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: Zweiter Gutachter:
Prof. Dr.-Ing. habil. B. Scholtes 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 Forschungsarbeiten 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..................................................................................
2.4.2
Definition of Possible Geometrical Boundary Conditions...................... 31
2.4.3
Overview of Previous Investigations......................................................
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
30 32
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
4.1
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
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
6.1
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....................................................................................
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.........................................
7
Evaluation of the Implementation of Geometry Specific
143
149
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.........................................
7.8
Summary of the Evaluation of the Calculation Program with
169
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 hcR3m hchR6
flat specimen with a thickness of T = 1.5 mm and a distance hole-edge of D = 20 mm FEM model* of a hollow cylindrical specimen with an outside radius of R = 3 mm 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 holedrilling 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 blindhole 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 selfequilibrating, 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 longrange 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 Elastic-plastic loading Machining Joining Founding Forming Heat-treating Coating
Manufacturing Processes bending, torsion, tension, compression drilling, grinding, milling, planning, turning adhering, brazing, soldering, welding shot peening, deep rolling ,drawing, forging, pressing, spinning hardening, case hardening, nitriding, quenching 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 xray 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 highspeed), 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 0.5 × d 0 .
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
Change in the residual stress condition when the strain gage bonding point is pretreated
Technique
•
Temperature response due to heating when drilling
•
Production of residual stresses due to drilling
•
Deviations from the ideal blind-hole shape
•
Eccentricity of hole
•
Incorrect depth setting and depth measurement
•
Influence of component edgings
•
Influence of neighboring holes
•
Influence of component geometry
•
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
•
Scatter of calculation coefficients and calibration curves
Residual Stresses
•
Error in system of calculation of stress gradients into depth
Drilling Technique
Boundary Conditions
Stress Condition
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 σ max and σ min for strain gage rosettes with finite gage length and gage width.
{2.1} σ max,min =
E (Δε 3 + Δε 1 ) E 2 ± 4 ATH 4 B TH
(Δε 3 − Δε 2 ) 2 + (Δε 2 − Δε 1° ) 2
22
In Fig. 2.3 the numbering of the single strain gages, as well as the direction and angle orientation of the principal stresses σ max and σ min , 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 Δε i {2.2} are the difference between the strain readings after the last drilling step to create the through-hole ε E (i ) and the initial strain readings ε 0( i ) before drilling the hole. {2.2} Δε i = ε E (i ) − ε 0(i )
Hole-drilling calibration constants for finite area strains ATH {2.3} and B TH {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]: r0 (1 + ν )S1 2 w(r1 − r2 ) 2
{2.3} ATH = −
2
{2.4} B
TH
− r0 3 2 ⎡ ⎤ (1 + ν )( S 2 − r0 S 3 ) + (1 − ν ) S1 ⎥ = ⎢ 2 w(r1 − r2 ) ⎣ ⎦
23
In these equations the geometry of the hole is considered in the hole radius r0 (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 r1 , r2 respectively (Fig. 2.4).
Fig. 2.4: Schematic definition of geometric parameters for hole and strain gage
The integrations constants S1 , S 2 , S 3 are listed in equations {2.5}, {2.6} and {2.7}:
x=r
x ⎤ 2 ⎡ {2.5} S1 = 2 ⎢arctan 0.5w ⎥⎦ x = r1 ⎣
x=r
2 x ⎡ ⎤ {2.6} S 2 = − w⎢ 2 2 ⎣ x + 0.25w ⎥⎦ x = r1
x = r2
⎤ 1 ⎡ x {2.7} S 3 = − w⎢ ⎥ 3 ⎣⎢ x 2 + 0.25w 2 2 ⎦⎥ x = r1
(
)
The orientation of the principal stresses σ max and σ min can be defined using the angle
ϕ (angle between strain gage 3 and σ max , Fig. 2.3). For this, an intermediate angle ϕ ∗ has to be calculated {2.8}.
24 {2.8} ϕ ∗ =
2Δε 2 − Δε 3 − Δε 1 1 1 Num arctan = arctan 2 2 Δε 1 − Δε 3 Den
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} Num < 0
Den > 0
⇒ ϕ = ϕ*
{2.10} Num < 0
Den < 0
⇒ ϕ = ϕ * − 90°
{2.11} Num > 0
Den < 0
⇒ ϕ = ϕ * − 90°
{2.12} Num > 0
Den > 0
⇒ ϕ = ϕ * − 180°
{2.13} Num = 0
Den > 0
⇒ ϕ = 0°
{2.14} Num < 0
Den = 0
⇒ ϕ = −45°
{2.15} Num = 0
Den < 0
⇒ ϕ = −90°
{2.16} Num > 0
Den = 0
⇒ ϕ = −135°
{2.17} Num = 0
Den = 0
⇒ rotation symmetric state of stress
2.3.2 Average Value over Hole Depth - Blind-Hole Case In many measurement applications where an average stress value is desirable (e.g. in
quality control tasks) the generation of a through-hole is not feasible because the component thickness is too big compared with the drilling tool. In addition, it is impossible to register analyzable strain changes after drilling a hole with a depth z of approximately the half hole diameter d0. Alternatively, it is possible to work with a bigger hole diameter or to drill a blind hole in the material. The latter leads to a complex stress state near the hole because the strains are always partially relieved in these cases. Therefore, an analytical solution does not exist for the average value in the blind hole case. Rather, a previous calibration is necessary to determine a relationship between a known stress state and the relieved strains after drilling the hole. The known stress state can be applied either experimentally using e.g. a tensile machine or numerically by calculating a finite element model. The calibration specimen or calibration model respectively has normally an ideal geometry, i.e. a thick wide plate. In practice, calibration constants A BH and B BH for the blind hole case are determined to substitute the calibration constants ATH and B TH in {2.1} [11].
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 d 0 obtaining the non-dimensional depth ξ {2.18}. In the following explanations, it is assumed that the calibration hole diameter d c ,0 equals the measurement hole diameter d 0 .
26 {2.18} ξ =
z d0
When carrying out the calibration for the hole drilling method using the MPA-Method, a hole of the diameter d c ,0 is drilled incrementally in a specimen subjected to a known external stress state with the calibration stresses σ c , x (ξ ) and σ c , y (ξ ) . The calibration stresses in the MPA-Method can be applied either in an experimental or numerical way. The relaxed calibration strains ε c, x (ξ ) and ε c, y (ξ ) are differentiated in order to calculate the calibration functions K x (ξ ) {2.19} and K y (ξ ) {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. dε (ξ ) dε c , x (ξ ) ⋅ σ c , x (ξ ) − c , y ⋅ σ c , y (ξ ) dξ dξ {2.19} K x (ξ ) = 1 2 σ c , x (ξ ) − σ c2, y (ξ ) E
[
{2.20} K y (ξ ) =
]
dε (ξ ) dε c , x (ξ ) ⋅ σ c , y (ξ ) − c , y ⋅ σ c , x (ξ ) dξ dξ
[σ E
ν
2 c, x
(ξ ) − σ c2, y (ξ )
]
To calculate the unknown residual stress depth distribution in a component, a hole with the hole diameter d 0 is also incrementally introduced. In this case the depth increments of the measurement Δz may be different to the calibration depth increments Δz c . The derived strain readings dε i (ξ ) / dξ of the three different strain gages as well as the calibration values K x (ξ ) and K y (ξ ) are used in {2.21} to {2.23} in order to calculate the stress values σ i (ξ ) in the direction of the respective strain gages.
{2.21} σ 1 (ξ ) =
dε (ξ ) ⎤ ⎡ dε (ξ ) E ⋅ ⎢ K x (ξ ) ⋅ 1 + ν ⋅ K y (ξ ) ⋅ 3 ⎥ 2 2 dξ dξ ⎦ K (ξ ) − ν K y (ξ ) ⎣ 2 x
27 {2.22}
σ 2 (ξ ) =
⎡ ⎛ dε (ξ ) dε 3 (ξ ) dε 2 (ξ ) ⎞⎤ dε (ξ ) E ⎟⎥ ⋅ ⎢ K x (ξ ) ⋅ 2 + ν ⋅ K y (ξ ) ⋅ ⎜⎜ 1 + − 2 2 dξ dξ dξ ⎟⎠⎦ K (ξ ) − ν K y (ξ ) ⎣ ⎝ dξ 2 x
{2.23} σ 3 (ξ ) =
dε (ξ ) ⎡ dε (ξ ) ⎤ E ⋅ ⎢ K x (ξ ) ⋅ 3 + ν ⋅ K y (ξ ) ⋅ 1 ⎥ 2 2 dξ dξ ⎦ K (ξ ) − ν K y (ξ ) ⎣ 2 x
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} σ max, min (ξ ) =
σ 1 (ξ ) + σ 3 (ξ ) 2
±
1 ⋅ (σ 1 (ξ ) − σ 2 (ξ ))2 + (σ 3 (ξ ) − σ 2 (ξ ))2 2
28 The angle ϕ ( z ) 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 P(H ) 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 p (h ) 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 rm and not the hole diameter d 0 as in the differential MPA-Method.
{2.25} H =
{2.26} h =
Z rm
with Z: depth from surface
z rm
with z: hole-depth
The relationship between the strain p (h ) and the stress P(H ) is given in {2.27}, where
Aˆ ( H , h) is the strain relaxation per unit depth or influence function to integrate.
{2.27} p (h) =
1 +ν E
∫
h
0
Aˆ ( H , h) P( H )dH
0≤H ≤h
The differential notation of the strain relaxation p (h ) {2.27} is unusable because the strain values pi 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 Pj is the equivalent uniform stress within the j − th hole depth increment and aij 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} pi =
1 +ν E
j =i
∑a j =1
⎡ p1 ⎤ ⎢p ⎥ 1 +ν {2.29} ⎢ 2 ⎥ = ⎢ p3 ⎥ E ⎢ ⎥ ⎣ p4 ⎦
ij
Pj
⎡ a11 ⎢a ⎢ 21 ⎢a31 ⎢ ⎣a 41
1≤ j ≤ i ≤ n
a 22 a 32 a 42
a33 a 43
⎤ ⎡ P1 ⎤ ⎥ ⎢P ⎥ ⎥⎢ 2 ⎥ ⎥ ⎢ P3 ⎥ ⎥⎢ ⎥ a 44 ⎦ ⎣ P4 ⎦
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 Pj at a specific increment j of a
30 hole. The loading steps of the calibration with the stress Pj and the resultant matrix of strain relaxations aij are
exemplarily shown in Fig. 2.6 for the case of four hole-
depth increments i . Here, Pj 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 aij can be numerically determined and the unknown residual stress distribution can then be then calculated using the measured strain distributions pi 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 d m , the hole diameter d 0 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 d m 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 d c ,0 , which is set after the calibration, and the diameter d 0 , 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 d 0 . 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 Δz c and the hole depth increments of the measurement Δz . 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 T > 0.4 × d m is considered to be thin whereas T < 1.2 × d m is considered to be thick. (e.g., thin T > 2.052mm and thick T < 6.156mm for a 062UM-120 strain gage rosette). The intermediate case T = 0.4...1.2 × d m 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 nonstandard. 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 T > d 0 [35] and T > 0.5 × d 0 [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 T > 2.5mm . 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 T = 1.34...2.0 × d m ( T = 1.88...2.811 × d 0 ). Considering the dimensions of the strain gage rosette (062RK-120) used in [40] the transition range of the specimen thickness is about T = 3.44...5.13mm . In a practical application, Schajer specifies in [34] the acceptable component thickness as being T > 1.2 × d m 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 T < 2.5 × d 0 . 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 T > 3 × d 0 . 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 D > 15 × d 0 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 W = 2 × D (s. Fig 2.9). A significant edge effect appears only by a width of W < 20mm or a distance D > 10mm
respectively. Considering specimens with
D = 5mm , 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 W = 3 × d m for the correct use of the H-Drill software. This means a component width of W = 15.4mm 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 Da = 5 × d 0 . Distances of Da ≥ 5 × d 0 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 R B = 40mm . Therefore, for a correct execution of the hole-drilling method he suggested a surface radius of a biaxial curved shape with R B ≥ 100mm whereas the other surface radius should in be all cases RB > 15mm , 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 geometryspecific 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 R ≥ 3 × d 0 with an evaluable depth of z ≤ R / 4d 0 (e.g. d 0 = 1.8mm correspond to R = 5.4mm and z ≤ 0.75mm ). 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 RB > 17.5 × d 0 . 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 R B = 5 × d 0 (with RB = 10mm and d 0 = 2mm ).
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. R > 25 × d 0 (with R > 50mm and d 0 = 2mm ) 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 simplyshaped 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
W > 10 - 20 x d0
W > 18 - 36 mm
T > 3 x d0
T > 5.4 mm
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
Min. Component Width Min. Component Thickness Min. Distance Between Hole and Component Edge
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 MPAMethod. 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 xaxis 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 ydirection. 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.
45
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.
46
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.
47
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.
48 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 highspeed 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-
49 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).
50 (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 Δε cΔ,σi c ( z ) , which is related to the calibration stress Δσ c {3.1}, is calculated after the calibration procedure using {3.2}. {3.1} Δσ c = σ max − σ min
here:
Δσ c = 210MPa − 50 MPa = 160MPa {3.2} Δε CΔσ,i c ( z ) = [ε cσ,imax ( z ) − ε cσ,imin ( z )] − [ε cσ,imax (0) − ε cσ,imin (0)] This procedure for the determination of the calibration strain Δε c ,i ( z ) 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
51 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
52 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 threedimensional 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.
53 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
54
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.
55 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 threedimensional 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.
56
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 this case, it is also assumed that the stress state is similar to that in a real component. This model combines all three geometrical parameters used in this investigation and bases on the geometry of the hcR6m model with an identical mesh of the hole region or measurement point (s. Fig. 3.10). Fig. 3.12 shows the shape of the Type 4 model with the main dimensions, the location of the measurement point, the approximate location of the strain gages of the hole drilling rosette, the approximate location of the control strain gage and the symmetric plane. Similar to the previous examples, only half of the geometry of the specimens was modelled, due to symmetry. The stress magnitudes at the hole volume before drilling, i.e. the resulting calibration stresses σc,x and σc,y, can be averaged over each drilling depth increment after each numerical calibration. The model contains a total of 92,596 elements; most of them are of the type C3D8. Nevertheless, some elements of the type
57 C3D6 were required in order to mesh the curvature of the cylinder near the measurement point. The abbreviation for this model is “hchR6m” and is based on the previous abbreviation of the experimental specimen with the addition of an “m” for “model”.
Fig. 3.12: 3D-model of the hollow cylindrical tensile specimen with an ø6 mm center through-hole
3.3.2
Material Model, Load Application and Boundary Conditions of the FEModels The evaluation methods of the hole-drilling method are based on the linear-elastic
theory. For this reason, the used material model is in all calculations, unless otherwise noted, linear-elastic with a Young Modulus of E = 210000 MPa and a Poisson’s Ratio of ν = 0.285. The loading and the boundary conditions of the models are shown schematically in Fig. 3.13. The displacement of all nodes in z-direction is always free for all four model
58 types. The calibration stresses for the models of Type 1 and 2 were applied using preloaded elements with the ABAQUS function “initial conditions, type = stress”. Thus, to satisfy the symmetry conditions and to ensure a uniform stress state in x-direction (and / or sometimes in y-direction) before simulating the drilling process, all nodes of the faces perpendicular to the xy-plane were fixed, i.e. they were assigned with zero displacement for the whole simulation (s. loose bearing symbols in Fig. 3.13).
Fig. 3.13: Boundary conditions of the models and load application
For the models with a center through-hole (Type 3 and 4) an external load was applied to the face in front of the symmetric plane. This external load generates, due to the notch effect, a stress concentration at the measurement point. The stress concentration can thus be separated in resulting calibration stresses σc,x and σc,y, which are averaged over the hole volume. The magnitude of the external loading was set comparing the control strain gage of the experimental calibration of the specimens of these types. Only the nodes on the face of the symmetric plane were fixed in order to satisfy the symmetry conditions.
3.3.3 Calculation of Strain The strain field near the hole is inhomogeneous due to the stress concentration. This
means that the measured strains of the hole-drilling rosette are an average value of the
59 strain field over the gage area ASG. Therefore, the strain evaluation of the FEM simulations should also consider the inhomogeneity of the strain field at the surface of the model by calculating average values over the location and dimension of the corresponding strain gage areas ASG. The main procedure of the strain calculation for one depth increment is illustrated in Fig. 3.14. On the left side, a model of a flat specimen with the hole area is shown. Assuming a calibration stress in x-direction σc,x, the limits of a strain field are defined considering the location of the virtual strain gage in x-direction (1). The dimensions of the defined strain field are slightly bigger than those from the strain gage. The single node strain values within the strain field are saved in a matrix in the form of r r r r r M = [x , y, ε x ] (2). Here, x and y are the xy-locations of the single surface nodes r within the strain field and ε x the corresponding node strains in x-direction. After this, the coefficients of a polynomial surface function ε x = f ( x, y ) are calculated using Gaussian elimination (3). The average strain value in x-direction ε x,m is finally calculated by numerically evaluating a double integral of the surface function ε x according to {3.3}. In this equation, the radial distances from hole center to strain gage ends r1 , r2 and the strain gage width w are the endpoints of the interval (s. Fig. 2.4). {3.3}
ε x,m =
1 ASG
r 2 0.5 w
∫ ∫ ε ( x, y ) x
r1 − 0.5 w
This calculation has to be repeated for each depth increment as well as for each calibration strain gage (εx and εy). In this work, the calculation of the average strain was automated using the numerical computing environment “Matlab” created by “MathWorks”. To simplify matters, a small algorithm was programmed in “Matlab”, in order to sort the output data taken from the ABAQUS result files of the type *.dat and to create the strain field matrices M for each calibration strain gage and each depth increment. It must be pointed out for the cylindrical models that the tangential strain εy was calculated using a cylinder coordinate system within the models. In this case, the y-
60 coordinate of the cylindrical models is specified as the segment of the circle which arises from the cylinder circumference.
Fig. 3.14: Calculation of average strain over the gage area ASG using the strain field near the hole
The calibration strains εx(z) and εy(z) are the principal strains because they are oriented parallel to the principal stresses (s. Fig 3.1). Hence, according to the relationships in Mohr’s circle, the strain values of strain gage no. 2, which is oriented ± 45° between strain gage no. 1 and no. 3, are calculated according to {3.4}.
{3.4}
ε2 =
εx +εy 2
This calculation is done in order to give an indication of the strain distribution in this direction in the results diagrams of Chap. 4. The calculated strain values of strain gage no. 2 are not taken for the subsequent calibration.
61
4 Results of Experimental and Numerical Calibration 4.1 Prefacing Remarks This chapter presents the results of the experimental calibration as well as selected results of the numerical calibration. The following individual geometrical parameters were varied: •
d0
hole diameter
•
T
specimen thickness (specimens of Type 1)
•
D
distance between hole and component edge (specimens of Type 1)
•
R
cylinder radius of solid cylinder (specimens of Type 2).
Additionally, the following combinations were taken into account in the calibrations: •
Thickness T and distance hole-edge D (specimens of Type 1)
•
Cylinder Radius R and thickness T (specimens of Type 2)
•
Thickness T and distance to center through-hole D (specimens of Type 3)
•
Cylinder Radius R, thickness T and distance to center through-hole D (Type 4 specimens)
Two types of diagrams are mostly used to illustrate the obtained calibration results. The first type of diagram compares the experimental calibration with the numerical calibration. Three strain distributions Δε measured (or calculated) from the strain gages of the hole-drilling rosette (strain gage no. 1, 2 and 3) are plotted over the hole depth z. The strain values are plotted until a depth of z = 1 mm. This depth exceeds a little the maximal analyzable depth of 0.5 × d 0 (with d0 = 1.8 mm) as mentioned in Chap. 2.23. The investigated specimens and the experimental hole diameter d0,exp as well as the simulated hole diameter d0,sim are indicated in the respective diagrams. The applied load is distinguished in a calibration tensile stress σc,x in the case of the specimens with a defined cross section (flat and cylindrical specimens of Type 1 and Type 2) and in an external applied load σx in the case of the specimens with a center through-hole (flat and cylindrical specimens of Type 3 and Type 4, respectively). In addition to these first type diagrams, a strain comparison table is included after each diagram. The tables list
62 the experimental and numerical absolute strain values as well as the difference of these values at selected hole depths. The second type of diagram shows the influence of geometrical parameters on the absolute values of the strain distributions of strain gage no. 1 and strain gage no. 3, respectively. The diagrams of this second type, which always include the strain distribution of the reference specimen, summarize the previous numerical results into one figure. Therewith, a first qualitative comparison of the geometrical influences on the hole-drilling results is established. With the exception of the diagrams in which the influence of the specimen thickness is demonstrated, the strain values are also plotted to a depth of z = 1 mm. A discussion of the influence of geometrical parameters is carried out in the following Chap. 5.
4.2 Reference Specimen and Variation of Hole Diameter - d0 The reference specimen is the one in which the geometrical boundary conditions for the hole-drilling method are not violated according to the differential stress calculation method MPA (s. Tab. 2.4). In addition, a hole diameter of d0 = 1.8 mm is defined as the reference hole diameter. Fig. 4.1 shows the calibration results of the reference specimen fT6D20. It is a typical hole-drilling calibration result for a tensile-loaded calibration specimen or model. Strain gage no. 3 is in the load direction of the tensile load; strain gage no. 1 is transverse to the load direction. Hence, the absolute strain values of strain gage no. 3 are the highest with negative algebraic sign whereas the absolute strain values of strain gage no. 1 are the lowest with positive algebraic sign.
63 100
sg 1
50
Δε [μm/m]
0
-50
sg 2 Experiment
-100
Simulation -150
·
___
fT6D20 (Type 1) σc,x = 160 MPa d0,exp = 1.80 mm d0,sim = 1.80 mm
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig. 4.1:
Comparison of experimental and numerical calibration strain depth distributions for reference specimen fT6D20 (T =6 mm, D = 20 mm, R = ∞)
Typical comparative strain values for reference steel specimens with a calibration stress of σc,x = 160 MPa at a hole depth of 0.2 mm, 0.5 mm and 1 mm are marked in the diagram of Fig. 4.1 with a dashed line box. The experimental and numerical values as well as the difference between them are listed in Tab. 4.1. ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
7
7
0
-36
-36
0
0.5
23
23
0
-104
-108
4
1.0
58
53
5
-199
-195
-4
Tab .4.1:
Comparison of experimental and numerical calibration strain values in selected hole depths for reference specimen fT6D20 (T =6 mm, D = 20 mm, R = ∞)
Fig. 4.2 and Fig. 4.3 show the influence of different dimensions of the hole diameter on the values of the strain depth distributions for strain gage no. 3 and strain gage no. 1, respectively. The results of these diagrams were calculated using the reference flat model with five different hole diameters. Both diagrams show that an enlargement of the hole diameter leads to a magnification of the relieved strain values.
64 0.0
Simulation of fT6D20 specimen with variable hole diameter d0 Strain gage no. 3 σc,x = 160 MPa
Δε [μm/m]
-50.0
-100.0
-150.0
d0 = 1.70 mm d0 = 1.75 mm d0 = 1.80 mm
-200.0
d0 = 1.85 mm d0 = 1.90 mm
Reference -250.0 0.00
0.20
0.40
0.60
0.80
1.00
1.20
z [mm]
Fig. 4.2:
Influence of hole diameter d0 on numerical calibration strain depth distribution (strain gage no. 3) for reference model fT6D20m 60
d0 = 1.90 mm d0 = 1.85 mm
Simulation of fT6D20 specimen with variable hole diameter d0 Strain gage no. 1 σc,x = 160 MPa
50
d0 = 1.80 mm d0 = 1.75 mm d0 = 1.70 mm
Δε [μm/m]
40
30
20
10
Reference 0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 4.3:
Influence of hole diameter d0 on numerical calibration strain depth distribution (strain gage no. 1) for reference model fT6D20m
4.3 Variation of Specimen Thickness - T The results for the experimental and numerical calibration for the flat specimens with a thickness T of 3 mm, 2 mm and 1.5 mm and D =20 mm = const. are shown in Fig. 4.4, Fig. 4.5 and Fig. 4.6, respectively. The numerically calculated strain distributions are in accordance with the experimentally measured depth distributions of the relieved strains, as can be seen in Tab. 4.2 (fT3D20), Tab. 4.3 (fT2D20) and Tab. 4.4 (fT1.5D20).
65 100
sg 1
50
Δε [μm/m]
0
-50
Experiment • Simulation ___
-100
sg 2
fT3D20 (Type 1) σc,x = 160 MPa d0,exp = 1.79 mm d0,sim = 1.80 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig. 4.4:
Comparison of experimental and numerical calibration strain depth distributions for specimen fT3D20 (T =3 mm, D = 20 mm, R = ∞)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
6
7
-1
-25
-36
11
0.5
23
24
-1
-92
-107
15
1.0
55
54
1
-186
-194
8
Tab. 4.2:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fT3D20 (T =3 mm, D = 20 mm, R = ∞) 100
sg 1
50
Δε [μm/m]
0
-50
-100
-150
-200
Experiment • Simulation ___
sg 2
fT2D20 (Type 1) σc,x = 160 MPa d0, exp = 1.78 mm d0,sim = 1.80 mm
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig. 4.5:
Comparison of experimental and numerical calibration strain depth distributions for specimen fT2D20 (T =2 mm, D = 20 mm, R = ∞)
66 ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
7
6
1
-37
-38
1
0.5
23
21
2
-113
-115
2
1.0
55
51
4.0
-210
-208
-2
Tab. 4.3:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fT2D20 (T =2 mm, D = 20 mm, R = ∞)
100
sg 1
50
Δε [μm/m]
0
-50
Experiment • Simulation ___
-100
sg 2
fT1.5D20 (Type 1) σc,x = 160 MPa d0,exp = 1.78 mm d0,sim = 1.80 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.6:
Comparison of experimental and numerical calibration strain depth distributions for specimen fT1.5D20 (T =1.5 mm, D = 20 mm, R = ∞)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
5
5
0
-38
-44
6
0.5
20
17
3
-123
-129
6
1.0
55
51
4
-219
-224
5
Tab. 4.4:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fT1.5D20 (T =1.5 mm, D = 20 mm, R = ∞)
The influence of the geometrical parameter thickness T on the calibration strains is demonstrated in Fig. 4.7 (sg. no. 3) and Fig. 4.8 (sg. no. 1). In these diagrams based on numerically determined data, a strain distribution of the calibration using a model with a thickness of T = 0.55xd0 = 1 mm is additionally included. The diagrams show strain distributions to a depth of z = 1.5 mm, in order to show the change of the slope a few drilling increments before the through-hole is introduced in the models with T = 1.5 mm
67 and T = 1mm. It can be seen that the strain distributions recorded with both strain gages for the calibration with the model fT3D20m with T = 1.67xd0 = 3 mm are almost equal to those from the reference model. Beginning with a thickness of T = 1.11xd0 = 2 mm, a reduction of the thickness leads to a change in the absolute values of the strain distributions compared to the reference. In the case of strain gage no. 3, the absolute strain values become higher with decreasing thickness. All strain distributions range in the same level. In the transverse case of strain gage no. 1, the absolute strain values decrease slightly with decreasing thickness. A few depth increments before the blindhole is transformed into a through-hole the slopes of the strain distributions of the models with the thickness T = 0.55xd0 = 1 mm and T = 0.83xd0 = 1.5 mm change and the values converge roughly to that of the reference calibration. 0
Simulation of Type 1 specimens with variable thickness Strain gage no. 3 σc,x = 160 MPa d0, sim = 1.8 mm
Δε [μm/m]
-50
-100
T = 1.11d0 = 2 mm -150
T = 1.67d0 = 3 mm Reference T = 3.33d0 = 6 mm
T = 0.83d0 = 1.5 mm -200 T = 0.55d0 = 1 mm -250 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z [mm]
Fig.4.7:
Influence of geometrical parameter T on numerical calibration strain depth distribution (strain gage no. 3)
68 120
Simulation of Type 1 specimens with variable thickness Strain gage no. 1 σc,x = 160 MPa d0, sim = 1.8 mm
100
Δε [μm/m]
80
60 Reference T = 3.33d0 = 6 mm 40
T = 1.67d0 = 3 mm T = 1.11d0 = 2 mm
20 T = 0.83d0 = 1.5 mm T = 0.55d0 = 1 mm 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z [mm]
Fig.4.8:
Influence of geometrical parameter T on numerical calibration strain depth distribution (strain gage no. 1)
4.4 Variation of Distance Hole-Edge - D The results for the experimental and numerical calibration for the flat specimens with a distance hole-edge of 10 mm, 5 mm and 2 mm and T = 6 mm = const are shown in Fig. 4.9, Fig. 4.10 and Fig. 4.11, respectively. Similar to the results for the variation of the thickness, the numerically calculated strain distributions in this calibration case reproduce well the experimentally measured depth distributions of relieved strain (s. Tab. 4.5, Tab. 4.6 and Tab. 4.7 for specimen fT6D10, fT6D5 and fT6D2, respectively). 100
sg 1
50
Δε [μm/m]
0
-50
-100
-150
-200
sg 2
Experiment • Simulation ___ fT6D10 (Type 1) σc,x = 160 MPa d0,exp = 1.90 mm d0,sim = 1.90 mm
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.9:
Comparison of experimental and numerical calibration strain depth distributions for specimen fT6D10 (T =6 mm, D = 10 mm, R = ∞)
69
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
6
8
-2
-32
-41
9
0.5
24
25
-1
-115
-121
6
1.0
59
57
2
-216
-215
-1
Tab. 4.5:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fT6D10 (T =6 mm, D = 10 mm, R = ∞) 100
sg 1
50
Δε [μm/m]
0
-50
sg 2
Experiment • Simulation ___
-100
fT6D5 (Type 1) σc,x = 160 MPa d0,exp = 1.85 mm d0,sim = 1.85 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.10:
Comparison of experimental and numerical calibration strain depth distributions for specimen fT6D5 (T =6 mm, D = 5 mm, R = ∞)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
5
7
-2
-32
-39
7
0.5
23
24
-1
-108
-114
6
1.0
57
55
2
-204
-204
0
Tab. 4.6:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fT6D5 (T =6 mm, D = 5 mm, R = ∞)
70 100
sg 1
50
Δε [μm/m]
0
-50
sg 2
Experiment • Simulation ___
-100
fT6D2 (Type 1) σc,x = 160 MPa d0,exp = 1.90 mm d0,sim = 1.90 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.11:
Comparison of experimental and numerical calibration strain depth distributions for specimen fT6D2 (T =6 mm, D = 2 mm, R = ∞)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
5
8
-3
-31
-40
9
0.5
21
26
-5
-109
-118
9
1.0
56
59
-3
-213
-207
-6
Tab. 4.7:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fT6D2 (T =6 mm, D = 2 mm, R = ∞)
In contrast to the variation of thickness, the variation of the distance hole-edge D appears to have only little influence on the calibration strains as can be seen in Fig. 4.12 for strain gage no. 3 and in Fig 4.13 for strain gage no. 1. In both diagrams, only the distribution for the model with D = 2 mm shows a slight deviation from the reference distribution, i.e. the absolute strain values of the calibration with model fT6D2m decrease for strain gage no. 3 and increase for strain gage no. 1. The distributions for model fT6D10m with D = 10 mm and model fT6D5m with D = 5 mm are superimposed on those from the reference model. All calibration curves show here a similar trend in the strain distribution.
71 0.0
Simulation of Type 1 specimens with variable distance hole-edge Strain gage no. 3 σc,x = 160 MPa d0, sim = 1.8 mm
Δε [μm/m]
-50.0
-100.0 Reference D = 11.11d0 = 20 mm -150.0 D = 1.1d0 = 2 mm D = 1.94 … 5.56d0 = 3.5 … 10 mm -200.0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.12:
Influence of geometrical parameter D on numerical calibration strain depth distribution (strain gage no. 3) 60.0
Simulation of Type 1 specimens with variable distance hole-edge Strain gage no. 1 σc,x = 160 MPa d0, sim = 1.8 mm
Δε [μm/m]
40.0
D = 1.1d0 = 2 mm
D = 1.94 … 5.56d0 = 3.5 … 10 mm 20.0 Reference D = 11.11d0 = 20 mm
0.0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.13:
Influence of geometrical parameter D on numerical calibration strain depth distribution (strain gage no. 1)
4.5 Combination of Parameter T and D Fig. 4.14 compares the experimental and numerical results of the calibration for specimen fT1.5D2, which is considered to be thin with T = 1.5 mm and at the same time, which has a small distance hole-edge with D = 2 mm.
72 100
sg 1
50
Δε [μm/m]
0
-50
sg 2
-100
Experiment Simulation
·
___
-150 fT1.5D2 (Type 1) σc,x = 160 MPa d0,exp = 1.90 mm d0,sim = 1.90 mm
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.14:
Comparison of experimental and numerical calibration strain depth distributions for specimen fT1.5D2 (T =1.5 mm, D = 2 mm, R = ∞)
Both experimental and numerical calibration strain distributions show accordance, although the absolute strain differences between experiment and simulation for strain gage no. 3 are higher than the previous comparisons in Chap. 4.3 and Chap. 4.4. The absolute values as well as the difference are listed for selected hole depths in Tab. 4.8. ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
5
4
1
-46
-41
-5
0.5
17
15
2
-137
-122
-15
1.0
54
50
4
-232
-209
-23
Tab. 4.8:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fT1.5D2 (T =1.5 mm, D = 2 mm, R = ∞)
The diagrams in Fig. 4.15 (strain gage no. 3) and Fig. 4.16 (strain gage no. 1) display the numerical results for previous calibrations of flat models with different combinations of parameters T and D. The results presented in the diagrams are calculated from the reference model fT6D20m, the “thin” model fT1.5D20m, the model with a “small” distance D fT6D2m and the “combined” flat model fT1.5D2m. The strain distributions are plotted to a hole depth of z = 1.5 mm. In this comparison, the geometrical parameter thickness has a major influence on the distribution of the calibration strain. In both diagrams the magnitudes of the relieved strains for the “thin”
73 models differ from those for the reference model with higher absolute values. Also the trajectories of the distributions of both “thin” models are more similar than those of the “thick” models. 0
Simulation of Type 1 specimens Strain gage no. 3 σc,x = 160 MPa d0, sim = 1.8 mm
Δε [μm/m]
-50
-100
fT6D2 -150 Reference fT6D20 -200 fT1.5D2 fT1.5D20 -250 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z [mm]
Fig.4.15:
Comparison of numerical calibration strain depth distributions (strain gage no. 3) for flat specimens with different combinations of geometrical parameters T and D
120.0
Simulation of Type 1 specimens Strain gage no. 1 σc,x = 160 MPa d0, sim = 1.8 mm
100.0
fT1.5D2
fT1.5D20
Δε [μm/m]
80.0 fT6D2 60.0 Reference fT6D20 40.0
20.0
0.0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z [mm]
Fig.4.16:
Comparison of numerical calibration strain depth distributions (strain gage no. 1) for flat specimens with different combinations of geometrical parameters T and D
When comparing the two “thin” models fTD1.5D20m and fT1.5D2m, a reduction of parameter D from D = 20 mm to D = 2 mm causes also (s. Chap. 4.4) a decrease of the absolute strain values as registered with strain gage no. 3. For strain gage no. 1, the
74 calibration with the “thin” model with D = 2 mm causes initially smaller absolute strain values and after a hole depth of z ≈ 1.12 mm they became slightly bigger.
4.6 Variation of Cylinder Radius - R 4.6.1 Solid Cylinder The results for the experimental and numerical calibration for the solid cylinder
specimens with a cylinder radius of 6 mm, 4 mm and 3 mm are shown in Fig. 4.17, Fig. 4.18 and Fig. 4.19, respectively. The numerically calculated strain distributions in this calibration case reproduce sufficiently the experimentally measured depth distributions of relieved strain as shown in Tab. 4.9, Tab. 4.10 and Tab. 4.11 for the mentioned specimens. The experimental results at the first drilling increments scatter until a hole depth of z ≈ 0.1 mm. 100
sg 1 50
Δε [μm/m]
0
-50
Experiment • Simulation ___
-100
sg 2
scR6 (Type 2) σc,x = 160 MPa d0,exp = 1.83 mm d0,sim = 1.80 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.17:
Comparison of experimental and numerical calibration strain depth distributions for specimen scR6 (R = 6 mm)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
7
8
-1
-26
-37
11
0.5
27
29
-2
-104
-115
11
1.0
71
67
4
-218
-210
-8
Tab. 4.9:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen scR6 (R = 6 mm)
75 100
sg 1 50
Δε [μm/m]
0
-50
Experiment • Simulation ___
-100
sg 2
scR4 (Type 2) σc,x = 160 MPa d0,exp = 1.82 mm d0,sim = 1.80 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.18:
Comparison of experimental and numerical calibration strain depth distributions for specimen scR4 (R = 4 mm)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
7
7
0
-17
-36
19
0.5
25
27
-2
-91
-114
23
1.0
70
66
4
-207
-211
4
Tab. 4.10:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen scR4 (R = 4 mm)
100
sg 1 50
Δε [μm/m]
0
-50
Experiment • Simulation ___
-100
sg 2
scR3 (Type 2) σc,x = 160 MPa d0,exp = 1.80 d0,sim = 1.80 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.19:
Comparison of experimental and numerical calibration strain depth distributions for specimen scR3 (R = 3 mm)
76 ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
12
5
7
-30
-32
2
0.5
33
20
13
-109
-109
0
1.0
70
54
16
-210
-208
-2
Tab. 4.11:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen scR3 (R = 3 mm)
0.00
Simulation of Type 2 specimens with variable solid cylinder radius Strain gage no. 3 σc,x = 160 MPa d0, sim = 1.8 mm
Δε [μm/m]
-50.00
-100.00
-150.00
Reference R=∞
-200.00 R = 1.67 … 3.33 d0 = 3 … 6 mm -250.00 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.20:
Influence of geometrical parameter R on numerical calibration strain depth distribution (strain gage no. 3) 80.00
Simulation of Type 2 specimens with variable solid cylinder radius Strain gage no. 1 σc,x = 160 MPa d0, sim = 1.8 mm
70.00
Δε [μm/m]
60.00 50.00
Reference R=∞
R = 2.22 … 3.33 d0 = 4 … 6 mm
40.00 30.00
R = 1.67d0 = 3 mm 20.00 10.00 0.00 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.21:
Influence of geometrical parameter R on numerical calibration strain depth distribution (strain gage no. 1)
77 The influence of the geometrical parameter R for the solid cylinder on the calibration strains is demonstrated in Fig. 4.20 (strain gage no. 3) and Fig. 4.21 (strain gage no. 1). It can be seen that the strain distributions in load direction (strain gage no. 3) obtained from calibration with the different solid cylinders are very close to each other. The absolute strain values are higher compared to the results of the reference model. In the case of the strains recorded with strain gage no. 1, the distributions for the models scR6m and scR4m, respectively, are close together and have higher absolute strain values compared to the distributions of the reference specimen. The distribution for strain gage no. 1 from the calibration with the model scR3m has lower absolute strain values than the results of the models with R =4 mm and R = 6 mm.
4.6.2 Hollow Cylinders Fig. 4.22, Fig. 4.23 and Fig. 4.24 compare the experimental and numerical results of the
calibration for the hollow cylindrical specimens hcR6, hcR4 and hcR3, respectively. In addition to the change of cylinder diameter, these specimens consider also a wall thickness of T = 1.25 mm = const. On the whole, the numerical results match adequately to the experimental results (s. Tab. 4.12, Tab. 4.13 and Tab. 4.14 for absolute strain values and differences between experiment and simulations of calibrations using specimen hcR6, hcR4 and hcR3, respectively).
100
sg 1
50
Δε [μm/m]
0 -50 -100
Experiment • Simulation ___
sg 2
-150
hcR6 (Type 2) σc,x = 160 MPa d0,exp = 1.86 mm d0,sim = 1.80 mm
-200 -250
sg 3
-300 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.22:
Comparison of experimental and numerical calibration strain depth distributions for specimen hcR6 (T = 1.25 mm, R = 6 mm)
78
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
7
78
-1.0
-49
-47
-2
0.5
22
25
-3
-161
-144
-17
1.0
66
67
-1
-264
-238
-26
Tab. 4.12:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen hcR6 (T = 1.25 mm, R = 6 mm)
100
sg 1 50
Δε [μm/m]
0
-50
sg 2
Experiment • Simulation ___
-100
hcR4 (Type 2) σc,x = 160 MPa d0,exp = 1.84 mm d0,sim = 1.80 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.23:
Comparison of experimental and numerical calibration strain depth distributions for specimen hcR4 (T = 1.25 mm, R = 4 mm)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
7
5
2
-32
-44
12
0.5
18
19
-1
-125
-138
13
1.0
54
56
-2
-238
-237
-1
Tab. 4.13:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen hcR4 (T = 1.25 mm, R = 4 mm)
79 100
sg 1 50
Δε [μm/m]
0
-50
Experiment • Simulation ___
-100
sg 2
hcR3 (Type 2) σc,x = 160 MPa d0,exp = 1.86 mm d0,sim = 1.80 mm
-150
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.24:
Comparison of experimental and numerical calibration strain depth distributions for specimen hcR3 (T = 1.25 mm, R = 3 mm)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
3
3
0
-39
-38
1
0.5
14
12
2
-126
-130
4
1.0
51
41
10
-228
-234
6
Tab. 4.14:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen hcR3 (T = 1.25 mm, R = 3 mm)
0
Simulation of Type 2 specimens with variable hollow cylinder radius Strain gage no. 3 σc,x = 160 MPa d0,sim = 1.8 mm
Δε [μm/m]
-50
-100
-150
Reference R=∞
R = 1.67d0 = 3 mm -200 R = 2.22d0 = 4 mm R = 3.33d0 = 6 mm -250 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.25:
Influence of geometrical parameter R and T on numerical calibration strain depth distribution (strain gage no. 3)
80 80
Simulation of Type 2 specimens with variable hollow cylinder radius Strain gage no. 1 σc,x = 160 MPa d0, sim = 1.8 mm
70
Δε [μm/m]
60
R = 3.33d0 = 6 mm
R = 2.22d0 = 4 mm
50 Reference R=∞
40
R = 1.67d0 = 3 mm 30 20 10 0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.26:
Influence of geometrical parameter R and T on numerical calibration strain depth distribution (strain gage no. 1)
The diagram in Fig. 4.25 compares (for strain gage no. 3) the strain distributions of the reference model with those of the different hollow cylindrical models. It can be seen that the distributions of the different hollow cylinder calibration are sufficiently close to each other and of higher absolute strain values than the reference. In the case of strain gage no. 1 (Fig. 4.26), the numerical strain distributions vary with varying hollow cylinder radius. The strain curve of the calibration with a hollow cylinder radius of R = 6 mm is above the reference curve. The strain curves of the hollow cylinder calibration fall with decreasing hollow cylinder radius.
4.6.3 Comparison of Solid Cylinders and Hollow Cylinders The following results summarize the previous results of Chap. 4.6.1 and Chap. 4.6.2.
This chapter is included in order to compare the calibration results of the solid cylinders with the calibrations results of the hollow cylinders, in which a specimen’s wall thickness of T = 1.25 mm is additionally considered. The diagram in Fig. 4.27 shows distributions registered with strain gage no. 3. The solid curves represent the calibration strain of the solid cylinders whereas the dashed lines represent those of the hollow cylinders. The dotted curve is the strain distribution of the reference model.
81 0
Simulation of Type 2 specimens comparison sc vs. hc Strain gage no. 3 σc,x = 160 MPa d0,sim = 1.8 mm
Δε [μm/m]
-50
-100
-150
Reference R=∞
-200
scR3 scR4 scR6
hcR3 hcR4 hcR6
-250 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.27:
Comparison of numerical calibration strain depth distributions between solid cylinders and hollow cylinders (strain gage no. 3)
One can see that the change of the cross section from a solid cylinder to a hollow cylinder causes an increase of the absolute strain values for the strain gage no. 3, which is parallel to the load direction. 70
Simulation of Type 2 specimens comparison sc vs. hc Strain gage no. 1 σc,x = 160 MPa d0, sim = 1.8 mm
60
Δε [μm/m]
50
scR6 scR4 hcR6 scR3
hcR4 40
hcR3
Reference 30
20
10
0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.28:
Comparison of numerical calibration strain depth distributions between solid cylinders and hollow cylinders (strain gage no. 1)
Fig. 4.28 shows the perpendicular case for strain gage no. 3 and has the same representation of curves as in the previous figure. It is visible that the strain distributions of the calibrations with the hollow cylinders are lower (for hcR6m slightly lower than
82 scR6m) and they disperse clearly from each other compared with the distributions with the solid cylinders.
4.7 Specimens with Center Through-Hole - Combined Geometrical Parameters This chapter presents the calibration results of the specimens of Type 3 and Type 4. The procedure of experimental calibration is described in Chap. 3.2.3 and Chap. 3.2.4. Due to the stress concentration caused by the center through-hole, which is present in both specimen types, plastic deformation may occur near the through-hole. In order to take into account the plastic deformations, the numerical calibration of the specimens of Type 3 and Type 4 uses an elastic-plastic material model with isotropic hardening instead of the linear-elastic material model of the previous examples. The procedure of loading in these numerical calibrations is similar to the experimental one, i.e. the simulation considers the different load steps cycles of 50 MPa, 210 MPa and 0 MPa. These loads are applied as external uniform loads (s. Fig. 3.13). Hence, the relieved strains of numerical calibrations are calculated using {3.2}.
4.7.1 Flat Tensile Specimens with a ø10 mm center through-hole The result of the experimental calibration as well as that of the numerical calibration of
the specimens of Type 3 is shown in Fig. 4.29 and Fig. 4.30 for the specimen fhT6D2 and fhT1.5D2, respectively. The experimental and numerical strain distributions are in accordance to each other. Tab. 4.15 and Tab. 4.16 listed the absolute strain values of experiment and simulation. The differences between both calibration methods are high, especially for strain gage no. 3, compared to the flat specimens without center throughhole. The resulting calibration loads were calculated from the model as an average value over the hole volume. Thus, the calibration stresses are σc,x = 251.5 MPa, σc,y =61.4 MPa for model fhT6D2m (s. Fig. 11.1 and Fig. 11.2 in appendix Chap. 11) and σc,x = 261.9 MPa, σc,y =60.2 MPa for model fhT1.5D2m (s. Fig. 11.3 and Fig. 11.4 in appendix Chap. 11).
83 50
sg 1
0
Δε [μm/m]
-50 -100 -150
Experiment • Simulation ___
sg 2
-200
fhT6D2 (Type 3) σx = 160 MPa d0,exp = 1.83 mm d0,sim = 1.80 mm
-250 -300
sg 3
-350 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.29:
Comparison of experimental and numerical calibration strain depth distributions for specimen fhT6D2 (T =6 mm, D = 2 mm, R = ∞)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
-3
-1
-2
-28
-49
21
0.5
-8
-2
-6
-159
-157
-2
1.0
4
13
-9
-323
-288
-35
Tab. 4.15:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fhT6D2 (T = 6 mm, D = 2 mm, R = ∞)
50 0
sg 1
-50
Δε [μm/m]
-100 -150
Experiment • Simulation ___
-200
sg 2
fhT1.5D2 (Type 3) σx = 160 MPa d0,exp = 1.87 mm d0,sim = 1.90 mm
-250 -300 -350
sg 3
-400 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.30:
Comparison of experimental and numerical calibration strain depth distributions for specimen fhT1.5D2 (T =1.5 mm, D = 2 mm, R = ∞)
84
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
-9
-9
0
-66
-60
-6
0.5
-23
-21
-2
-212
-190
-22
1.0
0
1
-1
-371
-335
-36
Tab. 4.16:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen fhT1.5D2 (T = 1.5 mm, D = 2 mm, R = ∞)
The strain distributions of the flat specimens with a center through-hole are plotted in one diagram together with the strain distributions of the flat reference specimen in order to see the influence of the introduced ø10 mm hole on the calibration results. Fig. 4.31 compares the numerical strain distribution calculated with strain gage no. 3 and Fig. 4.32 the distributions with strain gage no. 1. Compared to the reference calibration, the results of the calibration with the specimens of Type 3 differ in the strain magnitude and in the trajectory of the curves. The absolute strain values of the specimens with the center through-hole are clearly higher than the reference for strain gage no. 3 and clearly lower for strain gage no. 1. 0
Simulation of Type 3 specimens with T = 1.5 mm and T = 6 mm Strain gage no. 3 σx = 160 MPa d0,sim = 1.8 mm
-50
Δε [μm/m]
-100 -150
Reference fT6D20
-200 -250
fhT6D2
-300 fhT1.5D2 -350 -400 0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
z [mm]
Fig.4.31:
Comparison of numerical calibration strain depth distributions for flat models with ø10 mm center through-hole and different thickness T (strain gage no. 3)
85 110
Simulation of Type 3 specimens with T = 1.5 mm and T = 6 mm Strain gage no. 1 σx = 160 MPa d0,sim = 1.8 mm
90
Δε [μm/m]
70
Reference fT6D20
50
30 fhT6D2 10 fhT1.5D2
-10
-30 0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
z [mm]
Fig.4.32:
Comparison of numerical calibration strain depth distributions for flat models with ø10 mm center through-hole and different thickness T (strain gage no. 1)
The distributions of the calibration strains with the “thin” flat model with hole fhT1.5D2m run similar to the strain distributions of the “thin” flat model without hole fT1.5D20m compared to its “thick” model counterparts fhT6D2m and fTD6D20m. First of all, the strain values of the “thin” models are higher for strain gage no. 3 and lower for strain gage no. 1. Secondly, the slopes of the calibration curves of the “thin” models change with increasing depth to a greater extent than the calibration curves of the “thick” models.
4.7.2 Hollow Cylindrical Tensile Specimens with a ø6 mm Center Through-Hole The hollow cylindrical tensile specimens with an ø6 mm center through-hole combine
all three geometrical parameters which are considered in this investigation about the geometrical influence on the results of a hole-drilling measurement. In Fig. 4.33, the experimental strain distributions of the calibration on specimen hchR6 are confronted with the numerical ones. The numerical strain distribution for strain gage no. 1 reproduces enough the experimental distributions. In case of strain gage no. 3 the trajectory of the experimental distribution is qualitatively reproduced by the simulation with clearly lower absolute strain values. The absolute strain values as well as the difference between the experiment and simulation are listed in Tab. 4.17. One can see that the calculated differences in this case are the highest of all calibrations shown in these results chapter (Chap. 4).
86 100
sg 1
Δε [μm/m]
0
-100
sg 2
Experiment • Simulation ___
-200
hchR6 (Type 4) σx = 160 MPa d0,exp = 1.76 mm d0,sim = 1.80 mm
-300
sg 3
-400 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig.4.33:
Comparison of experimental and numerical calibration strain depth distributions for specimen hchR6 (T =1.25 mm, D = 2 mm, R = 6 mm)
ε1 [μm/m]
z [mm]
ε3 [μm/m]
experiment simulation difference experiment simulation difference
0.2
8
7
1
-38
-62
24
0.5
17
19
-2
-187
-186
23
1.0
72
62
10
-356
-308
-48
Tab. 4.17:
Comparison of experimental and numerical calibration strain values in selected hole depths for specimen hchR6 (T = 1.25 mm, D = 2 mm, R = 6 mm)
350
σc,x
hchR6 (Type 4) σx = 160 MPa d0 = 1.8 mm
300
σ [MPa]
250
200
150
100
σc,y
50
0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.34:
Stress distribution versus depth within hole volume before introducing the hole for numerical calibration of model hchR6m
87 The resulting calibration stresses σc,x and σc,y over the hole volume before drilling are plotted in Fig. 4.34. It can be see that the calibration stresses in the hole area are nonuniform in depth (s. also Fig. 11.5 and Fig. 11.6 in appendix Chap. 11).
0.00
Simulation of Type 4 specimens with T = 1.25 mm, D = 2 mm and R = 6 mm Strain gage no. 3 σx = 160 MPa d0, sim = 1.8 mm
-50.00
Δε [μm/m]
-100.00
-150.00
Reference fT6D20
-200.00 hcR6 -250.00
-300.00
hchR6
-350.00 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.35:
Comparison of numerical calibration strain depth distributions for the hollow cylindrical model with ø6 mm center through-hole with reference distributions (strain gage no. 3) 70.0
Δε [μm/m]
Simulation of Type 4 specimens 60.0
with T = 1.25 mm, D = 2 mm and R = 6 mm
50.0
Strain gage no. 1 σx = 160 MPa d0, sim = 1.8 mm
hcR6
hchR6
40.0
Reference fT6D20
30.0 20.0 10.0 0.0 -10.0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig.4.36:
Comparison of numerical calibration strain depth distributions for the hollow cylindrical model with ø6 mm center through-hole with reference distributions (strain gage no. 1)
Fig. 4.35 (sg. no. 3) and Fig. 4.36 (sg. no. 1) illustrate the difference of the calibration using the simulation of the hollow cylinder with the ø6 mm center through-hole
88 hchR6m compared to the reference calibration with the flat model and, additionally, the calibration for the hollow cylinder without the center through-hole hcR6m. The calibration strain distributions of the hchR6m model are different from the reference and the hollow cylinder model hcR6m. In the case of strain gage no. 3, the absolute strain values are clearly higher than the reference and the hcR6m distributions. For strain gage no. 1, the absolute strain values are lower than the two comparative distributions.
89
5 Discussion of Calibration Results 5.1 Prefacing Remarks In this chapter the results of the experimental calibration as well as the numerical calibration for the hole-drilling method are discussed by pointing out two main topics. The first topic deals with the question of whether the numerical calibration reproduces sufficiently the experimental strain distributions. The second topic concerns the influence of the component geometry on the results of a hole-drilling measurement: Questions that arise are e.g. in which cases the single geometrical limits listed in [2] (Tab. 2.4 of Chap.2.5) are sufficient or may be reduced and in which cases a geometry specific calculation could be an alternative. For the purpose of the discussion of these topics, this chapter is divided into four parts: The first part deals with the uncertainty of the calibration results shown in Chap. 4. For this reason the uncertainty of the numerical calibration as well as the uncertainty of the experimental calibration are discussed and visualized for some selected strain distributions. The second part of this discussion tends towards introducing the problematic of the stress calculation using commercially available evaluation software. The input data for this evaluation is selected from calibration results determined by using specimens with non-reference geometry and the purpose is to show in general the possible stress evaluation error if the boundary conditions are violated. The third part deals with the individual contribution of the investigated geometry parameters on the obtained results. This is done by calculating the strain and stress deviations or differences between distributions established with non-reference geometry calibrations and those established with the reference calibration. As a result, a possible recommendation is analyzed concerning the minimization of the geometry limit in the case of only one geometrical boundary condition being violated. The last part consists of demonstrating the possible reduction of the stress calculation error. For this purpose, hole-drilling measurements are simulated using models of
90 selected non-reference specimens and evaluating them with a simplified MPA algorithm in which the geometry specific calibration functions are included.
5.2 Calibration Uncertainty 5.2.1 Uncertainty in Numerical Calibration The quality of a finite element model depends mainly on the type of elements used, the
used finite element formulation, the mesh refinement, the boundary conditions and the material model. In the special case of the hole-drilling method the quality of the corresponding model itself can be proved as a first approximation comparing selected results with already existing theoretical solutions. In the best case, the numerical results should converge to the theoretical ones.
160
Analytical solution _______ Simulation fT1.5D20 • ▲
strain gage position
σrad
σx [MPa]
120
80
40
σtan 0
0.5 x d0 -40 0
1
2
3
4
5
distance x from hole center [mm]
Fig. 5.1:
Stress distribution near a through-hole: Comparison of results between a numerical calibration using model fT1.5D20 and analytical solution according to Kirsch
91 In case of the of the flat specimen models of Type 1 Kirsch’s analytical solution [16] for the stress state near a through-hole in a thin and infinitely wide plate may be used for the comparison. This procedure was used for finding an appropriate meshing of the models of Type 1. Models with different number of elements, which have a small thickness and a sufficiently wide broadness, were generated in order to obtain the stress values near the simulated through-hole. The results of the simulations were compared to each other in order to select one model which reproduces the analytical solution adequately and has an acceptable calculation time. The result of the selected model, which was the base of the used flat specimen models in this investigation, is shown in Fig. 5.1. The upper part of Fig. 5.1 shows schematically the loading conditions of the flat model, the mesh near the hole and the transition area between the hole mesh and the rectangular mesh of the rest of the model with the approximate location of the strain gage. The lower diagram shows the stress distribution radial σrad and tangential σtan to hole against the x-axis, which is parallel to the axis of strain gage no. 3 as can be seen in the picture above the diagram. The stress values are the average values extrapolated to the nodes of the elements along the x-axis. Clear deviations of max. 30 %-46 % are visible at the transition area near the hole (x = 0.9 … 1.26 mm) for the radial stress distribution σrad. It should be noticed that these high percental deviations arise due to the fact that the absolute stress values are very small and each small difference between them leads to a relatively high percental deviation. In the strain gage location (x = 1.78 … 3.35 mm) the differences are about 0.3-2.0 %, which can be considered as a good approximation of Kirsch’s solution. A similar observation is done with the distribution of the tangential stress σtan. The reason for the high stress deviation values near the hole could be the low mesh refinement at the transition area between the hole mesh and the rectangular mesh. The meshing of this transition area was necessary in order to design the further rectangular mesh which allows the generation of the models with different distance hole-edge D. This transition between the two different meshing shapes could be too sharp, i.e. the few elements at this area do not sufficiently reproduce Kirsch’s solution. Other sources for possible inhomogeneous stress distributions near the hole were not detected with the finite element post-processor. The flat specimens of Type 3 were generated using a mesh design similar to that of the Type 1 flat specimens. Therefore, the numerical uncertainty should be in the range as
92 the one from the models of Type 1. A quantitative statement about the quality of these models can be made by calculating the stress concentration factors at the edge of the ø10 mm center through-hole of these flat models and comparing them with theoretical values e.g. in [51]. The theoretical value (Ktg = 3.23) differs 8.8 % from that obtained with the fhT6D2m model and 2.9 % from that obtained with the fhT1.5D2m model.
3.25
theoretical value
3.2
Ktg [-]
3.15
fhT1.5D2m 3.1
3.05
3
fhT6D2m 2.95 0
1
2
3
4
5
6
Thickness [mm]
Fig. 5.2:
Stress concentration factor at the edge of a ø10 mm through-hole as a function of the specimen thickness (according to [51])
Fig. 5.2 shows the distribution of the stress concentration factor Ktg against the thickness for specimens with a ø10 mm center through-hole and different thickness. The value of Ktg increases with decreasing thickness. The theoretical value, similar to Kirsch’s solution, does not consider the specimen thickness and is plotted at T = 0 mm. Nevertheless, the Ktg values established with the finite element simulations converge with decreasing thickness to the theoretical value. This thickness dependency of the stress concentration factor is described in [51]. Thus, a stress variation in the thickness direction of a plate exists, as thick plates show lower maximum tangential stresses at the surface and higher tangential stresses at the midplane than the values determined with thin sheets. This observation can also be made in this investigation (s. also Fig. 11.1 and Fig. 11.3 in appendix chapter Chap. 11). The surface stress at the edge of the ø10 mm center through-hole is 476 MPa for the thick model fhT6D2m and 502 MPA for the thin model fhT1.5D2m. In contrast, the stress at the midplane is 526 MPa and 514 MPa for
93 models fhT6D2m and fhT1.5D2m, respectively. For this reason, it can be assumed, that the solutions established with the models of Type 3 reproduce the theoretical solution. In the case of the cylindrical specimens of Type 2 the models were generated only once, i.e. there was no selection of models with different number of elements. The quality of each cylindrical model is verified by comparing the stress concentration factor near the hole with the stress concentration factor taken from literature. Tab. 5.1 compares the maximal stress σmax,sim at the edge of the hole (hole-drilling measurement point) established with the hollow cylinder models hcR6m, hcR4m and hcR3m with the maximal stress σmax calculated using the stress concentration factor Ktn according to [51] and gives the deviation in percent. The geometrical parameters which are considered for the determination of Ktn are also listed in Tab. 5.1 and shown in Fig. 5.3.
Model
σmax,sim σmax = Ktn x σnom Deviation
Ktn
σnom
2R
2RI
d0
[MPa]
[MPa]
[%]
[-]
[MPa] [mm] [mm] [mm]
hcR6m
453
464
2.4
2.90
160
12
9.5
1.8
hcR4m
465
472
1.5
2.95
160
8
5.5
1.8
hcR3m
506
480
-5.1
3.00
160
6
3.5
1.8
Tab. 5.1:
Comparison of numerically established maximal stress σmax,sim and calculated maximal stress σmax according to [51]
Fig. 5.3:
Location of maximal stress σmax near a transverse circular hole in a hollow cylinder under axial tension
The deviations concerning the maximal stress at the edge of the hole (hole-drilling measurement point) between the cylindrical models used in this investigation and the literature solution are minimum 1.5 % and maximum 5.1 %. These deviations are possibly caused by the mesh design near the hole. In Fig. 11.7, Fig. 11.8 and Fig. 11.9 (appendix chapter Chap.11), the initial stress distribution in x-direction near the hole is plotted for model hcR6m, hcR4m and hcR3m, respectively. Although the initial stress is set with 160 MPa by preloading all elements of the model, small stress differences in
94 the range of 0.3 % to 1 % are observed near the hole. This stress inhomogeneity near the hole may cause, in terms of error propagation, the listed deviations in Tab. 5.1 after introducing the through-hole into the modeled specimens.
The same comparison could not be applied in case of the cylindrical model with an ø6 mm center through-hole hchR6m (specimen of Type 4). In this case, no theoretical solution for the calculation of a stress distribution near the edge of the hole-drilling measurement point was found in literature. A solution for the calculation of a stress concentration factor at the edge of the ø6 mm center through-hole, similar to the calculation summarized in Tab. 5.1, was not exactly possible because the stress concentration factors according to [51] are plotted up to the ratio d0/2R = 0.4. Considering the “big” center through-hole with d0 = 6 mm, the limit is exceeded with the ratio d0/2R = 0.5. A extrapolation of the concentration factor according to the diagram in the literature [51] determines a value, which is 10 % lower than the calculated value of σmax,sim/σnom= 3.9. Nevertheless, it is assumed that the hchR6m model has a similar numerical uncertainty as the hcR6m due to the fact that the model hchR6m of Type 4 was generated from the hcR6m model of Type 2. Both have the same mesh within the hole region (hole-drilling measurement point).
5.2.2 Uncertainty in Experimental Calibration As in all measurement applications, the strain measurement and consequently the
residual stress calculation using the hole-drilling method is always associated with errors and uncertainty. In Chap. 2.2.3 (Tab. 2.2) the influences and sources of error of the whole residual stress evaluation process are listed. Some of these influences and sources of errors are of importance only for the final steps of the residual stress analysis, i.e. the stress calculation after the introducing of the hole (e.g. by drilling) and the measuring of the relieved strains. To these influences one can include the selection of the evaluation method and indirectly the geometrical boundary conditions. This is due to the fact that the geometrical boundary conditions were specified according to the evaluation method. Every evaluation method, e.g. the differential MPA method, includes a specified amount of calibration functions or factors, which were determined using models or specimens with a simple geometry (e.g. a flat plate). Another influences or source of error can be related to the selection and use of the measurement
95 hardware. These are e.g. the strain gage technique or the drilling technique. Most of these influences can be minimized or mostly eliminated if the experimental techniques are correctly applied. At all events, some sources of errors remain and should be considered in an uncertainty analysis. In general, the measurement uncertainty u is defined as the interval between the correct mean value of a series of measurements x and the upper and lower limit [52]. It considers two components: the random component u z for the random deviations and the systematic deviations u s for the unknown systematic deviations. The measurement uncertainty can be calculated by the linear addition of both components or, by geometrical summation if both components are approximately of equal size {5.1}.
{5.1} u = u z + u s
or
u = u z2 + u s2
The random component u z can be calculated for a series of measurements under repetitive conditions with unknown standard deviation using {5.2}, in which the variable t is a distribution according to Student, n is the number of measurements and s is the empirical standard deviation.
{5.2} u z =
t n
s
{5.3} u s = ± a12 + a 22 ... + a n2
The systematic uncertainty u s , which exists in all measuring devices, can generally only be determined using reliable data from manufacturers or using appropriate experimental experience. A possible way to estimate the systematic uncertainty u s is by calculating the geometrical sum of all single values ai {5.3}. In the case of the analysis of uncertainty for the experimental calibration of the holedrilling method two points are to be taken into consideration. First of all, it must be considered that with the available equipment only one measurement can be performed for each calibration specimen. For example, each milling cutter has a certain
96 dimensional tolerance and introduces for this reason a hole with a specific diameter d0. An additional measurement requires another measuring point, which may have a different hole diameter d0. Thus, it is impossible to calculate the random uncertainty u z in terms of a statistical analysis because it is impossible to repeat the same calibration again. For this reason, the random uncertainty is quantified with zero in this investigation. The second component, the standard uncertainty u s of the experimental calibration, may be approximated by defining and quantifying the single uncertainties and subsequent calculation of u s according to {5.3}. For this work it is especially important that the uncertainty analysis should be applied on the experimental calibration strain distributions in order to quantify the uncertainty and to compare, from this point of view, the experiments with the simulations. Although the experimental calibrations in this investigation were carried out as accurately as possible, factors affecting the magnitude of relieved strains still remain. The following factors, which may be not exhaustive, were listed according to literature [33, 38, 53], and the experience made during the work: Strain measurement technique: The sources of measurement errors caused by the
strain gage technique can be found in the strain gages themselves, the measuring amplifier, the wires, etc. According to [54], the accuracy requirements for static stress analysis for CEA gage series are “moderate”, which means, in general, for the whole measuring chain that the accuracy is in the range of 2 % to 5 %. However, it should be noted that the resolution of the strain gages is approximately 1 µm/m. Hence, the uncertainty becomes greater, especially at small hole depths and small strain values. For the subsequent calculation, the strain gage uncertainty is quantified for all examples given in this discussion with a fixed value of 3.0 %, which is less than the average in the range of 2 % to 5 %. hole diameter d0: The dimension of the introduced hole diameter affects directly the
magnitude of the relieved strains. As it can be seen in Fig. 5.11 and Fig. 5.12 in a subsequent chapter Chap. 5.3.2, the absolute strain values increase with increasing hole diameter and vice versa. A change of 2.7 % in the dimension of the hole diameter (e.g. 1.80 mm to 1.85 mm) leads to a change of the strain values of approximately 5 %. A
97 change of 5.5 % in the dimension of the hole diameter (e.g. 1.80 mm to 1.90 mm) may even lead to a change of the strain values of approximately 10 %. This is of particular importance when comparing the experimental calibrations with the numerical calibrations in this work. In many cases, the experimentally introduced hole diameter does not coincide with the designed model hole diameter (s. results chapter Chap. 5). In the last statements it is assumed that the hole diameter is determined exactly. In practice, the hole diameter was determined using the microscope of the RS-200 after each experimental calibration. The hole diameter was measured by reading the scale division nsc in the microscope and by calculating it according to {5.4} The factors in this formula are the specific factors of the used microscope. They convert the scale division into the metric system.
{5.3} d 0 =
n sc × 0.158 × 25.4 102
in mm
This means that an error in the range of nsc = 1 is possible when measuring the hole diameter. For example, a hole diameter of nsc = 46 is approx. 1.80 mm and a hole diameter of nsc = 47 means approx. 1.85 mm. As already mentioned, a change of 0.05 mm of the hole diameter leads to a change of approx. 5 % in the relieved strain values. For the comparison of the experimental and numerical calibration it should be noticed that the previously described effect of a smaller or bigger experimental hole diameter on the strain values could be compensated by the measurement error of the determination of the hole diameter. For the subsequent calculations, the uncertainty of the hole diameter is quantified for each subsequent example individually depending on the measured hole diameter after the specific calibration. Establishing zero depth: The determination of z = 0 mm is in many cases difficult.
The reason is that the strain gage insulating rosette backing as well as the adhesive layer must be removed before the drilling process starts on the testing material. This rosette backing/adhesive layer is in the range of 0.1 mm. When working with the RS-200 holedrilling device, the user has to remove this polyimide/adhesive layer by making several attempts, i.e. drilling a very small increment, controlling the change in the drilling noise and inspecting wheter the material surface is visible when using the microscope. This
98 process of “listening and looking” [55] can lead to errors, especially if the surface is rough or the surface curvature is high. In practice, the effect of an erroneous zero depth is a shifting of the strain distributions along the depth axis z [38]. The uncertainty of the zero depth determination is quantified in this special case with 0.02 mm because this value is the smallest depth increment, which can be set with the depth setting micrometer of the RS-200 hole-drilling device, which was used for the calibration experiments. Setting up of hole depth increments Δz: After establishing zero depth with the depth
setting micrometer, the subsequent depth increments Δz are also adjusted to an accuracy of 0.02 mm. This error can possibly compensate the previous error of the zero depth determination so that it is not considered explicitly in the subsequent calculation. Eccentricity of the hole: Eccentricities of the holes as a result of an incorrect
adjustment will lead to a magnification or minimization of the calibration strain values, similar to the effect of the change of the hole diameter or the hole depth. The uncertainty related to an eccentricity less than 0.02 mm is quantified in [38] with less than 2 % r. However, this uncertainty value is related to the stress evaluation and not to the relieved strain of the calibration. This error is not considered in the subsequent calibration. Perpendicularity of the hole axis relevant to the surface: According to [53], the
uncertainty concerning the perpendicularity of the hole axis relevant to the surface is negligible for plane surfaces and unknown for bent surfaces. The experience in this work shows that in the case of the flat specimens, the correct perpendicular positioning of the milling cutter was set without difficulties. In the case of the cylindrical specimens, the perpendicular positioning was controlled particularly during the process of the alignment of the milling cutter over the hole-drilling rosette. This process may affect the subsequent drilling and leads to uncertainties, which were not investigated in this work. Irregularities in the hole shape: In contrast to the models, the experimentally
introduced hole during the calibration process shows deviations from the ideal blindhole shape. One reason is that the used milling cutters have a 45° bevel, which creates
99 consequently a bevel at the hole bottom. In order to quantify the uncertainty caused by the irregularities in the hole shape, a model with a 45° bevel at the bottom of each drilling increment was additionally generated. This additional model is, with exception of the hole region, identical to the reference model fT6D20. The difference between the strain distribution of the model with bevel Δεbevel and the strain distribution of the reference model Δεreference (with ideal blind-hole shape) is plotted in the diagram of Fig. 5.4 for strain gage no. 3. An axial load of σx = 160 MPa was applied on both models. At a hole depth of 0.02 mm the difference is 0.38 µm/m which is approx. 15 %. The maximal absolute difference is 4.3 μm/m which means 3.9 % at a hole depth of 0.5 mm. At a hole depth of 1 mm the difference is 3.1 µm/m which is approx. 1.6 %. In the case of strain gage no. 1, the maximal difference does not exceed 1 μm/m. In order to provide a uniform value for the subsequent calculation, the uncertainty is set with a value of 2 %.
Δεbevel - Δεreference [μm/m]
5
4
3
σx = 160 MPa strain gage no. 3
2
1
0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 5.4:
Strain difference between a simulation with a model which considers a 45°hole bottom bevel at each hole increment and the reference model with an ideal blind hole shape.
Magnitude of calibration stress - plastification: To avoid plastic strains as a
consequence of stress concentrations, the residual stresses in general and, specifically, the calibration stresses should not exceed approx. 60 % of the local yield stress as mentioned in Chap. 2.2.3. This request is fulfilled for most experimental calibrations, which use the flat specimens of Type 1 and the cylindrical specimens of Type 2. In the case of the specimens of Type 3 (flat specimens with ø10 mm center through-hole) and
100 especially of Type 4 (hollow cylindrical specimens with ø6 mm center through-hole), the notch effect is intensified due to the geometrical condition of the hole (measurement point) close to the center through-hole. This leads to plastification and, as a result, the deformations near the hole (measurement point) are inhomogeneous and not elastic. An example for this statement is shown in Fig. 11.10 (appendix Chap. 11). This figure shows the equivalent stress according to Mises, which is symbolized by a color plot over the hchR6m model. The actual hole depth is z = 1 mm and the model is externally loaded with 210 MPa, which is the maximum load in the experimental calibration. In the hole vicinity (measurement point), it can be seen, that in all grey colored sections of the model the equivalent stresses exceed 690 MPa. This value is, according to the manufacturer information, the minimal yield stress of the material used in this investigation. The observation is especially noticeable between the ø6 mm center through-hole and the measurement point. The resulting error was not investigated and for this reason, this effect is not quantified for the subsequent calculation of the measurement uncertainty. The plastification effect is ignored in this work in order to separate only the geometry effect during determination of the calibration functions. This means that the subsequent calculations using the MPA algorithm assumes linear-elastic material behavior and, for this reason, all FEM calibrations are elastic. Nevertheless, it should be mentioned that plastification problems can be significantly and several investigation are found in literature concerning this matter, e.g. [33, 56, 57].
Magnitude of calibration stress due to exact positioning of strain gage rosette near edges: The strain gage rosette is bonded manually to the measurement point of the
specimen and hence, it might be placed imprecisely. If the distance to specimen edges is sufficiently large and the stress distribution is mostly uniform, this uncertainty does not emerge. The situation is different for other specimens e.g. the flat specimens with the ø10 mm center through-hole and the hollow cylindrical specimen with the ø6 mm through-hole. Due to the stress concentration caused by the through-hole, the stress distribution is inhomogeneous in this region. The figures Fig. 10.1 to Fig. 10.4 (model fhT6D2m and fhT1.5D2m) and Fig. 10.5-Fig.10.6 (model hchR6m) show the stress variation with increasing distance D. Therefore, if the strain gage rosette is not placed exactly, the experimental calibration stress values may differ from those of the simulation. In the case of the hchR6 specimen for example, the hole center (measurement point) should be 2 mm away from the edge of the ø6 mm through-hole.
101 This distance is exactly modeled in the simulation but, in the case of the experiments, an uncertainty is existent. This uncertainty was not investigated in this work and is not considered for the subsequent calculation.
5.2.3 Graphical Representation of Uncertainty in Calibration Results In this chapter four calibrations are selected in order to compare the quality of the
numerical results with the experimental results from the perspective of the experimental uncertainty. Importance was attached to choosing calibrations which consider the different geometrical parameters for this graphical representation of uncertainty. The selected calibrations were carried out using the following specimen: •
fT6D20 - “reference flat specimen” (s. Fig. 4.1)
•
fT1.5D20 - “thin flat specimen” (s. Fig. 4.6)
•
hcR3 - “thin and curved specimen” (s. Fig. 4.24)
•
hchR6 - “thin, curved and near to edge specimen (s. Fig. 4.33)
The random uncertainty is set to zero as explained in the previous chapter. The systematic uncertainty is determined considering four single values which are discussed in the previous chapter Chap. 5.22. The first three values are the uncertainty from the strain gage technique astrain gage, the uncertainty from the hole diameter adiameter and the uncertainty of the hole shape ashape. These three values calculate the error bars parallel to the strain axis Δε according to {5.4}. The values of the strain gage technique uncertainty and the hole shape uncertainty are set for all four calculations with astrain gage
= 3 % and ashape = 2 %. The hole diameter uncertainty adiameter is set individually for
each calculation. The fourth value is the uncertainty of the determination of the hole depth with ahole depth = 0.02 mm and is plotted as an error bar parallel to the depth axis z {5.5}.
2 2 2 {5.4} u s ,ε = a straingage + a diameter + a shape
{5.5} u s , z = a holedepth = 0.02mm
in [%]
102 Fig. 5.5 shows the calibration results of the reference specimen fT6D20 (same diagram as Fig. 4.1) with the addition that the experimental uncertainty is symbolized with error bars. The calculated total systematic experimental uncertainty of the strain values (error bar in Δε-direction) is for this calibration 6.16 %. In this case the numerical hole diameter of the simulation d0,sim coincides with the experimental one d0,exp. Therefore, adiameter considers for this example the measurement error when determining the diameter with the microscope and is set to 5 %. For this example it can be held that the numerical strain distribution of strain gage no. 3 runs exactly within the range of the error bars. The strain distribution of strain gage no. 1 is, in deeper increments, slightly beneath the tolerance range. The same observation has been made in case of the calibration of the “thin flat specimen” fT1.5D20. The calculated total systematic uncertainty here is also 6.16 % with a hole diameter uncertainty adiameter = 5 %, which considers only the diameter measurement error (s. Fig. 5.6). Another example is the calibration result of the “thin and curved” specimen hcR3, which is plotted in Fig. 5.7 with the corresponding error bars. The hole diameter uncertainty of adiameter = 10 % considers, in this case, the difference between the experimental hole diameter d0,exp and the hole diameter of the model d0,sim. This leads to a calculated total strain uncertainty of 10.63 %. Similar to the two previous examples, the numerical strain distribution of strain gage no. 3 runs within the tolerance range and the distribution of strain gage no. 1 runs, for deeper increments, beneath the tolerance range.
103 100
sg 1
50
Δε [μm/m]
0
-50
sg 2 Experiment
-100
Simulation -150
·
___
fT6D20 (Type 1) σc,x = 160 MPa d0,exp = 1.80 mm d0,sim = 1.80 mm
-200
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig. 5.5:
Comparison of experimental and numerical calibration with reference specimen fT6D20. Uncertainty values: astrain gage = 3 %, adiameter = 5 %, ashape = 2 %, ahole depth = 0.02 mm
100
sg 1
50
Δε [μm/m]
0
-50
-100
-150
-200
Experiment • Simulation ___
sg 2
fT1.5D20 (Type 1) σc,x = 160 MPa d0,exp = 1.78 mm d0,sim = 1.80 mm
sg 3
-250 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig. 5.6:
Comparison of experimental and numerical calibration with specimen fT1.5D20. Uncertainty values: astrain gage = 3 %, adiameter = 5 %, ashape = 2 %, ahole depth = 0.02 mm
The last calibration example in Fig. 5.8 shows numerical and experimental strain distributions which differ at the first sight and also when the described uncertainty analysis is applied. This calibration was carried out using the hchR6 specimen, which highly violates the geometrical conditions by being “thin”, “curved” and having the measurement point “close to the edge”. The hole diameter uncertainty of adiameter = 5 % considers in this case the difference between the experimental hole diameter d0,exp and
104 the hole diameter of the model d0,sim. The total strain uncertainty is then 6.16 %. The numerical strain distribution for strain gage no. 1 mostly coincide with the experimental tolerance range whereas the numerical strain distribution for strain gage no. 3 coincide only at hole depths between 0.4 mm and 0.7 mm within the experimental tolerance range.
100 50
sg 1
Δε [μm/m]
0 -50
sg 2 -100
Experiment • Simulation ___
-150 -200 -250
hcR3 (Type 2) σc,x = 160 MPa d0,exp = 1.86 mm d0,sim = 1.80 mm 0.0
0.2
sg 3 0.4
0.6
0.8
1.0
1.2
-300
z [mm]
Fig. 5.7:
Comparison of experimental and numerical calibration with specimen hcR3. Uncertainty values: astrain gage = 3 %, adiameter = 10 %, ashape = 2 %, ahole depth = 0.02 mm 100
sg 1
Δε [μm/m]
0
-100
sg 2
Experiment • Simulation ___
-200
hchR6 (Type 4) σx = 160 MPa d0,exp = 1.76 mm d0,sim = 1.80 mm
-300
sg 3
-400 0.0
0.2
0.4
0.6
0.8
1.0
1.2
z [mm]
Fig. 5.8:
Comparison of experimental and numerical calibration with specimen hchR6. Uncertainty values: astrain gage = 3 %, adiameter = 5 %, ashape = 2 %, ahole depth = 0.02 mm
With the exception of the last example determined with the hchR6 specimen, the uncertainty analysis in this chapter shows that the numerical calibration almost
105 reproduces the experimental calibration within the calculated tolerance range. It must be pointed out that several experimental uncertainty sources, which were discussed in the previous chapter, are not considered in the uncertainty calculation. Especially in the case of the calibrations with the hchR6 specimen, uncertainties concerning plastification, the zero-depth determination, the perpendicularity of the hole axis relevant to the surface and the exact positioning of the strain gage rosette, may cause additional errors and, thus, contribute to a higher total strain uncertainty. Taking the results of this chapter into consideration, it can be said that a calibration is always associated with uncertainties, as many influencing factors affect the results. In the case of a calibration carried out with finite elements, the accuracy of the results depends on influencing factors like the selected type of elements, the refinement of the model, the design of the mesh, etc. The quality of the hole-drilling calibration model can be analyzed by comparing some specific results with existent solutions. Once an acceptable accuracy is achieved with a finite element calibration, the resulting calibration strain distributions run almost consistently and smoothly. In contrast to a numerical calibration, the experimental distributions may run within a tolerance range and are not always smooth due to the different sources of uncertainties described in this chapter. This is problematic inasmuch as the determined calibration strain distributions are used for the approximation of the calibration functions Kx and Ky of the MPA algorithm. A scattering or a shift of the strain distribution leads to errors in the determination of the calibration functions and consequently in the stress calculation. For this reason, an experimental calibration of the hole drilling method according to the MPA algorithm should be regarded as complementary to the numerical calibration in order to have comparative values and to gain experience. The proper determination of the calibration functions should be carried out using the results of the numerical calibration.
5.3 Stress Evaluation on Specimens with Non-Reference Geometry The calculation of stress depth distributions after a hole-drilling measurement is possible using different evaluation methods. Some of them are already implemented in commercial software. On the one hand, the evaluation algorithms of these programs
106 differ in the physical assumption of the strain relaxation after the removal of stressed material. On the other hand, geometrical boundary conditions are defined for each evaluation program, because the determination of these algorithms bases on calibrations using specimens or models which have an almost ideal shape, e.g. a thick and wide flat plate. The application of the hole-drilling method on components, which violate the specific boundary conditions, leads to stress errors, i.e. the calculated stress distribution value differs from the real one. In order to confirm this statement, four numerical calibration results (same examples as in Chap. 5.2.3) are selected as input strain distributions for the MPA software. Fig. 5.9 shows the stress distribution in x-direction σx calculated with the MPA software for the models fT6D20m (reference), fT1.5D20m and hcR3m. The uniform stress in x-direction within the hole volume is σc,xFEM = 160 MPa. The fourth example is shown in Fig. 5.10, which compares the calculated (with MPA software) stress distribution σx for the model hchR6m with the initial stress distribution in x-direction at the measurement point σc,x-FEM. 240 220
hcR3m
σx [MPa]
200
fT1.5D20m
180
σc,x-FEM
160
fT6D20m
140 120 100 0
0.2
0.4
0.6
0.8
1
z [mm]
Fig. 5.9:
Comparison of calculated residual stress σx with MPA software (differential Method) [2] and stress distribution σc,x-FEM at measurement point before drilling. Input strain distribution calculated from calibration simulations using models fT620m (reference), fT1.5D20m and hcR3m.
All four examples were smoothed using compensative spline functions with a strain smoothing factor of fil = 0.5 (s. Chap.6.2.4). In addition to the diagrams in Fig. 5.9 and Fig. 5.10, stress differences are listed for selected depth values in Tab. 5.2. The calculation of these percental differences is related to the uniform stress distribution
107 σc,x-FEM = 160 MPa for the models fT6D20m, fT1.5D20m and hcR3m. In the case of the model hchR6m, the related initial stress value depends on the hole depth.
350
σc,x-FEM 300
hchR6m
σx [MPa]
250
200
150
100
50 0
0.2
0.4
0.6
0.8
1
z [mm]
Fig. 5.10:
Comparison of calculated residual stress σx with MPA software (differential Method) [2] and stress distribution σc,x-FEM at the measurement point before drilling. Input strain distribution calculated from calibration simulation using model hchR6m
Stress deviation Δσx related to calibration stress σc,x-FEM [%] depth [mm]
fT6D20m
hchR6m
hcR3m
fT1.5D20m
0.0
34.9
5.6
16.9
-5
0.2
31.3
21.3
19.4
-2.5
0.5
16.6
31.3
20.0
0.6
1.0
-82.5
-22.5
-11.3
-3.1
Tab. 5.2:
(reference)
Stress deviation Δσx related to calibration stress σc,x-FEM for evaluated simulations fT6D20m, fT1.5D20m, hcR3m, hchR6m. All calculations were carried out with the differential MPA method
The four exemplarily calculated stress distributions show considerable differences to the initial stress distributions. The calculated result of the ft6D20m reference model shows a characteristic oscillating distribution for the evaluation of uniform stress using the MPA software. This oscillating distribution, which has its origin in the calculation formalism of the MPA software, depends, among other things, in the shape of the calibration functions Kx and Ky and the application of the previous smoothing. The topics concerning the execution of the evaluation formalism are discussed in the
108 following chapters (Chap. 6 and Chap 7). The reference model fT6D20 does not violate the geometrical boundary conditions and hence, the calculated differences related to the initial stress distribution σc,x-FEM = 160 MPa are small compared to the other calculations with a maximal difference of 5 % at the selected hole depths. At the same time, this result represents the current status of the stress evaluation using the differential MPA method. The other calculation results show clearly the fact that a violation of the boundary conditions specified for the MPA method leads to errors in the stress calculation. In the case of the three non-reference models, all calculations show higher deviations than the reference. The maximal deviation values are calculated for the hchR6m model, which is “thin”, “curved” and whose measurement point is “close to the edge” of the ø6 mm hole. The calculated distribution of this model also shows a different trend compared to the initial stress distribution. The stress differences as well as the distribution of the calculated result of the “curved” and “thin” model hcR3m are between the results of the hchR6m model and the fT1.5D20m model. The “thin” and “flat” fT1.5D20m model shows a max. 20 % stress deviation at the selected values and the stress results run up to a hole depth of approx. 0.5 mm approx. 35 MPa parallel over the distribution of the reference specimen.
240 220
hcR3m
σx [MPa]
200
fT1.5D20m
180 160
σc,x-FEM
fT6D20m
140 120 100 0
0.2
0.4
0.6
0.8
1
z [mm]
Fig. 5.11:
Comparison of calculated residual stress σx with H-Drill software (integral Method) [34] and stress distribution σc,x-FEM at the measurement point before drilling. Input strain distribution calculated from calibration simulations using models fT620m (reference), fT1.5D20m and hcR3m.
109 To finish this initial analysis, the same strain distributions of the models fT6D20m, fT1.5D20m, hcR3m and hchR6m are used as exemplary input data for the integral method implemented in the HDrill software [34]. The diagrams in Fig. 5.11 (fT6D20m, fT1.5D20m and hcR3m) and in Fig. 5.12 (hchR6m) show that a stress calculation on models with non-reference geometry with the integral method also leads to stress differences.
350
σc,x-FEM
300
hchR6m
σx [MPa]
250 200 150 100 50 0 0
0.2
0.4
0.6
0.8
1
z [mm]
Fig. 5.12:
Comparison of calculated residual stress σx with H-Drill software (integral Method) [34] and stress distribution σc,x-FEM at the measurement point before drilling. Input strain distribution calculated from calibration simulations using model hchR6m.
It should be noticed that the integral method requires a different depth increment distribution from the one used in these examples. A depth increment distribution with small depth increments leads to large errors according to [28, 58]. This problem is related to the matrix character of the calculation, in which a high spatial resolution causes a poorly conditioned calibration matrix and, consequently, leads to higher stress errors. For this reason, the stress values for a hole depth smaller than 0.15-0.17 mm are not plotted in Fig. 5.11 because some of them tend to extreme oscillation. Nevertheless, except the result of the reference model fT6D20m, which runs close to the initial uniform stress distribution of σc,x-FEM = 160 MPa, the other results show deviations from the initial stress distribution and are comparable, at first sight, to the results obtained with the MPA software. A further discussion on these results is not continued, as the integral method is not a subject of investigation in this work.
110 It can be concluded that the calculation of a residual stress depth distribution using current evaluation methods on components which violate the specific geometrical boundary conditions leads to considerable errors. The individual geometrical influences are discussed in the following chapters.
5.4 Influence of Geometrical Parameters on Results of Hole-Drilling Measurements 5.4.1 Influence of Hole Diameter - d0 This chapter is included in order to underline the importance of the hole diameter for the
evaluation algorithms of the hole-drilling method. When carrying out a hole-drilling measurement, the experimentally introduced hole diameter may differ depending on the elastic properties of the material and the used milling cutter. The different current evaluation programs of the hole drilling method (e.g. MPA, H-Drill) consider this fact by including a set of diameter specific calibration functions (or diameter specific factors) within the possible range of diameters that may be introduced for the particular strain gage rosette. If the dimension of the introduced hole diameter is between two calibration hole diameters, the calibration function (or the calibration factor) is calculated using linear interpolation between the two sets of diameter specific calibration functions. 0 z = 0.2 mm -50
Δε [μm/m]
-100 z = 0.5 mm -150
-200
Simulation of fT6D20 specimen influence of hole diameter d0 at selected hole depths Strain gage no. 3
-250
-300 1.65
1.7
1.75
z = 1.0 mm
1.8
1.85
1.9
1.95
d0 [mm]
Fig. 5.13:
Relieved Strain (strain gage no. 3) versus hole diameter at selected hole depths (reference model fT6D20m)
111 depth
strain gage no. 3: strain deviation [%]
[mm]
d0 = 1.70 mm d0 = 1.75 mm d0 = 1.80 mm d0 = 1.85 mm d0 = 1.90 mm
0.2
-9.9
-4.3
0.0
7.7
14.0
0.5
-10.2
-4.9
0.0
6.4
12.3
1.0
-9.8
-4.9
0.0
5.4
10.6
Tab. 5.3:
Strain deviation (strain gage no. 3) related to relieved strain of reference model fT6D20m with reference hole diameter of d0 = 1.8 mm
The influence of the hole diameter at selected hole depths on the relieved strain is shown in Fig. 5.13 and Fig. 5.14 for strain gage no. 3 and strain gage no. 1, respectively. These results are calculated from the reference model fT6D20m (s. in Chap. 4.4 Fig. 4.2 and Fig. 4.3). The linear dependency of the relieved strain Δε on the hole diameter d0 can be clearly seen in both diagrams where the strain values increase with increasing hole diameter. In addition to the diagrams, Tab. 5.3 and Tab. 5.4 give the strain difference related to the reference hole diameter d0 = 1.8 mm at the same hole depths. These differences are the basis for the calculation of the hole diameter uncertainty adiameter in the previously discussed chapter Chap. 5.23.
60 z = 1.0 mm 50
Simulation of fT6D20 specimen influence of hole diameter d0 at selected hole depths Strain gage no. 1
Δε [μm/m]
40
30
z = 0.5 mm 20
10 z = 0.2 mm 0 1.65
1.7
1.75
1.8
1.85
1.9
1.95
d0 [mm]
Fig. 5.14:
Relieved Strain (strain gage no. 1) versus hole diameter at selected hole depths (reference model fT6D20m)
112 depth
strain gage no. 1: strain deviation [%]
[mm]
d0 = 1.70 mm d0 = 1.75 mm d0 = 1.80 mm d0 = 1.85 mm d0 = 1.90 mm
0.2
-8.9
-4.1
0.0
5.4
10.3
0.5
-8.4
-4.0
0.0
4.9
9.4
1.0
-8.1
-4.0
0.0
4.3
8.4
Tab. 5.4:
Stain deviation (strain gage no. 1) related to relieved strain of reference model fT6D20m with reference hole diameter of d0 = 1.8 mm
An evaluation using the diameter specific strain distributions as input data for the MPA software calculates almost equal stress distributions. The maximal average difference of the stress calculation related to the reference calculated stress distribution is only 2.3 %. This can be interpreted as a calculation uncertainty. The hole diameter influences linearly the relieved strain. An implementation of (new) calibration functions or calibration factors into stress evaluation algorithms should always consider the possible range of the effectively introduced hole diameter by the experimental measurement.
5.4.2 Influence of Specimen Thickness - T The influence of the specimen thickness based on the previous numerical calibration
results (s. in Chap. 4.3 Fig. 4.7 and Fig. 4.8) is discussed in this chapter. In all examples, the models with different thickness T were loaded with a uniform tensile stress and had a constant hole diameter of d0 = 1.8mm. The first question is whether the established boundary condition of T = 3.33 x d0 [2] is adequate or may be minimized. For this purpose, the relieved strain Δε is plotted versus the specimen thickness T at selected hole depths in Fig 5.15 (strain gage no. 3) and Fig. 5.16 (strain gage no. 1). It can be seen that the strain distributions over the specimen
thickness
are
almost
constant
between
T = 3.33 x d0 = 6 mm
and
T = 1.67 x d0 = 3 mm. This observation is confirmed by the strain deviations listed in Tab. 5.5 (strain gage no. 3) and Tab. 5.6 (strain gage no. 1). The strain deviations between the results of T = 1.67 x d0 = 3 mm and the reference are small with values between 0.1-3.9 %. Thus, in this special case (uniaxial stress and violation of single boundary condition thickness) the boundary condition may be minimized.
113
The second question concerns the developing of the strain distributions if the thickness of the specimen exceeds the limit and consequently the strain values change. In the diagrams of Fig 5.15 and Fig. 5.16 it can be seen that for a decreasing thickness between T = 1.67 x d0 = 3 mm and T = 0.55 x d0 = 1 mm the strain values change nonlinearly without a consistent trend at the selected hole depths. When the specimen is perforated (at T = 1.0 mm and z = 1.0 mm), the absolute strain values reach a maximum which is clearly visible for strain gage no. 1 (transverse to load direction) with a difference of 90.1 %. To complete the discussion about the influence of the specimen thickness, the distribution of the strain differences over the hole depth are plotted in Fig. 5.17 and Fig. 5.18 for different thicknesses. The strain difference ΔεDIFF.(T,z) is calculated according to {5.6}. {5.6} Λε DIFF . (T , z ) = Λε (T , z ) − Λε ( REF , z ) The strain differences increase with increasing hole depth and with decreasing specimen thickness. First of all, it can be observed that the strain difference of the distribution with T = 1.67 x d0 = 3 mm runs close to zero for both strain gages. The distribution with T = 1.11 x d0 = 2 mm shows a visible difference to the reference distribution for strain gage no. 3 and small difference for strain gage no. 1. Especially in the case of the thin specimens with T = 1.5 mm and T = 1.0 mm the differences as well as the slope change of the distribution are high. The maximal strain difference is -60 μm/m at z = 0.7 mm = 0.7 x T for T = 1.0 mm and -30 μm/m at z = 0.9 mm = 0.6 x T for T = 1.5 mm. After introducing the through-hole, i.e. when the specimen is completely perforated, the absolute values of the strain differences decrease and increase for strain gage no. 3 (parallel to load direction) and strain gage no. 1 (transverse to load direction), respectively. These facts can be related to the loss of stiffness in the hole vicinity of the almost introduced or completely perforated through-hole, i.e. the effect of the blind hole, which causes a partial release of strains, is reduced gradually as the specimen is almost perforated. Finally, an elastic spring-back occurs after the perforation of the specimen. This causes finally a total relaxation of stress near the hole.
114 0 z = 0.2 mm -50
z = 0.5 mm
Δε(T)
-100
Influence of thickness T at selected hole depths Strain gage no. 3
-150
-200
z = 1.0 mm
-250 0
1
2
3
4
5
6
T [mm]
Fig. 5.15:
Relieved strain (strain gage no. 3) versus thickness at selected hole depths
depth
strain gage no. 3: strain deviation [%]
[mm]
T = 1 mm
T = 1.5 mm
T = 2 mm
T = 3 mm
T = 6 mm
0.2
61.1
21.2
6.2
-1.5
0.0
0.5
49.8
19.4
6.6
-0.7
0.0
1.0
20.0
15.3
6.7
-0.1
0.0
Tab. 5.5:
Models with different thickness: strain deviation (strain gage no. 3) related to relieved strain of reference model fT6D20m with reference thickness of T = 3.33xd0 = 6 mm
110
Influence of thickness T at selected hole depths Strain gage no. 1
100 90
Δε [μm/m]
80 70 60
z = 1.0 mm
50 40 30
z = 0.5 mm
20 10 z = 0.2 mm
0 0
1
2
3
4
5
T [mm]
Fig. 5.16:
Relieved strain (strain gage no. 1) versus thickness at selected hole depths
6
115 depth
strain gage no. 1: strain deviation [%]
[mm]
T = 1 mm
T = 1.5 mm
T = 2 mm
T = 3 mm
T = 6 mm
0.2
-92.7
-29.7
-6.3
3.9
0.0
0.5
-50.4
25.2
-7.9
1.7
0.0
1.0
90.1
-3.6
-3.0
1.2
0.0
Tab. 5.6:
Models with different thickness: strain deviation (strain gage no. 1) related to relieved strain of reference model fT6D20m with reference thickness of T = 3.33xd0 = 6 mm
10
Δε(T)−Δε(Ref.) [μm/m]
0
T = 1.67d0 = 3 mm
-10 T = 1.11d0 = 2 mm -20
-30
T = 0.83d0 = 1.5 mm
-40
Influence of thickness strain difference Strain gage no. 3
-50 T = 0.55d0 = 1 mm -60 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z [mm]
Fig. 5.17:
Influence of thickness: strain difference versus hole depth: strain difference of strain gage no. 3 is related to strain distribution of reference model Δε(Ref.)
50
Influence of thickness strain difference Strain gage no. 1
Δε(T)−Δε(Ref.) [μm/m]
40
T = 0.55d0 = 1 mm T = 0.83d0 = 1.5 mm
30
20
10
T = 1.11d0 = 2 mm
0 T = 1.67d0 = 3 mm -10
-20 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z [mm]
Fig. 5.18:
Influence of thickness: strain difference versus hole depth: strain difference of strain gage no. 1 is related to strain distribution of reference model Δε(Ref.)
116 To conclude, the last question is about the thickness influence on the stress calculation using the MPA software. For this purpose, the thickness specific calibration strain distributions are used as input data and evaluated into stress distribution with the MPA software. After this, the stress difference is calculated from the stress values σx(T) and σy(T) and the stress values of the reference specimen according to {5.7}. {5.7} σ DIFF . (T , z ) = σ (T , z ) − σ (REF ., z ) The input strain distributions for all examples consider values up to z = 1 mm, because the terminating condition of the MPA software evaluates up to a hole depth of ξ = z/d0 = 0.6 mm. In this special case with a constant hole diameter of d0 = 1.8 mm, the software terminates the evaluation at a hole depth of 1.08 mm. The results are plotted in Fig. 5.19 for the stress differences in x-direction and Fig. 5.20 for the stress differences in y-direction. In addition, Tab. 5.7 and Tab. 5.8 list the stress deviations (or differences) for the corresponding stress direction. Note that the values in Tab. 5.8 show the stress differences in transverse direction σy and are listed in MPa. This difference in the units is because some of the calculated σy(Ref.)-values are zero or close to zero. Hence, a calculation of a deviation in percent is impossible or leads to very high values.
150
σx(T)−σx(Ref.) [MPa]
Influence of thickness: stress difference σx using MPA-Method
T = 0.55d0 = 1 mm
100 T = 0.83d0 = 1.5 mm
50
T = 1.11d0 = 2 mm 0 T = 1.67d0 = 3 mm -50
-100
-150 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Fig. 5.19:
Stress difference σx(T) versus hole depth: Stress difference σx(T) is related to calculated stress distribution of reference specimen σx(Ref.) using MPA software
117 Stress deviation Δσx related to calculated reference stress σx(Ref.) depth [mm]
[%] T = 1 mm
T = 1.5 mm
T = 2 mm
T = 3 mm
0.0
67.1
23.0
6.6
2.0
0.2
62.2
22.4
7.1
0.6
0.5
28.6
19.3
8.1
0.6
1.0
-225.2
-8.4
6.5
1.3
Stress deviation Δσx related to calculated stress distribution (MPA software) of reference model σx(Ref.) for evaluated simulations with different specimen thickness
Tab. 5.7:
50
T = 0.55d0 = 1 mm
Influence of thickness: stress difference σy using MPA-Method
40
σy(T)−σy(Ref.) [MPa]
30
T = 0.83d0 = 1.5 mm
20 10 T = 1.11d = 2 mm 0 0 T = 1.67d0 = 3 mm
-10 -20 -30 -40 -50 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Fig. 5.20:
depth
Stress difference σy(T) versus hole depth: Stress difference σy(T) is related to calculated stress distribution of reference specimen σy(Ref.) using MPA software
Stress difference Δσy related to calculated reference stress σy(Ref.) [MPa]
[mm] T = 1 mm
T = 1.5 mm
T = 2 mm
T = 3 mm
0.0
40
12
2
-2
0.2
44
16
5
-1
0.5
1
13
6
0
1.0
-478
-32
-1
0
Tab. 5.8:
Stress difference Δσy related to calculated stress distribution (MPA software) of reference model σy(Ref.) for evaluated simulations with different specimen thickness
118 The differences for the calculated stress of the specimen with T = 1.67 x d0 = 3 mm are in both directions negligible as can be seen in the diagrams as well as in the deviation values in the tables. With decreasing thickness the calculated stress values increase and this
is
reflected
on
the
calculated
deviations.
For
the
specimen
with
T = 1.11 x d0 = 2 mm, the difference for σx is in average 7 % and for σy 3.5 MPa. In case of the thin specimens, the stress differences are considerable. For the specimen with T = 0.83 x d0 = 1.5 mm the differences for σx are for the selected hole depths on average approx. 18 % and 18 MPa for σy. The highest differences are determined for the thinnest specimen with T = 0.55 x d0 = 1 mm and especially at depth increments higher than z =0.6 mm the results are totally incorrect. The calculated stress difference is at the depths of z = 0, 0.02, 0.5 mm in average approx. Δσx = 53 %. But at a hole depth of z = 1 mm the difference is very high with Δσx = 225.2 % and Δσy = -478 MPa. Conclusion: The boundary condition “thickness” is set in [2] with T = 3 x d0. The
results in this chapter, which were all determined using uniformly loaded finite elements models under violation of only the single boundary condition T, show that the geometry limit may be minimized up to a thickness of T = 1.66 x d0. After this thickness, the relieved strains values increase with decreasing thickness. The trajectory of the strain distributions after a calibration on thin specimens depend upon whether the introduced hole almost perforates or completely perforates the specimen cross section. A stress calculation with the MPA software [2.44] leads consequently to higher stress values compared to a calculation using the strain distributions determined with the reference specimen. The stress differences are on average over the hole depth 18 % for T = 0.83 x d0 = 1.5 mm and about 53 % (225% at z = 1 mm) for T = 0.55 x d0 = 1 mm.
5.4.3 Influence of Distance Hole-Edge - D The results of Chap. 4.3 (Fig. 4.7 and Fig. 4.8) are taken as a basis for the discussion of
the influence of the distance from the hole center (measurement point) to the free edge of the specimen D. The boundary condition is set with D = 5 .. 10 x d0 according to the MPA specification [2]. Similar to the discussion of the previous chapter, the question to be answered about the distance influence is whether this specific limit may be reduced and what does it mean for a stress calculation with the MPA software. Therefore, the relieved strains at selected hole depths are plotted against the distance hole-edge in
119 Fig. 5.21 and Fig. 5.22 for strain gage no. 3 and strain gage no. 1, respectively. In both diagrams, the strain distributions of both strain gages at all selected hole depths run almost constantly in the range between D = 1.94 … 11.1 x d0 = 3.5 … 20 mm. Only at D = 1.11 x d0 = 2 mm and z = 1, a maximal strain difference related to the reference is calculated with approx. 4 % for strain gage no. 3 and approx. 2 % for strain gage no. 1. These relatively small strain differences lead to small differences of the calculated stress with the MPA software, as can be seen in Fig. 23. The stress differences σx(D,z) are calculated analogically as in {5.7}, with the difference that the stress distributions were dependet on the distance D. In addition, the stress deviations in per cent at selected hole depths are listed in Tab. 5.9.
0 z = 0.2 mm
-20 -40
Δε [μm/m]
-60 -80
z = 0.5 mm
-100 -120
Influence of Distance Hole-Edge at selected hole depths strain gage no. 3
-140 -160
z = 1.0 mm
-180 -200 0
2
4
6
8
10
12
14
16
18
20
Distance Hole-Edge [mm]
Fig. 5.21:
Relieved strain (strain gage no. 3) versus distance hole-edge at selected hole depths
120 60
z = 1.0 mm
50
Influence of Distance Hole-Edge at selected hole depths strain gage no. 1
Δε [μm/m]
40 z = 0.5 mm 30
20 z = 0.2 mm
10
0 0
2
4
6
8
10
12
14
16
18
20
Distance Hole-Edge [mm]
Fig. 5.22:
Relieved strain (strain gage no. 1) versus distance hole-edge at selected hole depths
50
Influence of distance hole-edge: stress difference σx using MPA-Method
40
σx(D)−σx(Ref.) [MPa]
30 20 D = 1.94 … 5.56 x d0 = 3.5 … 10 mm
10 0 -10 -20
D = 1.11 x d0 = 2 mm
-30 -40 -50 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Fig. 5.23:
Stress difference σx(D) versus hole depth: Stress difference σx(D) is related to calculated stress distribution of reference specimen σx(Ref.) using MPA software
Stress deviation Δσx related to calculated reference stress σx(Ref.)
depth [mm]
[%] D = 2 mm
D = 3.5 mm
D = 5 mm
D = 10 mm
0.0
-3.9
1.3
0.7
0.0
0.2
-3.2
1.3
0.6
0.0
0.5
-3.7
1.2
0.6
0.0
1.0
-9.7
2.6
1.3
0.0
Tab. 5.9:
Stress deviation Δσx related to calculated stress distribution (MPA software) of reference model σx(Ref.) for evaluated simulations with different distance hole-edge
121
The maximal stress difference in x-direction is calculated for D = 1.11 x d0 = 2 mm with approx. -5 % averaged over the hole depth and -9.7 % at z = 1 mm. The stress differences in y-direction are even smaller and, therefore, a graphical representation is not worthwhile. Within the scope of an additional investigation [59], FEM calculations were carried out using the models of the flat specimens with different distance hole-edge D. The models were loaded with a pure shear stress field of σc,x =160 MPa and σc,y = -160 MPa. For this loading case, the calculated stress deviation was in average 5 % for a model with D = 2 mm and T = 6 mm, which is similar to the deviations presented in this work for the uniaxial stress state. The stress deviation increases to a value of 20 % if the same model becomes “thin” with a thickness of T = 1.5 mm. In this work, the position of the CEA-06-062-UM strain gage rosette concerning the free edge of the specimen was always identical (s. also Fig. 3.1, Fig. 3.2 and Fig. 3.8). Thus, strain no. 3 is parallel to the free edge and strain gage no. 1 is perpendicular to the specimen edge with the following order: “specimens edge → hole → strain gage no. 1”. This order allows a positioning of the strain gage very near to the specimens edge with a minimum distance hole edge of D = 2 mm. If the order of the positioning of strain gage no. 1 is changed, e.g. “specimens edge → strain gage no. 1 → hole”, then the hole can not be introduced so closely to the specimen’s edge because of the length of the strain gage itself and the length of the soldering lug. The minimum distance hole-edge is then D = 9 mm. In such a positioning case, different FEM calculation with different loading conditions (uniaxial in x-direction, uniaxial in y-direction, biaxial plane stress, biaxial pure shear stress) show no significant differences of the relieved strains compared to the results with models with reference geometry. Conclusion: In contrast to the specimen thickness, the distance from the center of the
measurement point (hole center) to the specimen’s free edge does not influence the hole drilling results significantly. The maximum stress difference of approx. 5 % on average related to the reference is found for the results of the model with D = 1.11 x d0 = 2 mm. Thus, the boundary condition for this special case, which is set with min. D = 5 x d0 [2] may be minimized up to D = 1.94 x d0.
122 5.4.4 Influence of Cylinder Radius - R This chapter is based on the numerical results determined on cylindrical models (s.
Chap. 4.6) and is discussed similarly as in the previous examples. The boundary condition for the geometrical parameter radius of surface curvature is set with R = 3 x d0 according to [2]. A possible minimization of this boundary condition can be analyzed using the results in Fig. 5.24 and Fig. 5.25, which plot the relieved strain against the cylinder radius at selected hole depths for strain gage no. 3 and strain gage no. 1, respectively. These diagrams also distinguish between the relieved strain of the solid cylinders and the ones with the hollow cylinders. 0 sc hc
z = 0.2 mm -50
Δε [μm/m]
Ref. -100 sc
z = 0.5 mm
hc
Influence of Cylinder Radius at selected hole depths strain gage no. 3
-150
-200
sc
z = 1.0 mm
hc
/ /
-250 2
3
4
5
6
∞7
Cylinder Radius [mm]
Fig. 5.24:
Relieved strain (strain gage no. 3) versus cylinder radius at selected hole depths
strain gage no. 3: strain deviation [%]
depth
solid cylinder
[mm]
hollow cylinder
R = 3 mm
R = 4 mm
R = 6 mm
R = 3 mm
R = 4 mm
R = 6 mm
0.2
-10.8
-0.8
1.6
8.0
23.0
31.5
0.5
1.1
5.1
6.1
20.2
27.7
32.9
1.0
7.1
8.4
8.1
20.4
20.0
22.4
Tab. 5.10:
Cylindrical models: strain deviation (strain gage no. 3) related to relieved strain of reference flat model fT6D20m
123 70 sc 60 hc
Δε [μm/m]
50
Influence of Cylinder Radius at selected hole depths strain gage no. 1
z = 1.0 mm
40
Ref. 30 sc 20
z = 0.5 mm
10 z = 0.2 mm 0 2
hc sc hc
/ /
3
4
5
6
∞7
Cylinder Radius [mm]
Fig. 5.25:
Relieved strain (strain gage no. 1) versus cylinder radius at selected hole depths
strain gage no. 1: strain deviation [%]
depth
solid cylinder
[mm]
hollow cylinder
R = 3 mm
R = 4 mm
R = 6 mm
R = 3 mm
R = 4 mm
R = 6 mm
0.2
-34.2
-1.6
11.2
-61.2
-21.3
10.5
0.5
-12.9
16.6
25.2
-50.6
-16.9
8.4
1.0
2.3
24.8
27.2
-22.4
6.4
26.2
Tab. 5.11:
Cylindrical models: strain deviation (strain gage no. 1) related to relieved strain of reference flat model fT6D20m
Another illustration of the radius influence is plotted in the diagrams of Fig. 5.26 (strain gage no. 3) and Fig. 5.27 (strain gage no. 1), which show for all cylindrical models the distributions of the strain differences related to the reference strain distribution. The strain differences are calculated similarly to {5.6} using R instead of T as a variable. In addition to the diagrams, Tab. 5.10 (strain gage no. 3) and Tab. 5.11 (strain gage no. 1) list the strain deviation related to the reference strain of model fT6D20m (in %). First of all, it can be noticed that the difference between the results taken with the solid cylinder with R = 3.33 x d0 = 6 mm and the reference are approx. 6-8 % (max. 16 µm/m at z = 1 mm). This means that the boundary condition may not be minimized. After R = 6 mm (s. Fig. 5.25), the strain distributions in this case run almost constantly with decreasing cylinder radius. Furthermore, the addition of the parameter thickness, i.e. in
124 the case of the hollow cylinders, causes a magnification (approx. 10 % to 20 %) of the absolute strain values registered with strain gage no. 3. In the diagram of Fig. 5.26 the strain difference distributions for the solid cylinders with R = 6 mm and R = 4 mm run almost one upon the other and reach a maximal strain difference at z = 1 mm of 16 µm/m. The strain difference distribution of the solid cylinder with R = 3 mm runs approx. 2-5 µm parallel over the other two solid cylinder strain difference distributions. The strain difference distributions of the hollow cylinder compared to those of the solid cylinder behave almost similarly to the distributions of the thin flat specimens compared to thick flat specimens. Here, the strain differences increase with increasing hole depth up to a depth of z = 0.8 mm = 0.64 x T (with T = 1.25 mm). The maximal strain difference at this hole depth is -47 µm/m for the hollow cylinder with R = 6 mm. After this depth, the slope of the curves decreases and this thickness effect can be associated to the nearly perforated cross section of the hollow cylinder. The differences are at maximum 5-10 µm/m between the single distributions of the hollow cylinders (Fig. 5.26). The strains registered with strain gage no. 1 (transverse to the load direction) show significant strain deviations compared with the reference strain values. Already in the case of the solid cylinder with a radius of R = 3.33 x d0 = 6 mm the strain deviations at the selected hole depths are at maximum 26 %. A decrease of the cylinder radius causes a decrease of the absolute strain values of strain gage no. 1. The deviation is on average approx. -45 % for R = 1.67 xd0= 3 mm at the selected hole values. This curvature effect can be attributed to the position of strain gage no. 1 related to the influenced material section near the hole, which becomes smaller with decreasing radius. In Fig. 5.28, schematic sectional views of the cylinder specimens are shown. The position of the strain gage (s.g.) is symbolized with a gray line and the material section directly near the 1.8 mm hole (with z = 1 mm) is symbolized with a hatched area for all three examples. It can be seen that the length of the strain gage for the cylinder with R = 6 mm is completely over the cross-section material area which is near to the hole and about one half for the specimen with R = 3 mm. This assumption is verified by the plotted results over the sectional view of the cylinder models in Fig. 11.11 (scR6m), Fig. 11.12 (scR4m) and Fig. 11.13 (scR3m) in the appendix chapter Chap. 11. In these three examples the plotted strain is tangential to the cylinder surface with equal color scale. It can be seen that, in case of the models scR6m and scR4m, the strain gage is
125 almost completely over the orange marked surfaces. These orange surfaces symbolize strain values between 37.5-145.8 µm/m. In contrast, the strain gage of model scR3m is not completely over this orange marked area. A strain calculation, which is carried out by integrating the strain field over the strain gage dimensions, leads to smaller strain values in the case of model scR3m. Nevertheless, if a measurement should be carried out on a high curved surface, e.g. R = 3 mm, it is possible to use a smaller strain gage rosette. For example, a strain gage rosette, which allows the generation of a hole of max. d0 = 1 mm, reduces the described curvature effect. Moreover, in this case (R = 3 mm), the boundary condition “surface curvature” according to the MPA algorithm is then satisfied. The main disadvantage of a measurement using a smaller hole diameter is that the values of the relived strains are always smaller than the ones measured with a bigger hole diameter and consequently the measurement uncertainty has then a higher influence on the results.
10
scR3
scR4
Δε(R)-Δε(Ref.) [μm/m]
0
scR6
-10
-20
-30
Influence of cylinder radius hcR6 strain difference strain gage no. 3
-40
hcR4
hcR3
-50 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Fig. 5.26:
Influence of cylinder radius: strain difference versus hole depth: strain difference of strain gage no. 3 is related to strain distribution of reference model Δε(Ref.)
126 20 15
Δε(R)-Δε(Ref.) [μm/m]
scR6 10
scR4
hcR6
5 0
scR3
hcR4
-5 -10
hcR3
Influence of cylinder radius strain difference strain gage no. 1
-15 -20 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Fig. 5.27:
Influence of cylinder radius: strain difference versus hole depth: strain difference of strain gage no. 1 is related to strain distribution of reference model Δε(Ref.)
Fig. 5.28:
Schematic sectional view of cylindrical models and position of strain gage (strain gage no. 1) transverse to cylinder longitudinal axis
100
Influence of cylinder radius: stress difference σx using MPA-Method
σx(R)−σx(Ref.) [MPa]
75 hcR6 50
hcR4
hcR3
25 scR6
scR4
0 scR3 -25
-50 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Fig. 5.29:
Stress difference σx(R) versus hole depth: stress difference σx(R) is related to calculated stress distribution of reference specimen σx(Ref.) using MPA software
127
depth
Stress deviation Δσx related to calculated reference stress σx(Ref.) [%] solid cylinder
[mm]
hollow cylinder
R = 3 mm
R = 4 mm
R = 6 mm
R = 3 mm
R = 4 mm
R = 6 mm
0.0
-11.8
-7.2
-2.6
11.2
15.8
27.0
0.2
-3.8
7.1
7.1
24.3
34.0
39.1
0.5
2.5
8.1
8.1
30.4
29.8
29.8
1.0
-5.8
14.8
12.9
-20.0
-18.7
-38.7
Tab. 5.12:
Stress deviation Δσx related to calculated stress distribution (MPA software) of reference model σx(Ref.) for evaluated simulations with different cylinder radius R
The main question is concerns the calculated stress differences related to the reference stress distribution using the MPA software. For this purpose, the calculated stress differences are plotted in Fig. 5.29 and Fig. 5.30 for σx and σy, respectively. The stress differences are calculated similarly as in {5.7} with the difference that the stress distributions are dependent on R instead of T. In addition, the stress deviations (or differences) at selected hole depths are listed in Tab. 5.12 for σx (in %) and in Tab. 5.13 for σy (in MPa). The different units are chosen due to the fact that for σy some reference stress values are very small or zero with the result that, in the first case, the calculation leads to very high percental values, or, in the second case, the calculation is not possible because of the division with zero. The stress differences in x-direction as well as in y-direction are on the whole not very high for the solid cylinders with R = 6 mm and R = 4 mm. The maximal stress deviation in x-direction is approx.15 % (or less than 25 MPa) at z = 1 mm but on average over the hole depth the stress deviations are about 8%. Also in y-direction, the calculated stress differences are almost constant against the hole depth with a maximal stress difference less than 12 MPa. In contrast, the results taken with the scR3m have, on the whole, lower stress differences compared to the other two results with the solid cylinders. The stress deviation in x-direction for the solid cylinder with R = 3 mm is only at z = 0 mm 11.8 % but on average over the hole depth, the difference values are between -6 % to 2 %. Significant stress differences are calculated in y-direction with the results of the scR3m model: the absolute values of the stress differences increase from -11 MPa at z = 0 mm up to -100 MPa at z = 1 mm.
128 50 hcR3
σy(R)−σy(Ref.) [MPa]
25
hcR4 hcR6
0
scR6 scR4
-25 scR3 -50
-75
Influence of cylinder radius: stress difference σy using MPA-Method
-100
-125 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Fig. 5.30:
depth
Stress difference σy(R) versus hole depth: stress difference σy(R) is related to calculated stress distribution of reference specimen σy(Ref.) using MPA software
Stress difference Δσy related to calculated reference stress σy(Ref.) [MPa] solid cylinder
[mm]
hollow cylinder
R = 3 mm
R = 4 mm
R = 6 mm
R = 3 mm
R = 4 mm
R = 6 mm
0.0
-11
2
0
38
10
5
0.2
-22
-3
-3
41
16
12
0.5
-50
-10
-7
22
13
10
1.0
-100
-6
-2
-113
-43
-63
Tab. 5.13:
Stress difference Δσy related to calculated stress distribution (MPA software) of reference model σy(Ref.) for evaluated simulations with different cylinder radius R
In the case of the hollow cylinders, the stress difference in x-direction for the hcR6m model starts with a higher value of 27 % at z = 0 mm and has a maximum at z = 0.20.3 mm of approx. 40 %. After this maximum the stress difference shows a decrease up to -38.7 % at z = 1 mm. The distributions of the hollow cylinders with R = 4 mm and R = 3 mm of stress differences in x-direction at z = 0-0.3 mm run below the distribution of R = 6 mm. At z = 4 mm the stress difference curves in x-direction of the hollow cylinders are very close to each other and they diverge at z = 1 mm. In y-direction the distribution of the stress differences of the hollow cylinders with R = 6 mm and R = 4 mm shows a similar behavior of the curves with a small increase up to approx. at z = 0.3 mm and a decrease up to -63 MPa at z = 1 mm. The stress evaluation in ydirection on the hollow cylinder with R = 3 mm calculates a difference of approx.
129 40 MPa between z = 0 mm and z = 0.5 mm. After this depth, the values fall up to a stress difference of -113 MPa at z = 1 mm. Conclusion: The boundary condition “radius of surface curvature” is set in [2] with
T = 3 x d0. The results in this chapter were determined using uniformly loaded cylindrical finite elements model, which means that the models have a uniaxial curved shape instead of a biaxial curved shape (e.g. a shell or a sphere). They show that the boundary condition could not be minimized and should be even displaced to a higher value. The maximal stress deviations in x-direction are 12 % for the solid cylinder (39 % for the hollow cylinder) have already been calculated for the results with R = 3.33 x d0, which already satisfies the set boundary condition. In y-direction, which is transverse to the load direction, the maximum stress differences are significantly high for the results with R = 1.67 x d0. The difference values are approx. 100 MPa for the solid cylinder as well as for the hollow cylinder. The loss of cross section in the form of the hollow cylinders causes a similar thickness effect as in the case of the flat specimens.
5.5 Possible Reduction of Calculated Stress Differences using Geometry Specific Calibration Functions The previous chapters show that a stress evaluation on specimens with non-reference geometry leads in some cases to significant stress differences when using the differential MPA software for the hole-drilling method. This is due to the fact that the MPA software includes calibration functions which were determined using reference (flat) specimens. The possibility of the reduction of the calculated stress differences using geometry specific calibration instead of a reference calibration is discussed in general in this chapter. For this purpose hole drilling measurements are simulated using modified models of the already described models fT1.5D20m, hcR3m and hchR6m. The modification of the models only concerns the material model and the magnitude of the initial load, in order to distinguish between the simulated measurements and the calibration. Additionally, the material independent character of the calibration is thus demonstrated, i.e. one calibration can be used for measuring all kind of materials. The exemplarily simulated measurements were carried out on models which have typical
130 elastic parameters of an aluminum alloy (E = 70000 MPa and ν = 0.3). The applied uniform stress, which equals in the case of models ft1.5D20m and hcR3m the simulated residual stresses, is uniform over the hole depth with σx = 100 MPa. In case of model hchR6m, the stress distribution in x-direction σx(z) varies with the hole depth. The strain distributions of these simulated measurements are taken as input data for the evaluation of stress using the MPA algorithm, which is defined in Chap. 2.3.3.1 with the equations {2.18} to {2.23}. The calculation of the calibration functions Kx and Ky is done (a) by using the strain distributions of the reference specimen fT6D20m and (b) by using the strain distributions determined by the specific geometry of the simulated measurement. In both these calibration cases, the elastic constants as well as the loading magnitude are the ones described in Chap. 3.3.2 (E = 210000 MPa, ν = 0.285 and σc,x = 160 MPa). The selected calculation procedure in this chapter is a simplified version of the algorithm used in the MPA software. Some influencing factors are different or simply omitted. The main differences are: Order of polynomial approximation: In [2], the calibration strains are approximated
by a 4th polynomial order. This polynomial order is identified there as sufficiently precise. In this chapter, the calibration strains, which are used for the derivation dε c ,i (ξ ) / dξ in the calibration functions {2.19} and {2.20}, are similarly calculated by a
closed 4th order polynomial approximation using the strain distributions of the corresponding calibration model (reference model fT6D20m or one of the three geometry specific models). Because of the different calibration models, the calculated calibrations functions of the examples are inherently different compared to the calibration functions used in the MPA software. Approximation of simulated measured strain: The stress calculation using equations
{2.21} to {2.23} requires the approximation of the measured strain distribution. For these simplified examples, the measured strain is calculated, similar to the calibration functions, using a closed 4th order polynomial approximation, i.e. all simulated strain values of the measurement εi(ξ) are used for the approximation. In contrast, the MPA software uses a local polynomial approximation, which means that 3 adjacent strain
131 values of the measurement are approximated by a 2nd polynomial order [33]. This procedure has the advantage that the local trajectory of the strain distributions is described more precisely than in the case of a closed polynomial approximation. Selection of depth increments and maximal depth: The selected depth increments as
well as the selected maximal depth, which are used for the polynomial approximation, influence directly the calculated coefficients of the polynomial. This means that different depth increments as well as a different maximal depth lead to different calibration coefficients. In this chapter, all polynomial approximations (calibration strains and simulated measurement strains) are calculated up to a maximal depth of z = 1 mm. The used depth increments are: •
0.0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 for fT6D20m, fT1.5D20m and hchR6m
•
0.0, 0.04, 0.08, 0.14, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 for hcR3m.
Strain smoothing: The authors of the MPA algorithm suggest an initial smoothing of
the measured strain distribution in order to perform the subsequent strain differentiation with an appropriate accuracy [33]. This smoothing has an influence on the resulting stress distributions. The examples in this chapter were calculated without a smoothing of the measured strains. Calibration hole diameter vs. measurement hole diameter: The MPA software
provides a set of calibration functions related to a specific hole diameter. In cases in which the measured hole diameter diverge from one calibration hole diameter, the calibration function is calculated by linear interpolation between two sets of calibration functions. This calculation step is omitted in these examples because the calibration hole diameter is assumed to be equal to the measurement hole diameter with dc,0 = d0 = 1.8 mm. The next figures display the results for each simulated measurement, which are two stress distributions in x-direction. One was evaluated with the reference calibration (dotted lines) and one was evaluated with the geometry specific calibration (continuous lines). These calculated distributions are compared to the initial stress distribution
132 within the hole volume (dot and dash line) before starting the drilling process. Fig. 5.31 shows the calculated stress distributions calculated for the models fT1.5D20m (black lines) and hcR3m (gray lines) as well as the stress deviations (in per cent) related to the initial stress value of 100 MPa.
Fig. 5.31:
Calculated stress distribution and stress difference for models fT1.5D20m and hcR3m using a simplified MPA algorithm based on calibration functions which were determined with (a) reference model and (b) geometry specific model
The stress calculation based on the reference calibration lead in both cases to considerable stress differences. In the case of model fT1.5D20m, the stress differences are almost constant over the hole depth with values between 24 % and 30 %. The stress deviation for the model hcR3m starts at z = 0 mm with approx. 8 %, reaches a maximum of approx. 48 % at z = 0.8 mm and falls at z = 1 mm to a stress difference value of approx. 35 %. The stress evaluation using the geometry specific calibration minimizes the stress difference for a hole depth range of z = 0 - 0.6 mm. In the case of model fT1.5D20m, the stress difference is between z = 0 mm and z = 0.6 mm less than 6 %. The stress results of model hcR3m start at z = 0 mm with a stress difference of approx. 10 %, falls to a stress difference of approx. 0 % at z = 0.1 mm and increase slightly to a difference value less than 6 % at z = 0.6 mm. After z = 0.6 mm, both stress difference distributions increase highly up to values of approx. 100 % and 46 % for the models fT1.5D20m and hcR3m, respectively.
133 450 400 350
σx [MPa]
300
hchR6m (spec. calib.) 250
hchR6m (ref. calib.)
200 150
σ x (z)-FEM
100 50 0 0
0.2
0.4
0.6
0.8
1
z [mm]
Fig. 5.32:
Calculated stress distribution for model hchR6m using a simplified MPA algorithm based on calibration functions which were determined with (a) reference model and (b) geometry specific model 150
100
Diff σx [%]
hchR6m (ref. calib.) 50
hchR6m (spec. calib.)
0
-50
-100 0
0.2
0.4
0.6
0.8
1
z [mm]
Fig. 5.33:
Calculated stress difference for model hchR6m using a simplified MPA algorithm based on calibration functions which were determined with (a) reference model and (b) geometry specific model
In principle, the results obtained with model hchR6m are similar to the previous results, as can been seen in Fig. 5.32 and Fig. 5.33 for the absolute stress distribution and the distribution of the stress deviations, respectively. The initial stress distribution within the hole volume starts with a stress value of 90 MPa at z = 0 mm and increases almost linearly with increasing hole depth to a value of approx. 200 MPa at z = 1 mm (dot and dash line in Fig. 5.32). The stress deviations calculated with the reference calibration start at z = 0 mm with a stress difference of approx. 8 %, reach a maximum of 57 % at
134 z = 0.2 mm and fall up to a stress difference of approx. -105 % at z = 1 mm. In contrast, the stress differences with the geometry specific evaluation are less than 5 % between z = 0 mm and z = 0.5 mm. The distribution of the stress deviation shows a remarkable S-shaped oscillation between z = 0.7 mm and z = 1 mm with a maximal value of approx. 134 % at z = 0.8 mm and a minimum of -95 % at z = 0.9 mm. Conclusion: The results of the simplified version of the MPA algorithm in this chapter
show that the stress deviation using a geometry specific calibration is significantly minimized for hole depths up to 0.5 mm. At deeper hole increments, the stress distributions show relevant discrepancies to the initial stress distributions.
5.6 Summary of Discussion of Calibration Results The discussion of the calibration results covered two main topics which were the uncertainty of the calibration and the influences of the geometrical parameters on the stress evaluation using the hole drilling method and the differential MPA algorithm. The uncertainty of the numerical calibration was discussed comparing selected results of the developed models with results taken from literature. In the case of the experimental results, affecting factors of the experimental accuracy were listed in order to analyze the accordance between the numerical and the experimental calibration. The numerical and experimental results showed a fairly good accordance for the selected examples although not all described experimental uncertainty sources were taken into account for the comparison. In introducing examples of the second part of the discussion, it was demonstrated that the stress evaluation with actual stress calculation software on specimens with nonreference geometry leads to stress differences related to the reference evaluation. The influence of the geometry parameters thickness T, distance hole-edge D and cylinder radius R was analyzed concerning the evaluation of stress and the possible stress differences. The results of the calibrations of these uniaxially loaded specimens showed that the boundary condition thickness and distance hole-edge may be minimized
135 and redefined to T = 1.66 x d0 and D = 1.94 x d0, respectively. In the case of the cylinder radius, the boundary condition may not be minimized. The last subject of this chapter considered the reduction of stress differences using geometry specific calibration functions. The exemplary calculations with a simplified differential MPA algorithm showed that a reduction of the stress difference is, in principle possible, although the calculated distribution diverges at deeper hole increments from the initial distribution. The next chapters describe an implementation of geometry specific calibration functions into evaluation software using the MPA algorithm (Chap. 6). A parametric study of the previously described influencing factors on the results using the MPA algorithm is then carried out, in order to analyze a possible optimization of the stress calculation using geometry specific evaluation (Chap. 7).
136
6 Implementation of Geometry Specific Functions into Evaluation Program
Calibration
6.1 Prefacing Remarks The previous chapter shows, among other things, that the stress evaluation using specific hole-drilling calculation software on specimens with non-reference geometry leads to deviations from the initial stress state. Moreover, it was demonstrated that these stress deviations could be partially minimized, especially for near surface depth increments, by using a simplified differential MPA method together with geometry specific calibration functions. The exemplarily used evaluation software packages in Chap. 5.3 are valid for specific geometrical boundary conditions. Thus, there is no explicit consideration of the component’s shape when evaluating a part which violates the geometrical boundary conditions. This chapter describes the most important functional blocks of the implementation of geometry specific calibration functions into a hole-drilling evaluation program according to the differential MPA algorithm. For this purpose, a prototype program was developed using the numerical computing environment “Matlab” (MathWorks). The calculation process can be divided in two main parts which are, firstly, the modification and differentiation of the strain depth distributions and, secondly, the stress calculation. In the presented prototype program, the stress calculation can be optionally carried out by using three different sets of calibration functions: •
MPA standard: Calibration functions for a reference geometry according to [2]
were determined for a specific strain gage rosette and different hole diameters d0. They are implemented as fixed values within the program. •
MPA expanded: Calibration functions calculated in this work using the models
of the flat tensile specimens. They consider the hole diameter d0, the component thickness T and the distance hole-edge D. The calibration functions for the CEA-XX-062-UM rosette are also implemented as fixed values within the program. •
MPA specific: On-line calculation of specific calibration functions from
imported calibration strain distributions which are, possibly, determined using non-reference geometries.
137
Fig. 6.1:
Block diagram for the calculation of residual stress using the hole-drilling method and the differential MPA evaluation method
138 The input and output as well as the main functions of each block are described in the following subchapters. In order to provide an overview of the main functional blocks of the program, a rough sketch of the calculation process is shown in the diagram of Fig. 6.1. The aim of this chapter is to describe the background of the stress calculation for the subsequent parametric study in Chap. 7 and is not an exhaustive software description.
6.2 Modification and Differentiation of Measured Strain Distributions 6.2.1 Block 1 - Import of Input File Input: input file Output:
parameters of measurement and material; component geometry; measured strain depth distributions.
The first block of the program imports the input file for the stress calculation. This file is the result of the hole-drilling measurement and contains the measurement parameters and the registered strains at each drilled hole depth increment (s. Fig. 6.2). %Measurement fT1.5D20 Al % (0= Blind Hole; 1= Trough Hole) 0 (Strain Gage Rosette) 4 (Hole Diameter) 1.8 (Young-Modulus) 70000 (Poisson Ratio) 0.33 (Thickness) 1.5 (Distance Hole-Edge) 20 ************************************* z e1 e2 e3 0 0.0000 0.0000 0.0000 0.02 0.8426 -2.4529 -5.7483 0.04 1.6873 -5.2704 -12.228 0.06 2.5460 -8.3994 -19.344 0.08 3.4259 -11.807 -27.041 0.1 4.3329 -15.458 -35.249 ... 0.9 80.308 -159.63 -399.56 1 93.886 -164.38 -422.66 Fig. 6.2:
Input File: geometry parameters, material parameters and measured strain depth distribution
139
The first line between the percent signs of the input file is a commentary. The subsequent lines specify the shape of the hole (blind-hole vs. through hole), the used strain gage rosette, the hole diameter d0, the material parameters (E, ν) and the geometrical parameters T and D. The last lines after the asterisks list the drilled depth increments and the relieved strain of each strain gage.
6.2.2 Block 2 - Parameter Modification Input: parameters of measurement and material; component geometry Output:
modified parameters of measurement, material and geometry (optional)
The second block of the program shows the measurement parameters and allows optionally their modification within a special interactive program window (s. Fig. 6.3). This is useful in cases in which some parameters were wrongly stored in the input file or a new input file with different parameters should be created for subsequent calculations.
Fig. 6.3:
Program window: modification of parameters of measurement.
140 6.2.3 Block 3 - Strain Modification Input: measured strain depth distributions Output:
modified measured strain depth distributions (optional)
The third block of the program shows the measured strain depth distribution and allows optionally their modification within the interactive table on the left side of the program window in Fig. 6.4. The modification in this case means a manual change of the strain values or the omission of entire lines of the strain input.
6.2.4 Block 4 - Strain Smoothing Input: measured strain depth distributions Output:
smoothed strain depth distributions
Fig. 6.4:
Program window: modification of strain depth distributions and strain smoothing.
Measured strain depth distributions scatter due to the random uncertainty of the measuring technique. This fact should be especially considered if the strain distributions are differentiated in order to calculate residual stress depth distributions. In some cases, the scatter leads to numerical instability. For this reason, the objective of the smoothing algorithm is the minimization or omission of the statistical scatter. Outliers due to
141 systematic errors are not count as part of the scatter and should be eliminated before the actual calculation. In this work, two smoothing algorithms, according to [33], are implemented in the prototype program: a weighted average algorithm and an algorithm using compensative cubic spline functions. The first method, the weighted average algorithm, minimizes the scatter, preserves the local information of the distribution, ensures the continuity of the connection values, is numerically stable and it is simply to describe in detail. The smoothed strain value
ε ( z i ) is the weighted average {6.1} of three adjacent values ε ( z i −1 ) , ε ( z i ) , ε ( z i +1 ) and the intermediate averages ε i12 , ε i 23 {6.2}. The weight factors are set in [2.31] with fixed values {6.3}. {6.1} ε ( z i ) = ci −1ε ( z i −1 ) + ci12 ε i12 + ci ε ( z ) + ci 23ε i 23 + ci +1ε ( z i +1 ) {6.2} ε i12 = 0.5 × [ε (z i −1 ) + ε (z i )] {6.3} ci −1 = 0.1 ;
c12 = 0.2 ;
and
ci = 0.4 ;
ε i 23 = 0.5 × [ε (z i ) + ε (z i +1 )] c 23 = 0.2 ;
ci +1 = 0.1
The first value ε ( z1 ) as well as the final value ε ( z n ) are calculated considering only the next adjacent value {6.4} and the weight factors in {6.5} according to [2.31]. {6.4} ε ( z1 ) = c1ε ( z1 ) + c 2 ε (z 2 ) {6.5} c1 = 0.95 ;
c 2 = 0.05 ;
and
c n = 0.9 ;
ε (z n ) = c n ε (z n ) + c n −1ε (z n −1 ) c n −1 = 0.1
The second smoothing algorithm taken from [33] is described in detail in [60]. It bases on compensative cubic spline functions. In this calculation, the measured strain distribution is smoothed piecewise by cubic polynomial functions. The second derivations of the splines are set equal to zero at the endpoints of the interval. The advantage of this method is that it uses a compensating factor (smoothing factor fil) which can be manually set and allows the online adjustment of the smoothed strain distributions. As a matter of course, the consequences of the setting of the compensative
142 parameter should be verified by the user, as a large fil factor linearizes the trajectory of the strain distribution [33]. The right part of the program window of Fig. 6.4 shows exemplarily a diagram with measured strains (continuous lines) and the smoothed values (asterisks). The smoothing method can be selected beneath this diagram. A listing of both smoothing procedures can be found in the appendix Chap. 10.
6.2.5 Block 5 - Storage of Modified Input File Input: modified measurement parameters and modified strain distributions Output:
new input file
This optional function is necessary if the measured strains and the measurement parameters were modified and this information should be stored. A file can then be created by writing a new input file or overwriting the old input file.
6.2.6 Block 6 - Local Polynomial Approximation of Measured Strains Input: measured and smoothed strain distributions, hole diameter d0 Output:
differentiated strain values dεi(ξ)/dξ
In order to provide differentiable functions εi(ξ) (with ξ = z/d0) the measured and smoothed strain distributions can be approximated with polynomials. One possibility is the close approximation of the strain distribution, i.e. all measured points of the curve are simultaneously used for the determination of the coefficients. On the one hand, this has the advantage that only a small number of coefficients have to be calculated. On the other hand, the disadvantage of this procedure is the loss of local information due to the increase of the standard deviation. In addition, an overshoot may occur, e.g. in cases where the number of measurement points is small or the measured strains have an alternating distribution against the normalized hole depth. In order to avoid these disadvantages, an alternative procedure is proposed in [33]. Thus, the measured strain distributions are locally approximated using a quadratic ansatz {6.6}. For three adjacent points (εi-1, εi and εi+1) a system of equations {6.7} can be set up. Thus, the polynomial coefficients a0, a1 and a2 can then be determined by
143 means of least squares. The differentiated strain values dεi(ξ)/dξ, which are used in the stress equations {2.21} to {2.23}, are calculated using the derivation of the {6.8}. {6.6} ε i = a 0 + a1ξ + a 2ξ 2
ε i −1 = a 0i + a1iξ i −1 + α 2iξ i2−1 {6.7} ε i = a 0i + a1iξ i + α 2iξ i2
ε i +1 = a 0i + a1iξ i +1 + α 2iξ i2+1
{6.8}
dε i (ξ ) = a1 + 2a 2ξ i dξ
This procedure, which is implemented in the prototype program, is applied for all points within the strain distribution with the exception of the first and the last point. A program listing is included in the appendix Chap. 10.
6.3 Stress Calculation 6.3.1 General Aspect of Stress Calculation in Prototype Program The blocks “MPA Standard”, “MPA Expanded” and “MPA Specific” calculate the
stress distributions in the direction of the single strain gages (σ1, σ2, σ3) and the distributions of the principal stresses (σmax, σmin) according to the equations {2.21} to {2.24} (s. Chap. 2.3.1). The major difference of each block is the used sets of calibration functions. The blocks “MPA Expanded” and “MPA Specific” were developed during this work in order to provide two different calculation procedures which consider the component geometry.
6.3.2 Block 7 - Stress Calculation “MPA Standard” differentiated strain values dεi(ξ)/dξ; parameters of measurement and Input:
material Output:
σ1(z), σ2(z), σ3(z), σmax(z), σmin(z) (s. {2.21} to {2.24})
144 This block no. 7 (“MPA Standard”) uses the calibration functions of the original MPA algorithm. They are calculated according to {2.19} and {2.20} (s. Chap. 2.3.1). The 4th order polynomial coefficients of the calibration strain functions εc,x {6.9} and εc,y {6.10} were determined for different rosettes types using uniaxially loaded finite element models with σc,x = 100 MPa, σc,y = 0 MPa, E = 200000 MPa and ν = 0.3 [2]. {6.9} ε c , x = a 0 + a1ξ + a 2ξ 2 + a 3ξ 3 + a 4ξ 4
{6.10} ε c , y = b0 + b1ξ + b2ξ 2 + b3ξ 3 + b4ξ 4
The sets of calibration coefficients, which are included as fixed values within the program, consider five different hole diameters by means of the ratio dn {6.11}.
{6.11} d n =
dm d0
The implemented set of calibration strain functions covers hole diameters between 1.50…1.97 mm. In case of the hole-drilling rosette CEA-XX-062UM-120, the mean rosette diameter is dm = 5.13 mm. If the measured hole diameter (e.g. d0,v) is between two set hole diameters (e.g. d0,u and d0,w), the specific calibration coefficient av is calculated using linear interpolation {6.11} between the fixed calibration coefficients au and aw {6.12}.
{6.12} a v = au +
a w − au (d n,v − d n,u ) d n , w − d n ,u
with dn,u < dn,v < dn,w
6.3.3 Block 8 - Stress Calculation “MPA Expanded” Input: differentiated strain values dεi(ξ)/dξ, parameters of measurement and
material, component geometry parameters T and D Output:
σ1(z), σ2(z), σ3(z), σmax(z), σmin(z) (s. {2.21} to {2.24})
This block calculates the stress distributions identically as in the previous “MPA Standard” block. In contrast to the “Standard” block, the “Expanded” block signifies
145 that the calibration coefficients of the 4th order polynomials considers not only the hole diameter d0 but also the component thickness T and the distance hole-edge D for the hole-drilling rosette CEA-XX-062UM-120: •
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
The calibration coefficient sets are included as fixed values within the program and were determined in this work using the models of the flat tensile specimens of Type 1 as described in Chap. 3.3.1. It might happen that the geometrical parameters of the measurement (T, D, d0) do not exactly coincide with the fixed geometrical parameters of the calibration coefficient sets (Tc, Dc, dc,0). In this case, the actual calibration coefficients are determined by linear interpolation between the values of the calibration coefficients. This process is visualized with a schematic diagram in Fig. 6.5. Thus, two set of calibration coefficients C1(Tc, Dc, dc,0,u) and C2(Tc, Dc, dc,0,w) span two surfaces in a three dimensional diagram. The actual calculation coefficient aj(T ,D , d0) is then interpolated using the single values of C1 and C2.
Fig. 6.5:
Schematic illustration of determination of actual calibration coefficient aj using interpolation in block “MPA-Expanded”
146 The advantage of this calculation procedure with fixed values is that the user only has to include the geometry parameters T, D and d0 into the input file or in the parameter modification block (Block 2) in order to consider the shape of the component. The disadvantage is that the considered component geometry is limited by the sets of determined specific calibration functions. As a start, a three dimensional interpolation aj = f(T,D,d0) was chosen but in principle the computing environment MATLAB allows moreover an interpolation with more than three variables, e.g. if the geometry parameter curvature should be additionally considered. An exemplarily program listing of this interpolation procedure is included in the appendix Chap. 10. 6.3.4 Block 9 - Stress Calculation “MPA Specific” Input: differentiated strain values dεi(ξ)/dξ, parameters of measurement and
material,
parameters
of
specific
calibration,
calibration
strain
distributions, calibration stress distributions Output:
σ1(z), σ2(z), σ3(z), σmax(z), σmin(z) (s. {2.21} to {2.24})
The stress calculation in this block is identical to blocks 7 and 8. The calibration functions are calculated from the specific calibration files, which are previously imported and approximated in block 10 and block 11, respectively. In this case, “specific” means that the calibration was carried out using a component (or specimen) with a shape, which is identical or similar to the shape of the measured component. The geometry of the component can be complex and does not comply with the geometrical boundary conditions. Thus, the resulting calibration strains inherently consider the specific geometry situation near the measurement point. One possible application of this calculation procedure is the use for recurring measurements on complex shaped components. In such a case, the calibration files are determined once using, e.g. finite element models of the component, in which a holedrilling measurement point is modelled. The models should only consider the possible dimensional range of the introduced hole diameters, so that the approximated functions could be interpolated for the actual hole diameter of the measurement.
147 6.3.5 Block 10 - Import of Calibration Files Input: calibration files Output:
parameters of calibration material; calibration strain depth distributions and calibration stress depth distributions, hole diameters d0.
This block is activated if the stress calculation should be carried out using specific calibration functions. It imports one or more (max. five) calibration files in order to consider different hole diameters. An exemplarily calibration file is shown in Fig. 6.6. The calibration contains information about the used strain gage rosette, the calibration hole diameter and the elastic parameters of the calibration material. The data columns after the asterisk are the hole depth, the calibration strains εx and εy, and the calibration stress σx and σy. The calibration material parameters are passed to block 9 in order to calculate the calibrations functions Kx and Ky. %reference calibration fT6D20 % (Strain Gage Rosette) 4 (Hole Diameter) 1.8 (Young's Modulus) 210000 (Poisson´s Ratio) 0.285 (Mean Diameter Strain Gage Rosette) 5.13 ************************************* depth ey e2 ex sx sy 0.00 0.0000 0.0000 0.0000 160 0 0.02 0.4192 -1.319 -3.057 160 0 0.04 0.8413 -2.831 -6.504 160 0 0.06 1.2731 -4.508 -10.29 160 0 0.08 1.7190 -6.332 -14.38 160 0 0.10 2.1822 -8.283 -18.74 160 0 ... 0.90 43.642 -84.19 -212.0 160 0 1.00 51.006 -86.64 -224.3 160 0 Fig. 6.6:
Calibration File: calibration parameters, calibration strain depth distributions, calibration stress depth distribution
148 6.3.6 Block 11 - Polynomial Approximation of Calibration Strains Input: calibration strain distributions, calibration stress distributions, hole
diameters of calibration d0,c Output:
εc,x(ξ), εc,y(ξ), σc,x(ξ), σc,y(ξ)
The imported calibrations strain distributions, as well as the calibration stress distributions, are approximated in this block with polynomials using ξ = z/d0 and a close approximation. The polynomial order can be manually set within a program window and is, by default, of 4th and 1st order for the calibration strain distributions and calibration stress distribution, respectively. The approximated calibration coefficients of the functions εc,x(ξ), εc,y(ξ), σc,x(ξ), σc,y(ξ) as well as the calibration parameters are passed to block 9 in order to calculate the calibrations functions Kx and Ky.
6.3.7 Block 12 - Angle Calculation Input: σ1(z), σ2(z), σ3(z) Output:
φ(z)
The orientation angle φ between strain gage no. 3 and the maximal stress is determined by calculating an intermediate angle φ* {6.13} and by verifying the relationships {2.9} to {2.17} (s. Chap. 2.3.1).
{6.13} ϕ ∗ =
2 Δσ 2 − Δ σ 3 − Δσ 1 1 Num 1 = arctan arctan Den Δσ 1 − Δσ 3 2 2
In {6.13}, calculated stress values in the strain gage direction are used instead of measured strain values {2.8} for the determination of φ. This means that, although the measured strain distribution of the input file is constant, the value of the orientation angle may differ as the stress values are dependent on the previous calculation procedure (strain smoothing, strain approximation, etc.).
149 6.3.8 Block 13 - Visualization of Results Input: σmax(z), σmin(z), σ2(z), σ3(z), φ(z) Output:
diagrams of stress distributions and orientation angle φ(z)
This block is added in order to verify the stress results of the calculation before exporting the data into an output file. In Fig. 6.7, the resulting stress depth distributions as well as the distribution of the orientation angle against the hole depth of a hole drilling measurement are exemplarily shown within the diagrams of the program window.
6.3.9 Block 14 - Export and Storage of Result File σmax(z), σmin(z), σ2(z), σ3(z), φ(z) Input: Output:
result file
The calculated stress depth distributions as well as the distribution of the orientation angle against the hole depth can be stored in a result file which is exemplarily shown in Fig. 6.8.
Fig. 6.7:
Program window: plot of calculated stress depth distributions and orientation angle
150 STRESS CALCULATION ---- BLIND HOLE -----------------------------------Data Titel : Measurement fT1.5D20 Al Rosette Type : 062 UM Hole Diameter : 1.80 Young's Modulus : 70000 Poisson's Ration : 0.33
Depth [mm] 0.00 0.02 0.04 0.06 0.08 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Fig. 6.8:
SigMax [MPa] 116.75 113.16 110.95 110.57 111.51 113.03 120.89 126.25 126.82 122.97 116.03 107.78 99.557 94.134 90.911
SigMin [Mpa] 3.642 5.149 6.544 7.937 9.209 10.24 13.86 16.07 16.02 13.23 8.904 4.043 -2.09 -9.58 -19.3
Alpha [°] -0.144 -0.128 -0.125 -0.141 -0.147 -0.100 -179.7 -179.8 -0.191 -0.216 -179.9 -179.9 -0.119 -0.215 -0.455
Sig3 [Mpa] 116.75 113.16 110.95 110.57 111.51 113.03 120.89 126.25 126.82 122.97 116.03 107.78 99.557 94.132 90.904
Sig1 [Mpa] 3.642 5.149 6.545 7.938 9.210 10.24 13.86 16.07 16.02 13.23 8.904 4.043 -2.09 -9.58 -19.3
Result File: stress depth distributions and distribution of orientation angle against hole depth
151
7 Evaluation of the Implementation of Geometry Specific Calibration Files into Residual Stress Calculation Program 7.1 Prefacing Remarks In this chapter, the implementation of geometry specific calibration files into the differential MPA algorithm for the calculation of residual stress distributions after a hole drilling measurement is evaluated. For this purpose, exemplary calculations are carried out using the previously described prototype program in Chap. 6 in order to discuss the influencing factors of the algorithm on the stress calculation and to evaluate the two alternative calculation blocks which consider the component geometry. The first part of this chapter deals with the stress calculation using the MPA algorithm and illustrates especially the reason for the oscillating form of the calculated stress distributions (s. also Chap. 5.3). The next parts discuss influencing factors of the calculation algorithm on the stress evaluation by means of a parametric study. Therefore, the second part shows the influence of the local approximation on the calculated stress results. The influence of the initial strain smoothing is discussed in the third part. In the fourth part, the order of the polynomial approximation of the calibration strain distribution is discussed whereas the effect of the selection of the distribution of the depth increments is shown in the fifth part. The last part of this chapter compares results using the three different stress calculation options of the prototype program. These options are the stress calculation blocks MPA Standard, MPA Expanded and MPA Specific.
7.2 General Aspects of Calculation Chap. 5.3 shows, among other things, that small differences compared to the initial stresses are calculated, even if the geometrical boundary conditions are observed. This is the case for the calculation using the reference specimen. Moreover, especially in cases in which the initial stress depth distribution is uniform over the depth, the calculated stress distribution oscillates around the actual initial stress depth distribution.
152 The origin for the oscillating shape of the calculated stress distribution is found in the calculation itself. In order to visualize this problem exemplarily, the calibration strain distribution determined with the reference model fT6D20m is selected in order to calculate the stress distribution and compare it with the initial stress of σx = 160 MPA. The stress in x-direction is parallel to the direction of strain gage no. 3 and, therefore, the stress is calculated using {7.1} (s. in Chap. 2.3.3.1 equation {2.23}).
{7.1} σ 3 (ξ ) =
⎡ dε (ξ ) dε (ξ ) ⎤ E ⋅ ⎢ K x (ξ ) ⋅ 3 + ν ⋅ K y (ξ ) ⋅ 1 ⎥ 2 2 dξ ⎦ dξ K (ξ ) − ν K y (ξ ) ⎣ 2 x
Young’s modulus (E = 210000 MPa) and Poisson’s ratio (ν = 0.285) are constant values whereas the differentiated strain distributions dε1(ξ)/dξ and dε3(ξ)/dξ as well as the calibration functions Kx(ξ) and Ky(ξ) are variable against the depth. The distribution of the differentiated strain and the distribution of the calibration functions for this example are shown in Fig. 7.1 in diagram part a) and b), respectively. The strains were differentiated after carrying out a local polynomial approximation and were not initially smoothed. The calibration functions were calculated using the MPA Standard block by interpolating the calibration functions, as the hole diameter (d0 = 1.8 mm) of the selected reference model is between the nominal hole diameters of two sets of calibration functions. The calculated stress in x-direction is plotted in diagram part c). Additionally, values of the differentiated strains, the calibration functions and the calculated stress at selected depths are listed in Tab. 7.1. It can be seen that the distribution of the differentiated strains and the distribution of the calibration functions show an alternate shape. This is the main reason for the resulting alternate shape of the calculated stress distributions. The calculated stress distribution runs thus around the value of the initial stress. ξ [-]
dε3(ξ)/dξ [-]
dε1(ξ)/dξ [-]
Kx(ξ) [-]
Ky(ξ) [-]
σx [MPa]
0.0
0.000
-291 x10-6
55 x10-6
-0.39
-0.17
152
0.2
0.111
-393 x10-6
78 x10-6
-0.53
-0.36
156
0.5
0.277
-416 x10-6
111 x10-6
-0.54
-0.48
161
1.0
0.555
-219 x10-6
96 x10-6
-0.29
-0.40
155
depth [mm]
Tab. 7.1:
Stress calculation using MPA Standard block and reference model fT6D20m: differentiated strains, calibration functions, calculated stress σx at selected hole depths
153 200
a) differentiated strain
dε1/dξ
dεi/dξ [μm/m]
100 0 -100 z = 0.0 mm
z = 0.5 mm
z = 0.2 mm
z = 1.0 mm
-200 -300
dε3/dξ
-400 -500 0
0.1
0.2
0.3
0.4
0.5
0.6
ξ [−] -0.1
b) calibration functions
-0.2
Ky
Ki [-]
-0.3
-0.4
Kx -0.5 z = 0.0 mm
z = 1.0 mm
z = 0.5 mm
z = 0.2 mm
-0.6 0
0.1
0.2
0.3
0.4
0.5
0.6
ξ [−] 170
c) calculated stress
σx
160
150
z = 0.0 mm
z = 0.5 mm
z = 0.2 mm
z = 1.0 mm
140 0
0.1
0.2
0.3
0.4
0.5
0.6
ξ [−] Fig. 7.1:
Stress calculation using MPA Standard block and reference model fT6D20m: a) Distribution of differentiated strains, b) Distribution of calibration functions, c) Calculated stress σx
154 For this calculation example using the reference model, the average stress deviation over the hole depth is approximately 6 % with a maximal stress deviation of approximately 10 %. A similar observation is also made by the authors of the MPA method [2]. In conclusion, it should be taken into account that the stress calculation according to the differential MPA evaluation method implies a priori a level of uncertainty.
7.3 Influence of local polynomial approximation of Measured Strains The polynomial coefficients of the measured strain distributions are determined within the prototype program using a local approximation instead of a close approximation as was carried out with the simplified MPA algorithm in Chap. 5.5. This current chapter compares the resulting stress distributions using both approximation procedures. For this purpose, the exemplarily strain distributions in this chapter are the same as in Chap. 5.5. These are the simulated measurements using the “thin” model fT1.5D20, the “thin” and “curved” model hcR3m and the “thin” and “curved” model hchR6m with the measurement point “near the edge” of the ø6 mm center through-hole. The elastic constants of the models are E = 70000 MPa and ν = 0.285. The models are loaded externally with a stress of σx = 100 MPa. In case of the models fT1.5D20m and hcR3m the external load value equals the initial uniform stress at the measurement point. In the case of the hchR6m model, the initial stress distribution at the measurement point increases with increasing hole depth. The used approximation procedure of the measured strains is the only difference between the calculations. The other influencing factors are constant: the measurement strains were not initially smoothed; the calibrations functions are of 4th polynomial order and the selected distribution of depth increments is the same as in the simplified algorithm. The stress calculation for all three examples was carried out using the MPA Specific stress calculation block. In the case of this specific model of a flat specimen, a calculation with the MPA Expanded block leads to the same results. These examples, i.e. the FEM models and the selection of the influencing factors are also used in the next chapters, Chap. 7.4, Chap. 7.5 and Chap. 7.6. The only difference is the variation of the respective influencing factor.
155 200
close pol. approx.
fT1.5D6m 180
σx
160 140
local pol. approx.
120 100
σx-FEM = 100 MPa 80 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.2:
Stress distributions using MPA Specific block and model fT1.5D20m considering local or close polynomial approximation of the measured strain distributions.
Fig. 7.2 shows calculated stress distributions in x-direction for the simulated measurement with model fT1.5D20m. The black dashed line is the distribution calculated with local approximation and the grey continuous line is the one calculated with close approximation. It can be clearly seen that the stress deviations over the hole depth related to the initial stress distribution σx-FEM is, in this specific case of the thin flat specimen, less when using the local polynomial approximation of the measured strains instead of the close polynomial approximation. Especially from a depth of z = 0.6 mm, the calculation using local approximation is more accurate. Tab. 7.2 lists the stress deviations at selected depths related to the initial stresses for the examples discussed in this chapter. Stress Deviation related to initial stress [%] Depth [mm]
fT6D20m
hcR3m
hchR6m
local
close
local
close
local
close
approx.
approx.
approx.
approx.
approx.
approx.
0.0
-2
0
-60
-11
-54
0
0.2
-1
1
-17
0
0
1
0.5
1
3
0
3
1
5
1.0
0
99
0
46
-23
-56
Tab. 7.2:
Stress deviation using MPA Specific block and models fT1.5D20m, hcR3m and hchR6m. Consideration of local or close polynomial approximation of measured strain distributions
156 Fig. 7.3 and Fig. 7.4 show the calculated stress distributions for the models hcR3m and hchR6m, respectively. These calculations were carried out similarly to the previously described calculation of the fT1.5D20m model. With the exception of the initial depth increments z ≤ 0.2 mm for model hcR3m, the calculated stress distributions using local polynomial approximation show less stress deviations than the distributions calculated with the close approximation. In the case of the model hchR6m, the stress values calculated with the local approximation show less deviation for depth increments between of z = 0.1 mm and z = 0.8 mm. Both calculated distributions alternate for the final depth increments from z > 0.8 mm and have completely erroneous stress values.
160
close pol. approx.
hcR3m 140
local pol. approx.
σx
120 100
σx-FEM = 100 MPa 80 60
blind hole with d0
40 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.3:
Stress distributions using MPA Specific block and model hcR3m considering local or close polynomial approximation of the measured strain distributions.
The deviations at the first depth increments using the local polynomial approximation on the curved models hcR3m and hchR6m are caused because the local approximation uses only three strain values in order to determine the coefficients of the strain distribution of the measurement. At the first depth increments, the hole is not generated to the full extent of the hole diameter due to the curvature of the cylindrical specimens. Hence, at these initial depth increments of the simulation, less material is “removed”. This causes lower strain values and, thus, the local approximation at the initial increments calculates strain functions with lower slopes. Consequently, the stress values at these first depth increments are also lower then the initial values. A close
157 approximation averages this initial strain information because all strain values of the depth distribution contribute to the determination of the strain function.
450
local pol. approx.
hchR6m
400
close pol. approx.
350
σx
300 250 200 150 100
blind hole with d0
50 0 0.00
0.20
0.40
σx-FEM
0.60
0.80
1.00
1.20
z [mm]
Fig. 7.4:
Stress distributions using MPA Specific block and model hchR6m considering local or close polynomial approximation of the measured strain distributions
The high stress deviations between z = 0.8 mm and z = 1.0 mm of model hchR6m can be related to the principle of the differential calculation algorithm itself, which does not consider the partially relaxed stresses above the hole bottom. Deviations at the final depth increments are found in all three examples but in the case of the hchR6m model, the stress state is more complex than the other two examples: it has a biaxial initial stress state which increases against the hole depth. In conclusion, it can be said that the stress results using a local polynomial approximation of the measured strains show for the three presented examples in general less deviation from the initial stress distribution than a calculation using the close approximation. Deviations related to the local polynomial approximation appear at the initial depth increments for the curved models. In these cases, the stress values calculated at depth increments, in which the hole does not have an ideal blind hole shape with d0, should not be taken into account. For all three examples and for the two approximation procedures, stress deviations related to the calculation principle appear at the final depth increments.
158
7.4 Influence of Strain Smoothing The initial strain smoothing before the differentiation of the measured strain influences the stress results. To show this, the first example in this chapter uses the thin flat model fT1.5D20m which has an initial stress of σx = 100 MPa (E = 70000 MPa and ν = 0.33). Only the smoothing parameters are varied, the other influencing factors on the stress calculation are kept constant (local polynomial approximation, 4th order polynomials, equal distribution of depth increments as in the previous examples). First of all, Fig. 7.5 shows distributions of strain ε3 against the normalized depth using different smoothing procedures for the first three depth increments (z = 0 mm, 0.02 mm, 0.04 mm). These three values calculate directly the stress values at z = 0 mm, and z = 0.02 mm and contribute to the calculation of the following stress value, because of the local polynomial approximation. The diagram shows the distribution of the non-smoothed strain, the strain smoothed using the weighted average algorithm and strain distributions using compensative cubic splines functions with a smoothing factor of fil = 0.2, 0.5 and 1.0.
fil = 1.0
5
fil = 0.5 fil = 0.2
ε3 [μ/m]
0
no smooth -5
weight.
-10
z = 0.0 mm
z = 0.02 mm
z = 0.04 mm
-15 0
0.011
0.022
ξ [-] Fig. 7.5:
Strain ε3 against normalized depth using different smoothing parameters
It can be seen that the smoothing procedures changes the initial strain values. The nonsmoothed strain of strain gage no. 3 starts exactly at zero and decreases with depth. The strain distribution smoothed with the weighted average starts slightly beneath the nonsmoothed distribution whereas the distributions smoothed using compensative cubic
159 splines start above the non-smoothed distribution. In the case of the smoothing using compensative splines functions, the initial strain value increases with the increasing smoothing factor fil. In the diagram of Fig. 7.5, the different smoothed distributions run almost together at ξ = 0.022 but the change of the initial strain values causes a change in the slopes of the strain distributions. This means that, in this case of higher slopes of the strain distribution, the differentiated strains are determined also higher and consequently the calculated stress values increase. This is shown in Fig. 7.6 diagram a), in which the calculated stress in x-direction is plotted between z = 0.0mm and z = 0.04mm for this first example of model fT1.5D20m. The non-smoothed stress distribution starts at σx(0) = 98 MPa. The stress distribution calculated with the highest smoothing factor fil starts with the highest stress value of σx,
fil = 0.1(0)
= 152 MPa. With decreasing
smoothing factor, the initial stress values also decrease. The surface stress value calculated using the weighted average algorithm is 6 % lower than the non-smoothed surface stress value. The previous diagram a) is expanded in Fig. 7.6 in diagram b) up to a hole depth of z = 1.0 mm in order to show the whole resulting stress distributions using the different smoothing procedures. The stress distributions, which were smoothed with compensative splines functions, run almost equal to the non-smoothed stress distribution between z = 0.2 mm and z = 0.8 mm. These smoothed distributions disperse at depth increments of z > 0.8 mm. The highest stress deviation at the final depth increments are found on the distribution with the highest smoothing factor of fil = 1.0. The stress distribution, which was smoothed with the weighted average algorithm, shows almost a good accordance with the non-smoothed stress distribution with the exception of the values between z = 0.08 mm and z = 0.2 mm. At these depth increments the calculated stress values increase up to σx,weight(0.1) = 173 MPa. The reason for this stress deviation is that the depth distribution of the simulated measurement changes at z = 0.1 mm and has larger depth increments Δz = 0.1 mm instead of Δz = 0.02 mm. Consequently, this change increases the relieved strain values. For example, the smoothed strain value at z = 0.1 mm is averaged in the calculation using the non-smoothed values ε(0.08) = -27 μm/m, ε(0.1) = -35 μm/m and the significantly higher strain value of ε(0.2) = -82 μm/m. Finally, the strain differentiation
160 also leads also to higher values. The weighted average algorithm only considers the strain values without any information of the actual depth and should be applied in a combination of measured strains with uniform distribution of depth increments.
160
a) fil = 1.0
140
σx [MPa]
fil = 0.5 120
fil = 0.2 no smooth
100
weight.
80 0
0.02
0.04
z [mm]
180
b)
160
σx [MPa]
fil = 1.0 140
weight.
fil = 0.5
120
fil = 0.2 100
no smooth 80 0
0.2
0.4
0.6
0.8
1
z [mm]
Fig. 7.6:
Calculated Stress distribution using different smoothing parameters: a) depth z = 00.04 mm; b) depth z = 0-1.0 mm
The previous example shows the influence of the smoothing parameters on the stress calculation using the results of the fT1.5D20m model. This FEM result is inherently smooth and, consequently, less stress deviation is calculated using the non-smoothed strain distributions. The next example shows the importance of smoothing in practice. Fig. 7.7 shows the stress results taken from a hole-drilling measurement on a tempered
161 42CrMo4 steel specimen, whose surface was shot-peened. The input strain distributions file for the stress evaluation was the same for all three calculated stress distributions. The stress distribution with the black continuous line was calculated without smoothing. The distributions with the grey continuous line and the black dashed line were calculated using different smoothing factors fil = 0.1 and fil = 0.5, respectively. In order to have comparative values, a stress distribution on the same shot-peened surface was determined using X-ray diffraction and additionally included in Fig. 7.7 with a thin grey continuous line with circles. The effect of shot-peening related to the introduced residual stresses on metallic surfaces has been investigated in several works [61]. Thus, compressive residual stresses are commonly expected at the surface. Furthermore, a maximum of the compressive residual stress is often found beneath the surface of the peened material. The effective range of shot-peening on the surface of 42CrMo4 decreases at approximately z = 0.1-0.2 mm [61]. It can be seen that a stress evaluation without the initial smoothing of the measured strains leads to a significantly lower stress value at the surface (σx,drill.(0) = -47 MPa) than the one measured with X-ray diffraction (σx,diff.(0) = -666 MPa). Moreover, the non-smoothed stress distribution shows between z = 0.04 mm and z = 0.1 mm an oscillating trajectory whereas the distribution determined with X-ray diffraction has a maximum at z = 0.06 mm of σx,diff.(0.06) = 727 MPa. The smoothed stress distribution with fil = 0.1 starts at σx,drill.(0) = -497 MPa and the calculated distribution has a characteristic shape with a maximum of σx,drill.(0.08) = -657 MPa at z = 0.08 mm. In case of the calculated stresses with fil = 0.5, the distribution apparently shows a linear decreasing distribution starting at the surface with σx,drill.(0) = -670 MPa and showing no compressive stress maximum.
162 0
no smooth
σmin [MPa]
-200
-400
fil = 0.1 fil = 0.5
-600
X-Ray
-800 0.0
0.1
0.2
0.3
0.4
z [mm]
Fig. 7.7:
Stress distribution of a shot-peened 42CrMo4 steel specimen: Comparison of a hole drilling measurement with stress evaluation using different smoothing factors and an Xray diffraction measurement.
These two examples should underline, on the one hand, the effects of the initial strain smoothing after carrying out a real hole-drilling measurement. On the other hand, the residual stress distribution is not known before the measurement. Thus, the amount of smoothing to be used is difficult to define. In these cases, good engineering experience combined with knowledge of the origin and consequences of residual stresses is of especial importance.
7.5 Influence of Order of Polynomial Approximation of Calibration Strains The calculations in this chapter, in which the order of the polynomial approximation of the calibration strains is varied, are carried out using the results of the simulated measurements of models fT1.5D20m, hcR3m and hchR6m. All simulation parameters and the other influencing factors are constant (s. previous chapters). The results are shown in the following diagrams of calculated stress distributions: Fig. 7.8 shows the results calculated with model fT1.5D20m, Fig. 7.9 for model hcR3m and Fig. 7.10 for model hchR6m. In addition to the diagrams, Tab. 7.3 lists the calculated stress deviations related to the initial stress distribution σx-FEM for all three examples at selected hole depths.
163 140
3.th Order
fT1.5D20m σx-FEM = 100 MPa
6.th Order
2.th Order
120
σx [MPa]
5.th Order 4.th Order
100
80
60 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.8:
fT1.5D20m: calculated stress distribution using different orders of polynomial approximation of calibration strains
The results of the three examples show that a stress calculation using a 2th order polynomial approximation of the calibration strains leads to the highest stress deviations. The stress results using a 3th order polynomial approximation also do not reproduce sufficiently the initial stress distributions. The results using a 4th order, 5th order and 6th order polynomial approximation, respectively, reproduces on average the initial stress distribution with comparable results. The stress deviations using these three polynomial orders are on the whole lower than the stress deviations calculated with the 2th and 3th order polynomial. Problems appear at the initial depth increments for the curved models and at the final depth increments. Especially for model hchR6m, high stress deviations are calculated at the depth increments between z = 0.8 mm and z = 1 mm independently of the selected order of the polynomial approximation of the calibration strain (s. Fig. 7.10, part b)). The deviations at the initial depth increments of the curved models as well as the deviations at the final depth increments were discussed in the previous chapter Chap. 7.3).
164 140
hcR3m σx-FEM = 100 MPa
3.th Order
2.th Order
5.th Order
σx [MPa]
120
4.th Order 100
6.th Order 80
60 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.9:
hcR3m: calculated stress distribution using different orders of polynomial approximation of calibration strains
Depth
fT1.5D20m: Stress Deviation related to initial stress [%] th
[mm]
2 Order
3th Order
4th Order
5th Order
6th Order
0.0
-52
-28
-2
1
-2
0.2
-3
2
-1
-1
-1
0.5
16
0
0
1
1
1.0
-45
-96
-2
14
33
Depth
hcR3m: Stress Deviation related to initial stress [%]
[mm]
2th Order
3th Order
4th Order
5th Order
6th Order
0.0
-85
-73
-60
-60
-55
0.2
-21
-15
-17
-18
-18
0.5
18
0
0
1
1
1.0
-45
-183
0
0
-12
Depth
hcR6m: Stress Deviation related to initial stress [%]
[mm]
2th Order
3th Order
4th Order
5th Order
6th Order
0.0
-88
-71
-54
-52
-50
0.2
-6
3
0
-1
-1
0.5
9
-1
5
5
5
1.0
-124
-17
-23
-17
-22
Tab. 7.3:
Stress deviation using MPA Specific block and models fT1.5D20m, hcR3m and hchR6m. Consideration of different orders of polynomial approximation of calibration strains
165 240
a) hchR6m
4.th - 6.th Order
200
3.th Order
σx [MPa]
2.th Order
σx-FEM
160
120
80
40 0.00
0.20
0.40
0.60
0.80
1.00
z [mm]
1000
b) hchR6m
4.th , 5.th Order 6.th Order
500
σx [MPa]
2.th Order σx-FEM
0
3.th Order -500 0.00
0.20
0.40
0.60
0.80
1.00
1.20
z [mm]
Fig. 7.10:
hchR6m: calculated stress distribution using different orders of polynomial approximation of calibration strains; a) depth z = 0-0.8 mm; b) depth z = 0-1.0 mm
As a conclusion it can be said that a 4th order polynomial approximation of the calibration strain is sufficient in order to calculate residual stresses using the MPA evaluation algorithm. This statement can be also found in [2]. The calibration coefficients are calculated within the prototype program using close approximation. As a further prospect, the effect of a possible local polynomial approximation of the calibration strains should be verified. The way to solve this problem is the correct assignment of the locally determined calibration functions with the measured strains as both distributions, i.e. measured strains vs. calibration strains, may have different depth increment distributions in the measurement practice.
166
7.6 Influence of Selected Distribution of Depth Increments In this chapter the distribution of the depth increments of the measurement strain input file as well as the distribution of the calibration strain file, is alternately varied, in order to discuss its influence on the calculated stress distribution. The exemplarily used model in this chapter is model hcR3m. All simulation parameters and influencing factors are equal to the ones described in the previous chapter Chap. 7.4 and Chap. 7.5. The variants of the exemplarily used depth increment distribution are listed in Tab. 7.4. According to this, the initial variant V0 contain all strain values at the modelled depth increments of model hcR3m. The first variant V1 omits the initial strain values at z = 0.04 mm and z = 0.08 mm whereas the second variant V2 omits the initial strain values as well as the final strain values at z = 0.9 mm and z = 1.0 mm. Variant no. 3 omits only the final strain values.
Variants Initial Variant (V0) Variant no. 1 (V1) Variant no. 2 (V2) Variant no. 3 (V3) Tab. 7.4:
Distribution of depth increments [mm]
0.0 0.04 0.08 0.14 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0
-
-
0.14 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
-
-
0.14 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-
-
0.0 0.04 0.08 0.14 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-
-
Distribution of depth increments for different variants of measurement strains input files (M) or calibration strain files (C)
167 200 180
M_V1 M_V2
σx [MPa]
160 140 120 100 80
σx-FEM = 100 MPa
60
M_V0 M_V3
40 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.11:
Calculated Stress Distribution using different variants of depth increment distributions within the measurements input file; calibration file has initial variant V0
Fig. 7.11 plots the calculated stress results for model hcR3m using different variants of measurement input files. In this calculation, the calibration file has the initial variant V0. The distribution of variant V0 is the same result as in the previous chapter: it has significant stress deviations between depth increments of z = 0 mm and z = 0.2 mm. This calculated stress distribution sufficiently reproduces the initial stress of σxFEM = 100 MPa, between depth increments of z = 0.2 mm and z = 0.8 mm. A stress deviation of approx. 8 % is found at z = 0.9 mm. The calculations using the first (V1) and the second (V2) variants show also deviations at the initial depth increments with the difference that the surface stress values are approx. 90 % higher than σx-FEM. The reason for this is the significant change of the strain values, which occurs from z = 0.0 mm and z = 0.14 mm, that calculates a higher slope of the measurement strain functions. After z = 0.2 mm, both resulting stress distributions (V1 and V2) run almost equal to the one of the initial variant V0 (variant V2 runs, of course only, up to z = 0.8 mm). The results calculated with the variant no. 3, which run only up to z = 0.8 mm, are almost equal to that of the initial variant V0. In the diagram of Fig. 7.12, similar stress calculations are shown with the difference that in this case, the calibration file was varied according to the distribution of depth increments listed in Tab. 7.4.
168 250
C_V2 200
σx [MPa]
C_V3 150
C_V1 C_V0
100
σx-FEM = 100 MPa 50
0 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.12:
Calculated Stress Distribution using different variants of depth increment distributions within the calibration file; measurements input file has initial variant V0
The results of the stress distributions calculated using different depth distributions within the calibration file do not differ significantly to the distribution calculated with the initial calibration file of variant V0 at depths between z = 0.0 mm and z = 0.8 mm. All calculations show stress deviations at the initial depth increments. At the final depth increments only the distribution calculated with a calibration file of variant no. 2 shows significant deviations, 127 % at z = 0.9. The reason for this is because the approximation of the calibration functions is carried out with the variant of the depth distribution (V2), which contains the least strain values, i.e. the polynomial approximation does not reproduce accurately the calibration strain values at these depths. According to the results of this chapter, the selection of the distribution of the depth increments within the measurement input file or within the calibration file, respectively, does not significantly reduce the stress deviations at the initial depth increments as well as at the final depth increments. The spatial resolution at the initial depths should be small, i.e. the depth increments should be small and uniform, in order to avoid high strain changes which causes higher slopes of the strain functions.
169
7.7 Interpolation between Geometry Parameters In addition to the MPA standard block, which uses the original calibration functions determined in [2], the prototype program provides two alternative options for the calculation of residual stresses considering the shape of the component. The first option is realized in the MPA expanded stress calculation block. In this block, 4th order polynomial coefficients are included as fix values. These coefficients were determined for one strain gage rosette and considering the geometry parameters thickness T, distance hole-edge D and hole diameter d0. If the mentioned geometry parameters of a real component are between the geometry parameters considered in the fixed coefficients, the program calculates the new specific coefficients using linear interpolation. The second optional block calculates the calibration functions “online”, i.e. a calibration file can be imported in order to calculate the specific calibration functions. This procedure only makes sense, if the calibration file was determined using a model or specimen of the same shape or of a similar shape, respectively, as the shape of the measured component. In this case, this procedure is also independent of the used strain gage rosette. The objective of this chapter is to compare the results calculated using the MPA expanded block with the results using the MPA specific block. For this purpose, a model of a flat specimen was generated. The model has a thickness of T = 1.75 mm, a distance hole-edge of D = 3.13 mm and a hole diameter of d0 = 1.775 mm. This means that the geometry parameters of the simulated measurement were selected to be between the geometry parameters of the calibration coefficients of the MPA Expanded block (s. Chap. 6.3.2). The model of the simulated measurement has elastic material constants similar to those of aluminium (E = 70000 MPa and ν = 0.33) and was initially loaded with a biaxial uniform stress state of σx-FEM = 80 MPa and σx-FEM = 50 MPa. The same model geometry was used for the determination of the specific calibrations strains, in order to import it as a calibration input file into the MPA specific stress
170 calculation block. The model of this simulated calibration (with E = 210000 MPa and ν = 0.285) was uniaxially loaded with σc,x = 160 MPa. 130
MPA Expanded 110
σx [MPa]
MPA Specific 90
σx-FEM = 80 MPa 70
MPA Standard 50
30 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.13:
Calculated stress distributions in x-direction using the MPA Standard block, the MPA Expanded block and the MPA Specific block. Geometry of model: T = 1.75 mm, D = 3.13 mm and d0 = 1.775 mm
Fig 7.13 and Fig. 7.14 show the calculated stress results σx and σy, respectively. Both diagrams show three different calculated stress distributions: the one calculated with the MPA standard block (black continuous line), the one determined with the MPA expanded block (grey dashed line) and the one using the MPA Specific block (black dashed and dotted line). All three stress calculations were carried out using 4th order polynomial calibration functions, the same distribution of depth increments and with no smoothing of the measured strains. In addition, stress deviations of these calculations at selected hole depths are listed in Tab 7.5. For all three blocks, the calculated stress distributions in x-direction have a similar shape than the calculated stress distributions in y-direction. In both directions the calculated stress distributions run about the initial stress value σi-FEM. The highest deviations were expected and calculated using the MPA Standard block. In both stress directions, the distribution calculated with the MPA Expanded block runs almost equal to the MPA Specific block. At a depth of z = 0.8 mm and deeper, the results calculated with the MPA Expanded block deviate more from the initial stress values than the results calculated with the MPA Specific block.
171
130
110
σy [MPa]
MPA Expanded 90
MPA Specific 70
σy-FEM = 50 MPa
50
MPA Standard 30 0
0.2
0.4
0.6
0.8
1
1.2
z [mm]
Fig. 7.14:
Calculated stress distributions in y-direction using the MPA Standard block, the MPA Expanded block and the MPA Specific block. Geometry of model: T = 1.75 mm, D = 3.13 mm and d0 = 1.775 mm
Stress deviation related to initial stress σi-FEM
Depth
Deviation σx [%]
[mm]
Deviation σy [%]
Standard
Expanded
Specific
Standard
Expanded
Specific
0.0
-32
-7
-5
-30
-2
-1
0.2
14
-3
1
30
5
10
0.5
19
4
5
36
15
16
1.0
-14
52
16
-18
90
33
Tab. 7.5:
Stress deviation in x-direction and in y-direction: comparison of stress calculation blocks MPA Standard, MPA Expanded and MPA Specific
To summarize this chapter it can be said that both of the additionally provided stress calculations blocks, which consider the specific geometry parameters of the component, reproduce the initial stresses of the example model with less deviation than the calculation procedure based on the original calibration functions. The results determined with the MPA Expanded block show higher stress deviation at the final depth increments than the ones calculated with the MPA Specific block.
172
7.8 Summary of the Evaluation of the Calculation Program with Implemented Geometry Specific Calibration Functions The stress calculations carried out in this chapter were mostly based on strain results determined using models of specimens with non-reference shape. The sole exception was the use of the strain distributions of the reference model at the first part of this evaluation chapter in order to discuss general aspects of the stress calculation using the MPA algorithm and to show the resulting calculation uncertainty. In the case of the model of the flat reference specimen, the calculation uncertainty is on average approx. 6 %. The parametric study has shown, firstly, the influence of the local polynomial approximation of the measured strain compared to the close polynomial approximation. It was shown that a stress evaluation using local polynomial approximation determines on average a lower stress deviation as with close approximation. The possible problems at the initial and final depth increments were discussed. The second part of the parametric study has dealt with the effects and the importance of the initial smoothing of the measured strains. In the third part, the influence of the order polynomial approximation of the calibration strains was demonstrated. According to the shown result, a 4th order polynomial approximation is sufficient for the calculation of residual stresses using the MPA algorithm. The fourth part of the parametric study has discussed the influence of distribution of depth increments within the measurement input file or the calibration file. The last part compared the two provided calculation procedures which consider geometry specific calibration functions. Both calculation procedures (MPA Expanded and MPA Specific) lead to similar results: they reproduce in average the initial stress distributions with less stress deviation than the MPA Standard block, which uses the original reference calibration functions.
173
8 Conclusion This work investigated the geometry influence on the stress calculation according to the differential MPA evaluation algorithm for the hole drilling method, which is an accepted and widely used technique for the residual stress measurement. The available evaluation methods of the hole drilling method are based on a previous calibration of the method on a known stress state under reference geometrical conditions, which are realized usually with the FEM model of a thick and wide flat plate. Because of this, the violation of recommended geometrical restrictions, e.g. a measurement on a thin tube, influences the calculated residual stresses. In this work, exemplarily calculations with current evaluation software using geometrical non-reference specimens were carried out. The resulting stress distributions of the selected non-reference specimens show a significant overestimation of the initially presented stress distribution at the measurement point. In order to compensate these geometry effects some authors recommend the determination of geometry specific calibration functions. Following this recommendation, this work dealt with two main topics. The first topic was the systematic investigation of the individual contribution of the violation of single geometry parameters. For this purpose, calibration experiments as well as calibration FEM calculations were carried out, in which the following geometry parameters were systematically varied: the specimen’s thickness T, the distance between the measurement point and a free specimen’s edge D and the cylinder curvature radius R. The calibrations were compared by means of relieved strain deviations and calculated stress deviations, which were both related to the results obtained with the reference calibration. The influence of the single violation of geometry parameters can be summarized with the following statements: The boundary condition “thickness” is proposed with T = 3 x d0, according to the authors of the MPA algorithm. The results considering the geometry parameter thickness (and also the distance hole-edge) were determined using uniformly loaded flat specimens or flat finite element models, respectively. They showed that this geometry limit may be minimized up to a thickness of T = 1.66 x d0. Stress deviations of 18 % for T = 0.83 x d0 = 1.5 mm and about 53 % for T = 0.55 x d0 = 1 mm were determined on average over the hole depth.
174
The distance from the center of the measurement point (hole center) to the specimen’s free edge did not influence the hole drilling results as significantly as the specimen’s thickness. A maximum stress difference of approx. 5 % on average related to the reference was found for the results of the model with D = 1.11 x d0 = 2 mm. The boundary condition for this special geometry parameter, which is set with min. D = 5 x d0 in literature, may be minimized up to D = 1.94 x d0. The boundary condition radius of surface curvature is originally set with T = 3 x d0. The results considering this geometry parameter were determined using uniformly loaded cylindrical specimens or cylindrical finite elements models, respectively. It was shown that the boundary condition could not be minimized and should be even displaced to a higher value. The maximal stress deviations in x-direction were 12 % for the solid cylinder and 39 % for the hollow cylinder. These results were already calculated for the cylinders with R = 3.33 x d0, which already satisfies the set boundary condition. In ydirection, which was transverse to the load direction, the maximum stress differences were significantly high for the results with R = 1.67 x d0. An uncertainty analysis was carried out for the experimental calibration as well as for the numerical calibration. According to this uncertainty analysis, the experimentally determined calibration strains showed a tolerance range, as many influencing factors have an effect on the calibration results. The distributions of the numerical calibration strain ran within the determined tolerance range. In the case of the differential evaluation method, the proper determination of calibration functions should be carried out using the results of the finite element calibration, because there is almost no scattering. The scattering of the experimental calibration strain distribution could lead to high or low values of the calculated strain derivations, which are part of the equation of the calibration function. Thus, the stress calculation could lead to erroneous values. The second topic was the implementation of geometry specific calibration functions into an evaluation program using the differential MPA method. For this purpose, a prototype program was developed. This program allows the calculation of residual stresses according to the MPA method using three options. The first option uses the original calibration functions of the authors of the MPA method. The second option considers
175 geometry specific calibration functions, which are included as fixed polynomial coefficients. In this block, the fixed coefficients consider the geometry parameters thickness, distance hole-edge and hole diameter. The third option calculates the calibration functions “online”, i.e. the calibration strain distributions are imported and the calibrations functions are then calculated for these specific calibration strains. This procedure can be beneficial only if the geometry of the calibration model is equal or similar to the geometry of the measured component. The prototype program was evaluated by means of a parametric study. It was shown that the calculation procedure itself leads to small stress deviations even in the case of the evaluation of the reference specimen. Moreover, the importance of the strain smoothing, the importance of the used differentiation procedure and the influence of different selected depth distributions within the strain files were shown and discussed. The results obtained with the prototype program confirm previous investigations concerning the use of a 4th order polynomial approximation of the calibration strain distributions. To summarize the evaluation of the prototype program, it can be said that the calculation using geometry specific calibration functions minimizes on average the stress deviations compared to a calculation using the original reference calibration functions. Stress deviations still appear at the first depth increments of curved surfaces and at the final depth increments. The deviations at the near surface depth increments were caused due to the fact that the introduced hole did not have an ideal blind-hole (with d0) within the first increments whereas the deviations at the last increments were related to the general specification of the differential method, which does not consider stresses above the actual hole depth. The two presented calculation procedures determine similar results for the selected plate model with minor stress deviation using the MPA Specific block. The MPA Expanded calculation block with fixed calibration coefficients has the advantage that the user does not require a previous calibration. The disadvantage of this procedure is that it is always limited by the considered range of the geometry parameters implemented within the calibration coefficients. In this calculation block, three geometry parameters for one specific strain gage rosette are implemented within the calibration functions but it is
176 possible, in principle, to consider more geometry variables. This could be carried out at the cost of calculation accuracy as the evaluation becomes more complex. The MPA Specific block has the advantage that the exact geometry of the component, the used strain gage rosette and the stress conditions at the measurement point are considered during the determination of the calibration file, i.e. there is no need for the determination of the specific calibration coefficients by means of an interpolation between calibration coefficients, which consider single geometry parameters.
177
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-
Criteria,
Procedures,
183
10 Zusammenfassung in deutscher Sprache Titel:
Eigenspannungsanalyse an Bauteilen mit realen Geometrien unter Verwendung der inkrementellen Bohrlochmethode und eines differentiellen Auswerteverfahren
Eigenspannungen entstehen als Folge von Fertigungsprozessen in nahezu allen Bauteilen. In manchen Fällen kann dadurch das Bauteilverhalten je nach Höhe und Wirkrichtung der vorhandenen Eigenspannung günstig oder gar nachteilig beeinflusst werden. Aus diesem Grund ist die Nachfrage nach zuverlässigen und kostengünstigen Eigenspannungs-analysemethoden von besonderem Interesse in der Industrie. Die Bohrlochmethode ist in diesem Zusammenhang ein übliches und weit verbreitetes Messverfahren, das auf dem folgendem Prinzip beruht: Beim Einbringen eines Sackloches in die Bauteiloberfläche wird durch das Entfernen spannungsbehafteter Werkstoffbereiche das innere Gleichgewicht lokal gestört und es stellt sich in der Bohrlochumgebung
ein
neuer
Gleichgewichtszustand
ein.
Diese
Gleichgewichtsänderungen führen zu entsprechenden Verformungen, die mittels speziell für die Bohrlochmethode entwickelten Dehnungsmessstreifen (DMS) an der Oberfläche ermittelt werden können. Aus den gemessen Dehnungsänderungen lassen sich dann die im Bauteil vorhandenen Spannungen auswerten. Auswertemethoden, die u.a.
die
Bohrungstiefe
explizit
berücksichtigen,
um
z.B.
einen
Eigenspannungstiefenverlauf berechnen zu können, basieren auf vorher ermittelten Kalibrierfunktionen, die bei einem bekannten Spannungszustand aufgenommen worden sind. Die Ermittlung dieser Kalibrierfunktionen erfolgt dabei unter Einhaltung geometrischer Randbedingungen, wie z.B. der Bauteildicke, dem Abstand der Bohrung zur Bauteilberandung, dem Radius der Oberflächenkrümmung. Zahlreiche Bauteile aus der Praxis, an denen eine Eigenspannungsmessung mit dem Bohrlochverfahren sinnvoll erscheint, weisen oft Abweichungen von den in der Literatur angegebenen erforderlichen geometrischen Randbedingungen auf. Eine Eigenspannungsberechnung mit den kommerziell erhältlichen Auswertemethoden könnte z.B. zu einer
184 Überschätzung der Eigenspannungswerte führen. In solchen Fällen wird die Ermittlung von
geometriespezifischen
Kalibrierfunktionen
empfohlen,
um
die
Spannungsabweichung korrigieren zu können. In der vorliegenden Arbeit wird der Einfluss der Bauteilgeometrie auf die Spannungsberechnung nach dem differentiellen MPA Verfahren [2.44] für die Bohrlochmethode
untersucht.
Zuerst
soll
dabei
der
Einfluss
der
einzelnen
geometrischen Randbedingungen anhand von Dehnungs- und Spannungsabweichungen bezogen auf Ergebnisse, die mit der Referenzgeometrie erzielt worden sind, ermittelt werden. Die drei untersuchten geometrischen Parameter sind die Bauteildicke (T für engl. „Thickness“), der Abstand der Bohrung zum Bauteilrand (D für engl. „Distance“) und der Radius der Oberflächenkrümmung (R für engl. „Radius“). Als zweites Ziel sollen geometriespezifische Kalibrierfunktionen in dem differentiellen MPAAlgorithmus implementiert werden, um eine mögliche Spannungskorrektur zu zeigen. Für beide Zielsetzungen werden sowohl experimentelle Kalibrierungen als auch Kalibrierungen mittels FEM an Proben bzw. Modellen, welche die empfohlenen Randbedingungen
systematisch
verletzen,
durchgeführt.
Die
Ergebnisse
der
durchgeführten Kalibrierungen werden anhand einer Unsicherheitsanalyse diskutiert. Im letzen Teil der Arbeit wird die erste Version eines Auswerteprogramms nach der differentiellen
MPA-Methode
beschrieben
und
evaluiert.
Im
Gegensatz
zu
kommerziellen Bohrloch-Auswerteprogrammen ermöglicht das vorgestellte Programm die Berechnung von spezifischen Kalibrierfunktionen, welche die tatsächliche Bauteilgeometrie implizit berücksichtigen. Die erzielten Ergebnisse können wie folgt zusammengefasst werden: Die Randbedingung „Bauteildicke“ wird in [2.44] T = 3 x d0 empfohlen (d0 ist dabei der Bohrungsdurchmesser). Die Ergebnisse, die den Geometrieparameter Bauteildicke berücksichtigen (und auch den Parameter Abstand Bohrung-Kante) wurden mit einachsig belasteteten Flachzugproben bzw. Modellen ermittelt. Für diesen Fall zeigte sich, dass diese Randbedingung auf eine Dicke von T = 1.66 x d0 verkleinert werden kann. Für Proben mit T = 0.83 x d0 = 1.5 mm und T = 0.55 x d0 = 1 mm wurden im Durchschnitt Spannungsabweichungen bezogen auf die Referenzergebnisse von 18 % bzw. 53 % ermittelt.
185
Die Ergebnisse bezüglich des Parameters „Abstand Bohrung-Kante“ zeigen keinen signifikanten Einfluss auf die Kalibrierung im Vergleich mit dem der Kalibrierung unter Berücksichtigung der Bauteildicke. Die maximale Spannungsabweichung bezogen auf die Referenz beträgt durchschnittlich 5 % für das Modell der Probe mit D = 1.11 x d0 = 2 mm. Die Randbedingung „Abstand Bohrung-Kante“ kann daher von empfohlenen D = 5 x d0 auf D = 1.94 x d0 minimiert werden. Die Randbedingung „Radius der Oberflächenkrümmung“ ist ursprünglich mit D = 3 x d0 angesetzt. Die Ergebnisse unter Berücksichtigung dieses Parameters wurden unter Verwendung von einachsig belasteten zylindrischen Proben oder dessen FEM Modelle ermittelt. Es zeigt sich, dass die empfohlene Randbedingung nicht minimiert werden kann und sogar erweitert werden sollte. Schon bei einem Zylinderradius von R = 3.33 x d0 = 6 mm (die Randbedingung ist dabei erfüllt) betragen die maximale Spannungsabweichungen in Belastungsrichtung 12 % für Vollzylinderproben und 39 % für Hohlzylinderproben. Die Unsicherheitsanalyse wurde durchgeführt um die Güte der FEM Ergebnisse, bezogen auf die experimentellen Kalibrierergebnisse, zu vergleichen. Demzufolge zeigen die experimentell ermittelten Verläufe einen Toleranzband, welches aus den zahlreichen versuchstechnischen Einflussfaktoren verursacht wird. Die simulierten Ergebnisse verlaufen innerhalb des errechneten Toleranzbandes. Aus der Gewonnen Erfahrung in dieser Arbeit wird für die differentielle MPA-Auswertemethode eine Kalibrierung mittels FEM empfohlen, um die Streuung der Messdehnung zu vermeiden. Diese Streuung der gemessenen Kalibrierdehnung kann dazu führen, dass die berechnete Dehnungsableitungen der Kalibrierung, auf die das MPA-Verfahren basiert, zu hoch oder zu niedrig ausfallen und dadurch fehlerbehaftete Eigenspannungswerte berechnet werden. Das Berechnungsprogramm, das im zweiten Teil der Arbeit entwickelt worden ist basiert auf dem MPA-Verfahren und hat drei Optionen für die Spannungsberechnung. Die erste Option verwendet die originalen Kalibrierfunktionen des MPA-Verfahrens aus [2.44] („MPA Standard“), die unter Einhaltung der geometrischen Randbedingung an einer flachen und ausreichend dicken und breiten Platte ermittelt worden sind. Die
186 zweite Berechnungsoption („MPA Expanded“) berücksichtigt drei geometrische Parameter
(die
Bauteildicke,
Bohrungsdurchmesser)
in
Form
den
Abstand von
Bohrung-Kante
festgelegten
und
den
geometriespezifischen
Kalibrierfunktionen. Die dritte Berechnungsoption ermittelt die Kalibrierfunktionen während der eigentlichen Spannungsberechnung durch Import der spezifischen Kalibrierdehnungen und anschliessender Ermittlung der Kalibrierfunktionen („MPASpecific“). Diese Option ist nur dann sinnvoll, wenn die Geometrie der Kalibrierprobe bzw. des Modells ähnlich oder gleich der Geometrie des gemessenen Bauteiles ist. Das Programm wurde anhand einer Parameterstudie evaluiert. Es hat sich dabei gezeigt, dass eine geometriespezifische Berechnung im Durchschnitt die Spannungsabweichung im Vergleich mit der Berechnung unter Verwendung der Referenzkalibrierfunktionen minimiert.
187
11 Appendix 11.1 FEM Post Processor Plots
Fig. 11.1:
fhT6D2m: stress distribution σc,x before drilling
Fig. 11.2:
fhT6D2m: stress distribution σc,y before drilling
188
Fig. 11.3:
fhT1.5D2m: stress distribution σc,x before drilling
Fig. 11.4:
fhT1.5D2m: stress distribution σc,y before drilling
189
Fig. 11.5:
hchR6m: stress distribution σc,x before drilling
Fig. 11.6:
hchR6m: stress distribution σc,y (tangential to cylinder surface) before drilling
190
Fig. 11.7:
Inhomogeneity in hcR6m model: stress distribution in x-direction before drilling. Initial stress σc,x = 160 MPa
Fig. 11.8:
Inhomogeneity in hcR4m model: stress distribution in x-direction before drilling. Initial stress σc,x = 160 MPa
191
Fig. 11.9:
Inhomogeneity in hcR3m model: stress distribution in x-direction before drilling. Initial stress σc,x = 160 MPa
Fig. 11.10:
Plastification in hchR6m model: stress distribution in x-direction before drilling. Initial stress σc,x = 210 MPa
192
Fig. 11.11:
Location of strain gage no. 1 in hcR6m model: strain distribution tangential to cylinder surface. Hole depth z = 1 mm, Initial stress σc,x = 160 MPa
Fig. 11.12:
Location of strain gage no. 1 in hcR4m model: strain distribution tangential to cylinder surface. Hole depth z = 1 mm, Initial stress σc,x = 160 MPa
193
Fig. 11.13:
Location of strain gage no. 1 in hcR3m model: strain distribution tangential to cylinder surface. Hole depth z = 1 mm, Initial stress σc,x = 160 MPa
194
11.2 Program Listings 11.2.1 Smoothing using Weighted Average Algorithm % Smoothing of measured strains according to weighted means algorithm % B- strain data matrix, c-column function gw1 = gewichtfactor(B,c) [r1,c1]=size(B); index1=r1; for i = 1:index1 Tiefe1(i)=B(i,1); Eps901(i)=B(i,c); Y(i)=B(i,c); end; %Weight factors cim1=0.1; ci12=0.2; ci=0.4; ci23=0.2; cip1=0.1; for i = 2:index1-1 Yi12(i)=0.5*(Y(i-1)+Y(i)); Yi23(i)=0.5*(Y(i+1)+Y(i)); gw1(i)=(cim1*Y(i1)+ci12*Yi12(i)+ci*Y(i)+ci23*Yi23(i)+cip1*Y(i+1)); end; gw1(1)=(0.95*Y(1)+0.05*Y(2)); gw1(index1)=(0.9*Y(index1)+0.1*Y(index1-1));
11.2.2 Smoothing using Compensative Cubic Splines % Smoothing of measured strains according to compensative cubic splines % B- strain data matrix, c-column, fil - smoothing factor function yw =
CubicSpline(B,c,fil)
[r1,c1]=size(B); n=r1; for i = 1:n x(i)=B(i,1); y(i)=B(i,c); end; p=1000/fil^2; dx=x(3)-x(2); dxp=x(4)-x(3); dxm=x(2)-x(1); dy=y(3)-y(2); dym=y(2)-y(1); dxmm=.00001; if (dx == 0) dx=.00001;
195 end if(dxm == 0) dxm=.00001; end for i=2:(n-1) if (dxp == 0) dxp=.00001; end ak(i)=dxm -(6/ (p*dxm))*( 1/dxmm+1/dxm)-(6/(p*dxm))*( 1/dx+1/dxm); bk(i)=2*(dxm+dx)+ 6/(p*dxm^2) + 6/(p*dx^2) + (6/p)*(1/dx+1/dxm)^2; ck(i)=dx-(6/(p*dx))*(1/dx+1/dxm)-(6/(p*dx))*(1/dx+1/dxp); dk(i)=6/(p*dxm*dxmm); ek(i)=6/(p*dx*dxp); yk(i)=6*(dy/dx - dym/dxm); dxmm=dxm; dxm=dx; dx=dxp; dym=dy; if (i == (n-1)) continue else dy= (y(i+2)-y(i+1)); end if(i==(n-2)) continue else dxp=(x(i+3)-x(i+2)); end end ak(2)=0; dk(2)=0; dk(3)=0; ek(n-2)=0; ek(n-1)=0; ck(n-1)=0; zn(3)=ck(2)/bk(2); za(3)=ak(3); bk(3)=bk(3)-za(3)*zn(3); if (bk(3) == 0) bk(3)=.00001; end for i=4:n-1 zo(i)=ek(i-2)/bk(i-2); zn(i)=(ck(i-1)-za(i-1)*zo(i))/ bk(i-1); za(i)=ak(i)-dk(i)*zn(i-1); bk(i)=bk(i)-dk(i)*zo(i)-za(i)*zn(i); if (bk(i) == 0); bk(i) = .00001; end end ak(2)=yk(2)/bk(2); ak(3)=(yk(3)-za(3)*ak(2))/bk(3);
196 for i=4:n-1 ak(i)=(yk(i)-za(i)*ak(i-1)-dk(i)*ak(i-2))/bk(i); end yz(1)=0; yz(n)=0; yz(n-1)=ak(n-1); yz(n-2)=ak(n-2)-zn(n-1)*yz(n-1); for i=3:n-2 in=n-i; yz(in)=ak(in)-zn(in+1)*yz(in+1)-zo(in+2)*yz(in+2); end dxi=(x(2)-x(1)); if(dxi ==0) dxi=.00001; end yw(1)=y(1)-yz(2)/(p*dxi); for i=2:n-1 dxm=dxi; dxi=(x(i+1)-x(i)); if (dxi == 0) dxi=.00001; end yw(i)=y(i)-((yz(i-1)/dxm -(1/dxm+1/dxi)*yz(i) + yz(i+1)/dxi)/p); end yw(n)=y(n)-(yz(n-1)/(p*dxi));
11.2.3 Local Polynomial Approximation of Measured Strains % Local Polynomial Approximation of Measured Strains (2th order % polynomials) and Differentiation function [De11, De21, De31] = polynomialapproximation(ksi, e1, e2, e3, index); for i = 1:index % Surface Value of Differentiated Strain (z=0) if(i == 1) ek1 = polyfit(ksi(i:i+2), e1(i:i+2), 2); ek2 = polyfit(ksi(i:i+2), e2(i:i+2), 2); ek3 = polyfit(ksi(i:i+2), e3(i:i+2), 2); De1 = polyder(ek1); De2 = polyder(ek2); De3 = polyder(ek3); De11(i)=polyvalm(De1, ksi(i)); De21(i)=polyvalm(De2, ksi(i));
197 De31(i)=polyvalm(De3, ksi(i)); end % Final Value of Differentiated Strain (z=n) if( i== index) ek1 = polyfit(ksi(i-2:i), e1(i-2:i), 2); ek3 = polyfit(ksi(i-2:i), e3(i-2:i), 2); ek2 = polyfit(ksi(i-2:i), e2(i-2:i), 2); De1 = polyder(ek1); De2 = polyder(ek2); De3 = polyder(ek3); De11(i) = polyvalm(De1, ksi(i)); De21(i) = polyvalm(De2, ksi(i)); De31(i) = polyvalm(De3, ksi(i)); end % Values of Differentiated Strain (z>=0 & z
