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Deformation and fracture behaviour of a low-carbon dual-phase steel Mazinani, Mohammad 2006

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DEFORMATION AND FRACTURE BEHAVIOUR OF A LOW-CARBON DUAL-PHASE STEEL by MOHAMMAD MAZINANI B.Sc, Sharif University of Technology, 1989 M.Sc, University of Tehran, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Metals & Materials Engineering) THE UNIVERSITY OF BRITISH COLUMBIA December 2006 © Mohammad Mazinani, 2006 Abstract The primary goal of this study was to evaluate the effect o f martensite plasticity on the deformation and fracture behaviour of an intercritically annealed commercial low carbon (0.06 wt.%) dual phase steel. The volume fraction and the morphology (banded and almost equiaxed) of the martensite phase were systematically varied by control of the intercritical annealing temperature and the heating rate to this temperature. It was observed that the yield and tensile strengths were dependent on the martensite content but not on the martensite morphology. On the other hand, the true uniform strain, fracture strain and fracture stress were found to have a significant dependence on martensite morphology. A n Eshelby based model, which allowed for the calculation o f the stress in the martensite islands, was employed in order to rationalize the tensile properties of the dual-phase steel samples with different martensite contents and morphologies. In addition, by comparing the calculated stress in the martensite with an estimate of yield stress, it was possible to examine the conditions under which martensite plasticity occurs. The work hardening behaviour and the fracture properties of the steel samples were rationalized by the implications of martensite plasticity. For the cases where martensite showed significant plasticity (or co-deformed with the ferrite matrix), the void nucleation rate during post-necking deformation decreased considerably and hence, the final fracture properties were dramatically improved. The deformation of martensite in different dual-phase steel samples was examined both qualitatively (using optical micrographs of the undeformed and deformed sections of fractured i i tensile samples) and quantitatively (through image analysis of the microstructures before and after tensile deformation). The tensile stress-strain responses o f different dual-phase steel samples were modeled using the modified Eshelby method. This approach was found appropriate for modelling the stress-strain behaviour of the steels with equiaxed morphology and martensite contents below approximately 30%. In the case of banded morphology, the stress-strain behaviour of the steel sample with 17% martensite was successfully predicted by the model. However, the model overestimated the flow stress o f the steel with 30% martensite. For the martensite contents greater than 30%, the overestimation o f the flow stress of the steel samples with banded morphology was greater than that for the equiaxed samples. Finally, the void formation process during tensile deformation was examined quantitatively through image analysis of the fracture surface of the steels. The experimental results showed very little void growth during ductile fracture of the steel samples with 17% and 41% martensite. Modell ing the void formation process in these steels assuming no void growth stage resulted in the same observation. This confirmed the quantitative observation that void nucleation is the dominant effect during ductile fracture o f these steels. in Table of Contents Abstract '. ii Table of Contents iv List of Tables viii List of Figures ix Acknowledgements xv Dedication xvi Chapter 1. Introduction 1 Chapter 2. Literature Review 4 2.1. Dual-Phase Steels - Introduction 4 2.1.1. Advantages of Dual-Phase Steels 4 2.1.2. Production of Dual-Phase Steels 5 2.1.3. Chemistries of Dual-Phase Steels 7 2.2. Overview o f Typical Properties of Dual-Phase Steels 8 2.3. Characteristics of Microstructure 14 2.3.1. Ferrite-Martensite Microstructure 15 2.3.1.1. Martensite Content 15 2.3.1.2. Martensite Size, Morphology and Spatial Distribution 15 2.4. Properties of Constituent Phases 18 2.4.1. Ferrite Phase 19 2.4.2. Martensite Phase 21 2.4.2.1. Tempering of Martensite 23 2.5. Mechanical Properties of Dual-Phase Steels: Detailed Considerations 24 2.5.1. Strengthening Effect of Martensite 24 iv 2.5.2. Yielding Behaviour 25 2.5.3. Work Hardening Behaviour 25 2.5.4. Necking Condition (Conside're Criterion) 26 2.5.5. Fracture Behaviour 28 2.5.6. Deformation Behaviour of Martensite 30 2.6. Modell ing Stress-Strain Behaviour of Dual-Phase Steels 31 2.6.1. Empirical Approaches 32 2.6.2. Dislocation Models 33 2.6.3. Finite Element (FE) Method 34 2.6.4. Modell ing Based on Modified Eshelby Method 35 2.6.4.1. Eshelby Theory 36 2.6.4.2. Modified Eshelby Approach 37 . 2.6.4.2.1. Spherical Inclusion (Aspect Ratio = 1) 39 2.6.4.2.2. Ellipsoidal Inclusion (Aspect Ratio > 1) 41 2.7. Modell ing Ductile Fracture of Dual-Phase Steels 41 2.7.1. V o i d Nucleation - Criterion for V o i d Nucleation Process 42 2.7.2. V o i d Growth - Existing Models for V o i d Growth 43 2.7.3. V o i d Coalescence - Criterion for V o i d Coalescence 43 Chapter 3. Research Objectives 45 Chapter 4. Experimental Procedure and Microstructural Analysis 47 4.1. Starting Material , 47 4.2. Heat Treatment Cycles 47 4.2.1. Intercritical Annealing 47 4.2.2. Tempering Process 49 4.2.3. Production of Martensitic Steels 49 4.3. Estimation of Martensite Carbon Concentration 50 4.4. Sample Preparation for Optical Microscopic Observation 50 4.5. Tensile Testing 52 4.6. Image Analysis 54 v 4.6.1. Measurement of Ferrite Grain Size 55 4.6.2. Measurement of Martensite Content 55 4.6.3. Measurement o f Martensite Deformation 56 4.6.4. Quantitative Examination of V o i d Formation Process 59 Chapter 5. Experimental Results and Analysis 62 5.1. Intercritically Annealed Dual-Phase Steel Samples 62 5.1.1. Steel Microstructures 62 5.1.2. Intercritical Annealing Parameters 64 5.1.3. Discussion on Effect of Heating Rate 65 5.2. Mechanical Properties of Dual-Phase Steel Samples 66 5.2.1. Tensile Stress-Strain Behaviour 66 5.2.2. Tensile Properties 70 5.2.2.1. Y i e l d Strength and Ultimate Tensile Strength 70 5.2.2.2. Uniform Strain 71 5.2.2.3. Discussion on Deformation Behaviour (up to Necking Point) 73 5.2.2.4. Effect o f Tempering 76 5.2.2.5. R-Value . . . . . . . : : 78 5.2.2.6. Fracture Stress and Strain 80 5.2.3. Deformation of Martensite 82 5.2.3.1. Effect o f Martensite Content 82 5.2.3.2. Effect o f Martensite Morphology 86 5.2.3.3. Effect of Tempering Process , 88 5.2.3.4. Martensite Strength and its Plasticity 90 5.2.3.5. Unusual Fracture Behaviour 95 5.2.4. Observation on Microstructural Damage 96 5.2.4.1. Steel Micrographs 96 5.2.4.2. Quantitative Analysis of Voids 99 5.2.4.3. Discussion on Unusual Fracture Behaviour 106 5.3. Analysis o f V o i d Formation Process in Dual-Phase Steel Samples 108 v i Chapter 6. Modelling Results and Discussion 112 6.1. Micromechanics of Dual-Phase Steel Microstructures 112 6.1.1. Modell ing Load Transfer Process 112 6.1.2. Discussion on Martensite Plasticity 122 6.1.2.1. Equiaxed Martensite (in Steels with Heating Rate o f 1 °C/s) 123 6.1.2.2. Banded Martensite (in Steels with Heating Rate of 100 °C/s) 124 6.1.3. Implications on Yie ld Strength, Tensile Strength and Work Hardening 126 6.2. Modell ing Stress-Strain Response of Dual-Phase Steel Samples 129 6.2.1. Dual-Phase Steel Samples with Equiaxed Martensite Morphology 129 6.2.1.1. Elastic Martensite 129 6.2.1.2. Plastic Martensite 134 6.2.2. Dual-Phase Steel Samples with Banded Martensite Morphology 139 6.2.2.1. Elastic Martensite 139 6.2.2.2. Plastic Martensite 144 6.2.3. Estimated Plastic Strain in Martensite 145 6.2.4. Conclusion on Modell ing Stress-Strain Behaviour 146 Chapter 7. Summary, Conclusions and Future Work 148 7.1. Summary 148 7.2. Implications on Steel Design and Processing Route 151 7.3. Future Work 153 References 155 Appendix A . 164 v i i List of Tables Table 4.1. Chemical composition of the steel investigated 47 Table 4.2. Steel Chemistries and test conditions for production of martensitic steels 50 Table 4.3. The sample preparation procedure for carbon steels (recommended by Buehler Ltd.) '.' 52 Table 5.1. Summary of processing conditions and martensite island characteristics in different dual-phase steel microstructures 65 Table 5.2. The results of the measurements of martensite deformation at the necking point for the steel samples processed to have different martensite contents and morphologies 89 Table 5.3. The estimated carbon concentration and yield strength of martensite in the dual-phase steel samples with different martensite contents and morphologies. The measured yield stress (0.2% offset) of untempered and tempered martensite in selected number of steels is given for comparison 92 Table 5.4. Nucleation strain, uniform strain, initial void nucleation rate and initial major and minor radii o f voids first nucleated in some steel samples with banded martensite morphology. A l l strains are thickness strains ' 105 Table 6.1. The estimated stress in the martensite and the estimated martensite yield strength in dual-phase steel samples with different martensite contents and morphologies 123 Table 6.2. Fitting parameters in Equation 6.18 for two martensitic steels used for modelling 134 v i i i List of Figures Figure 2.1. Ultimate tensile strength and total elongation of dual-phase steels compared to other low alloy steels, rep lotted curves (Jacques, 1998) 5 Figure 2.2. A schematic diagram showing the production route o f a cold-rolled dual-phase steel 6 Figure 2.3. A schematic diagram showing the production route o f a hot-rolled dual-phase steel 7 Figure 2.4. Percent martensite in 1.5%Mn dual-phase steels as a function of the steel carbon concentration and intercritical annealing temperature (Speich and Mi l le r , 1979) 8 Figure 2.5. Fe-C phase diagram (for wt.%C = 0-1.2 and T = 200-1000 °C) showing the effects of steel carbon concentration and intercritical temperature on the amount of austenite formed and its carbon content (calculated diagram using Thermocalc software with the Fe-2000 database, licensed to U B C ) 10 Figure 2.6. The true stress-true strain curves of 1.5%Mn dual-phase steels with different carbon concentrations and martensite contents intercritically annealed at 760 °C (Figure 2.4) compared to the ferri tic steel (Speich and Mi l le r , 1979) 12 Figure 2.7. The yield strength and the ultimate tensile strength (UTS) o f 1.5%Mn dual-phase steels annealed at different temperatures as functions of percent martensite (Speich and Mil ler , 1979). For a given intercritical temperature, the data points from left to right correspond to 0, 0.06, 0.12, 0.16 and 0.20 wt.% carbon 12 Figure 2.8. Variation of the uniform elongation of the dual-phase steels in Figure 2.4 with the percent martensite (Speich and Mil le r , 1979) 13 Figure 2.9. The total elongation of dual-phase steels in Figure 2.4 as a function of the percent martensite (Speich and Mil ler , 1979) 14 ix Figure 2.10. The Effects of the starting microstructure and the heating rate to annealing temperature on the martensite size and morphology in a M n - M o dual-phase steel with 0.06% C, (a) hot-rolled steel, low heating rate (1 °C/sec), (b) cold-rolled steel, low heating rate (1 °C/sec), (c) hot-rolled steel, high heating rate (100 °C/sec) and (d) cold-rolled steel, high heating rate (100 °G/sec) (Huang, 2004) .: 16 Figure 2.11. The yield strength of martensite as a function of its carbon concentration, rep lotted curves (Leslie, 1981). Dashed lines indicate the lower and upper strength values 22 Figure 2.12. The Considere condition for necking and uniform strain 27 Figure 2.13. The fracture strain of dual-phase steels with 0.066-0.13%) carbon as a function of the martensite content and size, replotted curves (Balliger, 1982) 29 Figure 2.14. Definition of the secant Young's modulus for an elasto-plastic material 38 Figure 4.1. Gleeble sample and temperature zone with ±5 °C deviation from the goal temperature 49 Figure 4.2. The microstructure of a dual-phase steel with 18% martensite etched with 2% Nital (a) and aqueous sodium metabisulfite (b) 51 Figure 4.3. Subsize tensile specimen according to the A S T M E8, dimensions are in mm. The radius of fillet at the ends of the reduced section is 4mm 53 Figure 4.4. A typical dual-phase steel microstructure; (a) before and (b) after threshold adjustment 56 Figure 4.5. Different stages in quantitative analysis of a dual-phase steel microstructure in order to measure the average thickness of martensite islands 57 Figure 4.6. Microstructure of a fractured dual-phase steel sample close to the fracture surface; (a) before and (b) after threshold adjustment 60 Figure 4.7. A fractured tensile specimen (schematic) showing the image analysis parameters for the quantitative examination of the voids 61 x Figure 5.1. Microstructures of dual-phase steel samples produced with heating rates of 1 and 100 °C/s. (a) and (b) heating rate of 1 °C/s, held at intercritical annealing temperatures of 755 °C and 820 °C for 60 s, martensite contents measured to be 18% and 44%, respectively, (c) and (d) heating rate o f 100 °C/s, held at intercritical annealing temperatures of 730 °C and 780 °C for 60 s, martensite contents measured to be 17% and 41%, respectively 63 Figure 5.2. The engineering stress-strain curves o f dual-phase steel samples with different martensite contents, (a) samples with almost equiaxed martensite morphology produced with the heating rate of 1 °C/s and (b) samples with banded martensite morphology produced with the heating rate of 100 °C/s 67 Figure 5.3. The true stress - true strain curves of dual-phase steel samples with different martensite contents, (a) samples with almost equiaxed martensite morphology produced with the heating rate of 1 °C/s and (b) samples with banded martensite morphology produced with the heating rate of 100 °C/s. The dashed curves represent the post-necking deformation regime 69 Figure 5.4. The Y i e l d strength (ay,o.2%) and the tensile strength (UTS) of different dual-phase steel samples in longitudinal and transverse directions as a function of the martensite content for two different morphologies. Equiaxed and banded martensite morphologies correspond to the heating rates o f 1 °C/s and 100 °C/s, respectively 70 Figure 5.5. The yield strength to tensile strength ratio o f dual-phase steel samples as a function of the martensite content in longitudinal and transverse directions. Equiaxed and banded martensite morphologies correspond to the heating rates of 1 °C/s and 100 °C/s, respectively 71 Figure 5.6. The uniform strain of dual-phase steel samples with two different martensite morphologies as a.function of the martensite content in longitudinal and transverse directions. Equiaxed and banded martensite morphologies correspond to the heating rates of 1 °C/s and 100 °C/s, respectively 72 Figure 5.7. Microstructure of dual-phase steel samples with two orientations relative to the rolling direction, (a) and (b) 29% equiaxed martensite, longitudinal and transverse directions, respectively, (c) and (d) 41% banded martensite, longitudinal and transverse directions with aspect ratios of 5.5 and 4, respectively. Tensile direction is horizontal 75 Figure 5.8. Flow stress and work hardening rate o f dual-phase steel samples with equiaxed and banded martensite islands as a function of true strain illustrating the effect of martensite morphology on the Considere condition, (a) samples with 17-18%o martensite and (b) samples with 29-30%) martensite 77 x i Figure 5.9. The true stress - true strain curve of dual-phase steel sample with 17% martensite (banded morphology) before and after tempering at 500 °C for 60 minutes 78 Figure 5.10. The R-value of dual-phase steel samples with different martensite morphologies as a function of the percent martensite in longitudinal and transverse directions. Heating rates of 1 °C/s and 100 °C/s correspond to equiaxed and banded martensite morphologies, respectively 79 Figure 5.11. Unusual results showing the true fracture stress (a) and the true fracture strain (b) of different dual-phase steel samples in the longitudinal and transverse directions as a function of the percent martensite for two different martensite morphologies. Equiaxed and banded martensite morphologies correspond to the heating rates of 1 °C/s and 100 °C/s, respectively 81 Figure 5.12. The microstructure o f dual-phase steel sample with 17% martensite produced with the heating rate of 100 °C/s in the undeformed region (a) and the uniform deformation (tensile strain of 0.14) region (b) o f the fractured tensile specimen. Tensile loading direction is horizontal 83 Figure 5.13. Histogram showing the thickness of banded martensite islands in undeformed region and region with a tensile far field strain of 0.14 for a steel sample with 17%) martensite produced with a heating rate of 100 °C/s (corresponding microstructures shown in Figure 5.12) 84 Figure 5.14. The microstructure of dual-phase steel sample with 41% banded martensite produced with the heating rate of 100 °C/s in the undeformed region (a), the uniform deformation (tensile strain of 0.12) region (b) and near the fracture (tensile strain of 0.59) region (c) of fractured tensile specimen. Tensile loading direction is horizontal 85 Figure 5.15. Histogram showing the thickness of banded martensite islands in undeformed region and regions with the tensile far field strains of 0.12 and 0.59 for a steel sample with 41% martensite produced with a heating rate of 100 °C/s (corresponding microstructures shown in Figure 5.14) 86 Figure 5.16. The effect of martensite morphology on dimensional change and plastic strain of martensite in the uniform deformation region (with the tensile far field strains indicated) for two dual-phase steel samples with around 30% martensite; (a) and (b) samples with banded martensite, (c) and (d) samples with equiaxed martensite. Tensile loading direction is horizontal 87 x i i Figure 5.17. The effect of tempering on the martensite islands thickness in a quenched-tempered dual-phase steel sample with 17% banded martensite; tempering temperature and time: 500°C and 60 minutes. Tensile loading direction is horizontal 88 Figure 5.18. Martensite plastic strain at the necking point for different dual-phase steel samples, (a) effect of martensite content and (b) effect of martensite morphology and tempering 91 Figure 5.19. The true stress-true strain curves of 100% martensite steels with different carbon concentrations obtained from tensile test. Curves are plotted up to the necking point 93 Figure 5.20. Martensite plastic strain at the necking point for the dual-phase steel samples with equiaxed and banded martensite morphologies as a function of the martensite carbon concentration 94 Figure 5.21. Optical micrographs illustrating the void formation for sections parallel to the tensile axis on the banded steel samples with different martensite contents (produced using a heating rate of 100 °C/s); (a) 17% martensite, (b) 17% martensite after tempering at 500 °C for 60 minutes, (c) 25% martensite and (d) 41%o martensite 97 Figure 5.22. S E M secondary electron image of the deformed steel sample with 17% banded martensite produced by the high heating rate processing route (100 °C/s). Image was taken near the fracture surface representing the void formation which predominately occurred by •fracture of martensite islands : 98 Figure 5.23. The number of voids per unit area (a) and the area percent voids (b) in the steel samples with 17%> and 41 %> banded martensite (high heating rate, 100 °C/s) as a function o f the far field thickness strain 100 Figure 5.24. Variation of the major and minor radii o f voids as a function of far field thickness strain for two dual-phase steel samples with (a) 17% and (b) 41% banded martensite 102 Figure 5.25. The effect of tempering (500 °C for 60 minutes) on the number of voids per unit area (a) and the area percent voids (b) in the steel samples with 17% banded martensite (produced with the high heating rate, 100 °C/s) as a function of the far field thickness strain 104 Figure 5.26. The estimated area percent curves in the dual-phase steel samples with (a) 17%) and (b) 41% martensite compared to the corresponding experimental data 110 x i i i Figure 6.1. Calculated and experimental stress-strain curves of the investigated steel in the hot-rolled condition as the ferrite phase in dual-phase steel samples 114 Figure 6.2. The load transfer curves in a dual-phase steel sample with 18% spherical martensite calculated using different strain increments 119 Figure 6.3. Calculated ratios of martensite stress to ferrite stress in the dual-phase steel samples with different martensite contents and morphologies (aspect ratios).. 121 Figure 6.4. S E M micrograph showing martensite cracking in a dual-phase steel sample with 17%o banded martensite before necking point 128 Figure 6.5. The calculated stress-strain curves of dual-phase steels with different martensite contents compared to the experimental stress-strain curve of the ferrite phase 130 Figure 6.6. The calculated stress-strain curves of dual-phase steel samples with (a) 18%, (b) 29% and (c) 44% martensite (equiaxed morphology) compared to the experimental stress-strain curves 133 Figure 6.7. The calculated stress-strain curve of the steel sample with 44% plastic martensite (equiaxed morphology) compared to the experimental curve .136 Figure 6.8. The calculated stress-strain curve of the tempered dual-phase steel sample with 18%> plastic martensite (equiaxed morphology) compared to the experimental curve 138 Figure 6.9. The calculated stress-strain curve of a dual-phase steel with 20%> martensite as a function of the aspect ratio of the martensite islands 140 Figure 6.10. The calculated stress-strain curve of dual-phase steels with martensite aspect ratio of 5 as a function of the martensite content 141 Figure 6.11. The calculated stress-strain curves of the dual-phase steel samples with (a) 17%) martensite (aspect ratio of 4.2), (b) 30%> martensite (aspect ratio of 5.1) and (c) 41%) martensite (aspect ratio o f 5.5) compared to the experimental curves 143 xiv Acknowledgements I would like to express my sincere gratitude to my supervisor, Professor Warren J. Poole for his patience and all his guidance and constant encouragement throughout the course of this work. I would like to thank Stelco Inc. for providing the material for this research and their financial support and cooperation. The financial support from Natural Sciences and Engineering Research Council of Canada ( N S E R C ) is also appreciated. The comments and suggestions of Professors Matthias Mil i tzer and Daan Maijer on this thesis are also highly appreciated. The informal discussions with my colleagues in our research group, in particular Dr. Fateh Fazeli, are gratefully acknowledged. The last, but not the least, I am very much indebted to my parents from the first day of my life t i l l this achievement. xv Dedication This thesis is dedicated with Cove to my wife HAMI<LyEJ{ and to our children OfMI<D and<EMO<D XVI Chapter 1 - Introduction Dual-phase steels were developed in the late 1970's and early 1980's in response to the demand for high strength and highly formable steels. Research from this period is well documented in a series of conference proceedings and journal papers, examples of which can be found in the following references (Rigsbee and VanderArend, 1979; Lanzillotto and Pickering, 1982; Marder and Bramfitt, 1979; Speich and Mi l l e r , 1979; Davies and Magee, 1979; Speich, 1981; Embury and Duncan, 1981). In recent years, there has been a resurgence o f interest in dual-phase steels as their application has made significant inroads, particularly in the automotive sector. Dual-phase steels have a microstructure consisting of a hard second phase (mostly martensite sometimes with small amounts of bainite and/or retained austenite)(Rigsbee and VanderArend, 1979; Y i et a l , 1983; Pickering, 1992; Gladman, 1997). Due to their composite microstructure, dual-phase steels exhibit interesting characteristic mechanical properties such as continuous yielding, low yield stress to tensile strength ratios and high ductility which offer advantages compared with conventional high strength low alloy ( H S L A ) steels (Rigsbee and VanderArend, 1979; Lanzillotto and Pickering, 1982; Marder and Bramfitt, 1979; Speich and Mil ler , 1979; Davies and Magee, 1979; Speich, 1981; Embury and Duncan, 1981; Pickering, 1992; Gladman, 1997; Balliger and Gladman, 1981). Dual-phase steels can be produced industrially by two processing routes, either intercritical annealing of cold-rolled steels (Rigsbee et al., 1979; Klaar et al., 1990; Faral and Hourman, 1 Chapter 1. Introduction 1999), often performed in association with the galvanizing process, or by hot rolling followed by step cooling on the runout table (Waterschoot et al., 2001). The mechanical behaviour of dual-phase steels depends on the strength of the constituent phases as well as the martensite content (Lanzillotto and Pickering, 1982; Koo et al., 1980) and its morphology (Bag, et al., 1999; K i m and Thomas, 1981). The strength of the ferrite phase is mainly controlled by the steel chemistry, grain size (Pickering, 1992; Gladman, 1997; Pickering, 1978) and initial dislocation density (which may be affected by the presence of martensite in its microstructure) (Sherman, et al., 1981; Bourell and Rizk , 1983; Liedl et al., 2002). The strength of martensite depends primarily on its carbon content (Pickering, 1992; Leslie, 1981). Within the limits commercially used for steels, substitutional alloying additions have a minor effect (Leslie, 1981). In order to predict the overall deformation behaviour of dual-phase steels, one needs to have knowledge o f the properties o f the constituent phases and also the load transfer to the martensite phase when the steel is deforming. Given detailed knowledge of the load transfer process, it w i l l then be possible to understand the stress and strain partitioning between the two phases during deformation. Conventionally, dual-phase steels have had relatively high average carbon concentrations (0.1-0.2 wt.%) (Gladman, 1997) which leads to a relatively high martensite yield stress such that most experimental evidence shows that martensite plasticity is absent except for the regions very close to the fracture surface (Balliger and Gladman, 1981; Davies, 1978; Marder, 1982; Su and Gurland, 1987). Recently, there has been considerable interest in using lower carbon concentrations (below 0.1 wt.%>) to improve properties. In this case, it might be 2 Chapter 1. Introduction expected that martensite plasticity would be favored in achievement of enhanced mechanical properties. One of the main objectives of this research work was to examine under what conditions martensite plasticity occurs in dual-phase steels and to then consider the implications on the strength and ductility o f these materials. In this investigation, the deformation behaviour of a commercial low carbon (0.06 wt.%) dual-phase steel alloyed with molybdenum was studied experimentally. Dual-phase steel samples with a variety of martensite contents were produced through a series o f intercritical annealing treatments at different temperatures on cold-rolled starting material. Following the results of Huang (2004), steel microstructures were produced with two distinct martensite morphologies by controlling the heating rate to the intercritical temperatures. The deformation behaviour of martensite islands during tensile straining was investigated in dual-phase steel samples through image analysis o f the microstructures. In addition, the formation of microstructural damage during tensile deformation was also evaluated using the optical and scanning electron microscopes and then correlated to the overall macroscopic ductility measurements. Furthermore, attempts were made in this work to model the load transfer process in dual-phase steel samples with different martensite contents and morphologies in order to rationalize the mechanical properties. The results obtained from the load transfer model were used for modelling the stress-strain response of this low carbon dual-phase steel with taking into account the effect of martensite content and morphology and martensite plasticity. 3 Chapter 2. Literature Review In this chapter, the literature concerning dual-phase steels and their characteristic mechanical properties w i l l be reviewed. The literature review is comprised of the experimental observations regarding the deformation behaviour and fracture properties, and the modelling of the tensile stress-strain and ductile fracture behaviour of dual-phase steels. 2.1. Dual-Phase Steels - Introduction For the past several decades, efforts have been made to replace pearlite by martensite in the microstructure of plain carbon steels. In the following sections, it w i l l be shown that the presence of martensite in the microstructure can have a beneficial effect on the mechanical properties. Dual-phase steels have a microstructure consisting o f hard second phase particles (mostly martensite sometimes having small amounts of bainite and/or retained austenite) embedded in a ductile ferrite matrix. Due to their composite type microstructure, dual-phase steels exhibit interesting characteristic mechanical properties that are attractive for industry, particularly the automotive sector. 2.1.1. Advantages of Dual-phase Steels Continuous yielding behaviour, low yield strength to ultimate tensile strength ratios, high initial work hardening and relatively high ductility (Speich and Mi l le r , 1979; Speich, 1981, Balliger and Gladman, 1981; Rigsbee and VanderArend, 1979) are the interesting mechanical 4 Chapter 2. Literature Review properties of dual-phase steels compared to conventional high strength low alloy ( H S L A ) steels. Using dual-phase steels in automotive applications offers advantages over the conventional H S L A steels mainly due to the need in this industry for an improved safety and reduced weight of the vehicles. Figure 2.1 shows the tensile strength and total elongation of ferrite-martensite and bainite-martensite dual-phase steels compared to low alloy steels strengthened by solid solution and precipitation hardening (Jacques, 1998). A s can be observed in Figure 2.1, ferrite-martensite dual-phase steels with tensile strength and total elongation in the range of 600-900 M P a and 20-35%, respectively, exhibit a superior combination of strength and ductility when they are compared with other steels. 50 40 I 3 0 ro O) c o UJ 2 0 ro o 10 \ \ K Ferrite-Martensite \ \ I \dual-phase steels Solid-solution\ \ \ . strengthening \ ^ ^ N . Bainite-Martensite dual-phase steels Precipitations. strengthening 1 300 600 900 1200 1500 1800 Ultimate Tensile Strength (UTS), MPa Figure 2.1. Ultimate tensile strength and total elongation of dual-phase steels compared to other low alloy steels, replotted curves (Jacques, 1998). 2.1.2. P r o d u c t i o n o f D u a l - P h a s e S t e e l s Dual-phase steels can be produced industrially by two processing routes; either intercritical annealing of cold-rolled steels which is often done in association with the galvanizing process or accelerated cooling on the runout table following the hot rolling process. In the intercritical Chapter 2. Literature Review annealing process (Rigsbee and VanderArend, 1979; Davies, 1978; Klarr et al., 1990), the cold-rolled steel (with ferrite-pearlite microstructure) is first heated to the intercritical temperature range (ferrite-austenite phase region, between the lower and upper critical temperatures during heating, i.e. A d and A.3, respectively) where a certain amount o f austenite is formed and then cooled rapidly to room temperature (Figure 2.2). For hot rolled dual-phase steels (Waterschoot et al., 2001; Rigsbee et al., 1979), the hot rolling process takes place in the austenitic region and steel is then cooled into the ferrite-austenite temperature range where ferrite is formed. The remaining austenite transforms to martensite during the subsequent accelerated cooling on the runout table. The production route for hot-rolled dual-phase steels is shown in Figure 2.3. In Figure 2.3, A r ) and A r 3 are respectively the lower and upper critical temperatures during cooling. Figure 2.2. A schematic diagram showing the production route of a cold-rolled dual-phase steel. 6 Chapter 2. Literature Review Figure 2.3. A schematic diagram showing the production route of a hot-rolled dual-phase steel. 2.1.3. Chemistries of Dual-Phase Steels Dual-phase steels usually have 0.05-0.2%C and l - 2 % M n (Kot and Morris , 1979; Kot and Bramfitt, 1981), however, some other alloying elements, such as silicon, molybdenum and chromium may also be used (Speich, 1981; Waterschoot et al., 2001; Thomas and Koo , 1979; Messien et al., 1981). Carbon is the most important element affecting the hardenability of steel (Llewellyn and Hi l l i s , 1996) and indeed, the mechanical behaviour of dual-phase steel is to a large extent controlled by the carbon concentration through its effect on the volume fraction and strength of the martensite phase. Other alloying elements, such as manganese, silicon, chromium, molybdenum and niobium may be added to the dual-phase steels. Manganese is added because it is cost effective in promoting hardenability (Llewellyn and Hi l l i s , 1996). Si l icon is useful in preventing pearlite and carbide formation and also results in solid solution hardening of the ferrite phase (Thomas 7 Chapter 2. Literature Review and Koo , 1979; Llewellyn and Hi l l i s , 1996; Davies, 1979). Chromium and molybdenum improve hardenability, suppress pearlite formation and promote martensite formation (Waterschoot et al., 2001). Niobium refines the ferrite grain size and results in precipitation strengthening of the ferrite phase (Pickering, 1978). 2.2. Overview of Typical Properties of Dual-Phase Steels To illustrate the properties of dual-phase steels, it is useful to review the classic experimental study of Speich and M i l l e r (1979). They produced a series of dual-phase steels by systematically varying the carbon concentration of the steel and the intercritical annealing temperature. Figure 2.4 illustrates the amount of martensite (formed from austenite during a rapid cooling stage) in 1.5%Mn dual-phase steels as a function o f the steel carbon concentration for various intercritical annealing temperatures. 80 UJ l -55 z UJ I-< s Z UJ o UJ 0_ Intercritical -r x 780 C Temperature \ • - 780 °C A - 760 °C O - 740 °C 0 0.1 0.2 0.3 C A R B O N C O N T E N T O F S T E E L , pet Figure 2.4. Percent martensite in 1.5%Mn dual-phase steels as a function of the steel carbon concentration and intercritical annealing temperature (Speich and Miller, 1979). 8 * Chapter 2. Literature Review A n important observation in Figure 2.4 is that the same martensite content can be produced with significantly different carbon concentrations. The relationship between the amount of austenite phase (and its carbon concentration) formed during annealing at intercritical temperature and the overall carbon concentration of the steel can be understood with the assistance of the iron-carbon phase diagram. The portion of the iron-carbon phase diagram relevant to the current study is illustrated in Figure 2.5. In this Figure, two steels with 0.06wt.% and 0.12wt.% carbon and two different intercritical temperatures, i.e. 770 °C and 825 °C, are shown as possible examples. First o f all, the equilibrium carbon concentration of the austenite phase (y) formed at a fixed intercritical temperature is independent of the steel carbon content (e.g. 0.49wt.% and 0.24wt.% for both the steels at 770 °C and 825 °C, respectively). However, the fraction of austenite is a function of the steel carbon concentration, the higher the carbon concentration of the steel, the greater is the austenite content. According to the Fe-C phase diagram, the relative amount of austenite phase formed in the steels with 0.06wt.% and 0.12wt.% carbon at the intercritical temperature of 770 °C can be calculated by drawing a tie line at this temperature and using the lever rule as follows: Steel with 0.06wt.%C (point a): y = Q - Q 6 ~ Q - Q 1 5 x l 0 Q = 1 0 o / o 0.49-0.015 Steel with 0.12wt.%C (point b): v = ° ' 1 2 ~ ° ' 0 1 5 x l 0 0 = 22% 0.49-0.015 where 0.015 is the carbon concentration (in wt.%) of the ferrite phase at this temperature. As can be observed, an increase in the steel carbon concentration from 0.06wt.% to 0.12wt.% leads to an increase in the austenite (and martensite after rapid cooling) content from 10%> to 22%. 9 Chapter 2. Literature Review 1000 Ferrite (a) . 9 600 A 500 400 300 200 a + y a + F e 3 C (Cementite) 0.12 0 t Fe 0.06 0.2 0.4 0.6 0.8 1.0 Carbon Concentration (wt.%) 1.2 Figure 2.5. Fe-C phase diagram (for wt.%C = 0-1.2 and T = 200-1000 °C) showing the effects of steel carbon concentration and intercritical temperature on the amount of austenite formed and its carbon content (calculated diagram using Thermocalc software with the Fe-2000 database, licensed to UBC). The fraction of austenite formed during intercritical annealing depends also on the intercritical temperature. A s another example illustrating this effect, the amount of austenite formed at 825 °C can be calculated as follows: Steel with 0.06wt.%C (point a'): y = Q - Q 6 ~ Q Q 1 Q x 1 0 0 = 22% Steel with 0.12wt.%C (point b'): 0.24-0.010 0.12-0.010 0.24-0.010 x 100 = 48% 10 Chapter 2. Literature Review where 0.010 is the ferrite carbon concentration (in wt.%) at 825 °C. A s clearly seen, an increase in the intercritical annealing temperature from 770 °C to 825 °C results in an increase in the percent austenite (or martensite formed from austenite during the subsequent rapid cooling stage) for both the steels with 0.06wt.% and 0.12wt.% carbon. Another observation in Figure 2.5 is that for the formation o f a fixed austenite content (e.g. 22%) in steels with different carbon concentrations, the steel with lower carbon concentration must be annealed at higher intercritical temperature. A n important point here is that the carbon concentration of the austenite (and also resulting martensite) phase in the steel with lower carbon content (annealed at higher intercritical temperature) is lower. Figure 2.6 illustrates the tensile true stress-true strain curves of dual-phase steels with different carbon concentrations and martensite contents (all intercritically annealed at 760 °C, Figure 2.4) (Speich and Mil le r , 1979). A s can be observed, dual-phase steels show a continuous yielding behaviour, higher initial work hardening rates and greater tensile strengths compared to the ferritic steel. However, the presence of the martensite phase in the steel microstructure reduces the uniform strain. Furthermore, the yield strength (0.2% offset), the initial work hardening rate and the ultimate tensile strength o f dual-phase steels increase with an increase in the percent martensite from 25 to 60. A s shown in Figure 2.6, tensile properties of dual-phase steels are controlled by the martensite content. Figures 2.7 to 2.9 summarize the effect o f martensite content on strength, uniform elongation and total elongation of the same dual-phase steels as shown in Figure 2.4 annealed at different intercritical temperatures (Speich and Mi l le r , 1979). In Figure 2.7, the yield and tensile strengths of the steels are shown as functions of the percent martensite. 11 Chapter 2. Literature Review 2000 0 1 . . . • . . 1 0.000 0.030 0.060 0.090 0.120 0.150 0.180 0.210 TRUE STRAIN Figure 2.6. The true stress-true strain curves of 1.5%Mn dual-phase steels with different carbon concentrations and martensite contents intercritically annealed at 760 °C (Figure 2.4) compared to the ferritic steel (Speich and Miller, 1979). I 1 r % C of steels at each O - 740 °C 1 I I 0 20 40 60 PERCENT MARTENSITE Figure 2.7. The yield strength and the ultimate tensile strength (UTS) of 1.5%Mn dual-phase steels annealed at different temperatures as functions of percent martensite (Speich and Miller, 1979). For a given intercritical temperature, the data points from left to right correspond to 0, 0.06, 0.12, 0.16 and 0.20 wt.% carbon. 12 Chapter 2. Literature Review The steel samples annealed at each intercritical temperature indicated in Figure 2.7 have different carbon concentrations, i.e. 0.0 wt.%, 0.06 wt.%, 0.12 wt.%, 0.16 wt.% and 0.20 wt.% such that at fixed intercritical temperatures, higher martensite contents have been produced in steels with greater carbon concentrations. A s can be observed in this Figure, the yield strength (0.2% offset) and the tensile strength increase with increasing the percent martensite, but both are also functions of the steel carbon concentration and the intercritical annealing temperature. The variation of uniform elongation of the steels in Figure 2.4 with the percent martensite is given in Figure 2.8. It is clearly shown in this Figure that the uniform elongation o f the dual-phase steels annealed at different intercritical temperatures decreases with an increase in the martensite content. The percent martensite shows a similar effect on the total elongation (Figure 2.9) and the reduction in area (not shown), i.e. an increase in the percent martensite results in a decrease in the ductility. • - 780 °C A - 760 °C I 1 i i I 0 20 40 60 P E R C E N T M A R T E N S I T E Figure 2.8. Variation of the uniform elongation of the dual-phase steels in Figure 2.4 with the percent martensite (Speich and Miller, 1979). 13 Chapter 2. Literature Review 0 20 40 60 PERCENT MARTENSITE Figure 2.9. The total elongation of dual-phase steels in Figure 2.4 as a function of the percent martensite (Speich and Miller, 1979). To summarize, typical results from the literature show that the strength o f dual-phase steels increases with an increase in the percent martensite while their ductility decreases. 2.3. Characteristics of Microstructure Characteristic properties of dual-phase steels are directly related to their microstructures consisting of hard martensite islands in a ductile ferrite matrix. In order to understand the mechanisms through which these properties are achieved, knowledge of the dual-phase steel microstructures is needed. In the next sections, the ferrite-martensite steel microstructure w i l l first be examined and the parameters affecting this microstructure w i l l be evaluated and then, the individual phases wi l l be studied. 14 Chapter 2. Literature Review 2.3.1. Ferrite-Martensite Microstructure The microstructure of dual-phase steels is comprised of the martensite phase embedded in a ferrite matrix. A n y ferrite-martensite dual-phase steel is specified by its microstructural features such as the martensite content, size, shape and spatial distribution. In the following sections, the effect of these microstructural characteristics w i l l be summarized. 2.3.1.1. Martensite Content During the production route, a mixture of austenite and ferrite phases is formed in the steel and in the subsequent quenching stage, the austenite phase transforms to martensite ( if the cooling rate is sufficiently high). For a dual-phase steel with a fixed carbon concentration, the percent austenite formed depends on the annealing temperature and time. For a fixed hold time, the higher the intercritical temperature, the greater is the percent austenite (and resulting martensite after quenching). On the other hand, at a given intercritical temperature the austenite content is controlled by the overall carbon concentration o f steel. These effects can be observed in Figure 2.4 illustrating the simultaneous effects of the intercritical temperature and the steel carbon concentration on the percent martensite in an intercritically annealed dual-phase steel (with 1.5% Mn) for a fixed hold time of 60 minutes (Speich and Mi l le r , 1979). 2.3.1.2. Martensite Size, Morphology and Spatial Distribution The martensite islands in dual-phase steel microstructures can vary in size depending on the microstructure and deformation history of the starting material and the processing parameters. Huang (2004) has recently conducted a comprehensive investigation into the effect of starting material and heating rate to the intercritical annealing temperature on the martensite size and 15 Chapter 2. Literature Review morphology in a low carbon M n - M o dual-phase steel (the same steel as investigated in this work). As can be observed in the steel micrographs shown in Figure 2.10, there is a strong effect of the initial steel microstructure (cold and hot-rolled) and the heating rate on the final microstructures. Figure 2.10. The Effects of the starting microstructure and the heating rate to annealing temperature on the martensite size and morphology in a Mn-Mo dual-phase steel with 0.06% C, (a) hot-rolled steel, low heating rate (1 °C/sec), (b) cold-rolled steel, low heating rate (1 °C/sec), (c) hot-rolled steel, high heating rate (100 °C/sec) and (d) cold-rolled steel, high heating rate (100 °C/sec) (Huang, 2004). 16 Chapter 2. Literature Review For example, small martensite islands (almost equiaxed) are formed in the cold-rolled steel at the low heating rate (1 °C/sec) (2-10b) and in the hot-rolled steel at the high heating rate (100 °C/sec) (2.10c), relatively large martensite islands are formed in the hot-rolled steel annealed with the low heating rate (1 °C/sec) (2.10a) and finally, the martensite islands in the cold-rolled steel at the high heating rate (100 °C/sec) are banded (mostly ellipsoidal in shape, Figure 2.10d). Furthermore, there are reports in the literature regarding the dual-phase microstructures with fine and coarse martensite islands produced through different processing techniques ( K i m and Thomas, 1981; Bag et al., 1999). The combination of the processing route and the starting material also affects the martensite morphology in the resulting dual-phase microstructures. A s shown in Figure 2.10, the martensite islands in dual-phase steel can be either equiaxed or elongated ellipsoidal in shape depending on the starting material and the controlling parameters o f the heat treatment cycle. In addition, it has been reported that the martensite phase can be formed as fine fibrous, fine globular or even coarse islands during different processing routes ( K i m and Thomas, 1981; Bag et al., 1999). The spatial distribution of austenite is controlled by the initial microstructure, steel chemistry and processing parameters (Speich and Mil le r , 1979; Marder, 1982; Huang, 2004). Since martensite is formed during the quenching stage as a result of the austenite transformation, its distribution is mainly controlled by the austenite distribution. During the intercritical annealing of the steels with ferrite-pearlite starting microstructures, austenite nucleates preliminary at the pearlite colony boundaries and after pearlite dissolution, the subsequent growth of austenite occurs in the ferrite matrix (Speich et al., 1981; Garcia and 17 Chapter 2. Literature Review Deardo, 1981). In the cold-rolled steels, the recrystallization of the deformed ferrite affects the formation and distribution of austenite (Yang, et al., 1985). Therefore, much of the austenite phase in these cold-rolled dual-phase steels (intercritically annealed) can be distributed in the bands parallel to the initial rolling direction. In this case, the final steel microstructure at room temperature consists o f the martensite bands (Speich and Mi l l e r , 1979; Park et al., 1981; Marder, 1982) which are formed as a result of the austenite to martensite phase transformation during the subsequent rapid cooling stage. 2.4. Properties of Constituent Phases In dual-phase steels, ferrite can be viewed as the matrix and martensite as the reinforcing phase. The mechanical behaviour of dual-phase steels can be understood by comparing them to particulate composite materials (Chawla, 1998). When an external load is applied to these steels, the ferrite matrix transfers the load to the hard martensite which acts as an effective load-bearing component. This results in an increase in the load-bearing capacity of the steel and therefore, the strength of the steel is increased. Although the presence o f hard martensite in these steels results in a characteristic mechanical behaviour, it is obvious that the ferrite phase also plays an important role. For example, the yielding behaviour of dual-phase steels is controlled by the ferrite matrix (Liedl et al., 2002), as it is much softer than the martensite. To understand the mechanical properties of dual-phase steels, it is necessary to have knowledge of the role of the constituent phases in the microstructure as wel l as their properties. 18 Chapter 2. Literature Review 2.4.1. Ferri te Phase The flow stress o f ferrite depends on its chemical composition and its microstructural features. The flow stress of ferrite can be expressed in terms of various strengthening components. Different contributions to the flow stress of ferrite are as follows: 1- Lattice resistance. The first component of the flow stress o f ferrite shows the overall resistance of the crystal lattice to the dislocation movement. This intrinsic stress, rjj, can be described by the Peierls-Nabarro equation which represents the stress required to overcome the lattice resistance against dislocation movement (Hertzberg, 1996; Dieter, 1976). 2. Dislocation hardening. This component shows the dependence of the strength on the dislocation density (Hertzberg, 1996). A n increase in the stress required to cause the plastic flow due to the previous plastic deformation is known as work (or strain) hardening (Dieter, 1976). The dislocation hardening component of flow stress is proportional to square root of the dislocation density (p ) , i.e. = a G b M p " 2 in which a is a constant, G is the shear modulus, b is the Burgers vector and M is the Taylor factor. In the present context, plasticity in the ferrite matrix can occur due to accommodation of the martensite reaction as w i l l be discussed later. 3. Grain size strengthening. Grain boundaries serve as an effective barrier against dislocation motion, thus, a strengthening effect results from their presence in the microstructure. The grain size effect can be described by Hall-Petch equation (Hertzberg, 1996), i.e. c G B = a i + k y d " 1 / 2 which shows the yield strength of a polycrystalline material as a function o f its grain size, d (in this equation, k y is a constant). The detailed physical origin for this effect remains controversial (Meyers and Chawla, 1984), however, this equation provides a reliable method to describe this strengthening mechanism. 19 Chapter 2. Literature Review 4- Solid solution strengthening. The stress to cause dislocation motion can also be increased by the addition of solute atoms (Meyers and Chawla, 1984). In a solid solution (e.g. ferrite; a solvent lattice with at least one solute atom), the interaction between the stress fields o f the solute atom(s) and those of the dislocations causes a strengthening effect which is called solid solution hardening. Since the solute atoms can occupy either the atomic sites in the solvent lattice or the interstitial positions between the solvent atoms depending on their relative sizes, the resulting solid solutions can be of two types, i.e. substitutional and interstitial solid solution. In ferrite phase, carbon and nitrogen are the interstitial solute atoms whereas alloying elements such as silicon, manganese and molybdenum can be substituted for the iron atoms. The extent to which a crystal can be strengthened by solid solution depends on the solute concentration, c (Pickering, 1978), i.e. o s s = kc" where n is in the range of 1/2 to 2/3 and k is a constant. 5. Precipitation hardening. Precipitation of a second phase from the ferrite solid solution may result in an effective microstructural barrier to dislocation motion. The interaction between moving dislocations and the precipitates may come from multiple mechanisms (Meyers and Chawla, 1984; Hertzberg, 1996). In ferrite, different alloying elements such as niobium, vanadium and titanium may be precipitated as carbides and nitrides (or carbonitrides) and thus, may lead to precipitation hardening (Gladman, 1997; Leslie, 1981). The final strength of ferrite is the sum of these contributions. Usually, a linear addition law is used although non-linear addition laws have been considered in some cases (Charleux et al., 2001). 20 Chapter 2. Literature Review 2.4.2. Martensite Phase The main role o f martensite in a dual-phase steel is to carry part of the load that is externally applied to the steel. Accordingly, to understand the mechanism of strengthening effect of martensite in dual-phase steels, the parameters which may affect the martensite strength should be considered. Martensite is a metastable phase which is formed during rapid cooling o f austenite from high temperatures (Smallman and Bishop, 1999). Martensite is a supersaturated solid solution , o f carbon in ferrite with a crystal structure which is generally body-centered tetragonal (bet). A s martensite is usually very hard, it is used in dual-phase steels as reinforcement phase. Since a small amount of bainite and/or retained austenite may exist within the martensite structure (depending on its chemical composition and processing parameters), the hard constituent in dual-phase steels should be considered a second phase containing martensite with or without bainite and/or retained austenite. The martensite strength (and hardness) is mainly controlled by its carbon concentration (Pickering, 1978). The yield strength of martensite increases with an increase in its carbon content (Leslie and Sober, 1967; Leslie, 1981). Figure 2.11 shows the yield strength of martensite as a function of its carbon concentration (Leslie, 1981). There are two lines in this Figure which represent the upper and lower values of the yield strength. A s can be observed in Figure 2.11, the yield strength of a martensitic steel containing 0.2 wt.% carbon is at least 1000 M P a which increases to about 1265 M P a with an increase in its carbon concentration from 0.2 wt.% to 0.3 wt.%. 21 Chapter 2. Literature Review 1750 CD S 2 5 0 -0 -I , , , , , , , 1 0 0.1 0.2 0.3 0.4 Martensite Carbon Concentration (wt.%) Figure 2.11. The yield strength of martensite as a function of its carbon concentration, replotted curves (Leslie, 1981). Dashed lines indicate the lower and upper strength values. The strength of martensite depends also on the substitutional alloying elements such as manganese and silicon which cause the solid solution strengthening of the martensite structure, but this effect is usually considered to play a secondary role compared to the strong strengthening effect of carbon atoms (Pickering, 1978; Sinha, 1989). Martensite is often considered a hard and non-deformable phase, but the experimental results in Figure 2.11 show that the yield strength of martensite is in the range of 720-850 M P a for a carbon concentration of 0.1 wt.% C. In addition to lower yield strength, low carbon martensite may also exhibit a noticeable amount of plasticity after yielding. 22 Chapter 2. Literature Review 2.4.2.1 Tempering of Martensite Tempering is a heat treatment process through which martensite becomes softer and more ductile. Several processes may happen during tempering including crystal change from bet to bec, segregation of carbon atoms to the lattice defects, precipitation o f carbides (such as cementite), recovery and recrystallization and transformation o f retained austenite to martensite (Leslie, 1981; Speich and Leslie, 1972; Speich, 1969). The extent to which the deformation behaviour of martensite changes with tempering depends mainly on the tempering temperature and steel chemistry (Leslie, 1981). Since the mechanical behaviour of dual-phase steels depends on the martensite strength and tempering affects this strength, one may expect that the mechanical behaviour of dual-phase steels changes with tempering. Dual-phase steels are usually used in as-quenched condition in order to take the advantage of the strengthening effect of hard martensite. However, for those industrial production routes which employ a galvanizing treatment in the post-annealing process, the tempering of martensite may be an issue. A number of investigations have been conducted to characterize the tempering behaviour o f dual-phase steels (Panda et al., 1993; Jha et al., 1993; Samuel et al., 1987; Tavares et al., 1999; Davies, 1981). In these investigations, the tempering process has been found to alter the yielding behaviour, to decrease the tensile strength and to increase the ductility of dual-phase steels. Jardim et al. (1984) have evaluated the effect of tempering on martensite plasticity and fracture behaviour of a 0.08%C M n - S i dual-phase steel. Their fractographic observations have shown that the fracture i n as-quenched steel began at much lower far field strains compared with the quenched-tempered steel. Furthermore, they have found a transition in the martensite 23 Chapter 2. Literature Review deformation behaviour from elastic to plastic after tempering for 60 minutes at 300 °C. 2.5. Mechanical Properties of Dual-Phase Steels: Detailed Considerations In order to evaluate the mechanical behaviour o f dual-phase steels, one should take into account the complex interaction between the constituent phases during deformation. The fundamental problem is to understand how the load and deformation (or stress and strain) are partitioned between the phases when the material is subjected to an external load. 2.5.1. Strengthening Effect of Martensite The strengthening effect of martensite has been attributed to three main effects. First, the martensite phase can carry a substantial load which is transferred from the ferrite matrix. Second, martensite can affect the deformation behaviour (flow stress and work hardening behaviour) of the ferrite matrix by introducing so-called geometrically necessary dislocations (Ashby, 1970; Fleck et al., 1994). This is an effect in two-phase materials which results from the strain gradient between the constituent phases with different deformation behaviours. The geometrically necessary dislocations are required to accommodate this strain gradient and to allow the compatible deformation of constituent phases. Third, the presence of martensite can affect the ferrite phase due to the additional dislocations introduced into the ferrite structure in the vicinity of martensite islands as a result of the plastic strains associated with the martensite transformation (Bourell and Rizk , 1983; Crawley et al., 1981; Sherman et al., 1981). The density of additional dislocations introduced into the ferrite structure increases with increasing the martensite content (Sherman et a l , 1981), hence, one may expect that the 24 Chapter 2. Literature Review strengthening effect o f martensite on the ferrite phase becomes more pronounced as the martensite content increases. 2.5.2. Yielding Behaviour When a dual-phase steel is subjected to an external load, plastic flow starts in the ferrite matrix (Sakaki et al., 1983; Sarosiek and Owen, 1984). Thus, the yielding behaviour of dual-phase steels is controlled by the strength of the ferrite phase (Liedl et al., 2002; Chang and Preban, 1985) (see Section 2.4.1). If the ferrite were deformed in the absence of the martensite, its yielding behaviour would be discontinuous, however, it yields differently when used as the matrix in dual-phase steels. The yielding of ferrite phase in dual-phase steels is continuous suggesting that the initiation of plastic flow in the ferrite matrix occurs in a gradual and continuous manner (Sakaki et al., 1982; 1983). The continuous yielding of dual-phase steels can be attributed to the presence of internal stresses within the ferrite matrix originated from the transformation strains associated with the martensite transformation as well as the plastic incompatibility between the constituent phases (Sakaki et al., 1983; Gerbase et al., 1979). Internal stresses cause microyielding of the ferrite at regions around the martensite islands under relatively low applied stresses compared with the yield stress of bulk ferrite and consequently, plastic flow begins simultaneously in many regions within the ferrite matrix throughout the microstructure. 2.5.3. Work Hardening Behaviour Dual-phase steels exhibit a characteristic high initial work hardening rate (Speich and Mil ler , 1979; Gerbase et al., 1979; Sarosiek and Owen, 1984; Davies and Magee, 1979). 25 Chapter 2. Literature Review During deformation, the plastically deforming ferrite matrix transfers the applied stress to the load-bearing martensite phase. The dislocations generated in the ferrite matrix due to the martensite reaction (occurring during the rapid cooling) and the plastic incompatibility between the constituent phases also contribute to the work hardening o f dual-phase steels. The initial work hardening rate o f dual-phase steels increases with an increase in the martensite content (Speich and Mil le r , 1979; Ramos et al., 1979). This effect can be observed in Figure 2.6 which illustrates the stress-strain curves o f dual-phase steels with different martensite contents (approximately 25-60%). The work hardening behaviour is also affected by the martensite island size. According to Balliger and Gladman (1981) and Pickering (1981), the work hardening rate of dual-phase steels increases with a decrease in the martensite island size. The experimental results regarding the effect of martensite shape on the work hardening behaviour o f dual-phase steels are very limited. Sarosiek and Owen (1984) have shown that dual-phase steel with banded martensite morphology (mostly continuous) has almost the same work hardening behaviour as the homogenized dual-phase steel with smaller (noncontinuous) martensite islands. 2.5.4. Necking Condition (Considere Criterion) The onset of necking in a tension test occurs at the maximum load where the increase in the stress (due to a decrease in the specimen cross section area) becomes greater than the increase in the load-carrying ability of the material (due to work hardening) (Dieter, 1976). According to the Considere condition (which is valid for strain rate insensitive materials), the strength and 26 Chapter 2. Literature Review work hardening rate o f the material must fulfill the following requirement at a true strain equal to the uniform strain: d a where a and da/ds are the true stress and the work hardening rate, respectively. This equation specifies the condition at which the plastic instability or necking phenomenon begins during a tensile deformation. The Considere condition is schematically illustrated in Figure 2.12. Figure 2.12. The Considere condition for necking and uniform strain. Dual-phase steels show a relatively high uniform strain compared to the low alloy steels with ferrite-pearlite microstructures (Balliger and Gladman, 1981; Gladman, 1982; Rashid and Cprek, 1978). It has been shown that the higher work hardening rate o f dual-phase steels causes a delay in the onset o f necking which results in an enhanced uniform strain. The true uniform 27 Chapter 2. Literature Review strain of dual-phase steels typically decreases with an increase in the martensite content. Figure 2.8 clearly shows this effect in the dual-phase steels with 0.06 wt.% to 0.2 wt.%> C which are intercritically annealed at different temperatures (Speich-Miller, 1979). 2.5.5. Fracture Behaviour It has been shown by Balliger and Gladman (1981) that the replacement of pearlite in the steel microstructure with martensite phase (to make a dual-phase steel) has a detrimental effect on the total elongation (or fracture strain). This may be attributed to the fact that the hard and non-deformable martensite phase has an undesirable effect on the damage formation process during deformation. The martensite is usually a brittle phase (depending on its carbon concentration) which may fracture at high strain levels in the non-uniform (post-necking) deformation regime. On the other hand, decohesion of its interface with the ferrite matrix due to the plastic incompatibility may also occur in this deformation regime. Both these phenomena are reported in the literature as responsible mechanisms for ductile fracture of dual-phase steels (Shen and L e i , 1984; Jardim et al., 1984; Steinbrunner et al., 1988; Ahmad et al., 2000; Chawla et al., 1983; Sidjanin and Miyasato, 1989; Sarwar and Priestner, 1996). Although there is a hard and usually non-deformable martensite phase in the microstructure of dual-phase steels, these steels exhibit a ductile fracture behaviour (Marder, 1982; Su and Gurland, 1987; Steinbrunner et al., 1988; Ahmad et al., 2000). It has been observed that the ductile fracture in dual-phase steels occurs in three sequential stages, i.e. void nucleation, void growth and void coalescence resulting in a dimpled fracture surface (a characteristic of ductile fracture). Microvoids which are nucleated as a result of the decohesion o f the ferrite-martensite interface and/or martensite cracking, usually grow within the more ductile ferrite matrix 28 Chapter 2. Literature Review resulting in a ductile fracture pattern. It should be noted that this effect depends on the martensite content, size and morphology ( K i m and Thomas, 1981; Marder, 1982; Bayram et al., 1999). It has been shown in these investigations that the large, .banded and inter-connected martensite islands in dual-phase steels, compared to the fine and isolated martensite particles, result in poor fracture properties. The fracture behaviour of dual-phase steels is affected by the martensite content and size (Balliger, 1982; K i m and Thomas, 1981; Marder, 1982; Bayram et al., 1999; Samuel, 1983). Figure 2.13 illustrates the effect of martensite (here as second phase) content and size on the true fracture strain of dual-phase steels with 0.066 to 0.13 wt.% carbon (Balliger, 1982). 1.2 -r 1 -U— CO •I 0 .8-CO <D _ 0.6 -o ro 5 0 .4-i_ H 0.2 -0 -0 2 4 6 8 10 12 Mean Second Phase Island Diameter, d (um) Figure 2.13. The fracture strain of dual-phase steels with 0.066-0.13% carbon as a function of the martensite content and size, replotted curves (Balliger, 1982). A s can be observed in Figure 2.13, for a given percent martensite, the true fracture strain decreases with an increase in the mean diameter of the martensite islands. It is also shown that the martensite size effect becomes more pronounced as its content increases. In other words, as 29 V m = 30% Chapter 2. Literature Review t h e p e r c e n t m a r t e n s i t e g o e s u p , t h e e x t e n t t o w h i c h t h e m a r t e n s i t e s i z e a f f e c t s t h e t r u e f r a c t u r e s t r a i n b e c o m e s m o r e s i g n i f i c a n t . F u r t h e r m o r e , f o r a f i x e d m e a n d i a m e t e r o f m a r t e n s i t e i s l a n d s , t h e t r u e f r a c t u r e s t r a i n o f d u a l - p h a s e s t e e l i n c r e a s e s w i t h a d e c r e a s e i n t h e p e r c e n t m a r t e n s i t e f r o m 3 0 t o 1 5 . 2.5.6. Deformation Behaviour of Martensite A l t h o u g h t h e h a r d m a r t e n s i t e p h a s e ( u n t e m p e r e d ) i n d u a l - p h a s e s t e e l s u s u a l l y r e m a i n s e l a s t i c d u r i n g d e f o r m a t i o n ( a t l e a s t i n t h e u n i f o r m d e f o r m a t i o n r e g i m e ) , i t m a y d e f o r m p l a s t i c a l l y i n c e r t a i n c i r c u m s t a n c e s . D e f o r m a t i o n b e h a v i o u r o f m a r t e n s i t e d e p e n d s m a i n l y o n i t s c a r b o n c o n c e n t r a t i o n a n d t h e m a r t e n s i t e c a r b o n c o n c e n t r a t i o n i s c o n t r o l l e d b y i t s c o n t e n t a s w e l l a s t h e s t e e l c a r b o n c o n c e n t r a t i o n . A c c o r d i n g t o t h e l i t e r a t u r e , t h e m a r t e n s i t e p h a s e i n d u a l - p h a s e s t e e l s e x h i b i t s e l a s t i c d e f o r m a t i o n b e h a v i o u r d u r i n g s t r a i n i n g e x c e p t f o r s o m e a r e a s e x p e r i e n c i n g l a r g e a m o u n t s o f s t r a i n s ( B a l l i g e r a n d G l a d m a n , 1 9 8 1 ; K o r z e k w a e t a l . , 1 9 8 0 ; C h a w l a e t a l . , 1 9 8 3 ; J a r d i m e t a l . , 1 9 8 4 ) . F o r e x a m p l e , a c c o r d i n g t o B a l l i g e r a n d G l a d m a n ( 1 9 8 1 ) , p e a r l i t e a n d f e r r i t e i n a p e a r l i t i c s t e e l ( w i t h 0 . 1 2 % C a n d 14%> p e a r l i t e ) c a n c o - d e f o r m d u r i n g t e n s i l e s t r a i n i n g , h o w e v e r , w h e n p e a r l i t e i s r e p l a c e d w i t h m a r t e n s i t e , s t r a i n p a r t i t i o n i n g o c c u r s b e t w e e n t h e c o n s t i t u e n t p h a s e s , i . e . t h e m a r t e n s i t e p h a s e r e m a i n s e n t i r e l y e l a s t i c e x c e p t f o r r e g i o n s v e r y c l o s e t o t h e f r a c t u r e s u r f a c e . T h e e x p e r i m e n t a l r e s u l t s i n t h e l i t e r a t u r e c o n c e r n i n g d u a l - p h a s e s t e e l s w i t h p l a s t i c m a r t e n s i t e a r e v e r y l i m i t e d . J i a n g e t a l . ( 1 9 9 5 ) h a v e f o u n d m a r t e n s i t e p l a s t i c i t y i n a n i n t e r c r i t i c a l l y a n n e a l e d d u a l - p h a s e s t e e l w i t h 0 .12% c a r b o n t h r o u g h a n a l y z i n g t h e l n ( d o 7 d s ) - l n a c u r v e s . S h e n e t a l . ( 1 9 8 6 ) h a v e a l s o f o u n d m a r t e n s i t e p l a s t i c i t y i n t h e i r i n v e s t i g a t i o n o f d u a l -3 0 Chapter 2. Literature Review phase steels (0.09%, 0.23% and 0.29% C) through conducting a series of dynamic observations of martensite deformation using a scanning electron microscope. According to their experimental results, the macroscopic strain at which the martensite phase starts to deform plastically depends on the martensite content and carbon concentration. Su and Gurland (1987) have conducted a comprehensive examination on martensite plasticity in a dual-phase steel with 0.12% C. In this study, the plastic strain in the martensite phase was measured using a series of microgrid lines superimposed on the dual-phase steel microstructures. In this investigation, martensite plasticity was found to occur in the steels with martensite contents greater than 50-80%). 2.6. Modelling Stress-Strain Behaviour of Dual-Phase Steels A number o f modelling approaches have been employed in the literature in order to model the stress-strain response of these steels. The modelling techniques fall into three categories; i) empirical approaches in which the mechanical properties o f dual-phase steels are estimated using the results obtained from experiments, ii) dislocation models which correlate the flow stress of dual-phase steels with a number of microstructural variables such as the ferrite and martensite grain sizes, the martensite content and the dislocation density in the ferrite matrix and iii) continuum models (such as finite element and Eshelby models) in which the steel is considered as a continuum material and modeled based on the theories o f plasticity. In these modelling techniques, the martensite phase is assumed to be an elastic phase during deformation. In the following sections, these modelling approaches w i l l be reviewed in more detail. 31 Chapter 2. Literature Review 2.6.1. E m p i r i c a l Approaches The strength of dual-phase steels can be predicted by a series o f empirical equations which are developed based on the experimental results. In these equations, the strength of dual-phase steels is usually given as a function of several parameters including the martensite content, the carbon concentration of ferrite and/or martensite phases, the steel carbon concentration and the heat treatment temperature. The main idea in these approaches is to predict the tensile strength of dual-phase steel (with a fixed martensite content) from the strengths of both constituent phases using the rule of mixture. Bayram et al. (2002) have developed an empirical model to estimate the tensile strength o f M n - S i dual-phase steel (with 0.1% C) containing 25% martensite. Their estimation showed a ± 1 0 % difference compared to the experimental results. Chen and Cheng (1989) obtained an empirical expression for the tensile strength of dual-phase steels with 0.1 to 0.19 wt.%> carbon and the martensite content between 20% and 80%>. They found the tensile strength of these steels to be a function of the percent martensite as well as the carbon concentrations o f the steel and the ferrite phase. Speich and Mi l l e r (1979) have also attempted to empirically estimate the tensile strength o f dual-phase steels. They derived an equation for the estimation of the tensile strength of 1.5%Mn dual-phase steels with 0.06-0.29 wt.% C as a function of the ferrite tensile strength and the martensite content and carbon concentration. It should be mentioned that the empirical models are usually developed for dual-phase steels with given chemistries and for a number of fixed microstructural parameters and therefore, they may not be used as a modelling method to predict the mechanical behaviour of dual-phase steels with a variety of chemistries and microstructural aspects. 32 Chapter 2. Literature Review 2.6.2. Dislocation Models In these models, the dual-phase steel is considered a two-phase material containing a dispersion of hard and non-deformable particles in a ductile matrix. These models have been developed based on Ashby's theory of work hardening for dispersion-hardened alloys (Ashby, 1966). The main concept in this theory is the dislocation-dislocation as well as the dislocation-particles interactions which have been shown to greatly affect the flow stress and work hardening of dispersion-strengthened materials. Ashby's theory of work hardening was originally developed for a material (Cu-SiC.) containing small amounts o f hard particles which were sufficiently small in size (sub-micron), with the assumption that these particles only deform elastically and never fracture during deformation. In a dislocation model, the flow stress of the material is expressed by different components (Nan and Clarke, 1996) as follows: - the stress to make the dislocations pass the particles with a given average spacing - the stress component due to strain gradient effects associated with geometrically necessary dislocations - the stress component which accounts for the strengthening effect resulting from introducing an additional dislocation density into the matrix due to the mismatch strains generated during the phase transformation (e.g. austenite to martensite transformation in dual-phase steels) Several attempts have been made to model the flow stress and work hardening behaviour o f dual-phase steels using the dislocation approach (Lanzillotto and Pickering, 1982; Jiang et al., 1992; Jiang et al., 1995). In these examinations, the effect o f microstructural parameters such as 33 Chapter 2. Literature Review the martensite content and size, the ferrite grain size, the dislocation densities in the ferrite phase and the martensite particles spacing has been taken into consideration. The particle size has been shown to greatly affect the mechanical behaviour of the two-phase materials predicted by the dislocation models (Nan and Clarke, 1996). According to Nan and Clarke (1996), the stress-strain response of the materials with particles smaller than about 1 um is greatly dependent on the particle size. The stress-strain behaviour of materials with particles greater than 10 um, on the other hand, is almost independent of the particle size. For the cases where materials contain particles in the range of 1-10 um, dislocation and continuum mechanisms play a role. 2.6.3. Finite Element (FE) Method Finite element method is a continuum type of model which may be used to predict the stress-strain behaviour of dual-phase steel microstructures. In this modelling approach, the continuum material is divided into a finite number of elements and the mechanical response of each element is specified by its constitutive behaviour. The deformation behaviour of the complete system as an assembly of its elements is evaluated by considering the displacement compatibility, the equilibrium conditions of the externally applied load and the appropriate boundary conditions (Zienkiewicz and Taylor, 2000). To develop a finite element model for a dual-phase steel, the steel microstructure is first meshed so that the cell structure containing a finite number of elements is generated. Several attempts have been made to model the stress-strain behaviour of ferrite-pearlite steels as well as ferrite-bainite and ferrite-martensite dual-phase steels using the finite element 34 Chapter 2. Literature Review method (Nygards et al., 2001; Ishikawa et al., 2000; Huper et al., 1999; Al -Abbas i and Nemes, 2003). These preliminary attempts to model the stress-strain response o f dual-phase steels with a number of simplifications look promising, however, further investigation is needed to account for more detailed features of the actual steel microstructures. 2.6.4. Modelling Based on Modified Eshelby Method The Eshelby based modelling approach can be used to predict the stress-strain response of dual-phase steels. The main advantage of this modelling approach is that the two-phase material can readily be modeled as a heterogeneous system with a uniform distribution of the second phase. The volume fraction and shape of the second phase can be considered in this model. This modelling approach w i l l be discussed in more detail in the following sections and the equations for estimation of the stress (and strain) within the constituent phases w i l l also be given. The Eshelby model was initially developed to determine the elastic stress field of an ellipsoidal inclusion in an infinite homogeneous elastic media. In the original Eshelby model (Eshelby, 1957) which was developed for dilute systems, the inclusion is defined as a region within the matrix which undergoes a change of shape and size under the matrix constraint. This theory was then extended for non-dilute composite materials under an external load in which both the inclusion and matrix with different elastic properties ( E x c l u s i o n » E m a t r i x ) deform elastically (Eshelby, 1957; Clyne and Withers, 1993). The goal was to examine how the applied stress is distributed between the constituent phases. Weng (1990) extended the Eshelby approach to examine the case with matrix plasticity. In this modified version o f the Eshelby model, the inclusion is also allowed to deform plastically 35 Chapter 2. Literature Review during straining. In the following sections, the original Eshelby approach w i l l be reviewed first and then, the Weng modelling technique w i l l be described in detail. 2.6.4.1. Eshelby Theory The Eshelby theory can be understood with the help of a simple set of imaginary cutting, straining and welding exercises (Eshelby, 1957; Clyne and Withers, 1993). In this technique, an ellipsoidal region (called the inclusion) is cut and removed from the matrix and then subjected to a shape and size change (transformation strain, s T). After applying surface tractions in order to return it back to its original shape and size, the inclusion is replaced into the hole from whence it came. Once the surface tractions are removed, the matrix constrains the inclusion such that it cannot attain its transformed shape and size. If the constrained strain in the inclusion is assumed to be 8 ° , the inclusion stress can be calculated using Hooke's law: a I = C M ( s c - 8 T ) (2.2) where s - s is the net elastic strain and C M IS the stiffness tensor of the material (matrix). Eshelby found a relationship between the constrained strain and the transformation strain by introducing a parameter called the Eshelby ' S ' tensor: e c = S e T (2.3) therefore a , = C M ( S - l ) 8 T (2.4) Accordingly, for a given transformation strain, the stress in the inclusion can be calculated from the Eshelby S tensor. The Eshelby S tensor depends on the inclusion shape (aspect ratio) and 36 Chapter 2. Literature Review the Poisson's ratio of the matrix. The form of the S tensor for different inclusion shapes can be found elsewhere (Clyne and Withers, 1993; Brown and Clarke, 1975; Mura, 1982). 2.6.4.2. Modified Eshelby Approach The modified Eshelby approach has been developed to include the matrix (and inclusion) plasticity in the model in order to predict the overall stress-strain response of two-phase composite materials (Weng, 1990; Qiu and Weng, 1991). It is assumed in the model that the inclusion is uniformly dispersed in the matrix. The stress-strain behaviour o f the composite can be predicted using the stress-strain relations of both constituent phases. The plastic deformation of the constituents is taken into account by defining the secant moduli (Figure 2.14). For an elasto-plastic material with the stress-strain curve shown in Figure 2.14, the secant Young's modulus at an arbitrary point (oj, E i ) is defined as: E ; = = = _ J L _ (2.5) E ' + E - ^ + E P l + e * Since the Young 's modulus o f the material (E) is constant and its flow stress (c»i) depends on the plastic strain (ef ) , the secant modulus is only a function of the plastic strain. Accordingly, the secant modulus is variable and, indeed, decreases as the deformation proceeds. According to Equation 2.5, the secant Young's modulus of the phase r (refers to 0 and 1 for matrix and inclusion, respectively) is expressed as: E s r = (2.6) where E r is the Young's modulus,&PT is the plastic strain and rj r is the flow stress of phase r. 37 Chapter 2. Literature Review Figure 2.14. Definition of the secant Young's modulus for an elasto-plastic material. The secant shear modulus (\i s0) and bulk modulus ( K * ) can also be calculated using the following equations (Dieter, 1976) in which v* is the secant Poisson's ratio: A: = 2( l + v s r) E ! (2.7) (2.8) 3 ( l -2v s r ) The secant bulk modulus of each constituent is equal to its bulk modulus due to the plastic incompressibility (Weng, 1990), therefore 1 <X ^ - - ( - - v . ) — 2 2 r E , (2.9) At this stage, the stress partitioning coefficients, which are strongly dependent on the inclusion shape, can be determined. Although the mathematics behind the formulation needed to develop the appropriate equations for spherical inclusions is straightforward (Tandon and 38 Chapter 2. Literature Review Weng, 1988), it is greatly complicated in the case of ellipsoidal inclusions, however, the solutions for this type of inclusions are available in the literature (Bhattacharyya and Weng, 1994). 2.6.4.2.1. Spherical Inclusion (Aspect Ratio = 1) Two separate stress partitioning coefficients, i.e. b* and b", can be determined for spherical inclusions which correlate the stress in the composite with those in the constituent phases. These stress coefficients are functions o f the inclusion volume fraction (f) and the elastic properties of both phases. Assuming an appropriate equation for the distortional component of the Eshelby's S tensor (P*, which appears when the matrix phase starts deforming plastically), Weng has developed an analytical solution for the stress partitioning coefficients (Weng, 1990). When a two-phase material (e.g. a dual-phase steel) is subjected to an external load, both constituent phases deform elastically first and as the deformation proceeds, the softer matrix (ferrite) starts to deform plastically. A t the later stages of straining, the harder phase (martensite) may either remain elastic or start deforming plastically. The stress-strain relation of this material can be estimated using the modified Eshelby approach with taking into account the deformation behaviour of both constituents. i) Elastic matrix - Elastic Inclusion. For a dual-phase steel, this gives a simple solution since the elastic properties of the ferrite and martensite phases are identical. The elastic stress-strain relation of dual-phase steel is determined using Hooke's law (Weng, 1990). ii) Plastic matrix - Elastic Inclusion. To predict the stress-strain relation of a dual-phase steel at this stage (i.e. ferrite matrix yields while martensite remains elastic), the flow behaviour of 39 Chapter 2. Literature Review the ferrite phase must be known. The stress partitioning coefficients (bs0 and b ' ) for a dual-phase steel with a fixed martensite volume fraction (f) can be calculated using the elastic parameters of both constituents (|a.,and u.*) and the distortional component of S tensor, P*, (Weng, 1990). Since the flow behaviour of the ferrite matrix (i.e. the relationship between its flow stress, o 0 , and the plastic strain, ep0) is known, it can be shown (Weng, 1990) that [is0 and P* (and hence b* and b,) are only functions of the plastic strain in the ferrite matrix (e£) . Following the approach of Weng, the flow stress of a dual-phase steel can be estimated using the following equation for any given ferrite plastic strain, e£: a = - ^ a 0 (b*<l) (2.10) where o 0 and o are the flow stress of ferrite and dual-phase steel, respectively, and b* is the stress partitioning coefficient for the ferrite matrix. iii) Plastic matrix - Plastic Inclusion. The martensite phase in dual-phase steels begins to deform plastically when its yielding condition is reached (Weng, 1990). The yielding of martensite occurs when b,o=oy>1 (2.11) where o y , is the martensite yield stress and b[ is the stress coefficient for the martensite. In this stage of deformation, the stress-strain relation of martensite phase must also be known. Since both phases deform plastically in this stage, the flow stress of dual-phase steel can be calculated by solving two simultaneous equations (Weng, 1990). 40 Chapter 2. Literature Review 2.6.4.2.2. Ellipsoidal Inclusion (Aspect Ratio > 1) Estimation of the stress-strain relation of dual-phase steels with ellipsoidal martensite is similar to those with spherical martensite, provided that the appropriate components of the S tensor are used (Qiu and Weng, 1991). The flow stress o f these steels containing elastic martensite with an average aspect ratio greater than unity can be calculated using the following equation: * = q_o 0 (q_>l) (2.12) where q* is the stress partitioning coefficient. q* is a function of the martensite volume fraction, the components of Eshelby's S tensor and the elastic constants of both constituent phases. The components of the Eshelby's S tensor and the stress partitioning coefficient for ellipsoidal inclusions are given in Appendix A . A number o f investigations have been conducted to predict the stress-strain response of dual-phase steels using the modified Eshelby model (Rudiono and Tomota, 1997; Tomota et al., 1994; Bhattacharyya and Weng, 1996; Bhattacharyya et al., 1993; K i m , 1988). In these studies, the inclusion has been assumed to be spherical in shape with or without plasticity during deformation. However, no investigation was found in the literature dealing with a martensite morphology other than spherical. It seems that attempts should be made in order to model the stress-strain behaviour of dual-phase steels with ellipsoidal martensite. 2.7. Modelling Ductile Fracture of Dual-Phase Steels As mentioned previously, dual-phase steels exhibit ductile fracture behaviour with a characteristic dimpled fracture surface. The ductile fracture process in dual-phase steels, similar 41 Chapter 2. Literature Review to the fracture phenomenon in other ductile materials (Van Stone et al., 1985), occurs in three sequential stages, i.e. void nucleation, void growth and void coalescence (Thomason, 1990). It has been experimentally shown that inclusions (such as manganese sulfide in steels) and hard second phase particles in materials are suitable sites for the nucleation of microvoids during deformation (Thomason, 1990; LeRoy et al., 1981; Brown and Embury, 1973; Gladman et al., 1971). It is believed that the effect of hard and non-deformable martensite on the fracture behaviour of dual-phase steels is similar to that of the second phase particles in low alloy steels (e.g. cementite particles in spheroidized steels) (Balliger, 1982). Thus, one may expect that the ductile fracture models developed for the materials with second phase particles may be applicable for the case of dual-phase steels. Several attempts have been made to model the ductile fracture of materials containing second phase particles (Thomason, 1998; LeRoy et al., 1981; Thomason, 1968; Thomason, 1990). The first step in modelling is to consider the condition under which the voids are nucleated. 2.7.1. Void Nucleation - Criterion for Void Nucleation Process The necessary condition for void nucleation (by separation of the particle-matrix interface) is that enough energy be released by formation of a void to create new surface area (Goods and Brown, 1979). However, a supplementary condition is that a critical normal stress must be exceeded at the particle-matrix interface (LeRoy et al., 1981). LeRoy et al. have measured the nucleation strain in spheroidized carbon steels with 2-14% cementite particles by performing a series of tensile tests at low and high temperatures (LeRoy et al., 1981). However, they have reported that their estimation of the nucleation strain had a 42 Chapter 2. Literature Review limited sensitivity as the void formation was detected only when sufficient fraction of voids (greater than 0.001) were present. According to this experimental consideration, one may assume this critical fraction of voids as the basis for the definition of void nucleation strain (eN). Accordingly, the void nucleation strain can be defined as the strain at which voids with the area fraction o f 0.001 (or area percent of 0.1%) are nucleated (Kosco and Koss, 1993; Poruks et al., 2006). 2.7.2. Void Growth - Existing Models for Void Growth When a void is nucleated in a plastically deforming matrix, continuing plastic deformation of the matrix w i l l generally cause the void to undergo a volumetric growth and shape change (Thomason, 1990). Rice and Tracey (1969) have developed a model for the void growth process during ductile fracture. In the Rice and Tracey model, the spherical voids are assumed to nucleate in an infinite body o f a plastic material. The nucleated voids are then allowed to grow in a stress field so that they no longer remain spherical in shape. In modified versions of the Rice and Tracey model, the analytical solutions are given to calculate the rate of change in the radius of growing voids. B y integrating the resulting equations, the general expressions for the three principal radii o f the ellipsoidal voids w i l l be obtained (Le Roy et al., 1981; Thomason, 1990). 2.7.3. Void Coalescence - Criterion for Void Coalescence The last stage of ductile fracture is the void coalescence during which ductile fracture surfaces are formed by catastrophic coalescence of voids which are already nucleated and grown at the sites of inclusions and second phase particles (Thomason, 1990). The void 43 Chapter 2. Literature Review coalescence process has been studied by different authors in order to develop a model describing the condition under which this stage of the ductile fracture occurs (Brown and Embury, 1973; Thomason, 1968, 1993). Brown and Embury (1973) have postulated a simple geometric condition for the void coalescence. The main idea in this model is that the ductile matrix between the neighbouring voids cannot experience vary large amounts of local deformation as it is constrained by the surrounding materials. According to their model, the coalescence of voids occurs when the spacing of neighbouring voids becomes equal to their length. In this case, the slip planes are drawn between the voids and then, local deformation of the matrix material can take place in the spaces between the voids. This phenomenon is usually called local multiple necking (Brown and Embury, 1973). 44 Chapter 3 - Research Objectives The primary objective of this research is to evaluate the deformation and fracture behaviour of a commercial low carbon (0.06 wt.%) dual-phase steel produced through intercritical annealing. Recently, there has been considerable interest in producing dual-phase steels with lower carbon concentrations (below 0.1 wt.%>) than those conventionally studied in the literature (0.1-0.2 wt.%) to improve properties. In these low carbon dual-phase steels, the yield strength of martensite is relatively low so that there is a strong possibility for the martensite to start deforming plastically during straining. The presence of elasto-plastic low strength martensite in the dual-phase steel microstructures instead o f hard and non-deformable martensite may have beneficial effect on the mechanical properties, and this effect w i l l be examined in detail in this investigation. Important questions to explore in this investigation are i) under what conditions martensite plasticity would occur in this steel during tensile deformation, ii) how martensite content and morphology affect the onset of. martensite plasticity, i i i) in steel samples with martensite plasticity, how plasticity behaviour of martensite changes with its content and morphology and iv) how martensite plasticity affects the mechanical properties of this steel, particularly the strength and fracture behaviour. It is interesting to model the stress-strain response o f this low carbon dual-phase steel with a focus on the effect of martensite content and morphology as wel l as martensite plasticity. For the past few decades since dual-phase steels were commercially produced, there has been a lack 45 Chapter 3. Research Objectives of such a model with the ability to take into account the effects of these microstructural features. Finally, there have been limited attempts in this period to model the fracture behaviour o f dual-phase steels. It is useful to check the existing models (which have been developed for materials containing hard second phase particles) for their applicability to dual-phase steels. 46 Chapter 4 - Experimental Procedure and Microstructural Analysis This chapter describes the experimental procedures and the microstructural analysis used in this work to examine the tensile deformation behaviour of the 0.06%C dual-phase steel. The principal experiments and the analysis methods in this research were as follows: i) controlled heat treatment, ii) tensile testing, iii) measurement of martensite deformation and iv) quantification of void evolution. 4.1. Starting Material A cold-rolled (55%) low-carbon steel (DP600 steel supplied by Stelco Inc., Hamilton, ON) with an initial microstructure consisting of 8-10% banded pearlite embedded in a ferrite matrix was used as the starting material to produce dual-phase steel microstructures. The chemical composition of the steel investigated in this work is given in Table 4.1. Table 4.1. Chemical composition of the steel investigated. Element C M n Si M o T i Nb V Cr N i Cu P S A l wt.% 0.06 2.0 0.07 0.15 0.014 0.002 0.004 0.04 0.015 0.01 0.012 0.002 0.037 4.2. Heat Treatment Cycles 4.2.1. Intercrit ical Anneal ing Dual-phase steel samples were produced through a series of intercritical annealing processes followed by quenching in the Gleeble 3500 thermomechanical simulator. The 47 Chapter 4. Experimental Procedure and Microstructural Analysis Gleeble samples (200mmx50mmx 1.75mm) were first heated up to the intercritical temperatures with two different heating rates, 1 and 100 °C/s. In order to obtain nearly the same martensite contents at low and high heating rates, two series of annealing temperatures were chosen; 755, 780, 795, 820, 850 °C and 730, 740, 760, 780, 795, 810 °C for heating rates of 1 and 100 °C/s, respectively. The samples were quenched to room temperature after being held for 60 seconds at the intercritical temperatures (except for 850 °C where a hold time of 120 seconds was used). A pressurized mixture of helium gas and water (50 and 70 Psi , respectively) was used to quench the Gleeble samples during the cooling stage. The cooling rate by which the steel samples were quenched to room temperature was 260-280 °C/s. A series of preliminary heating and isothermal holding experiments was conducted (with heating rates of 1 and 100 °C/s and at temperatures between 730 °C and 850 °C) in order to measure the temperature gradient on the Gleeble sample during the intercritical annealing. Two K-type thermocouples (other than the control thermocouple at the centre) were used in each experiment to measure the sample temperature at different locations away from the center. Figure 4.1 shows a rectangular zone (50mmx30mm) at the center of the Gleeble sample at which deviation of the sample temperature from the goal temperature was below ±5 °C. A s can be observed, the gauge length (20mm) of the tensile samples and the test samples taken for metallographic examinations (20mmx5mm) are within this temperature zone for both sample orientations; parallel and perpendicular to the rolling direction (RD) or, longitudinal (L) and transverse (T) directions, respectively. 48 Chapter 4. Experimental Procedure and Microstructural Analysis f T Q R D y ••;>»-! . x ; J r H ! ! [ i f I'T i , - •-• •• ;•; : ;p .vr n : .x:\ *~ - •> i 1" T V - T ' I • -^^J^c^ii^zr^ _. -- ,---j-^ 44iH----* -i !• i t . j L—:—yv-.i .,/ v 1 Temperature Zone 0 50 mm 200 mm Figure 4.1. Gleeble sample and temperature zone with ±5 °C deviation from the goal temperature. 4.2.2. Tempering Process A selected number of dual-phase steel samples were tempered in order to evaluate the effect of tempering process on the strength of martensite phase and the deformation behaviour of the steel samples. Tempering was conducted at 300 °C, 400 °C and 500 °C for 60 minutes using a tube furnace with ±3 °C deviation from the tempering temperatures. The steel samples were air cooled to room temperature from the tempering temperatures. 4.2.3. Production of Martensitic Steels To determine the yield strength of the martensite as a function o f its carbon concentration, separate steels with the martensitic,microstructure were used. Furthermore, the flow behaviour of these martensitic steels was estimated to be the flow behaviour o f the martensite phase with almost the same carbon concentration in selected number of steel samples. The steel samples were first heated up to the austenitization temperatures and then quenched to room temperature after isothermal holding at austenitization temperatures. The chemistry of the steels used for these experiments together with the test conditions for the heat 49 Chapter 4. Experimental Procedure and Microstructural Analysis treatment cycles are summarized in Table 4.2. It should be noted that the 0.06C and 0.12C steels were heat treated in the Gleeble machine while the 0.3C steel was heat treated in a tube furnace with a brine solution (10% N a C l in water) as the quenching medium. Table 4.2. Steel Chemistries and test conditions for production of martensitic steels. Steel Chemistry (wt.%) Taustenitization (°C) Hold time (min.) Quenching 0.06C-2Mn-0.07Si-0.15Mo (DP600, Table 4.1) 1050 900 2 3 Water + Helium (pressurized mixture) 0.12C-1.6Mn-0.19Si-0.2Mo 1050 900 2 3 Water + Helium (pressurized mixture) 0.3C-1.5Mn-1.5Si 1000 20 Brine Solution 4.3. Estimation of Martensite Carbon Concentration To estimate the martensite strength, it was necessary to assess its carbon concentration. For each dual-phase steel sample with a fixed percent martensite ( V M ) , the equilibrium carbon concentration o f the ferrite phase (Cf) at the intercritical annealing temperature was estimated first using Thermocalc software with the Fe-2000 database. The carbon concentration of the martensite phase (Cm) was then calculated by considering a balance between the overall carbon concentration of the steel ( C D P = 0.06 wt.%) and that of the constituent phases, i.e.: C D p = V n C n + ( l - V m ) C f (4.1) 4.4. Sample Preparation for Optical Microscopic Observation After intercritical annealing, a slice 20mm in length and 5mm in width was cut from the centre of each Gleeble sample (Figure 4.1) for optical metallographic examination of the dual-50 Chapter 4. Experimental Procedure and Microstructural Analysis phase steel microstructures. After a grinding stage using different sand papers (from coarse to fine: 180, 320, 600, 1200 grit), samples were polished using two abrasive compounds (diamond suspension, 6 and 1 p.m). The polished samples were then etched in a solution prepared by adding 10 g sodium metabisulfite into 100 m L water (Vander Voort, 2001). This etching solution was found to be effective in revealing ferrite and martensite phases with two distinct contrasts in the dual-phase steel microstructures. Figure 4.2 illustrates the microstructure of a dual-phase steel with 18% martensite after being etched with two different etching solutions; 2% Nital (2 m L Nitric acid in 100 m L Ethyl Alcohol) and aqueous sodium metabisulfite. In the microstructure of the sample etched with 2%> Nital (Figure 4.2a), the grain boundaries are clearly revealed with nearly the same contrast as the martensite islands whereas the sodium metabisulfite solution has revealed the martensite as separate islands embedded in the ferrite matrix (Figure 4.2b). Thus, the 2% Nital solution was only used for the measurement of the ferrite grain size in the dual-phase steel microstructures. (a) (b) Figure 4.2. The microstructure of a dual-phase steel with 18% martensite etched with 2% Nital (a) and aqueous sodium metabisulfite (b). 51 Chapter 4. Experimental Procedure and Microstructural Analysis A special sample preparation procedure was used for quantitative analysis of the void formation process in the fractured tensile samples. This was mainly due to the fact that the usual sample preparation method was not appropriate for the metallographic examination of the voids, since it led to smearing of the voids. Thus, a special multi-stage sample preparation procedure recommended by the Buehler Ltd. was used. The details o f this sample preparation method are given in Table 4.3. Table 4.3. The sample preparation procedure for carbon steels (recommended by Buehler Ltd.). ^^Qondition S t a g e * \ ^ Surface Abrasive Type/Size Time (min.) Force per Sample (lb) Speed (rpm) Relative Rotation Planar Grinding Carbimet 320 grit Until Flat or Desired Depth 4 250 Contradictory Sample Integrity U L T R A - P O L 9 um M E T A D I SUPREME 1 4 4 200 Contradictory T E X M E T 1000 3 um M E T A D I SUPREME 2 4 200 Contradictory Final Polishing C H E M O M E T 1 M A S T E R P R E P2 3 8 150 Complementary 1 Polycrystalline diamond suspension 2 ' Alumina (0.05 urn) polishing suspension 4.5. Tensile Testing Tension tests were conducted using an M T S testing machine with a nominal strain rate of 2xl0" 3 s"1. Subsize tensile specimens were prepared according to the A S T M standard ( A S T M E8, 1988) with a gauge length of 20mm. The tensile specimen used in this work is shown in Figure 4.3. A n extensometer was used to measure the axial elongation of the tensile sample. 52 Chapter 4. Experimental Procedure and Microstructural Analysis 15 24 15 R4 1.75 Figure 4.3. Subsize tensile specimen according to the A S T M E8, dimensions are in mm. The radius of fillet at the ends of the reduced section is 4mm. The load-elongation graph obtained from each tensile test was converted to the engineering stress-strain curve by dividing the load and elongation by the original cross sectional area and the gauge length of the tensile sample, respectively: F F Engineering stress: Engineering strain: S = -A o W o t 0 A L L - L „ (4.2) (4.3) L L o o where F, w 0 , t 0 and L 0 are the tensile load, the original width (4mm), the original thickness (1.75mm) and the gauge length (20mm) of the tensile specimen, respectively. The true stress and the true strain (a = F / A and e = ln (L /L 0 ) , respectively, where A and L are the instantaneous cross sectional area and length of the tensile specimen) were calculated from the engineering stress and strain as follows: True stress: - a = S(e + l ) (4.4) True Strain: e = ln(e + l ) (4.5) Since Equations 4.4 and 4.5 are not valid beyond the maximum load in the non-uniform deformation region (because they are derived assuming volume constancy and in the non-53 Chapter 4. Experimental Procedure and Microstructural Analysis uniform deformation region the volume of deforming material is not constant), the true stress and strain at the fracture point were calculated using the actual dimensions of the fractured tensile specimen, i.e.: F f F f True stress at fracture: o f = — L- - —-— (4.6) A f w f t f True strain at fracture: e, = In—- = In A, 'w 0 O v w f t f j (4.7) where F f , w f and tf are the fracture load and the width and the thickness of the fractured tensile specimen, respectively. The dimensions of the tensile specimens at the fracture surface (i.e. W f and tf) were measured using images obtained from scanning electron microscopy (SEM). The width strain of the tensile specimens was measured using a width extensometer in order to measure the R-value of the steel samples. Having the axial and width plastic strains of the tensile specimen, the R-value was calculated (Hosford and Caddell, 1983) using the assumption of constant volume during plastic deformation, i.e.: R = — -^ = ^ — (4.8) e a + 6 w + 6 t = 0 (4.9) w h e r e8 a , 8 w a n d s t a r e axial, width and thickness plastic strains, respectively. The R-value of the steel samples was calculated at an axial strain of 15%. 4.6. Image Analysis Quantitative analysis of the dual-phase steel microstructures was carried out using Clemex image analysis software. The first and most important step in a successful and accurate image 54 Chapter 4. Experimental Procedure and Microstructural Analysis analysis o f a steel microstructure was to take a digital picture with the highest quality possible. For the measurement of the martensite content as well as the martensite dimensional change during deformation, the key point was to start with the dual-phase microstructures with two distinguishable phases and a minimum of etched ferrite grain boundaries. 4.6.1. Measurement of Ferrite Grain Size The ferrite grain size in the dual-phase steel samples was measured using the planimetric (or Jeffries's) procedure ( A S T M E l 12, 1988). To measure the ferrite grain size, a rectangle of known area was overlaid on a micrograph so that 200-250 ferrite grains (which were assumed to be equiaxed in shape) were counted in the field area. This was the sum of the grains completely within the known area plus one half of the number o f grains intersected by the circumference of the area. Accordingly, the number o f ferrite grains per unit area was calculated and then, the average diameter of ferrite grains in the field was estimated assuming the equiaxed ferrite grains to be spherical in shape. The grain size measurement for each steel sample was conducted on three different fields. 4.6.2. Measurement of Martensite Content Figure 4.4a shows a picture taken from a typical dual-phase microstructure. A s can be observed, the martensite islands and the ferrite matrix are clearly distinguishable in the microstructure. The microstructure of this steel after execution o f the first stage of the image analysis (manual adjustment of the gray threshold) is also illustrated in Figure 4.4b. It shows that this threshold adjustment has provided a digitized two-color image which appears to be appropriate for image analysis, because the martensite phase has been entirely substituted by 55 Chapter 4. Experimental Procedure and Microstructural Analysis objects that can easily be detected by the image analysis software. The measurement of martensite content was conducted on 20-30 different images for each dual-phase steel sample. These images were taken from different locations of the steel microstructure, mainly along the longitudinal and transverse directions, to take into account the effect of microstructural nonuniformity of the martensite islands. The percent martensite for each steel sample was reported as a mean value with a given deviation range. (a) (b) Figure 4.4. A typical dual-phase steel microstructure; (a) before and (b) after threshold adjustment. 4.6.3. Measurement of Martensite Deformation Tensile deformation o f martensite was examined via image analysis of the steel microstructures before and after tensile straining. Dimensional change of the martensite phase was measured through a series of thickness measurements of the martensite islands in different regions of the fractured tensile specimens. The thickness strain of martensite was calculated from the average thickness of martensite islands in the deformed region of the fractured tensile specimen compared to that of the undeformed (grip) section. 56 Chapter 4. Experimental Procedure and Microstructural Analysis Figure 4.5 shows how a dual-phase steel microstructure is treated using the image analysis software in order to measure the average thickness of the martensite islands. (a) (b) (c) (d) Figure 4.5. Different stages in quantitative analysis of a dual-phase steel microstructure in order to measure the average thickness of martensite islands. 57 Chapter 4. Experimental Procedure and Microstructural Analysis Figure 4.5b is an image obtained after applying the threshold adjustment function in the image analysis software. In the next step, the ferrite matrix has been removed from the image background (Figure 4.5c). A t this stage, the martensite phase is the only phase left in the image field. Then, a series of 25-30 vertical lines is overlaid on the image field (Figure 4.5d) covering almost all the details of the microstructure. A n appropriate function within the software is then applied in order to detect the points of intersections between the grid lines and the martensite islands (Figure 4.5e). The line segments left over from the intersections within the martensite islands represent the thickness of martensite islands at the point o f intersection. In Figure 4.5f, the martensite islands are removed so that the segments can easily be seen in the remaining image field. In the last stage of the image analysis, the length o f each segment (or object in the image analysis software) is measured. For each dual-phase steel sample, at least 30 different images from different locations through the thickness were examined corresponding to approximately 40,000 measurements o f martensite island thickness (the number of segments detected by the software in each image was around 1,200-1,400). Having obtained these measurements, the data was analyzed to calculate an average martensite island thickness. To calculate the average thickness, a weighted line length average was calculated as follows: n f n I • It? 1= 1 = 1 " = - g — (4.10) n t n n where n is the number of thickness measurements, tj is the measured thickness and t n is the n average thickness ( t n = ( ^ t j ) / n ) . Considering a weighting factor, as shown in Equation 4.10, i=l 58 Chapter 4. Experimental Procedure and Microstructural Analysis effectively weights the larger particles whose thicknesses could be more accurately measured and avoids the small islands to be over represented. The average level of strain in the martensite phase was then determined as: _ thickness i £mart. = m f t ^ V ^ initial J (4.11) where t i l l i t i a l is the average initial martensite island thickness and t d e f is the average martensite island thickness (as defined by equation [4.10]) for a given level o f far field strain. 4.6.4. Quantitative Examination of Void Formation Process The void formation process in the neck region of the fractured tensile specimens was also quantitatively examined using the image analysis software. The goal was to measure the dimensions (length and width) and area fraction of voids in the non-uniform deformation region of the tensile samples. The fractured tensile specimens were first cut in the longitudinal (axial) direction and then quantitatively analyzed after being polished using the Buehler special sample preparation procedure. The key point for image analysis of the voids, as mentioned already for the measurement of martensite content and martensite deformation, was to take a picture with the best quality possible. Figure 4.6a illustrates an optical micrograph showing the polished section of a fractured dual-phase steel sample (close to the final fracture point) with a given number of voids. Dark areas in this micrograph are voids with a distinct contrast compared to the background. Figure 4.6b also shows the micrograph after threshold adjustment and removal of the background. The image obtained from this adjustment is ready for the next stage of the image analysis. 59 Chapter 4. Experimental Procedure and Microstructural Analysis * • M r -w 150 (im \ (a) (b) Figure 4.6. Microstructure of a fractured dual-phase steel sample close to the fracture surface;.(a) before and (b) after threshold adjustment. The quantitative analysis of the voids was conducted within a series of slices in the neck area of the fractured tensile specimen (Figure 4.7) between the fracture surface and the region at which the non-uniform deformation starts. The void length, width, aspect ratio and area fraction were measured in each slice with a given thickness plastic strain. The uniform and fracture thickness strains o f the tensile sample (E u and S N , respectively) were calculated from the measured thickness of the sample in the undeformed region, t 0 (grip region), uniform deformation region, tu, and fracture surface, tn, i.e.: e u = l n ^ - , eN=hA (4-12) The area percent voids in slice-n, for instance, was calculated as follows: A„ %A„ - = — x l O O A „ + A ' n (4-13) where A n and A ' n are, respectively, the total area of voids and background (matrix) in this slice. 60 Chapter 4. Experimental Procedure and Microstructural Analysis The thickness strain of the tensile sample at a section where the slice-n is located (s n) was also calculated as: e n = l n ^ - (4-14) where tn is the slice length (or sample thickness at this section). uniform deformation region neck region Figure 4.7. A fractured tensile specimen (schematic) showing the image analysis parameters for the quantitative examination of the voids. In this chapter, a detailed description of the experimental techniques and the microstructural analysis used in this study was given. The next chapter w i l l present and describe the different results obtained from the experiments as well as the microstructural analysis. 61 Chapter 5 - Experimental Results and Analysis This chapter w i l l describe in detail the steel microstructures obtained from intercritical annealing experiments, and the mechanical behaviour of the steel samples with an emphasis on linking the observed macroscopic properties to the steel microstructures. 5.1. Intercritically Annealed Dual-Phase Steel Samples Using the results from the previous study of Huang (2004) as a starting point, dual-phase microstructures with a variety of martensite contents and morphologies were produced. In the following sections, the processing parameters, relationship between them and their effects on the resulting steel microstructures w i l l be summarized. 5.1.1. Steel Microstructures Figure 5.1 shows the effect of heating rate (1 and 100 °C/s) on the microstructure of the steel samples after annealing and quenching from different intercritical temperatures. The microstructures are mixtures of ferrite (white background) and martensite (dark gray islands) and it can readily be observed that the size, morphology and distribution of martensite are dependent on both the intercritical annealing temperature and the heating rate. Figures 5.1a and b show the microstructures of steel samples heated at 1 °C/s to the intercritical temperatures of 755 °C and 820 °C, respectively. In both cases, the martensite islands have an irregular shape but are almost equiaxed and the ferrite grain size is 5.4 (±0.4) u.m. The size of martensite 62 Chapter 5. Experimental Results and Analysis islands is approximately 1-5 um and they are inhomogeneously distributed, i.e. the islands are predominately observed on the ferrite grain boundaries. A n increase in the annealing temperature from 755 °C to 820 °C results in an increase in i) the martensite content from 18% to 44%>, ii) the size of the martensite islands and iii) the fraction of the ferrite boundary occupied by martensite islands. For the martensite contents greater than approximately 30%, an almost complete network of martensite islands forms on the ferrite boundaries. Figure 5.1. Microstructures of dual-phase steel samples produced with heating rates of 1 and 100 °C/s. (a) and (b) heating rate of 1 °C/s, held at intercritical annealing temperatures of 755 °C and 820 °C for 60 s, martensite contents measured to be 18% and 44%, respectively, (c) and (d) heating rate of 100 °C/s, held at intercritical annealing temperatures of 730 °C and 780 °C for 60 s, martensite contents measured to be 17% and 41%, respectively. 63 Chapter 5. Experimental Results and Analysis The situation is very different when the heating rate is increased to 100 °C/s. The morphology and the spatial distribution of the martensite are changed, i.e. the martensite appears as elongated islands in the ferrite matrix. A s shown in Figures 5.1c and 5.Id, as the intercritical annealing temperature is raised from 730 °C to 780 °C, i) the martensite content increases from 17% to 41%>, ii) the average thickness of the martensite islands changes from approximately 3 um to 6.9 um and iii) the aspect ratio of the islands increases from 4.2 to 5.5. For simplicity in the following sections, the morphology of the martensite for the low and high heating rate experiments w i l l subsequently be referred to as equiaxed and banded, respectively. 5.1.2. Intercritical Annealing Parameters The various combinations of heating rate, intercritical annealing temperature and hold time used in the annealing experiments are summarized in Table 5.1. These combinations were chosen to produce dual-phase steel samples with the martensite contents of approximately 20-80%. The microstructural characteristics of the martensite islands (aspect ratio, shape and size) are given in Table 5.1. A s can be observed, the heating rate to the intercritical temperature has an effect on the percent martensite, V m , (measured by image analysis) and the martensite morphology. For example, intercritical annealing at 780 °C for 60 s with low and high heating rates (1 and 100 °C/s) produced 29%> and 41%> martensite, respectively. The effect of heating rate on the martensite morphology was shown in the previous section. For both heating rates, an increase in the intercritical temperature leads to an increase in the martensite content, i.e. increasing the percents martensite from 18 to 83 and 17 to 77 for low and high heating rates corresponding to the increase in the intercritical temperatures from 755 °C to 850 °C and 730 °C to 810 °C, respectively. A s can be observed in Table 5.1, the hold time 64 Chapter 5. Experimental Results and Analysis at the intercritical temperature also has an effect. Although the steel sample was held at 810 °C for 60 s during the high heating rate annealing to produce around 80% austenite phase (which transforms to martensite during the subsequent rapid cooling stage), a 120 s hold time at a higher intercritical temperature (850 °C) was needed to form almost the same austenite content. Table 5.1. Summary of processing conditions and martensite island characteristics in different dual-phase steel microstructures. Heating Rate (°C/s) Intercritical Temperature (°C) Hold Time (sec.) V m (%) Aspect Ratio Martensite Morphology 755 60 18 « 1 ~ Equiaxed 780 60 29 = 1 ~ Equiaxed 1 795 60 32 - Network 820 60 44 - Network 850 120 83 - -730 60 17 4.2 Banded 740 60 25 4.7 Banded 760 60 30 5.1 Banded 100 780 60 41 5.5 Banded 795 60 52 5.8 Banded 810 60 77 - -5.1.3. Discussion on Effect of Heating Rate To understand the effect of heating rate on the martensite content and morphology, it is useful to consider the processing history of the initial steel microstructure. The starting steel had been cold rolled and was then intercritically annealed followed by water quenching. A n 65 Chapter 5. Experimental Results and Analysis important consideration is the carbon concentration and its spatial distribution at the end of the intercritical annealing cycle as this w i l l be inherited into the final microstructures after quenching. A s shown by Huang (2004) through an investigation on the same steel, the heating rate effect on martensite morphology arises due to the two limiting cases. A t low heating rates, ferrite recrystallization occurs prior to reaching the intercritical region and thus, the austenite phase forms, in addition to the austenite formation from the pearlite colonies, at the recrystallized ferrite grain boundaries as observed in Figure 5.1a and b. In contrast, when a high heating rate is employed, the ferrite does not have time to recrystallize prior to reaching the intercritical region. In this case, austenite nucleates primarily in the elongated pearlite colonies and then grows quickly into these regions resulting in the banded structure one observes in Figure 5.1c and d. 5.2. Mechanical Properties of Dual-Phase Steel Samples The tensile stress-strain behaviour and fracture properties o f intercritically annealed dual-phase steel samples with the microstructural characteristics summarized in Table 5.1 w i l l be examined in this section. The effect of martensite plasticity on the tensile deformation behaviour and ductile fracture of these samples w i l l also be evaluated. 5 . 2 . 1 . T e n s i l e S t r e s s - S t r a i n B e h a v i o u r The engineering stress-strain curves of the steel samples with different martensite contents and morphologies are shown in Figure 5.2 for samples tested parallel to the initial cold rolling direction. 66 Chapter 5. Experimental Results and Analysis 1000 ro 800 0.1 0.2 Engineering Strain (a) 0.3 0.4 1000 co 800 CL | 600 •4—» CO •I 400 0 CD c O) c UJ 200 0 0.1 0.2 0.3 - 0.4 Engineering Strain (b) Figure 5.2. The engineering stress-strain curves of dual-phase steel samples with different martensite contents, (a) samples with almost equiaxed martensite morphology produced with the heating rate of 1 °C/s and (b) samples with banded martensite morphology produced with the heating rate of 100 °C/s. 67 Chapter 5. Experimental Results and Analysis The true stress-true strain curves of the steel samples in Figure 5.2 are illustrated in Figure 5.3. The solid portion of the curves in Figure 5.3 represents the true stress and the true strain as measured from an extensometer. The dashed portion o f the curves represents a linear extrapolation between the necking point and the final fracture. It is important to note that the stress-strain curves shown in Figures 5.2 and 5.3 are for tensile specimens with a non-standard gauge length (20 mm) and since the post necking strain is a function of the gauge length of the test specimen (Christ, 1985), the total elongation of the test specimens cannot be directly compared to standard tests. A s can be observed in Figures 5.2 and 5.3, for both the equiaxed and banded morphologies, the yield stress and the ultimate tensile strength (or the true stress at the necking point) increase, as expected, with an increase in the percent martensite. Moreover, all the steel samples show a very high initial work hardening rate and no evidence of the yield point elongation (the yield elongation associated with the transition from elastic to plastic deformation behaviour in some metals and alloys), i.e. the typical observation for dual-phase steels. Comparison of Figures 5.3a and 5.3b shows that there is a difference in the large strain work hardening rates (i.e. measured as the slope of the stress-strain curve from the necking to the fracture point) for the samples with different martensite morphologies. For the samples with equiaxed martensite morphology (produced with low heating rate), the large strain work hardening rate was measured to be approximately 500-600 M P a and was found to be weakly dependent on the martensite content. In contrast, for the case o f elongated martensite islands (observed when a high heating rate was used), the large strain work hardening rate was found to be in the range of 400-675 M P a , increasing with an increase in the martensite content. 68 Chapter 5. Experimental Results and Analysis 1800 1500 Equiaxed Martensite £ 1200 $ 900 .2 600 300 83% 44% 18% 29% x Fracture 0.2 0.4 0.6 0.8 True Strain 1.2 1.4 (a) 1800 1500 900 in in <D 55 2 600 300 Banded Martensite 41% x Fracture 77% 17% 30% 0.2 0.4 0.6 0.8 True Strain 1.2 1.4 (b) Figure 5.3. The true stress - true* strain curves of dual-phase steel samples with different martensite contents, (a) samples with almost equiaxed martensite morphology produced with the heating rate of 1 °C/s and (b) samples with banded martensite morphology produced with the heating rate of 100 °C/s. The dashed curves represent the post-necking deformation regime. 69 Chapter 5. Experimental Results and Analysis 5.2.2. Tensile Properties 5.2.2.1. Yield Strength and Ultimate Tensile Strength Figure 5.4 summarizes the 0.2% yield strength and the ultimate tensile strength for all the conditions examined. In this Figure, the data for both parallel and perpendicular to the rolling direction (i.e. longitudinal and transverse, respectively) have been included. For both the yield and tensile strengths, all the data collapses into a simple linear function of the percent martensite. The solid lines, which represent the average yield and tensile strengths, can be expressed in terms of the percent martensite, V m (in %), as follows: r j y (0.2% offset) = 216 + 3 . 9 V m (MPa) (5.1) U T S = 615 + 2 . 3 V m (MPa) (5.2) Figure 5.4. The Yield strength (ay,b.2%) and the tensile strength (UTS) of different dual-phase steel samples in longitudinal and transverse directions as a function of the martensite content for two different morphologies. Equiaxed and banded martensite morphologies correspond to the heating rates of 1 °C/s and 100 °C/s, respectively. 70 Chapter 5. Experimental Results and Analysis The yield strength to tensile strength ratio for all dual-phase steel samples is illustrated in Figure 5.5 as a function of the martensite content. A s can be observed, this ratio increases from 0.45-0.5 to 0.65-0.7 with an increase in the martensite content from approximately 20% to 80%. 0.8 -i , ! 0.6 '</> c <D *Z 0.5 2 0.4 - • Equiaxed Martensite - Long. • Equiaxed Martensite - Trans. • Banded Martensite - Long. A Banded Martensite -Trans. 0.2 -! • , • , • , • 1 . 0 20 40 60 80 100 Percent Martensite, V m (%) Figure 5.5. The yield strength to tensile strength ratio of dual-phase steel samples as a function of the martensite content in longitudinal and transverse directions. Equiaxed and banded martensite morphologies correspond to the heating rates of 1 °C/s and 100 °C/s, respectively. 5.2.2.2. Uniform Strain The variation of the uniform strain of the steel samples with the martensite content is illustrated in Figure 5.6 for all the experimental conditions employed. A s can be observed in this Figure, the martensite morphology has an effect on the uniform strain with the experimental results showing that the steel samples with equiaxed martensite islands have a higher uniform strain compared to the samples with banded martensite morphologies (up to the martensite contents of approximately 50 %). Two curves are fitted for the uniform strain of 2 >: 0.3 -71 Chapter 5. Experimental Results and Analysis dual-phase steel samples with equiaxed and banded martensite morphologies as a function of the percent martensite (each curve represents the data points for the longitudinal and transverse directions): ^ Equiaxed Martensite: E u = 0.185 - 0.001 l V m (5.3) Banded Martensite: s u = 0.152 - 0.00065V m (5.4) For the percents martensite greater than 50, the two curves start to merge so that it is expected that the steel samples with both martensite morphologies and around 80% martensite show almost the same uniform strains (although the data point for the sample with 83%o equiaxed martensite in the transverse direction is away from both fitted curves). Figure 5.6. The uniform strain of dual-phase steel samples with two different martensite morphologies as a function of the martensite content in longitudinal and transverse directions. Equiaxed and banded martensite morphologies correspond to the heating rates of 1 °C/s and 100 °C/s, respectively. 72 Chapter 5. Experimental Results and Analysis 5.2.2.3. Discussion on Deformation Behaviour (up to Necking Point) As can be observed in Figure 5.4, the martensite morphology and the sample orientation have a negligible effect on the yield and tensile strength o f dual-phase steel samples. A l l the data falls within a scatterband of ± 5 % with respect to the best fit lines (i.e. the solid lines in the Figure). On the other hand, the results in Figure 5.6 show that the uniform elongation of the steel samples is a function of the martensite morphology. To elucidate the effects of martensite morphology and sample orientation on yield strength, one needs to have knowledge of the load transfer process in two-phase materials in which the matrix is much softer than the second phase. A s mentioned in the literature review chapter, when these materials are subjected to an external load, part o f the load is transferred to the harder phase with higher load-bearing capacity. The question now is how the load transfer process happens and what parameters affect this transition process. For dual-phase steels, there is no stress transfer to martensite phase in the elastic region since the elastic modulii of ferrite and martensite are the same. After the initial yielding of the matrix (as it has a lower yield stress than the second phase), the material enters a complex transitional region where plasticity spreads in the matrix and load is transferred into the non-deforming second phase (martensite in dual-phase steels). According to Brockenbrough and Zok (1995) and Bao et al. (1991), the load transfer process in two-phase materials depends on the relative modulii o f the constituent phases and the shape and volume fraction of the elastic second phase. In the next chapter, attempts w i l l be made to estimate the load transfer process in dual-phase steel samples with different martensite contents and morphologies (spherical and ellipsoidal). Using the results of 73 Chapter 5. Experimental Results and Analysis this estimation, it would be possible to rationalize the experimental results on the yield strength of steel samples shown in Figure 5.4. The results of the estimation of the load transfer process in the steel samples with different martensite morphologies can also be used for the rationalization of the negligible effect of sample orientation on the yield strength. This is because the relative morphology of the steel microstructures in terms of martensite islands can be controlled by the sample orientation with respect to the tensile loading direction. For the low heating rate samples with equiaxed martensite morphology, little difference was found between the microstructure of the samples tested parallel and perpendicular to the initial rolling direction. A n example of the longitudinal and transverse steel microstructures relative to the rolling direction for a steel sample (produced with low heating rate, 1 °C/s) with 29% equiaxed martensite is illustrated in Figures 5.7a and b. For the steel samples (annealed with high heating rate, 100 °C/s) with banded martensite, the aspect ratio of the martensite islands in samples tested parallel to the rolling direction was greater than that for the transverse direction. This can be observed in the longitudinal and transverse microstructures of the steel sample with 41%o banded martensite shown in Figures 5.7c and d. It can also be observed in Figure 5.4 that the dependence of the yield stress on the martensite content is lower than that reported in the literature for higher carbon dual-phase steels (Marder and Bramfitt, 1979; Su and Gurland, 1987; K o o et al., 1980). One might speculate that for higher carbon steels, the strengthening of ferrite phase due to the additional dislocations introduced to accommodate the martensite transformation would be greater due to the larger misfit strain for higher carbon martensite. 74 Chapter 5. Experimental Results and Analysis Figure 5.7. Microstructure of dual-phase steel samples with two orientations relative to the rolling direction, (a) and (b) 29% equiaxed martensite, longitudinal and transverse directions, respectively, (c) and (d) 41% banded martensite, longitudinal and transverse directions with aspect ratios of 5.5 and 4, respectively. Tensile direction is horizontal. The strength data in Figure 5.4 also shows that the tensile strength of the steel samples is almost independent of the martensite morphology. To rationalize this effect, one should consider that the tensile strength (the stress at necking point) is achieved when the Considere condition is reached. This can be observed in Figure 5.8 which illustrates the effect of martensite morphology on the stress-strain behaviour and the medium strain (8-16%) work hardening rate o f the steel samples with approximately 18%> and 30%> martensite. A s can be 75 Chapter 5. Experimental Results and Analysis observed, the flow stress of the steel sample with banded martensite is slightly greater than that of the sample with equiaxed morphology and this effect is more pronounced for the case o f 30% martensite. Figure 5.8 also shows that for both martensite contents the work hardening rate of the banded sample is considerably lower (10-15%) near the necking point) than that for the equiaxed sample. Accordingly, the uniform strain of dual-phase steel samples with banded martensite turns out to be lower than those with equiaxed martensite, because the Considere condition (doVds = a) for the onset o f necking, shown in Figures 5.8a and b, is fulfilled at lower strains, although the stresses at the necking points are almost the same. The results presented in Figure 5.8 for the fulfillment of the Considere condition are consistent with the data shown in Figure 5.6, that the true uniform strain is lower in steel samples with banded microstructures. It can also be observed in Figure 5.8 that the effect of martensite morphology on the work hardening rate is enhanced in the case of steel samples with 29-30%) martensite. It appears that the experimental results are not sufficient at this point in order to rationalize this effect. The results of the measurement of martensite plasticity (or strain partitioning between the ferrite and martensite phases) and the effect of martensite morphology on its plasticity might be helpful in understanding the work hardening behaviour of these steel samples. 5.2.2.4 Effect of Tempering Figure 5.9 shows the effect of tempering (at 500 °C for 60 minutes) on the stress-strain behaviour o f a steel sample with 17% banded martensite (the curves are plotted up to the necking points). The tempering process causes a transition from continuous to discontinuous yielding, a significant decrease in the initial work hardening rate and a reduced tensile strength. 76 Chapter 5. Experimental Results and Analysis 0 1 , , , , , , , 1 0 0.05 0.1 0.15 0.2 True Strain (b) Figure 5.8. Flow stress and work hardening rate of dual-phase steel samples with equiaxed and banded martensite islands as a function of true strain illustrating the effect of martensite morphology on the Considere condition, (a) samples with 17-18% martensite and (b) samples with 29-30% martensite. 77 Chapter 5. Experimental Results and Analysis The lower work hardening rate and tensile strength o f the tempered steel sample are attributed to the fact that the tempered martensite is softer and more ductile compared to the untempered hard martensite resulting in a lower strengthening effect. The appearance of the yield point elongation in the stress-strain curve after tempering may also be attributed to the absence of or limited plastic incompatibility which occurs between the constituent phases in the tempered sample during deformation, i.e. i f both phases plastically deform (this effect w i l l be discussed in more detail later). In addition, it is probable that the transformation strains due to the martensite transformation are relaxed during tempering. A s pointed out in Chapter 2, the plastic incompatibility between phases and the transformation strains are responsible for continuous yielding behaviour of untempered dual-phase steels. 1000 800 01 CL i . 600 in in a) i— 55 CD 3 400 200 'Steel with 17% Martensite (Bandedl Quenched 0+-0.05 0.1 True Strain 0.15 0.2 Figure 5.9. The true stress - true strain curve of dual-phase steel sample with 17% martensite (banded morphology) before and after tempering at 500 °C for 60 minutes. 5.2.2.5. R-Value The martensite morphology (or alternately the heating rate to annealing temperature) was found to affect the R-value of the dual-phase steel samples. The R-values of the samples with 78 Chapter 5. Experimental Results and Analysis different martensite contents and morphologies are given in Figure 5.10. A s shown in Figure 5.10, the R-value of the steel samples with banded martensite (high heating rate) is in the range of 0.8-1.0 whereas it is increased to 1.0-1.4 for the case of samples with equiaxed martensite islands (low heating rate), i.e. a minimum enhancement of about 25% in the R-value. R-value is usually attributed to the crystallographic texture of the material but the effect of martensite morphology may also be important. The steel microstructure with equiaxed martensite morphology was relatively more uniform with martensite islands having almost no strong preferred orientation whereas the martensite islands in the steel samples with banded morphology were preferentially formed parallel to the rolling direction. Future work is required in order to explain the effect of the microstructure on the R-value o f this dual-phase steel. 1.6 HR = 1 °C/s (Equiaxed Martensite) Long. & Trans., R-value - 1.0-1.4 1.2 3 1 0.8 cr: 0.4 HR = 100 °C/s (Banded Martensite) Long. & Trans., R-value - 0.8-1.0 20 40 60 Percent Martensite, V m (%) 80 100 Figure 5.10. The R-value of dual-phase steel samples with different martensite morphologies as a function of the percent martensite in longitudinal and transverse directions. Heating rates of 1 °C/s and 100 °C/s correspond to equiaxed and banded martensite morphologies, respectively. 79 Chapter 5. Experimental Results and Analysis 5 . 2 . 2 . 6 . F r a c t u r e S t r e s s a n d S t r a i n Figure 5.11 shows the variation of the true fracture stress and the true fracture strain with the martensite content. In this case, one observes the unusual results1 that both the true fracture stress and strain increase with an increase in the martensite content. These results are opposite of the previously reported results in the literature, such as those presented in Section 2.5.5. As can be observed in Figure 5.11, the martensite morphology also appears to have a significant influence on the fracture properties of this dual-phase steel. For the percents martensite below 50, the fracture stress of the steel sample with equiaxed martensite is about 100-150MPa higher than that o f the sample with banded martensite islands (Figure 5.11a). For martensite contents above 50%, this difference decreases such that for the martensite contents of approximately 80%>, the fracture stress is similar. This is probably due to the fact that the microstructures of the steel samples with such a high martensite content produced with low and high heating rates are more or less similar, i.e. approximately 20% ferrite islands embedded in a martensite matrix. The same trend can be observed for the unusual results regarding the true fracture strain shown in Figure 5.11b. Again, for the martensite contents lower than 50%, the true fracture strain of the steel samples with equiaxed martensite islands is 0.2-0.25 higher than that for the steel samples with banded martensite morphology, but the difference between the fracture strains becomes smaller as the martensite content increases in the range of approximately 50% to 80%. 1. According to the experimental results reported in the literature, the ductility of dual-phase steels decreases with increasing martensite content and further, as the strength increases the ductility decreases. The fracture properties of the dual-phase steel samples in this study are unusual as both the fracture stress and the fracture strain are enhanced with an increase in the martensite content. 80 Chapter 5. Experimental Results and Analysis 1.4 T 1.2 Figure 5.11. Unusual results showing the true fracture stress (a) and the true fracture strain (b) of different dual-phase steel samples in the longitudinal and transverse directions as a function of the percent martensite for two different martensite morphologies. Equiaxed and banded martensite morphologies correspond to the heating rates of 1 °C/s and 100 °C/s, respectively. 81 Chapter 5. Experimental Results and Analysis Two separate curves were fitted for the true fracture stress and strain of the steel samples with equiaxed and banded morphologies. The following equations represent the variation of the true fracture stress, a f , and the true fracture strain, Ef, as a function o f the percent martensite: Equiaxed Martensite: a f = 775 + 14 .7V m - 0 . 0 8 V m 2 (5.5) Banded Martensite: o f = 735 + 9 V m (5.6) and Equiaxed Martensite: e f = 0.495 + 0 . 0 1 1 8 V m - 0 . 0 0 0 0 7 V m 2 (5.7) Banded Martensite: e f = 0.33 + 0.0073V m (5.8) In order to rationalize the experimental results regarding the unusual fracture properties of this dual-phase steel, it is necessary to examine what is occurring at the microstructural level, particularly the possible plastic deformation of martensite during straining. The examination of the void formation process during the post necking deformation w i l l also provide useful information about the fracture mechanism of the steel samples. 5.2.3. Deformation of Martensite The deformation behaviour of martensite in dual-phase steel samples was found to be dependent on its strength (controlled by its content and tempering process) and morphology (equiaxed and banded martensite). In the following sections, the effect o f these parameters w i l l be discussed separately. 5.2.3.1. Effect of Martensite Content Figure 5.12 shows the optical micrographs illustrating the microstructure of a dual-phase steel sample with 17% martensite (banded morphology) at different regions of the fractured 82 Chapter 5. Experimental Results and Analysis tensile specimen, i.e. Figures 5.12a and b for the undeformed region (outside the gauge length) and the uniform deformation region with the far field strain o f 0.14, respectively. A close examination of these micrographs suggests that there is no readily distinguishable change in the thickness of the martensite islands before and after the sample deformation. The results obtained from the quantitative examination of the martensite deformation (through image analysis), as w i l l be given later, also showed that the martensite phase in this dual-phase steel sample remained elastic during uniform deformation. (a) (b) Figure 5.12. The microstructure of dual-phase steel sample with 17% martensite produced with the heating rate of 100 °C/s in the undeformed region (a) and the uniform deformation (tensile strain of 0.14) region (b) of the fractured tensile specimen. Tensile loading direction is horizontal. Figure 5.13 shows the results for the frequency distributions o f the martensite island thicknesses corresponding to an analysis of the regions shown in Figures 5.12a and b. A s can be observed in Figure 5.13, there is almost no difference between the size distribution curves of martensite islands in these two regions, confirming that there is no shape change for the martensite phase, i.e. the martensite phase remains elastic during tensile deformation. 83 Chapter 5. Experimental Results and Analysis 0 4 8 12 16 20 Martensite Island Thickness ((xm) Figure 5.13. Histogram showing the thickness of banded martensite islands in undeformed region and region with a tensile far field strain of 0.14 for a steel sample with 17% martensite produced with a heating rate of 100 °C/s (corresponding microstructures shown in Figure 5.12). Figure 5.14 shows a series of optical micrographs illustrating the change in microstructure as a function of the level of tensile deformation for a dual-phase steel sample with 41% banded martensite, i.e. Figures 5.14a, b and c from i) the undeformed region o f the tensile specimen, ii) the region of uniform deformation and iii) near the fracture surface, respectively. It can be observed in Figure 5.14 that the martensite islands in the uniform deformation region and near the fracture surface are qualitatively thinner than those in the undeformed region. The decrease in the thickness of the martensite islands is an indication of martensite plasticity. It is also clear from Figure 5.14 that the level of martensite deformation is larger in the region near the fracture surface indicating that the degree of martensite deformation scales with the level of far field deformation. In order to quantify this effect for different martensite 84 Chapter 5. Experimental Results and Analysis contents and as a function o f the far-field strain, a detailed metallographic study was conducted using image analysis. Figure 5.14. The microstructure of dual-phase steel sample with 41% banded martensite produced with the heating rate of 100 °C/s in the undeformed region (a), the uniform deformation (tensile strain of 0.12) region (b) and near the fracture (tensile strain of 0.59) region (c) of fractured tensile specimen. Tensile loading direction is horizontal. Figure 5.15 shows the results for the frequency distributions of the martensite island thicknesses in this steel sample corresponding to the regions shown in Figures 5.14a, b and c, i.e. undeformed, a far field tensile strain of 0.12 (uniform strain of tensile sample) and near the 85 Chapter 5. Experimental Results and Analysis fracture point where the true fracture strain was measured to be 0.59. Careful examination o f Figure 5.15 shows that the size distribution is shifted to smaller martensite island thicknesses when the far-field strain is increased as can be seen particularly wel l for the data taken near the fracture surface. 24 Steel Sample with 41% Martensite (Banded") Initial Microstructure Tensile Strain of 0.12 Tensile Strain of 0.59 4 8 12 16 Martensite Island Thickness (|im) 20 Figure 5.15. Histogram showing the thickness of banded martensite islands in undeformed region and regions with the tensile far field strains of 0.12 and 0.59 for a steel sample with 41% martensite produced with a heating rate of 100 °C/s (corresponding microstructures shown in Figure 5.14). 5.2.3.2. Effect of Martensite Morphology Figure 5.16 illustrates another series of steel microstructures showing the effect of morphology on martensite plasticity. In this Figure, the deformation of equiaxed martensite during uniform straining is compared with that of the banded martensite in two dual-phase steel samples with around 30% martensite. A s can be observed, there is a noticeable change in the thickness of martensite islands with banded morphology (Figures 5.16a and b) after the uniform deformation to a tensile strain of 0.13 whereas the martensite islands in the sample with 86 Chapter 5. Experimental Results and Analysis equiaxed morphology show no evidence of obvious martensite plasticity for a uniform strain o f 0.15 (Figures 5.16 c and d). This observation is consistent with the results of the quantitative examination of martensite deformation. The plastic axial strain of martensite in the steel sample with equiaxed martensite was measured to be close to zero (0.008) whereas the martensite islands in the banded sample underwent a considerable amount o f plastic deformation (0.082 or 62% of the corresponding far field strain). Figure 5.16. The effect of martensite morphology on dimensional change and plastic strain of martensite in the uniform deformation region (with the tensile far field strains indicated) for two dual-phase steel samples with around 30% martensite; (a) and (b) samples with banded martensite, (c) and (d) samples with equiaxed martensite. Tensile loading direction is horizontal. 87 Chapter 5. Experimental Results and Analysis 5.2.3.3. Effect of Tempering Process The effect of tempering as a heat treatment to lower the martensite strength is illustrated in Figure 5.17. It is clear in these micrographs that the martensite islands in the uniform deformation region are thinner than those in the undeformed region. Again, this qualitative observation is consistent with the microstructural measurement o f martensite deformation in this steel sample (martensite plastic strain was measured to be 0.06 or 40% of the sample strain). Results in Figures 5.12 and 5.13 and those represented in Figure 5.17 clearly show that the martensite phase in quenched sample remains mostly elastic during uniform straining, however, it deforms plastically after being tempered at 500 °C for 60 minutes. e = 0 e = 0.15! • '-1.-. "** •  . * . — _ . * •- ** > > ~—.*--sr d 25jmi (a) (b) Figure 5.17. The effect of tempering on the martensite islands thickness in a quenched-tempered dual-phase steel sample with 17% banded martensite; tempering temperature and time: 500°C and 60 minutes. Tensile loading direction is horizontal. A summary of all the experimental results regarding the level of martensite deformation measured at the necking point for different dual-phase steel samples is given in Table 5.2 (most results reported here are for the banded steel microstructures where the measuring approach used for the analysis was most convenient). It should be mentioned that all the strains in Table 88 Chapter 5. Experimental Results and Analysis 5.2 are thickness strains. Since the Poisson's ratio for plastic deformation is 0.5, the axial strain is twice the thickness strain. For all the samples in Table 5.2, the plastic strain in the martensite is less than the far field strain, as might be expected. However, examination of Table 5.2 in detail shows that there is a systematic decrease in the strain differential as the martensite content is increased. In Table 5.2, two further observations can be made, i) tempering leads to an increase in the relative amount of martensite plasticity as well as the far field strain and ii) for the martensite contents of 29-30%, the morphology has a significant effect on the deformation behaviour of the martensite. The steel sample produced with low heating rate containing equiaxed martensite islands shows less martensite plasticity than the sample produced with higher heating rate in which the martensite islands are elongated. Table 5.2. The results of the measurements of martensite deformation at the necking point for the steel samples processed to have different martensite contents and morphologies. Martensite Content, V m (%) Condition Martensite Morphology Far Field Thickness Strain (8fd.) Thickness Strain in Martensite (Smart.) Smart. / Sfd. 17 Quenched Band 0.072 ~ 0 0 !7 Quenched-Tempered Band 0.075 0.030 0.39 30 Quenched Band 0.066 0.041 0.62 29 Quenched Equiaxed 0.078 0.004 0.05 41 Quenched Band 0.061 0.051 0.85 52 Quenched Band 0.060 0.057 0.95 89 Chapter 5. Experimental Results and Analysis The results given in Table 5.2 are plotted in Figures 5.18a and b. In these Figures the errors associated with the measurements of the martensite deformation as well as the sample thicknesses are presented (as the error bars in both x- and y-axis). A s can be observed in Figure 5.18a, for the quenched steel sample with 17% martensite, the plastic strain in martensite islands is approximately zero and therefore, the plastic strain is almost entirely partitioned in the ferrite phase (i.e. the ferrite matrix is the only phase which deforms plastically). In the case of samples with 41%> and 52% martensite, on the other hand, there is almost no strain partitioning between the ferrite and martensite phases during deformation since the measured plastic strain in martensite is nearly the same as the far field strain o f the tensile specimen (dashed lines in Figures 5.18a and b show no strain partitioning between constituent phases). The deformation behaviour of the constituent phases in the dual-phase steel sample with 30% martensite is between the two extreme cases for 17%> and 41%> martensite, i.e. the martensite plastic strain is neither zero nor equal to that of the ferrite matrix but it is a fraction of the far -field sample strain. The effect of the martensite morphology and the tempering process on the deformation behaviour of martensite is shown in Figure 5.18b. These effects were already described with reference to the results presented in Table 5.2. 5.2.3.4. Martensite Strength and its Plasticity In the results given in Figures 5.12 to 5.17 and Table 5.2, it was shown that an increase in the martensite content as well as the tempering process resulted in martensite plasticity during deformation. This is, as pointed out in Section 2.4.2, because the yield strength of martensite phase decreases with increasing its content and by tempering. 90 Chapter 5. Experimental Results and Analysis in c <D ro c 'ro 06 o 0 5 ro 0.15 0.1 B 0.05 Steels with Banded Martensite • 17% Martensite • 30% Martensite A 41% Martensite • 52% Martensite 0.05 0.1 Far Field Thickness Strain 0.15 (a) c ro c 'ro l _ —^' CO o ro 0.15 0.1 •B 0.05 • 17% Banded Ms., quenched • 17% Banded Ms., tempered • 29% Equiaxed Ms., quenched O 30% Banded Ms., quenched 0.05 0.1 Far Field Thickness Strain 0.15 (b) Figure 5.18. Martensite plastic strain at the necking point for different dual-phase steel samples, (a) effect of martensite content and (b) effect of martensite morphology and tempering. 91 Chapter 5. Experimental Results and Analysis Table 5.3 shows the estimated martensite carbon concentration, the estimated martensite yield strength and the measured yield strength of martensite in different dual-phase steel samples. The carbon concentration of martensite in each steel sample was estimated using the Thermocalc software (Section 4.3) with the assumption that the rapid quench does not allow for substantial long range diffusion and thus, the ferrite-martensite microstructures at room temperature reflect the structure just after the intercritical annealing. Table 5.3. The estimated carbon concentration and yield strength of martensite in the dual-phase steel samples with different martensite contents and morphologies. The measured yield stress (0.2% offset) of untempered and tempered martensite in selected number of steels is given for comparison. Martensite Morphology V m (%) Cmart. (wt.%) (estimated) tfy.mart. ( M P a ) (estimated) tfy.mart. (MPa) (measured) C'y.tempered mart. (MPa) (measured) 18 0.31 1370 ± 7 0 12901 898' 29 0.20 1060 ± 7 0 - -Equiaxed 32 0.18 1020 ± 7 0 - -44 0.13 890 ± 70 968 2 869 2 83 0.07 720 ± 70 794 3 -17 0.33 1410 ± 7 0 12901 898 1 25 0.23 1140 ± 7 0 - -Banded 30 0.19 1040 ± 7 0 - -41 0.14 9 1 0 ± 7 0 968 2 869 2 52 0.11 830 ± 7 0 - -77 0.08 740 ± 70 - -1 0.30%C-1.5%Mn-1.5%Si steel, austenitized at 1000 °C for 20 min. then water quenched. 2 0.12%C-1.6%Mn-0.19%Si-0.2%Mo steel, held at 1050 °C for 2 min., cooled to 900 °C and held for 3 min. at this temperature and then water quenched. 3 0.06%C-2%Mn-0.07%Si-0.15%Mo steel, held at 1050 °C for 2 min., cooled to 900 °C and held for 3 min. at this temperature and then water quenched. * Tempering treatments were conducted at 500 °C for 60 min. 92 C h a p t e r 5 . E x p e r i m e n t a l R e s u l t s a n d A n a l y s i s T h e y i e l d s t r e n g t h o f m a r t e n s i t e i n T a b l e 5 . 3 w a s e s t i m a t e d from i t s c a r b o n c o n c e n t r a t i o n u s i n g t h e e x p e r i m e n t a l d a t a i n t h e l i t e r a t u r e ( L e s l i e , 1 9 8 1 ) , i . e . t h e y i e l d s t r e n g t h d a t a s h o w n i n F i g u r e 2 . 1 1 . T h e m a r t e n s i t e y i e l d s t r e n g t h i n s e l e c t e d n u m b e r o f s a m p l e s i n t h i s T a b l e w a s d e t e r m i n e d e x p e r i m e n t a l l y u s i n g s e p a r a t e s t e e l s w i t h f u l l y m a r t e n s i t i c m i c r o s t r u c t u r e s a n d t h e c h e m i s t r y a l m o s t s i m i l a r t o t h e m a r t e n s i t e p h a s e i n t h e c o r r e s p o n d i n g d u a l - p h a s e s t e e l s a m p l e s . T h e t e n s i l e t r u e s t r e s s - t r u e s t r a i n c u r v e s o f t h e s e s t e e l s u p t o t h e n e c k i n g p o i n t a r e s h o w n i n F i g u r e 5 . 1 9 . T h e s t e e l c h e m i s t r i e s a n d t h e h e a t t r e a t m e n t c o n d i t i o n s a r e g i v e n i n T a b l e 5 . 3 . 2400 -i 1 True Strain F i g u r e 5 . 1 9 . T h e t r u e s t r e s s - t r u e s t r a i n c u r v e s o f 1 0 0 % m a r t e n s i t e s t e e l s w i t h d i f f e r e n t c a r b o n c o n c e n t r a t i o n s o b t a i n e d from t e n s i l e test . C u r v e s a r e p l o t t e d u p t o t h e n e c k i n g p o i n t . A s s h o w n i n T a b l e 5 . 3 , a n i n c r e a s e i n t h e m a r t e n s i t e c o n t e n t r e s u l t s i n a d e c r e a s e i n t h e c a r b o n c o n c e n t r a t i o n o f t h e m a r t e n s i t e . T h e y i e l d s t r e n g t h o f m a r t e n s i t e a l s o d e c r e a s e s w i t h a n i n c r e a s e i n t h e p e r c e n t m a r t e n s i t e . F o r e x a m p l e , t h e m a r t e n s i t e y i e l d s t r e n g t h i n t h e s t e e l s a m p l e s w i t h 1 7 - 1 8 % m a r t e n s i t e ( w i t h t h e c a r b o n c o n c e n t r a t i o n o f 0 . 3 1 - 0 . 3 3 wt .%o) i s a b o u t 9 3 Chapter 5. Experimental Results and Analysis 500 MPa higher than that in the samples containing 41-44% martensite (with the martensite carbon concentration of 0.13-0.14 wt.%>). Furthermore, for a fixed percent martensite (or martensite carbon concentration), the yield strength of tempered martensite is considerably lower than that of the untempered martensite, e.g. a 30%> decrease in the yield strength of martensite phase was obtained after tempering the dual-phase steel samples with 17-18%) martensite at 500 °C for 60 minutes. Another observation in Table 5.3 is that the experimental results on the martensite yield strength are in good agreement with the estimated results from the literature. Figure 5.20 illustrates the measured strain ratio (the ratio of the plastic strain in martensite to the far field strain) at the necking point as a function of the martensite carbon concentration for the steel samples with banded and equiaxed morphology. 0.8 -. 0.6 -T J CO •c " 0.4 -0.2 -• 17% Martensite • 30% Martensite • 41% Martensite • 52% Martensite O 29% Martensite Equiaxed Steel (Low Heating Rate) \ \ Banded Steels \ (High Heating Rate) \ O 0 0.1 0.2 0.3 0.4 Martensite Carbon Concentration (wt%) Figure 5.20. Martensite plastic strain at the necking point for the dual-phase steel samples with equiaxed and banded martensite morphologies as a function of the martensite carbon concentration. 94 Chapter 5. Experimental Results and Analysis A s can be observed in Figure 5.20, for the case of banded morphology, the strain ratio increases with a decrease in the martensite carbon concentration (or with an increase in the percent martensite). In addition, when the carbon concentration of martensite is below approximately 0.1 wt.%, no or very little strain partitioning occurs between the ferrite and martensite phases. On the other hand, i f the martensite carbon concentration is equal to or greater than approximately 0.33 wt.%>, the martensite phase w i l l remain elastic during deformation (at least up to the necking point). For the case of steel samples with equiaxed martensite morphology, only one measurement was available due to the difficulties associated with the image analysis o f the corresponding steel microstructures. However, the experimental results clearly show that the morphology o f martensite has a strong effect on its deformation behaviour. It can be observed in Figure 5.20 that the strain ratio in the steel sample with 29%> equiaxed martensite is as low as 0.05 whereas it is significantly higher (0.62) in the case of steel sample with almost the same martensite content and carbon concentration (30%> and 0.19 wt.%>, respectively) but the banded morphology. 5.2.3.5. Unusual Fracture Behaviour The important questions here to explore are why the low carbon dual-phase steel investigated in this work showed unusual fracture behaviour (an enhanced combination of the fracture stress and strain compared to the conventional dual-phase steels in the literature) and whether or not martensite plasticity plays a role? To answer these questions, one needs to have knowledge about the ductile fracture mechanism in dual-phase steels and the relationship between the deformation characteristics of martensite and the fracture mechanism. It is possible 95 C h a p t e r 5 . E x p e r i m e n t a l R e s u l t s a n d A n a l y s i s t h a t m a r t e n s i t e p l a s t i c i t y h a s a n e f f e c t o n t h e f r a c t u r e p r o c e s s s i n c e i t i s e x p e c t e d t h a t t h e p r e s e n c e o f p l a s t i c m a r t e n s i t e i n t h e d u a l - p h a s e s t e e l m i c r o s t r u c t u r e s a f f e c t s t h e v o i d n u c l e a t i o n p r o c e s s ( e i t h e r b y c r a c k i n g o f m a r t e n s i t e p a r t i c l e s o r d e c o h e s i o n o f t h e i r i n t e r f a c e w i t h f e r r i t e m a t r i x ) d u r i n g t h e d u c t i l e f r a c t u r e . T h e p l a s t i c i n c o m p a t i b i l i t y b e t w e e n t h e c o n s t i t u e n t p h a s e s i s r e s p o n s i b l e f o r t h i s v o i d n u c l e a t i o n . It w a s a l r e a d y s h o w n t h a t a s t h e m a r t e n s i t e c o n t e n t i n c r e a s e s , i t s d u c t i l i t y i n c r e a s e s r e s u l t i n g i n a n i n c r e a s e i n t h e p o t e n t i a l t o c o - d e f o r m w i t h t h e f e r r i t e m a t r i x . I n o t h e r w o r d s , t h e p o s s i b i l i t y o f p l a s t i c d e f o r m a t i o n i n m a r t e n s i t e a n d c o -d e f o r m a t i o n b e c o m e g r e a t e r a s t h e m a r t e n s i t e c o n t e n t i n c r e a s e s . T h u s , s i n c e c o n s i d e r a b l e m a r t e n s i t e p l a s t i c i t y w a s f o u n d i n s t e e l s a m p l e s w i t h m o r e t h a n 1 7 - 1 8 % m a r t e n s i t e a n d t h e p l a s t i c i t y i n c r e a s e d w i t h a n i n c r e a s e i n t h e m a r t e n s i t e c o n t e n t , o n e m a y e x p e c t t h a t t h e n u c l e a t i o n o f v o i d s b e c o m e s m o r e d i f f i c u l t w i t h a n i n c r e a s e i n t h e m a r t e n s i t e c o n t e n t w i t h t h e f i n a l r e s u l t t h a t t h e f r a c t u r e s t r e s s a n d s t r a i n i n c r e a s e a s t h e p e r c e n t m a r t e n s i t e g o e s u p . O b v i o u s l y , f o r t h e f i n a l c o n c l u s i o n c o n c e r n i n g t h e e f f e c t o f m a r t e n s i t e p l a s t i c i t y o n t h e f r a c t u r e b e h a v i o u r o f t h e d u a l - p h a s e s t e e l , t h e e x p e r i m e n t a l e v i d e n c e s h o w i n g t h e e v o l u t i o n o f t h e m i c r o s t r u c t u r a l d a m a g e d u r i n g t h e p o s t n e c k i n g d e f o r m a t i o n s h o u l d b e i n c o r p o r a t e d w i t h t h e e x p e r i m e n t a l r e s u l t s o n m a r t e n s i t e p l a s t i c i t y . 5.2.4. Observation on Microstructural Damage 5.2.4.1. Steel Micrographs T h e e v o l u t i o n o f m i c r o s t r u c t u r a l d a m a g e d u r i n g t h e t e n s i l e d e f o r m a t i o n o f d u a l - p h a s e s t e e l s a m p l e s w a s i n v e s t i g a t e d t o e x a m i n e h o w t h e p l a s t i c d e f o r m a t i o n o f m a r t e n s i t e , w h i c h w a s e x p e r i m e n t a l l y s h o w n i n t h e p r e v i o u s s e c t i o n t o o c c u r i n s o m e s t e e l s a m p l e s , a f f e c t s t h e f r a c t u r e b e h a v i o u r o f t h e s t e e l s a m p l e s . T h e m i c r o g r a p h s i n F i g u r e 5 . 2 1 i l l u s t r a t e t h e p o l i s h e d 9 6 Chapter 5. Experimental Results and Analysis surface of the fractured tensile specimens in the steel samples with banded martensite morphology. (c) (d) Figure 5.21. Optical micrographs illustrating the void formation for sections parallel to the tensile axis on the banded steel samples with different martensite contents (produced using a heating rate of 100 °C/s); (a) 17% martensite, (b) 17% martensite after tempering at 500 °C for 60 minutes, (c) 25% martensite and (d) 41% martensite. In Figures 5.21a, c and d, the micrographs of the samples with different martensite contents are shown, i.e. the samples with 17%>, 25% and 41%> martensite, respectively. The progress of fracture phenomenon and also the effect of martensite content on the fracture pattern of these 97 Chapter 5. Experimental Results and Analysis samples can be qualitatively observed in this series o f micrographs. First o f all, both the number and area fraction o f voids decrease as the percent martensite increases. The average void size in the 17% martensite steel sample is in the range of 2-25 u.m whereas the voids in the sample with 25% martensite (with the average size of 2-35 u.m) are substantially larger. In addition, the banded morphology of the martensite phase in the sample with 17%> martensite is clearly evident in Figure 5.21a. The distribution of voids reflects the martensite band distribution since void nucleation was observed to occur mostly by fracture o f the martensite islands. Figure 5.22 clearly shows the voids which are predominately nucleated as a result of the particles fracture, although there was also evidence o f a limited amount o f void nucleation by decohesion of the interface between the ferrite matrix and the martensite particles. Figure 5.22. S E M secondary electron image of the deformed steel sample with 17% banded martensite produced by the high heating rate processing route (100 °C/s). Image was taken near the fracture surface representing the void formation which predominately occurred by fracture of martensite islands. Finally, the micrograph in Figure 5.21d shows that a limited number of voids are formed in the steel sample with 41%> martensite. The effect of tempering, which was shown in the 98 Chapter 5. Experimental Results and Analysis previous section to decrease the strength of martensite, on the void formation process in the steel sample with 17% martensite is shown in Figure 5.21b. Comparing Figures 5.21a and b, one can readily observe that the fracture strain is much larger in the tempered sample and that microstructural damage in the form of voids is considerably less for the tempered steel sample. 5.2.4.2. Quantitative Analysis of Voids The void nucleation and growth in the steel samples were quantitatively examined using the image analysis software. In Figure 5.21, the effect of martensite content and tempering (both affecting the martensite strength) on void formation process was clearly illustrated. The number of voids nucleated and their area fraction were measured at different levels of far field thickness strain (from near the necking point to the region close to the fracture surface) and their variation with the sample strain was also examined. The results for the steel samples with 17% and 41%> martensite (banded morphology) are shown in Figures 5.23a and b. A s can be observed, these samples show different void formation behaviours. First o f all , it appears that the number o f voids nucleated in the 17%> martensite steel sample linearly increases with the far field strain suggesting that the void nucleation in this sample is continuous. This is consistent with the observations of Gurland (1972) on a spheroidized 1.05% C steel. On the other hand, the void nucleation in the steel sample with 41%> martensite is different, i.e. a non-linear function of the strain in which most of the voids are nucleated in the regions which are closer to the fracture surface experiencing the far field strain of 0.37 to 0.47. This observation is also consistent with the results of LeRoy et al. (1981) on the spheroidized carbon steels who reported discontinuous void nucleation behaviour. 99 Chapter 5. Experimental Results and Analysis 2000 True Thickness Strain (b) Figure 5.23. The number of voids per unit area (a) and the area percent voids (b) in the steel samples with 17% and 41% banded martensite (high heating rate, 100 °C/s) as a function of the far field thickness strain. 100 Chapter 5. Experimental Results and Analysis As can be observed in Figure 5.23a, although the nucleation rate (the slope of the linear relationship) for 17% martensite steel is relatively high, the void nucleation rate for 41%> martensite steel is considerably lower over a wide range o f sample strain (0.11 to about 0.3), but the nucleation rate increases then significantly in the strain range o f about 0.37 to 0.47. The results illustrated in Figure 5.23b showing the area percent voids versus the far field strain confirm that an increase in the martensite content from 17%> to 41% has a considerable effect on the void formation process. A s can be observed in Figure 5.23b, for a given far field thickness strain, the area percent voids in the neck region of the sample with 17% martensite is much greater than that of the 41 %> martensite steel. Another observation in Figure 5.23b is that the area percent voids in the 17%> martensite steel increases linearly with increasing strain, consistent with the void nucleation behaviour. However, the area percent curve for the 41 %> martensite steel shows a different behaviour, i.e. a relatively low increase rate at the strain range of 0.11 to 0.3 and a higher increase rate in strain range of 0.37 to 0.47 (very close to the fracture strain). The experimental results for the void growth stage in the steel samples with 17% and 41% martensite are shown in Figure 5.24. These results represent the mean values of the major and minor semi-axes of the growing voids (elliptical in shape) in a strain range from the void nucleation to the final fracture. A s can be observed in Figures 5.24a and b, the voids first nucleated in both samples with 17%> and 41%» martensite are ellipsoidal. For the sample with 17%o martensite, the major and minor radii o f the voids (R\ and R 3 ) at the nucleation point are 3.6 and 2.2 um whereas these radii for the 41% martensite steel are 3.1 and 1.9 | im, respectively. 101 Chapter 5. Experimental Results and Analysis E cc ui •g o > CD $ 3 E <u CO 2 T3 C CD i_ O ST Banded Steel (17% Martensite) CD • Major Semi-Axis O Minor Semi-Axis 0.1 0.2 0.3 0.4 True T h i c k n e s s Strain (a) 0.5 0.6 E CO cc ui •g o > 3 3 E cu CO i o c fZ CD i o Banded Steel (41% Martensite) • — * A A , A A -A-A Major Semi-Axis A Minor Semi-Axis 0.1 0.2 0.3 0.4 True T h i c k n e s s Strain 0.5 0.6 (b) Figure 5.24. Variation of the major and minor radii of voids as a function of far field thickness strain for two dual-phase steel samples with (a) 17% and (b) 41% banded martensite. 102 Chapter 5. Experimental Results and Analysis A n important observation in Figures 5.24a and b is that the growth curves for both samples are linear with slopes very close to zero suggesting that there is almost no void growth in these samples. This shows that the fracture process in the steel samples with 17% and 41% martensite has occurred in two stages, i.e. void nucleation and void coalescence. Accordingly, it would be reasonable to assume that the fracture phenomenon in these steel samples is controlled by the void nucleation process. The effect of tempering (at 500 °C for 60 minutes) on void formation process in the steel sample with 17% martensite (banded morphology) is shown in Figures 5.25a and b. A s can be observed, tempering has a significant effect on the evolution o f microstructural damage in this sample. The effect of tempering appears to be similar to that o f the martensite content (Figure 5.23) where an increase in the percent martensite resulted in an enhanced fracture behaviour. It is clear in Figures 5.25a and b that the number of voids nucleated and the area percent voids are considerably lower in the case of tempered sample. In other words, tempering process has had a beneficial effect on delaying void formation in this sample. The true far field strain at which the voids are first nucleated (nucleation strain, 8N) can be estimated using the area percent curves (i.e. curves in Figure 5.23b for the steel samples with 17% and 41% martensite). For this estimation, an appropriate criterion for the void nucleation condition should be employed. A s pointed out in Chapter 2, a void nucleation condition has been reported in the literature as a criterion for definition of the void nucleation strain. According to this criterion, the strain at which 0.1 % area percent voids are nucleated has been chosen as the nucleation strain. 103 Chapter 5. Experimental Results and Analysis Figure 5.25. The effect of tempering (500 °C for 60 minutes) on the number of voids per unit area (a) and the area percent voids (b) in the steel samples with 17% banded martensite (produced with the high heating rate, 100 °C/s ) as a function of the far field thickness strain. 104 Chapter 5. Experimental Results and Analysis Table 5.4 illustrates the void nucleation strain for a selected number of dual-phase steel samples (with banded martensite morphology) estimated using this void nucleation criterion. The initial void nucleation rate (dN/ds) and the average radii of voids (Ri and R 3 ) first nucleated at the estimated nucleation strains are also given in Table 5.4. Table 5.4. Nucleation strain, uniform strain, initial void nucleation rate and initial major and minor radii of voids first nucleated in some steel samples with banded martensite morphology. All strains are thickness strains. V m (%) Condition Nucleation Strain ( 8 N ) Uniform Strain (Su) Initial Void Nucleation Rate (dN/de); , (#/mm2) Initial Radii of Voids ( L i m ) Ri R 3 17 Quenched 0.12 0.072 5645 3.6 2.2 17 Quenched- 0.18 0.075 453 4.1 2.6 Tempered 25 Quenched 0.13 0.071 3416 3.2 2.2 41 Quenched 0.17 0.061 412 3.1 1.9 As can be observed in Table 5.4, the nucleation strain in the untempered steel samples increases with an increase in the martensite content. For example, when the percent martensite increases from 17 to 25 and 41, the true thickness strain at which the void nucleation starts increases from 0.12 to 0.13 and 0.17, respectively. These results appear to be consistent with the observations on martensite plasticity (Table 5.2), i.e. no martensite plasticity in the steel sample with 17% martensite but a progressive amount of martensite plastic deformation in the steel samples with higher martensite contents such that martensite plasticity in the sample with 41%) martensite was found to be substantially high. If the deformation behaviour of martensite changes from elastic to plastic (e.g. as a result of an increase in its content or by tempering due to their effect on martensite strength), the plastic incompatibility between the ferrite and 105 Chapter 5. Experimental Results and Analysis martensite phases decreases and consequently, the voids may be nucleated at higher far field plastic strains. The results in Table 5.4 also show that tempering process has a considerable effect on the void nucleation strain in the steel with 17% banded martensite. Although the nucleation strain in the untempered sample with elastic martensite has been estimated to be 0.12, the void nucleation strain is 0.18 in the tempered sample with plastic martensite (Table 5.2). This is, as shown already, because tempering process may have the same effect as the martensite content on the strength and deformation behaviour of martensite. " . The curves in Figures 5.23a and 5.25a regarding the number o f voids as a function of the far field strain were quantitatively analyzed and the results are shown in Table 5.4. It can clearly be observed for the case of untempered steel samples that the void nucleation rate in the sample with 17%> martensite (with linear void nucleation behaviour) is greater than the initial void nucleation rates in the samples with 25%> and 41 %> martensite at which the majority of the voids are nucleated at later stages of the post-necking deformation. For example, an increase in the martensite content from 17%> to 41%> resulted in a considerable decrease in the void nucleation rate at the early stages of the void formation process from 5645 to 412 voids per unit area (#/mm2), respectively. Tempering also showed a large effect on the void nucleation rate, i.e. a decrease in the void nucleation rate from 5645 to 453 mm" 2 in the steel sample with 17%> martensite after being tempered at 500 °C for 60 minutes. 5.2.4.3. Discussion on Unusual Fracture Behaviour It is now possible to rationalize the unusual fracture properties o f dual-phase steel samples shown in Figure 5.11 by examination of the qualitative and quantitative analysis of the void 106 Chapter 5. Experimental Results and Analysis formation process with a focus on martensite plasticity. It is clearly shown in Figure 5.21 that the level of microstructural damage (both the number of voids and their area percent) is much greater in the steel samples with lower martensite contents, i.e. in the cases (e.g. the steel sample with 17% martensite) where the yield stress of martensite is high so that it remains elastic during deformation. For the case of steel sample with 17%> martensite, the results illustrated in Table 5.2 showing that there is no martensite plasticity during tensile deformation, confirms this observation. Similar observation can be made from the results shown in Figure 5.23 representing the quantitative analysis of the void formation process in the steel samples with 17%o and 41%> martensite. Accordingly, one may conclude that when the martensite content increases its yield strength decreases and consequently, it becomes more ductile so that it finally co-deforms with the ferrite matrix. In the cases where the martensite phase shows a considerable plasticity, the void nucleation is much more difficult compared to the cases where voids are nucleated as a result of the brittle fracture of elastic martensite islands. Hence, it is expected that the fracture stress and strain of dual-phase steel increase with an increase in the martensite content. As pointed out previously, this fracture behaviour is unusual with a reference to the results reported in the literature for dual-phase steels with higher carbon concentrations. In the cases where the decohesion of ferrite and martensite interface is responsible for the void nucleation, the situation can similarly be analyzed. In dual-phase steel samples with plastic martensite, the plastic incompatibility between the constituent phases is less severe due to the fact that the externally imposed strain can be accommodated at the interface and therefore, there would be less chance for the separation of the interface. 107 Chapter 5 . Experimental Results and Analysis The micrographs in Figures 5.21a and b along with the void formation curves in Figure 5.25 illustrate the effect of tempering on the evolution of microstructural damage in the steel sample with 17% martensite. A s can be observed in these Figures, the number and the area percent voids are considerably lower in the tempered steel sample suggesting that the effect of tempering on the void formation process is similar to that of the martensite content, i.e. its effect on lowering the strength of martensite (Tables 5.2 and 5.3). 5.3. Analysis of Void Formation Process in Dual-Phase Steel Samples As shown in Section 5.2.4.2 experimentally, there was almost no void growth stage during ductile fracture of the steel samples with low and high martensite contents, i.e. those containing 17% and 41% martensite (banded morphology). This results in a straightforward solution for the estimation of the area percent voids, i.e. a solution consisting o f i) an experimental condition under which the voids (which are assumed to be elliptical in shape with the major and minor radii of R i and R 3 , respectively) are first nucleated (0.1 % area percent criterion, Table 5.4), ii) the major and minor radii o f the voids almost independent o f the far field strain and i i i) the experimental results on the number of voids nucleated as a function of the strain (Figure 5.23a). Accordingly, the area percent voids (A v ) nucleated at any far field strain (s) can simply be expressed by the following equation: A v = 7 i R 1 R 3 N (5.9) where N is the number of voids per unit area nucleated at the sample strain of s (Figure 5.23a). In this equation, N is the only parameter which changes with strain. The variation of N with the far field strain can be estimated using the experimental results illustrated in Figure 5.23a, i.e.: 108 Chapter 5. Experimental Results and Analysis 17% Martensite Steel: N = 56458-573 (#/mm 2) (5.10) 41% Martensite Steel: N = 12.33exp(9.2s) (#/mm 2) (5.11) Accordingly, the area percent voids nucleated in these steel samples as a function of strain can be calculated using Equations 5.9 to 5.11 and the experimental results in Table 5.4 on the void nucleation strain ( S N ) and the void average size (R\ and R 3 ) . Figure 5.26a shows the estimated area percent voids in the dual-phase steel sample with 17% martensite as a function of the thickness strain (far field) with a comparison to the results obtained from the experiment. A s can be observed in Figure 5.26a, the area percent curve in this steel sample is linear representing a linear relationship between the number of voids nucleated and the strain (Equation 5.10). A good agreement can be observed in Figure 5.26a between the estimated and experimental results, although a considerable deviation is observed at larger strains due to the fact that the void growth was not considered. The estimated results for the void formation process in the steel sample with 41%> martensite are also shown in Figure 5.26b. A s observed in Figure 5.26b, the estimated area percent curve in this steel sample is exponential and, indeed, shows a very good consistency with the experimental results. In summary, void formation process in the dual-phase steel samples with 17%> and 41 %> martensite which was found experimentally to be independent of void growth, can successfully be predicted by a simple model developed using the experimental results on the number of voids nucleated at any far field thickness strain. 109 Chapter 5. Experimental Results and Analysis in •g o > -*—' c: <u o i— CD 0_ ro 0 Banded Steel (41% Martensite): fe N = 0.17 [R, = 3.1 um, R 3 = 1.9 um A Experimental — Estimated A ^ / /k. 0.1 0.2 0.3 0.4 True T h i c k n e s s Strain 0.5 0.6 (b) Figure 5.26. The estimated area percent curves in the dual-phase steel samples with (a) 17% and (b) 41% martensite compared to the corresponding experimental data. 110 Chapter 5. Experimental Results and Analysis The area percent voids nucleated in these steel samples during post-necking deformation regime was predicted in order to provide supporting evidence for the experimental results already obtained from the quantitative analysis of voids (Section 5.2.4.2). A s shown in Figure 5.26, this series of estimated results on the void formation process is consistent with the experimental results and, indeed, confirms that the void nucleation has been the dominant effect during the ductile fracture of these steel samples. Therefore, one may consider possible strategies to reduce the void nucleation with the final result of enhanced fracture properties. One possible option is martensite plasticity which was shown to significantly decrease the void nucleation rate (results in Table 5.2 and 5.4 for untempered samples with 17% and 41%> martensite) and hence, to cause a considerable increase in the fracture stress and strain (Figure 5.11). I l l Chapter 6 - Modelling Results and Discussion In this chapter, a model w i l l be developed based on the modified Eshelby approach to i) rationalize quantitatively the experimental observations on the mechanical behaviour o f dual-phase steel samples and ii) predict the stress-strain behaviour of the steel samples. 6.1. Micromechanics of Dual-Phase Steel Microstructures A s emphasized in the previous chapters, the mechanical behaviour of dual-phase steels is closely related to their composite type microstructures. It is thus expected that the macroscopic observations on the mechanical response of these steels can be rationalized through a micromechanical analysis of their two-phase microstructures. The load (or stress) transfer to the martensite phase is a process which occurs at the microstructural level when an external load is applied to the dual-phase steel. Micromechanical models can be employed to evaluate the load transfer process, and also to estimate the stress in the martensite phase. If the stress in the martensite phase is known, its mechanical response in dual-phase steel microstructure can be predicted. In the following section, the model w i l l be described in detail and the results for the dual-phase steels with different martensite contents and morphologies w i l l be summarized. 6.1.1. Modelling Load Transfer Process In order to estimate the load transfer to the martensite phase in dual-phase steels, the modified Eshelby model (Weng, 1990) has been used. This model was described in Chapter 2 112 Chapter 6. Modelling Results and Discussion (Section 2.6.4) with its application to modelling the load transfer process in two-phase materials. In modelling the load transfer process, the ferrite phase is assumed to be the only phase which deforms plastically during straining. B y comparing the estimated stress in the martensite phase with the martensite yield strength (summarized in Table 5.3 for dual-phase steel samples with different martensite contents and morphologies), one wi l l rationalize whether martensite plasticity occurs during deformation. To model the load transfer process in dual-phase steels with elastic martensite (both spherical and ellipsoidal morphologies), the flow behaviour o f the ferrite matrix must be known. In the present work, it was not possible to directly measure the stress-strain response of the ferrite phase. Instead, the as-hot-rolled steel with 8-10% pearlite and an average ferrite grain size similar to that of the ferrite phase (~ 5.4 urn) in dual-phase steel samples was taken as representative of the ferrite matrix. According to Gladman et al (1972), the addition of 10%> pearlite to a ferritic steel microstructure with the chemistry and the ferrite grain size similar to the steel investigated in this work results in approximately 0.2%> and 2% increase in the yield and tensile strength, respectively; i.e. there is a very small effect of pearlite (10%) on the strength o f the steel. The tensile stress-strain curve of the hot-rolled steel obtained from a tension test was considered as the stress-strain curve of the ferrite phase in all dual-phase steel samples. The flow stress of ferrite phase can be expressed using the Voce equation (Voce, Kocks and Mecking, 2003), i.e.: ° o = % +K 0 - a y , 0 ) [ l - e x p ( - k 0 eP) ] (MPa) (6.1) where o y o and o s o a re the yield strength and the saturation stress (true stress at very large 1*13 Chapter 6. Modelling Results and Discussion strains), respectively,ep0 is the plastic strain and k Q i s a constant.1 The constants, i.e. a y o , a s o a n d k 0 , were determined by fitting Equation 6.1 to the experimental stress-strain curve of the hot-rolled steel. Figure 6.1 shows the experimental stress-strain curve of the; steel compared to the fitted curve with values of 306 M P a , 605 M P a and 12.9, for a y o , a s o a n d k 0 , respectively. It should be noted that the hot-rolled steel showed a yield point elongation (Figure 6.1) which was not represented by equation 6.1. 1000 800 600 ti) 400 200 0 ro to CO CD CD 3 Hot-Rolled Steel as Ferrite Phase — Calculated (Eq. 6.1) — Experimental 0 0.05 0.1 0.15 True Strain 0.2 0.25 Figure 6.1. Calculated and experimental stress-strain curves of the investigated steel in the hot-rolled condition as the ferrite phase in dual-phase steel samples. It is now possible to consider the deformation of ferrite-martensite composite following the approach of Weng (1990). In the present work, however, another form o f constitutive model for the plastically deforming ferrite has been employed, i.e. different from what Weng used. 1. In Equation 6.1, the ferrite phase is represented as phase "0" while in the next set of equations, the martensite phase is assumed to be phase "1". 114 Chapter 6. Modelling Results and Discussion During the first stage of deformation, both the ferrite and martensite phases are elastic and since their Young's moduli are the same, i.e. -210 GPa (Hertzberg, 1996), there is no stress transfer to the martensite phase and consequently, stress and strain are uniformly distributed between the two phases. In this case, the stress in the ferrite and martensite phases can be calculated using Hooke's law: ° 0 = E 0 e 0 (6.2) <J, =E,e, (6.3) whereE 0 andE, are the Young's moduli of ferrite and martensite ( E 0 = E , = 210 GPa ) , and e 0 and E i are the elastic strains in ferrite and martensite, respectively. In the next stage of deformation, the ferrite phase deforms plastically while the martensite remains elastic. In this stage, the stress ratios (the stress in the dual-phase steel to the stress in the ferrite and martensite phases) are functions of the martensite morphology. For the case o f dual-phase steels with spherical martensite islands, the stress ratios for the ferrite and martensite phases can be determined using the following equations: Ferrite: — = -1 (b* < 1) (6.4) Martensite: — = — (bj > 1) (6.5) a , b| where a is the stress in dual-phase steel, b* and b* are the stress partitioning coefficients for the ferrite and martensite phases, respectively, i.e.: 0 [f+o-fKK^-LO+u* (6.6) 115 Chapter 6. Modelling Results and Discussion K = ^ (6.7) [ f + ( l - f ) P , 0 ] ( | i 1 - | i ' ) + ( i , 0 In Equations 6.6 and 6.7, \is0 is the secant shear modulus of the ferrite, P* is the distortional component o f the S Eshelby tensor, f is the martensite volume fraction and ft, is the shear modulus of the martensite phase, i.e. 2( l + v,) which gives the value of approximately 80.8 GPa assuming the Young's modulus ( E , ) and Poisson's ratio ( v , ) to be 210 GPa and 0.3, respectively. In order to calculate the stress coefficients b* and b | , one first calculates u,* and p*, which can in turn be determined from the secant Young's modulus and the secant Poisson's ratio of the ferrite. The secant Young's modulus of the ferrite phase, E * , for any given plastic strain, sp0, is: E* = = (6.9) E o ° "o E o o y o + ( a s o - a y o ) [ l - e x p ( - k 0 8 P ) ] Since the constants for the flow curve of the ferrite phase are known, the secant Young's modulus w i l l only be a function of the plastic strain, e?0 . The secant Poisson's ratio of the ferrite can also be determined from the secant Young's modulus as: *:-r(rv-)S (610) where the ferrite Poisson's ratio, v 0 , is assumed to be 0.3. The secant shear modulus (|x*) can be determined using Equations 6.9 and 6.10, i.e.: 116 Chapter 6. Modelling Results and Discussion ti= (6 .H) 2( l + v : ) Finally, the Eshelby's S tensor ($ s 0 ) for spherical martensite morphology can be calculated from the secant Poisson's ratio of the ferrite matrix (Weng, 1990; Clyne and Withers, 1993) as follows: P o 15 2 f4-5v!A l - v 0 j (6.12) Using Equations 6.4 and 6.5, the ratio of the martensite stress to the ferrite stress can be calculated for any arbitrary value of the ferrite plastic strain, ep0, i.e. ^•-v ( 6 1 3 ) A n important observation here is that the stress partitioning coefficients (b* and b") are functions of the martensite volume fraction, f, and the ferrite plastic strain, ep0 (through the relations shown in Equations 6.9 to 6.12). Since the martensite fraction is fixed for a dual-phase steel, the stress coefficients in the modified Eshelby model can be calculated for any given plastic strain in the ferrite phase. The stress ratio in dual-phase steels with ellipsoidal martensite islands can be calculated similarly. The corresponding equations are also functions of the martensite content and the ferrite plastic strain. Due to the complexity associated with the micromechanical analysis for the case of ellipsoidal inclusions, the equations needed for the calculations are given in Appendix A . 117 Chapter 6. Modelling Results and Discussion In order to evaluate the load transfer process to the martensite phase in dual-phase steels, the martensite to ferrite stress ratio was calculated. The model was implemented in Microsoft Excel™ and the calculations started with an initial value for the ferrite plastic strain (sp0) equal to zero. The stress ratio for this initial value of plastic strain was determined through the following sequential stages: 1) the secant Young's modulus of the ferrite phase, E * , was calculated using Equation 6.9. 2) the secant Poisson's ratio, v*, was then calculated from Equation 6.10. 3) the secant shear modulus, u.*, was determined from the calculated secant Young modulus and the secant Poisson's ratio, using Equation 6.11. 4) the Eshelby S tensor, , was determined using Equation 6.12 and the secant Poisson's ratio calculated from Equation 6.10. 5) the stress partitioning coefficients, b* and b ' , was then calculated for a given martensite volume fraction, f, using Equations 6.6 and 6.7. 6) the martensite stress to ferrite stress ratio ( — ) was finally calculated from Equation 6.13. The calculation was then conducted stepwise in order to obtain a smooth variation of the stress ratio over a range of plastic strain corresponding to the experimental plastic strains (tensile). To choose the strain interval, both the martensite content and morphology were taken into account. The strain increments of 0.01, 0.006, 0.003, 0.001 and 0.0005 were considered for the stepwise calculation o f the stress-strain relations of dual-phase steel samples with different martensite contents and morphologies. Figure 6.2 illustrates the results for the steel sample with 18% spherical martensite islands. 118 Chapter 6. Modelling Results and Discussion As can be observed in Figure 6.2, in the case of strain increments equal to or less than 0.001, the load transfer curves in the transition region do not change with a further decrease in the strain increment and also provide a smooth change in the load transfer parameter. The same result was observed for the samples with ellipsoidal martensite morphology where the saturation strain was 0.05 and the stress ratio was a function of the martensite content. In this case, the strain increment was found to be independent of the aspect ratio of the martensite islands. Accordingly, the strain increment of 0.001 was chosen for dual-phase steel samples with different martensite contents and morphologies. E b 2 -CO or (/> CD CO Steel with 18% Martensite (Spherical) AE = 0.001, 0.0005 -saturation / O f p r r = 2.5 - » - A E = 0.01 - ^ A e = 0.006 - » - A E = 0.003 -*-Ae = 0.001 - ^ A e = 0.0005 0.02 0.04 0.06 True Far Field Strain 0.08 0.1 Figure 6.2. The load transfer curves in a dual-phase steel sample with 18% spherical martensite calculated using different strain increments. 119 Chapter 6. Modelling Results and Discussion In order to calculate the far field plastic strain of dual-phase steel ( e p ) in Figure 6.2, the following relation (Tandon and Weng, 1988) was used: ' P = ( F J ~ I ^ ( 6 1 4 ) where E s is the secant Young's modulus and E is the Young's modulus (E = E 0 = E i = 210 GPa) of the dual-phase steel. This relation can easily be derived from Figure 2.14 for the case of uniaxial tensile deformation. In Equation 6.14, a is the stress in the dual-phase steel which can be calculated using Equation 6.4 for a given plastic strain in the ferrite phase. The secant Young's modulus of the dual-phase steel ( E s ) can be calculated from its secant shear modulus, u 5 , (Dieter, 1976), i.e.: E ' = - * *EL (6.15) where K s is the secant bulk modulus of the steel. Since the secant bulk modulus is equal to the bulk modulus (see Section 2.6.4.2), K s c a n be calculated using Equation 2.8 and the values for the Young's modulus and the Poisson's ratio of the constituent phases, i.e. 210 GPa and 0.3, respectively, i.e.: KS = K 0 = K , =175 G P a (6.16) The secant shear modulus o f the steel, u. s , can also be determined from the following equation: f(h-rO s s 1+- (6.17) In Figure 6.3, the calculated stress ratios ( — , here as G m a r t - ) are plotted as a function of °"ferr. the far field strain for the steel samples with different martensite contents and morphologies. It 120 Chapter 6. Modelling Results and Discussion should be noted that in all calculations the equiaxed and banded martensite islands were assumed to be spherical and ellipsoidal in shape, respectively. In addition, the martensite contents and aspect ratios correspond to the steel microstructures obtained from the experiments. 0 ' ' i • , , , , : , , , 1 0 0.05 0.1 0.15 0.2 0.25 0.3 True Far Field Strain Figure 6.3. Calculated ratios of martensite stress to ferrite stress in the dual-phase steel samples with different martensite contents and morphologies (aspect ratios). The results in Figure 6.3 can be summarized as follows: For the case o f spherical martensite, i) the stress ratio is unity at the yield point and it then increases reaching a plateau value of 2.5 after the transition strain o f approximately 2% (which is in agreement with the F E M calculations of Brockenbrough and Zok, 1995) and ii) the stress ratio is almost independent of martensite content (18-44%). On the other hand, for the case o f ellipsoidal martensite, the situation is different. It is clearly shown in Figure 6.3 that i) the 121 Chapter 6. Modelling Results and Discussion stress ratio is greater, i.e. 3-3.6 (a function of martensite aspect ratio ranging from 4.2 to 5.8 correspond to 17% to 52%> martensite, Table 5.1), ii) the stress ratio is dependent of the martensite content (in the range of 17-52%) and iii) the transition strain is greater than that of the spherical martensite, i.e. approximately 5%». 6.1.2. Discussion on Martensite Plasticity It was shown in the previous chapter that the martensite phase can deform plastically in some dual-phase steel samples depending on its content and morphology. It was also shown qualitatively that martensite plasticity has an important role in understanding the deformation and fracture behaviour o f dual-phase steels. It is now possible to examine martensite plasticity and its effect on the mechanical behaviour of dual-phase steels with reference to the numerical results obtained from the examination of the load transfer process. The effects of martensite content and morphology on martensite plasticity can also be evaluated using the results of the load transfer model. Finally, it w i l l be possible to use these observations to rationalize the unusual fracture behaviour of the low carbon dual-phase steel. The estimated stress in the martensite phase for steel samples with different martensite contents and morphologies is shown in Table 6.1. In the estimation of the martensite stress, the martensite phase was assumed to remain elastic during deformation. To calculate the stress in the martensite, the stress in the dual-phase steel was calculated first using Equation 6.4 for a given plastic strain in the ferrite phase. Since, according to Equation 6.1, the stress in the ferrite phase is known for any ferrite plastic strain, the martensite stress was calculated using the law of mixture for each dual-phase steel sample with a fixed percent martensite. Finally, the plastic strain in the dual-phase steel was determined using Equation 6.14. Table 6.1 presents the results 122 Chapter 6. Model l ing Results and Discussion for the model predictions for the stress in the martensite phase as a function of the plastic strain (i.e. at strain values of 1%, 2%, 5% and 10%). The estimated yield strength of martensite phase (from Table 5.3) as a function of its carbon concentration (or alternatively its content) is also given in Table 6.1. Table 6.1. The estimated stress in the martensite and the estimated martensite yield strength in dual-phase steel samples with different martensite contents and morphologies. Martensite Morphology V m (%) ' CTmart. (MPa) at plastic strain of D P steel: 0"y,mart. (MPa) 1% 2% 5% 10% Equiaxed 18 737 872 1111 1315 1300-1440 Equiaxed 29 762 904 1154 1352 990-1130 Network 44 807 962 1225 1405 820-960 Banded 17 812 997 1319 1594 1340-1480 Banded 30 884 1106 1489 1793 970-1110 Banded 41 950 1198 1618 1925 840-980 Banded 52 1031 1309 1761 2050 760-900 6.1.2.1. Equiaxed Martensite (in Steels with Heating Rate of 1 °C/s) The deformation behaviour o f martensite phase can now be rationalized using the results shown in Table 6.1. Plastic deformation of martensite w i l l be initiated when the stress transferred into it by the ferrite matrix reaches the yield stress of the martensite. For the dual-phase steel sample with 18%> equiaxed martensite, the model predicts that the stress in the martensite w i l l remain elastic for plastic strains in the range of 1-10% (note: at 10 %>, the stress 123 Chapter 6. Modelling Results and Discussion in the martensite is slightly greater than the lower bound of its yield stress). The possibility exists for the martensite to yield at larger strains, i.e. as the ferrite continues to work harden. A s the martensite content is increased and its yield stress decreases, strain at which plasticity can be initiated decreases. For 29% martensite, the data in Table 6.1 suggests that yielding of the martensite phase would be initiated for far field strains between 2 and 5 %. This is consistent with the experimental results presented in Table 5.2 where evidence for a small level o f martensite plasticity was experimentally measured. It is worth noting that once plasticity is initiated, the strain partitioning between the phases is controlled by the relative work hardening of the two phases (Unckel, 1937). For larger martensite contents (e.g. 44 %>), the morphology of the martensite changes as a continuous network is formed on the ferrite boundaries and the martensite flow stress is further decreased. Both of these factors favour plastic deformation and it is expected that the co-deformation of the two phases would occur from low strains (i.e. 1-2%) after which the load has been transferred into the martensite. This transition from elastic to plastic martensite is qualitatively similar to the observations of Su and Gurland (1987) although in their case, this transition occurred at higher martensite contents since their nominal chemistry was 0.12 wt.%> C , i.e. with a martensite flow stress considerably higher than the current case (0.06% C). 6.1.2.2. Banded Martensite (in Steels with Heating Rate of 100 °C/s) For the ease o f the steel samples with banded martensite morphology, the situation is different. In the case of steel sample with 17% martensite, the model predicts that yielding o f the martensite phase would occur in the plastic strain range of 5-10%. This is in disagreement with the experimental results presented in Table 5.2 where no evidence o f martensite plasticity 124 Chapter 6. Modelling Results and Discussion was observed for a far field strain of approximately 14%. There are three possibilities to explain this discrepancy, i.e. i) the stress in the martensite is over predicted, ii) the yield stress of the martensite is higher than the data from Leslie (1981) or i i i) the resolution of the experimental technique was insufficient to detect a small amount of martensite plasticity, or potentially a combination of these effects. Further work is required to definitively determine the source of this discrepancy. Nevertheless, the trend predicted by the model and the experimental results is consistent, i.e. yielding of martensite w i l l only occur after substantial plastic deformation. As the martensite content increases to 30%, yielding of martensite is predicted to occur at lower strains i.e. 1-2% which is in the transition region where load is rapidly being transferred into the martensite phase. This is consistent with the experimental results o f Table 5.2 which show substantial levels of martensite plasticity for this case. Again, it is worth emphasizing that the strain partitioning ratio between the ferrite and martensite phases when they are co-deforming w i l l be a function of their relative work hardening rates. For higher percents martensite (i.e. 41 and 52), the combination of i) a higher load transfer efficiency due to a larger aspect ratio of the islands and ii) the lower martensite yield stress indicates that martensite plasticity w i l l be initiated at applied strains of approximately 1%>. This is again consistent with the results from Table 5.2 which showed that the martensite strain was very close to the far field strain, i.e. the plastic strain in the martensite phase was measured to be 85% and 95%> of the far field strain, respectively. This lag was related to the transition strain required to transfer load into the martensite islands. 125 Chapter 6. Modelling Results and Discussion 6.1.3. Implications on Yield Strength, Tensile Strength and Work Hardening It was shown in Section 6.1.1 that the external load is transferred into the martensite islands over a range of far field strains of 2% and 5% for equiaxed and banded martensite morphologies, respectively. Since the yield strength is measured at considerably lower plastic strains (0.2% offset strain in the present case) before substantial load is transferred into the martensite phase, one may expect that martensite morphology has a limited effect on the yield strength of the dual-phase steel samples with equiaxed and banded morphologies. Thus, it may be speculated that the increase in the yield strength with increasing the martensite content (Figure 5.4) can be attributed to the strengthening effect of martensite phase on the ferrite matrix through introducing an additional density of dislocations due to the transformation strains associated with the austenite to martensite transformation, which would be expected to scale with the percent martensite formed. The negligible effect of the sample orientation on the yield strength can also be rationalized with the help of the load transfer model, since the strain necessary to transfer the load into the martensite islands is substantially greater than the yield strain and it would again be expected that yield strength would be controlled by the ferrite yield strength. The effect of martensite morphology on the tensile strength and work hardening behaviour of dual-phase steel samples is more complicated. A s shown in Section 5.2.2.3, for the steel samples with the same martensite contents, the necking occurs at lower strains when the martensite islands had banded morphology (Figures 5.8a and b), although the martensite morphology had a negligible effect on the true stress at the necking point. The lower necking strain of the steel samples with banded morphology was attributed to the fact that their work 126 Chapter 6. Modelling Results and Discussion hardening rate was considerably lower than that for the samples with equiaxed morphology. Furthermore, it can be observed in Figures 5.8 (for the samples with 17-18% and 29-30%) martensite) that the flow stress of banded samples is slightly greater than that of the equiaxed samples and this effect is more pronounced for the samples with higher martensite contents. This is because the load transfer into the martensite islands is larger in the steel samples with banded morphology and also, the load transfer is a function o f the martensite content, i.e. the greater the percent martensite, the larger is the load transfer to the martensite bands. In order to rationalize the work hardening behaviour of the steel samples (Figures 5.8a and b) and the effect of martensite content and morphology on this behaviour, martensite plasticity should be taken into account. For the steel samples with 17%> and 18%> martensite with banded and equiaxed morphologies, respectively (Figure 5.8a), the transition from elastic to plastic martensite was predicted to occur when the sample is highly deformed and hence, it is expected that martensite plasticity to be relatively small at the strains close to the necking point (for the case of sample with 17%> martensite, no martensite plasticity was found through experiment). The important point here is that although there has been little or no martensite plasticity in these steel samples, the steel sample with banded morphology showed a lower work hardening rate compared to the equiaxed steel. Martensite cracking during tensile deformation at relatively high strain values close to the necking point may be one possible explanation for this observation. It was found by examination of S E M micrographs of the steel samples with 17%> and 18% martensite that martensite cracking was more l ikely to occur in the sample with banded morphology at this strain range (Figure 6.4). Since the cracking of martensite islands 127 Chapter 6. Modelling Results and Discussion would result in a decrease in the work hardening rate of the steel, the lower work hardening rate of the sample with 17% martensite may be attributed to martensite cracking. Figure 6.4. S E M micrograph showing martensite cracking in a dual-phase steel sample with 17% banded martensite before necking point. In the case of steel samples with 29%> and 30% martensite (Figure 5.8b), however, martensite plasticity could be the main reason for lower work hardening rate of the banded sample since limited or no evidence was found concerning cracking of martensite in these samples. Both the modelling and experimental results showed that there was a greater propensity for martensite plasticity in the steel with 30%> banded martensite compared to the sample with 29%> equiaxed martensite. Since the strengthening effect of martensite in dual-phase steel microstructures decreases with its plasticity and this effect is more pronounced when martensite undergoes higher plasticity, it would be reasonable to assume that the lower work hardening rate of the banded steel is due to martensite plasticity which is much greater than in the equiaxed sample. 128 Chapter 6. Modelling Results and Discussion 6.2. Modelling Stress-Strain Response of Dual-Phase Steel Samples The stress-strain response of dual-phase steels can be predicted using the modified Eshelby model for both martensite morphologies. The modelling approach was described in Chapter 2, and for the present case, the model was developed in Section 6.1.1 in order to estimate the load transfer process for dual-phase steels with different martensite contents and morphologies. The equations resulting from modelling the load transfer process can be used for the prediction of the stress-strain relation of dual-phase steels. In the following sections, the modelling results w i l l be discussed for the case of steel samples with equiaxed and banded morphologies with and without martensite plasticity. 6.2.1. Dual-Phase Steel Samples with Equiaxed Martensite Morphology 6.2.1.1. Elastic Martensite The flow stress of a dual-phase steel with spherical martensite islands in a deformation regime where only the ferrite matrix deforms plastically can be calculated using Equation 6.4 (or Equation 2.10 in Chapter 2). A s shown previously, the flow stress o f dual-phase steel ( a ) in Equation 6.4 is only a function of the martensite content and the plastic strain in the ferrite matrix (e£) , thus, in order to determine the stress-strain relation o f a dual-phase steel with a fixed percent martensite, one can start with a given initial value for ep0. B y increasing ep0 (by a strain interval o f 0.001), the overall stress-strain response o f the steel at this deformation regime can readily be obtained. It should be mentioned that the flow stress is usually plotted as a function of the plastic far field strain, s p . Equation 6.14 can be used to calculate the plastic far field strain for any flow stress value calculated using Equation 6.4. 129 Chapter 6. Modelling Results and Discussion Figure 6.5 shows the calculated stress-strain curves of dual-phase steels with 10%, 20%, 30% and 40% martensite. A s can be observed, the modelling results illustrate the strengthening effect of spherical martensite islands in these steels. First o f al l , both the initial work hardening rate (at the true strain of 0.02 where the load transfer process is complete) and the flow stress of the ferritic steel (with the stress-strain curve obtained from the experiment) increase considerably with the addition of 10% martensite. For example, the work hardening rate (at the strain value of 0.02) for the ferritic steel was measured to be about 2.3 GPa whereas the model predicts it to be approximately 3.3 GPa when 10%o martensite is added. In addition, the calculated flow stress of the dual-phase steel with 10% martensite in the strain range of 0.1 to 0.2 is about 90 M P a greater than the flow stress of the ferritic steel in the same strain range. 1200 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 Figure 6.5. The calculated stress-strain curves of dual-phase steels with different martensite contents compared to the experimental stress-strain curve of the ferrite phase. 130 Chapter 6. Modelling Results and Discussion Quantitative examination of Figure 6.5 shows that the work hardening rate at the true strain of 0.02 (saturation strain) increases from 3.3 GPa to 4.5, 5.7 and 7 GPa, respectively, as the percent martensite increases from 10 to 20, 30 and 40. Furthermore, for the true strains between 0.1 and 0.2, an increase o f 90-95 M P a can be observed in the flow stress with each 10% increase in the martensite content. The results summarized in Figure 6.5 are qualitatively consistent with the results obtained from the experiments (Chapter 5) concerning the strengthening effect of martensite in dual-phase steels. In order to facilitate comparison with the experimental results, the model was implemented for the same martensite content as found in the dual-phase steel samples with equiaxed martensite morphology (assuming the equiaxed martensite islands to be spherical in shape). The modelling results for the steel samples with 18%>, 29% and 44%o martensite are shown in Figure 6.6 with a comparison to the experimental results. A s can be observed in Figure 6.6a, there is a good agreement between the predicted and experimental stress-strain curves of the steel sample with 18%> martensite. The difference between the estimated and measured flow stresses is less than 4%> for the entire range of the sample strain. The predicted stress-strain response of the sample with 29% martensite is illustrated in Figure 6.6b. A s can be observed, there is little difference between the predicted and measured flow stresses up to the true strains of approximately 0.04. Beyond this strain value, the two stress-strain curves start to diverge such that the stress differential reaches its maximum value of 5.5%> at the true strain of 0.15. The situation for the steel sample with higher martensite content is different. Figure 6.6c shows the predicted and experimental stress-strain curves o f the steel sample with 44%> martensite. 131 Chapter 6. Modelling Results and Discussion 1200 1000 Steel with 18% Martensite Calculated (Spherical Ms.) Experimental (Equiaxed Ms.) ™ 800 Ferrite 200 -0.05 0.1 0.15 True Plastic Strain 0.2 0.25 (a) 1200 1000 -Steel with 29% Martensite — Calculated (Spherical Ms.) — Experimental (Equiaxed Ms.) £ 800 CO Ferrite 200 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 (b) 132 Chapter 6. Modelling Results and Discussion 1200 -. 1000 H Steel with 44% Martensite — Calculated (Spherical Ms.) — Experimental (Equiaxed Ms.) CO CD £ 800 -5 2 400 -i ig 600 - Ferrite 200 -0 0 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 Figure 6.6. The calculated stress-strain curves of dual-phase steel samples with (a) 18%, (b) 29% and (c) 44% martensite (equiaxed morphology) compared to the experimental stress-strain curves. A s can be observed in Figure 6.6c, there is a substantial difference between two curves, i.e. the model has overestimated both the initial work hardening rate and the flow stress. Comparing the predicted stress-strain curve in Figure 6.6c with those in Figures 6.6a and 6.6b clearly shows that the deviation of the predicted stress-strain curve of this steel sample from the experimental curve is quite significant. This observation is not too surprising since the results in Table 6.1 show that the martensite phase in the steel with 44% martensite yields at the early stages of deformation and it is expected that it undergoes considerable plasticity as the deformation proceeds, i.e. the assumption of the model (elastic martensite) is no longer valid. In the next section, the modified Eshelby model developed for the cases with martensite plasticity w i l l be employed to examine its appropriateness in modelling the stress-strain response of dual-phase steels with elasto-plastic martensite. 133 Chapter 6. Modelling Results and Discussion 6.2.1.2. Plastic Martensite According to the results shown in Table 6.1, for the case o f steel sample with 44% equiaxed martensite, martensite plasticity may occur during the early stages o f deformation. Since the model overestimated the stress-strain response of this sample with the assumption of the elastic martensite, it is now useful to run the model in which the martensite phase is allowed to deform plastically. Martensite in dual-phase steels yields when the stress transferred into it reaches its yield stress. The condition under which martensite yielding occurs can easily be found using Equation 2.11. In this stage of deformation where both constituent phases are plastic, the experimental stress-strain relation of the martensite phase can also be expressed by the following equation (which is similar to Equation 6.1 for the ferrite phase): ° . = ° y . i +K. -a y , 1 )[ l -exp(-k 1 ej ' ) ] (MPa) (6.18) whereo y , anda s l are the yield strength and the saturation stress of martensite, respectively, k, is a constant and ep is the plastic strain in martensite phase. The fitting parameters in Equation 6.18 for the steels used in this section as the martensite phase are summarized in Table 6.2. Table 6.2. Fitting parameters in Equation 6.18 for two martensitic steels used for modelling. Steel/Condition rjy (MPa) CTS (MPa) k 0.3 wt.% Carbon1 Quenched-Tempered 898 1140 27 0.12 wt.% Carbon1 Quenched 968 1338 107 1 Steel chemistries and heat treatment conditions are given in Tables 4.2 and 5.3. 134 Chapter 6. Modelling Results and Discussion Both the ferrite matrix and the martensite phase deform plastically in this stage of deformation, thus, the flow stress of dual-phase steel is calculated by solving the following equations simultaneously (Weng, 1990): ° = ^ [K-3u4(l-P0)fb^n (6-19) a = -J r[o 1+3u*(l-K)(l-f)bJej'] (6.20) It is worth mentioning that Equation 6.19 is a modified form of Equation 6.4 in which the martensite phase is allowed to undergo plastic deformation. Since P„ is a parameter less than unity, it can clearly be observed in Equation 6.19 that martensite yielding results in a decrease in the flow stress of dual-phase steel. The overall stress-strain response of dual-phase steel with plastic martensite phase can be calculated in a same way as for the case of elastic martensite. Assuming an initial value for the ferrite plastic strain, ep0, the flow stress of ferrite, o 0 , the secant shear modulus, p.*, the Eshelby S tensor, P*, and the stress partitioning coefficients, b* and b*, w i l l be calculated. Accordingly, for a dual-phase steel with a fixed martensite content, f, there are only two unknown parameters in Equations 6.19 and 6.20, i.e. the flow stress of dual-phase steel, a , and the plastic strain in martensite phase, ejf ( a , in Equation 6.20 is only a function o f ep). Equations 6.19 and 6.20 can be solved using the solver function in the Excel program. The overall stress-strain curve of dual-phase steel can be calculated assuming an appropriate strain increment (i.e. 0.001) for the ferrite plastic strain. Figure 6.7 illustrates the predicted stress-strain curve of the dual-phase steel sample with 44% martensite. The estimated curve for this sample with elastic martensite islands is also 135 Chapter 6. Modelling Results and Discussion shown in Figure 6.7 for comparison. It should be noted that separate steel with 0.12 wt.%C and fully martensitic microstructure (Table 6.2) was considered as the martensite phase in this steel sample for which the martensite carbon concentration was estimated to be 0.13 wt.% (Table 5.3). The chemistry of this steel (slightly different from the steel investigated) and the conditions under which the heat treatment cycle was conducted are given in Table 5.3. The experimental stress-strain curve of this martensitic steel is shown in Figure 6.7 with the yield strength of 968 M P a . A s can be observed in Figure 6.7, the plastic deformation of martensite begins at the plastic true strain of 0.02 where, according to Equation 2.11, the condition for martensite yielding is reached. 1400 1200 03 Q. in in a) 1000 A 800 35 600 a> 3 400 200 0 Steel with 44% Martensite Elastic Martensite Plastic Martensite Ferrite Calculated (Spherical Ms.) Experimental (Equiaxed Ms.) 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 Figure 6.7. The calculated stress-strain curve of the steel sample with 44% plastic martensite (equiaxed morphology) compared to the experimental curve. It is clearly shown in Figure 6.7 that martensite plasticity leads to a decrease in its strengthening effect in the steel microstructure. However, the model still overestimates the 136 Chapter 6. Modelling Results and Discussion stress-strain response of this steel sample. This suggests that the lack o f martensite plasticity is not the only reason for such a significant overestimation of the stress-strain behaviour. The considerable discrepancy observed between the predicted and experimental stress-strain curves may be attributed to the fact that the microstructure of this steel is significantly different from what is assumed to be in the original Eshelby method and even in the modified version of the Eshelby approach, i.e. a composite microstructure consisting of small amount of reinforcing particles uniformly distributed in the matrix. Examination o f the microstructure of this steel sample illustrated in Figure 5.1b shows that it consists of a relatively high martensite content formed as a network at the ferrite grain boundaries. It is now useful to run the model for a case with lower martensite contents and considerable plasticity. The dual-phase steel sample with 18% martensite in the tempered condition was found suitable for this purpose. It can be shown using the estimated load transfer process described in Section 6.1.1 that a transition from elastic to plastic deformation behaviour can occur in this steel (tempered at 500 °C for 60 minutes). Using the results presented in Table 6.1, the stress in the tempered martensite in this sample can be estimated. To do this, one should note that the load transfer process in dual-phase steels does not change with tempering, hence, it would be reasonable to assume that the martensite stress in this steel sample to be similar to that in the untempered sample with 18%> martensite, i.e. 737, 872, 1111 and 1315 M P a at the plastic strains of 1%, 2%>, 5% and 10%, respectively (Table 6.1). On the other hand, the yield strength of martensite phase in this sample after tempering was measured to be 898 M P a (Table 5.3). This value is the yield strength of a separate martensitic steel in the tempered condition with carbon concentration of 0.30 wt.% which was considered the tempered martensite phase in 137 Chapter 6. Modelling Results and Discussion the steel sample containing 18% martensite with estimated carbon concentration of 0.31 wt.%>. B y comparing the martensite stress in this sample with its yield strength, one may expect that although the martensite phase may remain elastic at the early stages o f deformation, it starts deforming plastically at the later stages when the ferrite matrix is sufficiently work hardened. The predicted stress-strain response of the tempered steel sample with 18%> plastic martensite is illustrated in Figure 6.8. Two observations can be made from the modelling results in Figure 6.8: i) the modified Eshelby model has slightly overestimated the flow stress over a wide range o f plastic strain and ii) this model is unable to account for the yield point phenomenon which typically occurs in the stress-strain curve o f untempered dual-phase steels. 1400 1200 _ 1000 ro D_ — 800 in in CD 35 600 | CD i— H 400 200 0 Tempered Steel with 18% Plastic Martensite Martensite (tempered) Ferrite — Calculated (Spherical Ms.) — Experimental (Equiaxed Ms.) 0 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 Figure 6.8. The calculated stress-strain curve of the tempered dual-phase steel sample with 18% plastic martensite (equiaxed morphology) compared to the experimental curve. The modelling results in Figures 6.6 to 6.8 show that the prediction of the stress-strain response of dual-phase steels using the modified Eshelby approach is feasible. Good results 138 Chapter 6. Modelling Results and Discussion were obtained for samples with the martensite contents less than approximately 30% where i) the martensite phase is assumed to be elastic and ii) the microstructure consisted of martensite islands almost uniformly distributed in the ferrite matrix. 6.2.2. Dual-Phase Steel Samples with Banded Martensite Morphology The banded martensite islands in dual-phase steel samples produced with high heating rate were assumed to be ellipsoidal in shape with average aspect ratios greater than unity. Similar to the steel samples with equiaxed martensite morphology, the stress-strain behaviour of these samples can be predicted using the modified Eshelby model. 6.2.2.1. Elastic Martensite The flow stress of dual-phase steels with ellipsoidal martensite islands in the second stage of deformation (elastic martensite, plastic ferrite) can be calculated using Equation 2.12. In this Equation, the stress partitioning coefficient, q*, is a function of the martensite content, the components of Eshelby's S tensor and the elastic constants of both constituent phases (Appendix A ) . For a dual-phase steel with a fixed martensite content it can be shown (similar to the case of steels with spherical martensite) that q* is only a function of the ferrite plastic strain, ep0. The flow stress o f dual-phase steels with ellipsoidal martensite islands can be calculated in a similar way as described previously for the steels with equiaxed martensite morphology. Figure 6.9 shows the effect o f the aspect ratio of martensite islands on the calculated stress-strain response of a dual-phase steel with 20%> martensite. A s can be observed in Figure 6.9, the 139 Chapter 6. Modelling Results and Discussion strengthening effect of ellipsoidal martensite islands on the ferrite matrix is significant. The addition of 20% martensite into the ferritic steel microstructure results in an increase in the flow stress to an extent which depends on the aspect ratio of the martensite islands. In addition, the presence of martensite in the microstructure increases considerably the initial work hardening rate of the ferritic steel. The initial work hardening rate of steels also increases with increasing the aspect ratio of the martensite islands. 1200 -i 1000 ™ 800 CO CD 3 400 200 Steels with 20% Martensite ("Ellipsoidal") Aspect Ratio = 10 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 Figure 6.9. The calculated stress-strain curve of a dual-phase steel with 20% martensite as a function of the aspect ratio of the martensite islands. It is clear in Figure 6.9 that the presence of ellipsoidal martensite islands is more effective than the spherical martensite in strengthening the steel microstructures and this is strongly dependent on the martensite aspect ratio. This is, according to the modelling results in Figure 6.3, because the stress ratio in the steels with ellipsoidal martensite morphology is greater than that for the steels with spherical martensite islands, and for the case of steels with ellipsoidal martensite, the stress ratio increases with an increase in the martensite aspect ratio. 140 Chapter 6. Modelling Results and Discussion The effect o f martensite content on the predicted stress-strain behaviour of dual-phase steels with a fixed martensite aspect ratio of 5 is shown in Figure 6.10. B y comparing the predicted curves in Figure 6.10 with those shown in Figure 6.5 for the case o f spherical martensite morphology, it can be observed that the flow stresses (at the true strains of 0.1-0.2) for the steel samples with ellipsoidal martensite islands are higher, i.e. an increase of 130-140 M P a for every 10% increase in the percent martensite compared to 90-95 M P a for the case of spherical martensite islands. 1200 1000 H ™ 800 IS 600 i _ CO CD 2 400 200 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 Figure 6.10. The calculated stress-strain curve of. dual-phase steels with martensite aspect ratio of 5 as a function of the martensite content. The predicted stress-strain curves of steel samples containing 17%>, 30%o and 41%> banded martensite with average aspect ratios of 4.2, 5.1 and 5.5, respectively, are shown in Figure 6.11. As can be observed in Figure 6.11a, there is a good agreement between the estimated and experimental stress-strain curves of the steel sample with 17%> martensite. 141 Chapter 6. Modelling Results and Discussion Steel with 17% Martensite (Aspect Ratio = 4.2) Ferrite Calculated (Ellipsoidal Ms.) Experimental (Banded Ms.) 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 (a) Steel with 30% Martensite (Aspect Ratio = 5.1) Ferrite — Calculated (Ellipsoidal Ms.) — Experimental (Banded Ms.) 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 (b) 1 4 2 Chapter 6. Modelling Results and Discussion 1200 Steel with 41% Martensite (Aspect Ratio = 5.51 Ferrite 200 1 Calculated (Ellipsoidal Ms.) Experimental (Banded Ms.) 0 -I 0 0.05 0.1 0.15 True Plastic Strain 0.2 0.25 (c) Figure 6.11. The calculated stress-strain curves of the dual-phase steel samples with (a) 17% martensite (aspect ratio of 4.2), (b) 30% martensite (aspect ratio of 5.1) and (c) 41% martensite (aspect ratio of 5.5) compared to the experimental curves. For the case of steel sample with 30% martensite (Figure 6.11b), the calculated and experimental curves coincide up to the plastic strain of approximately 0.02, however, they start to diverge beyond this strain value. The stress differential between the stress-strain curves increases significantly as the deformation proceeds. The overestimation of the flow stress in the steel sample with 30%> martensite may be partly due to (similar to the case of sample with 44% equiaxed martensite) the invalid assumption (elastic martensite) that has been made in the model. Both the modelling results (Table 6.1) and the experimental results (Table 5.2) showed that the martensite phase in the steel sample with 30% banded martensite is plastic suggesting that the flow stress of this steel is lower than that for the case where martensite is elastic. 143 Chapter 6. Modelling Results and Discussion There is, however, a substantial difference between the calculated and measured flow stresses in the case of steel sample with 41% martensite (Figure 6.11c). It can be observed in Figure 6.11c that the stress differential increases as the far field strain increases reaching its maximum value of approximately 335 M P a at the true strain of 0.12. A n important observation here is that the overestimation of the flow stress is even greater than that for the steel sample with 44%o martensite with equiaxed martensite morphology. For the sample with 41%> martensite, the difference between the calculated and measured flow stresses at the true strain of 0.1 is approximately 320 M P a whereas the flow stress of the sample with 44% equiaxed martensite was predicted to be about 180 M P a greater than the measured value. Similar to the case of steel samples with equiaxed martensite morphology, part of this discrepancy is attributed to the strengthening effect of the martensite phase which decreases as a result of its plastic deformation. According to the experimental results shown in Table 5.2, the plastic deformation of martensite in this steel sample measured at the necking point is considerably high, i.e. the plastic strain of approximately 0.1 or 85% of the far field strain. 6.2.2.2. Plastic Martensite A s shown in the previous section, the modified Eshelby approach was found to be an appropriate method for modelling the stress-strain response o f the dual-phase steel sample containing 17%> banded martensite with no martensite plasticity. For the case of samples with higher martensite contents, e.g. the sample with 41%> martensite, where considerable martensite plasticity was measured (Table 5.2), the model is not applicable. There was a similar situation for the samples with equiaxed martensite morphology (Figure 6.6c) where martensite plasticity was proposed to be one possible reason for lowering the strengthening effect of martensite 144 Chapter 6. Modelling Results and Discussion islands. The application of the model for the 44% martensite steel (equiaxed) assuming martensite plasticity resulted in a decrease in the flow stress (Figure 6.7) but the difference between the calculated and measured flow stresses was still substantial. This significant discrepancy was then attributed to the steel microstructure which consisted of martensite islands formed mainly as a network at the ferrite grain boundaries. For the case of steel sample with 41%) banded martensite where the overestimation of the flow stress was even greater (Figure 6.1 l c ) , one may expect to make a similar observation. 6.2.3. Estimated Plastic Strain in Martensite It is now useful to compare the estimated plastic strain of martensite in a dual-phase steel sample with the corresponding experimental value given in Table 5.2. There are two sets of modelling results for the steel samples with 18%> martensite (quenched-tempered) and 44%> martensite (as-quenched) both with equiaxed morphology, and as it can be observed in Table 5.2, the only data available from the experiment is for the sample with 29% martensite (equiaxed morphology). However, the martensite plastic strain in the quenched-tempered steel sample with 17% banded martensite is available which can be assumed to be almost the same as that in the sample with 18% equiaxed martensite. This is a reasonable assumption because the carbon concentrations of martensite in these steel samples are almost the same and hence, they have the same yield strength (Table 5.3). Since the tempered martensite islands in both samples are soft enough to undergo considerable plasticity during deformation, one may assume that martensite plasticity in these samples is independent of the morphology. According to Table 5.2, the martensite plastic strain (axial) in the quenched-tempered steel sample with 17%> banded martensite at the true far field strain of 0.15 is 0.06. The martensite 145 Chapter 6. Modelling Results and Discussion plastic strain in the quenched-tempered sample with 18% equiaxed martensite, on the other hand, was estimated to be approximately 0.07 (this strain value was determined from Equations 6.19 and 6.20 which were used for estimation of the stress-strain response o f this steel sample). This shows that there is a relatively good agreement between the estimated and experimental results and the modified Eshelby model can successfully be used to predict the martensite plastic strain in a dual-phase steel sample in which the martensite phase is plastic. 6.2.4. Conclusion on Modelling Stress-Strain Behaviour In summary, the modified Eshelby model can successfully be used for the prediction of the stress-strain response of dual-phase steel samples with equiaxed morphology and martensite contents less than 30%. For the case of samples with higher martensite contents, e.g. the steel sample with 44%> martensite where the microstructures were considerably less uniform with regard to the martensite islands distribution, the modified Eshelby model was found to be inadequate to accurately predict the stress-strain behaviour. For the case of steel samples with plastic martensite, the modified Eshelby model was employed for both low and high martensite contents (18%> and 44%, respectively). Although the model slightly overestimated the flow stress of the steel sample with 18%> martensite, the estimated flow stress of the sample with 44%> martensite was substantially greater than the measured value. In the case o f steel samples with banded martensite morphology, the model was found to be appropriate for prediction of the stress-strain relation of the steel with 17%> martensite. For the steel sample with 30% martensite, the prediction from the model was very good at plastic 146 Chapter 6. Modelling Results and Discussion strains below 5%, however, there was an increasingly larger deviation for higher levels of plastic strain. Accordingly, it appears that one needs to use another modelling approach in order to predict the large strain stress-strain response of dual-phase steels with higher martensite contents and both equiaxed and banded morphologies. This approach, e.g. finite element model, should have the capability to take into account the detailed characteristics of the steel microstructures with regard to the non-homogeneity of their martensite islands distribution. Finally, it is useful to make a few comments on the industrially produced dual-phase steels and the modelling results that one might expect to achieve in these steels. A s indicated in the literature (Kot and Morris , 1979; Kot and Bramfitt, 1981), the martensite content in industrially produced dual-phase steels is typically in the range of 5-20%. According to the modelling results in the previous sections, the modified Eshelby model is an appropriate method for the prediction of the stress-strain behaviour of dual-phase steel samples containing 17-18%) elastic martensite with two different morphologies (equiaxed and banded). This shows that the modified Eshelby model is applicable to those dual-phase steels which are produced in the industrial scale with similar microstructures as the dual-phase steel samples in this study. 147 Chapter 7 - Summary, Conclusions and Future Work 7.1. Summary In this study, an attempt was made to i) produce dual-phase steel samples with a variety of martensite contents and morphologies using a commercial low carbon (0.06 wt.%) steel as the starting material through intercritical annealing followed by quenching, i i) evaluate the tensile deformation and fracture behaviour of the intercritically annealed dual-phase steel samples and to determine the critical processing parameters, i i i) predict the stress-strain response of the dual-phase steel samples using the modified Eshelby model and iv) model the void formation process in the dual-phase steel samples during tensile fracture. The tensile properties were qualitatively rationalized with an emphasis on martensite plasticity which was found to occur in some dual-phase steel samples. The micromechanical analysis developed based on the modified Eshelby model was employed for quantitative rationalization of the experimental results on the deformation and fracture behaviour of the dual-phase steel samples. The main results of this investigation can be summarized as follows: 1) The heating rate to the intercritical annealing temperature had a significant effect on the martensite content and morphology. At the same intercritical temperature, the high heating rate (100 °C/s) annealing cycle resulted in higher martensite content. A steel microstructure with roughly equiaxed martensite morphology was obtained through intercritical annealing using low heating rate whereas high heating rate annealing produced banded martensite morphology. For the steel samples produced with low heating rate (equiaxed morphology) and martensite 148 Chapter 7. Summary, Conclusions and Future Work contents greater than approximately 30%, an almost complete network of martensite islands formed on the ferrite grain boundaries. In the case of steel samples annealed with high heating rate (banded morphology), on the other hand, the elongated martensite islands formed parallel to the initial rolling direction of the steel with the average aspect ratio and thickness which increased as the percent martensite increased. 2) The dependence of the yield strength of the steel samples on martensite content was lower than that reported in the literature for the steels with higher carbon concentrations. This is, according to the literature, because the strengthening of ferrite matrix in higher carbon content steels is enhanced due to larger misfit strain associated with higher carbon martensite. 3) The martensite morphology (equiaxed and banded) and the sample orientation (longitudinal and transverse directions relative to the initial rolling direction of the steel) showed a negligible effect on the yield strength of the dual-phase steel samples. For both martensite morphologies, this was attributed to the load transfer process to the martensite phase which was shown to complete well beyond the 0.2%> plastic strain as the basis for the measurement of yield strength. 4) The ultimate tensile strength of the dual-phase steel samples was found to be almost independent of the martensite morphology. 5) It was shown that an increase in the martensite content can effectively lower its strength (through its effect on lowering the martensite carbon concentration) with a potential to result in a transition from elastic to plastic deformation behaviour. 6) The martensite phase in some dual-phase steel samples was found to undergo plastic deformation during tensile straining. Martensite plasticity increased with an increase in the martensite content such that for the case of high martensite content steel samples (e.g. the 149 Chapter 7. Summary, Conclusions and Future Work sample with 52% martensite), almost no strain partitioning was found between the martensite and ferrite phases during deformation. Martensite plasticity was greater for the steel samples with banded morphology as the load transfer to the elongated martensite islands was more effective. The tempering process effectively reduced the martensite strength and hence, resulted in a transition in the deformation behaviour of martensite phase from elastic to plastic. 7) The low carbon dual-phase steel investigated in this study showed unusual fracture properties during tensile deformation compared to the conventional dual-phase steels in the literature, i.e. both the fracture stress and the fracture strain increased with an increase in the martensite content. The unusual fracture behaviour of this dual-phase steel was attributed to martensite plasticity through its effect on lowering the void nucleation rate in the neck region resulted in an enhanced combination of the fracture stress and strain. 8) The work hardening rate of the dual-phase steel samples was found to be dependent on the martensite morphology. For the low martensite contents (i.e. 17-18%), the work hardening rate of the steel sample with banded morphology was lower than that for the equiaxed sample. Since the martensite phase in these samples was elastic, the lower work hardening rate of the steel sample with banded morphology was attributed to martensite cracking during deformation. For the case of steel samples with higher martensite contents (i.e. 29-30%>) where limited or no evidence was found for cracking of martensite, the onset of martensite plasticity at the uniform deformation regime was found to be responsible for the lower work hardening rate of the banded sample. 9) The modified Eshelby approach was found to be appropriate for modelling the stress-strain response of the dual-phase steel samples with equiaxed morphology when the martensite 150 Chapter 7. Summary, Conclusions and Future Work content was less than approximately 30%. The model slightly overestimated the flow stress when considering martensite plasticity in the steel sample with equiaxed morphology and 18% martensite. The modified Eshelby model was successfully employed for the prediction of the stress-strain behaviour of the steel sample with 17%> banded martensite. For the case of steel sample with 30%> banded martensite, the model prediction was very good at plastic strains below 5%, however, there was an increasingly deviation for higher strain values. 7.2. Implications on Steel Design and Processing Route According to the results of the present investigation, the overall carbon concentration of the steel, the martensite content and the processing route should be chosen properly in order to achieve the best combination of the strength and fracture properties. For the steel investigated with the overall carbon concentration of 0.06 wt.%, the intercritical annealing with low heating rate (1 °C/s) resulted in superior uniform strain, fracture stress, fracture strain and R-value. In addition, an increase in the martensite content resulted in an enhanced strength and fracture behaviour in dual-phase steel samples produced with both the low and high heating rates (in comparison with the conventional dual-phase steels in the literature with elastic martensite in which the ductility decreases as the martensite content increases, the fracture behaviour of the dual-phase steel samples in this study was unusual as both their fracture stress and fracture strain increased with increasing martensite content). In order to design a dual-phase steel with a given chemistry produced through a particular processing route, one should consider the actual application the steel might be chosen for, since different applications require a combination of different properties. For example, in the 151 Chapter 7. Summary, Conclusions and Future Work automotive industry where the demand is for high strength steels with high energy adsorption capacity at a low cost to be used as the automobile body, the low carbon dual-phase steel investigated in this study with relatively high martensite content may be a good option. Using high martensite contents in this steel with considerable plasticity during deformation leads to high strength and improved fracture properties. However, some additional considerations should also be taken into account in designing the chemistry of dual-phase steel and its associated processing routes. On the industrial scale, the processing routes typically involve hot rolling with coiling at moderately high temperature, followed by cold rolling and then intercritical annealing (usually associated with galvanizing). The carbon concentration of the steel here is critical. After hot rolling and coiling, one obtains a ferrite-pearlite microstructure with the carbon concentration affecting the amount of pearlite. The spatial distribution of carbon can be mechanically modified by varying the level of cold reduction (since the carbon is mostly found in the pearlite regions and the pearlite colonies co-deform with the ferrite matrix during cold rolling). The annealing conditions (particularly the heating rate to the intercritical annealing temperature) affect the morphology of austenite formation which subsequently determines the final martensite morphology. For rapid heating where austenite formation occurs before ferrite recrystallization, the morphology of the austenite phase w i l l be linked to the initial ferrite-pearlite morphology in the cold rolled steel. The combination of the steel carbon concentration and the intercritical annealing temperature wi l l determine the local carbon concentration in the martensite and its content and thus, control the possibility for martensite plasticity during subsequent ambient temperature deformation. 152 Chapter 7. Summary, Conclusions and Future Work The role o f substitutional alloying elements such as M o and Cr are also critical. These elements clearly delay the decomposition of austenite making it easier to form martensite upon cooling, however, they also control whether the ferrite phase recrystallizes • during heating (depending on the heating rate) to the intercritical annealing temperature prior to austenite formation. This w i l l then affect the final morphology and spatial distribution of martensite. In summary, it appears that it is possible to design combinations of steel chemistries and processing histories to tailor the microstructures and mechanical properties to the desired application. 7.3 Future Work Based on both the experimental and modelling results of this investigation, some additional investigations would be helpful to further understand the mechanical properties of low carbon dual-phase steels. The main areas for focus are: 1) Measurement of martensite plasticity in the steel samples with equiaxed morphology since the procedure used in this work was sensitive to the steel microstructures with fine particles and consequently, led to inaccurate results (especially for the low martensite steels) in most cases. Other techniques (such as Image Correlation System) might be appropriate for this purpose. 2) Evaluation o f the onset of martensite plasticity in dual-phase steel samples using the Bauschinger effect since the transition from elastic to plastic deformation behaviour of martensite results in a reduced strengthening effect of martensite. This should lead to a measurable decrease in the back stress generated in the ferrite matrix as a result of the presence of elastic and non-deformable martensite. 153 C h a p t e r 7. S u m m a r y , C o n c l u s i o n s a n d F u t u r e W o r k 3) E v a l u a t i o n o f t h e m i c r o s t r u c t u r a l t e x t u r e i n t h e d u a l - p h a s e s t e e l i n v e s t i g a t e d a n d t h e e f f e c t o f m a r t e n s i t e c o n t e n t a n d m o r p h o l o g y o n t h e t e x t u r e . T h i s w o u l d b e h e l p f u l f o r t h e r a t i o n a l i z a t i o n o f t h e R - v a l u e o f t h i s d u a l - p h a s e s t e e l w h i c h w a s f o u n d t o b e d e p e n d e n t o n t h e m a r t e n s i t e m o r p h o l o g y , b u t a l m o s t i n d e p e n d e n t o n t h e m a r t e n s i t e c o n t e n t . 4) M o d e l l i n g t h e s t r e s s - s t r a i n b e h a v i o u r o f d u a l - p h a s e s t e e l s w i t h d i f f e r e n t m a r t e n s i t e c o n t e n t s i n w h i c h m a r t e n s i t e p h a s e c a n d e f o r m p l a s t i c a l l y d u r i n g s t r a i n i n g . T h e f i n i t e e l e m e n t m e t h o d a n d t h e s h e a r l a g m o d e l a r e t h e p o s s i b l e o p t i o n s f o r t h i s p u r p o s e . 154 References Ahmad, E . , Manzoor, T., A l i , K . L . , Akhtar, J. I., 2000, J. Mat. Engng. Per/., V o l . 9, pp. 306-310. Al-Abbasi , F. M . , Nemes, J. A . , 2003, Int. J. Solids Struct., V o l . 40, pp. 3379-3391. 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According to Qiu and Weng (1991), the components of Eshelby's tensor, S- j k l , for an ellipsoidal inclusion are: 1 '1111 2 ( l - v * ) l - 2 v ! + 3 a 2 - 1 0 -s .2 0 / - 1 l - 2 v * + 3a 2 o 2 ( A - l ) ^2222 ~~ ^3333 8 ( l - v s 0 ) a 2 - l 4 ( 1 - 0 a 2 1 + • l - 2 vs0 j 0 4 ( a 2 - l ) g (A-2) O S Q S i 5 2 2 3 3 — ^3322 a 4 ( l - v s 0 ) 2 ( a 2 - l ) \-2v\ + j 0 4 ( a 2 - l ) g (A-3) ^2211 ~~ ^331 2 ( 1 - 0 a 2 - 1 4 ( l - v * ) a 2 1 + • 3a 2 a -1 g (A-4) O S Q S ^1122 - ^1133 1 2 ( l - < ) l - 2 v ! + 1 0 ~ 2 + -1 a 2 - l j 2 ( l - v * ) l - 2 v ! + ° 2 ( a 2 - l ) g (A-5) 1 a '2323 4 ( 1 - 0 2 ( a 2 - l ) • + l - 2 v 0 4 ( a 2 - l ) (A-6) O S Q S ^1212 — "1313 — 1 4 ( l - v s 0 ) l - 2 v ! ' -a 2 + l 1 o 2 a - l 2 1 - 2 V - * " + 1 ) o 2 1 a - l (A-7) where v* is the secant Poisson's ratio o f the matrix, a is the aspect ratio of the inclusion and g is given by the following equation for the inclusions with aspect ratios greater than 1: 164 Appendix A g = , 2 , , 3 / 2 ^ - 1)" 2 - cosh-a] (A-8) (a - 1 ) 2. The stress partitioning coefficient for ellipsoidal inclusions. The stress partitioning coefficient, q*, in Equation 2.12 of the text can be calculated from the following equation (Qiu and Weng, 1991; Bhattacharyya and Weng, 1994): q s „ = l - f 9 9 « s _1 1 ? C S _1 i I Zb 1 2 l 2 + i ^ 2 3 2 3 L [ ( S ^ 1 2 2 - S 2 2 3 3 ) ( 2 a 3 - a 4 +a 5a) + 5 2 S ; 2 1 2 + ^ / ( ^ - ^ ) 3 2 S 2 M , + j i ' 0 / G i I - u ' 0 ) 15a 2 2 3 3 A 3 2 ( S i m - S s 2 2 1 1 - l ) ( a , +a 2 ) + (S[ 1 2 2 - S s 2 2 2 2 +l)(2a 3 - a 4 - a s a ) ] (A-9) where f is the volume fraction of the inclusion, p., is the inclusion shear modulus, LI* is the secant shear modulus of the matrix and a, , a 2 , a 3 , a 4 , a 5 and a are given by the following equations: a, = 6 ( K , - K 0 ) G I , - ^ ) ( S S 2 2 2 2 + S 2 2 3 3 - 1 ) - 2 ( K 0 U , - K . u ^ + eK.Cu,-\i'0) (A-10) a 2 = 6 ( K , - K 0 ) ( L I , -n*)S; i 3 3 + 2 ( K 0 U , - K , ^ ) ( A - l l ) a 3 = - 6 ( K , - K 0 ) ( p , - ^ ) S 3 3 1 1 - 2 ( K 0 H , - K , ^ ) (A-12) a 4 = 6 ( K , - K 0 ) ( U . , - n ' ) ( S J N I - 1 ) + 2 ( K 0 H , - K ^ ^ ^ K , - K 0 ) (A-13) a 5 = l/[ss3322 - S * 3 3 3 + 1 - M J 1 , (A-14) a = 6 ( K , - K O ) 0 I , - j i*) [ (2SJ 1 3 ,S^ L L - ( S ; U I - 1 ) ( S ^ 2 2 + S * 3 3 3 -1)] + 2 ( K 0 U , - K , ^ 0 ) [ 2 ( S ; , 3 3 + S S 3 3 1 1 ) + (S,„ , -SL22 - S ; 3 3 3 ) ] - 6 ^ ( 1 1 , -M4)(S; n i - 1 ) - 6 H , ( K , - K 0 ) ( S S 2 2 2 2 + S S 2 2 3 3 - l ) - 6 K , m (A-15) In these Equations, K 0 and K , are the bulk moduli of the matrix and inclusion, respectively. In order to calculate the plastic strain in dual-phase steel ( e p ) using Equation 6.14 ( e p =[1/E S - 1 / E ] a ) , the secant Young's modulus of the steel ( E s ) should be determined first. 165 Appendix A The secant Young's modulus o f dual-phase steel is directly related to its shear modulus through the following equation in which v s ( = v 0 = v, = 0.3) is the-secant Poisson's ratio of the steel: E s ix s = ( A - 1 6 ) 2( l + v s ) The secant shear modulus of the steel, p. s, can be calculated using the following equation: H . 1 + F P ^ (A-17) where p* is expressed as: p* = - - + [2(a, + a, - a , ) + a. +a,a] (A-18) 166 

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