Institut für Werkstofftechnik Metallische Werkstoffe€¦ · Institut für Werkstofftechnik...

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

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

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

„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

5

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

6

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


4.2 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


5.2 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

7

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


6.2 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

8

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

9

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

10

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

11

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.

13

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.

14

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.

15

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

16

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

17

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.

18

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.

19

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

20

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

21

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

22

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

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

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