HDM method in the most general case of a blind hole with a non-uniform stress distribution. This correction is possible knowing eccentricity and eccentricity direction: hole-rosette eccentricity is a parameter used in the FEM studies forming the basis for the definition of the influence functions used in the HDM method. The stress profile can be reconstructed using a piecewise constant spline function, similar to the ASTM E standard representation) or any of the other methods defined in the HDM method (linear spline, cubic spline, Fourier series and power series) Correction using the special 6-element rosette The following figure 3 shows a 6-element rosette. The pattern is similar to the standard 3- element rosettes, but with diametrically opposed grids. Figure 3-6-element rosette for eccentricity correction It cannot be defined either as a type A or type B rosette, as per the ASTM standard, as the grids are positioned in opposed quadrants. The total resistance of the grid is therefore the sum of the resistance of the two diametrically opposed grids, just as variation of the total resistance, due to strain, is the sum of the variations of the two opposite grids. The balancing effect is due to the fact that if the hole is not perfectly concentric, taking any one of the three measuring directions (0 /45 /90 ), the grid closest to the hole measures a higher strain than it would have measured had the hole been concentric, whereas the diametrically opposed grid measures a lower strain. The variation in resistance is less on one grid but greater on the other, while the sum balances out, reproducing a measurement roughly equal to what would be obtained with a single grid if the hole were concentric. The 6-element rosette, therefore, connects with the acquisition system in precisely the same way as the standard 3-element rosette. The 6-element rosette is perfectly balanced in the case of eccentricity having only an effect of the first order on the single strain gauge measurement. In such a case, an eccentricity perpendicular to the grid would produce a null variation whereas an eccentricity parallel to the grid would produce a variation in line with the component of eccentricity in that direction. This applies only for small eccentricity values but is limited to a factor of / for higher eccentricities. AIAS TR-0:00 Page 59 of 70

60 8 Residual stress analysis software features The key features of any residual stress analysis software are the following: the possibility to choose the drilling and calculation depth increments in order to provide an accurate stress distribution profile while maintaining an acceptable level of uncertainty; a detailed presentation of strain and stress results, including: - tables of measured strains for every drilling depth, tables of the principal stresses and of the principal angle versus depth. - graphs of measured strains versus depth. - graphs of the principal stresses and principal angle versus depth. a recording of any measurement regarding hole shape (diameter, depth, eccentricity value and angle, etc.) memorization of details regarding the workpiece/component and strain gauge. Figure 33 - Hole-drilling software. Endmill Positioning Tool (left) and Drilling System Setup (right) (RSM by SINT Technology s.r.l.). Figure 33 shows some images drawn from a software for the management of the hole-drilling procedure. At the end of each drilling test, the measured strain data can be processed by an automatic evaluation program for calculating residual stresses (Figures 34 and 35). Figure 34 - Measured and interpoled strains versus depth. (EVAL by SINT Technology ) The evaluation program interpolates acquired data (best fitting), and on the basis of the curves calculated in this way, performs the processing by one of the methods available. Figure 34 shows the distribution of strain measured by the three grids. AIAS TR-0:00 Page 60 of 70

61 Figure 35 shows some images taken from the residual stress evaluation phase: the software allows analysis using the method described by standard ASTM E and a series of methods for evaluation of non-uniform residual stresses (for example, the Integral Method). Figure 35 - Residual stress evaluation: above analysis in accordance with ASTM E837-08, below stress analysis with the Integral Method. (EVAL by SINT Technology) AIAS TR-0:00 Page 6 of 70