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TRANSCRIPT
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Forschungsberichte
aus dem
Institut für Werkstofftechnik
Metallische Werkstoffe
der
Herausgeber: Prof. Dr.-Ing. B. Scholtes
Band 9
Enrique Garcia Sobolevski
Residual Stress Analysis of Components with Real Geometries
Using the Incremental Hole-Drilling Technique
and a Differential Evaluation Method
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Forschungsberichte aus dem Institut für Werkstofftechnik - Metallische Werkstoffe der Universität Kassel
Band 9
Herausgeber:
Prof. Dr.-Ing. B. Scholtes Institut für Werkstofftechnik
Metallische Werkstoffe Universität Kassel
Sophie-Henschel-Haus Mönchebergstr. 3
34109 Kassel
Die vorliegende Arbeit wurde vom Fachbereich Maschinenbau der Universität Kassel als Dissertation zur Erlangung des akademischen Grades eines Doktors der
Ingenieurwissenschaften (Dr.-Ing.) angenommen.
Erster Gutachter: Prof. Dr.-Ing. habil. B. Scholtes Zweiter Gutachter: Prof. Dr. rer. nat. A. Wanner
Tag der mündlichen Prüfung 20. Juli 2007
Bibliografische Information der Deutschen Nationalbibliothek
Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über
http://dnb.ddb.de abrufbar
Zugl.: Kassel, Univ., Diss. 2007 ISBN 978-3-89958-343-4
URN urn:nbn:de:0002-3438
© 2007, kassel university press GmbH, Kassel www.upress.uni-kassel.de
Umschlaggestaltung: Melchior von Wallenberg, Nürnberg Druck und Verarbeitung: Unidruckerei der Universität Kassel
Printed in Germany
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Vorwort des Herausgebers
Bei einer zunehmenden Verbreitung elektronischer Medien kommt dem gedruckten
Fachbericht auch weiterhin eine große Bedeutung zu. In der vorliegenden Reihe werden
deshalb wichtige Forschungsarbeiten präsentiert, die am Institut für Werkstofftechnik –
Metallische Werkstoffe der Universität Kassel gewonnen wurden. Das Institut kommt
damit auch – neben der Publikationstätigkeit in Fachzeitschriften – seiner Verpflichtung
nach, über seine Forschungsaktivitäten Rechenschaft abzulegen und die Resultate der
interessierten Öffentlichkeit kenntlich und nutzbar zu machen.
Allen Institutionen, die durch Sach- und Personalmittel die durchgeführten Forschungs-
arbeiten unterstützen, sei an dieser Stelle verbindlich gedankt.
Kassel, im Oktober 2007
Prof. Dr.-Ing. habil. B. Scholtes
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„Везде исследуйте всечасно,
Что есть велико и прекрасно,
Чего еще не видел свет.“
Михаил Васильевич Ломоносов
„Überall erforschet ohne Unterlaß
Was herrlich ist und wunderschön,
was die Welt noch nicht geseh’n.“
Michail Vasil’evič Lomonosov
Посвящается моей маме,
Александре Всеволодовне Соболевской-Глёкнер
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Contents
Symbols and Abbreviations.................................................................. 9
1 Introduction........................................................................................... 13
2 State of Knowledge................................................................................ 14
2.1 Residual Stress........................................................................................ 14
2.1.1 Definition and Origin of Residual Stress................................................ 14
2.1.2 Origin...................................................................................................... 15
2.1.3 Technological Relevance........................................................................ 16
2.1.4 Overview of Residual Stress Measurement Methods............................. 17
2.2 General Aspects of the Hole-Drilling Method........................................ 18
2.2.1 Principle.................................................................................................. 18
2.2.2 Development........................................................................................... 19
2.2.3 Scope, Influence and Restrictions........................................................... 20
2.3 Residual Stress Calculation..................................................................... 21
2.3.1 Average Stress Value over Specimen’s Thickness - Through-Hole
Case 21
2.3.2 Average Value over Hole Depth - Blind-Hole Case............................... 24
2.3.3 Residual Stress Depth Distribution......................................................... 25
2.3.3.1 Differential MPA Method....................................................................... 25
2.3.2.2 Integral Method....................................................................................... 28
2.4 Geometrical Boundary Conditions.......................................................... 30
2.4.1 Prefacing Remarks.................................................................................. 30
2.4.2 Definition of Possible Geometrical Boundary Conditions...................... 31
2.4.3 Overview of Previous Investigations...................................................... 32
2.4.3.1 Thickness................................................................................................. 32
2.4.3.2 Distance between Hole Center and Component Edge............................ 35
2.4.3.3 Radius of Surface Curvature................................................................... 37
2.5 Concluding Remarks on state of Knowledge.......................................... 39
3 Experimental and Numerical Methods of Investigation.................... 41
3.1 Prefacing Remarks.................................................................................. 41
3.2 Experimental Details............................................................................... 42
3.2.1 Material................................................................................................... 42
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3.2.2 Specimens................................................................................................ 43
3.2.3 Hole-Drilling Device and Calibration Equipment.................................. 48
3.2.4 Experimental Calibration Procedure....................................................... 49
3.3 Numerical Methods and Calibration....................................................... 51
3.3.1 Models and Simulation Procedure.......................................................... 51
3.3.2 Material Model, Load Application and Boundary Conditions of the
FE-Models............................................................................................... 57
3.3.3 Calculation of Strain................................................................................ 58
4 Results of Experimental and Numerical Calibration......................... 61
4.1 Prefacing Remarks.................................................................................. 61
4.2 Reference Specimen and Variation of Hole Diameter -d0..................... 62
4.3 Variation of Specimen Thickness – T..................................................... 64
4.4 Variation of Distance Hole-Edge – D..................................................... 68
4.5 Combination of Parameter T and D........................................................ 71
4.6 Variation of Cylinder Radius – R............................................................ 74
4.6.1 Solid Cylinders........................................................................................ 74
4.6.2 Hollow Cylinders.................................................................................... 77
4.6.3 Comparison of Solid Cylinders and Hollow Cylinders........................... 80
4.7 Specimens with Center Through-Hole -Combined Geometrical
Parameters...............................................................................................
82
4.7.1 Flat Tensile Specimen with a ø10 mm Center Through-Hole................ 82
4.7.2 Hollow Cylindrical Tensile Specimen with a ø6 mm Through-Hole..... 85
5 Discussion of Calibration Results........................................................ 89
5.1 Prefacing Remarks.................................................................................. 89
5.2 Calibration Uncertainty........................................................................... 90
5.2.1 Uncertainty in Numerical Calibration..................................................... 90
5.2.2 Uncertainty in Experimental Calibration................................................ 94
5.2.3 Graphical Representation of Uncertainty in Calibration Results............ 101
5.3 Stress Evaluation on specimens with Non-Reference Geometry............ 105
5.4 Influence of Geometrical Parameters on Results of Hole-Drilling
Measurements..........................................................................................
110
5.4.1 Influence of Hole Diameter - d0............................................................. 110
5.4.2 Influence of Specimen Thickness - T...................................................... 112
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5.4.3 Influence of Distance Hole-Edge - D...................................................... 118
5.4.4 Influence of Cylinder Radius - R............................................................ 121
5.5 Possible Reduction of Calculated Stress Differences using Geometry
Specific Calibration Functions................................................................
129
5.6 Summary of Discussion of Calibration Results...................................... 134
6 Implementation of Geometry Specific Calibration Functions into
Evaluation Program..............................................................................
136
6.1 Prefacing Remarks.................................................................................. 136
6.2 Modification and Differentiation of Measured Strain Distributions....... 138
6.2.1 Block 1 - Import of Input File................................................................. 138
6.2.2 Block 2 - Parameter Modification........................................................... 139
6.2.3 Block 3 - Strain Modification.................................................................. 139
6.2.4 Block 4 - Strain Smoothing..................................................................... 139
6.2.5 Block 5 - Storage of Modified Input File................................................ 142
6.2.6 Block 6 - Local Polynomial Approximation of Measured Strains.......... 142
6.3 Stress Calculation.................................................................................... 143
6.3.1 General Aspects of Stress Calculation in Prototype Program................. 143
6.3.2 Block 7 - Stress Calculation “MPA Standard”........................................ 143
6.3.3 Block 8 - Stress Calculation “MPA Expanded”...................................... 144
6.3.4 Block 9 - Stress Calculation “MPA Specific”......................................... 146
6.3.5 Block 10 - Import of Calibration Files.................................................... 147
6.3.6 Block 11 - Polynomial Approximation of Calibrations Strains.............. 148
6.3.7 Block 12 - Angle Calculation.................................................................. 148
6.3.8 Block 13 - Visualization of Results......................................................... 149
6.3.9 Block 14 - Export and Storage of Result File......................................... 149
7 Evaluation of the Implementation of Geometry Specific
Calibration Functions into Stress Calculation Program..................
151
7.1 Prefacing Remarks................................................................................. 151
7.2 General Aspects of Calculation.............................................................. 151
7.3 Influence of Local Polynomial Approximation of Measured Strains... 154
7.4 Influence of Strain Smoothing............................................................... 158
7.5 Influence of Order of Polynomial Approximation of Calibration
Strains...................................................................................................... 162
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7.6 Influence of Selected Distribution of Depth Increments......................... 166
7.7 Interpolation between Geometry Parameters......................................... 169
7.8 Summary of the Evaluation of the Calculation Program with
Implemented Geometry Specific Calibration Functions........................
172
8 Conclusion............................................................................................. 173
9 References............................................................................................. 177
10 Zusammenfassung in deutscher Sprache........................................... 183
11 Appendix................................................................................................ 187
11.1 FEM Post Processor Plots....................................................................... 187
11.2 Program Listings..................................................................................... 194
11.2.1 Smoothing using Weighted Average Algorithm..................................... 194
11.2.2 Smoothing using Compensative Cubic Splines...................................... 194
11.2.3 Local Polynomial Approximation of Measured Strains.......................... 196
11.2.4 Calculation of Geometry Calibrations Coefficients via Linear
Interpolation between Single Geometry Specific Coefficients...............
197
Acknowledgments……………………………………...……………... 201
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Symbols and Abbreviations
Latin Symbols
A(H,h) strain relaxation per unit depth
a1,...,n single values of uncertainty
aij matrix of strain relaxation
ASG strain gage area
ATH calibration constant for finite strain gage areas
BTH calibration constant for finite strain gage areas
D distance between hole-center and free edge or border of component
d0 hole diameter
DE distance between hole-center and edge of component
Den denominator
DH distance between two holes
dm mean rosette diameter
E Young’s modulus
fil compensating factor
H normalized depth from surface
h normalized hole depth (integral method)
Ktn concentration factor according to nominal stress
Kx calibration function in x-direction
Ky calibration function in y-direction
M strain matrix
n number of measurements
nsc number of scale divison
Num numerator
p(h) equi-biaxial plane strain
P(H) equi-biaxial plane stress
R cylinder outside radius
r0 hole radius
r1,2 radial distance from hole center to gage ends
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RB outside radius of sphere or sphere
RBI inside radius of sphere or sphere
ReS yield strength
RI cylinder inside radius
s standard deviation
S1,2,3 integration constants
s.g. strain gage
T thickness
t distribution according to Student
u uncertainty
us systematic uncertainty
uz random uncertainty
w strain gage width
W component width
z hole depth
Z depth from surface
Greek Symbols
Δεi relived strain
ε0(i) initial strain
ε1,2,3 strain in strain gage direction
εΕ(i) final strain
εx nodal strain value in x-direction
εx,m calculated average strain value in x-direction
φ orientation angle
φ∗ intermediate angle
ν Poisson’s ratio
σ1,2,3 stress in strain gage direction
σc,x calibration stress in x-direction
σc,y calibration stress in y-direction
σI,II,III kinds of residual stress
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σmax maximal calculated stress
σmax,min principal stresses
σmax,sim maximal stress (FEM)
σnom nominal stress
σRS residual stress
σx-FEM initial stress distribution at measurement point of FEM model
ξ normalized depth (differential method)
Abbreviations of Specimens and Models
The denominations of the used specimens and models are a combination of
abbreviations, which describe the main geometrical specifications:
D Distance between center of measurement point (hole) and specimen’s edge
f flat specimen
fh flat specimen with ø10 mm center through-hole
hc hollow cylindrical specimen
hch hollow cylindrical specimen with ø6 mm through-hole
m FEM model
R cylinder outside radius ref geometrical reference
sc solid cylindrical specimen
T specimen’s thickness
Examples:
fT1.5D20 flat specimen with a thickness of T = 1.5 mm and a distance hole-edge of D = 20 mm
hcR3m FEM model* of a hollow cylindrical specimen with an outside radius of R = 3 mm
hchR6 hollow cylindrical specimen with ø6 mm through-hole and an outside cylinder radius of R = 6 mm
* The FEM models are additionally indicated using the abbreviation “m” at the end of
the denomination.
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1 Introduction As a result of the manufacturing processes, residual stresses are found in nearly all
components. In some cases, the behavior of the component can be affected in a
beneficial or detrimental manner depending on the magnitude and direction of the
residual stresses. Thus, the demand for reliable and economically justifiable analysis
methods is of great significance in industry. One accepted and widely used technique
for measuring residual stresses is the incremental hole-drilling method. The basic hole-
drilling procedure involves drilling a small hole into the surface of a component at the
centre of a special strain gage rosette and measuring the relieved strains. The residual
stress distributions originally present at the hole location are then calculated from the
stepwise established strain distributions against the hole depth. The introduced blind-
hole relaxes the residual stresses only partially and, therefore, the available evaluation
procedures include, as a reference basis, a calibration of the method on a known stress
state under reference geometrical conditions. Because of this, the violation of
recommended geometrical restrictions could have an influence on the calculated
residual stresses, e.g. an overestimation of stress values. In this case, some authors
recommend the determination of geometry specific calibration functions.
This work investigates the geometry influence on the residual stress calculation
according to the differential MPA evaluation algorithm [1, 2] for hole-drilling
measurements. The first aim is to show the individual influence of the specimen
geometry by means of strain and stress deviations related to the results determined with
a reference specimen. The second aim is the implementation of geometry specific
calibration functions into the differential MPA evaluation algorithm in order to show a
possible correction of the calculated stress deviations. For both purposes, experimental
and finite element calibrations are carried out using specimens or models which
systematically violate the geometry parameters thickness, distance from the hole to the
specimen edge and the surface curvature. An uncertainty analysis discusses the quality
of the presented calibrations results. In the last part of the work, a prototype program for
the stress evaluation according to the MPA algorithm is described and evaluated. In
contrast to current software packages, the presented program calculates, additionally,
geometry specific calibration functions for the consideration of the actual component
shape.
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2 State of Knowledge
2.1 Residual Stress
2.1.1 Definition and Origin of Residual Stress Residual stresses (or locked-in stresses) are defined as those stresses which exist in a
solid body without the application of external forces or any other sources of load, such
as thermal gradients, gravity, etc. Residual stresses can be found in almost all rigid parts
independent of the material. All residual stress systems in a component are self-
equilibrating, i.e. the resultant forces and the moments that are produced inside the
component by residual stresses must be zero. [3, 4].
There are three kinds of residual stresses that are classified according to the range over
which they can be observed [4, 5, 6]:
• The first kind of residual stress σI is also called macroscopic stress. It is long-
range in nature and extends over at least several grains of the material, and usually
over many more.
• The second kind of residual stress σII is also called structural micro-stress. It
covers the distance of one grain or a part of a grain. It can be found between
different phases and has different physical characteristics. It can also be found
between embedded particles, such as inclusions and the matrix.
• The third kind of residual stress σIII ranges over several atomic distances within
the grain. It is equilibrated over a small part of the grain. This third type of residual
stress is inhomogeneous across the described areas of the material.
The relationship between the three different types of residual stresses is shown
schematically in Fig. 2.1. The total residual stress σRS (magnitude and direction) at a
particular point of a material is the result of the superposition of the locally existent
kinds of stresses.
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Fig. 2.1: Definitions of residual stresses of I., II., III. kinds [7]
2.1.2 Origin Main Groups Manufacturing Processes
Elastic-plastic loading bending, torsion, tension, compression Machining drilling, grinding, milling, planning, turning
Joining adhering, brazing, soldering, welding Founding -
Forming shot peening, deep rolling ,drawing, forging, pressing, spinning
Heat-treating hardening, case hardening, nitriding, quenching
Coating cladding, electroplating, galvanizing, plating, spraying, depositing Tab. 2.1: Origin of residual stresses after technological processes [4]
The origin of residual stresses is the consequence of inhomogeneous elastic or plastic
deformation caused by the previous technological treatment of structural parts or
components. This treatment can be either mechanical, thermal or chemical, or a
combination of these treatments. The generated residual stress depends on the geometric
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conditions of the treated technical part and on the parameters of the treatments and
applied processes. Tab. 2.1 lists possible origins of residual stresses in main groups and
some exemplary treatments or manufacturing processes. The residual stress distribution
caused by manufacturing is usually non-uniform over the component cross-section and
in some cases high stress gradients are characteristic.
2.1.3 Technological Relevance Residual Stress induced by manufacturing processes can influence the performance of a
component in a positive or negative way. Harmful effects are usually caused by tensile
residual stresses on the surface of a part since they are often the major cause of fatigue
failures, quench cracking and stress corrosion cracking. On the other hand, compressive
residual stresses on the surface induced e.g. by shot peening are usually desirable since
they increase both fatigue strength and resistance to stress corrosion cracking. In general
terms, residual stresses are beneficial when they are parallel to the direction of the
applied external load and of opposite sense, e.g. a compressive residual stress and a
tensile applied load. When considering the possible effects of residual stresses the need
of reliable and cost-effective analysis methods is of great practical interest. Three main
applications for residual stress measurements are mentioned in [8] exemplary for the
automotive industry:
1. Correcting design flaws: Using residual stress analysis, technical failure caused
by improper design can be corrected before it appears during later processing,
storage or in-service of the component.
2. Improving material properties: In many cases compressive residual stress is
intentionally induced to improve the lifetime of parts, e.g. by mechanical surface
treatment. At this stage, residual stress analysis can support the quality control of
the treatment process used.
3. Predicting behavior of working parts: Residual stresses are often considered
for simulation purposes in order to predict fatigue limits and deformation
following different manufacturing operations. The used numerical models
should be validated with proper residual stress measurements.
A recent example in which residual stress analysis plays a central role in a practical
context is shown in a collaborative research project [9] between partners from industry
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and a research institute. One main objective of the project was the minimization of
distortions of thin walled light metal die casting components that could be generated,
among other things, by manufacturing relevant residual stresses. Within the project the
whole manufacturing process was analyzed and the die casting was simulated with the
objective of optimizing design and individual steps of manufacturing. For this purpose
residual stress measurements in selected structural components were carried out in order
to evaluate processes such as heat treatment and to validate the die casting simulations.
2.1.4 Overview of Residual Stress Measurement Methods A variety of residual stress measurement techniques exist at the present time for
different analysis objectives, materials, kinds of residual stresses, part geometries, the
cost of the measurement and the prices of the equipment required. The main
characteristic of the methods is the physical principal of the measurement i.e. the
directly measured parameters. For this reason it is common in practice to use a
classification of four main groups [3, 6]:
1. Mechanical Methods: These methods are based on the measurement of strains
caused after the removing of stressed material. With these relieved strains the
stress state before material removing can be calculated using the elastic theory.
The hole-drilling method, the ring core technique, the bending deflection method
and the sectioning method are four of the most extensively used mechanical
methods in industry.
2. Diffraction Methods: The x-ray diffraction method and the neutron diffraction
method rely on elastic lattice strains by detecting the changes in the interplanar
spacing of polycrystalline material based on knowledge of the incident
wavelength λ and the change in the Bragg scattering angle Δθ. The stress state is
calculated using the lattice strains and the elastic theory.
3. Magnetic Methods: The magnetic methods measure either the Barkhausen
noise (analysis of magnetic domain wall motion) or the magnetostriction
(measurement of permeability and magnetic induction) of ferromagnetic
materials. The stress state depends on the interaction between magnetization and
the strain state of these materials.
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4. Ultrasonic Methods: These techniques are based on the variations in the
velocity of ultrasonic waves, which can be related to the residual stress state of
the material they fly through.
5. Calculation (Modeling) of Residual Stress: Although this is not a
measurement method, the numerical calculation, e.g. using finite elements, is of
particular importance for the analysis of the origin, influence and relaxation of
residual stress on components. The modeling of residual stresses is carried out,
among other things, with the aim of saving costs and time-consuming
measurements.
At all events, according to an industrial survey [10] the hole-drilling method and the x-
ray diffraction method are the most widely used techniques for measuring residual
stresses in practice.
2.2 General Aspects of the Hole-Drilling Method
2.2.1 Principle The hole-drilling method is a mechanical method for measuring residual stresses and it
is standardized in the ASTM E 837 [11]. Other publications with reference character are
the technical note TN 503 [12] and the NPL guide No. 53 [13]. The principle of the
hole-drilling method is visualized in Fig. 2.2. The basic hole-drilling procedure involves
(1) the drilling of a small hole of diameter d0 into the surface of a stressed material. The
hole can be drilled with mechanical drills and milling cutters (low-speed and high-
speed), with an air-abrasive system or with an electrical discharge machining system.
The stress equilibrium is locally disturbed due to this intervention whereby a new
equilibrium is reached. This change is measured (2) usually radial to the hole with
special strain gage rosettes (dm mean rosette diameter) in the form of relieved strains.
Another possibility is the measurement of the full strain field using optical methods, e.g.
laser interferometry, holography, moiré interferometry and strain mapping. The residual
stresses originally present at the hole location (3) are then calculated from these strain
values using the elastic theory either as an average over the drilling depth from the total
strains upon reaching the final depth, or as a depth distribution from the strain
distribution over the depth as established by incremental drilling.
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Fig. 2.2: Principle of the hole-drilling method
2.2.2 Development Many scientists have worked, and continue to research, on the different aspects of the
hole drilling method as e.g. the drilling process itself, the strain measurement and the
evaluation methods. Some important contributions for the hole-drilling method using
electrical strain gage rosettes and conventional drills or mills are mentioned here.
The hole-drilling method was first proposed by Mathar in 1933 [14, 15]. He measured
diametral changes of a drilled through hole over the cross-section of a plate with a
mechanical extensometer and calculated the average stresses using Kirsch’s theoretical
solution [16] for the stress state in an infinite thin plate. In 1950, Soete and
Vancrombrugge used electrical-resistance strain gages instead of the former mirror
extensometers [17] and thus improved the accuracy of the measurements. In 1956,
Kelsey developed an evaluation method to analyze residual stress depth distributions by
incrementally drilling a blind hole [18]. Kelsey’s method was the basis for later
differential algorithms. The hole-drilling method became a systematic and easily
reproducible procedure for measuring residual stresses after the improvements made by
Rendler and Vigness in 1966 [19], who defined among other things the geometry of the
ASTM E837 standard hole drilling rosette. The practicability of high-speed drilling to
provide a stress-free drilling method was shown in the investigations of Flaman in 1980
[20]. Using finite elements calculations, Schajer developed, among other things, two
practical applications for the evaluation of in-depth stress distributions based on an
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integral approach [21]: the Powerseries Method in 1981 [22] and the Integral Method in
1988 [23, 24]. In 1993 Kockelmann and Schwarz proposed a differential method that
allows the determination of inhomogeneous residual stress depth distributions with
reduced calibration complexity [1, 25].
2.2.3 Scope, Influences and Restrictions The hole-drilling method is, in comparison to other residual stress measuring
techniques, a common, cheap, fast and popular method which covers the following
scope:
• Type of Measured Stresses: The method determines macro residual stresses.
Most of the in-depth evaluation algorithms provide a solution to determine an
elastic plane stress state. However, to avoid local yielding because of the stress
concentration due to the hole, the maximal magnitude of measured residual
stresses should not exceed 60-70 % of local yield stress ReS [1].
• Material Condition: The method is applicable in general to all groups of
materials. Firstly, the materials should be isotropic and the elastic parameters
should be known. Secondly, the analyzed materials should be machinable, i.e.
the boring of the hole should not prejudice the measured strains.
• Resolution: The local resolution of the method is dependent on the equipment
used. Laterally, the resolution ranges in the area of the produced hole diameter.
The minimal infeed or boring increment is in the region of some µm, whereas
the maximal analyzable depth of the hole does not exceed 05.0 d× .
On the whole, the hole-drilling method is simple; disturbing influences and sources of
error have to be taken into account by eliminating or compensating them. Several
publications deal with problems that may arise when using the hole-drilling technique,
e.g. in [26, 27, 28].
The possible disturbing influences and sources of error are mentioned, investigated and
classified into five main groups in [29]. Tab. 2.2 lists the sources of these errors. The
authors consider the influences of the strain gage technique, the influences of the
drilling technique as well as most of the influences of the stress condition as not being
of importance, if these techniques are correctly applied. Possible adverse effects should
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be considered for all influences related to the geometrical boundary conditions, the
plasticization near the hole and the calculation errors of stress gradients into depth.
Strain Gage
Technique
• Change in the residual stress condition when the strain gage
bonding point is pretreated
• Temperature response due to heating when drilling
Drilling Technique
• Production of residual stresses due to drilling
• Deviations from the ideal blind-hole shape
• Eccentricity of hole
• Incorrect depth setting and depth measurement
Boundary Conditions • Influence of component edgings
• Influence of neighboring holes
• Influence of component geometry
Stress Condition
• Influence of multi-axiality
• Influence of stress gradients
• Influence of orientation of strain gages
• Plasticization as a result of the notch effect of the hole
Determination of
Residual Stresses
• Scatter of calculation coefficients and calibration curves
• Error in system of calculation of stress gradients into depth
Tab. 2.2: Disturbing influences and sources of error of the hole-drilling method [29]
In this work, the influences of the component geometry are investigated in detail.
Therefore, the problems regarding the geometrical boundary conditions as well as an
overview of previous investigations are, respectively, mentioned and discussed in
Chapter 2.4.
2.3 Residual Stress Calculation
2.3.1 Average Stress Value over Specimen’s Thickness - Through-Hole Case The through-hole case is the basic calculation procedure for the hole-drilling method.
This solution is based on the solution of the stress state in an infinite thin plate [16]. An
equation {2.1} calculates average values over the thickness of a thin plate (using
Young’s Modulus E) of the principal stresses maxσ and minσ for strain gage rosettes
with finite gage length and gage width.
{2.1} 2122
2313
minmax, )()(42
4)(
°Δ−Δ+Δ−Δ±Δ+Δ
= εεεεεε
σ THTH BE
AE
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In Fig. 2.3 the numbering of the single strain gages, as well as the direction and angle
orientation of the principal stresses maxσ and minσ , are defined for a strain gage rosette.
Fig. 2.3: Hole- drilling strain gage rosette CEA-XX-062UM-120 with angle definition and direction of
principal stresses
The relieved strains iεΔ {2.2} are the difference between the strain readings after the
last drilling step to create the through-hole )(iEε and the initial strain readings )(0 iε
before drilling the hole.
{2.2} )(0)( iiEi εεε −=Δ
Hole-drilling calibration constants for finite area strains THA {2.3} and THB {2.4} are
used in equation {2.1} according to the procedure proposed by [30]. An equivalent
procedure using trigonometric relationships is described in [31]:
{2.3} ( )
)(21
21
12
0
rrwSr
ATH−
+−=
ν
{2.4} ⎥⎦⎤
⎢⎣⎡ −+−+
−−
= 132
0221
20 )1()
23)(1(
)(SSrS
rrwr
BTH νν
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23
In these equations the geometry of the hole is considered in the hole radius 0r (Fig. 2.4).
The geometry of the strain gages is included in the strain gage width w and the radial
distances from hole center to strain gage ends 21 , rr respectively (Fig. 2.4).
Fig. 2.4: Schematic definition of geometric parameters for hole and strain gage
The integrations constants 321 ,, SSS are listed in equations {2.5}, {2.6} and {2.7}:
{2.5} 2
15.0
arctan21rx
rxwxS
=
=⎥⎦⎤
⎢⎣⎡=
{2.6} 2
1
222 25.0
rx
rxwxxwS
=
=⎥⎦⎤
⎢⎣⎡
+−=
{2.7} ( )
2
1
2223 25.031
rx
rxwxxwS
=
=⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−=
The orientation of the principal stresses maxσ and minσ can be defined using the angle
ϕ (angle between strain gage 3 and maxσ , Fig. 2.3). For this, an intermediate angle ∗ϕ
has to be calculated {2.8}.
-
24
{2.8} DenNumarctan
212arctan
21
31
132 =Δ−Δ
Δ−Δ−Δ=∗
εεεεε
ϕ
The angle ϕ can be designated after verifying the relationships {2.9} to {2.17} between
the numerator Num and denominator Den in {2.8}.
{2.9} 0Den ⇒ *ϕϕ =
{2.10} 0Den ⇒ °−= 180*ϕϕ
{2.13} 0=Num 0>Den ⇒ °= 0ϕ
{2.14} 0
-
25
2.3.3 Residual Stress Depth Distribution It is possible to determine a residual stress depth distribution by introducing the hole
incrementally i.e. drilling the hole in consecutive steps and recording strain values for
every depth. There are several methods of evaluation for the calculation of the residual
stress depth distribution, which differ basically in the physical assumption that leads to
the strain relaxation. The most common methods are:
• the differential method [1, 2, 18]
• the integral method [21, 23, 24]
• the powerseries method [22]
• the average stress method [32]
For example, in differential methods it is assumed that the registered strain values at a
specific depth z depend only on the stress values at this depth z. In contrast, integral
methods consider simultaneously the contribution of all stresses at all depths to the
measured strain for a specific depth z. As can also be seen in the blind hole case
(chapter 2.3.2) a previous calibration is needed according to the specific evaluation
method. The actual evaluation algorithms also take into consideration the ideal plate
geometry for the calibration specimen or calibration model respectively.
In the next chapters, the differential MPA method and the integral method are described
in detail.
2.3.3.1 Differential MPA Method The differential MPA-Method was developed in 1993 by Kockelmann and Schwarz in
the Material Research Laboratory (Materialprüfungsanstalt MPA) at the University of
Stuttgart [1, 2, 25]. It is based on the works of Kelsey [18] and Kockelmann and König
[33]. Fig. 2.5 resumes for the MPA-Method the relationship between the calibration
(upper part) and the stress calculation over the measured strains (lower part). To reduce
the amount of possible combinations the hole depth z is normalized over the hole
diameter 0d obtaining the non-dimensional depth ξ {2.18}. In the following
explanations, it is assumed that the calibration hole diameter 0,cd equals the
measurement hole diameter 0d .
-
26
{2.18} 0d
z=ξ
When carrying out the calibration for the hole drilling method using the MPA-Method,
a hole of the diameter 0,cd is drilled incrementally in a specimen subjected to a known
external stress state with the calibration stresses ( )ξσ xc, and ( )ξσ yc, . The calibration
stresses in the MPA-Method can be applied either in an experimental or numerical way.
The relaxed calibration strains ( )ξε xc, and ( )ξε yc, are differentiated in order to calculate
the calibration functions ( )ξxK {2.19} and ( )ξyK {2.20}. Note that in the case of a
calibration, only two strain gages, which are positioned in direction of the two
(principal) calibration stresses, are required The calibration is independent of the
material, i.e. the elastic parameters E and ν of the calibration specimen or model are
not necessarily equal to the material of the measured component.
{2.19} [ ])()(1
)()(
)()(
)(2,
2,
,,
,,
ξσξσ
ξσξ
ξεξσ
ξξε
ξycxc
ycyc
xcxc
x
E
dd
dd
K−
⋅−⋅=
{2.20} [ ])()(
)()(
)()(
)(2,
2,
,,
,,
ξσξσν
ξσξ
ξεξσ
ξξε
ξycxc
xcyc
ycxc
y
E
dd
dd
K−
⋅−⋅=
To calculate the unknown residual stress depth distribution in a component, a hole with
the hole diameter 0d is also incrementally introduced. In this case the depth increments
of the measurement zΔ may be different to the calibration depth increments czΔ . The
derived strain readings ( ) ξξε dd i / of the three different strain gages as well as the
calibration values ( )ξxK and ( )ξyK are used in {2.21} to {2.23} in order to calculate
the stress values ( )ξσ i in the direction of the respective strain gages.
{2.21} ⎥⎦
⎤⎢⎣
⎡⋅⋅+⋅⋅
−=
ξξε
ξνξξε
ξξνξ
ξσd
dK
ddK
KKE
yxyx
)()()()(
)()()( 312221
-
27
{2.22}
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+⋅⋅+⋅⋅
−=
ξξε
ξξε
ξξε
ξνξξε
ξξνξ
ξσd
dd
dd
dK
dd
KKK
Eyx
yx
)()()()(
)()(
)()()( 23122222
{2.23} ⎥⎦
⎤⎢⎣
⎡⋅⋅+⋅⋅
−=
ξξε
ξνξξε
ξξνξ
ξσd
dKd
dK
KKE
yxyx
)()()(
)()()(
)( 132223
Fig. 2.5: Differential MPAII Method: Relationship between Calibration and Calculation of Stress
The maximal and minimal stresses are then calculated easily by using the relationships
in Mohr’s circle {2.24}.
{2.24} 2232
2131
minmax, ))()(())()((21
2)()()( ξσξσξσξσξσξσξσ −+−⋅±+=
-
28
The angle ( )zϕ is calculated for each increment similar to {2.8}, with the difference that the equation uses the stress values in strain gage direction σ1,2,3 instead of strain
values ε1,2,3.
2.3.3.2 Integral Method The basic formulae of the integral method shown in this chapter are taken from
Schajer’s 1988 work [23, 24] and are based on the previous works of Bijak-Zochowski
[21]. The method is implemented among other things in the commercial residual stress
evaluation programme H-Drill [34]. To give a brief explanation of the integral method,
an equi-biaxal plane stress state ( )HP at a normalized depth from surface H {2.25} is assumed (using almost the same nomenclature as in [23, 24]). Thus, the global strain
response ( )hp in the actual normalized hole depth h {2.26} would be equal for all strain gages. In the case of the integral method the normalizing factor is the strain gage
mean radius mr and not the hole diameter 0d as in the differential MPA-Method.
{2.25} mrZH = with Z: depth from surface
{2.26} mrzh = with z: hole-depth
The relationship between the strain ( )hp and the stress ( )HP is given in {2.27}, where
),(ˆ hHA is the strain relaxation per unit depth or influence function to integrate.
{2.27} dHHPhHAE
hph
)(),(ˆ1)(0∫
+=
ν hH ≤≤0
The differential notation of the strain relaxation ( )hp {2.27} is unusable because the strain values ip are recorded in practice after each drilling step or increment i and not
continuously. Therefore, {2.27} can be approximated by the discrete form in {2.28}
where jP is the equivalent uniform stress within the thj − hole depth increment and ija
is the strain relaxation due to a unit stress within increment j of a hole of i increments
-
29
deep. {2.29} is an equivalent matrix notation in which four hole depth increments are
given as examples.
{2.28} jij
jiji PaE
p ∑=
=
+=
1
1 ν nij ≤≤≤1
{2.29}
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
4
3
2
1
44434241
333231
2221
11
4
3
2
1
1
PPPP
aaaaaaa
aaa
Epppp
ν
Fig. 2.6: Integral method: loading steps of calibration and matrix of strain relaxation
The calibration of the integral method is usually done by FEM calculation because it is
almost impossible to apply experimentally the stress jP at a specific increment j of a
-
30
hole. The loading steps of the calibration with the stress jP and the resultant matrix of
strain relaxations ija are exemplarily shown in Fig. 2.6 for the case of four hole-
depth increments i . Here, jP is applied e.g. as a pressure load on specific depth sector
j of the inner surface of the hole, which is schematically represented for each loading
step with a geometrical two dimensional hole which has been halved. Hence, each value
of ija can be numerically determined and the unknown residual stress distribution can
then be then calculated using the measured strain distributions ip in the inverted form
of {2.28}.
2.4 Geometrical Boundary Conditions
2.4.1 Prefacing Remarks The geometrical conditions of the component play, in some cases, an important role
when carrying out a hole drilling measurement. One of the first problems that may arise
is the application of the strain gage rosette at the desired measurement point. This
means that the component geometry at the measurement point should allow the bonding
of the strain gage rosette, e.g. the width of the component should be greater than the
dimensions of the rosette. Moreover, the geometrical conditions around the
measurement point should not constrict the drilling tool or device when introducing the
hole.
Apart from these practical problems, different geometrical parameters influence the
evaluation of a hole-drilling measurement. The most important geometrical parameters
are the strain gage mean diameter md , the hole diameter 0d and the hole depth z ,
because they directly influence the calibration strain as well as the measured strain. For
example, a set of calibration strains is always valid for a specific strain gage rosette and
a specific hole diameter. The strain gage mean diameter md is treated as a constant
parameter, because the calibration is normally carried out for one specific strain gage
rosette. In contrast, the hole-diameter can differ between the diameter 0,cd , which is set
after the calibration, and the diameter 0d , which is determined after the measurement.
Evaluation algorithms contain a limited set of calibration functions or calibration
constants for specific hole-diameters, which range between the maximum and the
-
31
minimum possible (measurement) hole diameter 0d . In this case, an interpolation of the
specific calibration strains, calibration functions or calibration factors respectively is a
common procedure [24, 33]. The same difficulties appear concerning differences
between the calibration hole depth increments czΔ and the hole depth increments of the
measurement zΔ .
In addition, the calibration of the hole-drilling method for the different residual stress
evaluation algorithms is normally carried out using, as already mentioned, the geometry
of an ideal thick and wide plate. For this reason, errors in stress calculation could appear
when evaluating the residual stress after a hole-drilling measurement on a real
component e.g. a part with a thin cross section and a curved shape. In some works, a
geometry-specific calibration is suggested in order to minimize the calculation errors (s.
Ch. 2.4.3). This means that in these cases some geometrical values should be treated as
an additional parameter in the evaluation.
Thus, the main focus of this work is the investigation of the influence of the component
geometry on the residual stress calculation and a correction of potential errors. In the
following chapters geometrical parameters are listed and an overview of previous
researches dealing with the geometry influence on the results of the hole-drilling
method is given.
2.4.2 Definition of Possible Geometrical Boundary Conditions Real components are characterized by a complex shape compared to the shape of the
calibration specimens or calibration models respectively, on which the hole-drilling
evaluation methods are based. Although some hole-drilling measurements are carried
out in simple geometrical parts, e.g. in sheet metal, many measurement points of real
components are characterized by a combination of two or more geometrical conditions.
Tab 2.3 lists geometric parameters that may influence the results of the hole-drilling
measurement, if the defined limit of these parameters is under-run.
-
32
Symbol Description
T Component Thickness
D Distance between hole center and free edge or border of component
DE Distance between hole center and edge of component
DH Distance between two holes
W Component Width
R Uniaxial curved shape: Outside radius of shell or cylinder
RI Uniaxial curved shape: Inside radius of shell or cylinder
RB Biaxial curved shape: Outside radius of shell or sphere
RBI Biaxial curved shape: Inside radius of shell or sphere Tab. 2.3: Geometrical parameters which can influence a hole-drilling measurement
2.4.3 Overview of Previous Investigations The influence of geometry on residual stress analysis has already been investigated by
several researchers, who basically discuss the individual effect of the component
thickness T, the distance between the hole and the component edge D (or the
component width W) and the radius of surface curvature R (s. Fig. 2.7).
2.4.3.1 Thickness In ASTM 837 [11] a definition of thick and thin specimens is already considered. Thus,
a specimen with a thickness mdT ×> 4.0 is considered to be thin whereas mdT ×< 2.1
is considered to be thick. (e.g., thin mmT 052.2> and thick mmT 156.6< for a
062UM-120 strain gage rosette). The intermediate case mdT ×= 2.1...4.0 is not within
the scope of the standard. According to ASTM 837, the stresses are assumed to be
uniform throughout the hole depth. Otherwise the result should be considered as non-
standard. A measurement on a thin specimen should be carried out and evaluated using
the procedure for the through-hole case. In the case of drilling a hole in a thick
specimen, the standard provides calibration data for different strain gage rosette types.
-
33
Fig. 2.7: Selected types of geometrical boundary conditions for the hole-drilling technique
In earlier works Altrichter [35] and Motzfeld [36] pointed out possible sources of errors
when carrying out a hole-drilling measurement. They both described the procedure for
the through-hole case and gave limits of thickness with 0dT > [35] and 05.0 dT ×>
[36]. At that time, no standardized strain gage rosettes were used and the dimensions of
e.g. the hole-diameter were generally bigger then in modern hole-drilling
measurements. Motzfeld used e.g. a self-made hole-gage assembly with a hole diameter
of d0 = 6mm and a sheet thickness of T = 12mm respectively [36]. Using such
experimental configuration the total relieved strain is the sum of the strain relaxation
due to the applied stress, the strains due to drilling operation and the strain due to
localized plastic flow as established in [37], where the alignment of the hole-gage
assembly was modified to a recommended range of nondimensional hole diameter of
-
34
λ = d0/rm = 0.5...1.2, which corresponds approximately to the suggested values in the
later TN-503 [12].
The influence of the specimen thickness on the results of a residual stress vs. depth
calculation using an in-depth evaluation method was shown by Münker in [38]. He
calculated, among other things, the stress difference in percentage which arises after
comparing the hole-drilling results of thin specimens in relation to a reference thick
specimen. Fig 2.8 shows the stress error in percent vs. the specimen thickness taken
from [38]. The calculations were made using FEM models of different thickness which
were loaded with an equibiaxal plane stress condition σII / σI = 1 and an equibiaxal pure
shear stress condition σII / σI = -1. The stress evaluation was carried out according to the
powerseries method. The stress differences are negligible for thick specimens with
mmT 5.2> . For thin specimens the maximal difference is more than 20% for the
models with plane stress condition and more than 10% for the models with pure shear
condition. Münker pointed out that for the investigation of thin components a specific
calibration which considers the component thickness is necessary.
Fig. 2.8: Stress difference against the component thickness T according to [38]
The relationship between the calibration coefficient for the integral in-depth evaluation
method and plate thickness was investigated numerically as well as experimentally by
Aoh and Wei in [39] and [40]. Thus, the calibration coefficients vary with the thickness
-
35
of the flat specimen so that the accuracy of residual stress analysis can be improved if
appropriate calibration coefficients are chosen to match the component thickness. For
this purpose, a transition range between thin and thick plates is given with
mdT ×= 0.2...34.1 ( 0811.2...88.1 dT ×= ). Considering the dimensions of the strain
gage rosette (062RK-120) used in [40] the transition range of the specimen thickness is
about mmT 13.5...44.3= . In a practical application, Schajer specifies in [34] the
acceptable component thickness as being mdT ×> 2.1 for the correct calculation of
residual stresses using the H-Drill software.
In his work in the area of metal forming, Haase [41] underlines the need to adapt the
calibration to the component thickness, if the component is thin with 05.2 dT ×< . The
developed evaluation program there is based on the integral method and has
implemented several sets of calibration coefficients for different metal sheet thickness.
Kockelmann and König [33], who developed a parent version of the differential in-depth
evaluation MPA-Method, defines the limit of component thickness for a correct stress
calculation as being 03 dT ×> . He pointed out that a measurement on thin structures
could lead to misinterpretations because global stress changes and deformation
additionally influence the measurement results.
2.4.3.2 Distance between Hole Center and Component Edge The ASTM 837 standard as well as the TN-503 do not contain recommendations
dealing with limits of the distance between the hole and the component edge. Altrichter
[35] and Motzfeld [36] set the limits of the distance hole-edge with 015 dD ×> for a
correct calculation of a through-hole measurement.
Münker [38] investigated, analogical to the thickness, the influence of the distance
between the hole and the component edge by varying the model width DW ×= 2 (s.
Fig 2.9). A significant edge effect appears only by a width of mmW 20< or a distance
mmD 10> respectively. Considering specimens with mmD 5= , the maximal
difference is c. 7.5% for the models with plane stress condition and c. 12.5% for the
models with pure shear condition. Schajer [34] gives the minimal component width
-
36
with as mdW ×= 3 for the correct use of the H-Drill software. This means a component
width of mmW 4.15= for a 062UM-120 strain gage rosette.
Fig. 2.9: Stress difference against the distance hole-edge D according to [2.35]
Preckel [42] carried out investigations in a 2 mm thick metal sheet plate using different
loading conditions (equibiaxal plane stress condition and equibiaxal shear stress
condition). He defined the minimal distance from the border of the hole to the
component edge as being 05 dDa ×= . Distances of 05 dDa ×≥ yield to a maximum
relative error of less then 10% (s. Fig 2.10). Similar observations were made by König
[33].
-
37
Fig. 2.10: Maximum Relative Error against the distance hole-edge Da according to [33]
2.4.3.3 Radius of Surface Curvature Preckel calculated in [42] the effect of the surface curvature with a finite element
analysis using a sphere model. According to this, the maximal relative error is c. 15% in
a specimen with surface curvature radius of mmRB 40= . Therefore, for a correct
execution of the hole-drilling method he suggested a surface radius of a biaxial curved
shape with mmRB 100≥ whereas the other surface radius should in be all cases
mmRB 15> , to provide an appropriate application of the strain gage rosette. Also, in the
case of the surface curvature, Haase [41] underlines the need to implement a geometry-
specific calibration in the evaluation algorithms.
According to Kockelmann and König [33], the correction of the influence of the surface
curvature cannot be given in general and it is possible only in specific cases. To avoid
errors greater than 20% the surface radius should be 03 dR ×≥ with an evaluable depth
of 04/ dRz ≤ (e.g. mmd 8.10 = correspond to mmR 4.5= and mmz 75.0≤ ).
In [43], Zhu and Smith developed an analysis for interpreting relaxed strain data from
the incremental hole drilling method and an integral in-depth evaluation for components
-
38
with curved surfaces. For this purpose, the geometry of an 8mm diameter steel was
explicitly considered in three dimensional FE calculations, which provide the stiffness
coefficients for the evaluation method. The developed analysis was tested for 8mm
diameter hot forged steel round bars. According to the paper, the obtained results
showed a good correlation when compared with x-ray diffraction measurements.
The influence of a complex geometry on the results obtained with the hole-drilling
method was investigated in several works by Montay, Lu et al. Using FE calculations
and an integral evaluation method the hole-drilling method was adapted to the following
three dimensional shapes: (1) a sphere [44], (2) a cylinder [45] and (3) a crankshaft [46].
The calibration coefficients show a dependence on the curvature radius for the spherical
models as well as for the cylindrical models. According to [2.41], the surface of a
sphere can be considered to be plane, if the sphere radius is 05.17 dRB ×> . As shown in
Fig. 2.11, the maximal calculated error on numerical calibration coefficients is c. 35%
for a specimen with a sphere radius of 05 dRB ×= (with mmRB 10= and mmd 20 = ).
Fig. 2.11: Calculated error of calibration coefficients over the hole-depth for different sphere radius R
[44]
-
39
Fig. 2.11: FE mesh around a hole in the fillet of a crankshaft for the calculation of geometry-specific
calibration coefficients for the hole-drilling method [46]
In case of the investigation with the cylindrical model [45], the authors suggest a
cylinder radius of c. 025 dR ×> (with mmR 50> and mmd 20 = ) to be considered as
plane, because no significant variation in the calibration coefficients depending on the
cylinder radius was observed. In [46] the hole-drilling method was adapted in the fillet
of a crankshaft model (s. Fig. 2.12) and as a result, geometry-specific calibration
coefficients were calculated in order to solve residual stress evaluation in complex
shapes.
2.5 Concluding Remarks on State of Knowledge The measurement and analysis of residual stress is of great interest in the field of
material engineering as well as for industrial objectives. Therefore, the hole-drilling
method provides a versatile, easy to handle and economic measurement solution. Like
other measurement techniques, the hole drilling method is influenced by different
parameters which can restrict in special cases the use of the method. One possible origin
of measurement errors is the influence of the geometry of a component with complex
shape. This is due to the fact that most of the evaluation algorithms include calibration
functions or coefficients, which were determined by using the geometry of a simply-
shaped specimen or model. One possible approach to this problem is the definition of
geometrical limits as established by different studies. Tab. 2.4 summarizes the
geometrical boundary conditions for the adequate use of the differential MPA-Method
[2] and is based mainly on the previous investigations of Kockelmann and König [33].
-
40
Another possibility proposed by different researchers is the implementation of a
geometry-specific calibration on the evaluation algorithms. Examples of such an
implementation on evaluation algorithms based on the integral method have been shown
in different works. A possible correction of the geometry influence using a differential
evaluation method (e.g. MPA-Method) has not yet been investigated.
Boundary Condition Reference Value e.g. d0 = 1.8 mm
Min. Component Width W > 10 - 20 x d0 W > 18 - 36 mm
Min. Component Thickness T > 3 x d0 T > 5.4 mm
Min. Distance Between Hole and Component
Edge D > 5 - 10 x d0 D > 9 - 18 mm
Min. Distance Between Two Holes DH > 5 x d0 DH > 9 mm
Min. Radius of Surface Curvature R > 3 d0 R > 5.4 mm Tab. 2.4: Reference values for geometrical boundary conditions according to the MPA-method [2]
-
41
3 Experimental and Numerical Methods of Investigation
3.1 Prefacing Remarks The general procedure of the experiments, as well as that of the numerical calculations
in this investigation, is based on the calibration principle of the differential MPA-
Method. This allows the analysis of the influence on the results of a hole-drilling
measurement caused by:
• individual geometrical parameters T, D or R using flat and cylindrical tensile
specimens respectively, or,
• a combination of these geometrical parameters using the same specimens and,
additionally, specimens of complex shape which produce a stress state similar to
that in a real component.
The aim of the experimental calibration is to provide evaluation data in order to
compare and verify the numerical models. For this reason, the specimen’s material, the
specimen’s shape, the used equipment and the calibration process are described in the
first part of this chapter. Ultimately, the objective of the numerical calculations is the
systematic investigation of the geometry influences on the hole-drilling results and the
calculation of geometry specific calibration functions. Hence, the second part of this
chapter specifies the different numerical models used in this investigation as well as the
numerical calibration procedure.
Fig. 3.1: Calibration coordinate system and position of strain gages
-
42
Fig. 3.1 shows the conventions used for the experimental investigation as well as for the
numerical calibration concerning the coordinate system, the strain gage rosette and the
loading conditions. The point of origin is in both cases, experimentally and numerically,
the hole center at the surface. The experimental calibration is carried out in a uniaxial
testing machine, i.e. the x-axis is defined as the load direction with σc,x as the
corresponding calibration stress. Thus, strain gage no. 3 is positioned parallel to the x-
axis or load direction whereas strain gage no. 1 is parallel to the y-axis. The geometry of
the FEM-models and their loading conditions are based on the experiments. The
numerical calibrations distinguish a stress component σc,x in x-direction and σc,y. in y-
direction. These conventions are effective within the whole work.
3.2 Experimental Details
3.2.1 Material The specimens were made of fine grain structural steel S 690 QL, which is water
quenched and tempered. The chemical composition as established after an analysis
using sparking spectroscopy is listed in Tab. 3.1.
C Si Mn P Cr Mo Fe
0.180 % 0.599 % 0.943 % 0.00724 % 0.740 % 0.255 % balance Tab. 3.1: Chemical composition of S690 QL
E [MPa] Rp0,2 [MPa] ReH [MPa] Rm [MPa] Elongation [%]
Experiment 210000 747 - 833 14
[47] - 690 690 790-940 16 Tab. 3.2: Mechanical proprieties of S690 QL
The hole-drilling method is based on elastic equations and, consequently, yielding and
plastic strains respectively must be avoided during the calibration. However, stress
concentrations, three times greater than the nominal or calibration stress, arise in the
vicinity of the hole. This could lead to erroneous strain measurements, if the local stress
concentration exceeds the yield stress. On the other hand, the experimental calibration
-
43
requires the highest possible stress values in order to measure evaluable calibration
strains. With a minimum yield stress of ReH = 690 MPa (s. mechanical properties in
Tab. 3.2) the chosen specimen material S 690 QL offers the possibility to apply
maximal calibration stresses of approx. 210 MPa without local yielding near the hole.
Before carrying out the experiments, the specimens were submitted to a stress relief
annealing to avoid detrimental effects due to residual stresses induced by the
manufacturing. This process was carried out at a temperature of approx. 600 °C during
1.5 h in a protective-gas tube furnace of the type “Labotherm R50/300/13”. Finally, the
specimens were cooled down slowly inside the furnace.
3.2.2 Specimens Four different types of specimens were used:
Type 1: Flat tensile specimens with different thickness (s. Fig. 3.2).
Fig. 3.2: Flat tensile specimen
The flat tensile specimens allow an individual investigation or a combination of the
geometrical parameter T and D by using specimens with different thickness and by
bonding the strain gage rosette with a defined distance between the hole center and the
specimen’s edge.
The following flat tensile specimens with different geometrical parameters were
investigated:
-
44
• Specimens with a variable thickness of T= 1.5; 2.0; 2.5; 3.0; 4.0; 6 mm and a
constant distance of D = 20 mm.
• Specimens with a variable distance of D = 2.0; 5.0; 10; 20 mm and a constant
thickness of T = 6 mm.
• Thin specimens with T=1.5 mm and a minimal distance of D = 2 mm.
The reference specimen has a thickness of T = 6 mm with a strain gage rosette bonded
on the center of the specimen with a resulting hole-edge distance of D = 20 mm. In this
case, assuming a maximal hole diameter of d0 = 2 mm, the reference specimen satisfies
the criteria for the geometrical boundary conditions as established in Tab. 2.4.
The individual flat tensile specimens are identified by an abbreviation which includes
both parameters T and D. For example, the abbreviation “fT1.5D2” refers to the flat
specimen with a thickness of T = 1.5 mm and a distance of D = 2 mm.
Type 2: Solid and hollow cylindrical tensile specimens with different cylinder
radius (s. Fig. 3.3).
The solid cylindrical specimens allow the investigation of the geometrical parameter R,
the outside radius of a uniaxial curved shape. Three solid cylinders with R = 3; 4; 6 mm
and a constant gage length of l = 40 mm were manufactured. The strain gage rosette was
always bonded in the center of the specimen (s. Fig 3.3 specimen scR6), i.e. the
geometrical parameter D does not fall below the boundary condition.
The hollow cylindrical specimens allow the investigation of the combination of the
geometrical parameter R and T. They are based on the geometry of the solid cylindrical
specimens with the difference that the wall thickness is always constant with
T = 1.25 mm, which is considered to be thin.
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Fig. 3.3: Solid and hollow cylindrical tensile specimen
The abbreviations of the cylindrical specimens consider shape of the cross-section of
the specimen (“sc” solid cylinder and “hc” hollow cylinder) as well as the outside
radius R.
Type 3: Flat tensile specimens with a ø10 mm center through-hole and different
thickness (s. Fig. 3.4).
With this type of specimen, a stress state similar to that in a simple flat component can
experimentally be simulated. Considering the notch effect, a biaxial stress state before
drilling at the measurement point can be applied.
The geometry of these specimens base on that of the flat specimens of Type 1 and,
additionally, have a through-hole of ø10 mm in the center. The strain gage rosette is
bonded near the edge of the ø10 mm through-hole with D = 2 mm. A thick specimen
with T =6 mm and a thin specimen with T =1.5 mm were manufactured.
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Fig. 3.4: Flat tensile specimens with a ø10 mm center through-hole and different thickness
In addition, a single strain gage is placed axial to the centerline of the ø10 mm hole (s.
Fig. 3.4) in order to control the load application and correlate the control strain readings
with the subsequent simulation.
The specimens are identified with an abbreviation which includes the thickness (T) of
the specimen and the distance to the edge of the ø10 mm hole (D). For example
“fhT1.5D2” refers to the flat specimen with center hole, a thickness of T = 1.5 mm and
a distance D = 2 mm.
Type 4: Hollow cylindrical tensile specimens with a ø6 mm center through-hole (s.
Fig. 3.5).
Similar to the Type 3 specimens a through-hole of ø6 mm is introduced in the center of
a hollow cylindrical specimen and over its whole cross-section in order to
experimentally simulate a stress state which can be found in a component.
This specimen bases on the geometry of the hollow cylindrical specimen “hcR6” and
provides, on the other hand, a combination of the three geometrical parameters, which
are investigated in this work: a wall thickness of T = 1.25 mm, a distance to the edge of
the ø6 mm through-hole of D = 2 mm (when bonding the strain gage rosette
accordingly) and a curvature radius of R = 6 mm.
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Fig. 3.5: Hollow cylindrical tensile specimen with a ø6 mm center through-hole
Analog to the Type 3 specimens, a single strain gage is placed axial to the centerline of
the ø6 mm hole (s. Fig. 3.5) in order to control the load application and correlate the
control strain readings with the subsequent simulation.
The abbreviation for this type of specimen is “hchR6” and means a hollow cylindrical
specimen with a center hole and an outside radius of R = 6 mm.
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3.2.3 Hole Drilling Device and Calibration Equipment
Fig. 3.6: Experimental calibration set-up
The hole in the measurement point on the specimen was introduced using the high-
speed drilling device “RS-200 Milling Guide” of the company “Measurements Group”.
The high-speed turbine generates, according to the manufacture’s specifications, a
rotation of the cutting tool of up to 300.0000 r.p.m., which ensures that almost no
plastic deformation in the vicinity of the hole occurs due to the drilling process. Special
milling cutters with titanium-nitrite coating were used for drilling. Their diameter is
ø1.6 mm and leads to a typical hole diameter of approx. d0 = 1.8 mm. The relieved
strains were measured using the strain gage rosettes of type CEA-06-062-UM (also
“Measurements Group”). The strain gages were connected in a (Wheatstone) quarter
bridge and were completed to a full bridge by using an additional temperature
compensation strain gage and the amplifier “DMCPlus” of “HBM”. During calibration
the values of the different strain gages were displayed on-line using data acquisition
software programmed with “Testpoint” of the company “Keithley”. After each depth-
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increment, the strain values were recorded. The final calibration data was written to an
ascii- file.
The experimental calibration set-up is shown in Fig 3.6. The tensile specimens were
loaded using a universal testing machine of the type Zwick Z100 which has a 100 kN
load cell. For the specimens without a center through-hole the calibration stress was
calculated with σc = F / A (F: applied uniaxial force; A: specimen’s cross-section area).
In the case of the specimens of Type 3 and 4, the applied load was set using a control
strain gage. The hole-drilling device was jointed using fixing screws to the lower
clamping jaw of the testing machine in order to place the milling cutter perpendicular to
the clamped specimen. For this purpose, a special mount was designed. The mount
allows height and lateral adjustment in order to arrange the hole-drilling device and,
especially, the milling cutter over the measurement point.
3.2.4 Experimental Calibration Procedure Before starting with the calibration the strain gage rosettes were applied at the desired
place on the specimen according to the recommendations in [48, 49]. The steps of the
calibration procedure are (s. Fig 3.7):
(1) Engage specimen to clamping jaws by loading and unloading with a small stress
magnitude.
(2) Zero-balance the strain gage circuits.
(3) Apply a load σc,x,min (50 MPa) and measure strain (before drilling).
(4) Apply a load σc,x,max (210 MPa) and measure strain (before drilling).
(5) Unload specimen. Arrange the hole-drilling device over the measurement point
and establish zero depth.
(6) Apply a load σc,x,min (50 MPa) and measure strain (z = 0) in order to verify step
3.
(7) Apply a load σc,x,max (210 MPa) and measure strain (z = 0) in order to verify step
4.
(8) Unload specimen. Introduce first drilling increment (e.g. z = 0.02 mm).
(9) Apply a load σc,x,min (50 MPa) and measure strain at first increment (e.g.
z = 0.02).
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(10) Apply a load σc,x,max (210 MPa) and measure strain at first increment (e.g.
z = 0.02).
… Repeat steps 8 to 10 using sufficient depth increments until reaching the
maximum hole depth.
(n-1) Apply a load σc,x,max (210 MPa) and measure strain at maximum hole depth.
(n) Unload specimen and measure the hole diameter.
Fig. 3.7: Experimental calibration procedure
The relieved calibration strain )(, zc
icσε ΔΔ , which is related to the calibration stress
cσΔ {3.1}, is calculated after the calibration procedure using {3.2}.
{3.1} minmax σσσ −=Δ c here:
MPaMPaMPac 16050210 =−=Δσ
{3.2} )]0()0([)]()([)( min,max
,min
,max
,,σσσσσ εεεεε icicicic
ciC zzz −−−=Δ
Δ
This procedure for the determination of the calibration strain )(, zicεΔ is proposed in
different works [38, 41] and should ensure that the calibration stress is only dependent
on the applied uniaxial calibration load. Using stress differences, other sources of load
are omitted from the calculation. In general, undesirable sources of load could be the
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prestress caused by some small inexactness when fixing the specimen to the clamping
jaws in the form of bending or torsion or residual stresses in the specimen caused by
manufacturing.
After finishing the calibration, the hole diameter is measured using the microscope
assembly which is provided with the “RS-200 milling guide”.
3.3 Numerical Methods and Calibration
3.3.1 Models and Simulation Procedure The geometry of all three-dimensional finite elements models was generated using the
pre-processor “MSC.Patran”. The simulation itself was carried out by means of the
finite element code ABAQUS Standard [50]. For this purpose, a PC was used with a
“Pentium 4” processor of 3.4 GHz and a RAM of 4.0 GB. In all cases, the drilling of the
hole was simulated by applying the birth and death option to the elements forming the
hole volume. All models are based on the geometry and the strain gage rosette
arrangement of the specimens of the experimental calibration (s. Ch 3.2.2). Thus, four
types of models were generated:
Type 1: 3D-models of flat tensile specimens with different thickness (s. Fig. 3.8).
The models of the flat tensile specimen allow, similar to the experimental calibration, an
individual investigation or a combination of the geometrical parameter T, D and, in
addition, the consideration of different hole diameters d0. For this, a prototype model
was first generated. The models with different geometrical parameters T, D and d0 were
modelled by deleting and rearranging the elements of the prototype model.
The following geometrical parameters were considered for the models of the flat
specimens:
• Thickness: T = 1; 1.5; 2; 3; 6 mm
• Distance hole-edge: D = 2; 3.5; 5; 10; 20 mm
• Hole diameter d0 = 1.7; 1.75; 1.8; 1.85; 1.9 mm
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The combination of all geometrical parameters leads to 125 models of the flat tensile
specimens. The number of elements varies between 26,460 for the models with
T = 1 mm and 41,580 for the models with T = 6 mm. The elements are three-
dimensional solid elements of the type C3D8 (8-node linear brick). Only some elements
in the hole volume are of the type C3D6 (6-node linear triangular prism). Due to
symmetry, only the half geometry of the specimens was modelled, i.e. the left side of
the models is regarded as the symmetric plane. The main dimensions of the models as
well as the approximate location of the strain gages are shown in Fig 3.8.
Fig. 3.8: 3D-model of a flat tensile specimen
The model fT6D20m with T = 6 mm and D = 20 mm is defined as the reference model,
because it satisfies the criteria for the boundary conditions (s. Tab 2.4). The reference
hole diameter is specified with d0 = 1.8 mm for this investigation.
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The abbreviation of the models is the same as the specimen with the addition of an “m”
for “model” at the end. For example, “fT1.5D2m” refers to the model of the flat
specimen with T = 1.5 mm and D = 2 mm.
Type 2: 3D-models of solid and hollow cylindrical tensile specimens with different
cylinder radius (s. Fig. 3.9 and Fig. 3.10).
In the case of the solid cylinder, three different models were generated which consider
three different cylinder radius R = 3, 4, 6 mm. The hollow cylinders were generated out
of them by deleting the corresponding elements and creating a tube profile. All hollow
cylindrical models have a thickness of T = 1.25 mm. Fig 3.9 shows exemplarily the
shape and the main dimensions of the models of the hollow cylindrical tensile
specimens. For all cylindrical models a hole diameter of d0 = 1.8 mm was modelled.
Fig. 3.9: 3D-models of hollow cylindrical tensile specimen whit different cylinder radius R
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Fig. 3.10: Detailed views of the hole region in the hcR6m model
Only a quarter of the cylinder needs to be modeled due to symmetry. Fig. 3.10 shows
exemplarily for the hcR6m model two detailed views of the hole region. In this picture,
there is schematically illustrated the hole volume, both symmetric planes and the
approximate location of the strain gages in x- and y-direction.
The majority of the elements used are of the Type C3D8. In some regions of the models
C3D6 elements were needed in order to create a congruent mesh of the cylinder
curvature. In Tab. 3.1 the number of elements for each cylindrical model is listed.
Models scR6m hcR6m scR4m hcR4m scR3m hcR3m
No. Elements 70,746 46,410 47,760 33,348 33,988 26,284 Fig. 3.3: Total number of elements for the different models of the cylindrical tensile specimens.
The abbreviation for the different cylindrical models is the same as for the specimens
with the addition of an “m” at the end.
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Type 3: 3D-models of flat tensile specimens with a ø10 mm center through-hole
and different thickness (s. Fig. 3.11).
The models of this type of flat tensile specimen utilize the notch effect near the ø10 mm
center through-hole, which causes a non-uniform bi-axial stress state. In a real
component it is assumed that approximately such a stress state exists.
Two specimens were modelled, a “thin” one with T = 1.5 mm and a “thick” one with
T = 6 mm. The measurement point is modelled with a distance of D = 2 mm to the edge
of the ø10 mm through-hole. The stress magnitudes before drilling at the hole volume,
i.e. the resulting calibration stresses σc,x and σc,y, can be averaged over the depth after
each numerical calibration. Fig. 3.11 shows the main dimensions of these Type 3
specimens, the location of the measurement point (with a variable hole diameter of
d0 = 1.7; 1.8, 1.9 mm) and the approximate location of the strain gages. Additionally,
the models of Type 3 are longer than the models of Type 1 in order to get strain
readings in the region of the control strain gage. The number of elements varies between
47,120 for the models with T = 1.5 mm and 76,570 for the models with T = 6 mm. With
the exception of some C3D6 elements in the hole volume, the elements are three-
dimensional solid elements of the type C3D8. Because of the symmetry, only half of the
geometry of the specimens was modelled.
Two abbreviations based on the experiments are used for the two different models:
“fhT1.5D2m” for the “thin” model and “fhT6D2m” for the thick model.
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Fig. 3.11: 3D-model of a flat tensile specimen with a ø10 mm center through-hole
Type 4: 3D-model of the hollow cylindrical tensile specimens with a ø6 mm center
through-hole (s. Fig. 3.5).
Similar to the models of Type 3, this model of a cylindrical tensile specimen utilizes the
notch effect near the ø6 mm center through-hole, which causes a bi-axial stress state
with a different depth distribution at the measurement point. In t