determination of full range stress -strain curves by means...

Determination of full range stress-strain curves by

means of instrumented indentation

Andreas De Smedt

Promotoren: prof. dr. ir. Wim De Waele, dr. ir. Stijn Hertelé

Masterproef ingediend tot het behalen van de academische graad van

Master in de ingenieurswetenschappen: werktuigkunde-elektrotechniek

Vakgroep Mechanische Constructie en Productie

Voorzitter: prof. dr. ir. Patrick De Baets

Faculteit Ingenieurswetenschappen en Architectuur

Academiejaar 2012-2013

Acknowledgement

This thesis is the results of the cooperation and help of different people. Therefore I want to express my

gratitude to a number of people in particular.

In the first place I would like to thank my mentor, dr. ir. Stijn Hertelé for his daily assistance and a lot of

help. I would also like to thank Prof. dr. ir. Wim De Waele, Timothy Galle, Koen Van Minnebruggen

and Matthias Verstraete, all from the Department of Mechanical Construction and Production at Ghent

University, for their advice.

Finally I would like to thank Prof. Didier Chicot and Xavier Decoopman from université De Lille 1 for

their help with the experimental testing.

Permission to loan

The author gives permission to make this master dissertation available for consultation and to copy

parts of this master dissertation for personal use.

In the case of any other use, the limitations of the copyright have to be respected, in particular with

regard to the obligation to state expressly the source when quoting results from this master dissertation.

De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van

de masterproef te kopiëren voor persoonlijk gebruik.

Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de

verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze masterproef.

Ghent, June 2013

Promotor

Prof. dr . ir. W. De Waele

Promotor

dr. ir. S. Hertelé

Author

A. De Smedt

Overview

Determination of full range stress-strain curves by means of instrumented indentation

Promoters: prof. dr. ir. Wim De Waele, dr. ir. Stijn Hertelé

Thesis submitted to achieve the degree of Master in Engineering: electromechanical engineering

Department Mechanical Construction and Production

Chairman: prof. dr. ir. Patrick De Baets

Faculty of Engineering, Ghent University

Academic Year 2012-2013

Summary

This master dissertation handles about instrumented indentation. Instrumented indentation is a hardness

test which can be applied to reconstruct stress-strain behavior and has as advantage that it can be used to

qualify the stress-strain properties locally.

This thesis starts with an extensive literature review which describes all the common methods available.

One particular method, interesting for pipeline steels, is discussed more in detail and is successfully

validated by finite element simulations.

A parametric finite element model was developed which allows for reverse modeling. The model was

validated based on own experiments and experiments available by literature. A parametric study was

executed as well.

Keywords: instrumented indentation, Stress-strain behavior, local properties

Overzicht

Het bepalen van het volledige spanning-rek diagram via geïnstrumenteerde indentatie

Promotoren: prof. dr. ir. Wim De Waele, dr. ir. Stijn Hertelé

Masterproef ingediend tot het behalen van de academische graad van

Master in de ingenieurswetenschappen: werktuigkunde-elektrotechniek

Vakgroep Mechanische Constructie en Productie

Voorzitter: prof. dr. ir. Patrick De Baets

Faculteit Ingenieurswetenschappen en Architectuur

Academiejaar 2012-2013

Samenvatting

In dit proefschrift wordt geïnstrumenteerde indentatie onderzocht. Dit is een hardheidstest die gebruikt

kan worden om het trek-rekdiagram te reconstrueren en heeft als voordeel dat lokale

materiaaleigenschappen kunnen opgemeten worden.

Deze thesis start met een uitgebreide literatuurstudie waarin de meest toegepaste methodes besproken

worden. Een bepaalde methode, interessant voor pijplijn stalen is verder onderzocht en tevens ook

succesvol gevalideerd door eindige elementen simulaties.

Een parametrisch eindige elementen model is opgesteld het oog op invers modelleren. Het model is

gevalideerd door eigen experimenten en experimenten beschikbaar in de literatuur. Ook werd een

parameterstudie uitgevoerd.

Trefwoorden: geïnstrumenteerde indentatie, trek-rekdiagram, lokale materiaaleigenschappen

Determination of full range stress-strain curves bymeans of instrumented indentation

Andreas De Smedt

Supervisor(s): Wim De Waele, Stijn Hertele

Abstract— It is possible to reconstruct the stress-strain curve by an in-strumented hardness test. Instrumented indention testing allows to qualifythe local stress-strain properties of heterogeneous microstructures such aswelds. In this test the displacement of the indenter and its force are contin-uously measured. All common methods available were listed in this masterdissertation. One particular method, interesting for pipeline steels, was dis-cussed more in detail. A parametric finite element model was developedwhich allows for reverse modeling. This model was optimized and a para-metric study was executed.

Keywords—Instrumented indentation, Stress-strain behavior, local prop-erties

I. INTRODUCTION

Knowledge of the material properties (stress-strain behavior)is essential for the design of new and the qualification of existingstructures. An instrumented indentation test (IIT) can, oppositeto a tensile test, qualify local properties (Fig. 1). IIT is a hard-ness test during which execution the force on the indenter andthe displacement of the indenter are recorded. Many differentmethods are available in literature to reconstruct the stress-straincurve out of instrumented indentation test. One straightforwardmethod which provides the basic framework is explained below.To obtain average properties it is preferred to work on micro-scale. For the indentation body a Vickers indenter is chosen.This kind of indenter requires lower forces than a Brinell typeindenter for the same indentation depth. Because of the geomet-rical similarity between the height of the indenter and its pro-jected surface, IIT for sharp indenters is simplified. Differentmethods available by literature for sharp indentation can also becombined to eliminate the potential presence of multiple solu-tions.

Fig. 1. An instrumented indentation test can be used as a substitute for a tensiletest

II. CURVE-FITTING METHOD

A. Force-depth curve

The evolution of an indentation force-depth curve dependson the material, indenter type and maximal force of displace-ment used but generally follows a characteristic pattern. A typi-cal force-depth curve is characterized by the curvature C during

loading, the maximum imprint depth hm and the slope S duringunloading (Fig. 2).

Fig. 2. Force-depth curve is characterized byC, hm and S, curve generated fora strongly strain hardening steel

B. How to reconstruct the stress-strain curve

The material in the vicinity of the indentation does not remainperfectly flat. From Fig. 3 it can be seen that the response ofthe surface depends Ramberg-Osgood’s strain hardening expo-nent nRO. This increases the complexity of developing adequatedefinitions for the representative stress. Several researchers de-veloped equations to predict pile-up and sink-in for certain ma-terial groups. Due to the many approximations which need to bemade, a point-to-point relationship is limited in accuracy.

Fig. 3. Contact surface between indenter and material depends on the strain-hardening behavior (expressed here as Ramberg-Osgood’s strain hardeningexponent)

However it is possible by curve fitting methods to write theindentation parameters S, C and hm as functions of the param-eters of a constitutive law describing the stress-strain curve. De-spite the limitation of having to respect a constitutive law, this is

the most common IIT method for stress-strain curve reproduc-tion.

C. Dimensionless correlations between the force-depth curveand stress-strain curves of steel (Ramberg-Osgood model)

Given that the Youngs modulus is known for steel, only theRamberg-Osgood parameters (strain hardening exponent nRO

and yield strength σ0) of the stress-strain curve remain un-known. Two dimensionless functions suffice to solve the prob-lem. In the first dimensionless-relation (Fig. 4) C/σr is a func-tion of E∗/σr. σr is the stress which corresponds with a plastictrue strain of 0.033. For this specific stress Fig. 4 becomes in-dependent of nRO. E∗ is the reduced Young’s modulus whichtakes Young’s modulus and Poisson coefficient of both the in-denter and the material into account. The second dimensionlessrelationship (Fig. 5) shows S/(E∗hm) as a function of E∗/σr.From this figure, nRO can be estimated. Both dimensionless re-lations were successfully validated especially for pipeline steelswith σ0 between 400MPa and 700MPa and nRO between 5 and25. Further, both functions were validated for 54 specific casesprovided by literature and very good agreement between simu-lations and the dimensionless functions was obtained. As can beseen from Fig. 5 these curve-fitting functions need to be slightlyredefined for low strain hardening materials (high nRO). Allsimulated materials yield a unique force-depth curve. This in-dicated that no more dimensionless correlations are needed forconvergence to a unique result.

Fig. 4. With known C and E∗, σ0.033 can be found using [1]

Fig. 5. With known S, E∗ and hm, nRO can be found using [1]

III. REVERSE MODELING

A. Computational model

In the majority of the available literature the indentation testis finite element modeled as axisymmetric. This reduces thecalculation time and simplifies the analysis but has the draw-back that for force controlled conditions a minimal tip radiusfor sharp indenters of 10 µm is needed for convergence reasons.For displacement controlled conditions this drawback does notapply. A parametric Python script was set up which creates andprocesses axisymmetric simulations using Abaqus. The gener-ated output was qualified based on C, hm, S and the calculationtime. These four parameters were automatically extracted outof the generated force-depth curves by a Matlab script. Thisscript was validated by examining the 54 different cases men-tioned above. Based on this four parameters it was found thatlinear elements were needed with size of 1µm near the indentertip yield converging numerical solutions. For forces up to 40N,the radius and height of the specimen should be at least 4mm.This parametric script was validated based on experimental re-sults listed in literature as well as by own experimental tests. Cwas always slightly overestimated.

B. Parametric study

First, it was found that friction has a major influence on S.In contrast with literature more pile-up was obtained for higherfriction coefficients. Second, the influence of tip blunting whichoccurs with frequent use was investigated and it was concludedthat tip radius should be carefully monitored. Finally, as men-tioned in literature, high nRO produces pile-up and low nRO

sink-in.

IV. CONCLUSION

Instrumented indentation testing allows for the constructionof stress-strain curves using the recorded indentation force-depth curve as an input. Summarizing an extensive literaturereview, it is advised to estimate the parameters of a constitutivelaw rather than deduce a stress-strain curve in point-wise terms.For the determination of properties for steels, two reported di-mensionless relations stand out in simplicity and have been fur-ther examined. One of these is not in full correspondence withfinite element results and should be investigated in closer detail.

A parametric finite element model that allows for inversemodeling of instrumented indentation tests has been developed.Although very good correspondence with experiments is foundfor the unloading stiffness, the loading curvature is systemati-cally overestimated. Hence, further model optimization and ex-perimental validation is needed.

ACKNOWLEDGMENTS

The author is very grateful for all the advise and help fromhis promoter Stijn Hertele and would also like to acknowledgeof the support of Prof. Didier Chicot and Xavier Decoopman.

REFERENCES

[1] M. Dao, N. Chollacoop, K.J. Van Vliet, T.A. Venkatesh, and S. Suresh,“Computational modeling of the forward and reverse problems in instru-mented sharp indentation,” Acta Materialia, vol. 49, no. 19, pp. 3899 –3918, 2001.

Het bepalen van het volledige spanning-rek diagramvia geınstrumenteerde indentatie

Andreas De Smedt

Supervisor(s): Wim De Waele, Stijn Hertele

Abstract—Een geınstrumenteerde indentatie test maakt het mogelijk omhet volledige spanning-rek diagram te reconstrueren. Heterogene struc-turen, zoals lansnaden, kunnen met deze test bestudeerd worden. Bij dezetest wordt de verplaatsing van, en de kracht op de indenter continu opgeme-ten. De meest voorkomende methodes worden opgelijst in deze thesis.Een bepaalde methode, interessant voor pijplijnstalen, wordt dieper onder-zocht. Ook wordt een parametrisch eindig elementen model ontworpen. Ditmodel maakt invers modeleren mogelijk. Het model wordt geoptimaliseerden tevens wordt een parameter studie uitgevoerd.

Keywords—geonstrumenteerde indentatie, spanning-rek diagram, lokaleeigenschappen

I. INLEIDING

Kennis van de materiaal eigenschappen (spanning-rek dia-gram) is essentieel voor het ontwerp van nieuwe en de beoordel-ing van bestaande structuren. Via een geonstrumenteerde inden-tatie test (IIT) is mogelijk, in tegenstelling tot een trekproef, onde lokale materiaaleigenschappen op te meten. IIT is een hard-heidstest waarbij de kracht en de verplaatsing van de indentercontinu opgemeten worden. Er zijn veel verschillende methodeste vinden in de literatuur en ee n van hen wordt nader bestudeerd.Om uitgemiddelde materiaaleigenschappen op te meten wordter op microschaal gewerkt. Als indrukkingslichaam is vooreen Vickers indenter gekozen, hiervoor zijn lagere krachtenvereist in vergelijking tot een Brinell indruklichaam om totdezelfde indrukking te komen. Voor indenters die eindigen opeen punt en dus het geprojecteerde oppervlak lineair toeneemtmet de indrukkingsdiepte zijn eenvoudigere methodes beschik-baar. Voor dit soort indenters kunnen ook verschillende meth-odes beschreven in de literatuur gecombineerd worden. Dit ver-groot de kans op unieke karakterisatie van het materiaal.

Fig. 1. IIT kan gebruikt worden ter vervanging van een trekproef

II. CURVE-FITTING METHODE

A. kracht-indrukking kromme

De kracht-diepte kromme hangt af van de gebruikte inden-ter, opgelegde maximale kracht of verplaatsing en natuurlijkook van de materiaaleigenschappen. Alle curves hebben weldezelfde karakteristieke vorm, deze wordt bepaald door dekromming tijdens het laden C, de maximale indrukking hm dehelling S tijdens het ontladen (Fig. 2).

Fig. 2. Kracht-indrukking curve is gekarakteriseerd door C, hm and S, curveopgemeten voor een sterk verstevigend materiaal

B. Hoe het spanning-rek diagram reconstrueren

Het materiaal in de directe omgeving van de indenter tijdensde indrukking blijft niet perfect vlak. Fig. 3 toont dat het gedragvan het oppervlak functie is van Ramberg-Osgood’s koudverste-vigings exponent nRO. Deze afhankelijkheid maakt het moeil-ijk om de spanning onder de indenter te bepalen. Verschei-dene onderzoekers ontwikkelden benaderingen die toelaten omde hoeveelheid pile-up en sink-in te voorspellen. Hiertoe dienenechter meerdere vereenvoudingen gemaakt te worden. Daaromis een relatie die de opgemeten kracht en indrukking puntsgewijsomzet naar een spanning en een rek, het spanning-rek diagramdus, gelimiteerd in nauwkeurigheid.

Fig. 3. Het contactoppervlak tussen het indruklichaam en het materiaal hangtaf van de verstevigingsfactor (Hier benaderd vmet een Ramberg-Osgood’swet)

Desondanks is het mogelijk om, gebaseerd op curve fittingmethodes, de indrukkingsparameters S,C en hm te schrijven alsfunctie van een constitutieve wet die het spanning-rek diagrambenaderd. Ookal dient het materiaalgedrag steeds door een wetbenaderd te worden is dit de meest toegepaste methode.

C. Dimensieloze relaties tussen de kracht-indrukking curveen het spanning-rek diagram voor staal (Ramberg-Osgoodmodel)

Omdat voor staal Youngs modulus gekend is dienen alleende Ramberg-Osgood paameters (verstevigings exponent nRO

en vloeigrens σ0) bepaald te worden. Twee dimensieloze re-laties voldoen om het probleem volledig te karakteriseren. Inde eerste dimensieloze relatie (Fig. 4) is C/σr functie vanE∗/σr. σr is de spanning die overeenkomt met een plastischware rek van 0.033. Voor deze bepaalde rek wordt Fig. 4 on-afhankelijk van nRO. E∗ is de gereduceerde Young’s modu-lus en hangt af van Young’s modulus en Poisson coefficient vanzowel het materiaal als de indenter. In de tweede dimensielozerelatie (Fig. 5) is S/(E∗hm) functie van E∗/σr. Uit dezerelatie kan nRO bepaald worden. Beide dimensieloze relatieswerden succesvol getest voor pijpleiding staal met σ0 tussen400MPa en 700MPa en nRO tussen 5 en 25.Beide functies wer-den gevalideerd voor 54 verschillende gevallen uit de literatuuren heel goeie overeenkomst werd bereikt. Het tweede verband(Fig. 5, [1]) dient wel lichtjes bijgesteld te worden voor dezelage verstevingings staalsoorten.

Fig. 4. C enE∗ zijn bekend en dus kan σ0.033 gevonden worden, gebruikmak-end van [1]

Fig. 5. S, E∗ en hm zijn bekend en dus kan nRO gevonden worden, gebruik-makend van [1]

III. INVERS MODELEREN

A. Eindig elementen model

In de overgrote meerderheid van de aanwezige literatuurwordt IIT axisymmetrisch gemodeleerd. Dit haalt de rekentijd

naar beneden en vereenvoudigd de analyse maar heeft als keerz-ijde dat in het geval de indenter kracht gestuurd is een mini-male afrondingsstraal van 10 µm voor de tip vereist is. Voorverplaatsingsgestuurde indentatie dient deze voorwaarde niet inrekening te worden gebracht. Een parametrisch Python script isopgesteld dat axisymmetrisch simulaties aanmaakt en uitvoert,hierbij gebruik makend van Abaqus. Voor de beoordeling vande gegenereerde output werd gebruik gemaakt van C, hm, S ende rekentijd. Deze vier parameters werden automatisch uit degegenereerde kracht-indrukking kromme gehaald door middelvan een Matlab programma. Dit programma werd gevalideerdvoor dezelfde 54 gevallen als bovenstaand. Gebaseerd op de4 eerder vermelde parameters werd gevonden dat lineaire ele-menten met afmetingen van 1µm in de buurt van de indenter tipnodig waren om convergentie te bereiken. Voor krachten tot 40Ndient de hoogte en breedte van het proefstuk minstens 4mm tebedragen. Het parametrisch script werd gevalideerd door exper-imentele testen geraporteerd in de literatuur alsook door eigenexperimentele proeven, C was was altijd lichtjes overschat.

B. Parameterstudie

Eerst werd de invloed van de wrijving onderzocht, deze hadeen grote invloed op S In tegenstelling tot de literatuur werdvastgeteld dat meer pile-up optrad bij hogere wrijvings coeffi-cienten. Ten tweede diende de afrondingstraal van de tip vande indenter zorgvuldig opgemeten te worden. Tenlaatste werd,net als in de literatuur, voor hoge nRO pile-up waargenomen envoor lage nRO sink-in.

IV. CONCLUSIE

geınstrumenteerde indentatie laat toe om het spanning-rekdiagram te reconstrueren, gebruik makend van de opgemetenkrachten en indrukking. Uit een uitgebreide literatuurstudie vol-gde dat het moeilijk was om opgemeten kracht en indrukking omte zetten naar een spanning en rek. Het is beter was om de con-stitutieve wet die het spanning-rekdiagram benaderd te bepalendan de volledige curve. Twee eenvoudige dimensieloze relatieswerden naar voren geschoven, deze maken het mogelijk om dede materiaaleigenschappen van staal (σ0 and nRO) te bepalen.Een van beide is echter niet in volledige overeenstemmig met re-sultaten die verkregen door eigen eindige elementen simulatiesen dient verder onderzocht te worden.

Een parametrisch eindig elementen dat inverse modeleringtoelaat werd ook opgesteld. Goede overeenkomst met exper-imentele resultaten werd verkegen maar kromming tijdens in-drukking werd steeds overschat. Daarom dient het model verdergeoptimaliseerd te worden en dient meer experimentele validatieuitgevoerd te worden.

DANKWOORD

De auteur is zeer dankbaar voor alle hulp en advies van zijnpromotor Stijn Hertele en ook voor de steun van Timothy Galle,Prof. Didier Chicot en Xavier Decoopman.

REFERENCES

[1] M. Dao, N. Chollacoop, K.J. Van Vliet, T.A. Venkatesh, and S. Suresh,“Computational modeling of the forward and reverse problems in instru-mented sharp indentation,” Acta Materialia, vol. 49, no. 19, pp. 3899 –3918, 2001.

Contents

Symbol list 1

1 Introduction 21.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Structure of the master dissertation . . . . . . . . . . . . . . . . . . . . . . . . 3

I Literature review 4

2 Structural analysis of indentation measurements 52.1 General concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Force-depth curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Constitutive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Spherical Indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Pile-up and sink-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Contact radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Stress and strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Approximation for Young’s modulus . . . . . . . . . . . . . . . . . . . . 102.2.5 Mystical materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Sharp Indenter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Geometrical similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Pile-up and sink-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 Force-depth curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.4 Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Dual Indenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Most appropriate second indenter . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Methods to obtain stress-strain behaviour by means of IIT 193.1 Spherical indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Method 1: Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Method 2: Approximation of the penetration curve . . . . . . . . . . . . 213.1.3 Method 3: Dimensionless analysis based on finite element analysis . . . 22

3.2 Sharp Indenter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Method 4: 0.29 plastic strain . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Method 5: Dimensionless correlations between the force-depth and the

stress-strain curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Dual Indenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Method 6: Representative strain . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Method 7: Work ratio and loading curvature . . . . . . . . . . . . . . . 30

i

ii

3.3.3 Method 8: Purely elastic and elastic perfectly plastic component . . . . 323.3.4 Method 9: Le’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Considerations for IIT testing 414.1 Micro or nano-indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Sharp or spherical indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Dual or single indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

II Parametric FEA-model 43

5 Computational model 445.1 Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1.1 Parameter file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1.2 Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.1.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1.4 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1.5 Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.6 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.7 Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.8 History output request . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.9 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.10 Interaction Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.11 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.12 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Matlab model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Optimization and validation of the developed model 546.1 Element and mesh density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Force on the indenter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.3 Mesh size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4 Validation based on literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.5 Validation based on experimental tests . . . . . . . . . . . . . . . . . . . . . . . 626.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 Parametric study 667.1 Influence of friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.2 Influence of tip bluntness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.3 Mystical materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.4 Applicability of Method 9 for pipeline steels . . . . . . . . . . . . . . . . . . . . 727.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8 Conclusion and future work 74

Appendices 76

A SCAD-paper 77

List of Figures 87

iii

List of Tables 90

Symbol list

Ac Contact areaac Contact radiusC Loading curvature or Kick’s constantd Diameter indenterE Young’s modulus specimenE∗ Reduced Young’s modulus, takes elasticity indenter into accountEi Young’s modulus indenterεr Representative indentation strainεy The strain of the material when it starts to yieldγ Angle between the axis of the indenter and the contact line taken at contact depthh Indentation depthhc Contact heighthm Maximum indentation depthhr Residual indentationµ Friction coefficient between material and indenterν Poisson modulus specimennH Hollomon strain hardening coefficientνi Poisson modulus indenternL Ludwik strain hardening coefficientnRO Ramberg Osgood strain hardening coefficientP Indentation forcep Perpendicular pressure on the indenterPm Maximum indentation forcepm Meyer’s hardnessR Indenter radiusr Radius of the circular intersection of the indenter and a plane parallel to the surfaceS Unloading stiffnessσ Stress in the case of a tensile testσ0 Yield strengthσUTS Tensile strengthθ Half cone angleuz Displacement under the indenterWt Total workWp Plastic workWe Elastic work

1

Chapter 1

Introduction

1.1 Context

Knowledge of the material properties (stress-strain behavior) is essential for the design of newand the qualification of existing welded structures. Traditionally, material behavior is docu-mented by a stress-strain curve. In certain circumstances, when the local material propertiesneed to be tested or when the structure is in operation and destructive testing is not an optionother solutions need to be found. An instrumented indentation test is an interesting alternativefor these certain cases.

In an instrumented indentation test a hard object is pressed into the material under inves-tigation. The way the material deforms is measured. By instrumenting this kind of test (forceas a function of imprint) not only the hardness is found, but it is also possible to reconstructthe stress-strain properties.

1.2 Problem

There are many different methods available to reconstruct the stress-strain curve out of aninstrumented hardness test. However yet, none of them is outstanding and widely applied.Some authors make use of a spherical indenter, others use a Vickers indenter and other evena flat punch.

All methods are based on the same theory but cohesion lacks.Many different methods available can be used for a wide range of materials, from ceramics

over copper to steel. There’s no method available which is designed specifically for steel. Al-though instrumented indentation is very interesting for weld investigation and more particularwelds made out of steel.

Many methods are very complex and work for simulations but have not been experimentallyvalidated. Because of this and the lack of one outstanding method, instrumented indentationtesting is not yet used by industry as a substitute for classic tensile tests.

A hardness test can be used as a replacement for a tensile test but also the opposite istrue. If a hardness test is executed is possible to find by reverse modeling the stress-strainparameters needed to generate the data measured by the hardness test.

There is need for a thorough literature review which lists all the different methods availableand a qualifies them. This study should enable to get insights into the problem.

A simple and well explained framework is needed. This framework should be specific forsteel welds which are the main scope of my study. It should be validated experimentally andeasy to apply, back to the basic principles of a hardness test.

A parametric model needs to be developed which allows to reconstruct the stress-straincurve by reverse modeling.

2

3

1.3 Structure of the master dissertation

This dissertation consists of two parts. Literature review is based on the literature availableand the Parametric FEA discusses a parametric model. The first part is divided into threedifferent sections. Structural analysis of indentation measurements (Chapter 2) gives a generalframework which is needed to understand all the different methods to reconstruct the stress-strain curve. A distinction is made between spherical indenters and sharp indenters. A thirdpossibility, the combination of two sharp indenters, called dual indenters is discussed as well.With this framework it is possible to understand the third chapter, Methods to obtain stress-strain behavior by means of instrumented indentation (Chapter 3) where 9 different methodsavailable are discussed with the same distinction as in the second chapter. Considerationsfor IIT testing (Chapter 4) is an overview and gives a proposal for a simple framework toreconstruct the stress-strain curve.

As regards part two, Computational model (Chapter 5) deals with how to use the finiteelement model. Of course the computational model needs to be validated. This is doneby experimental results described in literature as well as by own experimental tests in thechapter: ”Optimization and validation of the developed moded (Chapter 6)”. Here, differentelements and mesh size effects are also discussed. Parametric study (Chapter 7) investigatesthe influence of friction and tip bluntness. Several links with the literature are made as well.Conclusions are drawn in the final chapter.

Part I

Literature review

4

Chapter 2

Structural analysis of indentationmeasurements

2.1 General concepts

2.1.1 Force-depth curve

All instrumented indentation methods use the force-depth or penetration curve in order tocharacterize the material (Fig. 2.1). The loading continues until the maximum indentationdepth hm is reached. The relation between the force and imprint during loading is very oftendescribed by Kick’s law Eq. (2.1) although several alternatives exist (see section 3.1.2).

P = Ch2 (2.1)

The total deformation consists of an elastic and a plastic part. When the indenter isremoved and the material bounces back, the elastic deformation is recovered. If the indenter isfully removed, thus when the material is completely unloaded, the resulting indentation depthis hr. The penetration curve during unloading can be approximated by Eq. (2.2) with B, hrand m, the curve fitting parameters.

P = B (h− hr)m (2.2)

By differentiating Eq. (2.2) and evaluating at hm the unloading stiffness S is obtained.

Figure 2.1: Penetration curve

5

6

2.1.2 Constitutive models

Generally the indentation consists of three different stages: elastic, elastic-plastic en fullyplastic stages. The stress in the elastic area is always prescribed by the linear relationship Eq.(2.3).

σ = Eε, σ ≤ σ0 (2.3)

When the plastic strain develops after exceeding the yield stress (εp = 0 for σ ≤ σ0), theLudwik-equation may be used (Eq. (2.4)).

σ = σy + kεnL (2.4)

Because the elastic and elastic/plastic deformation stages in steels generally occur at verylow indentation loads only the plastic deformation stage is often investigated (Lee et al. [1],Nayebi et al. [2] and Kucharski et al. [3]). This is called the rigid-plastic material model. Theelastic strain is set equal to zero and Ludwik’s law is often reduced to the Hollomon powerlaw equation (Eq. (2.5)).

σ = kεnH (2.5)

Sometimes the Ramberg-Osgood equation Eq. (2.6) is used (Kucharski et al. [3]). It’sparticularly suited in vicinity of the yield point and again includes the elastic regime.

ε =σ

E+K

( σE

)nRO

(2.6)

In this master dissertation only steel is considered and thus a fixed Young’s elasticitymodulus equal to 210GPa is set forward. In order to reduce the number of independentparameters the reduced elasticity modulus, Eq. 3.9, is used which takes into account thematerial properties of both the indenter and the material.

E∗ =

((1− ν2

)E

+

(1− ν2i

)Ei

)−1

(2.7)

Ei and νi represent Young’s modulus and Poisson coefficient of the indenter while E andν represent the Young’s modulus and Poisson coefficient of the substrate.

2.2 Spherical Indentation

The material can be characterized by several different indenter configurations, the fist one tobe discussed is a Brinell indenter with radius R. A stress-strain curve is characterized by astress and a strain while a force-depth curve is characterized by a force and imprint depth. Ifone wants to link both curves a represented strain and stress need to be defined as a functionof the penetration force P and imprint depth h.

2.2.1 Pile-up and sink-in

One of the main issues in the case of spherical indentation is how to characterize the repre-sentative strain, because there is no direct link between the depth of imprint and the contactradius ac. As a matter of fact it depends on whether pile-up or sink-in occurs next to thecontact area. For instance, in Fig. 2.2 the material surrounding the indenter is piling up. Thiscauses the contact depth hc to become larger than h. γ is the angle between the z-axis and theradius of the indenter taken at contact depth, uz is the displacement under the indenter and

7

r is the radius of the circular intersection of the indenter and a plane parallel to the surfaceat depth uz. Alternatively if hc is less than h sink-in occurs.Whether pile-up or sink-in occurs is governed by the strain hardening coefficient (Ma et al.[4]) (Fig. 2.3). Low strain hardening materials, such as high strength steels, have a tendencytowards pile-up.

Figure 2.2: Pile-up influences the contactradius (Jeon et al. [5])

Figure 2.3: Pile-up and sink-in depends onnH (Ma et al. [4])

Kim et al. [6] investigated this relation in more detail. They noticed an almost linearrelationship between hpile/hc (Fig. 2.4) and the strain hardening exponent nH at fixed ratioof indentation depth to indenter radius hm/R (Fig. 2.4) and a quadratic relationship betweenhpile/hc and hm/R for different nH(Fig. 2.5). Out of both numerical observations Eq. (2.8)was proposed and experimentally validated.

hpilehc

= 0.131(1− 3.423n+ 0.079n2

)(1 + 6.258

hmR− 8.072

(hmR

)2)

(2.8)

2.2.2 Contact radius

Hertz found Eq. (2.9) to predict the contact radius as a function of h [7]. However it is notused any due to a lack of accuracy. This accuracy is key to the determination of representativestress and strain.

Figure 2.4: Linear Relationship betweenhpile/hc and nH (Kim et al. [6])

Figure 2.5: Quadratic relationship betweenhpile/hc and hm/R (Kim et al. [6])

8

ac(h) =√Rh (2.9)

Hay and Wolff (Collin et al. [8]) introduced a method to estimate the contact radiusbased on the Hertzian theory. The initial unloading stiffness S (Fig. 2.1) should be calculatedfirst. S can for example be calculated by deriving an approximation (curve-fitting) for theexperimentally measured unloading curve.

S(h) =dP

dh=

2γEac1− ν2

(2.10)

γ = 1 +2ac3πR

(EiEs

(1− 2ν)(1 + ν)EiE (1− ν2) + (1− ν2i )

)(2.11)

ac is the contact radius and R is the radius of the indenter. When Eqs. (2.10) and (2.11)are solved for the contact radius ac, Eq. (2.12) is obtained.

ac(h) =3πR(

√∆(h)− 1)

4B(2.12)

with:

∆(h) = 1 +4B(1− ν2s )S(h)

3πRE(2.13)

B =

(EiE (1− 2ν)(1 + ν)

EiE (1− ν2) + (1− ν2i )

)(2.14)

The stress during loading is approximated by a discrete number of representative stresses.To calculate the representative stress the contact radius is needed. For each representativestress the material needs to be unloaded to obtain the unloading stiffness S as can be seenfrom Eq. (2.12). This method is limited in application due to the a/R value, which shouldbe less than 0.25( corresponding with a representative strain less than 0.05). As will beproven in section 2.2.5 a certain indentation depth is needed in order to distinguish mysticalmaterials which limits the feasibility of Eq. (2.12). Collin [9] found by numerical investigation acorrection factor η defined in Eq (2.15) that allowed his earlier defined method to be expandedfor higher impressions. Taking the correction factor η into account the earlier defined contactradius is transformed into anew according to Eq. (2.15).

η(h, P ) = −0.0194− 0.158h

Rln(

h

R) + 0.000194 ln2(

P

E∗R2) (2.15)

anew = a− η(h, P )R (2.16)

Lee et al. [1], Ahn et al. [10] and Jang et al. [11] use the contact height, defined in Eq.(2.17), instead of the contact radius. This relationship is only valid if sink-in occurs.

hc = hm − 0.75(hm − hi) (2.17)

2.2.3 Stress and strain

The strain under the indenter is not constant because the displacement varies along the depthaxis under the indenter. This variation becomes more pronounced with increasing penetrationdepth. Only a small portion of the material reaches the extreme strain levels (Beghini et al.[12]).

Tabor [7] experimentally derived a sine-function to obtain a representative strain value,with K1 generally taken as 0.2. γ is represented in Fig. 2.2.

9

Figure 2.6: Tangent function for the representative strain provides the best approximationfor higher nH (Jeon et al. [5])

ε = K1

(acR

)= K1 sin(γ) (2.18)

Nowadays more commonly used is the tangent function (Kim et al. [13], Jeon et al. [5]).Jeon et al. [5] proved, using a physical criterion, that a representative strain using the tangentfunction (Eq. (2.19)) best describes the material behaviour.

ε = α tan(γ) (2.19)

Of course the coefficients of the sine and the tangent-function also play an importantrole, but Jeon et al. proved that the accuracy of the predicted nH only depends on thefunction, sine or tangent, and not on the coefficient. As you can see in Fig. 2.6 the tangentfunction best described the material the best in the whole strain hardening range. The tangentfunction remains in agreement with tensile stress-strain results with increasing strain hardeningexponent nH unlike the sine function which causes a deviation.Kim et al. [6] were able to explain why the tangent function is more appropriate. At largedepth the true strain defined by the tangent-function increases more rapidly compared to thesine-function but the definition for the true stress (Eq. (2.22)) is the same for both methods.Generally materials show a slow increase in true stress in the high strain range and the tangentindeed gives the lowest slope for the stress-strain curve in the high strain area.

The tangent-function can also be analytically validated. The displacement under the in-denter is given by Eq. (2.20). With uz and r defined in Fig. 2.2.

uz = h− (R−√R2 − r2) (2.20)

The tangent-function is obtained by differentiating the displacement in the depth direction,Eq. (2.21). By evaluating r at ac Eq. (2.19) is obtained.

ε =

(α√

1− (r/R)2

)( rR

)= α tan(γ) (2.21)

Ahn et al. [10] and Jang et al. [11] noted that the best agreement between the experimentaltensile and indentation data is obtained for α equal to 0.1. More recently, Jeon et al. [5])found by numerical investigation that α equal to 0.14 gives better results.

The true stress is always defined in the same way. It depends on the mean pressure (orMeyer’s hardness) and can be expressed as in Eq. (2.22) (Jeon et al. [5], Kim et al. [13],Tabor [7]):

σ =

(1

ψ

)pm =

(1

ψ

)(P

πa2c

)(2.22)

10

In an indentation test the bulk material experiences a triaxial stress state whereas in atensile test the stress state is uniaxial. That’s why the plastic constraint function ψ is setconstant, namely 3.

2.2.4 Approximation for Young’s modulus

Le [14] found a near to linear relationship (Fig. 2.7 and Eq. (2.23)) involving the reducedelasticity modulus and the Kick’s constants at two different indentation depths hm/R = 0.1and hm/R = 0.3. By using a least squares fitting procedure, coefficients B1 and B2 aredetermined as functions of E/σ0. The yield strength σ0 can thus be estimated when theYoung’s modulus is known.

Figure 2.7: Linear relationship makes it possible to estimate Young’s Modulus Le [14]

E∗

C0.3= B1

(E∗

C0.1

)B2

(2.23)

2.2.5 Mystical materials

It is possible that different materials exhibit the same penetration or force-depth curve (Beghiniet al.[15]). Those materials are called mystical. A material is in fact not only characterizedby its penetration curve but also by its strain field evolution in the sub indenter region (Fig.2.8, 2.9).

Figure 2.8: Strain field AL 6082-T6 (Begh-ini et al. [15])

Figure 2.9: Strain field mystical equivalent AL6082-T6 (Beghini et al. [15])

The strain field is strongly influenced by the friction between the material and the indenter.It’s very hard to estimate the actual friction coefficient (it varies between 0.1 and 0.3 (Cao etal. [16]) and it is typically taken as 0.15 for friction between metal and diamond (Bucaille etal. [17])). The friction between the indenter and substrate is often modelled by the Coulombfriction law (Le [14]). Unlike the strain field, the penetration curve isn’t influenced by friction.

11

So two different materials could give similar penetration curves (Fig 2.10) but different strainfields (e.g. two kinds of aluminium Table 2.1, Fig. 2.8 and 2.9).

Figure 2.10: Mystical materials showing indistinguishable loading curve

Table 2.1: Mystical materials (Beghini et al. [15])

Beghini et al. [15] showed that the two different types of aluminium can be distinguishedif the indentation is deep enough, exceeding 16% of the indenter diameter. However manydeveloped equations are not valid for large penetration depths and unpredictable frictional ef-fects become more important. There’s a trade off between reducing the possibility of uniquelyidentifying mystical materials and reducing the error due to friction effects. In general, fric-tional effects increase with indentation depth and with strain hardening exponent nH (Le [14]),they decrease rapidly with σ0/E and the indenter radius. Le [14] investigated the influence offriction on the total work to elastic work ratio Wt

We(Fig. 2.1) and Kick’s constant C. The de-

viations in C due to friction vary between −4.2% and 2.5% and the work ratio varies between−4.3% and 5.6%, both for hm/R (Fig. 2.1) equal to 0.3, and the variation of indentation loadis larger than 8%.

Noteworthy, Le [14] also investigated the influence of indenter compliance by means offinite element analysis (FEA). He observed that deviations of Kick’s constant were the lowestfor soft materials like lead and that the influence of the indenter compliance on C increaseswith increasing indentation load.

2.3 Sharp Indenter

2.3.1 Geometrical similarity

For sharp indenters, defining a representative stress and strain, with as purpose to immedi-ately convert a force-depth curve into a stress-strain curve, is highly challenging due to thegeometrical similarity of this type of indentation. This means that an IIT at large depth isessentially a magnified picture of a IIT at a small depth. Thus average pressure or hardnessremains constant during the loading process. Hardness is calculated by the average contact

12

pressure, defined as the load divided by the contact area Ac. The contact area is proportionalto the square of the penetration depth. As an illustration, the tip angle of a Rockwell indenter(Fig. 2.11) is 120◦ and thus is θ equal to 60 ◦. The contact area is given by Eq. (2.24) and

Figure 2.11: Sharp indenter (Bucaille et al. [17])

thus depends only on h2

Ac = (h tan(60◦))2 = 3h2 (2.24)

The average contact pressure Eq. (2.25) is reduced to a constant (Tho et al. [18]) when Kick’slaw Eq. (2.1) is taken into account.

H = pave =P

A=Ch2

3h2=C

3(2.25)

As a consequence, methods based on a sharp indentation are confined to providing linksbetween model parameters of a force-depth curve and a stress-strain curve rather than betweentheir distinct data points.

Since the average pressure is independent of indentation depth, the representative strainhas a unique value. Dao et al. [19] found out that the stress-strain curves of materials havingthe same Kick’s constant C exhibit the same true stress for a plastic strain prescribed by Eq.(2.26).

ε = 0.105 cot θ (2.26)

This relation was than used for deriving dimensionless quantities for parametric finite elementanalysis. Very often this particular strain is chosen as representative strain. The correspondingstress is given in Eq. (2.27).

σr = RεnH =σ0εn0

(ε0 + εr)nH = σ0

(1 +

E

σ0εr

)nH

(2.27)

Although the representative strain is independent of the load it is weakly influenced by thetip radius. This relation is used in section 3.2.2.

2.3.2 Pile-up and sink-in

Notwithstanding the simplicity of Eq. (2.25) one should still bear in mind that the contactarea is influenced by pile-up (e.g. Fig. 2.12) and sink-in and thus the constant hardness,function of Eq. 2.24, is merely an approximation. This is illustrated in Fig. 2.12 for an APIX65 steel (Lee et al. [20]).

The behaviour of the underlying material can be classified into three different areas eachcharacterized by a unique characteristic strain (Fig. 2.13). The inner-zone is rigid-perfectlyplastic and is described by the classical slipline theories. It’s surrounded by an elastic-plasticzone. Giannakopoulos et al. [21] revealed by numerical studies that the radius of the hemi-spherical shape which marks plasticity is equal to Eq. (2.28).

13

Figure 2.12: Due to pile-up and sink-in hardness depends slightly on the load (Lee et al. [20])

c =

√3Pmax2πσ0

(2.28)

Figure 2.13: Material behaviour classification under the indenter (Giannakopoulos et al. [21]). Note the presence of pile-up, typical for low strain-hardening materials.

The true contact area can be found by investigating the surface with a scanning electronmicroscope or by a empirical approximation as for spherical indentation. For sharp indentersthe ratio of hr to hm (Fig. 2.1) indicates whether or not pile-up (hr/hm ≥ 0.875) or sink-in(hr/hm ≤ 0.875) occurs (Giannakopoulos et al. [21]).

2.3.3 Force-depth curve

The force-depth curve, Fig. 2.14, depends on the indenter used but generally follows thecharacteristic pattern described in Fig. 2.1. For a spherical indentation the curve depends onthe radius R of the indenter, for a sharp indenter is depends on the indenter angle used andalso on the tip radius. Indeed, a sharp indenter is never infinitely sharp and the tip radius hasan effect on the penetration curve if the indentation is less than R/40.

Besides the parameters explained in section 2.1.1 the force-depth curve Fig. 2.14 can alsobe characterized by the plastic work Wp and the elastic work We done during loading. Thetotal work Wt is the sum of both.

14

Figure 2.14: Penetration curve

Wt and We are less sensitive to scatter because they are calculated based on all measureddata points rather than a single value. Tho et al. [18] found a linear relationship betweenhr/hm and WP /WT Eq. (2.15).

Wp

Wt= 1.2973

hrhm− 0.298 (2.29)

Because of this linear relation Wp/Wt is often used instead of hr/hm for sharp indentation.

Figure 2.15: Linear Relationship Wp/Wt and hc/hm (Tho et al. [18])

This relation is only valid for Wp/Wt > 0.5 (Fig. 2.15) which corresponds to E∗/σ0 ≥60 and 0.0 ≤ nH ≤ 0.6. Most engineering metals and alloys fall in this range of materialproperties. For instance, a high strength steel with E equal to 210GPa and σ0 equal to800MPa has a E∗/σ0 of approximately 250. I will consider in my master dissertation mainlypipeline steels with nH mostly limited to 0.1. Thus can Eq. (2.15) be applied.

2.3.4 Residual Stresses

Instrumented indentation can, besides material properties, also provide information aboutresidual stresses present in the material. Jang et al. [22] investigated the possibility of extract-ing the residual stresses out of the penetration curve. Tensile stresses facilitate the indenterpenetration into the material causing the strain to be higher for the same amount of stress.The opposite takes place in the case of a compressive stress state (Fig. 2.16). This anomalywas further investigated by Lee [20].

15

Figure 2.16: For compressive stresses the loading curve is steeper compared to stress-free state(Kim et al. [13])

2.4 Dual Indenters

2.4.1 Most appropriate second indenter

As indicated in section 3.2.2, one force-depth curve for sharp indenters doesn’t necessarilyprovide enough information to uniquely identify the material. A solution was found by inves-tigating the behaviour of the surface neighbouring at the residual imprint [23]. However, thissolution is not interesting in the field and the method is also not yet extensively worked out.A more practical engineering solution is provided by using two indenters.

One of the two indenters is always the axial symmetric Berkovich equivalent, having ahalf cone angle of 70.3◦. The other one can be freely chosen but there are some restrictions.Too small angles should be avoided because the diamond tip can be damaged. The materialmoreover behaves differently for small angles, it’s cut rather than plastically deformed. Also,the indentation is not self-similar any more. From comparison between Figs. 2.17 and 2.18,it can be seen that friction forces (represented by the coefficient of friction ν) have a stronginfluence for small indenter angles. This is also reflected in Eq. 2.30, where F presents thenormal force, p the contact pressure on the indenter as can been seen from Fig. 2.11. Onemay notice from Eq. (2.30) that without friction p becomes the hardness. Bucaille et al. [17]and several others (eg. Tho et al. [18]) concluded that The effect of friction is negligible forindenters with half-angle θ higher than 60◦.

F = πpa2c

(1 +

µ

tan θ

)(2.30)

Bucaille et al. [17] noticed that Kick’s constant is strongly affected by the exact angle ofthe indenter. A difference of 1◦ induces a difference of 10% for C (Fig. 2.19, in which D issteel).

The main advantage of working with two indenters is the increased ability to uniquelyreconstruct the material out of the penetration curve (Chollacoop et al. [24]). Whereas forone indenter angle (θ = 70, 3◦) two materials may yield the same penetration curve, thesecurves are not the same for the second indenter (Fig. 2.20). The ability to uniquely identifythe correct material behaviour increases with an increasing difference between the angles ofthe two indenters.

In rare cases it’s still possible that two materials have the same penetration curve for thetwo different indenters. For specific dual indenters (θ1 = 70.3◦, θ1 = 80◦) and (θ1 = 70.3◦,θ1 = 63.14◦), only low strain materials (n ≤ 0.2) with σ0/E ≈ 0.01 may become mysticalmaterials. For a common dual indenter combination (Vickers or Berkovich cone equivalentθ1 = 70.3◦, Rockwell indenter θ1 = 60◦), very little realistic mystical materials with low strain

16

Figure 2.17: For a half cone angle of 42.3◦,friction strongly reduces the amount ofpile-up (Bucaille et al. [17])

Figure 2.18: The influence of friction isless for higher half cone angles, in this case70.342.3◦(Bucaille et al. [17])

Figure 2.19: θ significantly influencesKick’s constant (Dao et al. [19])

Figure 2.20: Mystical materials discovered bydual indenter combination (Chollacoop et al.[24])

hardening have been found (Le et al. [23]). Because in this master dissertation the Young’smodulus is known, the chance of discovering a mystical material is further reduced.

2.4.2 Residual Stresses

Residual stresses have already been discussed in section 2.3.4, but with the assumption thatfor the same material also a force-depth curve without residual stresses is available. In somecases this is not possible because the whole surface experiences residual stresses σres (eg. acoating or a weld). Yan et al. [25] offers a method to determine the material properties but theequi-biaxial residual stresses (Fig. 2.21) and Young’s modulus need to be known in advance.The stress and strain field introduced by instrumented indentation strongly interacts withthe residual stress-strain field. Some materials and residual stress combinations can resultin a similar response (Fig. 2.22). Yan used three different indenters (70.3◦, 63.14◦ and 60◦)to obtain the best result. For each angle he plotted and derived an equation for C/σr asa function of lnE/σr (Fig. 2.23 and Eq. (2.31). The coefficients of Eq. (2.31) depend onthe ratio σres/σ0; Hence, an initial guess for the yield strength needs to be made before therepresentative stress can be obtained. Once the representative stress and strain are known foreach angle, the strain hardening exponent can be calculated out of Eq. (2.32). With Kick’sconstant, Young’s modulus and nH known σ0 can be approximated from figures like Fig. 2.23given by Yan et al. [25]. When this is done for each indenter angle the average yield strength

17

Figure 2.21: definition of equi-biaxialresidual stress (Yan et al. [25])

Figure 2.22: Residual stresses can cause twomaterials to become mystical (Yan et al. [25])

can be calculated. Depending on the difference between the guessed and the calculated yieldstress a new initial guess for the yield stress needs to be made. The loop continues until theguessed and the calculated yield strength are identical.

Figure 2.23: Dimensionless equation in order to obtain σ0 (Yan et al. [25])

C

σr(εr)= a1

(ln

E

σr(εr)

)3

+ a2

(ln

E

σr(εr)

)2

+ a3

(ln

E

σr(εr)

)+ a4 (2.31)

σr(εr) = R

(2σr(εr)

E+ 2εr

)n(2.32)

Table 2.2: Validation of the method proposed by Yan et al. [25]

2.5 Conclusion

Just like a tensile test generates a stress-strain curve, an instrumented indentation test gen-erates a force-depth curve. Unlike for spherical indenters, the shape of force-depth curves

18

generated by sharp indenters is independent of the maximum force applied by the indenter.During the indentation the material does not remain flat; it starts to pile-up or sink-in basedon the strain hardening exponent nH . Different approximations exist to predict this materialbehavior. For spherical indenters a representative stress and strain can be defined. It is shownthat the representative strain best corresponds with tan(γ).

There is no guarantee for a unique correspondence between material properties and force-depth curve. Different materials may have the same force-depth curve. Although this non-unique relation may occur, it is not very likely. If a second indentation test is executed witha different indenter, a higher likelihood for uniqueness is obtained. The bigger the differencebetween the angles of both indenters, the more chance for uniqueness. However the halfincluded angle cannot be lower than 50◦ to diminish the influence of friction.

Chapter 3

Methods to obtain stress-strainbehaviour by means of IIT

3.1 Spherical indentation

3.1.1 Method 1: Transformation

In this method [1, 6, 8, 26] several relations are listed which enable to find for each experi-mentally measured point of the force-depth curve its corresponding place on the stress-straincurve (Fig. 3.1). The materials considered in literature are always supposed to obey a simplepower law (Hollomon) equation Eq. (2.5 or Ludwik’s equation Eq. (2.4)).

As stated before, the displacement of the indenter h is not sufficient to be used as aparameter in the representative strain because it doesn’t take pile-up or sink-in into account.Collin et al. [8] validated the modified Hertzian Eq. (2.12) numerically by the finite elementsoftware Cast3m. The average precision is greater than 2 % as can be seen from Fig. 3.2which gives the contact radius in function of the indentation depth. A second cause of erroraffecting the contact radius is the deformation of the indenter. Collin et al. [8] [26] foundbased on a numerical study a polynomial function Eq. (3.1) depending on the indenter radiusR, the contact radius ac, the load P and Young’s modulus of the indenter Ei to calculate thedeformation of the indenter δ. If the experimentally measured force-depth curve is comparedwith a numerically generated force-depth curve for C22-steel this correction should be takeninto account as can been seen in Fig. 3.3.

δ

R= B1f1 +B2f2 + C11f1f2 + C22f

22 (3.1)

Figure 3.1: Flowchart Method 1

Figure 3.2: Numerical validation of the es-timated contact radius ac (Collin et al. [8])

19

20

Figure 3.3: Experimental Ph-curve requires a correction for the indenter deformation (Collinet al. [8])

.

f1 =P

πR2

1− ν2iEi

(3.2)

f2 =P

πa21− ν2iEi

(3.3)

With an adequate expression for the main parameter describing the material, the contactradius,the representative strain can be calculated.

Lee et al. [1] validated the above mentioned method experimentally by using Eq. (2.17) fortwo kinds of steel: X20CrMo12.1V and SA-213 T23. The results are summarized in table 3.1.J-Y Kim et al. [6] validated the procedure for steel SCM415 (Fig. 3.5) and other materials byusing Eq. (2.8) for the contact depth. Their results are summarized in Table 3.1. A possiblereason for the high error in yield stress is the presence of Luders strain (Kim et al. [6]). Boththe yield stress and the tensile strength are derived from the flow curve. From the ultimatetensile strain, the ultimate tensile stress can be calculated. Assuming Hollomon constitutivelaw, the ultimate tensile strain is equal to the strain hardening coefficient. The yield stress iscalculated by the intersection of the flow curve and a line with as slope the elastic modulusand a strain offset of 0.2 % from the origin (Fig. 3.4).

Figure 3.4: Reconstruction of the stress-strain curve (Kim et al. [6])

Figure 3.5: Transformation of the Ph-curve gives good results (Kim et al. [6])

From Fig. 3.6 it can been seen that the absolute error of the yield strength is independentof the yield to tensile ratio. There’s a positive correlation between the yield to tensile ratioand the error on the tensile strength.

21

Yield strength (MPa) Tensile Strength (MPa)

Reference Material Tensile IIT Error % Tensile IIT Error %

[6] P91 569.7 558.7 -1.9 772 807.5 4.6S45C 372.9 336.4 -9.8 883.2 843.7 -4.5

SCM21 290.2 314.9 8.5 626.5 609.3 -2.7SCM415 237.4 230.1 -3.1 616.3 620.7 0.71

SUJ2 306.8 270.2 -12 907.7 862.2 -5.0SKD11 243.4 270.1 1.1 923.1 880.1 -4.7SKD61 348.9 361.8 3.7 896.5 882.2 -1.6

SKS3 366.4 330.9 -9.7 781.5 810.6 3.7API X65 451.8 522.9 16 610.8 630.3 3.2API X70 529.9 550.9 4.0 782.2 770.9 -1.4

[1] X20 534.6 566.3 5.9 859.8 837.2 -2.6T23 497.1 500.8 0.74 675.4 638.4 -5.4

Average absolute error 7.2 3.4

Table 3.1: Experimental Validation Method 1

Figure 3.6: Error as a function of σ0/σUTS (algorithm of [6, 1])

3.1.2 Method 2: Approximation of the penetration curve

Different theoretical approximations [26, 3, 12] have been developed in order to minimize thedifference between the theoretical and the experimental indentation curve. Instead of method1 the parameters describing the stress-strain properties are obtained and not the whole curve.The material properties are derived from the theoretical approximation that yields the bestsimilarity with the experimental curve (Fig. 3.7).

One of the main advantages of this procedure is that the contact radius should not becalculated and the material should not be unloaded. Collin et al. [26] already proposed anumerical approximation Eq. (3.4) for the force-depth curve. Note that this approximationdiffers from Kick’s law, Eq. (2.1).

P

ER2=

(h

R

)Aexp(−B) with A,B = f(σ0, E, n) (3.4)

Kucharski et al. [3] found a more sophisticated equation to describe the loading part ofthe force-depth curve as a function of material properties. He expressed the loading curve as

22


a weighted average for a powerlaw hardening model and a perfectly plastic response.Using Eqs (3.5) and (3.6) the theoretical penetration curve can be analytically expressed.

Eq. 3.5 was experimentally validated (Fig. 3.8), the relative error between Eq. (3.5 and theexperimental loading part of a force-depth curve did not exceed 1%.

P

ED2=

4∑k=1

Ak

(h

D

)k/2(3.5)

Ak =6∑i=1

6∑j=1

αijkσi−10 nj−1 (3.6)

Figure 3.8: Comparison between Eqs. (3.5) and numerical/experimental results (Beghini etal. [12])

Beghini et al. [12] constructed, by finite element analysis, a library of force-depth curvesbelonging to a wide range of steels (100 ≤ σy ≤ 1 000MPa and 0 ≤ n ≤ 0.5).

An optimization algorithm (Beghini et al. [12]) scans the library for the other materialproperties σ0 and nH in order to minimize χ, Eq. (3.7). The relative errors are within 4% forσ0 and 2% for nH . Beghini et al. [15, 12] proved that this method Eqs. (3.5), (3.6) and (3.7)could distinguish the two different earlier discussed mystical materials (Table 2.1).

χ(E, σ0, n) =M∑m=1

(P th(hexpm , E, σ0, n)− P expm

)2(3.7)

3.1.3 Method 3: Dimensionless analysis based on finite element analysis

The difference between this method [14, 27, 28, 29, 16], (Fig. 3.9) and the method 2 discussedin section 3.1.2 which also made use of reverse analysis is the way the material propertiesare derived. Beghini et al. compared force-depth curves generated by finite element analysis

23

(FEA) while in this method (Le [14] and Ogasawara et al. [27]) the properties are estimatedby FEA-relations between the parameters of force-depth curve and the stress-strain curve. Toderive those relations, dimensionless analysis comes on the scene. Because the indentationresponse is calculated by finite element analysis, no challenging indentation parameters likecontact radius should be approximated. This is an advantage of method 3 to method 1. Collinet al. [26] came to the same conclusion: ”However, for more complex behaviour laws inverseapproach will be the more adequate method”.


The indenter force can be written as (Le [14] and Dao et al. [19]):

P = P (h,E, ν, Ei, νi, σ0, n,R) (3.8)

The number of independent parameters is decreased by using the reduced elasticity modulusEq. (3.9), which takes into account the material properties of both the indenter and thematerial. Eq. (3.8) reduces to Eq. (3.10).

E∗ =

((1− ν2

)E

+

(1− ν2i

)Ei

)−1

(3.9)

P = P (h,E∗, σ0, n,R) (3.10)

The∏

-theorem in dimensional analysis gives:

P = E∗h2f1(σ0E∗ , n,

h

R) (3.11)

And finally by using Kick’s law Eq. (2.1) the first dimensionless function is derived:

P

E∗h2= f1(

σ0E∗ , n,

h

R) =

C

E∗ (3.12)

It’s more precise to use with work instead of the force. This takes all calculated points of thepenetration curve into account resulting in a less significant resulting error due to scatter. Byintegrating Eq. (3.11) the total indentation work can be calculated.

Wt =

∫ hm

0Pdh = E∗h2mf2(

σ0E∗ , n,

hmR

) (3.13)

24

The unloading force Pu can also be written as a dimensionless relationship (Le et al. [14]).

Pu = Pu(h, hm, E∗, σ0, n,R) (3.14)

Dimensional analysis gives:

Pu = E∗h2f3(h

hm,σ0E∗ , n,

hmR

) (3.15)

A fourth dimensionless function can be derived by setting Eq. (3.15) equal to zero at the endof the unloading process, when h=hr.

hrhm

= f4(σ0E∗ , n,

hmR

) (3.16)

The elastic indentation work can be calculated as:

We =

∫ hm

hr

Pudh = E∗h2mf5(E∗

σ0, n,

hmhr

) (3.17)

Combining Eq. (3.13) and Eq. (3.17) gives

Wt

We= f6(

E∗

σ0, n,

hmR

) (3.18)

Assuming E∗ is known Eq. (3.18) still depends on two unknown material quantities: σ0and n. The unknown quantities are reduced further (Le [14]) by calculating the indentationparameters for h = 0.1R. Eqs (3.19) and (3.20) are obtained. Equation (3.19) is visualised anda linear relationship between Wt/We and E∗ is noticed(Figs. 3.10 and 3.11), which significantlyreduces the difficulty of solving Eqs. (3.19) and (3.20). To apply this method unloading athR = 0.1 is needed.

Wt

We= f7(

E∗

C, n,

hmR

) (3.19)

Wt

We= f8(

(Wt

We

)hR=0.1

, n,hmR

) (3.20)

Additional information like the hardness, Eq. (3.21) can also be obtained by applying thismethod.

E∗√CH

= f9(E∗

C, n,

hmR,We

Wt) (3.21)

By means of least squares fitting procedure (Le [14]),the coefficients of Eqs. (3.19) and(3.20) are formulated as cubic polynomial functions of n and parabolic polynomial functionsof hm/R. The exact equations can be found in [14]. The strain hardening exponent and nHar found by solving Eqs. (3.19) and (3.20), σ0 from Eq. (2.23) and the Vickers hardness Hfrom Eq. (3.21).

Ogasawara et al. [27] developed a different method. Estimating the material propertieswas done by solving three dimensionless equations. Eq. (3.22), Fig. 3.12 at h/R = 0.1 andEqs. (3.23) and Eq. (3.24), Fig3.13 at h/R = 0.3.

WthmR

=0.1

h3σ(1 + 0.94n)=

(1

1.69 E∗σ(1+0.94n)

+1

125.7

)−1

(3.22)

WthmR

=0.1

h3σ(1 + 1.34n)=

(1

0.97 E∗σ(1+1.34n)

+1

37.2

)−1

(3.23)

25

Figure 3.10: f7 fixed n (Le [14]) Figure 3.11: f7 fixed hm/R (Le [14])

Figure 3.12: Dimensionless relation (Oga-sawara et al. [27])

Figure 3.13: Dimensionless relation (Oga-sawara et al. [27])

S2(1 + 1.34n)

E∗(h− h(P=0.1Pmax))= −2.2 ∗ 105

(E∗

σ(1 + 1.34n)

)+ 0.518

(E∗

σ(1 + 1.34n)

)+ 1.33 (3.24)

The representative stress in each of the given dimensionless relations can be found byevaluating Eqs. (3.25) and (3.26) at the two different indentation depths. Once the stress-strain parameters are known one can reconstruct the stress-strain curve by applying a Hollomonlaw.

ε = 0.033h

R+ 0.00616 (3.25)

σ = k(ε+σ

E)n (3.26)

Cao et al. [16] derived dimensionless equations that allow to reconstruct the representativestress and strain. Just like Ogasawara the respresentative stress and strain need to be evaluatedat two different indentation depths, Eqs. (3.27) and (3.28).

εr = f(We

Wt) (3.27)

σr = f(We

Wt, C) (3.28)

26

By evaluating Hollomon’s law at the two different depths the unknown material propertiescan be obtained. If Young’s modulus is known in advance nH can be calculated from Eq.(3.29) and σ0 from Eq. (3.30).

nH =

ln(σ h

R=0.03

σ hR

=0.15

)

ε2ε1

ε1 = ε hR=0.15 +

σ hR=0.15

E, ε2 = ε h

R=0.03 +

σ hR=0.03

E(3.29)

σ0 =

(σ hR=0.03

(Eε2)n

) 11−n

(3.30)

Zhao et al. [28] wrote Eq. (3.12) as a function of the representative stress and evaluatedthe result at two different indentation depths ( hR = 0.13, 03). Together with Eq. (3.31) thematerial properties can be obtained.

S

hE∗ = f10

(E∗

σr, n

)(3.31)

Jiang et al. [29] could link the amount of pile-up or sink-in as a function of the yield strainεy and nH , generated by curve fitting. This function allowed to derive an analytic expressionfor the total work as a function of E, εy and n. They also developed and expression for theratio of the force on the indenter at different indentation depths. With Young’s modulus apriori known and Eqs. (3.32) and (3.33) the stress-curve can be reconstructed.

Wt = f11(E, εy, nH) (3.32)

PacR=0.7

PacR=0.6

= f12(εy, n) (3.33)

An overview of the material characterization by parametric FEA by Le [14], Zhao et al.[28], Jiang et al. [29], Cao et al. [16] and Ogasawara et al. [27] is given in Table 3.2. Le [14].gives the best approximation for nH while Cao et al. [16] gives better approximation for σ0.Although it is dangerous to judge the different methods by Table 3.2, since all studies useddifferent materials to validate and optimize their method. It would be better if one validatedall the different methods for one kind of reference material.

Table 3.2: Results Method Le [14]

27

3.2 Sharp Indenter

3.2.1 Method 4: 0.29 plastic strain

This method (Giannakopoulos et al. [21], Fig. 3.14) has been established by finite elementsimulations along with experiments. It can be applied for Vickers, Rockwell and Berkovichtype indenters indenters. In Eq. (3.34) σ0.29 is the stress corresponding with a strain equal to0.29 and C1 and C2 are constants that depend on the kind of indenter (Vickers, Rockwell orBerkovich). If pav/σ0 exceeds the limits the response is either elastic (lower limit) or elastic-perfectly plastic (upper limit). Note that this method is only applicable for very ductilematerials.


C =P

h2= C1σ0.29

(1 +

σ0σ0.29

)(C2 + ln(

E∗

σ0)

), 0.5 ≤ pm

σ0≤ 3.0 (3.34)

Of course pile-up and sink-in should be taken into account but a relative simple equation(simulations) is used to calculate the true contact area Eq. (3.35), pm/E

∗ can be calculatedby Eq. (3.36). In order to solve Eq. (3.36) hr/hm can be obtained from Eq. (3.37).

Amaxh2m

= C1 + C2(pm) + C3(pm)2 + C4(pm)3 + C5(pm)4 + +C6(pm)5 (3.35)

hrhm

=Wp

Wt= 1− C1

PavE∗ (3.36)

σ0.29 − σ00.29E∗ = 1− 0.142

hrhm− 0.975

(hrhm

)2

(3.37)

The reduced Young’s modulus Eq. (3.9) can be calculated out of Eq. (3.38). For steelyoung’s modulus is already known in advance and it can be used to check the accuracy of theformula.

E∗ =

√π

2√Amax

(dP

dh

)(3.38)

Finally the strain hardening exponent is approximated as followed:

nH = lnσ0.29 − σ0

5(3.39)

Recognizing the simplicity of this method, it will not be able to distinguish mystical ma-terials (Venkatesh et al. [30]).

28

Figure 3.15: f1 becomes independent of n for εR equal to 0.033 (Dao et al. [19])

3.2.2 Method 5: Dimensionless correlations between the force-depth andthe stress-strain curve

Tho et al. [18] derived in the same way as for spherical indentation three dimensionless Eqs(3.40), (3.41) and (3.42) to calculate the three independent properties E∗, σ0 and n. C/σ0 isused instead of C/E. The equations are independent of the indenter angle (radius) θ. Theindenter angle is fixed as 70.3◦, this is the axisymmetric equivalent of Berkovich indenter.

C

σ0= f1

(E∗

σ0, n

)(3.40)

Wp

Wt= f2

(E∗

σ0, n

)(3.41)

dP/dh |hmChm

σ0E∗ = f3(

E∗

σ0, n) (3.42)

Tho et al. [18] also concluded the non-uniqueness of the solution. He plotted for a certainforce-depth curve Eqs. (3.41) and (3.42). (Fig. 3.17). There are multiple intersectionsindicating several solutions and thus mystical materials. Two nuances need to be made. Allsolutions have a different elasticity modulus indicating that the method will converge in casethe elasticity modulus is known. Secondly, the indentation in not that deep (4.5 µm) and inorder to provide a unique solution deep indentation is advised, see section 2.2.5.

Dao et al. [19] continued in the same way as Tho et al. [18] did, but instead of usingσ0 to make C dimensionless he used a representative stress σr. This representative stressσr corresponds with a randomly chosen representative strain εr on the stress-strain curve.Although, for a specific representative strain εr, depending on the indenter angle, f1 (Eq.(3.40)) becomes independent of the strain hardening factor. This can be seen from Fig. 3.15for a Berkovich or indenter with εr equal to 0.033. Dao et al. [19] also provided an expressionfor f3 (Eq. (3.42)) depending on the representative stress, Fig. 3.16. If Young’s modulus isknown you can determine sigma0 and nH out of Figs. 3.15 and 3.16.

Bucaille et al. [17] approximated this representative strain given a certain half cone angleθ by Eq. (3.43).

εr = 0.105 cot θ (3.43)

Dao et al. [19] calculated the true contact area from Eq. (3.44) and the reduced elasticitymodulus from Eq. (3.38). Next, both were used to calculate the representative stress corre-sponding to that particular specific strain. Therefore he used function (3.45) which he noticedto be independent of the strain hardening. Last he calculated the strain hardening exponentwith Eq. (2.27).

29

Figure 3.16: f3 as a function of the representative stress (Dao et al. [19])

Figure 3.17: Solution of f2 and f3 (Tho et al. [18])

1

E∗

(P

Ac

)= f4

(hrhm

)(3.44)

C = σrf5

(E∗

σr

)(3.45)

Le and al. [23] found that this method doesn’t give unique results and investigated this non-uniqueness in more detail and found a linear relationship between two dimensionless functions:S/(Chm) and Wt/Wp (We = Wt−Wp) (See Fig. 3.18). With only two dimensionless functionsresulting it’s impossible to determine the material properties (E∗, σ0 and n) uniquely or todistinguish mystical materials. Ma et al. [4] addressed this problem by importing anotherindependent quantity Eq. (3.46), namely the ratio of the additional residual area and the areaof the indentation profile (relative to the original surface) at hm. The additional residual area∆A is positive in the case of pile-up ∆A and negative for sink-in. (Fig. 3.19 ) It’s very difficultif not impossible to compute ∆A. Hence, another dimensionless function has to be found ormultiple indenters are needed.

∆A

A= f6(

Y

E, n) (3.46)

30

Figure 3.18: Linear Relationship (Le [23])

Figure 3.19: Additional Residual Area (Ma et al. [4])

3.3 Dual Indenters

3.3.1 Method 6: Representative strain

Bucaille et al. [17] extended the method of Dao et al. [19] towards multiple indenters. Thecontact area, elasticity modulus are calculated in the same way as Eqs (3.44) and (3.38). Iftwo different types of indenters are used, this will result in two different representative strains,one for each indenter Eq. (2.26). As a consequence two different representative stresses canbe calculated Eq. (3.45). Those two stresses immediately give an indication of the strainhardening exponent. The yield strength and strain hardening exponent are calculated withEq. (2.27) using the two different stress-strain couples, one for each indentation (Fig. 3.20).

Chollacoop et al. [24] went on with Bucaille’s but changed the way how to calculate theyield strength if no strain hardening would be observed. The method was experimentallyvalidated for two kinds of aluminium using indenters with semiangles equal to 50◦ and 70.3◦.A maximum error of 30% on the yield strength was obtained. For a single indenter the errorwas 60% for this kind of material. The problem for Bucaille’s multi-set P-h curves approachis that for a half included indenter angle 50◦ the friction between indenter and the specimencan have a significant effect on the P-h curve (Luo et al. [31]).

3.3.2 Method 7: Work ratio and loading curvature

In his algorithm, Swaddiwudhipong [32] used the axisymmetric Berkovich equivalent typeindenter and a Rockwell indenter, 60◦ half tip angle (Fig. 3.21). These indenters are sufficientlydifferent to distinguish mystical materials, and both angles are big enough to be able to neglectthe effect of friction. This method doesn’t adopt the representative strain but requires theunloading curvature generated by both indenters and the ratio of the plastic work to the totalwork Wp/Wt.

E∗/σ0 and n are calculated from Eqs. (3.49), (3.50) and (3.51), σ0 from Eq. (3.47) or

31

Figure 3.20: Flowchart Method 6, f1 = Eq. (3.40)


32

(3.48). Once σ0 is known, one can calculate E∗ and thus E.

C

σ0|60◦= f1,60◦

(E∗

σ0, n

)(3.47)

C

σ0|70.3◦= f1,70.3◦

(E∗

σ0, n

)(3.48)

Wp

Wt|60◦= f2,60◦

(E∗

σ0, n

)(3.49)

Wp

Wt|70.3◦= f2,70.3◦

(E∗

σ0, n

)(3.50)

C60◦

C70.3◦=

f1,60◦(E∗σ0, n)

f1,70.3◦(E∗σ0, n) (3.51)

Eqs (3.49), (3.50) and (3.51) intersect at one point across the domain (Fig. 3.22) provingthat a unique set of solutions exists. This method is validated, Table 3.3.2,yielding errors upto 2.4% for σ0 and up to 100% for n. The error does not depend on the the strain hardeningcoefficient nH , Fig. 3.23.

Figure 3.22: The algorithm of Swaddiwudhipong et al. [32] yields unique solutions, (blue =Eq. (3.49), green = Eq. (3.50) and red Eq. (3.51))

3.3.3 Method 8: Purely elastic and elastic perfectly plastic component

Luo et al. [31](Fig. 3.3.3) split the elastic-plastic behaviour during loading (Fig. 3.25 and Eq.(3.52)) into a purely elastic component Eq. (3.53) and an elastic perfectly plastic componentEq. (3.54).

Pep,L = (1−Wt)Pe +WtPepp,L (3.52)

Pe(E∗, h) = Ce(E

∗)h2 = C0E∗h2 (3.53)

The elastic-plastic material response during unloading (Fig. 3.26 and Eq. (3.55)) consistsof a full contact straight line Eq. (3.56) and a purely elastic curve Eq. (3.57) WL and WUL

are function of the mechanical properties of the materials and represented in [31].

33

Figure 3.23: There is no clear relation between error and nH(algorithm of [32])


34

Young’s modulus (GPa) Yield strength (MPa)


[32] Al6061 69 68.6 -5.8 275 276.9 0.69Al7075 70.1 69.9 -0.29 500 505.8 1.2

Steel 210 212.7 1.3 500 488.2 -2.4Ion 180 180.3 0.17 300 293.7 -2.1

Zinc 9 8.7 -3.3 300 306.8 2.3


Strain hardening exponent nH

Material Tensile IIT Error %

Al6061 0.05 0.049 2Al7075 0.122 0.118 0.0327869

Steel 0.1 0.107 -7Ion 0.25 0.255 -2

Zinc 0.05 0 100

Average absolute error 23

Table 3.3: Validation Method 7, Swaddiwudhipong et al. [32]

Figure 3.25: Loading (Luo et al. [31]) Figure 3.26: Unloading (Luo et al. [31])

Pepp,L(E∗, σ0, h) = Cepp,L(E∗, σ0)h2 (3.54)

Pep,UL = (1−We)Pem +WePfc (3.55)

Pem = Pe −∆P = C0E∗h2 − (C0E

∗h2m − Pm) (3.56)

Pfc = S(h− hm) + Pm (3.57)

In order to obtain all material properties uniquely a third equation is needed, namely Eq.(3.58). Eq. (3.58) could be improved by not merely using the maximum and residual depthbut the whole unloading curve. However this is highly challenging because secondary effectsoccur during unloading. This equation is also the weakest point of the method. It doesn’tsufficiently relate the whole unloading curve to the mechanical properties (Luo et al. [31]).

hrhm

= f

(E∗

σ0, n

)(3.58)

The equations to relate E, σ0 and n to Pe, Pepp,L, Wt, We and hr/hm are calculated usingleast squares fitting method. A cost function is defined and by reverse analysis the material

35

properties are obtained. The results are represented in Table. 3.3.3. For completeness theerror as a function can of the strain hardening coefficient nH can be found in Fig. 3.27.

Young’s modulus (GPa) Yield strength (MPa)


[31] Al 70 71 1.4 500 493 -1.4Ti 110 111 0.91 600 592 -1.3Fe 180 181 0.56 400 442 10.5Ni 207 210 1.4 800 861 7.6

Steel 210 214 1.9 900 902 0.22


Strain hardening exponent nh

Material Tensile IIT Error %

Al 0.15 0.16 6.7Ti 0.1 0.11 10Fe 0.25 0.22 -12Ni 0.4 0.38 -5

Steel 0.3 0.3 0

Average absolute error 6.74

Table 3.4: Validation Method 8

Figure 3.27: There appears to be a negative correlation between the strain hardening coefficientand its error (algorithm of [31])

3.3.4 Method 9: Le’s method

The instrumented indentation equations are generally very complex when all material parame-ters are involved. Due to the self-similarity of sharp indenters, several indentation parametersare independent of the indentation depth and depend on the material properties and the in-denter geometry. Thus, the use of indentation parameters (namely C, S, Wt/We) instead of astress-strain property may allow formulating simpler and clearer dimensionless functions Eq.(3.59), (3.60) and (3.61) (Le et al. [23], [33]).

S

Chm= Ks1(n)

E∗

C+Ks2(n) (3.59)

36

Wt

We= Kw1(n)

E∗

C+Kw2(n) (3.60)

hmhe

= Kh1E∗

C+Kh2(n) (3.61)

Besides the simpler dimensionless functions, Le et al. also investigated the duality betweenthe corresponding parameters obtained by different indenters.

E∗

C60= Dec1(n)

E∗

C70.3+Dec2(n) (3.62)

(S

Chm

)60

= Dsc1(n)

(S

Chm

)70.3

+Dsc2(n) (3.63)

(WT

We

)60

= Dw1(n)

(Wt

We

)70.3

+Dw2(n) (3.64)

(hmhr

)60

= Dh1(n)

(hmhr

)70.3

+Dh2(n) (3.65)

By solving Eq. (3.60) (or Eqs (3.59), (3.61)), Eq. (3.62) and Eq. (3.64) (or Eqs (3.63),(3.65)) the reduced Young’s modulus and n are calculated. One can always choose which ofthe dependent dimensionless indentation parameters are used. The last results were obtainedusing Eqs (3.60) and (3.63). From now on only equations in terms of Wt

Wewill be considered.

Le [34] also noticed that when E/σ0 and C/σ0 are known, the strain hardening exponentn can be determined according to Eq. (3.66). This inspired him to use C/σ0 instead of n.

E∗

C= f

(E∗

σ0, n

)(3.66)

By applying this simplification Wt/We is written in function of E∗/σ0 and C/σ0. On alogarithmic scale Wt/We varies linearly with E∗/σ0 and C/σ0 (Fig. 3.28), this is defined inEq. (3.67).

C

Y= kw

(E∗

σ0

)1.031(Wt

We

)Gw

(3.67)

The same simplification is applied to Eq. (3.62) giving Eq. (3.68). By writing C/σ0 infunction of E∗/σ0 Eq. (3.67) becomes Eq. (3.69), which depends only on E∗/σ0.

lnC60

σ0= Dy

c1

(E∗

σ0

)lnC70.3

σ0+Dy

c2 (3.68)

ln

(WT

We

)60

= Dyw1

(E∗

σ0

)ln

(Wt

We

)70.3

+Dyw2

(E∗

σ0

)(3.69)

The hardness can be written in two different ways, depending on the strain hardeningexponent (Fig. 3.30 and Eq. (3.70)) or depending on E∗/σ0 (Fig. 3.31 and Eq. (3.71)) .

E∗√CH

= KH1(n)E∗

C+KH2(n) (3.70)

lnE∗√CH

= GH1

(E∗

σ0

)ln

(WT

We

)70.3

+GH2

(E∗

σ0

)(3.71)

There are two different ways to solve the problem. One is function of n (Eqs (3.60), (3.64)and (3.70)) and the other one is function of C/σ0 (Eqs (3.67), (3.69) and (3.71)). The best

37

Figure 3.28: Linear fits θ = 70.3◦ (Le [34])Figure 3.29: C60/σ0 i.f.o. C70.3/σ0 (Le[34])

Figure 3.30: Linear fit 1 (Le [34]) Figure 3.31: Linear fit 2 (Le [34])

results are obtained when both methods are combined, see Fig. 3.32. This method has beenexperimentally validated (Table 3.3.4).

There is no clear relation between the strain hardening exponen nH and the different errors,Figs. 3.33 and 3.34.

3.4 Conclusion

For sharp indenters geometrical similarity exists. This disables the conversion of each pointon a force-depth curve to a point on the stress-strain curve. This limitation does not apply tospherical indentation. However, for spherical indentation other difficulties appear. Dependingon the strain hardening exponent the material piles up or sinks in along the indenter. Withoutknowing the exact contact surface it is not possible to convert the force on the indenter intoa stress. Several researchers developed equations to predict pile-up and sink-in for certainmaterial groups. Due to the many approximations which need to be made, a point-to-pointrelationship is limited in accuracy.

Method 5, section 3.2.2, gives a way to reconstruct the stress-strain curve and provides alsoall the basic understanding needed to apply many other methods. It reports three existingrelations (Eqs. (3.40) to (3.42)) between the parameters describing a stress-strain curve andthose of a force-depth curve. Because Le [23] proved that two of them are dependent (Fig.3.18) only two correlations remain (Eqs. (3.40) and (3.42) and Figs. 3.15 and 3.16). Providedthat Young’s modulus is known, this suffices to determine the yield strength and the strainhardening exponent.

38

Figure 3.32: Flowchart Method 9 (f1,f2,f3,f4,f5,f6)= Eqs. (3.64), (3.69), (3.60), (3.67), (3.70),(3.71)

Figure 3.33: Error nH as a function of strain hardening coefficient nh (algorithm of Le [34])

39

Figure 3.34: Error H and σ0 as a function of strain hardening coefficient nh (algorithm of Le[34])

40

Yie

ldst

ren

gth

(MP

a)H

ard

nes

s(G

pa)

Str

ain

har

den

ing

exp

onen

tnh

Ref

eren

ceM

ate

rial

Ten

sile

IIT

Err

or%

FE

AII

TE

rror

%T

ensi

leII

TE

rror

%

Le

[34]

Cop

per

1011

.53

15.3

0.89

020.

8669

-2.6

0.5

0.47

87-4

.3A

lum

iniu

m20

19.1

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.40.

1313

0.13

260.

990.

150.

1477

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Gol

d38

39.9

5.0

0.33

450.

3272

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0.22

0.21

22-3

.5L

ead

1011

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10.2

0.03

450.

0343

-0.6

00.

050.

070.

4S

ilve

r60

62.5

64.

30.

6077

0.59

67-1

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0.27

0.26

53-1

.7T

un

gste

n55

057

7.66

5.0

1.45

21.

4768

1.7

0.00

50.

0186

272

Iron

300

284.

1-5

.32.

225

2.21

950.

250.

250.

2489

-0.4

4T

itan

ium

123

027

2.48

18.5

1.00

61.

0048

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Ste

el1

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489.

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978

1.98

330.

270.

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icke

l80

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7.17

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3214

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0.4

0.38

84-2

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teel

290

081

1.13

-9.9

5.93

76.

0214

1.4

0.3

0.29

22-2

.6T

itan

ium

260

054

5.51

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2.17

22.

205

1.5

0.1

0.10

858.

5A

lum

iniu

mA

lloy

500

465.

95-6

.81.

841.

8646

1.3

0.12

20.

1258

3.1

Ti-

6A

l-4V

830

773.

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235

3.26

96-1

.00.

150.

154

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Zin

c30

029

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1.1

0.05

0.07

9559

Sil

icon

6000

6103

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1.7

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025

0.01

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ate

rial

1771

5.61

692.

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548

2.56

280.

580.

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-11

Mate

rial

1887

2.47

953.

999.

32.

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2.36

693.

10

0.01

3in

f.M

ate

rial

1965

9.4

577.

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169

3.27

55-3

.40.

2038

0.24

520

Mate

rial

2069

1.8

625.

25-9

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187

3.18

6-0

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0.19

130.

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Mate

rial

2182

5.5

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Mate

rial

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0.16

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Mea

nE

rror

7.9

1.6

inf.

Tab

le3.

5:V

alid

atio

nM

eth

od

9L

e[3

4]

Chapter 4

Considerations for IIT testing

4.1 Micro or nano-indentation

Nano-indentation has the advantage that lower forces are needed and micro-films can be tested.Lower forces also means less elastic deformation of the indenter and the machine and thus moreexact results (although several different methods exist to take the deformation of the indenterinto account (Ambriz et al. [35])). Because of the lower indentation forces, not the material butthe micro structure is tested. This implies different results for the same material. To obtainaverage material properties it is preferred to work on micro-scale. The low indentation depthof nano-indentation decreases the ability to uniquely characterize the material. In literaturefew authors mention the scale of their IIT tests.

4.2 Sharp or spherical indentation

For both indenters, instrumented indentation tests have been developed. Many authors publishon both topics at the same time. Spherical indentation is more appropriate for higher forces.The carbide ball is more robust compared to a sharp tip. In contrast, a diamond tip willdeform less than a carbide ball and thus give more exact results. A sharp indenter cuts thematerial rather than presses and thus lower forces are needed to characterize the material.This implies that low force micro-indentation machines can be used. Rockwell indentation isthe only test that calculates hardness out of indentation depth. However to execute a correctRockwell-hardness test the material also needs to be preloaded. None of the discussed methodscan be applied if the material is preloaded.

4.3 Dual or single indentation

For sharp and spherical indentation mystical materials may occur. The chance of observingthis phenomenon is reduced by performing different indentations on the same material and thusextracting more information out of the material. None of the methods developed for sphericalindentation takes mystical materials into account [15]. Because of geometrical similarity mys-tical materials are more likely to occur for sharp indentation. The methods developed can beeasily extended to dual indenters.

4.4 Conclusion

Because average material properties are wanted, micro indentation will be applied. For steelonly two material parameters remain unknown: the yield strength σ0 and the Hollomon orRamberg-Osgood strain hardening factor. And thus can method 5 for sharp indenters be

41

42

applied. It provides two straightforward dimensionless correlations which can be used toreconstruct σ0 and n. This method is tested for Aluminium (Appendix A, SCAD-paper).Note that for some sorts of steel those two remaining correlations may become dependent aswell, Fig. 2.20. Although this very high yield strength is not representative for the steelswhich for I will apply instrumented indentation (mainly pipe line steels X60-X80 with yieldstrength between 400 and 700MPa and high Ramberg-Osgood hardening exponents nRO ≥ 5).However, if I would encounter mystical materials, the one remaining correlation of method 5could still be expanded by examining another independent and more complex correlation orby performing a second test.

Part II

Parametric FEA-model

43

Chapter 5

Computational model

As explained in section 3.1.3 and 3.2.2, methods exist to link stress-strain to force-depthparameters. These methods work in two directions: it is possible to predict the stress-strainparameters out of those describing the force-depth curve and vice versa. A computation modelis required to extract the force-depth parameters out of the stress-strain curve. It can be usedfor reverse modeling. The first four steps in the computational model, Fig. 5.1, are executedby Abaqus, the last one by Matlab.

Figure 5.1: Different steps model discussed in this chapter

5.1 Abaqus

5.1.1 Parameter file

A parametric Python script allows to create and analyze finite element models in AbaqusStandard without making use of the graphical interface. As can be seen in Fig. 5.2 the scriptconsists out of three different functions: self.preProcess(), self.submitJobs(), self.postProcess(). When all these steps are succeeded, the force-depth curve is generated. The most importantparameters describing the model can be found in the parameter file. This allows to createparametric studies with a minimum of user effort. The repetitive part of the model is in factprogrammed in the script. This saves the user a lot of time.

When defining the parameter file, two major decisions need to be made, Table 5.1. Firstof all, the kind of indenter needs to be defined. Secondly the user needs to decide if hewants to model a force or a displacement controlled loading condition. It is preferable to usedisplacement controlled but this will be discussed further in more detail, section 7.2. Theoffset is the initial distance between the indenter and the specimen defined in the assembly.When you want to model force controlled, the indenter and the specimen should make contactbefore the force can be applied. The initDisplacement is thus the sum of the offset andinitial penetration of the specimen into the material. For force controlled conditions the

44

45

Figure 5.2: The parametric script consists of three different functions

typeIndenter ? Vickers, Berkovich, Rockwell Rtip [mm] , heightIndenter [mm]Brinell radiusBall [mm]

howControlled ? displacement displacement, initDisplacement [mm], offset [mm]force force [N], initDisplacement [mm], offset [mm]

Table 5.1: Specific parameters of the parameters file

initial displacement should be at least 10nm higher than the offset between the indenterand the specimen. Nevertheless if the indenter is displacement controlled, the indenter andthe specimen should not come into contact. In this case, it’s best to equalize the initialdisplacement and the offset. All this is summarized in table 5.1. The general parameterswhich should always be defined are summarized in Table 5.2.

By parameterfile.preProcess() the Abaqus input-file is generated, Fig. 5.2 and more indetail Fig. 5.3. The function self.preprocess() is discussed in more detail by walking thoughall the different steps, Fig. 5.3.

5.1.2 Parts

Different kinds of sharp indenters can be modeled as an equivalent conical indenter whichhas the same projected area. This conical equivalent enables to model the instrumentedindentation test axisymmetric. In the majority of the available literature [3, 4, 12, 14, 18, 19,26, 31, 32, 33, 34, 36, 37, 38], the axisymmetric model was used as well. Although some authorsuse also a 3D model as well [4, 39, 40]. The indenter itself is modeled as an axisymmetricanalytical rigid wire. Its deformation is incorporated by defining the reduced Young modulusE∗, Eq. (3.9). The indenter is made of diamond which is very hard and the plastic deformationwill be limited. However, by versatile use the indenter tip will get worn out. The tip acerbityshould be carefully monitored as indicated in section 7.2. Although modeling the indenteras rigid is an assumption, it is justified by many different authors [4, 18, 23, 24, 31, 41].The indenter (Fig. 5.4) is characterized in the input script by its type (Vickers, Berkovich,Rockwell, Brinell). A sharp indenter requires its height, heightIndenter and the radius, Rtipof the tip. A spherical indenter is characterized by its radius, radiusBall. A reference point

46

B Half width specimen [mm]H Height specimen [mm]

frictionCoefficient Friction coefficent between indenter and specimenE Young’s modulus specimen [MPa]nu Poisson modulus specimen

matSpecimen Plastic true stress-strain data of the specimen [%, MPa]Lfree, Lstruct Mesh partition dimensions [mm]

fineBias, coarseBias Mesh density properties [mm]elemCode Element-type (CAX4, CAX4R, CAX8, CAX8R)

Convergence-parameters For each step individually defined

Table 5.2: General parameters of the parameter-file

Figure 5.3: Different subroutines included in self.preProcess()

47

refPoint, at the centreline where the indenter is clamped in the machine (indicated in Fig. 5.4by ”RP”), is added.

Figure 5.4: Parameters defining a sharp indenter

The test-material, specimen (Fig. 5.5) is modeled as an axisymmetric deformable shell. Apartition sketch is also added which divides the specimen into 5 different areas, this is neededto mesh the part in a deliberate matter (see section 5.1.3). The width B, height H and thepartition dimensions can be easily accessed through the input file.

Figure 5.5: Parameters defining specimen

48

5.1.3 Mesh

Where the indenter comes into contact with the specimen a fine mesh is needed. The localmesh density influences the amount of increments needed for convergence. The finer the mesh,the more increments needed and thus the more points describing the force-depth curve. Toobtain a finer mesh at the contact area, the specimen is defined into five different areas, Fig.5.5. The smallest square has the finest mesh and this is where the indenter makes contact.The dimensions of the finely meshed partition can be changed by varying Lstruct, Fig. 5.6.Only quad elements are allowed and the mesh technique is structured. The elements are thussquares with dimensions determined by fineBias. Both squares at the right side and the oneon the left below of the specimen are also meshed structured. The dimensions of the squareare determined by coarseBias.

Figure 5.6: H = B = 1mm, Lfree = 0.4mm, Lstruct = 0.1mm, 0.2mm, 0.3mm, fineBias =0.01mm and coarseBias = 0.05

The transition from the finest mesh to the more coarse mesh is done by defining a biaswith size fineBias near the finest meshed region and a coarseBias close the the coarse part.This transition is meshed free and again, are only quad elements allowed, Fig. 5.7.

Figure 5.7: H = B = 1mm, Lfree = 0.2mm, 0.4mm and 0.6mm, Lstruct = 0.1mm, 0.2mm,0.3mm, fineBias = 0.01mm and coarseBias=0.05

Linear CAX4 and quadratic CAX8 elements are possible. Each one can also be used withreduced integration, CAX4R4 and CAX8R. The kind of element is defined by elemCode.

5.1.4 Materials

The specimen material is characterized at two different levels: elastic and plastic. In bothareas the material is considered as isotropic. Since the application covers steel, the elastic areais defined by Young’s modulus E and Poisson’s ratio ν, 206.98 GPa and 0.3 respectively. Thematerial behavior in the plastic area is defined by true stress-strain data. This data should besaved as a text-file in the folder Materials. By defining matSpecimen in the parameter file asthe name of the text file holding the material behavior, the plastic area is defined.

49

5.1.5 Section

As mentioned before, the specimen is divided into five different partitions. The material needsto be declared to each partition individually.

5.1.6 Assembly

The indenter is positioned in the first quadrant with an offset of the origin. The specimen islocated in the second quadrant with the top left corner in the origin. Therefore no translationin the assembly is required.

5.1.7 Steps

Four steps are needed: Initial, makeContact, applyForce, removeForce. A distinction shouldbe made between force and displacement controlled conditions.

In the case of force controlled conditions, the indenter moves in the second step makeCon-tact over a distance defined by initDisplacement to make contact with the specimen. Thiscontact is needed for convergence reasons. The second step applyForce puts the load at thereference point refPoint of the indenter. The force is defined in the input-script as: force. Thetime-period, initial increment, minimum and maximum increment and the maximum numberof increments are user defined. The default values are: 1, 10−3, 10−7 and 1, 109.

For displacement controlled conditions the indenter is not brought into contact duringmakeContact. Instead, it is just moved so that the distance between the indenter and thespecimen is zero. During applyForce the indenter moves down over a distance displacement andduring removeForce it returns to its initial position. The names of the steps and convergenceparameters are similar as for force controlled conditions.

5.1.8 History output request

The instrumented indentation test is characterized by the depth and the force during the com-pression and elastic recovery of the material, corresponding with the last two steps applyForceand removeForce respectively. For each increment of both steps the depth and the force arewritten down for further postprocessing (section 5.2).

5.1.9 Interaction

The indenter is the master surface and the specimen is the slave surface in a surface to surfacecontact. This contact is created during the initial step and propagates through makeContact,applyForce and removeForce.

5.1.10 Interaction Properties

Only tangential behavior is defined. The friction effect is taken into account as a penalty andis assumed to be isotropic. Almost every author [4, 14, 42] uses the same friction coefficient fordiamond-steel contact: 0.15. For half included tip angles higher than 50◦ it is often neglected[17, 19, 31, 24].

5.1.11 Load

For force-controlled conditions one load has to be specified, namely FapplyForce. It is cre-ated during applyForce and modified during removeForce. It is mechanical and defined as aconcentrated force taking place at the reference point on the indenter. The direction of theforce is the negative y-direction and its amplitude is force. The force on the indenter increases

50

linearly. During removeForce the force decreases linearly until it is fully removed. If the testis displacement controlled no force is imposed.

5.1.12 Boundary conditions

Three different boundary conditions are implemented, Fig. 5.8: BCmakeContact, onlyMo-tionZ, specimenEncastre. For force controlled conditions the initial y-displacement, whichcauses penetration of the indenter into the specimen, is applied by a boundary condition. Fordisplacement controlled, this boundary condition exists too but its amplitude is zero. Howevertwo more boundary conditions are added, BCimprint during applyForce and BCremoval dur-ing removeForce. Displacement controlled is more accurate compared to force control becauseonly forces are generated during applyForce, removeForce. The extraction of S and C is alsoonly based on those two steps and thus all information generated during the simulation isused.

The second boundary condition onlyMotionZ allows the indenter to move only in the y-direction. Thirdly the right and bottom side of the specimen are encastred.

Figure 5.8: Different boundary conditions applied to specimen

5.2 Matlab model

As explained in section 2.1.1, the force-depth curve during loading is often approximatedby Kick’s constant C. Unloading is approximated by the unloading stiffness S. These twoparameters will be used to quantify the correctness of the developed finite element model. Toextract C and S out of the force-depth curve, which is generated by Abaqus, Matlab is used.This Matlab script can also be used to examine experimental data, as in section 6.5. Thescript is explained in Fig. 5.9.

For calculating S, C, Hm and the calculation time, Matlab needs two files: The outputfile and the message file. The location of both files are made clear to Matlab by definingname1. After the output file is converted to an array, the maximal depth can be extracted.The curvature C of the loading part is extracted by the Nelder-Mead method. In this methodan unconstrained nonlinear minimization of the sum of squared residual with respect to thevarious parameters is performed. To the first part of the unloading curve a third degreepolynomial is fitted. By differentiating, the polynomial S is obtained. The message file is usedto extract the calculation time.

The correctness of the Nelder-Mead method is investigated by a case study of 53 differentmaterials, varying E between 10GPa and 210GPa, σ0 between 30MPa and 3000MPa and nHequal to 0.1, 0.3 and 0.5. The lowest correlation coefficient occurred for E=170GPa, σ0=300

51

Figure 5.9: Algorithm of the matlab script which extracts hm, S, C and CPU-time

52

and nH=0.3. It was 0.99984. This high correlation between the data and its approximationindicates the soundness of the approximation. To calculate the unloading stiffness only built-inMatlab functions were used. These were not tested for correctness.

For completeness, the force-depth curve and Matlab-approximation were given for three dif-ferent kinds of pipeline steels. They all have a yield-strength of 400MPa but different Ramberg-Osgood strain-hardening coefficients, section 2.1.2: 5 (σ0/σUTS=0.5), 15 (σ0/σUTS=0.85) and25 (σ0/σUTS=0.93), Fig. 5.10. This relative low yield strength is chosen because it will causehigh deformations. High deformations are required because they cause high deformation ofthe elements and they are thus challenging for the Abaqus script, this is discussed further.

Figure 5.10: Engineering stress-strain curves tested materials

These materials give force-depth functions shown in Fig. 5.11. This figure also displaysthe constants extracted by Matlab (C, S) showing a very good accordance. The loading partis approximated by Kick’s law and the unloading by a linear fit with the extracted unloadingstiffness as slope.

Figure 5.11: Validation of Matlab model

Because all three are steels with equal Young’s modulus, the unloading stiffness is similar.This is also indicated by Giannakopoulos et al. [21] who showed that S depends on E (Eq.(3.38)) given an equal indentation depth.

53

5.3 Conclusion

A axisymmetric parametric model intended for reverse modeling is developed. The points ofinterests can be accessed through the parameter script. First, the parametric script is convertedinto the input file. In this step the model is defined and the job created. By executing thejob and the parametric script the field output is generated. The force on the indenter and itsdisplacement are saved as output. This output file is next analyzed by Matlab and C, hm andS are calculated. The Matlab program is validated for 54 different cases.

Chapter 6

Optimization and validation of thedeveloped model

6.1 Element and mesh density

As explained in section 5.1.3, two different density element classes exist: linear elements, CAX4and a quadratic elements CAX8. The influence of the kind of element will be investigatedby comparing the output of the parametric model for different kinds of elements and meshdensities.

As explained before the indentation is modeled axisymmetric, which implies that the spec-imen is modeled as a cylinder. During this study, the radius and the height of the cylinder is4mm, Lstruct is 0.1mm and Lfree is 0.4mm. The coarse bias is always 5 µm but the fine biasis varied. The fine one is 1µm, the medium one is 2.5µm and the coarse bias is 5µm. Thecorresponding numbers of elements and nodes can be found in table 6.1.

Nodes

Mesh density Elements Linear Quadratic

coarse 7120 7308 21735medium 8616 8827 26269

fine 19841 20117 60074

Table 6.1: Different meshes used for validation

In Fig 6.1 the mesh density fine can be found.

Figure 6.1: Where the indenter comes into contact with the specimen, the mesh is very fine

The different elements and meshes will be tested for convergence with three different kindsof steel with σ0 equal to 400MPa and nRO equal to 5, 15 and 25. Note that this is thesame material as in section 5.2. Throughout the remainder of this and the next chapter these

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three materials will always be used. As mentioned in section 4.4 this study focuses on theconical equivalent of the Vickers indenter. Besides widely applied, this indenter has a bighalf included angle (70.3 ◦) and thus can the influence of friction be neglected. The Vickersindenter is used in all simulations of this chapter and the next. The influence of friction is onlyneglected in this chapter. The tip radius is taken as 1µm. Both parameters are discussed insection 7. The indentation is displacement controlled with a maximum indentation depth of25µm, this corresponds with a force of 38.3N, 27.9N and 26.6N for strain hardening coefficientnRO 5, 15 and 25 respectively. The results will be qualified based on Kick’s constant C andunloading stiffness S. The error for a certain element type and mesh will always be qualified aspercentage deviation of the result using the element with the highest number of nodes. This isthe quadratic element with the finest mesh. It is considered as the reference for other results.

From Fig. 6.2, which shows the error of C as a function of mesh density and element, canbe concluded that the errors decrease with mesh density which indicates that convergence isobtained. For a medium mesh density the error is lower than 1%. For the the highest meshdensity the error between the linear and quadratic element is only 0.02% for nRO equal to5 and 25. For nRO=15 no simulation convergence was obtained and the result is thus notdisplayed.

Figure 6.2: The linear element with finest mesh gives good results

A similar trend is visible for the unloading stiffness S, see Fig. 6.3. The errors are highercompared to Kick’s constant. For the highest mesh density the error of the linear elementrelative to the quadratic element is only 0.05% for nRO equal to 5 and 0.2% for nRO equal to25. Again, is for nRO=15 no convergence was obtained.

When the calculation time is taken into account, one can notice that it increases stronglywith increasing mesh density, Fig. 6.4. The calculation time is also systematically higher forlower strain-hardening materials. The calculation time is about three times higher for thequadratic element compared to the linear one. The low error (less than 0.2%) for S and Cbetween the linear element with finest mesh density and quadratic element and the significantlower calculation times promotes the use of the linear element with highest mesh density.

The output of the finite element model depends besides the kind of element and the meshdensity also on the integration scheme used. The result for nRO equal to 5 for full and reducedintegration is displayed in Fig. 6.5. The relative error is always calculated based on the resultsobtained by quadratic elements, highest mesh density and without reduced integration. Theerror on C is lower for reduced integration. It is not possible to make a comparison forthe linear element with highest mesh density because it didn’t converge. The error for thequadratic element with reduced integration and finest mesh amounts 0.07%.

For the unloading stiffness S the error is, except for medium mesh density, always higher

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Figure 6.3: Good results are obtained for S using the finest linear element

Figure 6.4: Calculation time increases strongly with increasing mesh density

Figure 6.5: For nRO equal to 5 reduced integration reduces the calculation error

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for reduced integration. The error of the quadratic element with reduced integration and finestmesh is 0.17%.

Figure 6.6: For nRO equal to 5 reduced integration doesn’t reduce the calculation error

Although reduced integration is used, the calculation time, Fig. 6.7 is still higher thanwith full integration. This is explained by the observation that more iterations and incrementsare needed to obtain convergence. There were much more convergence problems for reducedintegration. None of the simulations with reduced integration for low strain hardening mate-rials (nRO =15, 25) converges. Because of the many convergence problems and the increasedcalculation time, reduced integration is not further considered.

Figure 6.7: For nRO equal to 5 reduced integration doesn’t reduce the calculation error

6.2 Force on the indenter

Because of geometrical similarity rules for sharp indenters, the hardness is independent ofthe imprint depth. This implies that Kick’s constant should be independent as well. Indeed,both are function of P/h2. As discussed in section 4.4, indentation with sharp indenters ispreferred and more precise: Method 5, section 3.2.2 is selected for further investigation. In thismethod the parameters of the power law describing the stress-strain relationship are functionof C/σr (Fig. 3.15) and S/hm (Fig. 3.16). In order to use graphs 3.15 and 3.16, for C andS calculated from force-depth curves with different forces as those represented on the graph,

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both dimensionless functions should become independent of the force. This dependence isinvestigated in more detail, Figs. 6.8 and 6.9. Because displacement controlled is preferredto model in Abaqus (section 7.2), not the force, but the displacement is varied. The erroris showed as a function of the gathered value at highest displacement (35µm for nRO=5 and25µm for nRO=25).

Figure 6.8: C does not strongly depends on the depth of imprint

Figure 6.9: Higher imprints are needed for f3 to become independent

With higher forces more increments are required for convergence and thus the calculationtime increases, Fig. 6.10. It is clear that S/hm depends more on the force compared to C.Because of the relative high error for f3, it is preferred to create both graphs for a certaindisplacement/force and when be applied, use an experimental force-depth curve generated forthis force/displacement. During all the simulations performed in this chapter a displacementof 25µm is applied.

6.3 Mesh size

The specimen is always arbitrarily modeled as a cylinder with the same height as the radius.The ISO 6507 gives a guideline for the minimum distance between two hardness tests. ISO6507 prescribes that the distance between two indentations should at least be three timesthe diameter of the Vickers imprint. This norm can be used to make an initial guess for the

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Figure 6.10: Calculation time varies almost linearly with indenter displacement

necessary radius of the specimen. From Fig. 6.11 it can been seen that when 25N is appliedon a Vickers indenter the radius of influence is about 0.2mm.

Figure 6.11: Radius of imprint for a Vickers equivalent conical indenter and a force of 25N isabout 0.2mm for a steel with yield strength equal to 400Mpa.

Following ISO 6507 the radius of the speimen should at least be 0.3mm. For safety, aminimal radius of 0.6mm is taken. To investigate the influence of the mesh size, the radiusand the depth are simultaneously varied (0.6mm, 1.2mm, 2mm, 4mm, 6mm and 8mm) andthe error of C (Fig. 6.12) and S (Fig. 6.13) are compared to the biggest mesh size (8mm)investigated.

With increasing specimen dimensions the calculation time increases as well, Fig. 6.14.It can be concluded that at least a radius of 4mm is needed for accurate results with

occurring forces lower than 40N (nRO=5: 0.042% for C and 1.48% for S, nRO=25: 0.048%for C and 1.3% for S). For this validation parameter study the specimen will be modeledas a cylinder with a height and radius of 4mm. Although it is advised for higher precisionrequirements to use a bigger mesh.

6.4 Validation based on literature

After defining the force, the mesh density and specimen size, the model can be validated.Luo et al. [31] executed and documented two experimental instrumented indentation tests on

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Figure 6.12: With bigger mesh the error on C decreases

Figure 6.13: S depends strongly on mesh size

Figure 6.14: Calculation time increases strongly with increasing mesh size.

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nickel, one with a Vickers and one with a Rockwell indenter, Fig. 6.15. The material behaviorof the nickel was approximated by a Hollomon law (E=207GPa, σ0=800MPa and nH=0.4).

Figure 6.15: Validation by literature [31] for nickel (nH=0.4)

Chollacoop et al. [24] also documented experimental force-depth curves (Fig. 6.16) for AL7075T651, again the material behavior is prescribed by a power law (E=70.1GPa, ν =0.33,σ0=500MPa and nH=0.0122).

Figure 6.16: Validation by literature [24] for aluminium

Mata et al. [42] documented two force-depth curves as well (Fig. 6.17). Both steels, SAF2507 and AISI 329, are prescribed by a power law (SAF 2507: E=200GPa, σ0=675MPa andnH=0.19, AISI 329: E=190GPa, σ0=525MPa and nH=0.17).

As mentioned before, no author mentioned the exact friction coefficient and bluntnessof the indenter tip for the tests performed. The force-depth curves generated by myModelneglected the effect of friction and the tip bluntness was estimated as 1 µm. In section 7.2and 7.1, these two parameters are investigated in more detail.

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Figure 6.17: Validation by literature [42] for steel

Next, a detailed case study is conducted with as purpose to reconstruct Figs. 3.15 and 3.16.The same material data-set was used as in section 5.2. The result can be found in Figs. 6.18and 6.19. As can be seen, good similarity between the simulations and the curves provided byDao et al. [19] (solid lines on Figs 6.18 and 6.19) is found. Fig. 6.18 shows that C/σR indeedbecomes fairly independent of nH for εR equal to 0.033. The functions visible on the graphsextracted from Dao et al. [19], were fitted to finite element simulations, the friction coefficientis set equal to zero and an infinite sharp tip were set forward.

Figure 6.18: Independence of nH and good agreement with literature

6.5 Validation based on experimental tests

At the university of Ghent no instrumented test set-up is available. At the mechanics labora-tory of the Ecole centrale de Lille, a CSM micro hardness tester (Fig. 6.20) is available. Thedisplacement of the indenter is force controlled and an indentation test consists of 4 steps.First a displacement is imposed to the indenter. As soon as the indenter registers resistance,(an indication that it comes into contact with the specimen) the force is applied. The rate of

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Figure 6.19: Good agreement as well

force increase with time can be chosen. When the maximal force is obtained the indenter loadis fixed for 15 seconds. This delay is needed to reduce the amount of creep when the indenteris removed. The rate of force removal is user controlled as well. It is advisable to take therate of force removal equal to rate of force appliance and let one test (force appliance + fixedposition + force removal) last for one minute. In case macros are tested, they should be planparallel and have a diameter of 4cm. However, it is advisable to test non-embedded specimenssince no epoxy between the specimen and positioning table is advisable. The force is limitedto 22N.

At the Centre Arts et Metiers ParisTech de lille a MTS XP nano Indenter (Fig. 6.21) isavailable. This machine is more advanced than the above mentioned micro hardness tester.The indentation tests are computationally programmed. The machine decides by itself whento perform the tests. The vibration should be lower than a certain degree which means thatthe machine mainly works at night. When using this machine there is no chance to take yourspecimen after the tests back home. You should leave it inside the machine and pick it up thenext day. In high load mode the machine can apply up to 10N. The diameter of the macro’sis restricted to 3cm. Only macros for which the specimen is supported by metal contact, areaccepted.

Figure 6.20: CSM micro hardness tester Figure 6.21: MTS XP nano indenter

I used this machine for experimental validation of myModel, Fig. 6.5. The differencebetween the experimental result (Fig. 6.5) and those out of literature (Figs. 6.15, 6.16 and6.17) is that in Fig. 6.5 the compliance of the machine is included. The most applied methodis those of Oliver-Pharr, explained by Ambriz et al. [43]. The correction factor is obtained

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by evaluation Eq. 3.38 for different forces on the indenter but for the same material (dP/dhmis the total compliance, of the machine and the specimen). By displaying the inverse of the

total compliance as a function of A−1/2c for different forces, extrapolating and evaluating the

best linear approximation at the origin, the machine compliance is obtained. At the originAc is zero and thus only the compliance of the machine remains. The disadvantage of thismethod is that Ac should be known. Eq. (2.17) is often used to approximate Ac. A matlabscript (IITCompliantie is developed, this script generates the machine compliance out thedifferent force-depth curves. The machine compliance is taken into account by decreasing themeasured value proportional to the correction factor. This caused the whole curve to moveto the left and does not influence C or S. This correction is already applied to the resultsfollowed from literature. Still, the same trend is observed for each validation: Kick’s constant isoverestimated. Although for appliance of Method 5, Figs. 6.18 and 6.19 good results betweenliterature and experimental tests were obtained. In this method the reduced Young’s modulus,Eq. (3.9) is included. The Young’s modulus and the Poisson coefficient of the indenter are1140GPa and 0.07 respectively.

Figure 6.22: Again an overestimation of C

6.6 Conclusion

Out of a convergence study it has followed that a mesh consisting of the linear elements sizeof 1µm near the contact area provides a good trade-off between accuracy and computationaltime. This assumption was made by evaluating the error of C, hm and S. Reduced integrationprovides a lower error but the calculation time increases and many convergence problems occur.

Based on geometrical similarity C was found to be independent of the maximum force ordisplacement. In contrast S/hm was found to be dependent of hm. This indicates that Fig.3.16 does depend on the maximum force or displacement.

For parametric studies, a radius and height of the indenter equal to 4mm are advised.However, for higher precision purposes, bigger dimensions (implying longer calculation times)are advised.

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The parametric script has been validated by experimental tests documented in literature.Good results for S have been obtained but C is systematically overestimated. The curvesdocumented in literature were always corrected for machine compliance. Machine compliancecan thus not be the reason for the overshoot. Figs 3.15 and 3.16 could be reproduced with goodcorrespondence and without convergence problems. Validation based on an own experimentaltest (performed at Centre Arts et Metiers ParisTech de lille) gave an overestimation of C aswell.

Chapter 7

Parametric study

7.1 Influence of friction

The influence of friction is investigated for three different steels (σ0=400MPa and nRO=5, 15,25). The error for nRO= 5 and 25 is arbitrarily defined relative to the case of zero friction andradius and height of the specimen equal to 4mm. Because for nRO= 15, zero friction and samespecimen size, the simulation did not converge, the error for nRO=15 is displayed relative tothe case of zero friction for a mesh with radius and height equal to 8mm. As explained insection 6.3 this has a minor effect. First, as can be seen in Fig. 7.1, the error always staysbelow two percent. It is remarkable that µ=0 and µ=0.25 shows more correspondence withµ=0 than µ=0.15 and µ=0.25. Second, the error of the unloading stiffness S is in general alot higher, Fig. 7.2. It is not likely to allocate the high error for nRO for µ=0 and 0.15 tothe different mesh sizes. As shown in Fig. 6.13, the error between a mesh size of 4 and 8mmfor nRO= 5, 25 was about 1%. In Fig. 7.2 the difference is 10%. Finally, absence of frictionslightly increases the calculation time (Fig. 7.3).

Figure 7.1: Error C due to friction is always lower than 2%

In Fig. 7.4 the force-depth curve for nRO=15 for different friction coefficients is shown.This figure indicates that the error between the parametric script and experimental (section6.5) test can not be due to friction.

In Fig. 7.5 the influence of friction on the residual imprint size and thus on the hardnessis investigated. It can be seen that with high friction, the material surface flows more alongthe indenter. This observation is opposite to what has been described by Bucaille et al. [17].Bucaille et al. observed lower pile-up for high friction for a Vickers indenter, Fig. 2.18.

For completeness the imprint is also shown for nRO=5 (converted to a power law nH=0.20),

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Figure 7.2: High influence of friction

Figure 7.3: Calculation time is not strongly influenced by friction

Figure 7.4: Influence of friction on the force-depth curve is to low to explain difference betweenparametric script and literature/experimental (Figure generated for nRO=15)

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Figure 7.5: Friction doesn’t oppose the material from piling-up along the indenter, nRO=25.

which is a relatively high strain hardening material and thus sink-in occurs. Fig. 7.5 corre-sponds with nRO=25 (converted to a power law nH=0.043), a low strain hardening material.For this material pile-up occurs. This is the same trend as observed in Fig. 6.11.

Figure 7.6: Sink in occurs for high strain hardening materials, nRO=5

Figs. 7.7 and 7.8 give the deformation of the mesh for low friction and high frictionrespectively. Friction prevents the elements to slide along the indenter. This is the samebehavior as Mata et al. [42] described.

7.2 Influence of tip bluntness

Besides friction, tip bluntness has a major influence on the force-depth curve. An indenteris never infinitely sharp and with time it becomes blunter. A Berkovich indenter is easierto manufacture compared to a Rockwell indenter since only three faces need to intersect.This implies that Berkovich indenters are sharper than Vickers indenter. For comparison, theindenter used for experimental validation (Fig. 6.5) was a Berkovich indenter with tip radius20nm. The finest Rockwell indenter available by ST instruments bv has a tip radius of 10µm,

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Figure 7.7: Deformation mesh for nRO=5,µ=0.05

Figure 7.8: Deformation mesh for nRO=5,µ=0.25

the finest Vickers indenter available by ST instruments bv has a tip radius of about 60-80 nm.The error caused by blunting on Kick’s constant C is given in Fig. 7.9 (note the logarithmicscale). Reference to this error is the corresponding simulation with an infinitely sharp indenter.Increasing tip blunting causes higher forces, Fig. 7.10. Unlike C, the error of S (Fig. 7.11)is higher for high hardening materials. From Figs. 7.9 and 7.11 it can be concluded that theradius of the tip should be carefully monitored.

Figure 7.9: Increasing error with bluntness for all materials

The calculation time (Fig. 7.12) slightly increases with tip blunting, this effect is strongerfor low strain hardening materials.

As explained in chapter 5, the user can choose for force or displacement controlled con-ditions. The contact definitions are different for both modules. For force control an initialoverlap (10 nm) is needed. With this overlap the indenter ’feels’ the influence of the materialand the force can be applied. For displacement controlled this overlap is unnecessary. Thisreduces the error because the initial overlap causes pre-strain and as a consequence higherKick’s coefficient C (Chollacoop et al. [37]). As a second consequence of force control the tipradius needs to be at least 10µm for convergence reasons. It is thus only possible to model aBerkovich indenter by the displacement controlled condition.

7.3 Mystical materials

Both Ma et al. [4] and Chollacoop et al. [24] mentioned the same three mystical materials,Table 7.1 and Fig. 2.20. These three materials have also the samen C although S is slightlydifferent (Fig. 7.13 and Table 7.2).

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Figure 7.10: Higher forces needed with increasing tip bluntness

Figure 7.11: Higher dependence tip bluntness for higher strain hardening materials

Figure 7.12: Higher dependence tip bluntness for higher strain hardening materials

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Youngs modulus [GPa] Yield strength [MPa] Strain hardening exponent nHMaterial 1 200 2,360 0.0Material 2 200 2,000 0.1Material3 200 1,240 0.3

Table 7.1: Mystical materials from literature [4] are confirmed

hm [µm] Pmax [N] S [N/mm] C [MPa]

Material 1 5 3.98 6,697.16 161,219.3Material 2 5 4.00 6,272.835 160,051.4Material3 5 3.97 5,999.94 159,249.2

Table 7.2: Exact value of the force-depth curve parameters

Figure 7.13: Mystical materials from literature are confirmed for C [4]

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7.4 Applicability of Method 9 for pipeline steels

In section 6.4 Figs. 3.15 and 3.16 have been reproduced with the parametric model. Those twofigures allow to reconstruct the constitutive law (Ramberg-Osgood or Hollomon) describingthe material behavior if Young’s modulus is known. Both graphs are now regenerated but fortypical pipeline steels (σy= 400, 500, 600 and 700MPa, nRO=5 (σ0/σUTS ≈ 0.5), 10 (σ0/σUTS≈ 0.75), 15 (σ0/σUTS ≈ 0.85), 20 (σ0/σUTS ≈ 0.9) and 25 (σ0/σUTS ≈ 0.93), E always equalto 206.9GPa). Displacement controlled condition were used with a maximum displacementof 25 µm. Only 12 out of the 20 cases converged and are displayed on Figs. 7.14 and 7.15(Table 7.4). The approximation made by Dao et al. [19] is represented as well. For low strainhardening steels, Fig. 7.14 remains independent of nRO for a representative strain εr equal to0.033 as well.

E [GPa] σ. [MPa] nRO hm [µm] S [N/mm] C [MPa] Pmax [N]

210 500 5 25 34781.15 73755.77 46.16210 500 10 25 37958.31 59908.36 37.43210 500 15 25 39335.49 55749.09 34.97210 600 5 25 34283.71 85341.14 53.38210 600 10 25 37359.7 69717.3 43.41210 600 15 25 38766.07 65080.42 40.72210 700 5 25 33756.33 96392.82 60.12210 700 10 25 36768.92 79045.49 49.44210 700 15 25 38162.03 73926.66 46.04210 700 20 25 38934.8 71447.27 44.56

Table 7.3: Force-depth parameters belonging to the discussed pipeline steels

Figure 7.14: The approximation given by Dao et al. [19] satisfies for f1 (Eq. (3.40)

From Fig. 7.15 can be seen that even very low strain hardening materials still can bedistinguished in theory. However, the most frequently used pipeline steels have nRO between15 and 25. For these strain hardening exponents not only many convergence problems occurredbut there is also a sensitivity. This phenomenon should be investigated further. The relationbetween both dimensionless quantities (solid line), given by Dao et al. [19] becomes lessaccurate for pipeline steels. However this could also be due to the difference in force appliedon the indenter. Dao derived the approximation for a maximum force of 3N and during mysimulations with the displacement controlled conditions, forces up to 60N occurred. This could

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also justify the good correspondence between simulations and literature of Fig. 7.14. In section6.2 was proven that C depended less than S/hmax on the force applied on the indenter.

Figure 7.15: New approximations need to be made for f3 (Eq. 3.42)

7.5 Conclusion

In a first parametric study friction was found to have a major influence on S. Pile-up for highRamberg-Osgood coefficient was observed and sink-in for lower Ramberg-Osgood coefficients.It was found that friction did not oppose the material from sliding along the indenter as docu-mented in literature. The opposite phenomenon was observed. Second, tip radius has a majorinfluence on C (and, to a lesser extent, on S) and should for that reason be carefully moni-tored. Third, steels which were categorized mystical gave the same force-curve for simulationexecuted by the parametric script as well.Method 9 has been evaluated for low strain hardening steels (such as modern pipeline steels)and satisfies. However, the relation f2 (Eq. (3.42)) given by Dao et al. [19] should be furtherevaluated especially for this kind of material. The sensitivity of f3 (Fig. 7.14 and Eq. (3.40))is very low for nRO > 15 which is the scope for modern pipeline steels. The possibility toobtain σ0.33 (Fig. 7.15 and Eq. (3.42)) without having to know nRO is very interesting. Thismakes it possible to compare the strength of the weld with the base material rather easily.

Chapter 8

Conclusion and future work

Starting from an extensive literature review it has been found best to estimate the parametersof a constitutive law describing the stress-strain behavior instead of the entire stress-straincurve. Method 9 gives a basic framework to reconstruct the stress-strain curve. The force-depth curve is characterized by curvature during loading C, maximum imprint hm and theunloading stiffness S. The stress-strain curve is characterized by Young’s modulus E, yieldstrength σ0 and a strain hardening exponent n. For steel only σ0 and n remain unknown. Theparameters describing the force-depth curve can been written as a function of those describ-ing the stress-strain curve. Two relations standout in simplicity: C/σr (f1, Eq. (3.40)) andS/(hmE

∗) (f3, Eq. (3.42)) both function of E∗/σr. For εr equal to 0.033 the first relationbecomes independent of n. This independence was tested two times, first for a wide range ofmatarials (54 cases) and a second time especially for low strain hardening steels. Two times itwas found, just like in literature, that C/σ0.033 can be written only as a function of E∗/σ0.033.This allows to calculate σ0.033 easily and σ0.033 can be used to quantify the strength of theweld relatively to the base material.

The second relation does depend on n and after determining σR n can be easily found.These relations have been validated by finite element simulations and good accordance wasobtained. Then, this method have been closer investigated for low strain hardening steels5 < nRO <25 (or 0.5 < σ0/σUTS < 0.93) and again very good results were obtained. Notonly has the same trend been observed as in literature, the method also appears applicablefor materials with very low strain hardening. However, the correlations given in literature forS/(hmE

∗) did not perfectly satisfy and are lowly sensitive to n for low strain hardening. Moreresearch should be undertaken to compose a better correlation between low strain hardeningand the result of an instrumented indentation test.

Besides the relation between parameters of the stress-strain curve and those of a stress-strain curve, reverse modeling has been considered as well. A parametric script has beendeveloped which allows to model the indentation test axisymmetric. Different parameters(type indenter, mesh dimensions, force, ... ) can be varied easily. A mesh convergence studyhas been successfully executed for indenter forces up to 40N. This was based on the error ofC, S and the calculation time. The radius of the axisymmetric specimen should be at least6mm for forces lower that 40N. During this study some convergence problems occurred. Theseshould be investigated in closer detail. Although there is very good correspondence betweenS from simulations and from experimental tests (most of which were found in literature, oneof which was performed at the Centre Arts et Metiers ParisTech de lille), C appears to be sys-tematically overestimated. In order to use the parametric model with full confidence, it shouldgive better correspondence with the experimental results. Further research and validation isneeded.

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Appendices

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Appendix A

SCAD-paper

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INSTRUMENTED INDENTATION FOR DETERMINATION OF FULL

RANGE STRESS-STRAIN CURVES

A. De Smedt1, S. Hertelé2, M. Verstraete2, K. Van Minnebruggen2 and W. De Waele2

1 Ghent University, Belgium

2 Ghent University, Laboratory Soete, Belgium

Abstract: One common method for the determination of full range stress-strain curves by instrumented indentation is presented and validated for an aluminium alloy. This method relates properties describing the indentation force-depth curve with those describing the uniaxial stress-strain curve as traditionally obtained from a tensile test. The first aim of this paper is to explain the basic concepts of instrumented indentation. Next, the analysis method is presented and validated. This study ends with discussing the uniqueness of the obtained solution. It is concluded that accurate determination of stress-strain behaviour can be realized, but for certain materials two indentations are needed.

Keywords: instrumented indentation, indenter, stress-strain, mystical materials

1 INTRODUCTION

Imagine one wants to obtain the local stress-strain properties of a metal in a non-destructive manner and with no restriction on specimen size and shape. This is made possible using an instrumented indentation test (IIT). In this paper the stress-strain curve of an aluminium alloy 6061-T6511 will be reconstructed by IIT [1, 2]. This alloy is widely used in commercial construction applications.

IIT is a hardness test during which execution the force on the indenter and the displacement of the indenter are recorded. After specific data analysis, this results in a force-depth curve (Figure 1). This curve is characterised by a number of parameters that are discussed in section 2. The stress-strain curve of a metal is typically characterised by its Young’s modulus, strain hardening exponent and yield strength. This will be elaborated in section 3. Several methods have been proposed to relate the parameters of an indentation force-depth curve with those of a stress-strain curve. Some brief remarks are given in section 4. The curve fitting method used in this paper is fairly generic and widely used for the reconstruction of stress-strain curves. This is the topic of section 5. In section 6 the uniqueness of the obtained solution is discussed.

2 INDENTATION FORCE-DEPTH CURVE

The evolution of an indentation force-depth curve depends on the indenter type used but generally follows a characteristic pattern. This pattern is explained by means of an example, shown in Figure 2, obtained for the aluminium alloy 6061-T6511 using a Berkovich type indenter [2]. Such an

Figure 1: Concept of IIT

indenter has the same projected area as a Vickers indenter (or a cone with a half tip angle of 70.3°) but is characterized by three facets instead of four.

Figure 2: Indentation force-depth curve for an aluminium alloy 6061T6511 obtained using a Berkovich type indenter

The loading phase of the indentation response is described by Kick’s law:

(1)

For this specimen Kick’s constant C is 27.6 GPa. The (un)loading rate and maximum load are user defined and for the sample considered, these parameters are 4.4 N/min and 3 N respectively. At the moment of maximum load, the maximum imprint depth hmax equals 10.19 µm.

Afterwards the material is unloaded, the indenter removed and the imprint recovers elastically. The unloading phase can be approximated by Eq. (2) in which B, m and hr are curve fitting parameters [3].

( ) (2)

The unloading stiffness S is defined as the slope of the unloading curve near the maximum indentation depth. It can be calculated by differentiating Eq. (2) and evaluating at hmax:

|

( ) (3)

Much of the information obtained by one single instrumented indentation test is described by the four parameters introduced above: hr, hmax, C and S. For some IIT methods the work performed during loading WT (area underneath the indentation force-depth curve during the loading phase) and unloading WE (area underneath the indentation force-depth curve during the unloading phase) are required rather than hr and hmax. For sharp indenter types, WT and WE contain the same information as hr and hmax and are less sensitive to scatter because they are calculated based on all measured data points rather than a single value.

3 STRESS-STRAIN CURVE

The linear-elastic region of a stress-strain curve is characterized by the Young’s modulus E. Further, the Poisson coefficient is introduced to describe lateral restraint effects during elastic (un)loading.

In instrumented indentation of metals, the stress-strain curve in the plastic area (i.e. stress exceeds the yield strength ) is typically approximated by a Hollomon type equation [4]:

0

1

2

3

0 5 10

Fo

rce

(N

)

Depth h (µm)

(4)

With R the strength coefficient and the strain hardening exponent. The stress in the plastic area can be rewritten by Eq. (5) when elastic strain is given by εy, and the plastic component of strain by εp.

( ) (

)

(

)

(5)

This stress-strain model is described by ( ) or ( ) with the ‘representative’ stress

corresponding to a particular ‘representative’ strain as explained in section 5.

Figure 3 plots the true stress-strain curve of the aluminium alloy investigated in this paper determined by traditional tensile testing [2]. Its model parameters are E = 66.8 GPa, = 0.8 and = 278 MPa.

Figure 3: Stress-strain curve of the aluminium alloy 6061-T6511

4 RELATIONSHIP BETWEEN INDENTATION FORCE-DEPTH AND STRESS-STRAIN CURVES

A point-to-point systematic relationship between an indentation force-depth curve and a stress-strain curve would lead to a non-destructive method to determine the stress-strain properties of a steel. Such relationship has only been developed for spherical indentation (Brinell type indenter) [5-7], thereby making use of numerous profound assumptions and approximations. For sharp indentation (Vickers, Berkovich or Rockwell type indenters), however, finding an immediate link between both curves is highly challenging due to the geometrical similarity of this type of indentation [8]. This means that an IIT at large depth is essentially a magnified picture of an IIT at a small depth. As a consequence, methods based on a sharp indentation are confined to providing links between the model parameters of an indentation force-depth curve and a stress-strain curve rather than between their distinct data points.

The material in the vicinity of the indentation does not remain perfectly flat, which increases the complexity of characterising the deformation and developing adequate definitions for the representative stress and strain. Two different indentation responses may occur; pile-up and sink-in as illustrated on Figure 4.

0

100

200

300

400

0.00000 0.02000 0.04000 0.06000

Tru

e S

tress (

MP

a)

True Strain (%)

0 2 4 6

Figure 4: Pile-up (Left) and sink-in (right) influences the accuracy of stress and strain calculations

Multiple authors [5-7, 9] developed equations to predict pile-up and sink-in for certain material groups. The tendency towards pile-up increases with decreasing strain hardening exponent and

ratio [10]. Due to the many approximations which need to be made, a point-to-point relationship

is limited in accuracy.

5 INSTRUMENTED INDENTATION ANALYSIS BY MEANS OF CURVE FITTING

In curve fitting methods, the indentation parameters are written as functions of the unknown stress-strain properties. This enables the reconstruction of an indentation force-depth model curve out of the stress-strain model curve and vice versa. A selected curve fitting method for sharp indentation is applied to the indentation force-depth curve of Figure 2. This specific method is valid for common engineering materials: E from 10 to 210 GPA, from 30 to 3000 MPa and from 0 to 0.5, with Poisson’s ratio fixed at 0.3.

The influence of elastic deformations of both the indented metal and the indenter is captured by the reduced Young’s modulus E

* defined as:

(

)

(6)

with i the Poisson coefficient of the indenter and Ei its Young’s modulus. For a diamond indenter these values equal 0.07 and 1140 GPa respectively. The Young’s modulus E of the aluminium alloy is 66.8 GPa and thus is the reduced Young’s modulus E

* equal to 70.4 GPa.

Next, three different universal dimensionless functions [11] are used to link both curves. Each author has its own functions but in general they have the same structure, given by the equations below [12-16].

∏ (

)

(7)

∏ (

)

(8)

∏ (

)

(9)

These functions are expressed as a function of the representative stress corresponding with a

plastic strain of 0.033, rather than the yield strength . The choice of representative strain simplifies

the solution to Eqs. (7)-(9), since with this value ∏ was empirically observed to be independent of

(Figure 5). is further also denoted as . By solving Eqs. (7), (8) and (9) the unknown material properties can be obtained. To illustrate this principle the functions will be solved graphically.

Given that the Young’s modulus is known for aluminium alloys, only the Hollomon model parameters ( and ) of the stress-strain curve remain unknown. Thus only two dimensionless functions are

needed to solve the problem. Because Le et al. [13] proved that Eq. (8) and Eq.(9) are dependent for sharp indentation, Eq. (7), Figure 5 and Eq. (8), Figure 6 are used to solve the problem.

Figure 5: as a function of indentation parameters [11]

Figure 6: as a function of indentation parameters [11]

With C and E* a priori known (27.6 GPa and 70.4 GPa respectively) a straight line with slope equal

to C/E* can be drawn on Figure 5. From the intersection of this line with the function ∏ , the

representative stress is determined as 337.4 MPa. S and hmax are 4500 kN/m and 10.19 µm respectively and thus from Figure 6 it follows that = 0.088. The yield stress can be calculated using Eq. (5), resulting in σy = 278 MPa.

Figure 7 shows the resulting stress-strain curve which is in good approximation with the one obtained from the tensile test (10% error on and 2% error on ) .

Figure 7: IIT has the potential to provide good approximations for stress-strain behaviour

6 UNIQUENESS OF THE SOLUTION

If the Young’s modulus is not known in advance, it is impossible to reconstruct the stress-strain curve uniquely by sharp indentation, given the dependence of Eqs. (8) and (9). It is clear from Figure 8 that different materials may yield the same indentation force-depth curve for sharp indentation, independent of the dimensionless relations used. Such materials are referred to as ‘mystical materials’. By using a second and different indenter (e.g. a 60° cone, see Figure 8) in a so called ‘dual indenter method’, different materials are uniquely distinguishable. Several dual indenter methods exist [2, 13, 17-20] but with other dimensionless functions than Eqs. (7)-(9). Apart from adopting a second indenter, increasing the penetration depth decreases the likeliness that the force-depth curve is the same for the different materials. However, with increasing penetration depth, unpredictable and therefore undesired frictional effects become increasingly pronounced.

250

300

350

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Tru

e S

tress (

MP

a)

True Strain (%)

Tensile Test

IIT

0 2 4 6

Figure 8: Impossibility to uniquely reconstruct the stress-strain curve with only one Berkovich indenter [2]

7 CONCLUSION

Instrumented indentation allows for the construction of uniaxial stress-strain curves using the recorded indentation force-depth curve as an input. This paper has illustrated a common method, based on curve fitting and applied for sharp indenters. A case study indicates that satisfactory results can be obtained. Transforming an indentation force-depth curve into a uniaxial stress-strain curve is challenging due to poorly quantifiable phenomena such as pile-up and sink-in, and the existence of mystical materials. It is shown that for uniquely determining the stress-strain properties two indentations with different indenters may be necessary. Both indentations can then be analysed using the single indentation method explained in this paper. This leads to a sufficient number of independent equations to solve for the stress-strain curve parameters.

8 ACKNOWLEDGEMENTS

The authors would like to acknowledge the advice of Prof. Didier Chicot from Université De Lille 1.

9 REFERENCES

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Fo

rce

(m

N)

Depth h (µm)

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List of Figures

2.1 Penetration curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Pile-up influences the contact radius (Jeon et al. [5]) . . . . . . . . . . . . . . . 72.3 Pile-up and sink-in depends on nH (Ma et al. [4]) . . . . . . . . . . . . . . . . . 72.4 Linear Relationship between hpile/hc and nH (Kim et al. [6]) . . . . . . . . . . 72.5 Quadratic relationship between hpile/hc and hm/R (Kim et al. [6]) . . . . . . . 72.6 Tangent function for the representative strain provides the best approximation

for higher nH (Jeon et al. [5]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Linear relationship makes it possible to estimate Young’s Modulus Le [14] . . . 102.8 Strain field AL 6082-T6 (Beghini et al. [15]) . . . . . . . . . . . . . . . . . . . . 102.9 Strain field mystical equivalent AL 6082-T6 (Beghini et al. [15]) . . . . . . . . 102.10 Mystical materials showing indistinguishable loading curve . . . . . . . . . . . . 112.11 Sharp indenter (Bucaille et al. [17]) . . . . . . . . . . . . . . . . . . . . . . . . . 122.12 Due to pile-up and sink-in hardness depends slightly on the load (Lee et al. [20]) 132.13 Material behaviour classification under the indenter (Giannakopoulos et al. [21]) 132.14 Penetration curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.15 Linear Relationship Wp/Wt and hc/hm (Tho et al. [18]) . . . . . . . . . . . . . 142.16 For compressive stresses the loading curve is steeper compared to stress-free

state (Kim et al. [13]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.17 For a half cone angle of 42.3◦, friction strongly reduces the amount of pile-up

(Bucaille et al. [17]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.18 The influence of friction is less for higher half cone angles, in this case 70.342.3◦(Bucaille

et al. [17]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.19 θ significantly influences Kick’s constant (Dao et al. [19]) . . . . . . . . . . . . 162.20 Mystical materials discovered by dual indenter combination (Chollacoop et al.

[24]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.21 definition of equi-biaxial residual stress (Yan et al. [25]) . . . . . . . . . . . . . 172.22 Residual stresses can cause two materials to become mystical (Yan et al. [25]) . 172.23 Dimensionless equation in order to obtain σ0 (Yan et al. [25]) . . . . . . . . . . 17

3.1 Flowchart Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Numerical validation of the estimated contact radius ac (Collin et al. [8]) . . . 193.3 Experimental Ph-curve requires a correction for the indenter deformation (Collin

et al. [8]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Reconstruction of the stress-strain curve (Kim et al. [6]) . . . . . . . . . . . . . 203.5 Transformation of the Ph-curve gives good results (Kim et al. [6]) . . . . . . . 203.6 Error as a function of σ0/σUTS (algorithm of [6, 1]) . . . . . . . . . . . . . . . 213.7 Flowchart Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.8 Comparison between Eqs. (3.5) and numerical/experimental results (Beghini

et al. [12]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.9 Flowchart Method 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.10 f7 fixed n (Le [14]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.11 f7 fixed hm/R (Le [14]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

88

89

3.12 Dimensionless relation (Ogasawara et al. [27]) . . . . . . . . . . . . . . . . . . . 253.13 Dimensionless relation (Ogasawara et al. [27]) . . . . . . . . . . . . . . . . . . . 253.14 Flowchart Method 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.15 f1 becomes independent of n for εR equal to 0.033 (Dao et al. [19]) . . . . . . . 283.16 f3 as a function of the representative stress (Dao et al. [19]) . . . . . . . . . . . 293.17 Solution of f2 and f3 (Tho et al. [18]) . . . . . . . . . . . . . . . . . . . . . . . 293.18 Linear Relationship (Le [23]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.19 Additional Residual Area (Ma et al. [4]) . . . . . . . . . . . . . . . . . . . . . . 303.20 Flowchart Method 6, f1 = Eq. (3.40) . . . . . . . . . . . . . . . . . . . . . . . . 313.21 Flowchart Method 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.22 The algorithm of Swaddiwudhipong et al. [32] yields unique solutions, (blue =

Eq. (3.49), green = Eq. (3.50) and red Eq. (3.51)) . . . . . . . . . . . . . . . . 323.23 There is no clear relation between error and nH(algorithm of [32]) . . . . . . . 333.24 Flowchart Method 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.25 Loading (Luo et al. [31]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.26 Unloading (Luo et al. [31]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.27 There appears to be a negative correlation between the strain hardening coeffi-

cient and its error (algorithm of [31]) . . . . . . . . . . . . . . . . . . . . . . . . 353.28 Linear fits θ = 70.3◦ (Le [34]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.29 C60/σ0 i.f.o. C70.3/σ0 (Le [34]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.30 Linear fit 1 (Le [34]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.31 Linear fit 2 (Le [34]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.32 Flowchart Method 9 (f1,f2,f3,f4,f5,f6)= Eqs. (3.64), (3.69), (3.60), (3.67),

(3.70), (3.71) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.33 Error nH as a function of strain hardening coefficient nh (algorithm of Le [34]) 383.34 Error H and σ0 as a function of strain hardening coefficient nh (algorithm of

Le [34]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 Different steps model discussed in this chapter . . . . . . . . . . . . . . . . . . 445.2 The parametric script consists of three different functions . . . . . . . . . . . . 455.3 Different subroutines included in self.preProcess() . . . . . . . . . . . . . . . . . 465.4 Parameters defining a sharp indenter . . . . . . . . . . . . . . . . . . . . . . . . 475.5 Parameters defining specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.6 H = B = 1mm, Lfree = 0.4mm, Lstruct = 0.1mm, 0.2mm, 0.3mm, fineBias =

0.01mm and coarseBias = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.7 H = B = 1mm, Lfree = 0.2mm, 0.4mm and 0.6mm, Lstruct = 0.1mm, 0.2mm,

0.3mm, fineBias = 0.01mm and coarseBias=0.05 . . . . . . . . . . . . . . . . . 485.8 Different boundary conditions applied to specimen . . . . . . . . . . . . . . . . 505.9 Algorithm of the matlab script which extracts hm, S, C and CPU-time . . . . 515.10 Engineering stress-strain curves tested materials . . . . . . . . . . . . . . . . . 525.11 Validation of Matlab model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.1 Where the indenter comes into contact with the specimen, the mesh is very fine 546.2 The linear element with finest mesh gives good results . . . . . . . . . . . . . . 556.3 Good results are obtained for S using the finest linear element . . . . . . . . . 566.4 Calculation time increases strongly with increasing mesh density . . . . . . . . 566.5 For nRO equal to 5 reduced integration reduces the calculation error . . . . . . 566.6 For nRO equal to 5 reduced integration doesn’t reduce the calculation error . . 576.7 For nRO equal to 5 reduced integration doesn’t reduce the calculation error . . 576.8 C does not strongly depends on the depth of imprint . . . . . . . . . . . . . . . 586.9 Higher imprints are needed for f3 to become independent . . . . . . . . . . . . 586.10 Calculation time varies almost linearly with indenter displacement . . . . . . . 59

90

6.11 Radius of imprint for a Vickers equivalent conical indenter and a force of 25Nis about 0.2mm for a steel with yield strength equal to 400Mpa. . . . . . . . . . 59

6.12 With bigger mesh the error on C decreases . . . . . . . . . . . . . . . . . . . . 606.13 S depends strongly on mesh size . . . . . . . . . . . . . . . . . . . . . . . . . . 606.14 Calculation time increases strongly with increasing mesh size. . . . . . . . . . . 606.15 Validation by literature [31] for nickel (nH=0.4) . . . . . . . . . . . . . . . . . 616.16 Validation by literature [24] for aluminium . . . . . . . . . . . . . . . . . . . . 616.17 Validation by literature [42] for steel . . . . . . . . . . . . . . . . . . . . . . . . 626.18 Independence of nH and good agreement with literature . . . . . . . . . . . . . 626.19 Good agreement as well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.20 CSM micro hardness tester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.21 MTS XP nano indenter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.22 Again an overestimation of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.1 Error C due to friction is always lower than 2% . . . . . . . . . . . . . . . . . . 667.2 High influence of friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.3 Calculation time is not strongly influenced by friction . . . . . . . . . . . . . . 677.4 Influence of friction on the force-depth curve is to low to explain difference

between parametric script and literature/experimental (Figure generated fornRO=15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.5 Friction doesn’t oppose the material from piling-up along the indenter, nRO=25. 687.6 Sink in occurs for high strain hardening materials, nRO=5 . . . . . . . . . . . . 687.7 Deformation mesh for nRO=5, µ=0.05 . . . . . . . . . . . . . . . . . . . . . . . 697.8 Deformation mesh for nRO=5, µ=0.25 . . . . . . . . . . . . . . . . . . . . . . . 697.9 Increasing error with bluntness for all materials . . . . . . . . . . . . . . . . . . 697.10 Higher forces needed with increasing tip bluntness . . . . . . . . . . . . . . . . 707.11 Higher dependence tip bluntness for higher strain hardening materials . . . . . 707.12 Higher dependence tip bluntness for higher strain hardening materials . . . . . 707.13 Mystical materials from literature are confirmed for C [4] . . . . . . . . . . . . 717.14 The approximation given by Dao et al. [19] satisfies for f1 (Eq. (3.40) . . . . . 727.15 New approximations need to be made for f3 (Eq. 3.42) . . . . . . . . . . . . . . 73

List of Tables

2.1 Mystical materials (Beghini et al. [15]) . . . . . . . . . . . . . . . . . . . . . . . 112.2 Validation of the method proposed by Yan et al. [25] . . . . . . . . . . . . . . . 17

3.1 Experimental Validation Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Results Method Le [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Validation Method 7, Swaddiwudhipong et al. [32] . . . . . . . . . . . . . . . . 343.4 Validation Method 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Validation Method 9 Le [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1 Specific parameters of the parameters file . . . . . . . . . . . . . . . . . . . . . 455.2 General parameters of the parameter-file . . . . . . . . . . . . . . . . . . . . . . 46

6.1 Different meshes used for validation . . . . . . . . . . . . . . . . . . . . . . . . 54

7.1 Mystical materials from literature [4] are confirmed . . . . . . . . . . . . . . . . 717.2 Exact value of the force-depth curve parameters . . . . . . . . . . . . . . . . . . 717.3 Force-depth parameters belonging to the discussed pipeline steels . . . . . . . . 72

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determination of full range stress -strain curves by means...

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