Optimising mechanical behaviour of new advanced steels based on fine non-equilibrium microstructures

Optimising mechanical behaviour of new advanced steels

based on fine non-equilibrium microstructures

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op [09, 12, 2015] om [10:00] uur

door

Farideh HAJYAKBARY

Master of Science in Metallurgy and Materials Engineering, University of Tehran, Tehran, Iran

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. J. Sietsma

Copromotor: Dr. M. J. Santofimia

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof. dr. ir. J. Sietsma, Technische Universiteit Delft, the Netherlands, promotor Dr. M. J. Santofimia, Technische Universiteit Delft, the Netherlands, copromotor Dr. G. Miyamoto, Tohuku University, Tohoku, Japan

Onafhankelijke leden:

Prof. dr. I.M. Richardson, Technische Universiteit Delft, the Netherlands Prof. dr. ir. L.A.I. Kestens, Ghent University, Ghent, Belgium

Prof. dr. Kip Findley, Colorado School of Mines, Colorado, USA

Dr. D. S. van Bohemen, Tata Steel Research Development and Technology, IJmuiden, the Netherlands

This research was carried out under the project number M41.10.11437 in the framework of the Research Program of the Materials innovation institute (M2i) in the Netherlands (www.m2i.nl).

ISBN 978-94-6295-398-7

Copyright © 2015, Farideh HajyAkbary f.hajyakbary@gamil.com

All rights reserved. No part of the material protected by this copy right notice may be reproduced or utilized in any form or by any means, electronically, including photocopy, recording or by any information storage and retrieval system, without permission from the author.

This thesis is dedicated to

my little angel Arshida

Contents

iv TABLE OF CONTENTS

1 Introduction --- 1

1.1 Research aims --- 3

1.2 Content of the thesis--- 3

1.3 References --- 5

2 Effects of specimen size on the tensile behavior of steels --- 7

2.1 Introduction --- 8

2.2 Mathematical modelling of the effective parameters on the crosshead displacement --- 10

2.3 Experimental procedure --- 12

2.4 Results and discussion --- 15

2.5 Conclusions --- 23

2.6 References --- 24

3 An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density in Lath Martensitic Structures --- 27

3.1 Introduction --- 28

3.2 Calculation of dislocation density --- 29

3.3 Experimental procedure --- 34

3.4 Results and discussion --- 36

3.5 Conclusions --- 47

3.6 References --- 48

4 Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Quenching and Partitioning Steel --- 51 4.1 Introduction --- 52 4.2 Experimental procedures --- 53 4.3 Results --- 53 4.4 Discussion --- 63 4.5 Conclusions --- 68 4.6 References --- 69

5 Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Quenching and Partitioning Steel --- 71

5.1 Introduction --- 72

5.2 Theoretical calculation of the yield strength of the constituent phases --- 72

5.3 Experimental procedure --- 74

5.4 Results --- 77

5.5 Discussion --- 88

5.6 Conclusions --- 92

5.7 References --- 93

6 Optimising Mechanical Properties of a 0.3C-1.5Si-3.5Mn Quenching and Partitioning Steel --- 95

6.1 Introduction --- 96

6.2 Experimental procedure --- 96

6.3 Results and discussion --- 98

6.4 Conclusion --- 101

6.5 References --- 102

7 Conclusions and recommendations --- 103

7.1 Conclusions --- 104

Contents

Summary---107

Samenvatting---109

Acknowledgements---113

List of publications---115

1

CHAPTER 1

1

Introduction

Car industry, as a main consumer of steels, is interested in steels with high strength and high formability to reduce the fuel consumption and increase passenger safety [1]. Therefore, a significant research effort has been directed towards the development of Advanced High Strength Steels (AHSS) which have a good combination of high strength and ductility. The microstructure of AHSS consists of at least two different phases, one hard phase like martensite or bainite and one soft phase such as ferrite or retained austenite [2]. The AHSS grades that are currently being applied or are under increased investigation by steel researchers can be categorized into three generations [3], as presented in Fig. ‎1-1. The first generation of AHSS contains fairly low alloy steels, with a multiphase microstructure that is primarily ferritic-based. These steels are well established and they are currently the most applied AHSS which results, apart from their improved strength and formability, from their low price. This generation includes dual phase (DP) steels, transformation induced plasticity (TRIP) steels, complex phase (CP) steels and martensitic steels. The second generation of AHSS have excellent mechanical properties, but they are highly alloyed steels, resulting in a significant cost increase. This generation involves high strength steels such as austenitic twinning induced plasticity (TWIP) steels and lightweight steels with induced plasticity (L-IP) [4]. Academic and industrial researchers are interested to develop the third generation of AHSS with strength and ductility at the same levels that are exhibited by the second generation but with lower alloying levels [5]. To develop the third generation of AHSS, attention has been paid to processes which deliver a hard bainitic or martensitic matrix containing a dispersion of retained austenite. These microstructural components are phases that are formed in non-equilibrium conditions and/or remain in the steel microstructure under metastable conditions [6].

Introduction

2

Fig. ‎1-1 Elongation-strength relationships for different grades of steels [7].

One of the most promising heat treatments for the development of the third generation of AHSS is the Quenching and Partitioning (Q&P) process. A schematic of the Q&P treatment is illustrated in Fig. ‎1-2. The Q&P process involves rapid quenching of an austenitic microstructure to a temperature lower than the martensite-start temperature (Ms) to form a

controlled fraction of martensite. Here, the martensite which is formed during the initial quenching is called initial martensite. The process is followed by an isothermal treatment, either at or above the initial quenching temperature. This isothermal holding process is called partitioning process and it is aimed to allow carbon to partition from supersaturated martensite to austenite and stabilizes austenite. The treatment is ended by quenching the microstructure to room temperature [8]. Secondary martensite may form during the final quenching, if some parts of austenite do not become stable enough [9]. Moreover, bainite may form by decomposition of austenite during the isothermal treatment [10]. Accordingly, the Q&P microstructures can be composed of initial martensite, bainite, secondary martensite and retained austenite and, depending on the final microstructure, varying ranges of mechanical properties can be achieved. In this matter, the key to optimise the mechanical properties of these steels and make them commercialized is understanding the relation of the microstructural and mechanical properties. This can be achieved by investigation of the contributions of dislocations, precipitations, morphology and chemical composition of the phases on their independent mechanical behavior as well as studying the synergistic influence of the phases on the ductility and strength of the Q&P steels.

The Q&P process is a complicated heat treatment and the final microstructure is sensitive to the exact temperature profile and therefore also to the temperature gradients. At the laboratory scale, an accurate control of the heating process and avoiding temperature inhomogeneity within the specimen is possible by heat treating small specimens. The evaluation of the mechanical properties of these small specimens can be performed by using microtensile tests. Using the microtensile test for mechanical property characterization creates concerns about whether the measured mechanical properties are influenced by the specimen dimensions. Therefore, it is important to study the relation of the results of the microtensile tests with the performance of the material at a macro-scale.

Chapter 1

3

Fig. ‎1-2Scheme of the Q&P process. Here, ɣ is initial austenite, M1 is initial martensite, M2 is secondary

martensite, B is bainite, RA is retained austenite, Ci is carbon content of the steel, Cɣ is carbon content of

austenite and CM1 is carbon content of initial martensite.

1.1 Research aims

The aims of the research described in this thesis are three-fold: (1) To correlate small-scale and conventional tensile test methods. (2) To investigate microstructural development during the Q&P process.

(3) To identify the contributions of (non-equilibrium) microstructural components on the mechanical properties of the Q&P steels, in order to deliver AHSS with superior mechanical properties.

1.2 Content of the thesis

In line with the research aims, the thesis is divided into three parts: chapter 2 investigates the influence of specimen size on the tensile behaviour of steels, chapters 3 and 4 outline the characterization of the microstructural properties of Q&P microstructures developed in a 0.3C-1.6Si-3.5Mn (wt.%) steel with non-homogenous chemical composition and chapters 5 and 6 discuss the relation between the tensile properties and the microstructural properties.

Chapter 2 studies the influence of the specimen geometry on the tensile behaviour of steels. This is done by tensile testing of miniature and standard specimens from different grades of steels. Moreover, a model is developed to determine the elastic strain of the miniature specimens from the crosshead displacement of the tensile test machine.

Chapter 3 introduces an improved method to measure dislocation density of a lath martensitic steel by applying an X-ray diffraction profile analysis method. The proposed method is would choice due to the considered range of the Fourier length. This method leads to a dislocation density that is in good agreement with the dislocation density determined based on the dislocation strengthening.

Chapter 4 investigates the relations between the Q&P process parameters, the local chemical composition and the microstructural properties. A comprehensive microstructural analysis of the constituent phases, including their volume fractions and chemical compositions, is performed. The interactions between bainite formation, carbide

Introduction

4

precipitation and carbon partitioning process on the microstructural development is discussed.

Chapter 5 studies contributions of dislocations, precipitates, morphology and chemical composition of the constituent phases on their independent yield strength. The influence of instability of austenite on the yield strength of the Q&P microstructures is investigated by applying in-situ X-ray diffraction. Moreover, the synergistic influence of the phases on the ductility and strength of the Q&P microstructures is analysed.

Chapter 6 discusses the key microstructural parameters which result in developing Q&P microstructures with a good combination of tensile strength and ductility. Furthermore, the mechanical properties of the developed microstructures are compared with other types of AHSS.

Chapter 7 summarizes the main conclusions of the project and provides recommendations for future research.

Chapter 1

5 1.3 References

[1] R. Kuziak, R. Kawalla and S. Waengler, “Advanced high strength steels for automotive industry”, Archives of Civil and Mechanical Engineering, vol. 8, pp. 103-117, 2008.

[2] E. De Moor, P. J. Gibbs, J. G. Speer, D. K. Matlock and J. G. Schroth, “Strategies for third generation advanced high-strength steel development”, AIST Transactions, Iron & Steel Technology Transactions, vol. 7, pp. 133-144, 2010.

[3] S. Keeler and P. Ulintz, “Advanced high strength steels solve glowing demands for formability”, Met. Form., vol. 45, pp. 24-28, 2011.

[4] L. Samek, E. Arenholz, R. Schneider and J. Gentil, “Influence of the thermal processing on the microstructure and mechanical properties of a high-performance high-manganese steel”, Metal Conference, Czech Republic, 2012.

[5] A. Grajcar, R. Kuziak and W. Zalecki, “Third generation of AHSS with increased fraction of retained austenite for the automotive industry”, Archives of Civil and Mechanical Engineering, vol. 12, pp. 334-341, 2012.

[6] D. K. Matlock and J. G. Speer, “Design considerations for the next generation of advanced high strength sheet steels”, The Conference of Korean Institute of Metals and Materials, Korea, 2006.

[7] J. N. Hal, “Evolution of Advanced High Strength Steels in Automotive Applications”, Great Design in Steels Seminar, 2011.

[8] D. V. Edmonds, K. He, F. C. Rizzo, B. C. De Cooman, D. K. Matlock and J. G. Speer, “Quenching and partitioning martensite—A novel steel heat treatment”, Mater. Sci. Eng. A, vol. 438–440, pp. 25–34, 2006.

[9] J. Mola and B. C. De Cooman, “Quenching and Partitioning (Q&P) processing of martensitic stainless steels”, Matal. Mater. Trans. A, vol. 44, pp. 946-967, 2013.

[10] A. J. Clarke, J. G. Speer, M. K. Miller, R. E. Hackenberg, D. V. Edmonds, D. K. Matlock, F. C. Rizzo, K. D. Clarke and E. De Moor, “Carbon partitioning to austenite from martensite or bainite during the quench and partition (Q&P) process: A critical assessment”, Acta Materal., vol. 56, pp. 16-22, 2008.

Introduction

7

CHAPTER 2

2

Effects of specimen size on the tensile

behavior of steels

*

Abstract

The effect of the specimen’s parallel length on the tensile behavior of four different grades of steels was studied. The steels that were object of this analysis were one interstitial free steel, two dual phase steels with different fraction of ferrite and one martensitic steel. Miniature specimens were tested in two different geometries, with parallel lengths of 4 mm and 3 mm. The measurement of elastic strain in the miniature specimens was done by means of the crosshead displacement of the tensile-test machine. Since the elastic elongation of the fillet zones of the tensile specimen and machine compliance were recorded along with the elastic elongation of the tensile specimen as the crosshead displacement, measuring the elastic strain by this method led to an overestimation of strain. A mathematical model for calculating the elastic elongation of the fillet-zones of a dog-bone tensile specimen and the machine compliance as a function of the applied load was proposed. The subtraction of the fillet-zones elongation and the machine compliance from the crosshead displacement allowed the calculation of the elastic elongation of miniature specimens, leading to values in agreement with strains measured via digital image correlation. Comparing the tensile behaviour of the miniature specimens and A80 standard specimens showed that reducing the specimen parallel length did not influence the observed yield stress and tensile strength of the steel. The fracture strain of the miniature specimens was higher than of the standard ones. A correction method was applied to correct the fracture strain of the miniature specimens.

Keywords: Micro-tensile test, Crosshead displacement, Fillet-zones, Machine compliance Yield strength, Tensile strength, Fracture strain.

*

This chapter is based on a scientific paper:

F. HajyAkbary, M. J. Santofimia and J. Sietsma, Elastic strain measurement of miniature tensile specimens, Experimental Mechanics, vol. 54, pp. 165-173, 2014.

Effects of specimen size on the tensile behavior of steels

8

2.1 Introduction

Nowadays, the development of new fabrication technologies and miniaturized products which restrict the specimen dimensions results in an increased use of miniaturized tests for studying mechanical properties of materials. Additionally, analysing miniature specimens instead of standard ones saves material and time for both industrial and academic researchers.

Generally, tensile specimens used in miniaturized tests are dog-bone shaped. A dog-bone tensile specimen can be divided into five zones: the parallel-zone, the two fillet-zones and the two grip-zones as shown in Fig. ‎2-1. Dimensions of miniature specimens deviate from ASTM standards, the parallel length of miniature specimens being in the range from 1 mm [1] to several millimetres [2, 3]. This situation naturally invites concern as to whether the geometries/dimensions of the miniature specimens have any influence on the experimental results and, if so, how strong these influences are [4].

According to an investigation by Zhao et al. [5] reducing the parallel length has no influence on the yield stress of ultra-fine grained copper, but produces an increase on ductility. For the case of steel, there is a lack of research considering the influence of the specimen parallel length on the mechanical behavior during microtensile testing. Therefore it is essential to do a comprehensive study on the tensile behavior of steels with miniature dimension.

Another difficulty of application of miniature specimens to characterize the mechanical properties is that measuring the precise elastic strain is challenging. The elastic strain of standard specimens can be determined by using clip-on extensometers, but this option is difficult in miniature specimens due to their small dimensions [6]. At present, there exist non-contacting strain measuring systems such as laser and video extensometers [7] but their high price and complex set-up limit their application. Therefore, using extensometers is not a common method for elastic strain measurements of miniature specimens.

There are two main alternatives to the use of extensometers for determining the elastic strain of miniature specimens: the application of Digital Image Correlation (DIC) and the measurement of the strain from the crosshead displacement that is recorded by the tensile-test machine. The DIC method consists of the measurement of the strain of the specimen during testing by comparing, pixel by pixel, images of the specimen before and after elongation [8]. This technique requires the use of a high resolution camera, followed by data post-processing by using the corresponding software. On the other hand, the measurement of the elastic strain of miniature specimens from the crosshead displacement does not need any extra equipment and data processing. However, one drawback of using crosshead displacement for elastic strain measurements is that the elongation of the fillet-zones of the specimen and the tensile machine parts are included in the crosshead displacement. Therefore, this method overestimates the elastic strain of miniature specimens, as will be detailed further on.

Chapter 2

9

Fig. ‎2-1 Scheme of a tensile-test machine and a dog-bone specimen which are modelled by five series of springs.

Although elongation of the fillet-zones of miniature specimens during tensile tests has been reported by different researchers [5], most of the published investigations considered the fillet-zones as rigid items when measuring the strain from the crosshead displacement [9]. To eliminate the influence of the fillet-zones elongation on the measured elastic strain, Koubaa et al. [10] defined the initial length in the strain calculation as the total length of the parallel-zone and the two fillet-zones. The strain measured by their proposed approach is in better agreement with the strain calculated using finite element analysis than the strain measured by dividing the crosshead displacement by the initial length of the parallel-zone. However, their proposed method underestimates the strain, since the strain in the fillet-zones is smaller than in the parallel-zone.

A tensile-test machine is not a monolithic part and it consists of different parts like the machine frame together with measuring and fixturing devices. The machine components are not rigid and they deform elastically in tension. These elongations, which are known as machine compliance, are included in the recorded crosshead displacement [11]. The elastic elongation of miniature specimens is relatively small and the machine compliance has significant effect on the crosshead displacement. Therefore, the machine compliance should be precisely considered when the elastic strain of miniature specimens is to be measured from the crosshead displacement. The ASTM standard for tensile testing of single filament materials determines the machine compliance by assuming the tensile-test machine and specimen as two linear springs which are connected in series [12]. According to this standard, the machine stiffness depends on the specimen stiffness and dimensions. However, experimental measurements of the machine stiffness reveal that the machine stiffness is a function of the applied load and it is independent of the specimen properties [13].

In this chapter, miniature and standard specimens of four different types of steels were tested in tension. The influence of the specimen dimension on the yield strength, the tensile strength and the fracture strain was studied. Furthermore, a new correction method was established to determine the elastic strain of miniature specimens from the crosshead displacement. This was done by the calculation of the fillet-zones elastic elongation and the machine compliance and subtracting their values from the recorded crosshead displacement. This correction method was used to calculate the elastic strain in the parallel-zone of the miniature specimens. Resulting values were in good agreement with the elastic strains measured from DIC method.

Effects of specimen size on the tensile behavior of steels

10

2.2 Mathematical modelling of the effective parameters on the crosshead displacement A tensile-test system consists of tensile specimen and tensile-test machine. The elastic elongation of the tensile-test system components during the tensile test can be considered as the elongation of a series of springs. In this chapter, the tensile test system is modelled by five springs in series: two for the tensile-test machine, two for the fillet-zones and one for the parallel-zone of the tensile specimen (Fig. ‎2-1). In the current approach, each arm of the tensile-test machine is modelled by a spring with stiffness of 2𝐾𝑚. A factor 2 is included to

simplify the calculation procedure so the total stiffness of the tensile-test machine can be considered as a single spring with stiffness of 𝐾𝑚. The apparent stiffness that is displayed by

the tensile-test system (𝐾𝑎𝑝𝑝) is calculated as: 1 𝐾𝑎𝑝𝑝 = 1 𝐾𝑚+ 1 𝐾𝑝+ 2 𝐾𝑓, ‎ 2-1

where 𝐾𝑚, 𝐾𝑝 and 𝐾𝑓 are the stiffness of the tensile-test machine, the stiffness of the

parallel-zone and the stiffness of one fillet of the fillet-zones of the tensile specimen, respectively. The total elongation of the tensile-test system is recorded as crosshead displacement by the tensile-test machine. This recorded displacement is defined here as the apparent elongation of the tensile specimen (∆𝑙𝑎𝑝𝑝) and it can be calculated by:

∆𝑙𝑎𝑝𝑝= ∆𝑙𝑚+ ∆𝑙𝑝+ 2∆𝑙𝑓, ‎2-2

where ∆𝑙𝑚, ∆𝑙𝑃 and ∆𝑙𝑓 are the elongation of the tensile-test machine parts, the elongation

of the parallel-zone and the elongation of one of the fillet-zones of the tensile specimen, respectively.

2.2.1 Elastic elongation of the fillet-zones

In this section, a model is developed to calculate the elastic elongation of the fillet-zones. It is known that the elastic strain can be calculated from the Hooke’s Law:

𝜀 = 𝜎

𝐸 , ‎2-3

where ε, σ and E are the elastic strain, the stress and the Young’s modulus of the material, respectively. The elastic strain of one fillet of the fillet-zones (𝜀𝑓) at location x, which is the

distance between the boundary of the grip-zone and the fillet-zone (Fig. ‎2-2), is determined as:

𝜀_𝑓 = 𝐹 𝐸𝑑𝑤𝑓(𝑥),

‎

2-4

where F is the applied force and d is the specimen thickness. Here, 𝑤𝑓(𝑥) is the specimen

Chapter 2

11

Fig. ‎2-2. Scheme of a fillet-zone of a dog-bone tensile specimen.

The function 𝑤𝑓(𝑥) is defined as:

𝑤_𝑓(𝑥) = 𝑤_𝑔− 2 √𝑟2_{− (𝑟 − 𝑥)}2_, ‎2-5

where 𝑤𝑔 is the specimen width at the boundary of the fillet-zone and the grip-zone and r is

the fillet-zone radius. The elongation of a fillet-zone is determined as:

∆𝑙_𝑓= ∫ 𝜀₀𝑟 𝑓𝑑𝑥 , ‎2-6

Thereupon, the elongation of a fillet-zone is calculated by substituting Eq. ‎2-4 and Eq. ‎2-5 into Eq. ‎2-6, as:

∆𝑙_𝑓= ∫ 𝐹 𝐸𝑑× 𝑑𝑥 𝑤_𝑔− 2√𝑟2_{− (𝑟 − 𝑥)}2 𝑟 0 . ‎ 2-7

The ratio of the elongation of a fillet-zone to the parallel-zone is:

∆𝑙𝑓 ∆𝑙𝑝

=

= 𝛼

where 𝑤𝑃 and 𝑙𝑃 are the width and the length of the specimen in the parallel-zone,

respectively. The parameter 𝛼 is a geometrical coefficient and it is independent of the applied force and the material.

2.2.2 Machine compliance

Elastic elongation of components of a tensile-test machine during the tensile test can be modelled by the elongation of a spring with stiffness 𝐾𝑚. It is well established that the

stiffness of a tensile-test machine (𝐾𝑚) is a non-linear function of the applied force and it is

independent of the specimen type and geometry [13]. The machine compliance (∆𝑙𝑚) is

Effects of specimen size on the tensile behavior of steels 12 ∆𝑙_𝑚 = 𝐹 𝐾_𝑚, ‎ 2-9

where F is the applied force. Since 𝐾𝑚 is a function of the applied load, it can be concluded

from Eq. ‎2-9 that the compliance of a tensile-test machine is a function of the applied load. The compliance function is invariant for different specimens and it can be used to calculate the machine compliance at a certain value of the applied force.

2.2.3 Method to calculate the parallel-zone strain

The elastic elongation of the parallel-zone can be calculated from the crosshead displacement by taking the following steps:

a) The first step is specifying the machine compliance function. In this matter, a specimen is tested by the tensile-test machine while the reference elongation through its parallel-zone, ∆𝑙_𝑝𝑟𝑒𝑓, is recorded by a direct method such as DIC. The machine compliance, at a certain value of force, is calculated by combining Eq. ‎2-2 and Eq. ‎2-8 and substituting the corresponding values of the crosshead displacement (∆𝑙𝑎𝑝𝑝) and the reference elongation

(∆𝑙_𝑝𝑟𝑒𝑓) in:

∆𝑙_𝑚 = ∆𝑙_𝑎𝑝𝑝− (1 + 2𝛼)∆𝑙_𝑝𝑟𝑒𝑓. ‎2-10

The geometrical coefficient 𝛼, is computed by considering the specimen dimensions in Eq. ‎2-8. Finally, the compliance function (∆𝑙𝑚) of the tensile-test machine is determined by

plotting the machine compliance-force (∆𝑙𝑚𝑣𝑠. 𝐹) diagram.

b) Then, the corrected elongation of the parallel-zone (∆𝑙𝑝𝑐) for every tensile specimen at

a certain value of the applied load and apparent elongation is determined by combining Eq. ‎2-2 and Eq. ‎2-8 and substituting the machine compliance function in the following equation:

∆𝑙_𝑝𝑐 ₌ 1

(1 + 2𝛼)(∆𝑙𝑎𝑝𝑝− ∆𝑙𝑚).

‎

2-11

The corrected elastic strain within the parallel-zone of miniature specimens (𝜀𝑝𝑐) is

expressed as: 𝜀_𝑝𝑐 ₌ 1 (1+2𝛼)( ∆𝑙𝑎𝑝𝑝−∆𝑙𝑚 𝑙𝑝 ). ‎ 2-12 Equation ‎2-10 is independent of the material type and it can be determined for every tensile specimen at a certain value of the applied load and apparent elongation.

2.3 Experimental procedure

In this work, the tensile behavior of four different steels: e. g. one interstitial free steel (IF), two dual phase steels with different fraction of ferrite (DP1000 and DP600) and one martensitic steel (M1400), was studied using specimens with different sizes and geometries. The key points of the experimental procedure are given in this section.

Chapter 2

13

Table ‎2-1. Nominal dimensions of the standard and miniature specimens (mm)

Specimen Width Thickness Parallel length Grip area Overall length Fillet radius T.120 20.0 1.0 120.0 30.0×40.0 230.0 15.0

T.4 1.0 1.0 4.0 4.0×2.0 10.0 1.0

T.3 0.8 0.8 3.0 3.5×3.0 10.0 0.5

Dog-bone miniature specimens with two different dimensions and standard A80 tensile specimens were tested in tension. The effect of the specimen dimensions on the elastic elongation was studied by analysing the elastic strain measurement of the miniature specimens, via DIC, and the standard specimens. Moreover, the elastic strain of the miniature specimens was measured from the crosshead displacement and the results were compared with the elastic strain which was measured by DIC. A mathematical model was developed to correct the elastic strains of the miniature specimens which were measured from the crosshead displacement. This was done based on the elastic strain measurement of the miniature specimens from steel M1400. This model included the subtraction of the fillet-zones elongation and the machine compliance from the crosshead displacement. The proposed correction method was experimentally validated with miniature specimens from different types of steels (DP1000, DP600 and IF).

2.3.1 Specimens geometry

The specimen dimensions were listed in Table ‎2-1. In the appellation of the specimens, T refers to the tensile specimen and the next number shows the specimen parallel length in millimetre. The dimensions of the miniature specimens satisfied some of the ASTM standard requirements. The standard indicates that the ratio of the parallel length to the parallel width was 4 and the radius of the fillet-zone is equal or greater than the width of the zone [14]. Also, to ensure that the specimen failure will occur within the parallel-zone, the standard specifies a ratio of the grip width to the parallel width equal or higher than 1.5 [15]. Miniature specimens were machined from sheets using an electro discharge machine while the specimen axis was perpendicular to the rolling direction.

2.3.2 Tensile testing

Three specimens were tested for each group of geometries and steel, except for the M1400 steel in which only one specimen was tested due to slippage difficulties. All the standard and miniaturized tensile tests were done until failure. Standard specimens were tested with a "Schenk Trebel tensile-test machine 100KN". A uniform elongation region with initial length of 80 mm was considered as the gauge length of the standard specimens. The strain of the standard specimens (εps), within the gauge length, was measured using a

contact extensometer.

Miniaturized tensile tests were performed using a "Deben Microtest 5KN Tensile Stage". There was no control on the stress which was applied to the miniature specimens during fixing them in the tensile machine. For the miniaturized tests, the apparent strains (𝜀𝑎𝑝𝑝)

were determined by dividing the recorded crosshead displacement by the initial length of the parallel-zone of the specimen. Furthermore, the reference elastic strain (𝜀_𝑝𝑟𝑒𝑓) of the miniature specimens was measured by DIC technique. The tensile tests were repeated for the T.3 and T.4 specimens from DP and IF steels by using "Shimadzu AG-X 50KN" tensile test

Effects of specimen size on the tensile behavior of steels

14

machine, while the applied force to fix the specimens in the tensile machine was kept below the yield strength of the specimens.

The apparent strain rates of the standard and miniaturized tests were calculated by dividing the crosshead velocity by the initial length of the parallel-zone of the specimen. The standard and miniaturized tensile tests were performed at apparent strain rate of 2×10-3 s-1 at room temperature with the exception of the IF T.4 miniature specimens, which were tested at apparent strain rate of 4×10-5 s-1. The reason of testing the IF miniature specimens at lower strain rate was that its limited elastic elongation occurred in a few seconds. On the other hand, the DIC method requires that a camera makes consecutive images of the deforming specimen in a certain time interval, which was 6 seconds in the current research. Therefore, in the case of IF miniature specimens, a lower strain rate was required to take an adequate number of images for accurate determination of the elastic strain. Before the tensile tests, the flat surfaces of the miniature specimens were ground using 1200 grit SiC papers.

2.3.3 Digital image correlation

The Digital Image Correlation (DIC) method was applied for measuring the elastic strain of the T.4 miniature specimens within the parallel-zone. DIC is an optical method that determines the elongation of an object during the mechanical tests. With this technique a mathematical correlation analysis is used to calculate the strain of the specimen from a series of consecutive digital images of the specimen surface [16]. To obtain accurate results with the DIC, the specimen needs to have a recognizable speckle pattern on its surface.

In this research, to guarantee a proper speckle pattern, the specimen surface was painted with a white spray and then a random black pattern was finely created with a black spray. An Oxford camera recorded images from the full parallel-zone with a resolution of 1024×768 pixels. Finally, the "digital image correlation and tracking" toolbox of the MATLAB code was used for strain calculations. The accuracy of the DIC technique for determining strain depends on the minimum detectable displacement, which is the spatial size of a pixel in an image. The spatial size of a pixel can be calculated by dividing the specimen dimension to the camera resolution [17]. In this study, the minimum displacement that can be characterized for the T.4 miniature specimens was given by dividing the parallel length of the specimen (4 mm) by the vertical resolution of the image (1024), leading to 4×10-3 mm. Therefore, the detection strain limit of the DIC was considered 10-3 (mm/mm) and only strains in the range from 10-3 (mm/mm) to the yield point were determined by the DIC technique.

Chapter 2

15

Fig. ‎2-3 (a) 3D model of the one fillet of the fillet-zones with the applied boundary conditions and (b) the elastic elongation distributions in the fillet-zone (elongation scale is in mm).

2.3.4 Finite element modelling

The proposed model for calculating the elastic elongation of one fillet of the fillet-zones (Eq. ‎2-7) was validated by finite element simulation of the elastic elongation of a fillet-zone. Finite element simulations were done using the commercial code ABAQUS 11.6-1. A 3D model of one fillet of the fillet-zones of the T.4 miniature specimen was developed. The elastic properties of the material in the simulation were taken from the measurement performed on the M1400 standard specimens, with Young’s modulus and the Poisson’s ratio of 210 GPa and 0.3, respectively. The fillet-zone was modelled by solid element C3D8R which is an 8-node linear brick with reduced integration and hourglass control. As it is shown in Fig. ‎2-3a, to simulate the tensile test, the left side of the fillet-zone was encastered while the right side of the fillet was elongated. The fillet-zone was elongated by 0.02 mm. This is equal to the elongation of the parallel-zone of the T.4 miniature specimen when it was deformed by the elastic strain limit (0.005 mm/mm) of the M1400 standard specimen.

2.4 Results and discussion

Fig. ‎2-4 presents the engineering stress-strain graphs of the miniature specimens and standard specimens for all groups of steels. The tensile curves of miniature specimens were measured by using Deben machine. During fixing the miniature specimens to the tensile machines, there was no control on the stress which was applied to the specimens. If the applied force exceed the yield strength of the tested specimen, the yield strength and tensile strength measurements would be affected. The strain of the miniature specimens was measured based on the crosshead displacement of the tensile machine. According to this figure, the elastic strain of miniature specimens is higher than standard specimens. Furthermore, the yield strength and tensile strength as well as fracture strain increases by reducing the specimen gauge length. The influence of the microtensile measurements on the tensile behaviour of steels is discussed in this section.

2.4.1 Elastic Strain Measurement of Miniature Tensile Specimens

Results on the calculation of the elastic strain based on the proposed model and verification of this methodology are presented and discussed in this section.

Effects of specimen size on the tensile behavior of steels

16

Fig. ‎2-4 Engineering stress-strain diagrams of miniature specimens with 3 (T.3) and 4 (T.4) mm parallel length and standard specimens (T.120) with parallel length of 120 mm corresponding to steels (a) IF, (b) DP600, (c)

DP1000 and (d) M1400.

 Experimental measurement of the elastic strain

The elastic strain-stress (𝜀ps− 𝜎) graphs of the standard and the apparent elastic

strain-stress (𝜀app− 𝜎) graphs of the miniature specimens for steels M1400, DP1000, DP600 and IF

are illustrated in Fig. ‎2-5. Additionally, it presents the reference elastic strain-stress (𝜀pref−

𝜎) curves of the miniature specimens that were measured by DIC. Fig. ‎2-5 shows that the reference elastic strain-stress curves of the miniature specimens and the elastic strain-stress of the standard specimens are in excellent agreement. This confirms that the specimen geometry has no effect on the actual measured elastic elongation of material and the elastic slope of the standard and reference miniaturized tests is independent of the specimen geometry. Furthermore, the apparent elastic slope of the miniaturized tests, which were determined from apparent elastic strain-stress (εapp− σ) graphs, are lower than the

standard ones. This indicates that the fillet-zones elongation and machine compliance strongly increase the crosshead displacement within the miniaturized tests.

 Calculation of the fillet-zones elongation and the machine compliance

To evaluate the accuracy of the developed model, the elastic elongation of one fillet (∆𝑙𝑓)

of the fillet-zones of the T.4 miniature specimen from M1400 was simulated by finite element analysis. The distribution of the elastic elongation in the fillet-zone is shown in Fig. ‎2-3b and it indicates that the elastic elongation is not uniform in this zone, contrary to the assumption of the uniform elongation in the parallel-zone and the fillet-zones which was done by Koubaa et al. [10].

Chapter 2

17

Fig. ‎2-5 Elastic strain-stress curves of (a) M1400 (standard and T.4 miniature specimens), (b) DP1000 (standard and T.4 miniature specimens), (c) DP1000 (standard and T.3 miniature specimens), (d) DP600 (standard and T.4

miniature specimens) and (e) IF (standard and T.4 miniature specimens) steels. The apparent elastic strain (𝜀𝑎𝑝𝑝), the reference elastic strain (𝜀𝑝𝑟𝑒𝑓) and the corrected elastic strain (𝜀𝑝𝑐) of the miniature specimens were

determined from the crosshead displacement, DIC and the proposed method, respectively. The elastic strain of the standard specimens (𝜀𝑝𝑠) was determined with an extensometer.

The elongation of the fillet-zone was computed from the finite element simulation and the proposed model (Eq. ‎2-7) and the results of both calculations are illustrated as the force-elastic elongation curves in Fig. ‎2-6. The results show that the proposed mathematical model calculates the fillet elongation accurately.

Effects of specimen size on the tensile behavior of steels

18

Fig. ‎2-6 The elongation-force curves of one fillet of the fillet-zones of T.4 miniature specimen from steel M1400 calculated by FEM (dashed line) and Eq. ‎2-7 (dotted line).

Substituting the T.4 miniature specimens dimensions from Table ‎2-1 in Eq. ‎2-8, the parameter 𝛼 is found to be equal to 0.19 for these specimens. Furthermore, the ratio between the elastic elongation of one fillet of the fillet-zones and the parallel-zone was determined for the standard specimens as 𝛼 = 0.09. The low value of 𝛼 for the standard specimens in comparison to 𝛼 = 0.19 for the T.4 miniature specimens shows that the fillet-zones elongation has a much smaller effect on the crosshead displacement of the standard specimens. For the T.4 miniature specimens from all four groups of steels, the reference elongation of the parallel-zone (∆𝑙𝑝𝑟𝑒𝑓) was measured by DIC. The machine compliance, at

different levels of ∆𝑙_𝑝𝑟𝑒𝑓, can be determined by using Eq. ‎2-10 and subtracting the elastic elongation of the fillet-zones and parallel-zone of the specimen from the crosshead displacement. The machine compliance vs. applied force diagram is illustrated in Fig. ‎2-7. For all the tested steels variations of the machine compliance versus the applied force follows the same trend and it is independent of the material type. By interpolation of the T.4 miniature specimen from M1400 data, the compliance function of the tensile-test machine was expressed as a bilinear curve by:

∆𝑙_𝑚 = { 0 𝐹 < 150 𝑁 8.1 × 10−5₍𝑚𝑚 𝑁 ) 𝐹 − 0.0096(𝑚𝑚) 150 ≤ 𝐹 < 450 𝑁 2.1 × 10−4₍𝑚𝑚 𝑁 ) 𝐹 − 0.0678(𝑚𝑚) 450 ≤ 𝐹 < 1200 𝑁 ‎ 2-13

For forces lower than 150 N, the machine compliance is insignificant and its value is assumed zero. For all the studied steels, the contributions of the fillet-zones elongation and the machine compliance to the crosshead displacement of one T.4 miniature specimen are illustrated in Fig. ‎2-8. In this figure, all the specimens were deformed equivalently (2×10-3 mm) by using DIC method. In this fihure the elastic elongation of the fillet-zones and the machine compliance were determined by using Eq. ‎2-7 and Eq. ‎2-10, respectively.

Fig. ‎2-8 shows that the elongation of the fillet-zones is equivalent for all the steels and as it was discussed in the section ‘’Elastic elongation of the fillet-zones’’, the ratio of the elongation of the one fillet of the fillet-zones to the elongation of the parallel-zone is independent of the material type.

Chapter 2

19

Fig. ‎2-7 Machine compliance-applied force curve of the T.4 miniature specimens from IF, DP 600, DP1000 and M1400. The solid line is interpolating of the M1400 miniature specimen data.

Fig. ‎2-8 Contribution of the parallel-zone elongation (∆𝑙𝑝𝑟𝑒𝑓), the fillet-zones elongation (2∆𝑙𝑓) and machine

compliance (∆𝑙𝑚) on the crosshead displacement in the tensile testing of the T.4 miniature specimens. The

parallel-zone elongation of all the steels was the same (2×10-3 mm) and the applied forces to create this elongation were recorded.

It can be recognized that stronger specimens deform elastically up to a higher load and thereby the machine compliance, which is function of the applied force, is larger for these specimens. This figure also indicates that, for all the steels, the machine compliance forms the main contribution on the crosshead displacement and its influence on the elastic strain measurement should be precisely considered.

 Validation of the proposed model

The correction procedure to calculate the elastic strain in the parallel-zone of the miniature specimens was developed by inserting the machine compliance function (Eq. ‎2-13) and the α value, 0.19 for the T.4 miniature specimens and 0.14 for the T.3 miniature specimen, in Eq. ‎2-12. Then the parallel-zone strain of the miniature specimens at different levels of the crosshead displacement and the applied force were calculated.

Effects of specimen size on the tensile behavior of steels

20

Table ‎2-2 Apparent elastic slope (𝐸𝑎𝑝𝑝), reference elastic slope (𝐸𝑟𝑒𝑓) and corrected elastic slope (𝐸𝑐) of the miniature specimens and elastic slope of the standard specimens (𝐸𝑠). The relative error of apparent elastic

slope (𝜂app) and corrected elastic slope (𝜂c) were calculated based on the elastic slope of the standard specimens. In this table, the elastic slope and relative error are given in GPa and percentage, respectively.

Specimen 𝑬𝒔 𝑬𝒓𝒆𝒇 _𝑬𝒂𝒑𝒑 _𝜼app _𝑬𝒄 _𝜼c T.4 M1400 203 204 55 73 204 0.5 T.4 DP1000 212 212 51 76 235 10 T.3 DP1000 212 215 50 76 220 4 T.4 DP600 202 200 98 52 187 7 T.4 IF 173 188 88 96 165 5

As it can be seen in Fig. ‎2-5 the corrected strain-stress (𝜀𝑝𝑐 − 𝜎) graphs of the miniature

specimens and the strain-stress (𝜀𝑝𝑠− 𝜎) graphs of the standard specimens are in good

agreement and both miniature and standard geometries show the same elastic slope. For each type of steel, the elastic slope of the standard specimens (𝐸s) was measured from standard elastic strain-stress curves and presented in Table ‎2-2. Moreover, this table includes the apparent elastic slope (𝐸app), the reference elastic slope (𝐸ref) and the corrected elastic slope (𝐸c) of the miniature specimens which were determined from apparent elastic strain-stress curves, reference elastic strain-stress curves and corrected elastic strain-stress curves, respectively. The relative error of apparent elastic slope (𝜂app) and corrected elastic slope (𝜂c) were calculated based on the elastic slope of the standard specimens. Although the relative error of the apparent elastic strain is around 50-96% the relative error of corrected elastic strain is less than 10%. These results show that this model can be considered as a reliable method for calculating the elastic strain of the miniature specimens from the crosshead displacement.

2.4.2 The influence of the specimen geometry on the yield and tensile strength

Fig. ‎2-4 shows that the yield and the tensile strength of steels increase by reducing the parallel length. Since engineering stress-strain curves of miniature specimens were parallel to each other in the plastic zone, the increase in the strength is due to the fact that the force which was applied to fix the miniature specimens on the tensile test machine was not become zero at the initial stage of the tensile. In this sense, by reducing the specimen size (cross section area) the applied stress during the fixing the specimen results in higher stress enhancement. This explanation was verified by tensile testing of T.4 and T.3 miniature specimens from IF and DP 600, while the applied force during fixing the specimens was kept very low. The results of the engineering stress-strain curves are shown in Fig. ‎2-9. As it can be seen the yield strength and tensile strength of the specimens are independent of the specimen gauge length.

Chapter 2

21

Fig. ‎2-9 Engineering stress-strain diagram of miniature specimens with 3 (T.3) and 4 (T.4) mm parallel length and standard specimens (T.120) with parallel length of 120 mm corresponding to steels IF and DP600. The applied force during the fixing the specimens on the tensile test machine was kept below the yield strength.

2.4.3 The influence of the specimen geometry on the fracture strain

According to Fig. ‎2-4 and Fig. ‎2-9, the fracture strain of the specimens increased with a reduction of the specimen parallel length. Similar increase in the fracture strain with decreasing the parallel length of miniature specimens have been reported by different researchers. This can be related to the fact that post necking elongation is concentrated in the necking region and it is independent of the specimen geometry [17]. On the other hand, to calculate the fracture elongation, the measured deformation is divided by the initial parallel length. Since the initial parallel length in miniature specimens is much lower than in standard ones, the measured post necking strain in the miniature specimens is higher than in standard specimens. In view of this problem, ISO developed the international standard ISO 2566 to eliminate the effect of the specimen parallel length on the fracture strain and enable a better comparison of data generated from different specimen geometries. This method is based on the Oliver formula, which is now has been widely used for conversions of fracture strain. For specimens having the parallel length to width ratio of 4, as in the case of this work, the Oliver equation is expressed as [18]:

𝐴_𝑟 = 1.74 [√𝑆0 𝐿₀ ]

0.4

𝐴, ‎2-14

where 𝐿0 and 𝑆0 are the gauge length and cross section area of the standard specimen,

respectively. 𝐴 is fracture strain of the miniature specimen and 𝐴𝑟 is the corrected strain on

gauge length L0. Eq. 2-14 has been used in [19] to correct the influence of the specimen size

Effects of specimen size on the tensile behavior of steels

22

Fig. ‎2-10 Comparison between fracture strains measured from miniature and standard specimens and converted values calculated according to the Oliver’s equation.

Fig. ‎2-11 Overview of tensile strength and fracture strain of standard and miniature specimens. In the microtensile tests, the applied force during fixing specimens is kept below the yield strength.

Substituting the dimensions of standard specimen from Table ‎2-1 into Eq. ‎2-14 the following equation is obtained for correcting the measured fracture strain of miniature specimens:

𝐴𝑟 = 0.55𝐴. ‎2-15

The fracture strain of the miniature specimens with 3 mm and 4 mm gauge length were corrected and compared with the values obtained from standard specimens in Fig. ‎2-10. The corrected fracture strains of DP600 and DP1000 steels are in good agreement with the fracture strains of the standard specimens. For the M1400 miniature specimens, the corrected fracture strain is 9% while the fractures strain of the standard specimens was 5%. The difference between the corrected fracture and the strain of the standard specimens can be attributed to effect of the machine compliance on the crosshead displacement. Considering that the machine compliance is a function of the applied force the influence of the machine compliance on the strain is more significant in case of materials with high strength like M1400 than in material like DP600 and DP1000. The corrected fracture strains of IF miniature specimens is lower than the standard ones. This was expected, since the international standard ISO 2566 explained here can be only applied to low carbon steels while IF steel is carbon free.

Chapter 2

23 To illustrate the effect of the specimen geometry on the tensile properties, the fracture strain-tensile strength curves of the standard and miniature specimens are shown in Fig. ‎2-11. The tensile behavior of the steels is not affected by specimen dimensions and the corrected fracture strain of miniature specimens gives an estimation of failure strain of the standard specimens. Therefore, the miniature specimens can be used to determine the tensile behaviour of steels. However, it is required to correct the influence of the machine compliance on the measured elastic strain, if the strain is measured from the crosshead displacement.

2.5 Conclusions

The influence of the specimen geometry on the tensile behaviour of different types of steels (M1400, DP1000, DP600 and IF) is studied and the results are summarised as following:

 The specimen geometry has insignificant influence on the actual measured elastic strain of the materials. Measurements recorded with the crosshead displacements on the miniature specimens displayed higher strain as a result of the effect of the elastic strain of the fillet-zones and the machine compliance.

 A mathematical model is proposed to calculate the elastic strain of the fillet-zones and the machine compliance. The mathematical model is experimentally evaluated for miniature specimens from different types of steels and different dimensions. For each type of steel, the calculated elastic strain and the strain measured on the standard specimens are in excellent agreement and consequently the proposed model can be used for calculating the elastic strain of the miniature specimens from the crosshead displacement.

 The yield strength and tensile strength does not change by reducing the parallel length from 120 mm to 3 mm.

 The fracture strain of the miniature specimens is higher than the standards values. This is a result of the calculation method, since the fracture strain is calculated by dividing the elongation by the initial gauge length, which is smaller for miniature specimens. Since the post-uniform deformation is independent of the specimen parallel zone, the measured fracture strain is higher in miniature specimens. The Oliver equation was applied for correcting the fracture strain of the miniature specimen to the strain obtained from standard ones. It was found that this equation accurately corrected the fracture strain of the low carbon steels, but failed for the case of martensitic and interstitial free steels.

Effects of specimen size on the tensile behavior of steels

24

2.6 References

[1] R. Z. Valiev, A. V. Sergueeva and A. K. Mukherjee, “The effect of annealing on tensile deformation behavior of nanostructured SPD titanium”, Scr. Mater., vol. 49, pp. 669–674, 2003.

[2] X. Sun, A. Soulami, K. S. Choi, O. Guzman and W. Chen, “Effects of sample geometry and loading rate on tensile ductility of TRIP800 steel”, Mater. Sci. Eng. A, vol. 541, pp. 1-7, 2012. [3] A. M. Korsunsky, G. D. Nguyen and K. Kim, “The analysis of deformation size effects using multiple gauge length extensometry and the essential work of rupture concept”, Mater. Sci. Eng. A , vol. 423, pp. 192–198, 2006.

[4] Y. H. Zhao, Y. T. Zhu, X. Z. Liao, Z. Horita and T. Langdon, “Tailoring stacking fault energy for high ductility and high strength in ultrafine grained Cu and its alloy”, Appl. Phys. Lett., vol. 89, pp. 121906-1–1121906-3, 2006.

[5] Y. H. Zhao, Y. Z. Guo, Q. Wei, T. Topping, A. M. Dangelewicz, Y. T. Zhu, T. G. Langdon and E. J. Lavernia, “Influence of specimen dimensions and strain measurement methods on tensile stress–strain curves”, Mater. Sci. Eng. A, vol. 525, pp. 68-77, 2009.

[6] K. J. KarisAllen and J. R. Matthews, “Low damping absorbers and the determination of load-displacement data for pre-cracked charpy specimens”, ASTM STP, vol. 1248, pp. 232-245, 1995.

[7] C. B. Hurchill, J. A. Shaw and M. A. Iadicola, “Tips and tricks for characterizing shape memory alloy wire: part 2-fundamental isothermal responses”, Exp. Tech., vol. 33, pp. 51-62, 2009.

[8] F. Hild and S. Roux, “Digital Image Correlation: from displacement measurement to identification of elastic properties–a review”, Strain, vol. 42, pp. 69–80, 2006.

[9] A. V. Sergueeva, J. Zhou, B. E. Meacham and D. J. Branagan, “Gage length and sample size effect on measured properties during tensile testing”, Mater. Sci. Eng. A, vol. 526, pp. 79-83, 2009.

[10] S. Koubaa, R. Othman, B. Zouari and S. El-Borgi, “Finite-element analysis of errors on stress and strain measurements in dynamic tensile testing of low-ductile materials”, Comput. Struct., vol. 89, pp. 78-90, 2011.

[11] K. J. KarisAllen and J. Morrison, “The determination of instrumented impact machine compliance using unloading displacement analysis”, Exp. Mech., vol. 29, pp. 152-156, 1989. [12] M. L. Meier and A. K. Mukherjee, “The onset of tensile instability”, National Aeronautics and Space Administration, pp. 361-378, 2002.

[13] S. R. Kalidindi, A. Abusafieh and E. El-Danaf, “Accurate characterization of machine compliance for simple compression testing”, Exp. Mech., vol. 37, pp. 210-215, 1997.

[14] O. N. Pierron, D. A. Koss and A. T. Motta, “Tensile specimen geometry and the constitutive behavior of Zircaloy-4”, J. Nucl. Mater., vol. 312, pp. 257–261, 2003.

[15] M. Maringa, “Dimensioning of dog bone specimens and numerical analysis of the effects of different fillet radii, clamp area and pinhole loading on the stresses in such specimens”, Afr. J. Sci. Technol. Sci. Eng. Ser., vol. 5, p. 60–72, 2004.

[16] Z. Tang, J. Liang, Z. Xiao and C. Guo, “Large deformation measurement scheme for 3D digital image correlation method”, Opt. Las. Eng., vol. 50, p. 122–130, 2012.

[17] R. Cintrón and V. Saouma, “Strain measurements with the digital image correlation system vic-2D”, University of Colorado, 2008.

Chapter 2

25 [18] D. A. Oliver, “Proposed new criteria of ductility from a new law connecting the percentage elongation with size of test‐piece”, Archive proceedings of the institution of mechanical engineers, vol. 115, pp. 827-864, 1928.

[19] D. N. Hanlon; S. M. C. van Bohemen and S. Celotto, “Critical assessment 10: tensile elongation of strong automotive steels as function of test piece geometry”, Mater. Sci. Technol., vol. 31, pp. 385-388, 2015.

Effects of specimen size on the tensile behavior of steels

27

CHAPTER 3

3

An Improved X-ray Diffraction Analysis

Method to Characterize Dislocation

Density in Lath Martensitic Structures

*

Abstract

An improved X-ray diffraction line profile analysis method is developed to determine dislocation density of lath martensitic steels. This method combines the modified Warren-Averbach (MWA) and the modified Williamson-Hall (MWH) methods. The developed method is stable under different initial conditions and leads to unique values for the dislocation density, the effective outer cut-off radius of the dislocations (Re) and the dislocations

distribution parameter (M). Dislocation structure of lath martensite in a steel, in the as-quenched as well as tempered conditions, are characterized by using the proposed method. The calculated dislocation density is compared with the values obtained from the MWH method by considering a constant value for M. It was found that both methods provide dislocation densities in the range of the values calculated from the dislocation strengthening component of the yield strength.

Keywords: Dislocations, X-ray diffraction, Martensite, Mechanical characterization

*

This chapter is based on a scientific paper:

F. HajyAkbary, J. Sietsma, A. J. Bӧttger and M. J. Santofimia, An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density in Lath Martensitic Structures, Material Science and Engineering A, vol. 639, pp. 208-218, 2015.

An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density

28

3.1 Introduction

Knowledge of the microstructural properties and their effects on the yield strength is required to tailor the mechanical properties of steels. Dislocation density as a crucial factor influencing the yield strength can be determined by applying Transmission Electron Microscopy (TEM) and X-ray diffraction (XRD) line profile analysis [1, 2]. It should be kept in mind that in an inhomogeneous system such as martensitic steel, where the dislocation density varies from place to place within a grain on a sub-micron scale, the XRD method gives a macroscopic average value while TEM gives a microscopic local value [3]. Furthermore, in the case of martensitic microstructures with a high density of dislocation (higher than 1014 m-2) applying TEM is difficult. This is because of the complicated image contrasts from the sample [2]. Therefore, especially at high dislocation densities, like in highly deformed metals or in martensitic steels, XRD offers a promising alternative. However, the quantification of the dislocation density from an XRD pattern of broadened peaks is not straightforward.

The modified Williamson-Hall (MWH) method [4] is known as an accessible method in XRD line profile analysis. This method determines the dislocation density if the dislocations distribution parameter (M) is known. It is argued that M depends on the effective outer cut-off radius of the dislocations (𝑅𝑒) and the dislocation density (𝜌) [2]. No direct method has

been used to determine M and it can only be obtained from 𝑀 = 𝑅𝑒√𝜌 [5] in which 𝑅𝑒 is

calculated from the MWA method. This means that the MWH equation includes two unknown parameters, 𝜌 and 𝑀. Therefore, the MWH approach has been applied under the assumption of a fixed value for M, as a qualitative method in limited number of research [6, 7].

An alternative method for XRD line profile analysis is the modified Warren-Averbach (MWA) which has been widely used to determine the dislocation density and the effective outer cut off radius of the dislocations [2, 3, 8, 9]. Generally, this method is used by assuming the strain function of the dislocations (the Wilkens function) as a logarithmic function of (𝑅𝑒/𝐿) [10] where 𝐿 is the Fourier length. The strain function

phenomenologically describes the dislocation-dislocation correlations that appear in high order Fourier coefficients [11]. More details about 𝐿 and strain function are given in section ‎3.2.2. This approach was applied by Movaghar et al. [9] to study the influence of severe plastic deformation on the dislocation structure of a martensitic steel. Furthermore, the same approach was used to evaluate the dislocation density of a 11Cr-0.1C (wt.%) martensitic steel after annealing [12]. It should be recognized that calculation of the dislocation density by assuming a logarithmic strain function is applicable only at small L values (𝐿 < 2.88 𝑅𝑒) [5]. Although this method has been used widely, it is not a robust

method. The reason is that 𝑅𝑒 is unknown and it is not possible to determine the relevant

range of L. Additionally, this approach depending on the assumed range of L gives different 𝑅𝑒 and dislocation density. An alternative approach in the MWA method is defining the

strain function for whole ranges of L, e. g. 𝐿 < 2.8816 𝑅𝑒 as well as 𝐿 > 2.8816 𝑅𝑒 [10].

Although this approach is valid for any L, it is not robust also and the results depend on the initial conditions and the assumed range of L. In conclusion, for the determination of the dislocation density, none of the XRD line profile analysis methods has been generally accepted.

Chapter 3

29 A quantitative value for the dislocation density can also be calculated from the dislocation strengthening component of the yield strength. The dislocation strengthening component is determined by subtracting the contributions of the other strengthening components, including lattice friction stress, the solid solution strengthening, the grain boundary strengthening and the precipitation strengthening, from the total yield strength. Subsequently, the dislocation density can be estimated from the relation between dislocation strengthening and dislocation density. Since the yield strength is a bulk property, the obtained dislocation density value represents the bulk dislocation density. In this sense, the calculated dislocation density can be used to validate the dislocation density determined by using the X-ray diffraction analysis methods.

In the present chapter, an improved approach has been developed for XRD line profile analysis by combining the MWH and MWA methods. This approach provides an expression for the Fourier coefficients that is valid for any range of L and is stable under different initial conditions. The developed method is applied to determine the dislocation density, the effective outer cut-off radius of the dislocations (Re) and the dislocations distribution

parameter (M) of lath martensite in a steel under the as-quenched as well as tempered conditions. The calculated dislocation densities are compared with the values obtained from the MWH method by considering a constant value for M. It will be shown that the calculated dislocation densities from XRD analysis methods are in a good agreement with the values that are obtained from contribution of the dislocation strengthening to the yield strength. 3.2 Calculation of dislocation density

3.2.1 Analysis of XRD peak broadening by Modified Williamson-Hall Method

The XRD peak broadening caused by strain has long been used to characterize dislocation density. For isotropic materials, the Williamson-Hall equation approximates the dislocation density from X-ray peak broadening as [13]:

∆ 𝐾 ≅𝛼𝑠

𝐷 +𝑁𝑏√𝜌 𝐾, ‎3-1

where ∆𝐾 is the peak width, 𝑁 is a constant (0.263), 𝛼𝑠 is the shape factor, 𝐷 is the

crystallite size, 𝐾 is the magnitude of the diffraction vector, 𝑏 is the magnitude of the Burgers vector and 𝜌 is the dislocation density. Here, 𝛼𝑠 is given 0.9 under the assumption of

spherical crystals with cubic symmetry [14] (thereafter 𝛼𝑠 is assumed 0.9) and 𝐾 is obtained

by 𝐾 = 2𝑠𝑖𝑛𝜃/𝜆, in which 𝜃 and 𝜆 are the diffraction angle and the wavelength, respectively. In principle this equation is valid for each {hkl} reflection and the dislocation density is obtained by fitting Eq. ‎3-1 to a plot of ∆𝐾 versus 𝐾. However, in cases of strong strain anisotropy, such as observed in lath martensitic steel, ∆𝐾 is not a linear function of 𝐾 [2]. In these applications, this method overestimates the dislocation density [3].

Ungar et al. [4] developed a modified Williamson-Hall (MWH) method by accounting the influence of the strain anisotropy. To do this, they defined a scaling parameter, 𝐶̅, which is called the average contrast factor of dislocations. The MWH equation is written as [6]: