AIAS TR_01_2010

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The hole-drilling strain gauge method for the measurement of uniform or non-uniform residual stresses

AIAS TR-01:2010 Page 1 of 70

Working Group on Residual Stresses

A.Ajovalasit, M.Scafidi, B.Zuccarello, University of PalermoM.Beghini, L.Bertini, C.Santus - University of Pisa

E.Valentini, A.Benincasa, L.Bertelli SINT Technology s.r.l.

AIAS TR01:2010

The hole-drilling strain gaugemethod for themeasurement of uniform or non-uniform residual

stresses

Revision: 02.09.2010

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PREFACE

This test method is the result of work by the AIAS Working Group on Residual Stresses over theperiod from 2006 to 2010.

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-gaugemethod. 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 inindustry to determine near-surface residual stresses.

The technical standard on the subject (ASTM E 837-08), 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-uniformstresses, however, the static nature of the method means that it is impossible to evaluateresidual stresses in many practical cases.

While acknowledging the progress that has been achieved thanks to the ASTM E837-08standard, the purpose of this guide is to go a step further, integrating new methods of correctingand calculating residual stress values with the considerations set out in the ASTM standard.

This method presents detailed instructions for the test reports and provides considerationsregarding uncertainty analysis in residual stress measurement.

The contributions presented herein reflect the results of the work carried out on these subjectsby Italian researchers both in the theoretical-experimental field and in design and constructionof new measurement instruments.

Thanks go to the researchers of the University of Palermo, the University of Pisa and thecompany SINT Technology srl for the invaluable contributions they have given both to thescientific 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 2010

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CONTENTS

1 INTRODUCTION ......................................................................................................................... 72 SCOPE ........................................................................................................................................... 73 REFERENCED DOCUMENTS .................................................................................................. 74 SYMBOLS ..................................................................................................................................... 85 PRINCIPLE OF MEASUREMENT ......................................................................................... 106 PRACTICAL ISSUES ASSOCIATED WITH THE MEASUREMENT ............................... 13

6.1 APPLICABILITY OF THE METHOD................................................................................................. 136.1.1 PARAMETERS OF THE MATERIAL ......................................................................................... 136.1.2 ACCESSIBILITY OF THE MEASUREMENT AREA ..................................................................... 146.1.3 EFFECT OF NON-UNIFORMITY AND PLASTICITY ................................................................... 14

6.2 STRAIN GAUGE ROSETTE SELECTION .......................................................................................... 146.2.1 ROSETTE DESIGNS................................................................................................................ 146.2.2 ROSETTE DIMENSIONS ......................................................................................................... 156.2.3 OTHER FACTORS INFLUENCING SELECTION ......................................................................... 16

6.3 SURFACE PREPARATION AND INSTALLATION.............................................................................. 186.3.1 SURFACE PREPARATION....................................................................................................... 186.3.2 CHOICE OF ADHESIVE. ......................................................................................................... 18

6.4 STRAIN-MEASUREMENT INSTRUMENTATION .............................................................................. 186.5 ALIGNMENT. .................................................................................................................................. 196.6 PERPENDICULARITY ..................................................................................................................... 216.7 EFFECTS OF THE FILLET RADIUS AT THE BOTTOM OF THE HOLE. ............................................. 226.8 HOLE SPACING .............................................................................................................................. 246.9 DISTANCE FROM GEOMETRIC DISCONTINUITIES........................................................................ 246.10 ZERO DEPTH DETECTION.............................................................................................................. 24

6.10.1 ELECTRICAL CONTACT DETECTION ..................................................................................... 246.10.2 OBLIQUE OBSERVATION OF DRILLING ................................................................................. 25

6.11 HOLE-PRODUCING TECHNIQUES.................................................................................................. 256.11.1 HIGH-SPEED DRILLING ......................................................................................................... 266.11.2 MEDIUM-SPEED DRILLING ................................................................................................... 276.11.3 LOW-SPEED DRILLING.......................................................................................................... 276.11.4 ABRASIVE JET MACHINING .................................................................................................. 276.11.5 ELECTRO-CHEMICAL MACHINING........................................................................................ 286.11.6 HIGH-SPEED ORBITAL DRILLING .......................................................................................... 28

6.12 DRILLING CUTTERS....................................................................................................................... 286.13 VERIFICATION OF THE DRILLING PROCESS................................................................................. 306.14 SELECTION OF DRILL DEPTH INCREMENTS ................................................................................. 306.15 MEASUREMENT OF STRAIN........................................................................................................... 30

6.15.1 EFFECT OF THE TURBINE AIR SUPPLY TEMPERATURE.......................................................... 306.15.2 HEAT GENERATED DURING THE DRILLING PROCESS............................................................ 30

6.16

MEASUREMENT OF HOLE DIMENSIONS AND ECCENTRICITY

...................................................... 31

6.17 FINAL HOLE DEPTH MEASUREMENT CHECK ............................................................................... 326.18 PRACTICAL EXAMPLE OF APPLICATION...................................................................................... 33

7 RESIDUAL STRESS ANALYSIS TECHNIQUES .................................................................. 347.1 STANDARD ASTME837-08: GENERAL ........................................................................................ 35

7.1.1 STRAIN GAUGE ROSETTES.................................................................................................... 357.1.2 STRAIN RELIEF IN PROXIMITY TO THE HOLE ........................................................................ 357.1.3 NUMERICAL VALUES OF aAND b..................................................................................... 367.1.4 SENSITIVITY OF THE METHOD.............................................................................................. 36

7.2 STANDARD ASTME837-08: CALCULATION OF RESIDUAL STRESSES........................................ 38

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7.2.1 THIN WORKPIECE ................................................................................................................. 387.2.2 THICK WORKPIECE............................................................................................................... 387.2.3 RESIDUAL STRESS UNIFORMITY TEST .................................................................................. 397.2.4 CALCULATION OF UNIFORM RESIDUAL STRESSES ............................................................... 397.2.5 CALCULATION OF NON-UNIFORM RESIDUAL STRESSES ....................................................... 407.2.6 INTERMEDIATE THICKNESS WORKPIECE .............................................................................. 45

7.3 CALCULATION OF NON-UNIFORM RESIDUAL STRESSES.OTHER METHODS.............................. 457.3.1 INTEGRAL METHOD ............................................................................................................. 457.3.2 INCREMENTAL STRAIN METHOD (ALSO KNOWN AS THE SCHWARZKOCHELMANN METHOD)

487.3.3 HDMMETHOD .................................................................................................................... 497.3.4 NON-UNIFORM RESIDUAL STRESSES WITH AN OFF-CENTRE HOLE....................................... 50

7.4 CORRECTION FOR PLASTICITY (ELASTIC RELAXATION OF STRESSES) ..................................... 527.4.1 CORRECTION WITH A 3-ELEMENT ROSETTE......................................................................... 537.4.2 CORRECTION WITH A SPECIAL 4-ELEMENT ROSETTE........................................................... 55

7.5 CORRECTION FOR ECCENTRICITY ............................................................................................... 567.5.1 CORRECTION FOR ECCENTRICITY: THROUGH HOLE............................................................. 577.5.2 CORRECTION BY HDM TECHNIQUES ................................................................................... 597.5.3 CORRECTION USING THE SPECIAL 6-ELEMENT ROSETTE ..................................................... 59

8 RESIDUAL STRESS ANALYSIS SOFTWARE FEATURES ............................................... 609 TEST REPORT ........................................................................................................................... 62

9.1 CONTENTS OF THE TEST REPORT................................................................................................ 629.1.1 GENERAL ............................................................................................................................. 629.1.2 PRESENTATION OF THE RESULTS ......................................................................................... 63

10 UNCERTAINTY ANALYSIS .................................................................................................... 6410.1 SUMMARY OF THE SOURCES OF UNCERTAINTY .......................................................................... 6410.2 CORRECTION OF THE MAIN ERRORS AFFECTING MEASUREMENT............................................. 6410.3 EVALUATION OF UNCERTAINTIES ON STRESSES ......................................................................... 66

11 REFERENCES ............................................................................................................................ 68

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INDEX OF FIGURES

Figure 1 - Symbols used in this publication. (On the left the symbols necessary for determiningthe state of stress, on the right the symbols used for correct definition of the geometry of therosettes). 10

Figure 2 - Relaxation of residual stresses after hole-drilling. 11Figure 3 - Diagram of the measurement chain using a high-speed air turbine. 12Figure 4 - Designs of strain gauge rosettes recommended by standard ASTM E837-08. 15Figure 5 - On the left a CW numbering scheme, on the right a rosette with CCW gauge

identification. 15Figure 6 - Hole drilling apparatus with a high speed air turbine (MTS 3000 - SINT Technology) 20Figure 7 Hole drilling device: on the left alignment, on the right rotation of the drilling head. 21Figure 8 - Checking the vertical perpendicularity of the hole-drilling tool. 21Figure 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. 23Figure 10 - 2D (left) and 3-D (right) BEM models for studying the effects of the hole-bottom fillet

radius. 23Figure 11 - Identifying the zero cutter depth by an electrical connection. 25Figure 12 - Types of holes that can be produced with the techniques studied by Flaman: 26Figure 13 - High speed drilling technique 26Figure 14 - Medium-speed drilling technique. 27Figure 15 - High-speed orbital hole-drilling 28Figure 16 - High-speed orbital hole-drilling technique. Detail of the cutting tool 28Figure 17 - Cutters used for high-speed drilling 29Figure 18 - Hardness ranges for which the three types of cutters are recommended 29Figure 19 - Measurement of hole diameter and eccentricity 31Figure 20 - Off-centre hole, parameters necessary for calculating hole-rosette eccentricity 32Figure 21 - Instrument for measuring hole depth 32Figure 22 - Graphical test of through-thickness stress uniformity (ASTM E837-08) 39Figure 23 - Schwarz Kochelmann method. 48Figure 24 - On the right, calibration functions Kx and Ky for the HBM rosette shown on the left. 49Figure 25 -. Symbols used in the HDM method. 50Figure 26- Assumed material constitutive law: bilinear isotropic hardening 53Figure 27- Ratio between the measured relaxed strains versus plasticity factor 54Figure 28 .HBM 4-element Rosette 0/90/157,5/225 (Left), Angles between gauges (Right) 56Figure 29: (a) Principal Angle (least squares minimisation); (b) Reconstruction of measured strain

versus angle. 56Figure 30 Equi-biaxial Stress Field: difference between the values of strain measured in the

absence (above) and presence (bottom) of eccentricity (e=0.1 mm) 57Figure 31 - Notations relating to a rosette with an off-centre hole 57Figure 32 - 6-element rosette for eccentricity correction 59Figure 33 - Hole-drilling software. Endmill Positioning Tool (left) and Drilling System Setup (right)

60Figure 34 - Measured and interpoled strains versus depth. 60Figure 35 - Residual stress evaluation: above analysis in accordance with ASTM E837-08, below

stress analysis with the Integral Method. 61

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INDEX OF TABLES

Table 1 - Symbols. 10Table 2 - Typical dimensions of type A, B and C rosettes described by standard ASTM E837-08. 16Table 3 - Rosettes produced by HBM and Vishay Measurement Group. 17Table 4 - Maximum and minimum workpiece thicknesses and hole diameters, and drilling depths

recommended by standard ASTM E837-08. 22Table 5 - Residual stress calculation methods: principal features. 34Table 6 - Numerical values of coefficients a and b provided by standard ASTM E837-08 for type

A, B and C rosettes for uniform stress evaluations with through holes and blind holes. 36Table 7 - Convention used for placement of angle (ASTM E837-08). 38Table 8 - Coefficients a and b for type A rosettes for non-uniform residual stress evaluations

(ASTM E837-08). 41Table 9 - Coefficients a and b for type B rosettes for non-uniform residual stress evaluations

(ASTM E837-08). 42Table 10 - Coefficients a and b for type C rosettes for non-uniform residual stress evaluations

(ASTM E837-08). 43Table 11 - Coefficients a and b of the integral method for type A, B and C rosettes. 47Table 12 - Errors due to hole-rosette eccentricity for some types of rosette considered in standard

ASTM 837-08 58Table 13 - Contributions of uncertainty in residual stress measurement. 65

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1 Introduction

Residual stresses are present in almost all structures. They may be caused by manufacturingprocesses or may be created during the life of a mechanical component. Residual stresses areoften a predominant factor contributing to structural failure, particularly of structures subject toalternating service loads or corrosive environments.

The effect on properties can also be beneficial, in which case residual stresses are createdpurposely to improve the behavior of a material, for example, the compressive stressesproduced by shot peening. In either case, it is important to determine the residual stresses inorder to be able to foresee static resistance and fatigue strength.

The hole-drilling method is a practical, inexpensive and widely used method for determiningresidual stresses near the surface of a component to be analysed. It can be applied to a widerange of materials.

It involves attaching a three-element strain rosette to the surface, drilling a hole in a series ofdepth increments through the centre of the rosette, and measuring the strains that are producedreflecting the stress relaxation which takes place with the removal of material.

2 ScopeThis test method specifies an incremental hole-drilling procedure for determining residual stressprofiles near the surface of an isotropic linearly elastic homogeneous material. The test methodis applicable also to plastic materials and composite materials: these materials present adifferent mechanical behavior from that of metal materials and also require particular attention inthe choice of hole-drilling procedure.

The test method may be considered semi-destructive because the damage that it causes islocalized 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 ASTME837-08 [1], may also be applied in cases where: a) residual stresses vary with depth, b) thereis 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 yieldstress, 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 standardholds and if the thickness is between 0.4 D and 1.2 D the results have to be consideredapproximate.

3 Referenced documents

Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain GaugeMethod, ASTM E837-08.

Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain Gauge

Method, ASTM E837-01. 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 PracticeGuide No.53, National Physical Laboratory, UK, 2002.

LU J., Handbook of Measurement of Residual Stresses, Society for ExperimentalMechanics, Fairmont Press, Lilburn, GA, 1996, Chapter 2.

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4 Symbols

The diagrams shown in Figure 1 are useful for understanding the majority of the symbols listedin Table 1.

Symbol Definition Units

a Calibration constant for isotropic stresses

b Calibration constant for shear stresses

jka Calibration matrix for isotropic stresses

jkb Calibration matrix for shear stresses

D Gauge circle diameter mm

GL Grid length mm

GW Grid width mmR1 Distance from the centre of the rosette to the internal edge of the grid mm

R2Distance from the centre of the rosette to the external edge of thegrid

mm

W Rated resistance of the strain gauge rosette

D0 Diameter of the drilled hole mm

E Youngs modulus MPa

Ep Plastic modulus of proportionality MPa

r Strain hardening ratio of the material

Poissons ratio

j Number of drilled hole depth steps

k 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 45shear stress MPa

Qk 45shear stress within hole depth step k MPa

q Uniform 45shear strain m/m

qk 45shear 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

P Regularization factor for P stresses

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Q Regularization factor for Q stresses

T Regularization factor for T stresses

Angle measured clockwise from r to max direction

Relieved strain for uniform stress case m/m

r Relieved strain measured by the gauge, in radial direction m/m

1,2,3 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

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 Sensitivity merit index

Biaxiality ratio

C Plasticity corrective coefficient

f(C) Dimensionless load parameter

X1,X2 Hole radiuses measured in x direction mm

Y1,Y2 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

Eccentric angle

p(hj), q(hj),t(hj),

p, q and t values calculated for the hole depth steps by the integralfunctions proposed by Schajer

MPa

A(H,hj), B(H,hj) Influence functions of the integral method

Kx, Ky Numerical/experimental calibration functions

j(11)

,j(33)

,j(13)

Influence functions describing the state of stress (HDM)

Kj(11)

,Kj(33)

,Kj(13)

Coefficients for the calculation of strains (HDM)

Objective function

u(x) Uncertainty tied to factor x

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ci Weight of uncertainty associated with parameter x

Uc(y) Total uncertainty associated with the measurement

k Normal distribution of uncertainty coverage factor

y Quantity measured in the test

U Extended uncertainty associated with the measurement

V Result of the test

Table 1 - Symbols.

Figure 1 - Symbols used in this publication. (On the left the symbols necessary for determining the stateof 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 thecentre of a special strain gauge rosette, and measuring the relieved strains. The maximumdepth of hole is approximately equal to 0.4 D.

The single measurements represent the average values of surface strain in the area of the gridscaused by relaxation of the stresses and the value of the readings is more sensitive torelaxation of the material the closer they are taken to the surface. This sensitivity decreases asthe depth increases until it reaches zero. The residual stresses originally present at the holelocation are then calculated from the measured strain values.

The relieved strains depend on the stresses that originally existed at the boundaries of thedrilled hole (the residual stresses are assumed to act uniformly over the in-plane region aroundthe rosette and to vary only through the thickness of the material) and are not affected by thestresses beyond the hole boundary.

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It is always preferable to drill the hole in small increments of depth, recording the measuredstrains and hole depth at each increment.

It is advisable that the drilling system for the incremental method is automatic and electronicallycontrolled: for example, Figure 3 shows a typical diagram of the measurement chain using ahigh-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 allowsconsiderations to be made on the uniformity of the stresses.

The basic method described in ASTM E837-08 and presented in Section 7.2 is strictly validwhere the stresses do not exceed approximately 50% of the yield strength. In these cases theexperimentally derived strain calibration coefficients experimentally developed from testspecimens 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 fromincremental strain data using the so-called integral method [2]. With this method, thecontributions to the total measured strain relaxation of the stresses at all depths are consideredsimultaneously. It will be examined in greater detail in Section 7.3.1.

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6 Practical issues associated with the measurement

There are two major factors that influence uncertainty associated with the measurementsobtained 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 thestrain measurements.

These factors will be considered separately in the following sections. Some of the practicalissues are considered below, and recommendations on the analysis methods are presented inSection 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.

6.1 Applicability of the method

Hole-drilling is a semi-destructive technique with relatively low sensitivity and can analyseresidual stress profiles in proximity to the surface of a material. It is the least expensive andmost widely used technique for measuring residual stress.

6.1.1 Parameters of the material

A component on which the test for determining residual stress is to be carried out should bemade of an isotropic material and the properties of the material should be known.

If possible, values for Youngs modulus (E) and Poissons ratio () experimentally determinedon a sample of the material under investigation should be used, particularly for non-standardalloys 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 areroughly considered to be in the 1 - 4% range and can therefore contribute significantly to theoverall uncertainty in the measurement.

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6.1.2 Accessibility of the measurement area

It is necessary to be able to access the areas of the component to be analysed both in order toapply 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 sufficientinformation, 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.

6.1.3 Effect of non-uniformity and plasticity

Standard ASTM E837-08 is applicable to residual stress profile determinations where thestresses may be uniform or non-uniform through the thickness of the component underinvestigation.

In addition, the test method provides accurate results if the stresses are less than approximately50% of the yield stress.

There are many circumstances where these requirements are not met, for example, residualstress measurements on a shot peened surface, close to a weld or a hole. This does not meanthat the hole-drilling technique cannot be applied, but numerical corrections are required to takeaccount of these effects.

For example, the welding process generates high residual stress values that may reach andeven exceed the yield strength of the base metal being welded, and in this case the twoprincipal 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 inSection 7.

The error in residual stress measurements due to the effect of localized yielding has beenanalysed 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 relationshave been proposed to correct the value of stresses, clearing obtained results of the effect ofplasticity.

The influence of plasticity is discussed in detail in Section 7.

6.2 Strain gauge rosette selection

6.2.1 Rosette designs

A number of commercial strain gauge rosette designs are available, designed specifically for thehole-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 atthe centre of the gauge circle.

Standard ASTM E837-08 describes the three strain gauge designs which are shown in Figure 4.

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Figure 4 - Designs of strain gauge rosettes recommended by standard ASTM E837-08.

Standard ASTM E837-08 distinguishes rosettes also by the arrangement of the measurementgrids: the numbering scheme can follow a clockwise (CW) convention if a clockwise rotation isnecessary to go from grid 1 (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 1 and 3:whereas the position of grid 2 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 allgrids in a single quadrant) is used for measurements near an obstacle, such as a fillet radius orweld, and type C for situations where high strain sensitivity and high thermal stability arerequired.

The type C rosette consists of six grids forming three pairs, with radially and tangentially alignedgrid axes. The opposed grids (for example, 1T and 1R in Figure 4) are to be wired in half-bridgeconfigurations.The type C gauge has increased sensitivity (varying from +70% to +140%) inrelation 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 sixstrain gauges (connected to three measurement channels).

Table 2 shows the typical geometric dimensions of type A, B and C rosettes described by

standard ASTM E837-08. A variety of sizes and types of strain gauge currently produced byHBM and Vishay Measurement Group are presented in Table 3.

6.2.2 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.),

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the depth required for the residual stress analysis (larger gauges are more suitable fordetermining the stress profile at greater depths whereas smaller gauges are suitable for anear-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 1.5 1.57 mm. This size of gauge is capable of providing useful residual stress data to a depth ofapproximately 1 mm.

It should be noted that the experimental errors associated with the measurements from smallstrain gauges (hole eccentricity, control of depth, etc) are higher than those associated with thecorresponding measurements with larger gauges.

However, the larger strain gauges should be selected with caution because of the size of drillsrequired and the large amount of material to be removed during the drilling process.

Table 2 - Typical dimensions of type A, B and C rosettes described by standard ASTM E837-08.

6.2.3 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.

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6.3 Surface preparation and installation

Installation of the strain gauge rosette should be carried out by qualified personnel inaccordance with the strain gauge and adhesive manufacturers instructions. [4].

The instructions provided by the UNI 10478 standards [5-9] should be followed for correctinstallation of strain gauges.

Surface-preparation and gauge-installation procedures must be of the highest quality as theyhave 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-preparationand gauge-installation procedures.

6.3.1 Surface preparation

To ensure a high-quality bond between the strain gauge and the component, the surface mustbe properly prepared.

This is particularly important when using the incremental hole-drilling technique as the strainsmeasured are generally very small (typically only several m/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 10478-3 standard suggests a surface roughness (Ra) in the 2.0 4.0 m range forgauge bonding with a cyanoacrylate-based adhesive [6].

However, it is recommended that mechanical abrading be avoided as much as possible if theincremental hole-drilling method is to be used for determining near-surface stresses [10-11]

Surface abrasion influences only the range of depth nearest the surface and the importance of itdepends on the residual stress gradients and the measurement requirements.

It should be noted that extremely rough surfaces must be avoided due to ambiguity inestablishing the zero depth for incremental hole-drilling [12].

ASTM E837-08 also recommends restricting surface preparation to those methods that havebeen demonstrated to induce no significant residual stresses (particularly for workpieces thatcontain sharp near-surface stress gradients).

6.3.2 Choice of adhesive.

The simplest, quickest and most common method of bonding the strain gauge to the specimenis to use a conventional cyanoacrylate adhesive.

These adhesives consist of a single component with a short cure time (1-2 minutes), and arerealtively easy to use.

If the surface of the component is particularly rough, it is important that the chosen adhesive fillsthe asperities and irregularities to achieve a good bond. In such cases, a more viscous, two-

component epoxy adhesive may be more suitable.

6.4 Strain-measurement instrumentation

It is important that the instrumentation chosen for strain measurement is calibrated and suitableto be used for this application.

ASTM E837-08 stipulates that the instrumentation for recording of strains should have a strainresolution of 1 m/m and that stability and repeatability should also be 1 m/m.

Generally, most modern strain-measurement instrumentation has the required resolution andstability for measuring the small strains in incremental hole drilling.

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However, the following minimum requirements are believed to be advisable for incrementalhole-drilling applications: strain resolution of 0.25 m/m, stability 0.5 m/m, repeatability 0.5m/m.

With the more conventional rosettes (types A and B) a three-wire quarter bridge circuit shouldbe used (self-temperature-compensating for as far as regards apparent thermal strain of theleads) 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 10 and 50acquisitions) be made for every measurement interval.

ASTM E837-08 recommends checking the integrity of the gauge installation by applying a smallload to induce strains and evaluating the mechanical hysteresis of the strain gauges forming therosette. The standard also recommends visual inspection of the rosette installation.

For the strain gauge installation, however, it is advisable to refer to the preliminary checksspecified by the standard UNI 10478-3 [7].

6.5 Alignment.

Eccentricity between the hole and gauge centre can introduce significant errors into themeasurement of residual stresses.

Alignment between these centres is normally achieved with the aid of a microscopeincorporating a reticle in the focus of the objective, the centre of which should coincide with thecentre of the endmill for drilling the hole.

After installation of the strain gauge rosette, the mechanical part of the measurement system ismoved close to the point where the measurement is to be made, and is positioned so that thestrain gauge centering marks are within the field of view of the microscope. Two adjustmentsset at 90 to each other are used for centering until the microscope reticle coincides with thestrain gauge centering marks.

A typical alignment and air turbine drilling system is shown in Figure 6. In this setup, themicroscope is incorporated in the measurement system and is not taken off duringmeasurements: all that is necessary is a rotation of the drilling head as it is aligned with themicroscope (Figure 7).

The drilling tool is fitted in front of the microscope after the alignment procedure. In othermeasurement 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 theuncertainty in positioning the tool holder (in the region of a few microns) to be taken intoaccount.

ASTM E837-08 states that the centre of the drilled hole should be aligned concentric with thestrain gauge circle to within 0.004 D.

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This corresponds to a substantial error in depth in the typical increments that are used inincremental measurements: its effect will depend on the orientation between the angle axis andthe rosette configuration [12].

It is important that the drilling system be checked before any test to avoid any errors caused bythe 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 threemagnetic 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 thestrain gauge grid endloops to protect the grids.

The need for this margin limits the maximum allowable diameter of the drilled hole D0.

The recommended minimum hole diameter is 60% of the maximum allowable diameter.

Table 4 indicates the maximum and minimum diameters recommended for standardized, typeA, B, and C rosettes.

Table 4 - Maximum and minimum workpiece thicknesses and hole diameters, and drilling depthsrecommended by standard ASTM E837-08.

As indicated in Section 7.1.4, it is important to note that as the ratio of D0/D increases, thesensitivity of the method increases in approximate proportion to (D0/D)

2.

Consequently, larger holes are recommended to achieve higher sensitivity.

Drilling diameters between 1.6 and 2.0 mm are normally used for rosettes with grids from 1.5 1.57 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 residualstresses 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 variesbetween 4% and 20% of the hole diameter D0; whereas with electrical-discharge machining(EDM) or abrasive jet machining techniques the fillet radius can reach values greater than 30%.

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This technique can be applied when analysing conductive materials and providing the airturbine conducts electricity [17]. Figure 11 shows zero depth detection by the electrical contacttechnique.

Figure 11 - Identifying the zero cutter depth by an electrical connection.

The advantages of this method are the simplicity in determining the initial contact, the short timerequired (a few seconds), the low cost (no auxiliary equipment is needed except for an electricallead) and automation of the method (managed by the electronic control system andmeasurement instrumentation software) [18,19,20]

The measurement system shown in Figure 11 has an automatic procedure for determining theinitial drilling point, removing the strain gauge backing and positioning the end mill cutter incontact with the workpiece metal surface.

6.10.2 Oblique observation of drilling

The technique consists in carrying out oblique observation of the drilling process through a minivideo camera, magnifying eyeglass or a microscope. The device should be held close to thehole location and cold light reflected from the strain gauge backing makes it possible to detectthe thinning and subsequent elimination of the strain gauge backing. A cold light source does notgenerate significant heat, whereas use of conventional inspection lamps may introduce undesiredthermal strains [12].

This technique for determining the zero position provides a less accurate detection of zerodepth than the electrical contact method. It may be applied to all types of materials and not justmaterials 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 beminimized [12].

6.11 Hole-producing techniques

The two key factors to be considered in selecting the hole-producing technique are thefollowing:

introduction of additional residual stresses during the machining process;

the ability of the technique to produce geometrically well-defined holes. In fact, calculation ofresidual stresses with one of the techniques available requires a cylindrical hole with a flatbottom.

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M.T.Flaman and J.A.Herring [21] studied four different techniques which were comparedquantitatively on the basis of induced stresses and hole geometry and qualitatively in terms ofportability and ease of use.

In addition, a fifth drilling technique, the orbital hole-drilling technique, was introduced and laterstudied by the aforementioned M.T.Flaman [22].

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 12 showing the geometric characteristics of the holes that canbe made by the four techniques studied by M.T.Flaman.

Figure 12 - 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 13 - High speed drilling technique

These methods for residual stress measurement are described and analysed in detail in thefollowing sections.

6.11.1 High-speed drilling

High-speed drilling was first used by M.TFlaman [21] employing an air turbine drilling systemrotating at speeds of up to 400,000 rpm (Figure 13). The typical cutting tool is an inverted-conetungsten carbide cutter, which produces a circular hole with straight sides and a flat bottom.

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In addition, abrasive jet machining cannot be used for determining non-uniform residualstresses as it does not allow sufficient control of hole depth and diameter. It is notrecommended for the less hard materials. [12]

6.11.5 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 residualstress although convexities are formed on the bottom of the hole, as can be seen for type D inFigure 12, which can influence the measured value of residual stress.

Use of these hole machining processes is limited to electrically conductive materials: thepresence of high electric discharges that generate stresses on the surface layers of the materialplus the presence of chemical agents can cause problems for protection of the strain gaugegrids. These factors have prevented development and diffusion of these techniques inproducing holes for the measurement of residual stresses. [12]

6.11.6 High-speed orbital drilling

Another technique available for measurement of residual stresses by the hole-drilling method ishigh-speed orbital drilling. It was first introduced by Flaman [22].

With this technique, the drill is deliberately offset from the centre of the strain gauge and thehole is drilled with an orbital motion. The diameter of the cutting tool is smaller than the diameterof the hole (figures 15 and 16).

Figure 15 - High-speed orbital hole-drilling Figure 16 - High-speed orbital hole-drillingtechnique. Detail of the cutting tool

The orbital drilling technique is an effective method for drilling hard, highly abrasive materialssuch as spring and bearing steels and cast aluminium alloys with a silicon content greater than6% (for example AlSi9Cu3 and AlSi7Mg).

With the orbital drilling technique the removal and extraction of chips is facilitated and moreefficient. A further advantage are greater drilling diameters.

6.12 Drilling cutters

For high-speed drilling the recommended drill for most materials is the inverted-cone tungstencarbide type. An inverted-cone polycrystalline diamond coated cutter can be used for hardermaterials.

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Figure 20 - Off-centre hole, parameters necessary for calculating hole-rosette eccentricity

while hole eccentricity and orientation are:

2

)( 21 XXeX

=( 4 )

2

)( 21 YYeY

=( 5 )

22

YX eee += ( 6 )

and the eccentric angle is expressed as:

=

180arctan

X

Y

e

e

( 7 )

6.17 Final hole depth measurement check

After removing the strain gauge, the final hole depth can be measured using a conventionaldepth gauge. A depth measuring instrument like the one shown in Figure 21 can be used.

Figure 21 - Instrument for measuring hole depth

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Any difference from the expected hole depth (recorded during drilling by the micrometer gaugeof the drilling apparatus) should be taken into consideration.

Cutter wear, the grip between the tool holder and cutter shank, and inadequate stiffnessbetween the component and the drilling apparatus can all contribute to hole depth errors.

6.18 Practical example of application

Automatic residual stress measurement systems are generally used having the advantage ofenabling 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 verticaladvance,

set-up of the test parameters. For example:

o Hole depth: 2.0 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 anelectrical contact that is made with removal of the strain gauge backing film and bringingthe 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 threestrain gauge readings.

Hole-drilling procedure example:

tool: 1.6 mm. diameter, inverted cone, surface-treated, tungsten carbide endmill,

speed of rotation (typical): from 350,000 to 400,000 rpm,

feed rate: 0.2mm/min,

depth increment: 0.05 mm,

delay time: 5 seconds.

Typical results are presented in section 8 (Residual stress analysis software features)

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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.

7.1 Standard ASTM E837-08: general

The ASTM E837-08 standard is the procedure that can be used for measuring residual stressesin 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 rosettesspecified in the standard (A, B and C). Nevertheless, it is possible to extend the standard by re-calculating 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 25% of the yield stress.

However, in practice satisfactory results are achieved providing the residual stresses do notexceed 60% of the material yield stress.

7.1.1 Strain gauge rosettesFigure 4 shows the geometry of the strain gauge rosette and the preferred numbering for thedirection of the principal stresses.

The centres of the three radially oriented gauges are D/2 from the gauge target and the centreof the hole.

Although, in theory, the angles between the strain gauges can be chosen arbitrarily, the majorityof commercially available rosettes are rectangular with gauges oriented at 0/45/90. The typesof strain gauge rosette standardized by ASTM E837-08 are presented in Table 2.

In the ASTM type A rosette design (gauges in two quadrants, ie, at 0/225/90), gauge 2 (or b)has been transposed to be diametrically opposite its original position to give more samplingabout 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 thegauge 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 opposedcircumferential and radial grids. Compared to the other rosettes, this design provides greatersensitivity and accuracy.

7.1.2 Strain relief in proximity to the hole

Considering the state of uniform stress in proximity to the hole, schematically illustrated inFigure 2 surface strain relief is tied to residual stresses , ,x y xy by the following relationship:

( 8 )

The two calibration constants a and b are dimensionless, almost independent of theproperties of the material, and vary with hole depth, as indicated in Table 2.

In the case of a through-hole in a thin workpiece, a is independent of the Poissons ratio.

Whereas, considering the case of non-uniform stresses within depth, the surface strain reliefassociated with the hole depth step j ( jk1 ) is tied to the relieved principal stresses by the

following relationship:

( ) ( )

2sin1

2cos2

1

2

1+

+

+

+= xy

yxyx

r bE

bE

aE

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7.2 Standard ASTM E837-08: calculation of residual stresses

7.2.1 Thin workpiece

For a thin workpiece or a through hole (thicknesss 0.4 D plane stress state conditions) thestresses are considered uniform in the depth direction and only a single reading of strains 1, 2and 3 is required.

The following quantities can be calculated from the measured strains:

( )

2

31 +=p

( )

2

13 =q

( )

2

2 213 +=t( 18 )

where p is the hydrostatic strain component and q and t are the shear strain components.

The stress components P, Q and T are calculated from p, q and t with the following equations:

)1(2P

y

+=

+=

a

Epx

b

Eqy=

=

2Q

x

b

Etxy == T

( 19 )

Finally, the principal stresses are calculated using:22, TQPMINMAX += ( 20 )

The angle , which the maximum principal stress max forms with the direction of strain gauge 1,(measured clockwise for the CW rosettes and counterclockwise for CCW rosettes), is calculatedwith the following equation:

=

Q

Tarctan

2

1

( 21 )

The direction of the angle is defined by Table 7, dependent on signs T and Q.

Q>0 Q=0 Q

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7.2.3 Residual stress uniformity test

In the case of a thick workpiece it is necessary to verify that the residual stresses are uniformwithin 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 oftheir values at the hole depth corresponding to 0.4 D,

plotting these percent strains versus hole depth (D0/D). These graphs should yield datapoints very close to the curves shown in Figure 23.

Data points that are separated from the curves in Figure 23 by more than 3% indicate eithersubstantial stress non-uniformity through the material thickness, or strain measurement errors.

Figure 22 - Graphical test of through-thickness stress uniformity (ASTM E837-08)

In either case, the measured data are not acceptable for the residual stress calculationsdescribed in the ASTM E837-08 standard.

This graphical test is not a sensitive indicator of stress field uniformity. Workpieces withsignificantly non-uniform stress fields can yield percentage relieved strain curves substantiallysimilar to those shown in Figure 22.

The main purpose of the test is to identify grossly non-uniform stress fields and strainmeasurement errors. This stress uniformity test may be applied only to thick workpieces.

7.2.4 Calculation of uniform residual stresses

When working with thick workpieces, all eight sets of 1, 2, 3 measurements are used forcalculating the magnitude and direction of the principal stresses.For each of the hole depths corresponding to the eight sets of 1, 2, 3, measurements, the

numerical values of the calibration constants a and b , corresponding to the hole depth anddiameter, 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 combinationstrains p, q and t are calculated using the following formulas:

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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 1/16 inch (5.13 mm) nominal size rosette: if a 1/32 inch (2.56mm) rosette is used, all hole and stress depths in the tables should be multiplied by 0.5; if a 1/8

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in. (10.26 mm) rosette is used, they should be multiplied by 2. Since the tabulated numbersrefer to a nominal hole diameter of 2 mm, the numbers have to be adjusted once the actual holediameter is measured and be multiplied by the following corrective factor: (actual diameter/nominal diameter)2.

Table 9 - Coefficients a and b for type B rosettes for non-uniform residual stress evaluations

(ASTM E837-08).

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Table 10 - Coefficients a and b for type C rosettes for non-uniform residual stress evaluations (ASTM

E837-08).

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depth step considered: with this procedure the integral equations seen above can easily beevaluated, 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

j

jjii PapE

1

,1

=

=i

j

jjii QbqE1

,

=

=i

j

jjii TbtE1

,

( 39 )

where n indicates the hole depth step considered and jia , and jib , indicate the relieved strains

due to unit stresses P, Q and T at depth j for hole depth step i.

The jia , coefficients are related to the functions ),( hHA as follows:

( )

=i

i

H

H

iji dHhHAa

1

,, ( 40 )

Discrete formulation of the problem therefore implies solution of a linear system with a lowertriangular matrix of coefficients.

Figure 24 - Drilling depths: physical interpretation of coefficients jia , .

With the aid of a finite element calculation, coefficients jia , have been determined by

calculating the following functions

( ) ( )=H

i dHhHAhhA0

,,

( 41 )

by which coefficients jia , are evaluated as:

( ) ( )ijijji

hHAhHAa ,, 1, = ( 42 )

Functions A and B have been provided for ratios D0/D equal to 0.3, 0.4 and 0.5 forcalculation depth h between 0.05 and 0.50.

The coefficients are obtained by interpolation for different D0/D ratio and calculation depth hvalues.

The values of the coefficients proposed by G. S. Schajer for calculating residual stresses by theintegral method are indicated in Table 11.

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Table 11 - Coefficients a and b of the integral method for type A, B and C rosettes.

7.3.1.1 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 straindata can be smoothed to reduce noise.

At a smaller number of calculation increments, combination strains p, q and t are calculatedfrom the smoothed strain data.

Cumulative strain relaxation functions ( A and B ) are calculated (for the measured holediameters), by interpolation, from the values of the coefficients of the triangular matricesprovided.

Coefficients jia , and jib , 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

j

jjii PapE

1

,1

=

=i

j

jjii QbqE1

,

=

=i

j

jjii TbtE1

,

( 43 )

Residual stresses and residual stress orientation (max, min, ), for each calculationincrement, are obtained from the corresponding combination stresses.

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7.3.2 Incremental strain method (also known as the SchwarzKochelmannmethod)

The incremental strain method, proposed by T.Schwarz and H.Kochelmann [27] in 1993, isbased on measurement of the strain rate during the drilling operation (Figure 23).

Figure 23 - Schwarz Kochelmann method.

The method involves a preliminary stage of experimental/numerical determination of therelaxation functions defined as:

( 44 )

( 45 )

where x and y are the strains measured respectively by the strain gauge grids oriented parallelto the loading direction and perpendicular to the loading direction in the case of uniaxial loading.

Figure 24 shows relaxation functions Kx and Ky calculated for an HBM rosette, type 1-Y61-1.5/120S for dm/d0=3.

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Figure 24 - 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 thefollowing formulas:

( 46 )

( 47 )

( 48 )

The principal stress values can be calculated by the following equation:

( 49 )

( 50 )

This method can be applied only for HBM rosettes as the numerical/experimental values offunctions Kx and Ky have been calculated only for this type of rosette.

Although the residual stress results obtained with this method may agree with those evaluatedwith the integral method, it must be pointed out that the method is approximate because it doesnot take account of the change in hole geometry with depth (but only of the residual stress inthe removed stratum of material).

7.3.3 HDM Method

The HDM Method [28, 29, 30, 31, 32, 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 ),( hHA and ),( hHB .

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 themodel of rosette used,

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Observing Figure 25 and considering an isotropic linearly elastic homogeneous material, therelationship between measured strain and related stresses can be described as follows:

( 51 )

where the influence functions A1A9 depend on the properties of the material, hole depth andeccentricity and rosette geometry.

Each of these influence functions can be described with a double power series, the coefficientsof which have been calculated by a finite-element regression analysis of surface displacementsfor every particular configuration of the problem under examination (Poissons ratio, ratiobetween 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 ofequations seen above by an inverse formulation, in order to determine the state of residualstress existing in the component.

Supposing that each stress component may be described with a series of functions, thefollowing expressions can be obtained:

( 52 )

where:

J11, J12 and J13 are the degrees of freeedom of the stress field,

represent the functions used to describe the stress state,

are constant coefficients determined by the least squares method for bestreconstructing the experimental strain measurements.

By combining the two systems of equations, it is possible to obtain the following relationship:

( 53 )

where i = 1,2n = number of hole-drilling steps.

Knowing the form of the influence functions, the integrals in the equations can easily beanalytically solved, and therefore the whole relationship is reduced to a linear system of 3nequations in which J11+J12+J13 are unknowns, that can be simplified to:

( 54 )

This system can be solved directly when 3n=J11+J12+J13, and with the least squares methodwhen 3n> J11+J12+J13.

The latter analysis technique is better because it reduces the influence of random experimentalerrors.

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Nevertheless, the liberty granted in selecting the j functions used to describe the state ofresidual stress implies introduction of some new parameters, the definition of which influencesthe accuracy of the result (for example, in the case of the power series, the maximumsuperscript at which to stop expansion). To obtain the optimum solution with the hole-drillingmethod, such parameters can be selected by automatic methods, for example, by geneticalgorithms [36].

The choice of these parameters is very important as it actually corresponds to the level offlexibility that is given to the representation of the state of stress, and therefore, to the ability toreproduce even highly complex stress functions. A low level of flexibility can lead toapproximate solutions whereas too high a flexibility can cause excessive sensitivity tomeasurement error, which is always present in acquired strain data.

The criterion of optimization follows the principle that the strains obtained from calculatedresidual stresses have to reconstruct the acquired state of strain with the same accuracy of themethod. Any inferior accuracy leads to an approximate solution, whereas a higher accuracyresults in reproducing the measurement errors that are normally associated with a highinstability of the results.

For this reason, the following objective function is defined:

( 55 )

where 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 theerror 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 serialmethods employ an exhaustive algorithm:

the genetic optimization algorithm makes it possible to position base points randomly in theinterval (0, zmax), where zmax is the maximum hole depth, collapses neighbouring base pointsif their distance is less than a threshold (the threshold is set at 5% of zmax) and thenidentifies 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 J11+J12+J13 is verified).

7.4 Correction for plasticity (elastic relaxation of stresses)

The local plastic deformation which occurs around the drilled hole can introduce significanterrors in the calculation of residual stresses: standard ASTM E837-08 actually specifies a limitto 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 HBMhas produced a special 4-element rosette (Figure 27) 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 inthe principal directions.

To use a special rosette with 4 elements at 0/90/157,5/225 (for example HBM 1-VY61-1.5/120S. Figure 28): the fourth grid allows the principal directions to be calculatedapproximating the stress profile with the use of a fourth-order Fourier series development.

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7.4.1 Correction with a 3-element rosette

The hole-drilling method in accordance with standard ASTM E837-08 poses a limit related tothe maximum measureable residual stress: for the relationships between strains and stressesdefined by the standard to be valid, the measured stress components should not exceed about50% of the yield stress of the material, in absolute value.

The local plastic deformation which occurs around the drilled hole, must be considered in orderto determine correct residual stress values.

In the case of uniform stress through the thickness of a workpiece, it is possible to correct theresidual stress values and to estimate the actual value of residual stresses in a component[3,47]In recent studies [49] a more accurate FE elasticplastic parametric analysis has beendeveloped and a more accurate, and general, correcting procedure is provided which can beapplied to all the rosettes available on the market.

Figure 26- Assumed material constitutive law: bilinear isotropic hardening

To correct stresses for the effect of local yielding it is necessary to know the yield stress andstrain hardening ratio of the material (R) (Figure 26):

E

ER

plastic= (56)

Eplastic indicates the ratio between deformation and displacement in the plastic field.

The von Mises equivalent stresses

yxyxeq += 22

was assumed to quantify the effect of biaxiality and a dimensionless plasticity factor f isintroduced:

ieqY

ieqeqf

,

,

= (57)

where eq,i is the equivalent residual stress producing the onset of plasticity in the 2D case, andY is the material yield stress.The condition of f=0 represents the highest residual stress that still does not produce plasticity,while f=1 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 approximateonset of plasticity given by the plane Kirsch solution [15]. For the correction algorithm, it is

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necessary to consider the biaxial stress ratio . The ratio between the measured relaxed strains

along the principal directions x/y depends on the stress ratio but it is almost unaffected bythe plasticity factor, as shown in Figure 27.

As a consequence, the biaxiality ratio can be approximated by the ratio between theelastically calculated residual stress components x, el, y, el

elx

ely

elx

y

,

,

== (58)

Figure 27- Ratio between the measured relaxed strains versus plasticity factor

The equivalent residual stress at the plasticity onset can be expressed as a function of thebiaxiality ratio , according to the plane stress Kirsch solution [15].

+=

3

21

, Yieq(59)

The acquired strain can be used to obtain the as elastically-evaluated residual stresses and forthe elastically-evaluated equivalent stress definition:

elyelxelyelxeleq ,,2,

2,,

+= (60)

The related elastically-evaluated plasticity factor is obtained by the equation:

ieqY

ieqeleq

elf,

,,

= (61)

If a significant plasticity is produced, the elastically-calculated plasticity factor is larger than theactual plasticity factor. As the plasticity is not expected to play a significant role for a plasticityfactor near 0, it follows that fel

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The parameters W and depend on the biaxial ratio , the hardening ratio of the material R,the hole depth Z, the hole diameter D0 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 1 and 3 grids aligned with the known (or at least assumed) principal residual stressdirections and the signals are set to 0.

The relaxed strains 1, 2, 3, are measured after the hole, the x axis is chosen parallel tothe grid measuring the maximum absolute value; therefore, if | 1|>| 3|, then x=1 and

y=3, otherwise x=3 and y=1. The elastically-evaluated residual stresses x,el, y,el are determined.

The elastically-calculated biaxiality ratio el is assumed to be an accurate approximation ofthe actual residual stress biaxiality ratio

The elastically-evaluated plasticity factor fel can be obtained with the equivalent residualstress at the yield onset eq,eI

If fel=0, no correction is required and the elastically-evaluated residual stresses areassumed as the actual residual stresses; otherwise, the correction is needed

Calculation of the parameters W and Calculation of the actual f from fel by inverting equation (62):

Wff

elf +=

To solve this unelementary equation, the NewtonRaphson algorithm is recommended which

gives an accurate numerical approximation of the plasticity factor, hereafter called f

Estimation of the equivalent residual stresseq

obtained from the plasticity factor f

ieqYfieqeq ,

, += (63)

Calculation of the principal residual stress componentsx

, y obtained from eq :

21

1

+=

eqx,

xy = (64)

7.4.2 Correction with a special 4-element rosette

As previously observed, a limitation in this procedure is that of requiring the relaxed strainsalong the principal directions x and y.

This implies prior knowledge of the principal directions of the residual stress distribution, whichcan 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 todeduce the relaxed strains that would be read if the grids were oriented along the principaldirections 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 28).Assuming the principal directions are unknown, there are four strain readings in four directions:

a=(= 0);b=(= 90);c=(= 180-22:5);d=(= 180+45) (65)while the four values to be obtained are '

0 ,'

2 ,'

4 and according to the formula:

[ ] [ ])(4cos)(2cos)( '4'

2

'

0 ++++= (66)

Inversion of the calculation is not linear; the principal angle is calculated by solving a least

squares minimisation problem. Figure 29a shows the determination of the principal angle andFigure 29b shows the reconstruction of measured strain versus angle.

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a aa a ab b ac cJ J J= + +' ' '

(68)

b ba a bb b bc cJ J J= + +' ' '

(69)

c ca a cb b cc cJ J J= + +' ' '

(70)

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 theformulas used for a centered rosette.

If eccentricity between the hole and rosette is neglected, the principal stresses 1 and 2 areaffected by the following errors:

1

1

'

11001

=E (71)

2

2

'

21002

=E (72)

where'1 and '2 are the calculated principal stresses, that is, without taking account of eccentricity;

1 and 2 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 unfavourableconditions of (orientaton of the greater principal strain) and (orientaton of the eccentricity) isconsidered.

For a given relative eccentricity e'%= (e/r1)%, and with other conditions being the same,0/45/90rosettes are more unfavourable than 0/225/90rosettes.

In the most unfavourable conditions of 2/1 (2/1=-1) the errors relating to the type A and Brosettes considered in standard ASTM E837, calculated for the minimum hole diameter are:

ASTM type A rosettes (0/225/90): %6,5 '1

eE

(73)

ASTM type B rosettes (0/45/90): %1,8 '1

eE (74)

The formulas stated above therefore make it possible to determine the upper error limit, in the

most unfavourable conditions of , and 2/1.

ASTMtype ofrosette

Average diam. ofthe rosette

D=2r(mm)

Distance of the inneredge of the grid from

the centre of therosette

r1(mm)

Hole-centreeccentricity

e(mm)

Per centrelative

eccentricity

e'%=(e/r1)%

Error relating tomaximum

stress

E1%

Type A -1/320/225/90

2.57 0.89 0.025 2.81 16%

Type A -1/160/225/90

5.13 1.77 0.025 1.41 8%

Type B-1/160/45/90

5.13 1.77 0.025 1.41 11%

Table 12 - Errors due to hole-rosette eccentricity for some types of rosette considered in standard ASTM837-08

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Table 12 provides the error limit calculated by formulas (73) and (74) for some types of rosettesspecified by standard ASTM E837.

If the orientation of the cross of the principal stresses is known, the rosette can be set in aknown angular position in relation to the cross. Thus it is possible to reduce the hole-rosetteeccentricity 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 whenusing the standard centre-hole rosette formulas.

7.5.2 Correction by HDM techniques

Eccentricity can be corrected by the HDM method in the most general case of a blind hole witha non-uniform stress distribution. This correction is possible knowing eccentricity andeccentricity direction: hole-rosette eccentricity is a parameter used in the FEM studies formingthe 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 theASTM E837-08 standard representation) or any of the other methods defined in the HDMmethod (linear spline, cubic spline, Fourier series and power series).

7.5.3 Correction using the special 6-element rosette

The following figure 32 shows a 6-element rosette. The pattern is similar to the standard 3-element rosettes, but with diametrically opposed grids.

Figure 32 - 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 gridsare positioned in opposed quadrants.The total resistan

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