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Strengthening mechanisms in some heat-treated low alloy steels Malik, Lalit Mohan 1972

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STRENGTHENING MECHANISMS IN SOME HEAT-TREATED LOW ALLOY STEELS BY LALIT MOHAN MALIK B. Tech. (Hons.), Indian I n s t i t u t e of Technology, Kharagpur, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of METALLURGY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1972 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and s tudy . I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood that copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date i i ABSTRACT The microstructures and substructures of several heat-treated low a l l o y steels have been studied using x-ray l i n e p r o f i l e a n a l y s i s , electron micro-scopy and electron d i f f r a c t i o n . Y i e l d strengths of the steels at 20°C have been determined a f t e r d i f f e r e n t heat treatments, and have been related to structure i n terms of theories for various strengthening mechanisms. iFor tempered low-carbon and medium-carbon martensites, d i s l o c a t i o n sub-structures o r i g i n a t i n g with the martensite transformation were found to be responsible for a s i g n i f i c a n t part of the strength, i n contrast with the findings of the previous i n v e s t i g a t o r s . The balance of that part of the y i e l d strength which varied with tempering temperature i n these materials was a t t r i b u t e d to dispersed cementite p a r t i c l e s . When the d i s l o c a t i o n sub-structure took the form of subgrains, as i n heavily tempered specimens, the data were consistent with a Langford-Cohen model of subgrain strengthening, i n which strengthening i s rela t e d to the r e c i p r o c a l of subgrain s i z e . An age-hardening copper-bearing s t e e l was also studied i n the aged, and i n the cold worked-and-aged conditions. In the peak-aged state, almost a l l the copper was present as coherent c l u s t e r s , which produced strengthening associated with the necessity to shear them and produce new i n t e r f a c e s . When the s t e e l was overaged, incoherent p r e c i p i t a t e s were present, but t h e i r con-t r i b u t i o n i s shown to be i n s i g n i f i c a n t i n view of the large r a t i o of t h e i r spacing to t h e i r diameter. In cold-worked-and-aged specimens, the substruc-ture contribution to strength was s i g n i f i c a n t , but was e s s e n t i a l l y indepen-dent of aging time. i i i TABLE OF CONTENTS Page 1. INTRODUCTION 1 1.1 Nature of the Problem 1 1.2 Strengthening Mechanisms-Theory 5 1.2.1 S t r a i n Hardening ( D i s l o c a t i o n - D i s l o c a t i o n Inter-action 5 1.2.2 S o l i d Solution Strengthening (Dislocation-Solute Interaction) 5 1.2.3 Age Hardening (Dislocation-Coherent P r e c i p i t a t e Interaction) 6 1.2.4 Dispersion Hardening (Dislocation-Incoherent P a r t i c l e Interaction) 8 1.2.5 Grain and Subgrain Boundary Hardening 10 1.3 The Iron-Copper System 15 1.4 Scope of the Present Work 18 2. EXPERIMENTAL PROCEDURE 20 2.1 Materials Supply and Preparation 20 2.2 Specimen Preparation and Heat Treatment 20 2.2.1 Medium Carbon Steel (0.42% C, 1.1% Mn) 20 2.2.2 Low Carbon Steel (0.11% C, 0.9% Mn) 22 2.2.3 Iron-Copper-Nickel A l l o y 23 2.2.4 Ferrovac E Iron 25 2.3 Tensi l e Testing and Hardness Testing Procedures 25 2.3.1 Medium Carbon Steel (0.42% C, 1.1% Mn) 25 2.3.2 Low Carbon Steel (0.11% C, 0.9% Mn) 27 2.3.3 Fe-Cu-Ni A l l o y and Ferrovac E Iron 28 i v Page 2.4 Metallography 2.4.1 O p t i c a l Microscopy 28 2.4.2 Electron Microscopy 28 2.4.3 Determination of Volume Fraction of P r e c i p i t a t e s .. ,30 2.4.4 Ca l c u l a t i o n of I n t e r p a r t i c l e Spacing 30 2.4.5 Measurement of Grain Size and Subgrain Size 32 2.5 X-Ray D i f f r a c t i o n 33 3. X-RAY LINE BROADENING ANALYSIS 38 3.1 P r i n c i p l e s Underlying X-Ray Line Broadening Analysis 38 3.2 Stokes Correction 40 3.3 Evaluation of Domain Size and Nonuniform L a t t i c e Strains . 45 3.4 Computer Program 48 3.5 Williamson and Smallman Analysis 49 4. RESULTS AND DISCUSSION 53 4.1 Tensile and Hardness Tests 53 4.2 Replica Electron Microscopy 56 4.3 X-Ray D i f f r a c t i o n 63 4.3.1 General 63 4.3.2 Discussion of X-Ray Results i n Tempered Martensites (Carbon Steels) 64 4.3.3 Discussion of X-Ray Results f o r Pure Iron and Fe-Cu-Ni A l l o y 73 4.3.4 R e l i a b i l i t y of the X-Ray Line Broadening Analysis . 76 4.4 Transmission Electron Microscopy of Tempered Iron-Carbon Martensites 77 4.4.1 S t r u c t u r a l Changes due to Tempering of Medium Carbon Martensite 77 V 4.4.2 S t r u c t u r a l Changes due to Tempering of Low Carbon Martensite 82 4.4.3 Subgrain Size 85 4.4.4 Selected Area Electron D i f f r a c t i o n 85 4.5 Transmission Electron Microscopy of the Fe-Cu-Ni A l l o y and Pure Iron 95 4.5.1 Observations of Structure 96 4.5.2 Analysis of Structure 101 4.6 Strengthening Mechanisms i n Tempered Martensites 109 4.6.1 Contributions from Dispersed Cementite P a r t i c l e s .. ,109 4.6.2 Contribution from Random Dislocations 110 4.6.3 Contribution from Subgrain Boundaries 113 4.6.5 Comparison with Previous Work 124 4.7 Strengthening Mechanisms i n the Copper Bearing Steel 126 4.7.1 Strength i n the Underaged Condition 126 4.7.2 Contributions of Incoherent e P r e c i p i t a t e s 126 4.7.3 Contribution of Coherent Clusters and Substructure. 129 4.7.4 Matrix Strength 135 5. SUMMARY AND CONCLUSIONS 138 APPENDIX A 141 REFERENCES 144 v i LIST OF FIGURES Figure Page 1 Fe-Cu Phase Diagram (Speich et al"*^) 17 2 Flow Sheet f o r Specimen Preparation i n the Fe-Cu-Ni A l l o y . 24 3 Flow Sheet for Specimen Preparation i n Ferrovac E Iron ••• 26 4 Stokes Correction f o r Instrumental Line Broadening < (Warren? 6) 41 5 Y i e l d Strength vs Tempering Temperature f o r the Medium and Low Carbon Tempered Martensites 57 6 Y i e l d Strength vs Aging Time f o r the Fe-Cu-Ni A l l o y 57 7 Replica Electron Micrograph of the Medium Carbon Martensite Tempered at 350°C 59 8 Replica Electron Micrograph of the Medium Carbon Martensite Tempered at 500°C 59 9 Replica Electron Micrograph of the Medium Carbon Martensite Tempered at 700°C 60 10 • Replica Electron Micrograph of the Low Carbon Martensite Tempered at 300°C 60 11 Replica Electron Micrograph of the Low Carbon Martensite Tempered at 450 C 61 12 Replica Electron Micrograph of the Low Carbon Martensite Tempered at 600 C 61 13 Edge-to-Edge P a r t i c l e Spacing, D, vs Tempering Temperature fo r the Medium and Low Carbon Martensite 62 14 V a r i a t i o n of L a t t i c e S t r a i n with L a t t i c e Distance i n Medium Carbon Martensite Tempered at D i f f e r e n t Temperatures 68 15 V a r i a t i o n of Domain Size C o e f f i c i e n t with L a t t i c e Distance i n Medium Carbon Martensite Tempered at D i f f e r e n t Temperatures 68 16 V a r i a t i o n of L a t t i c e S t r a i n with L a t t i c e Distance i n Low Carbon Martensite Tempered at D i f f e r e n t Temperatures 69 17 V a r i a t i o n of Domain Size C o e f f i c i e n t with L a t t i c e Distance i n Low Carbon Martensite Tempered at D i f f e r e n t Temperatures 69 v i i Figure Page 18 V a r i a t i o n of L a t t i c e S t r a i n with L a t t i c e Distance i n Fe-1.8% Cu-1.3% Ni A l l o y (Series B) and Cold Rolled Ferrovac E Iron 70 19 V a r i a t i o n of Domain Size C o e f f i c i e n t with L a t t i c e Distance i n Fe-1.5% Cu-1.3% Ni A l l o y (Series B) and Cold Rolled Ferrovac E Iron 70 20 D i s l o c a t i o n Density vs Tempering Temperature i n the Medium and Low Carbon Tempered Martensites 72 21 Transmission Electron Micrograph of the Medium Carbon Martensite Tempered at 250 C 79 22 Transmission Electron Micrograph of the Medium Carbon Martensite Tempered at 400 C 79 23 Transmission Electron Micrograph of the Medium Carbon Martensite Tempered at 550°C 80 24 Transmission E l e c t r o n Micrograph of the Medium Carbon Martensite Tempered at 600°C 80 25 Transmission E l e c t r o n Micrograph of the Medium Carbon Martensite Tempered at 700 C 81 26 Transmission Electron Micrograph of the Low Carbon Martensite Tempered at 300°C 81 27 Transmission Electron Micrograph of the Low Carbon Martensite Tempered at 400 C 83 28(a) Transmission Electron Micrograph of the Low Carbon Martensite Tempered at 500°C 83 28(b) Transmission Electron Micrograph of the Low Carbon Martensite Tempered at 550 C 84 29 Transmission Electron Micrograph of the Low Carbon Martensite Tempered at 700°C 84 30(a) Electron D i f f r a c t i o n Pattern of the Medium Carbon Martensite Tempered at 600°C 90 30(b) Selected Area f o r the Pattern i n Figure 30(a) 90 31(a) Electron D i f f r a c t i o n Pattern f o r the Medium Carbon Martensite Tempered at 650 C 91 31(b) Selected Area f or the Pattern i n Figure 31 (a) 91 32 Electron D i f f r a c t i o n Pattern from an Area surrounding the sub-boundary, marked by an arrow i n F i g . 25(Medium Carbon Martensite, Tempered at 700°C) 92 v i i i Figure Page 33 Electron D i f f r a c t i o n Pattern from the Area Shown i n Figure 28(a) (Low Carbon Martensite Tempered at 500°C) ... 92 34(a) Electron D i f f r a c t i o n Pattern f o r the Low Carbon Martensite Tempered at 650°C 93 34(b) Selected Area f o r the Pattern i n Figure 34(a) 93 35 Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged f or 100 hours (Series A) 97 36; Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Solution-Treated and Cold-Rolled 50 percent 98 37 Transmission Electron Micrograph of Ferrovac E Iron, Quenched and Cold-Rolled 50 percent 98 38 Electron D i f f r a c t i o n Pattern from the Area Shown i n Figure 36 (Cold-Rolled Fe-Cu-Ni Alloy) 99 39 Electron D i f f r a c t i o n Pattern from the Area Shown i n Figure 37(Cold-Rolled Ferrovac E Iron) 99 40' Transmission Electron Micrograph of the. Fe-Cu-Ni A l l o y , Aged f o r 10 hours, (Series B) 102 41 Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged f or 100 hours (Series B) 102 42(a) Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged f o r 10 hours (Series B) 103 42(b) Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged f or 10 hours (Series B) 103 • 43(a) Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged for 100 hours (Series B) 104 43(b) Electron D i f f r a c t i o n Pattern from the Area Shown i n Figure 43(a) 104 44 Plot of l o g ( P a r t i c l e Diameter) vs Aging Time f or the Fe-Cu-Ni Alloys 108 45 Plot of Percent P r e c i p i t a t i o n vs Aging Time f or the Fe-Cu-Ni A l l o y s 108 -1/2 46 P l o t of (a -a ) vs t for the Medium and Low Carbon y.s. or Tempered Martensite '. 116 47 Plot of (a -a ) vs t for the Medium and Low Carbon y.s. or Tempered Martensites 116 Figure Page 1/2 48 Plot of a vs f . f o r the Fe-Cu-Ni A l l o y (Series A). 131 y.s. 2 J 1/2 49 Plot of (a -a ) vs f„ ' for the Fe-Cu-Ni A l l o y y.s. t 2 J (Series B) 131 X LIST OF TABLES Table Page I Composition of Materials Investigated 21 II T e n s i l e Results for the Medium Carbon and the Low Carbon Tempered Martensites 54 III T e n s i l e Results f o r Fe-Cu-Ni A l l o y and Ferrovac E Iron ... 55 IV Dispersion Parameters f o r the Medium Carbon and the Low Carbon Tempered Martensites 58 V L a t t i c e S t r a i n s , Domain Sizes, D i s l o c a t i o n Densities and Configurations for Tempered Medium Carbon Steel 65 VI L a t t i c e S t r a i n s , Domain Sizes, D i s l o c a t i o n Densities and Configurations f o r Tempered Low Carbon Steel .66 VII L a t t i c e S t r a i n s , Domain Size, and D i s l o c a t i o n Configuration i n Fe-Cu-Ni A l l o y (Series B) and i n Cold Rolled Ferrovac E Iron . . 67 VIII Anomalies i n the P o s i t i o n of Peak S t r a i n i n S t r a i n vs Lat-t i c e Distance Plots 75 IX R e l i a b i l i t y of Computer Analysis 75 X Subgrain Sizes i n Tempered Martensites 86 XI Summary of Selected Area D i f f r a c t i o n Patterns 86 XII e Phase P a r t i c l e Size and Extent of P r e c i p i t a t i o n 105 XIII Dispersion Parameters and Subgrain Sizes i n the Fe-Cu-Ni A l l o y 105 XIV Dispersion Strengthening i n Tempered Martensites I l l XV Strengthening Contribution of Random Dislo c a t i o n s Arrays i n Tempered Martensites 114 XVI Strengthening Contributions i n the Medium Carbon Tempered Martensite 119 XVII Strengthening Contributions i n the Low Carbon Tempered Martensite 120 XVIII Strengthening Contributions i n Series A, Fe-Cu-Ni A l l o y .. 136 x i Table Page XIX Strengthening Contributions i n Series B, Fe-Cu-Ni A l l o y .. 136 A l Flow Stress and C e l l Size Data for Cold Drawn Iron Wire (Embury et a l 4 9 ) 141 A2 Y i e l d Stress and Corresponding Subgrain Sizes i n Pure Aluminum (Ball94) 142 A3 Room Temperature Y i e l d Strength and the Corresponding Sub-grain Diameters for Pure Aluminum a f t e r Various Thermo-mechanical Treatments (Sahoo ) 143 ACKNOWLEDGEMENT The author wishes to express h i s sincere gratitude to Dr. J.A. Lund, fo r h i s advice and assistance during the course of t h i s i n v e s t i g a t i o n . Thanks are also extended to Dr. F. Weinberg f o r c r i t i c a l l y reviewing the manuscript and making many use f u l suggestions. H e l p f u l discussions with other f a c u l t y members and fellow graduate students are also g r a t e f u l l y acknowledged. The assistance of t e c h n i c a l s t a f f i s greatly appreciated. F i n a n c i a l assistance i n the form of National Research Council Scholar-ship i s g r a t e f u l l y acknowledged. INTRODUCTION 1.1 Nature of the Problem Low a l l o y s t e e l s are of enormous importance as materials of engineering. They f a l l i n t o two major groups: i ) medium to high carbon s t e e l s (> 0.25% C) i i ) low carbon steels (< 0.2 % C) The former category responds very desirably to a heat treatment c o n s i s t i n g of a u s t e n i t i s i n g , quenching and tempering. The product of quenching i s a hard tetragonal martensite. The substructure of t h i s martensite i s influenced by carbon content, and gradually changes from laths containing a high density of d i s l o c a t i o n s to internally-twinned plates as the carbon content i s increased from about 0.5 wt.pct. to 1.0 wt.pct. Generally, medium-carbon low a l l o y roartensites contain some twinned p l a t e s , but the amount i s too small to be of any s i g n i f i c a n c e . A f t e r tempering, the s t e e l i s strong ( y i e l d strength i n _2 range of 700-2000 MNm ) and has u s e f u l d u c t i l i t y . One problem with t h i s class of low a l l o y s t e e l s i s that most of the shaping of a part must be done p r i o r to heat treatment, yet the heat treatment i t s e l f can introduce d i s t o r t i o n and shape changes due to the s t r a i n associated with the ma r t e n s i t i c transforma-t i o n . Low carbon s t e e l s respond rather d i f f e r e n t l y to heat treatment. The pro-2 duct of quenching from austenite* il**' a cubic (or nearly cubic) martensite, the useful strength and ductility of which are l i t t l e i f any better than a fine grained product of air cooling from austenite. Moreover, the low carbon steels have i n t r i n s i c a l l y low hardenability so that only small sections can be heat treated in this way. Accordingly, much of the use of this group of steels is in the hot worked condition (e.g. the common structural steels), without any heat treatment. However, several alloying elements, i f present even in relatively small concentrations, perform one or both of two functions which may make a low carbon steel usefully respond to heat treatment i) they improve the hardenability of the steel, i i ) they contribute to the formation of a fine matrix-strengthening Tpre-cipitate when the cubic martensite is aged at a suitable temperature. In this type of low alloy steel, the i n i t i a l heat treatment s t i l l consists of quench-ing from austenite, and this is followed by aging (sometimes referred to as tempering) at temperatures in the range of 450-650°C. Because the martensitic product of the quenching reaction is not hard and b r i t t l e , these low alloy steels can be shaped in the quenched condition. Subsequent aging treatment does not produce distortion. Compared with medium and high carbon steels-, the heat-treatable low carbon steels have lower available yield strengths, typically -2 in the range of 500-700 MNm . However, this i s a significant improvement over -2 the 200-400 MNm yield strength of hot rolled or normalised structural s.teels. Moreover, the heat-treatable low carbon steels are suitable for welding, have typically low ductile-to-brittle transition temperatures, and have good atmo-spheric corrosion resistance. Of the low alloy steels in common commercial use, only the age-hardening members of the low carbon steel group have been fu l l y developed since 1950. Many of the other common s t r u c t u r a l , machinery and,tool s t e e l s which f a l l into the low a l l o y category are very much older i n o r i g i n , and procedures f o r hardening these a l l o y s by heat treatment were discovered hundreds of years ago. However, a detai l e d knowledge of the structure of steels has had to await the discovery and use of such powerful a n a l y t i c a l techniques as trans-mission electron microscopy, and th i s knowledge i s s t i l l being accumulated. Since 1950, basic information gained about the r e l a t i o n s h i p between structure, compostion and heat treatment of steels has led ra p i d l y to the development of new st e e l s and new thermomechanical processing techniques. As a r e s u l t , struc-tures and machine components can now be designed i n s t e e l s which possess much higher u s e f u l y i e l d stress (after f a b r i c a t i o n and heat treatment) than those which were a v a i l a b l e as recently as 15 years ago. The structures of heat treated low a l l o y s t e e l s are complex and have been the subject of many previous in v e s t i g a t i o n s based on o p t i c a l and electron microscopy, x-ray d i f f r a c t i o n and e l e c t r i c a l r e s i s t i v i t y measurements. The possible contributors to the y i e l d strength of low a l l o y s t e els a f t e r heat treatment may be summarized as follows: (a) P e i e r l s s t r e s s , (b) Forest d i s l o c a t i o n s introduced either by mechanical working or as a consequence of s t r a i n accompanying phase transformations (e.g. the martensitic transformation i n stee l . ) (c) I n t e r s t i t i a l and s u b s t i t u t i o n a l solutes, always present i n s t e e l s . (A major contribution may be expected from i n t e r s t i t i a l solutes which are pre-sent beyond t h e i r equilibrium s o l u b i l i t y l i m i t . ) (d) Coherent zones, clusters or p r e c i p i t a t e s formed during the e a r l i e r stages of tempering or aging. (e) A dispersion of hard, incoherent, second phase p a r t i c l e s , introduced either by powder metallurgy techniques or by phase transformations. 4 (f) Grain, subgrain or twin boundaries. An i n d i c a t i o n of the amount of disagreement that exists i n the l i t e r a -ture concerning the r e l a t i v e importance of each of the above contributions to the y i e l d strength of quenched and tempered medium carbon s t e e l s i s obtained 1 2 3 by comparing the conclusions of Cox , Smith and Hehemann , Hyam and Nutting 4 and Tyson . Cox^, on the basis of transmission electron microscopy alone, claimed that carbon i n s o l i d s o l u t i o n i s responsible for most of the high strength 2 of tempered martensite. Smith and Hehemann , a f t e r making c e r t a i n assumptions regarding the substructure, concluded i n marked contrast to Cox"'' that the spacing of the cementite p a r t i c l e s and the l a t h s i z e (small dimension of the martensite lath) are the only important variables during low temperature tempering. On the other hand, for heavily tempered structures, Hyam and 3 Nutting concluded that the y i e l d strength was p r i m a r i l y a function of f e r r i t e 4 grain s i z e , and Tyson argued that the y i e l d strength was determined p r i m a r i l y by the spacing of the incoherent cementite p a r t i c l e s (dispersion strengthening). There i s also disagreement as to the major sources of strength i n age-5 6 hardening, copper-bearing s t e e l s . Thus F u j i i et a l and Pattanaik chose to explain the y i e l d strength of s l i g h t l y overaged a l l o y s i n terms of dispersion hardening (Orowan mechanism) whereas Hornbogeri'' emphasized the r o l e of coherent clusters of copper atoms. There are two sources of the aforementioned controversies: (a) There have been d i f f i c u l t i e s i n properly i d e n t i f y i n g the pertinent s t r u c t u r a l parameters. Thus, i n heat treated medium carbon s t e e l s , the density and d i s t r i b u t i o n of d i s l o c a t i o n s have been poorly characterised by the techniques used (primarily electron microscopy). (b) D i f f e r e n t models and theories of strengthening predict different. 5 contributions f o r a given value of a s t r u c t u r a l parameter, as noted below. (c) The a d d i t i v i t y of c e r t a i n combinations of strengthening mechanisms has often been assumed without a good t h e o r e t i c a l foundation. 1.2 Strengthening Mechanisms-Theory 1.2.1 S t r a i n Hardening ( D i s l o c a t i o n - D i s l o c a t i o n Interaction) I t i s generally predicted^'9,10 t j i a t j i n f - C > c > a n ( j b.c.c. metals, the strengthening contribution of random d i s l o c a t i o n i s proportional to the square root of the d i s l o c a t i o n density. Thus i f T i s a flow stress contribution 8 9 (defined as half the t e n s i l e flow stress) ' due to the d i s l o c a t i o n s , then T = aubplg where p i s the shear modulus, b i s the Burgers vector and p the d i s l o c a t i o n density. The constant a has been t h e o r e t i c a l l y estimated to be 0.4 by Bailey 9 11 8 and Hirsch and 0.34 by L i . For i r o n , Keh and Welssmann experimentally determined the value of a to be 0.4; considering the assumptions that were made i n a r r i v i n g at a t h e o r e t i c a l value, the agreement with experiment i s good. Converting the above expression to a t e n s i l e stress contribution, cr = 2avibpJs (1) where = 2x = the contribution of d i s l o c a t i o n s to the t e n s i l e strength. 1.2.2 S o l i d Solution Strengthening (Dislocation-Solute Interaction) Several theories have been formulated to explain s o l i d s o l u t i o n strengthen-ing i n metals. Each theory assumes a d i f f e r e n t type of i n t e r a c t i o n between the solute atom and a moving d i s l o c a t i o n ; e.g. s i z e i n t e r a c t i o n , modulus i n t e r -6 action, etc. In low a l l o y s t e e l s s u b s t i t u t i o n a l elements are present i n small concentrations. Their t o t a l contribution to strength can be estimated by add-12 13 ing the r e s u l t s from i n d i v i d u a l Fe-X systems ' , where X i s the s u b s t i t u t i o n a l 12 a l l o y i n g element, since i t has been w e l l demonstrated by Lacy and Gensamer that such contributions are add i t i v e . The presence of i n t e r s t i t i a l s i n st e e l s i s , however, of greater import-ance because they cause tetragonal d i s t o r t i o n of the l a t t i c e and thus cause greater hardening than s u b s t i t u t i o n a l a l l o y i n g elements. A number of studies have been made to investigate the strength dependence on the carbon content'C' . 1 U 1/3 Di f f e r e n t dependences; v i z . , c » C" arid C > have been proposed by L e s l i e and Sober"^, Fleischer"'""' and Winchell and Cohen"'" ^  r e s p e c t i v e l y . Experimentally, 17-20 20 maximum support i s found for Fl e i s c h e r ' s model and, according to Wert's observations, the v a r i a t i o n i n the y i e l d strength of f e r r i t e with carbon i n s o l i d s o l u t i o n can be expressed as a(MNm-2) = 0 q + 1 . 5 9 x 1 0 3 (wt.pct. C)^ 0 2 ) According to Equation 2, therefore, 100 ppm. of carbon i n s o l i d s o l u t i o n -2 should strengthen the matrix by 159 MNm 1.2.3 Age Hardening (Dislocation-Coherent P r e c i p i t a t e Interaction) I t i s generally agreed that when small coherent p r e c i p i t a t e s are present a f t e r age hardening, d i s l o c a t i o n s must overcome the coherency s t r a i n s , and 21 shear the p a r t i c l e s . In the absence of coherency s t r a i n s , K e l l y and Nicholson have proposed a simple theory, which for the case of small s p h e r i c a l disordered p r e c i p i t a t e s , involves an energy balance between the work done i n creating a new in t e r f a c e and i n moving the d i s l o c a t i o n through the p r e c i p i t a t e . Thus, i f Y i s . the energy per unit area of the i n t e r f a c e produced during the shearing process, the applied shear s t r e s s , T, necessary to move the d i s l o c a t i o n i s evaluated from 2 xb D = ybd s s were i s the mean planar centre-to-centre, nearest-neighbour i n t e r p a r t i c l e spacing, d g i s the mean planar p a r t i c l e diameter and b i s the Burgers vector. S u b s t i t u t i n g the value of D g i n terms of volume f r a c t i o n of p r e c i p i t a t e s , f, (see Section 2.4.4, Equation 12) the above expression becomes = Y f 1 / 2 (3a) b ( l - f ) 22 Using a Taylor f a c t o r of 2 for i r o n , Equation 3(a) for small values of f becomes 2 Y f 1 / 2 (3) coh b where a , i s that contribution to the y i e l d strength which i s due to coher-coh ent p r e c i p i t a t e s . In d e r i v i n g Equation (3), i t has been assumed that the Burger's vector of a d i s l o c a t i o n has the same magnitude i n both the matrix and the p r e c i p i -tate, and that the s l i p planes i n the matrix and i n the p r e c i p i t a t e are par-21 a l l e l to each other. This need not be the case always . Accordingly, the 23 surface energy y should include the energy of any m i s f i t d i s l o c a t i o n pro-21 duced at the i n t e r f a c e , and the energy of the dipole produced 24 A n s e l l , using a d i s l o c a t i o n pile-up model, also a r r i v e d at e s s e n t i a l l y the same r e l a t i o n s h i p although the value of the p r o p o r t i o n a l i t y constant i s 25 d i f f e r e n t . A n s e l l and Lenel go on to claim that the pile-up model i s ap^ p l i c a b l e to the case of a hard, noncoherent dispersed phase as w e l l . In the model, y i e l d i n g occurs when the shear st r e s s due to a piled-up group of d i s -locations i s s u f f i c i e n t to fracture or p l a s t i c a l l y deform the dispersed sec-8 ond phase p a r t i c l e s . Furthermore, the model predicts that y i e l d strength i s proportional to the r e c i p r o c a l square root of the i n t e r p a r t i c l e spacing and that i f the p a r t i c l e s i z e i s less than a c r i t i c a l s i z e (~^~ where a y. s. a i s the y i e l d strength of the a l l o y ) , then for a constant volume frac-y • s • tio n the y i e l d strength i s constant and independent of p a r t i c l e s i z e . Ex-perimentally, l i t t l e support has been found f o r these p r e d i c t i o n s , and 21 6 K e l l y and Nicholson and Pattanaik have provided e f f e c t i v e c r i t i c i s m of the model. 1.2.4 Dispersion Hardening (Dislocation-Incoherent P a r t i c l e Interaction) i The e a r l i e s t , and experimentally the most commonly supported theory for the strengthening e f f e c t of noncoherent second phase p a r t i c l e s , i s due 2 6 to Orowan . He proposed that, i f the applied stress i s large enough, the matrix d i s l o c a t i o n can bypass p a r t i c l e s by bending and bowing out between them leaving behind c i r c u l a r d i s l o c a t i o n loops around the p a r t i c l e s . The y i e l d strength i s determined by the c r i t i c a l configuration of the by-pass-ing d i s l o c a t i o n , which was assumed to be a semicircular loop of a diameter equal to the i n t e r p a r t i c l e spacing, D. Once again using a Taylor factor-of 2, the extra strengthening e f f e c t of noncoherent p a r t i c l e s , c a l l e d the •~ , . 21 Orowan stress , a i s or a = — (4a) or bD where T i s the l i n e tension of the curved d i s l o c a t i o n . K e l l y and Nichol-21 son , using Nabarro's estimate, of l i n e tension, rewrote Equation (4a) as yb 1 1 1 D (4) a = — . -Tr (1 + -—-) — in — or IT 2 1-v D 2b where v i s Poisson's r a t i o f o r the matrix. 27 28 Recently Ashby ' has objected to Equation (4) on two counts. F i r s t l y , due to the mutual i n t e r a c t i o n between the bowed-out segments around a p a r t i c l e , he suggests that the logarithmic term should be replaced ds by In—, where r i s the inner cut-off radius of a d i s l o c a t i o n and has been r o o taken to be equal to 2b i n Equation (4). Secondly, he has shown that the c r i t i c a l configuration of a loop, beyond which i t becomes unstable and ex-pands without further increase i n st r e s s , i s not a semi c i r c l e of diameter D. If 2<f> i s the angle subtended by the bowed-out segments of a d i s l o c a t i o n at the p a r t i c l e , then at the c r i t i c a l configuration,^, instead of being zero (a semicircular loop) has a f i n i t e value. With the a p p l i c a t i o n of 2 8 this c o r r e c t i o n , Ashby obtains a modification of Equation (4), which.can be written for a t e n s i l e stress as yb 1 , 1 N 1 , . rds.T , ,D j (1 + -z~) £ cost). £n [ ^ ( 1 + (~ - Dsifl*.] (5) a or TT o In the l i t e r a t u r e , maximum support i s found for Equation (4). The 29 30 resu l t s of Byrne et a l and of Dew Hughes and Robertson with Al-Cu 31 32 a l l o y s , and those of Gregory and Grant and Hansen with S.A.P. type 33 a l l o y s , a l l provide a good f i t with Equation (4). Ashby's own r e s u l t s with copper si n g l e c r y s t a l s containing a s i l i c a dispersion were found to 28 be consistent with Equation (4). Ashby replotted the data of Ebeling 34 35 and Ashby and of Jones and K e l l y according to a simpler version of Equation (5). The slope observed was about 14% higher than t h e o r e t i c a l l y predicted. Even t h i s agreement could only be claimed by taking 'r ', the inner cut-off radius of a d i s l o c a t i o n , as an adjustable parameter. A value 36 of r Q = 4b has been used by Ashby, although H i r t h and Lothe have calcu-lated that for metals, r would have a maximum value of 2b. Jones and o 10 35 K e l l y had e a r l i e r p l o t t e d t h e i r r e s u l t s according to both Equations (4)and the simpler version of (5) and they claimed an equally good f i t i n both cases. 35 Once again, Jones and K e l l y used d i f f e r e n t value of r Q i n the two equations. I t i s evident, therefore, that although Ashby's modifications may be theoreti-c a l l y w e l l founded, they do not explain experimental r e s u l t s any better than 21 the e a r l i e r Kelly-Nicholson statement of the Orowan theory. 37 38 In the l i t e r a t u r e , there i s electron microscopic evidence ' which suggests that the by-passing of p a r t i c l e s can also be accomplished by cross 33 s l i p . Ashby has explained t h i s by considering the second phase p a r t i c l e s as stress concentrators. When a d i s l o c a t i o n i s pressed against a p a i r of p a r t i c l e s , a distance D apart, by an applied shear s t r e s s , T, then the d i s -l o c a t i o n segment pressed against the p a r t i c l e experiences a short range back s t r e s s , x^, from the p a r t i c l e . For equilibrium, TB ' d s = T * D - ' or T * " T " t (6) where d i s the mean planar p a r t i c l e diameter. Generally, D > d and hence s s the length of the d i s l o c a t i o n pressed against the p a r t i c l e i s subjected to a stress which i s greater than the applied shear stress by a factor equal to -^j— . A n s e l l has put t h i s argument i n mathematical form and has shown s that when D 10d s, then cross s l i p w i l l occur more r e a d i l y than by-passing by the Orowan mechanism. 1.2.5 Grain and Subgrain Boundary Hardening H a l l " ^ , P e t c h ^ and o t h e r s ^ have shown that the t e n s i l e y i e l d stress or flow s t r e s s , a, of p o l y c r y s t a l l i n e i r o n i s r e l a t e d to the grain s i z e , £, by an equation of the type a = a + k Z~1/Z (7) o g where a and k are material parameters of l i t t l e t h e o r e t i c a l s i g n i f i c a n c e , o g In most i n t e r p r e t a t i o n s , a and k are considered to be a measure of the o g ' f r i c t i o n s t r e s s ' of the matrix and the 'grain boundary strength', respec-t i v e l y . Several attempts have been made to e s t a b l i s h a t h e o r e t i c a l basis f o r the Hall-Petch r e l a t i o n s h i p (Equation 7). The e a r l i e s t explanations, due 39 40 42 to H a l l , Fetch and C o t t r e l l , envisaged a p i l e up of di s l o c a t i o n s at grain boundaries. As the r e s u l t of the stress concentration at the hend of a p i l e up, p l a s t i c flow was nucleated on the other side of the grain boundary. Experimentally, there i s no support for t h i s model; i . e . , there have been no d i r e c t observations of p i l e ups at grain boundaries i n pure 43 metals. Conrad has t r i e d to derive the Hall-Fetch r e l a t i o n on the basis 44 of a work hardening model. But th i s approach predicts a strain-indepen-dent value of a , and an increase i n the magnitude of the 'Fetch slope' k o» & g 45 with s t r a i n , neither of which i s commonly observed. J.C.M. L i has pro-posed a s i m i l a r theory wherein grain boundary ledges are considered to be d i s l o c a t i o n sources. This theory c o r r e c t l y p r edicts that the Petch slope should be independent of temperature as long as the ledge density i n grain boundaries, m, i s not affected. Furthermore, by incorporating the e f f e c t of. dynamic recovery, the theory c o r r e c t l y predicts a decrease i n the value of the Petch slope at large s t r a i n s . However, i t i s d i f f i c u l t to make any quantitative comparison between the theory and experiments, since the ledge density 'm' cannot be established accurately to within less than an order of magnitude. 41 Anderson et a l have tabulated relevant y i e l d s t r e s s - g r a i n size data 12 for i r o n from 13 sources. The constant was found to have a wide range -3/2 -3/2 of values i n d i f f e r e n t i n v e s t i g a t i o n s , ranging from 0.45 MNm to 1 MNm 41 Anderson et a l pointed out that most of these studies covered only a nar-row range of grain s i z e , which permitted acceptable f i t s to be obtained to Equation (7). Their own r e s u l t s , on specimens having grain sizes which -1/2 ranged from approximately 3.5ym to 200ym, gave a a versus I p l o t which -1/2 was .not l i n e a r but was concave towards the I axis. At the same time, for each set of k and a values obtained from d i f f e r e n t sources i n the l i t e r a -g o ture, t h e i r own values f o r and over a corresponding range of grain 41 sizes were very s i m i l a r . This deviation of the Anderson et a l data from the Hall-Petch r e l a t i o n has been a t t r i b u t e d to the necessity of using d i f f e r -ent, heat treatments to get d i f f e r e n t grain s i z e s . Thus, there were varying degrees of segregation of solute atoms at grain boundaries associated with the,variation of grain s i z e . Solute atoms tend to a l t e r the grain boundary ledge structure and thus change the value of the Petch slope. Other r e l a t i o n s h i p s between y i e l d stress and grain s i z e have been pro-posed. Conrad^ has derived an I r e l a t i o n s h i p whereas B a l d w i n ^ favours a I r e l a t i o n s h i p . Hutchison and Pascoe^, a f t e r an elaborate s t a t i s t i -c a l analysis of t h e i r r e s u l t s on copper-base a l l o y s , found support for an il ^ r e l a t i o n . 49 ' 50 45 It has been suggested by Embury et a l , Warrington and J.C.M. L i that i n the presence of a c e l l u l a r d i s l o c a t i o n substructure, the y i e l d s,tress i s c o n t r o l l e d by the si z e of t h i s substructure, t, i n a manner analogous, to the grain s i z e . The term " c e l l u l a r substructure" i s taken here to imply c e l l s produced by cold working as well as the substructure which r e s u l t s from recovery a f t e r cold work. Such a d e f i n i t i o n of c e l l u l a r substructure i s consistent with the terminology used widely i n the l i t e r a t u r e . To quote 13 51 J.C.M. L i , "...polygonization i s found to be only a s p e c i a l form of sub-boundary formation. Most sub-boundaries are formed even during deformation, frequently c a l l e d ' c e l l w alls', the structure of which i s not very w e l l defined. During annealing, these c e l l walls become sub-boundaries showing regular d i s l o c a t i o n structures." Thus, the empirical r e l a t i o n f o r sub-boundary or c e l l w a l l strengthening may be stated as -1/2 cr = a + k • t ' (8) o t The value of k f o r iron i s observed to l i e i n the range 0.3 to 0.45 -3/2 MNm . This range i s lower than that of k indicated above for grain boundaries, which would suggest that the exact mechanisms by which a c e l l u l a r substructure and grain boundaries r e s i s t p l a s t i c deformation are d i f f e r e n t . The f i r s t t h e o r e t i c a l model to extend the v a l i d i t y of the Hall-Petch 52 r e l a t i o n to sub-boundaries was proposed by Gay, Hirsch and K e l l y on the 53 basis of an Eshelby et a l c a l c u l a t i o n for p r e d i c t i n g the stress concentra-t i o n at a c e r t a i n distance ahead of the p i l e up. Y i e l d i n g i s assumed to occur when the magnitude of the stress concentration i s enough to form a new d i s l o c a t i o n loop and to expand i t to a c e r t a i n c r i t i c a l s i z e . The app-roach i s e s s e n t i a l l y the same as those of pile-up theories f o r grain boundary strengthening, and the same objections apply. 45 L i has modified h i s theory for grain boundary strengthening to ex-p l a i n strengthening due to sub-boundaries. He f i r s t points out that the free energy of formation of a ledge i n a small-angle boundary i s rather high. Hence, he proposes that ledges may not ex i s t at a l l i n low angle boundaries. Instead i t i s suggested that d i s l o c a t i o n s are generated from p a r t i a l l y - p i n n e d boundaries without any p i l e up. The stress required to generate the d i s l o c a t i o n i s shown to be a function of misorientation across 14 the boundary and of the extent of pinning i n the boundary. The d i s l o c a -t i o n so generated has to move i n the presence of the s t r a i n f i e l d s of other similarly-generated d i s l o c a t i o n s . The stress required to move the d i s l o c a -t i o n i s greater than that required to generate i t , and th i s argument leads L i to the expression a = a + ( § e } 1 / 2 t - i / 2 9 o 2TT(1-V) Trb v ' where 6 = "misorientation" across the sub-boundary and t i s the subgrain s i z e . Equation (9) predicts that the Petch slope f o r subgrains should be dependent upon the misorientation across the sub-boundary. However, B a l l has shown that f o r i r o n " ^ and aluminum^that t h i s i s not the case. Li^~ * has,attributed B a l l ' s observation to the presence of solute atoms at the sub-boundaries. However, according to L i ' s own statement, solute atoms influence only the stress required to generate a d i s l o c a t i o n , and th i s stress i s less than that required to propagate a d i s l o c a t i o n . Thus L i ' s theory does not adequately explain the apparent i n s e n s i t i v i t y of the Petch slope to the average angle of misorientation across the boundary. 55 Recently, Langford and Cohen found that t h e i r structure observations and t e n s i l e r e s u l t s for cold drawn wire were not consistent with a H a l l -Petch type of r e l a t i o n s h i p . Instead, a good l i n e a r f i t was found between the, flow stress and the r e c i p r o c a l of the c e l l s i z e ' t ' . Langford and Cohen went on to provide a t h e o r e t i c a l j u s t i f i c a t i o n for the observed r e l a -tionship. In t h e i r approach, l i k e that of L i ' s , i t i s assumed that the applied stress required to generate a d i s l o c a t i o n at a sub-boundary i s sm^ll and that the strengthening e f f e c t arises from the stress required to expand each d i s l o c a t i o n loop across the s l i p plane of the subgrain or the c e l l u n t i l 1 5 i t i s assimilated into the sub-boundary at the perimeter. Thus the s t r a i n associated with the passage of each d i s l o c a t i o n i s 2 where N i s the length of the c e l l and the f a c t o r / arises from the pre-ferred o r i e n t a t i o n i n cold drawn i r o n wire. Next, the energy per unit v o l -ume expended i n forcing the expanding d i s l o c a t i o n loop against the ' l a t t i c e f r i c t i o n ' , a , i s equated to the energy per unit volume required to create a d i s l o c a t i o n length equal to the perimeter of the s l i p plane. Thus one. obtains P T (a - a ) Ae = 1.54^ (10a) O AN where P = perimeter of the s l i p plane. T = l i n e tension of the d i s l o c a t i o n and AN = volume of the subgrain or c e l l . A f t e r s u b s t i t u t i n g the appropriate values of P, A and N, and using 1 2 T - -^ yb , Equation (10a) reduces to a = a + 3.0 (10) o t Experimentally, the observed slope of a a vs. t ^ p l o t was 5.9 yb, as com-pared to the t h e o r e t i c a l value of 3.0 yb. This difference was a t t r i b u t e d by Langford and Cohen to the f a c t that 3.0 pb i s the r e v e r s i b l e energy to expand the d i s l o c a t i o n and thus represents a lower l i m i t only. 1.3 The Iron-Copper System Since the age-hardening s t e e l used i n the present i n v e s t i g a t i o n con-16 tains copper, a b r i e f review of previous studies of the e f f e c t s of copper i n iron are presented below. The i r o n - r i c h end of the iron-copper phase diagram has been r e l i a b l y established by Wreidt and Darken"^ and by Speich et al"'''. Figure 1 shows 57 the diagram as determined by Speich et a l . The eutectoid temperature. and composition are seen to be 850°C and 2.65 wt.pct. Cu re s p e c t i v e l y . The s a l i e n t feature of the diagram, however, i s the marked decrease i n the s o l i d s o l u b i l i t y of copper i n a-iron below 850°C, which s i g n i f i e s the age-harden-ing p o t e n t i a l of these a l l o y s . The S phase, which p r e c i p i t a t e s during aging a f t e r s o l u t i o n treatment, i s almost pure copper. , The age hardening c h a r a c t e r i s t i c s of Fe-Cu a l l o y s have been studied by 58 7 Hornbogen and co-workers ' . They concluded that, as a r e s u l t of the high concentration of vacancies a f t e r s o l u t i o n treatment, s p h e r i c a l clusters of copper atoms, which are coherent with the matrix, s t a r t growing at the aging temperature without any incubation period. I t has been speculated that: nucleation of these clusters may be associated with I n d i v i d u a l vacancies or vacancy c l u s t e r s . I t i s suggested that the c l u s t e r s then transform into the f.c.c. e l a t t i c e a f t e r they have attained a c r i t i c a l s i z e . F i n a l l y , a f t e r prolonged aging treatment, the s p h e r i c a l e-phase p r e c i p i t a t e s assume a rod shape. Two possible reasons for t h i s shape change were suggested; anisotropy of surface energy, and reduction of s t r a i n energy. It should be appreciated that copper clusters could not be d i r e c t l y observed by these workers since iron and copper have s i m i l a r s c a t t e r i n g factors f o r x-rays and electrons. In addition, the copper atom has about the same size as the i r o n atom so that l i t t l e coherency s t r a i n can be ex-pected around a c l u s t e r . Thus, i t i s impossible to detect the clusters by transmission electron microscopy, by x-ray d i f f r a c t i o n or by electron d i f -1 8 f r a c t i o n . Indirect evidence of the existence of c l u s t e r i n g , however, comes from several sources: (a) the f i r s t appearance of e p a r t i c l e s of measurable si z e (b) hardening which occurs before any v i s i b l e e p r e c i p i a t e i s observed 59 (c) observations made with Fe-Au alloys , where the clusters can ,be detected due to the differences i n atomic s i z e s and atomic s c a t t e r i n g fac-tors f or i r o n and gold. The strength of age-hardened iron-copper a l l o y s has been investigated by Pattanaik^ and by F u j i i et al~*. In both studies, the major source of strength was i d e n t i f i e d as Orowan hardening; i . e . , the requirement that-glide d i s l o c a t i o n s by-pass e p a r t i c l e s by bowing around them i n the s l i p plane. Hornbogen^ has suggested that strengthening i n the peak-aged con-d i t i o n i s caused by incoherent p r e c i p i t a t e s as w e l l as by c l u s t e r s , but he made ho attempt to assign a separate contibution to each source. 1 . 4 Scope of the Present Work: This i n v e s t i g a t i o n was undertaken i n an attempt to resolve some of the controversy which exists concerning the contribution of d i f f e r e n t strengthen-ing mechanisms to the y i e l d strength of heat-treated low a l l o y s t e e l s . To characterise d i s l o c a t i o n substructure i n these materials, the power-f u l technique of x-ray l i n e broadening analysis has been added to electron microscopy. Previous use of the x-ray method has been e s s e n t i a l l y confined to studies of deformed metals and a l l o y s , yet the method should be generally applicable to the analysis of substructure, regardless of i t s o r i g i n . Four materials were chosen for the i n v e s t i g a t i o n . In order of increas-ing complexity of structure a f t e r heat treatment, these were: Iron - 0.006 wt.pct. C (Ferrovac E iron) 19 - Low carbon s t e e l ; 0.11 wt.pct. C, 0.9 wt.pct. Mn. - Iron - 0.006 wt.pct. C - 1.8 wt.pct. Cu - 1.3 wt.pct. N i . - Medium carbon s t e e l ; 0.42 wt.pct. C, 1.1 wt.pct. Mn The copper-bearing a l l o y i n t h i s group represents a new class of low-cost, age-hardening s t e e l s which are of great p o t e n t i a l f o r s t r u c t u r a l and fastener a p p l i c a t i o n s . Although there have been several previous studies of the response of copper-bearing steels to heat treatment, there i s con-siderable doubt : i n the l i t e r a t u r e as to the major source of strengthening i n the optimally aged a l l o y . I t was hoped to resolve this question i n the present work, as w e l l as to use the structure and properties of the Fe-Cu-Ni a l l o y to shed some l i g h t on the d i f f e r e n t contributions to strength i n the more complex medium carbon s t e e l . Because the copper-bearing s t e e l (In common with Ferrovac E i r o n ) i s amenable to cold work p r i o r to aging, It i s possible to introduce a d i s l o c a t i o n substructure by deformation, and to compare i t s r o l e with that of the substructure induced by the martensi-t i c transformation i n steels of higher carbon content. 20 2. EXPERIMENTAL PROCEDURES 2.1 Materials Supply and Preparation The low carbon and medium carbon s t e e l s were obtained from l o c a l sup-p l i e r s i n the form of hot r o l l e d bars of 41 mm and 50 mm diameter, respec-t i v e l y . Ferrovac E i r o n was a v a i l a b l e as hot r o l l e d s t r i p , - 5 mm thick, and as 32 mm bar. Results of analysis on these materials aregiven i n Table I. The Fe-Cu-Ni a l l o y was prepared by vacuum induction melting i n a fused magnesia c r u c i b l e . Raw materials used for the a l l o y were Ferrovac E i r o n , e l e c t r o l y t i c copper and e l e c t r o l y t i c n i c k e l r e spectively. The molten metal was poured into a mild s t e e l s p l i t mold, under a vacuum of approximately lOym, which yielded a slab casting of dimensions 250 x 62 x 16 mm. The slab was homogenized by heating i n helium at 1000°C for 24 hours. Samples were taken from top and bottom of the slab for chemical analysis, with the res u l t s shown i n Table I. 2.2 Specimen preparation and Heat Treatment 2.2.1 Medium Carbon Steel (0.42% C, 1.1% Mn) A portion of the h o t - r o l l e d s t e e l bar was m i l l e d to a rectangular cross-Table I Composition of Materials Investigated ( A l l figures i n weight percent) M a t e r i a l C Mn S i Mo S P Cu Ni N Cr Co A l , Mg | i ; T i , V Medium Carbon S t e e l 0.42 1.11 0.22 0.13 0.07 0.011 0.25 0.1 Trace Trace Low Carbon S t e e l 0.11 0.9 0.06 0.008 0.25 0.08 0.03 Trace Fe-Cu-Ni A l l o y 0.0038 1.8 1.30 0.0025 Ferrovac E Iron 0.005 0.0005 0.0008 section of 40 mm x 25 mm. For x-ray d i f f r a c t i o n studies, s l i c e s approxi-mately 2 mm thick were cut transversely from this section by sawing. The s l i c e s were further surface-ground to render t h e i r faces f l a t , the f i n a l thickness being - 1.5 mm. Discs approximately 1 mm thick were s i m i l a r l y s l i c e d transversely from the o r i g i n a l bar and surface ground to a thickness' of approximately 0.75 mm. Each di s c was then cut into four pieces to provid' specimens f or heat treatment and subsequent electron microscopy. For ten-s i l e t e s t i n g , double-shouldered round specimens of reduced gage section 11.5 mm x 3.2 mm were machined from the o r i g i n a l bar. A l l specimens (x-ray, microscopy and t e n s i l e ) were austenitised at 850°C f or 1/2 hour i n a neutral s a l t bath consisting of a e u t e c t i c mixture of sodium and barium chlorides, and were then quenched i n iced brine. The quenched specimens were tempered i n potassium nitrate-sodium n i t r a t e s a l t baths f o r 1 hour at one of ten equally spaced temperatures i n the range of 250-700°C. The specimens were quenched i n water a f t e r tempering. 2.2.2 Low Carbon Steel (0.11% C, 0.9% Mn) Specimens used f o r x-ray diffractometry were obtained i n the same man-ner as for the medium carbon s t e e l . However, for electron microscopy, the lower hardenability of t h i s s t e e l necessitated the use of thinner specimens i n order to ensure complete transformation. Accordingly, 1 mm thick discs were s l i c e d transversely from the o r i g i n a l bar and then surface ground ;to 0.63 mm thickness. Each di s c was further polished mechanically to approx-mately 0.4 mm thickness before heat treatment. The same l i m i t a t i o n of hard-e n a b i l i t y prohibited the use of standard t e n s i l e specimens. The a u s t e n i t i s i n g treatment was ca r r i e d out i n a v e r t i c a l tube furnace at 920°C for 1/2 hour under a dynamic vacuum of 20um of mercury. I t was 23 found necessary to keep some zirconium s t r i p i n the heating zone to prevent decarburization at the surface of the specimens. Quenching of specimens was accomplished by pinching o f f the rotary pump and l e t t i n g cracked ammonia into the tube. This forced open the s e a l at the bottom of the tube and simultaneously closed an e l e c t r i c a l c i r c u i t , r e s u l t i n g i n the dropping of the specimens into an iced brine bath kept d i r e c t l y below the tube. F i n a l l y , the specimens were tempered f o r 1 hour, i n neutral s a l t baths, at one of ;10 equally spaced temperatures i n the range 250-700°C. 2.2.3 Iron-Copper-Nickel A l l o y F a b r i c a t i o n and heat treatment steps used for the Fe-Cu-Ni a l l o y s p e c i -mens are summarized i n the flow sheet of Figure 2. P r i o r to each hot r o l l i n g step, the a l l o y was heated i n a i r to the hot r o l l i n g temperature of 1000°C. The cast slab was soaked for 2 hours at 1000°C to ensure uniformity of temperature. Subsequent retention times i n the furn-ace, between r o l l i n g passes were only 10 to 15 minutes. Surface cleaning of hot r o l l e d s t r i p was effected by a sequence of ; buffing and p i c k l i n g operations. . A 10 pet. H^SO^ so l u t i o n was used f o r pick-l i n g . T e n s i l e specimens were blanked from s t r i p using a pneumatically-acti-vated punch and die set. The reduced gage section of the specimens was 5 mm wide by 20 mm long. Two d i s t i n c t groups of specimens were derived from the processing i n -dicated i n Figure 2. Series 'B' specimens were cold r o l l e d (50% reduction) after s o l u t i o n treatment but before aging, whereas Series 'A' specimens were aged d i r e c t l y a f t e r s o l u t i o n treatment. The mechanical processing treatments were arranged such that a l l Series 'A' specimens were of the 24 Cast slab 250 x 62 x 16 mm 1000 C, 24 hours (homogenisation) Hot r o l l to 5.6 mm at 1000 C Surface clean Cut rectangles 25 x 38 x 1.5 mm x-ray d i f f r a c t i o n specimens for s o l u t i o n treatment and aging * (Series A) Hot r o l l to 2.8 mm at 1000°C Surface Clean Cold r o l l to 1.5 mm Cold r o l l to 0.9 mm Cut Rectangles 25 x 38 x 3 mm f o r x-ray work Solution Treat Cold r o l l 50% to 1.5 mm Age x-ray d i f f r a c t i o n Cut Coupons - r\ specimens 19 mm x l 9 m m x • T>\ (Series B) 0.9 mm (Electron Microscopy) and s t r i p 75 mm x 19 mm x 0.9 mm (Tensile specimens) Blank Tensile specimens from the 75 mm long s t r i p s Solution treat and age the Electron microscopy and Tensile specimens (Series A) Cold r o l l to 3 mm Cold r o l l to 1.8 mms Cut Coupons 19 mm x 19 mm St r i p s 75 x 19 mm Solution Treat Cold r o l l 50% to 0.9 mm Blank Tensile Specimens from s t r i p s - , Age •J-Electron Microsopy and Tensile Specimens (Series B) Figure 2 Flow Sheet for Specimen Preparation i n the Fe-Cu-Ni A l l o y 25 same thickness as the equivalent Series 'B' specimens a f t e r the f i n a l aging operation. Solution treatment consisted of soaking i n vacuum at 1000°C f or 1/2 hour, followed by quenching i n iced brine. The furnace and quenching procedures used were the same as were described above for the hardening of low carbon s t e e l . Aging was carried out at 500°C i n a neutral s a l t bath, using 'Sen Pak' s t a i n l e s s s t e e l envelopes to protect those specimens which were aged f o r more than 2 hours. For Series 'A', the aging times were 0, 0.16, 0.5, 1, 2, 5, 10 and 100 hours. For Series 'B', the times were 0, 0.16, 0.5, 1, 2, 10, 30, 60 and 100 hours. 2.2.4 Ferrovac E Iron Processing of pure i r o n specimens from the o r i g i n a l 5 mm t h i c k s t r i p i s summarized i n Figure 3. Specimens were austenitised by heating to 970°C f or 15 minutes i n the v e r t i c a l tube furnace (as used f o r the low carbon steel)> and were then quenched into iced brine. Some specimens were cold r o l l e d 50% af t e r quenching, and were tested and examined i n the cold worked condition. Others were tested i n the as-quenched state. No tempering or a r t i f i c i a l aging treatment was applied to pure i r o n specimens. 2.3 Tensile Testing and Hardness Testing Procedures 2.3.1 Medium Carbon Steel (0.42% C, 1.1% Mn) The as--machined and heat treated samples were l i g h l y polished with emery paper and f i n a l l y electropolished In a chrome-acetic s o l u t i o n to give a bright f i n i s h . A f t e r eliminating any specimens containing obvious quench cracks, 26 Ferrovac E Iron S t r i p 5 mm t h i c k x 23 mm wide Cut Rectangle -< 38 mm x 23 mm x 3 mm A u s t e n i t i s e , and quench Cold r o l l 50% to 1.5 mm x-ray specimen (cold r o l l e d ) Cold r o l l -to 3 mm thickness •+ Cold r o l l to 18 mm Cut Coupons 23 mm x 19 mm and s t r i p s 19 mm x 75 mm Aus t e n i t i s e and quench Cold r o l l 50% to 0.9 mm Blank t e n s i l e specimens from s t r i p s Electron Microscopy and Tensile Specimens (Cold r o l l e d ) Cold r o l l to 0.9 mm Cut coupons 23 mm x 19 mm and s t r i p s 75 mm x 19 mm Blank Tens i l e Specimens Au s t e n i t i s e and quench Electron Microscopy and Tens i l e Specimens (as quenched) Figure 3 Flow Sheet for Specimen Preparation i n Ferrovac E Iron 27 two t e n s i l e specimens were tested at 0.011 min on an Instron machine for each tempered condition, with good r e p r o d u c i b i l i t y . From each load-elonga-t i o n p l o t , true stresses and corresponding true p l a s t i c s t r a i n s were compiled f o r a number of points within the homogeneous p l a s t i c flow range. These were plotted on a log-log basis and extrapolated to a true p l a s t i c s t r a i n of 0.002 i n order to determine the value of ^ (0-2 pet. o f f s e t y i e l d stress) re-ported i n t h i s work. The object of t h i s procedure was to eliminate the hetero-geneous y i e l d i n g portion of the s t r e s s - s t r a i n curve, and thus to provide a more consistent method of assessing an o f f s e t s t r e s s . 2.3.2 Low Carbon Steel (0.11% C, 0.9% Mn) As mentioned before, the hardenability of t h i s s t e e l i s very low. To obtain a completely martensitic structure throughout the cross-section of a t e n s i l e specimen would have required that the thickness of a specimen was approximately 0.5 mm or l e s s . Such t h i n specimens are known from experience to d i s t o r t badly when quenched and the d i s t o r t i o n has an unpredictable e f f e c t on t e n s i l e r e s u l t s . The y i e l d stress i s p a r t i c u l a r l y influenced. Therefore, i t was decided to use hardness as a measure of strength f o r the low carbon s t e e l . Testing was done on a Vickers hardness t e s t i n g machine using a load of 5 kgms on the diamond indenter. The x-ray d i f f r a c t i o n specimens were used for t h i s purpose. A minimum of f i v e p a i r s of hardness readings were taken 60 from each specimen, with very l i t t l e observed scatter.. D e l i s l e and G a l i b o i s had previously reported y i e l d stress and hardness data for a tempered 0.05% C s t e e l . Using these published r e s u l t s the present hardness data were converted to y i e l d stress values. For specimens tempered between 250 and 350°C, extra-polations had to be made based on the abovementioned work and from standard 28 Hardness-U.T.S. tables 2.3.3 Fe-Cu-Ni A l l o y and Ferrovac E Iron The f l a t t e n s i l e specimens were mechanically polished on emery papers down to 000 g r i t immediately a f t e r they were blanked; i . e . , a f t e r quench-ing from the a u s t e n i t i s i n g temperature and cold r o l l i n g i n the case of both Series 'B' and cold r o l l e d Ferrovac E specimens, and p r i o r to heat t r e a t -ment f o r Series 'A' and the as-quenched Ferrovac E i r o n specimen. Once they had been f u l l y heat treated, specimens were electropolished f or 1/2 hour each i n a chrome-acetic s o l u t i o n so that the thin oxide layer that had formed during heat treatment was removed. At least two specimens were tested on the Instron machine f o r each thermomechanical treatment with good r e p r o d u c i b i l i t y . The s t r a i n rate used was 0.0125 min ^. Autographic load-elongation plots obtained from the Instron machine were analysed the same way as the pl o t s f or the medium carbon s t e e l (Section 2.3.1). Once again, the 0.2 pet. o f f s e t y i e l d stress values reported are thos# obtained from log-log p l o t s . 2.4 Metallography 2.4.1 O p t i c a l Microscopy Specimens to be examined were hand polished on a s e r i e s of emery papers using flood l u b r i c a t i o n with kerosene. The specimens were then lapped using diamond paste. The etching reagent used was 2% N i t a l (2% n i t r i c acid i n ethyl a l c o h o l ) . 2.4.2 Electron Microscopy 29 A two-stage r e p l i c a technique was the basis of determinations of the carbide p a r t i c l e spacing i n the medium and low carbon s t e e l s . A negative c e l l u l o s e acetate r e p l i c a was dry-stripped from the polished and etched specimen surface, followed by chromium shadowing and carbon deposition on the acetate paper. The acetate was next dissolved away i n acetone and the r e p l i c a remaining was examined i n an H i t a c h i HU-11A electron microscope operated at 50 kv. To in v e s t i g a t e the nature of substructure generally and the s i z e of copper p a r t i c l e s i n the copper-bearing s t e e l , transmission electron micro-scopy was performed on t h i n f o i l specimens. Specimens for microscopy were f i r s t chemically polished to a thickness of 75 to 100 ym i n a s o l u t i o n of the following composition: HN03 HC1 H 3 P ° 4 CH3C00H 50 c.c. 20 c.c. 20 c.c. 100 c.c. , In order to avoid roughening of the f o i l s , the s o l u t i o n had to be vigorously s t i r r e d and discarded a f t e r thinning a s i n g l e specimen. The 6 2 f p i l s were thinned further by e l e c t r o p o l i s h i n g using the Bollman techni-que. E l e c t r o p o l i s h i n g was carried out at 18 to 22 v o l t s i n a chrome-acetic s o l u t i o n of the following composition: Cr0 3 : 25 gms. CH3C00H : 135 c.c. Water : 7 c.c. The t h i n f o i l s were examined i n the microscope at 100 kv, using a high r e s o l u t i o n stage. A gold standard was used to determine the camera constant for electron d i f f r a c t i o n a n a l y s i s . 30 2.4.3 Determination of Volume Fr a c t i o n of P r e c i p i t a t e s In the tempered martensite structures obtained from medium carbon and low carbon s t e e l s , i t could be assumed that a l l carbon was present as Pe^C. The volume f r a c t i o n of the carbide, ' f , was calculated from a knowledge of the carbon content of the s t e e l , the Fe-C phase diagram, and the densi-t i e s of i r o n and cementite (7.87 and 7.40 gms/c.c. r e s p e c t i v e l y ) . The res-pective values of ' f for the medium and low carbon s t e e l s of the present work were found to be 0.066 and 0.0162. The t h e o r e t i c a l volume f r a c t i o n of £ p r e c i p i t a t e s i n the heat-treated iron-copper a l l o y was calculated i n the same manner as above. The e q u i l i -l i b r i u m s o l u b i l i t y of copper i n i r o n at 500°C, as obtained by extrapolating the data of Speich et al"''', i s less than 0.03 weight pet. The calculated value of ' f for the Fe-Cu-Ni a l l o y was 0.016. However, on the basis of transmission electron microscopy i t was l a t e r evident that a l l copper was not p r e c i p i t a t e d under the conditions of aging used i n the present work. Therefore, the actual volume f r a c t i o n of copper p r e c i p i t a t e s for a given heat treatment, f ^ , was calculated from the number and s i z e of p r e c i p i t a t e s observed i n t h i n f o i l s by transmission microscopy. 2.4.4 C a l c u l a t i o n of I n t e r p a r t i c l e Spacing For purposes of t e s t i n g theories of dispersion strengthening, or i n the estimation of the t h e o r e t i c a l strength contribution due to dispersed incoherent p r e c i p i t a t e p a r t i c l e s , i t i s e s s e n t i a l to know the planar i n t e r -p a r t i c l e spacing. S p e c i f i c a l l y , i n the c a l c u l a t i o n of dispersion strength-ening by the Orowan model, the important parameter i s the edge-to-edge spa-4 cing, D, of nearest neighbour p a r t i c l e s i n a s l i p plane. In p r i n c i p l e ; i f 31 any two of the following three parameters are known, then D i s e a s i l y com-puted: i ) the volume f r a c t i o n ' f of p r e c i p i t a t e , i i ) the true mean p a r t i c l e diameter, d , and v i i i ) the planar mean free path, A. The mean free path 'X' can be determined metallographically according to 63 the r e l a t i o n s h i p derived from geometry by Fullman x -where i s the number of p a r t i c l e s i n t e r s e c t i n g a unit length of a random l i n e i n a polished section. In the Fullman derivation i t i s assumed that the p a r t i c l e s are uniform and s p h e r i c a l . Several r e l a t i o n s h i p s have been proposed between A, f and d^, each assuming a p a r t i c u l a r geometrical model for the d i s t r i b u t i o n of s p h e r i c a l 64 second phase p a r t i c l e s . Two of these, due to Edelson and Baldwin and to Kocks^^, re s p e c t i v e l y , assume a random d i s t r i b u t i o n and predict s i m i l a r values of the i n t e r p a r t i c l e spacing, D, for a given f and A. In the pre-64 sent work, Edelson and Baldwin's r e l a t i o n s h i p has been used, according to which: 2d A = T F " ( 1 _ F } and / 2d 2 D =7WL- U-*> '(12) s V 3f where i s the centre to centre planar spacing. Also D = D - d s s where d i s the mean planar p a r t i c l e diameter and i s equal to J-| d^. 32 In the case of carbon s t e e l s , the planar mean free path. X, was f i r s t determined f o r each heat treated condition using at le a s t s i x enlarged rep-l i c a e l e ctron micrographs. Once X was known, the edge.-to-edge cementite p a r t i c l e spacing was e a s i l y computed from a knowledge of the volume f r a c -t i o n of cementite, f, and the above r e l a t i o n s . For the age-hardening s t e e l , the average true diameter of e p a r t i c l e s was determined by transmission electron microscopy. The o r i g i n a l photo-graphs, taken at 85,000 to 140,00 magnifications, were furt h e r enlarged 2.5 times, and at le a s t four micrographs were analysed to evaluate the value of d^ f o r a given specimen. Knowing d^ and the volume f r a c t i o n of p r e c i p i t a t e s , f, the i n t e r p a r t i c l e spacing was evaluated using the r e l a t i o n s given above. 2.4.5 Measurement of Grain Size and Subgrain Size Estimates of grain s i z e and subgrain s i z e were based on measurements from micrographs and transmission e l e c t r o n micrographs, r e s p e c t i v e l y . The mean l i n e a r intercept method was used. I f n i s the number of grains occupy-ing a length of planar intercept I, then the mean l i n e a r Intercept t" i s given by I t has been pointed out that the mean l i n e a r i n t e r c e p t t" i s less than the mean diameter of the subgrains or grains i n the plane of se c t i o n -ing, t 1 , and that the mean diameter of the subgrains or grains i n a planar s e c t i o n , t ' , i s less than the true diameter of the subgrains or grains i n the aggregate, t . Cahn et a l ^ , assuming a tetrakaidekahedral shape f o r the grains, have suggested that the mean radius of subgrains or grains, should be given by the r e l a t i o n 33 t _ 1.675 ~2 ~ — B — (13a) where B i s the r a t i o of grain surface to grain volume. Furthermore, Smith 6 8 and Guttman have shown that B i s twice the number of intercepts of bound-aries along a unit length of random l i n e . Therefore B = ~ (13b) Combining Equations. (13a) and (13b) t = 1.675 t" (13) The mean l i n e a r intercept, t " , was evaluated from ten or more photo-micrographs in v o l v i n g at l e a s t 400 intercepts. The value of t obtained i n th i s way i s believed to be accurate within ±3%. 2.5 X-Ray D i f f r a c t i o n Specimens for X-ray d i f f r a c t i o n work, a f t e r heat treatment, were l i g h t l y polished on a series of emery papers u n t i l a l l surface blemishes had been removed. Generous amounts of kerosene were used to provide l u b r i c a t i o n for the p o l i s h i n g . The surface to be examined was then lapped using 5ym diamond paste. This was followed by e l e c t r o p o l i s h i n g i n a chrome-acetic so l u t i o n for an hour, which removed approximately 50um from the surface of the spe c i -men, and thus provided a surface which was believed to be free of any deformed layer due to e a r l i e r p o l i s h i n g . An x-ray l i n e broadening analysis o r i g i n a l l y due to Warren and Aver-69 1 bach was used to estimate the non-uniform l a t t i c e s t r a i n (such as that caused by l i n e defects) and the small c o h e r e n t l y - d i f f r a c t i n g domain s i z e i n the matrix of the heat treated a l l o y s . The technique, discussed more f u l l y i n the next chapter of the thesis, involves, (i) accurate scanning of 34 two orders of r e f l e c t i o n s from the same plane (hk£) of the specimen under examination, ( i i ) removing the instrumental l i n e broadening from the observ-ed p r o f i l e s by scanning a reference specimen under conditions i d e n t i c a l to those for the specimen under examination, and ( i i i ) separating q u a n t i t a t i v e l y the l i n e broadening due to l a t t i c e s t r a i n s from that due to coherently d i f -f r a c t i n g domains. In the present work, which involved only b.c.c. matrixes, (110) and (220) r e f l e c t i o n s were used. The corresponding values of the Bragg angle, 6, using CoK^ r a d i a t i o n are approximately 26° and 62°. The reference specimen used was of the same s t e e l as the specimen under inves-t i g a t i o n . I t had e a r l i e r been annealed at 700°C for 48 hours and slowly cooled to room temperature, thus giving a specimen r e l a t i v e l y free of l a t -t i c e s t r a i n s and with a large domain s i z e . The apparatus used for x-ray d i f f r a c t i o n was a P h i l i p s x-ray generator with a v e r t i c a l goniometer coupled to a step scanning device, and a high speed d i g i t a l p r i n t e r to record the output from the counter and the timer on a paper s t r i p . Cobalt r a d i a t i o n , generated at 38 KV and 18 ma , was used throughout the work. An i r o n f i l t e r located at the tube s h i e l d was employed to eliminate the r a d i a t i o n . The p a r t i c u l a r p o s i t i o n of the f i l t e r was chosen on the basis of the manufacturer's recommended prac-t i c e when the specimen to be examined and the f i l t e r to be used are of the same material. A set of 2-degree divergence and scatter s l i t s combined with a 0.1 mm receiving s l i t , were found to be suitable f or the angular range and specimen s i z e used i n t h i s i n v e s t i g a t i o n . The x-ray pulses were detected by a proportional counter. The proportional counter was preferred over a s c i n t i l l a t i o n counter because of i t s greater s e n s i t i v i t y (smaller resolution time) and i t s a b i l i t y to be used i n conjuntion with a pulse height analyser. On the control panel f o r the goniometer, the high tension 35 and pulse height analyser were adjusted by a procedure described i n the P h i l i p s instrument manual whereby the peak-to-background r a t i o i s maximised. Pulses from the pulse shaper were attenuated by a fac t o r of 8, and then were counted by the counter. Before s t a r t i n g the d i f f r a c t i o n work, the goniometer was checked f o r alignment and ca l i b r a t e d using a s i l i c o n standard. The step-scanning of any peak was preceded by a long range continuous scan. A goniometer speed of 0.25° (20) min \ a chart speed of 1 cm. min ^ and a time constant of 4 seconds were employed f o r t h i s purpose. From the peak p r o f i l e so obtained, a v i s u a l estimate was made of the 26 l i m i t s of the r e f l e c t i o n ; i . e . where the t a i l s of the r e f l e c t i o n had completely merged with the background. This 20 range f o r both r e f l e c t i o n s from a l l specimens never exceeded 8°; v i z . 48° to 56° f o r the (110) r e f l e c t i o n and 120° to 128° for the (220) r e f l e c -t i o n . Next, a step-scan procedure was used i n which the goniometer advanced automatically i n steps of 0.02° (20) s t a r t i n g at the low angle l i m i t f o r that r e f l e c t i o n . At each step, the timer output was printed as the time required ( i n seconds) to receive 20,000 x-ray pulses. A preset count meas-urement made was employed to achieve a constant r e l a t i v e s t a t i s t i c a l error f o r every r e s u l t . The number of x-ray pulses received at each step, 20000, was chosen on the basis of C l e g g ' s ^ work wherein he showed that a t o t a l of 20,000 counts has a 96% p r o b a b i l i t y of being within an err o r of 1.4%. Also the choice of voltage, milliamperage, s l i t s and pulse height analyser were such that the time required to receive 20,000 pulses always exceeded one second thereby keeping the error i n the timer and the s t a t i s t i c a l error i n the counter to approximately the same magnitude of about one percent. 36 Thus, the output from the p r i n t e r consisted of a set of 401 numbers fo r each r e f l e c t i o n , each number representing the inverse of the x-ray i n -te n s i t y d i f f r a c t e d at 0.02° steps across the (110) and (220) r e f l e c t i o n s from each specimen. The numerical data so obtained, however, had to be manually treated f o r the presence of c e r t a i n overlapping r e f l e c t i o n s from cementite or copper, as observed from the continuously scanned peak p r o f i l e . These r e f l e c t i o n s from the p r e c i p i t a t e s are l i s t e d below: Cu (111) at 20 = 50.7° Fe 3C (121) at 26 = 50.4° Fe 3C (211) at 26 = 54° The Cu (111) r e f l e c t i o n was detected only i n the reference specimen of the Fe-Cu-Ni a l l o y . The cementite r e f l e c t i o n s were observed i n tempered medium carbon martensite and i t s reference specimen. In the case of low carbon s t e e l , the cementite r e f l e c t i o n s were barely discerned. The c o n t r i -butions to i n t e n s i t y from these overlapping r e f l e c t i o n s were removed by smoothing out the trace on the continuously scanned peak p r o f i l e i n the appropriate regions. This was next used as a guide to correct the corre-t sponding values of times i n the p r i n t e r output. The approximate nature of t h i s correction f o r the presence of p r e c i p i t a t e r e f l e c t i o n s Is not expected to have any s i g n i f i c a n t e f f e c t on the accuracy of the analysis since the associated 26 values are such that, the d i f f r a c t e d X-ray i n t e n s i t y i s only s l i g h t l y higher than the background i n t e n s i t y . No overlapping r e f l e c t i o n s , due e i t h e r to Fe^C or Cu, were observed i n the v i c i n i t y of (220) r e f l e c t i o n s , and hence no such correction was required for the high angle l i n e . The modified numerical data for (110) and (220) r e f l e c t i o n s , i n con function with corresponding data from the reference specimen, was punche 69 on to data cards and subjected to a Warren and Averbach analysis on a computer to y i e l d values for root mean square (r.m.s.) l a t t i c e s t r a i n and for the coherently d i f f r a c t i n g domain s i z e . The s a l i e n t features of the 70 program, o r i g i n a l l y written by Clegg , are given i n Section 3.4. 38 3. X-RAY LINE BROADENING ANALYSIS 3.1 P r i n c i p l e s Underlying X-Ray Line Broadening Analysis Nonuniform l a t t i c e s t r a i n and a small coherently d i f f r a c t i n g domain si z e (also referred to as ' p a r t i c l e ' s i z e and ' c r y s t a l l i t e ' size) are known 71 to cause x-ray l i n e broadening. Scherrer f i r s t showed that the pure x-ray d i f f r a c t i o n broadening, 8 , due to a small c r y s t a l l i t e domain s i z e , p, i s given by the equation KX C14) p p.cost) where K i s a constant approximately equal to unity, X i s the wavelength of the r a d i a t i o n employed and 9 i s the Bragg angle for the p a r t i c u l a r r e f l e c -t i o n Involved. The quantitv 8 i s d i f f e r e n t from the breadth B of a d i f f r a c -- p t i o n l i n e as a c t u a l l y observed under experimental conditions. 3^ i s the pure breadth of a hypothetical r e f l e c t i o n free of a l l broadening due to the experimental method employed i n observing i t . On the other hand, broadening due s o l e l y to nonuniform l a t t i c e s t r a i n , 72 g g, was shown by Stokes and Wilson to be given by: g = 2 e h k £ t a n 0 (15) where e, , . i s the s t r a i n i n (hk£) d i r e c t i o n and 6 i s the Bragg angle. hk£ Since any observed l i n e p r o f i l e w i l l have some broadening, b, due to the-39 non-perfect nature of the x-ray o p t i c s , the above two equations cannot be put to d i r e c t use. To circumvent t h i s problem, the instrumental broaden-ing i s evaluated by observing the breadth of the f;ame r e f l e c t i o n from a well-annealed material having a large c r y s t a l l i t e s i z e and with n e g l i g i b l y small l a t t i c e s t r a i n s . Thus, i n p r i n c i p l e , i f B and b are experimentally known, one could evaluate 3 or 3 and substitute i n the above two equa-. p s H tions to obtain c r y s t a l l i t e s i z e or non-uniform l a t t i c e s t r a i n . Even i f the two sources of l i n e broadening are present simultaneously, one could make use of t h e i r d i f f e r e n t dependences on 6 or on A and thus, by scanning more than one r e f l e c t i o n , evaluate both the domain s i z e and the l a t t i c e s t r a i n . In p r a c t i c e , however, the exact determination of 3 or 3 ( or 3 when p s both sources of broadening are present) from the two experimentally deter-mined quantities B and b i s not easy. The re l a t i o n s h i p between B, b and 3 depends on the form of the mathematical function which gives the best f i t with the p r o f i l e of the d i f f r a c t i o n l i n e . I f I i s the x-ray i n t e n s i t y at 2 2 any point a distance x from the peak p o s i t i o n and i f I = 1^/(1 4- k x ); i . e . the p r o f i l e has a Cauchy form, then several workers have shown that 3 = B - b S i m i l a r l y , i f the l i n e p r o f i l e has a Gaussian form i . e . I = I exp 2 2 73 (-k x ), then according to Warren 2 2 2 3 = B - b These two r e l a t i o n s have no un i v e r s a l v a l i d i t y since the x-ray l i n e p r o f i l e s can r a r e l y be described as completely Cauchy or Gaussian curves. These uncertain assumptions are necessitated once again to rel a t e 3 , 3 and 3 . s 40 This d i f f i c u l t y has been overcome by various investigators by express-ing the experimental p r o f i l e s as a Fourier s e r i e s . Since i t can be shown that the x-ray l i n e p r o f i l e observed from a specimen i s a convolution of the functions representine instrumental broadening and broadening due to nonuniform l a t t i c e s t r a i n and small domain s i z e , a l l that i s required i s to unfold the convolution and extract the l a t t e r function, o r i g i n a l l y buried inside the convolution. A rigorous method, inv o l v i n g complex integration? 74 and d i v i s i o n s has been presented by Stokes . The end r e s u l t of Stokes cor-rection f or instrumental broadening i s a Fourier s e r i e s describing the pro-f i l e due to i n t r i n s i c broadening from the specimen. Furthermore, i t has been emphasized that the c o e f f i c i e n t s i n t h i s Fourier s e r i e s are functions of nonuniform l a t t i c e s t r a i n and of the coherently d i f f r a c t i n g domain s i z e . Thus, i n t h i s approach, i t i s possible to obtain quantitative estimates for both l a t t i c e s t r a i n and domain s i z e , without making any assumption re-garding the shape of the l i n e p r o f i l e s . Excellent reviews of the subject have been given by Warren^ and C l e g g ^ and the present study i s based on these two works. A b r i e f d e s c r i p t i o n of the mathematical formulation i n the above two works i s presented i n the following section. 74 3.2 Stokes Correction 76 The p r i n c i p l e of the correction has been described by Warren i n terms of the three curves of Figure 4. The curve f(y) represents the i n -t r i n s i c broadening from the specimen and t h i s i s the function to be evalu-ated. The curve g(z) represents the e f f e c t of instrumental broadening only such as that obtained from the reference specimen. The curve h(x) i s the l i n e p r o f i l e experimentally observed from the specimen, and i t contains Figure 4 Stokes Correction f o r Instrumental Line Broadening (Warren ). 42 both the i n t r i n s i c broadening and the instrumental broadening. The r e l a t i o n between the three curves i s obtained by considering an element of area g(z)dz on the curve for instrumental broadening only. Due to the presence of i n t r i n s i c broadening from the specimen, the area i s . spread by the function f ( y ) . At a displacement y, the ordinate i s the con-t r i b u t i o n dh(x) at the p o s i t i o n x = z + y on the h(x) curve. Since the.or-dinates i n the two curves are proportional to the peak areas, dh(x) = f(y) g(z)dz A where A i s the area of the f(y) curve. Substituting y = x - z, the ordinate of the h(x) curve i s given by h(x) = j- f g(z)f (x - z)dz (16) Equation (16) shows that the observed p r o f i l e from the sample i s a convolu-ti o n of the function representing the i n t r i n s i c broadening of the specimen and the instrumental broadening. To unfold the convolution, i n order to obtain f ( y ) , l e t the three func-tion f ( y ) , g(z) and h(x) be expressed as Fourier ser i e s i n an i n t e r v a l -a / 2 to +a / 2 . f(y) = g F(n)exp ( - 2 T T i n y/a) (17a) g(z) = E, G(n')exp ( - 2 T r i n'z/a) (17b) h(x) = J„ H(n")exp ( -27Tin"x/a) (17c) Since h(x) i s the broadest p r o f i l e of the three, the i n t e r v a l -a / 2 to +a /2 should be large enough to include everything i n the h(x) curve. 43 Substituting Equations (17a), (17b) and (17c) into (16), +a/2 h(x) = x / ^,G(n ,)exp(-2Trin'z/a) ^F(n)exp [-2Trin( x-z)/a]dz A / n xi -a/2 The l i m i t s f o r the i n t e g r a l are so chosen because g(z) i s non-vanishing only i n t h i s i n t e r v a l . Combining the terms invoving z, ; +a/2 h(x) = E, EG(n')F(n)exp(-2Trinx/a) / exp [-2iTi (n' -n) z/a] dz A n n j -a/2 +a/2 Since / exp[-2Tri(n'-n)z/a]dz = a for n' = n -a/2 o for n' 4 n, the equation s i m p l i f i e s to h(x) = f EG(n)F(n)exp(-2TTlnx/a) A n Comparing with Equation (17c) and comparing c o e f f i c i e n t s | G(n)F(n) = H(n) The m u l t i p l i c a t i o n factor — can be omitted since only shapes, of the curves influence the analysis. The Fourier c o e f f i c i e n t of the f(y) curve i s then given by the r e l a t i o n , F(n) = (18) In general, the three c o e f f i c i e n t s i n Equation (18) are complex i . e . have r e a l and imaginary parts. Let su f f i x e s ' r' and ' i ' denote r e a l and imaginary. Equation (18) then becomes r M , . . . n Hr(n) + iHi(n) Fr(n) + .iFi(n) = 7 ^ ) + i G 1 ( n ) 44 M u l t i p l y i n g the numerator and denominator on the r i g h t hand side by the complex conjugate of the denominator and equating the r e a l and imagi-nary parts, we obtain r? *\ _ Hr(n)Gr(n) + Hi(n)Gi(n) Gi: (n) + Gi (n) and (19) _,. f v Hl(n)«Gr(n) - Hr(n)»Gl(n) Fi(.nj = r • • — Gr (n) + Gi(n) The i n t r i n s i c broadening function f(y) can then be synthesized to give f(y) = ^{Fr(n)cos2imy/a + Fi(n) sin2Trny/a} (20) The c o e f f i c i e n t s Hr(n), Gr(n), Hi(n) and Gi(n) are evaluated by using the Fourier transform i n t e g r a l s f or the observed curves h(x) and g(z). Thus, using Stokes correction, Fr(n) a n d F i ( n ) and hence the i n t r i n s i c broadening p r o f i l e can be determined. However, to carry out the transform integrations an o r i g i n has to be chosen for the sin0 axis. The centroid of the p r o f i l e was chosen f o r t h i s purpose since Wagner^ has shown that i t reduces the o s c i l l a t i o n s i n the r e a l c o e f f i c i e n t s and also makes the imaginary c o e f f i c i e n t s small. Symmetry i n the peak p r o f i l e also renders the imaginary c o e f f i c i e n t s small. The doublet, on the other hand, causes asymmetry on the high angle sides of p r o f i l e s and thus i t i s desirable to correct for the con-t r i b u t i o n of the K component of the r a d i a t i o n used. This can be achieved a 2 either by using a c r y s t a l monochromator, or, as was done i n the present' 7 8 study, by a step-wise procedure known as the Rachinger correction . This procedure makes the assumption that the i n t e n s i t y of the K component i s a2 one ha l f that of K , except that i t i s h a l f the height and i s s h i f t e d to-. " l 4 5 wards large angle by A (26) = 2 tan6 AX/X where AX = X(a 2> - X(a^). 3.3 Evaluation of Domain S i z e and Nonuniform L a t t i c e Strains From a specimen having nonuniform l a t t i c e s t r a i n s and small c r y s t a l -l i t e ' domain s i z e , the d i f f r a c t e d x-ray power per unit length of d i f f r a c -t i o n l i n e (hOO) at a given Bragg angle, 9, i f given by the r e l a t i o n ^ : q(28) = K ( 6 ) | - |N(t) [cos2TTJ 1t - c o s 2 7 T h X(t) - s in2Trj ±t-sin2TrhX(t) ] (21) where i s a function of the angle 26, h i s the order of the r e f l e c t i o n (hOO), X(t) i s a function of the l a t t i c e s t r a i n and N(t)/N^ i s a function of the domain s i z e f o r integer values of the harmonic number, t. To obtain X(t) and N(t)/N^ experimentally, they are combined i n the following equa-tions A(t) = ^^-cos2TrhX(t) 1 B(t) = ^ ^ s i n 2 7 T h X ( t ) 1 From Equation (21) the function K(6) can be eliminated since angular corrections due to the L o r e n t z - p o l a r i s a t i o n and atomic sc a t t e r i n g factors were applied to a l l the r e f l e c t i o n s . The v a r i a b l e can be redefined i n terms of the distance x on (26) axis of the experimentally recorded x-ray d i f f r a c t i o n p r o f i l e ; x being i n units of s i n 9 . In addition, as shown by Wagner^, the harmonic number, t, i s related to the true l a t t i c e distance L (in a d i r e c t i o n perpendicular to the d i f f r a c t i n g planes) through an a r t i -f i c i a l l a t t i c e parameter. Thus, the t h e o r e t i c a l x-ray i n t e n s i t y of Equa-tio n (21) can be expressed i n the form of the following equation 46 q(x) = E [ A ( L ) C O S 2 T T L Y L + B (L^i^TrL- 2^-] (22) where A(L) = ^ | ^ c o s 2 T T h X(L) 1 and B (L) = ^^sin2TrhX(L) Comparing Equations (20) and ( 2 2 ) , t h e i r s i m i l a r i t y i s immediately noticed where A(L) corresponds to Fr(n ) , the l a t t e r being q u a n t i t a t i v e l y estimated through Equation (19) and the Fourier transform r e l a t i o n s h i p s . Next, to estimate domain s i z e and l a t t i c e s t r a i n , l e t N„9^  be de-noted as the Domain Size C o e f f i c i e n t , A P ( L ) , and cos2-rrhX(L) as an order-dependent s t r a i n c o e f f i c i e n t , A (Ljh 1). Thus A(L,h) = A P(L)-A S(L,h) ? VT 2 2 or £nA(L,h) = £nA P(L) - ^ n L C a Warren^ has shown that the above equation, though derived f o r (hOO) r e f l e c t i o n s , i s equally applicable to any (hk£) r e f l e c t i o n s when h i s r e -/"~2~ 2 2 placed by hQ =\Jh + k + I . The s t r a i n , e, i s then measured i n the appropriate (hk£) d i r e c t i o n . Thus the equation can be rewritten as ? 2 > A 2 2 £nA(L,h Q) = £nA P(L) - ^ " e ^ £ (23) a It can be shown that at L '= 0, A P(L) = A S(L,h Q) = 1, and hence, A(L,h o) = 1. Therefore, the experimentally determined values of F r ( n ) , (equivalent to A ( L } h o ) ) , before being inserted i n Equation (23), are normalised to unity f o r L = 0. To u t i l i z e Equation (23) , v a r i a t i o n s i n h^ are required. In p r i n c i p l e 47 any two or more r e f l e c t i o n s could be used. He^wever, metals always exhibit anisotropy, which makes i t e s s e n t i a l to use d i f f e r e n t orders of the same r e f l e c t i o n . The value of the s t r a i n s so obtained would then be i n the d i r e c t i o n i . e . , perpendicular to the d i f f r a c t i n g planes. In the pre-sent study of b.c.c. ferrous a l l o y s , the (110) and (220) r e f l e c t i o n s were 2 used, the respective values for h being 2 and 8. Thvjs two se r i e s of values of A(L,h Q) are obtained by experiment f o r a range of values of L, one for each r e f l e c t i o n . From these, to i n t e r p r e t Equation (23) , a set of s t r a i g h t l i n e s can be drawn with values of £nA(L,h Q) 2 as.ordinates and the two values of h as abscissae, each chosen value of o 'L' r e s u l t i n g i n one l i n e . The slope of each of these l i n e s i s equal to , 2 2 2 - — 2—> from which values of mean square s t r a i n e are obtained for a a range of values of L. This root mean squared s t r a i n i s measured i n a d i r e c -t i o n normal to the r e f l e c t i n g planes, averaged over the length L, squared and averaged over a l l regions i n the specimens. The l a t t i c e s t r a i n data 2 1/2 i s f i n a l l y expressed as a graph of root mean square s t r a i n (e ) versus 2 1/2 L, the l a t t i c e distance. It i s observed i n these plots that (e ) r i s e s quite sharply with decreasing L and the s t r a i n values near L = 0 are quite uncertain. By contrast, i t was observed that, f o r specimens which had been rather heavily aged or tempered, the s t r a i n was zero from L = 0 to some f i n i t e value of the l a t t i c e distance (Section 4.3.3). I t could not be decided i f this was a c h a r a c t e r i s t i c of the polygonised structure or due to some u n r e l i a b i l i t y of the analysis for small values of L. To avoid such d i f f i c u l t i e s , i t i s customary to quote s t r a i n values at some a r b i t r a r y l a t t i c e distance. In the present study, s t r a i n values at 100 A° l a t t i c e distance were used. 2 2 From the intercepts of £nA(L,h ) versus h plo t s at h = 0, a serie s o o o 48 of values of £nA (L) are obtained, converted to A (L), normalised to unity forL = 0 and plotted as a function of L. Such a curve i s generally ex-79 pected to be a smooth one and Bertaut has shown that the negative r e -c i p r o c a l of the i n i t i a l slope at L = 0 i s the domain s i z e i t s e l f . Since A P(L) has already been normalised to unity a L = 0, i t implies that, i n the i d e a l case, the intercept on the L axis of the tangent to the A P(L) versus L; curve at L = 0 gives the domain s i z e d i r e c t l y . However, i n prac-t i c e , due to d i f f i c u l t i e s involved i n accurate measurement of background, a region of negative curvature, c a l l e d the "hook e f f e c t " , occurs i n this p l o t at or near L = 0. The domain s i z e obtained by considering the slope at L = 0, therefore, would have no phy s i c a l meaning. To overcome this d i f -f i c u l t y , an approximate value f o r domain s i z e i s f i r s t c alculated from twice the area under the curve of A P(L) versus L, and then a Better e s t i -mate i s obtained from the negative r e c i p r o c a l slope at a distance from the o r i g i n equal to 20 pet. of the domain s i z e f i r s t c a lculated. In f a c t , how^ ever, due to the presence of small o s c i l l a t i o n s , i t was found best to judge the p l o t t e d curve on i t s own merits and to measure the negative r e c i p r o c a l slope at an appropriate point to avoid errors due to the hook e f f e c t . 3.4 Computer Program The computation and p l o t t i n g described i n the previous section were carried out on an IBM 360/67 computer, using a program written by C l e g g ^ . The s i g n i f i c a n t tasks performed are l i s t e d below: (a) Corrected for the angular dependence of x-ray i n t e n s i t y , i n c l u d -ing that of atomic s c a t t e r i n g f a c t o r . (b) Converted the data into corresponding i n t e n s i t y values at equal 49 i n t e r v a l s of sinQ along the abscissa. (c) Subtracted the background i n t e n s i t y . 78 (d) Applied the Rachinger correction to remove the i n t e n s i t y con-t r i b u t i o n from the K component of the K doublet. a2 a (e) Calculated the centroid of the r e f l e c t i o n and rescaled the abscis-sa to the centroid as o r i g i n . (f) Carried out the integrations and complex d i v i s i o n s to obtain Fourier c o e f f i c i e n t s as a function of l a t t i c e distance, L, f o r each r e f l e c -t i o n , and normalised the values to unity for L = 0. (g) Used the Fourier c o e f f i c i e n t s to c a l c u l a t e the root mean square 2 1/2 l a t t i c e s t r a i n as a function of l a t t i c e distance i . e . (e ) versus L, and the domain s i z e c o e f f i c i e n t A^(L) as a function of L, as described i n Section, 3.3. 2 l/2 (h) Plotted (e ) versus L and A^(L) versus L as graphical outputs. ( i ) Calculated the coherent c r y s t a l domain s i z e from the l a t t e r graph according to the two c r i t e r i a discussed i n Section 3.3, The domain sizes and s t r a i n s so obtained were interpreted, by means 80 of an analysis due to Williamson and Smallman , i n terms of the density and configuration of d i s l o c a t i o n s . 80 3.5 Williamson and Smallman Analysis The b a s i c premise of t h i s analysis i s that i n a cold worked metal con-taini n g a d i s l o c a t i o n substructure, the same di s l o c a t i o n s are responsible for both the l a t t i c e s t r a i n and the small domain s i z e that cause x-ray l i n e broadening. Furthermore, the r e l a t i v e amounts of l a t t i c e s t r a i n and p a r t i c l e s i z e broadening are determined by the manner i n which d i s l o c a t i o n s are grouped; i . e . , i n high energy p i l e ups, i n low energy small angle bound-50 a r i e s , or as i n d i v i d u a l random d i s l o c a t i o n s . According to Williamson and Smallman, the domain s i z e , p, and the d i s -location density associated with i t , p^, are r e l a t e d as follows P ' = % (24) P P 2 This r e l a t i o n s h i p assumes that the c r y s t a l has cube-shaped domains, there being 'n' d i s l o c a t i o n s along each face of a domain. S i m i l a r l y the l a t t i c e s t r a i n , e, and the d i s l o c a t i o n density associ-ated with i t , p^ , are r e l a t e d by the expression P ' - - ^ (25) Fyb £n(r/r ) o where E i s Young's modulus f o r the matrix u i s the shear modulus, b i s the Burgers vector of a d i s l o c a t i o n i n the matrix r / r ^ i s the r a t i o of the radius of the volume of c r y s t a l containing the d i s l o c a t i o n to the d i s l o c a t i o n core radius, which arises from the usual 4 expression for d i s l o c a t i o n energy. (Its value i s customarily taken as 10 ) F i s a d i s l o c a t i o n i n t e r a c t i o n factor . Substituting appropriate values for i r o n i n Equation (25) , one obtains p ' = 1.97 | x 1 0 1 6 (26) s F To u t i l i z e Equation (24) and (26) we i n i t i a l l y assign a value of unity to both n and F. This implies that there i s only one d i s l o c a t i o n per do-main w a l l and that no d i s l o c a t i o n i n t e r a c t i o n takes place; i . e . , the s t r a i n f i e l d s of the d i s l o c a t i o n s do not overlap. The respective d i s l o c a t i o n den-s i t i e s become P = \ (27) P D 2 51 and p g = 1.97e 2 x 1 0 1 6 (28) If i t i s i n fact found experimentally that = p g , then the assump-tio n that n and F equal unity i s e s s e n t i a l l y correct. The actual d i s l o c a -tion density, p, then i s equal to p^ or p g and d i s l o c a t i o n s are taken to be d i s t r i b u t e d at random i n the specimen. I f p > p , then a pile-up configuration i s indicated. Both F and n s p are greater than unity and F = n. Substituting Equations (27) and (28) i n f o (24) and (26) r e s p e c t i v e l y , and recognising that both the p a r t i c l e s i z e and s t r a i n are associated with the same true d i s l o c a t i o n s d e n s i t i e s , the value of n or F may be obtained as follows I I s s n pP = PP = P s " IT = ~ 2 P s n - — P P P s 1/2 and n = (—) P n U n ( 2 9 ) or p = (p p ) s p In a ' p i l e up' model, n i s the average number of d i s l o c a t i o n i n the p i l e up. F i n a l l y i f p^ > p g , then a polygonized substructure i s indicated. Since the energy of each d i s l o c a t i o n i s reduced i n a polygonized array, i t P s s i g n i f i e s a value of F less than unity Thus p = — and p > p r S * However, n remains greater than unity and so the actual d i s l o c a t i o n density p would even be greater than p . hus, under conditions of poly-P gonization, P > P„ > P p p s Under such conditions, Williamson and Smallman have suggested the following 52 r e l a t i o n f o r p P = p Zn s r 7 10 x 3 2 ^ / In 3 x 10 (30) It i s d i f f i c u l t to solve Equation (30) by t r i a l and error. Also the equation can* at best, give only crude estimates of p. Accordingly, i n the present work no d i s l o c a t i o n d e n s i t i e s have been estimated i n cases where a polygonized array was indicated by the data. 53 4. RESULTS AND DISCUSSION 4.1 Tensi l e and Hardness Tests Tables II and III summarize the y i e l d strength, ultimate t e n s i l e strength and t o t a l elongation values for the medium carbon s t e e l , the copper bearing s t e e l and Ferrovac E i r o n , i n various heat-treated conditions. The y i e l d strength values correspond to a 0.2% o f f s e t flow stress and were determined by the procedure described e a r l i e r i n Section 2.3.1. At least two spec i -mens were tested for each heat-treated condition. In the case of the copper--2 bearing s t e e l , with one exception where the difference was ~ 20 MNm , dup-l i c a t e r e s u l t s f o r y i e l d stress and ultimate t e n s i l e strength always agreed -2 within ~ 7 MNm . In the case of the medium carbon martensite, duplicate _2 test values f o r y i e l d stress agreed within 20 MNm . These maximum d i f f e r -ences, with the one exception noted, represent only ± 1 to 3 percent v a r i a -t i o n about the mean values quoted i n the Tables. Table II also contains Vickers hardness values f o r the low carbon tempered martensite, which were subsequently converted to y i e l d strength 60 values using the data of D e l i s l e and Galibois f o r 0.05% C tempered mar-tensite, The conversion i s believed to be r e l i a b l e i n the tempering tempera-ture range 400-700°C since the absolute y i e l d strength values i n the present 81 work compared favourably with Kunitake's reported values f o r a 0.13% C 54 Table II Tensi l e Results for the Medium Carbon and the Low Carbon Tempered Martensites. Medium Carbon Steel (0.42%C. 1.1%Mn) Low Carbon Steel (0.11% C, 0.9% Mn) Tempering Temperature °C 0.2% o f f s e t flow s t r e s s , MNm"2. Ultimate Tens i l e Strength, MNm-2 T o t a l Elongation i n percent Hardness DPH, Load 5 Kgms. Estimated Y i e l d Strength MNm-2 250 1600 1860 12 355 1040 300 1530 1750 13 349 1010 350 1450 1630 13 315 900 400 1300 > 1410 14 296 830 450 1230 1340 14 260 780 500 1040 1170 16 239 700 550 900 1090 17 215 600 600 840 1010 19 195 520 650 690 880 23 176 440 700 540 780 27 157 360 T Table III Tens i l e Results for Fe-Cu-Ni a l l o y and Ferrovac 'E' Iron. Series A: Quenched and Aged Series B: Quenched, Cold Rolled and Aged Aging Time at 500°C, hours 0.2% o f f s e t Flow stress MNm"2 Ultimate Ten s i l e Strength MNm""2 T o t a l Elongation i n percent 0.2% o f f s e t Flow st r e s s MNm"2 Ultimate T e n s i l e Strength MNm"2 T o t a l Elongation i n percent 0 285 395 23 635 690 = 3 0.16 380 520 20 685 760 11 0.5 495 635 20 705 800 13 1 525 675 20 700 830 13 2 525 685 19 685 815 13 5 505 670 16 665 780 14 10 475 635 18 635 760 13 30 585 690 11 60 565 675 11 100 375 490 16 540 655 11 Pure Iron 160 200 29 310 340 = 3 (As-quenched) 56 martensite tempered at temperatures greater than 500°C. However, the values for 250°C, 300°C and 350°C tempering can be considered approximate only, since d i r e c t conversion from a v a i l a b l e experimental values was not possible. The y i e l d strength vs tempering temperature data of Table I I , for the medium carbon and the low carbon martensites, are plo t t e d i n Figure 5. Simi-l a r l y , from Table I I I , the y i e l d strength of the Fe-Cu-Ni a l l o y has been plo t t e d as a function of aging time i n Figure 6. I t i s observed from these plo t s that y i e l d strengths of the medium carbon and low carbon tempered mar-tensites decrease continuously as the tempering temperature i s increased from 250 to 700°C, the tempering time being f i x e d at 1 hour. In the Fe-Cu-Ni a l l o y , on the other hand, an age hardening peak was observed at approximately 1 - 2 hours f o r Series A, and at approximately 1/2 hour f o r Series B (cold r o l l e d p r i o r to aging). 4.2 Replica Electron Microscopy Table IV contains the measured and derived values of various dispersion parameters f o r the. medium carbon and the low.carbon tempered martensites. Values of the planar mean free path, A, were determined from r e p l i c a e l e c -tron micrographs of which several representative examples are reproduced, i n Figures 7 - 12, Figure 13 i s a p l o t of the planar edge-to-edge cementite p a r t i c l e spac-ing for both the medium and the low carbon martensites as a function of tem-pering temperature. In both cases, the spacings show a gradual increase at low tempering temperatures and sharp r i s e at higher tempering temperatures i n the range studied. This behaviour probably r e f l e c t s the exponential, dependence of d i f f u s i o n rate (of carbon) on temperature. I t i s also clear from F i g . 13 that the carbide spacing increases much more rap i d l y with i n -1500 -100 -700 -300 200 Figure 5 400 600 TEMPERING TEMPERATURE , °C 800 Y i e l d Strength vs Tempering Temperature f or the Medium and Low Carbon Tempered Martensites. 280* 1 1 1 0 10"' 10° io' io 2 AGING TIME , hr Figure 6 Y i e l d Strength vs Aging Time for the Fe-Cu-Ni A l l o y . Table IV Dispersion Parameters f o r the Medium Carbon and the Low Carbon Tempered Martensites Medium Carbon Steel Low Carbon Stee 1 Tempering -Temperature A D d D A D 0 D °C s s s s 250 0. 27 0.085 0 .023 0.06 0. 75 0. 115 0. 015 0. 10 300 0. 29 0.09 0 .025 0.065 0. 79 0. 125 0. 016 0. 11 350 0. 30 0.095 0 .026 0.07 0. 87 0. 135 0. 017 0. 12 400 0. 32 0.10 0 .028 0.075 0. 91 0. 14 0. 018 0. 125 450 0. 35 0.11 0 .030 0.08 1. 06 0. 165 0. 021 0. 15 500 0. 39 0.125 0 .034 0.09 1. 62 0. 255 0. 032 0. 22 550 0. 44 0.14 0 .038 0.10 2. 30 0. 36 0. 046 0. 315 600 0. 51 0.16 0 .044 0.115 3. 40 0. 53 0. 07 0. 46 650 0. 83 0.26 0 .072 0.19 4. 20 0. 65 0. 085 0. 57 700 1. 32 0.415 0 .114 0.30 7. 80 1. 22 0. 155 1. 06 A l l numbers i n microns 59 I JiL Figure 7 Replica Electron Micrograph of the Medium Carbon Martensite Tempered at 350°C. Figure 8 Replica Electron Micrograph of the Medium Carbon Martensite Tempered at 500°C. 60 c c < • C\ v . c c c Figure 9 Replica Electron Micrograph of the Medium Carbon Martensite Tempered at 700°C. Figure 10 Replica Electron Micrograph of the Low Carbon Martensite Tempered at 300 C, ure 11 Replica Electron Micrograph of the Low Carbon Martensite Tempered at 450°C. • > ,;V-1C « V • • — T V — ' 4-gure 12 Replica Electron Micrograph of the Low Carbon Martensite Tempered at 600^C. 0 1 — - — ' J 1— 1 I I 200 400 600 800 TEMPERING TEMPERATURE , °C Figure 13 Edge-to-Edge P a r t i c l e Spacing, D, vs Tempering Temperatu For the Medium and Low Carbon Martensite. 63 creasing tempering temperature for the low carbon s t e e l than for the medium carbon s t e e l . Thus, the p a r t i c l e spacing i s as low i n the medium carbon s t e e l a f t e r tempering at 550°C as i t i s i n the low carbon s t e e l a f t e r tem-pering at only 250°C. 4.3 X-Ray D i f f r a c t i o n 4.3.1 General As mentioned i n Section 3.3, for each specimen the output from the com-puter consisted of graphs of l a t t i c e s t r a i n and domain size c o e f f i c i e n t ver-sus the l a t t i c e distance, with two values for the domain s i z e . Some t y p i c a l plots for the three a l l o y systems are shown i n Figures 1 4 - 1 9 . I t was sometimes observed that at large values of l a t t i c e distance, L, the domain s i z e c o e f f i c i e n t and s t r a i n values began to r i s e . This anomaly i s believed to be associated with i n s t a b i l i t i e s i n the values of the Fourier co-e f f i c i e n t s i n this region and thus has been ignored. However, the f i r s t of the two values of domain s i z e reported by the computer i s subject to a large error when the anomaly i s present. The second computer value of the domain si z e i s also subject to error i n cases where small o s c i l l a t i o n s were present i n the p l o t at r e l a t i v e l y small values of L. Therefore, i n a l l cases, i t was decided to determine manually the slope of the computer plo t of domain s i z e c o e f f i c i e n t versus l a t t i c e distance at an approximate L value of 0.2p2, where P 2 i s the domain size rs determined i n the computer by the second c r i t e r i o n . Generally, the hook e f f e c t did not extend upto 0.2p 2 and was e a s i l y avoided. In most cases, the manually determined value of domain size was close to p 2 > 2 1/2 The domain size values, p, and root mean square s t r a i n values ( ( e ) ) at 100A° are summarized i n Tables V, VI, and VII for the three a l l o y systems 64 under investigation and for pure iron specimens. Also included in the ta-bles are dislocation densities and their configurations as obtained, where 80 possible, from the Williamson and Smallman analysis (see Section 3.5). Data for the Series A copper-bearing steel specimens are not included, for reasons to be discussed. 4.3.2 Discussion of X-Ray Results in Tempered Martensites (Carbon Steels) It is observed from Tables V. and VI, and from Figures 14 to 17, that for medium carbon and low carbon tempered martensites, the domain size i n -creases and the lattice strain at 100A° decreases as the tempering tempera-ture is increased from 250°C to 700°C. There are two exceptions to this generalization in the case of the low carbon tempered martensite (Table VI); viz. lattice strains for specimens tempered at 300 and 350°C, and domain size for specimens tempered at 500 and 600°C. No explanation can be offered for the origin of these exceptions, although they may reflect the extent to which an individual result of the x-ray analysis is unreliable. Considering the dislocation densities and configurations indicated in medium carbon tempered martensite, i t is noticed that upto a tempering temper-ature of 600°C, the dislocation densities calculated from domain size (p ) P and from lattice strain(p ) are very nearly equal. In cases where p > p , s s p the value of n, the number of dislocations per domain wall or in each pile up, has a maximum value of only 1.15. On the other hand, when p > p , a P s polygonized configuration is indicated from the dislocation substructure. Again, the difference in the two values never exceeds 20%, and i t can be concluded that dislocations are distributed quite randomly after tempering at temperatures upto 600°C. Small differences in the values of p^ and p g can be attributed to two causes: (a) there is the possibility of slight errors Table V L a t t i c e Strains, Domain Sizes, D i s l o c a t i o n Densities and Configurations for Tempered Medium Carbon S t e e l Tempering Temperature °C Domain Size ! 1 A ° p' A V , 3 cm/ cm R.M.S. s t r a i n , (eZ)UZ x 100 V cm/1 3 zm. n True D i s l o c a t i o n density, p D i s l o c a t i o n Configuration 250 350 24.5 X i o 1 0 0.325 20 .7 X i o 1 0 20 X i o 1 0 Random . 300 400 18.7 X i o 1 0 0.297 17 .4 X i o 1 0 17 X i o 1 0 Random 350 500 12.0 X i o 1 0 0.270 14 .4 X i o 1 0 1.09 13 X i o 1 0 Random 400 640 7.3 X i o 1 0 0.221 9 .6 X i o 1 0 1.15 8 X i o 1 0 Random 450 765 • 5.1 X i o 1 0 0.146 4 .2 X i o 1 0 4 X i o 1 0 Random 500 1015 2.9 X i o 1 0 0.131 3 .4 X i o 1 0 1.08 3 X i o 1 0 Random 550 1225 2.0 X i o 1 0 0.114 2 .6 X i o 1 0 1.13 2.3 X i o 1 0 Random 600 1380 1.6 X i o 1 0 0.086 1 .5 X i o 1 0 1.5 X i o 1 0 Random 650 1640 1.1 X i o 1 0 0.047 0 .4 X i o 1 0 Polygonized ON Table VI L a t t i c e Strains, Domain Sizes, D i s l o c a t i o n Densities and Configurations f or Tempered Low Carbon S t e e l Tempering Domain Temperature Size °C V . A ° 3 P P 2 R.M.S. s t r a i n , p , cm/cm" ( e 2 ) 1 / 2 x 100 cm/ cm n True D i s l o c a t i o n D i s l o c a t i o n Configuration density, p 250 300 350 400 450 500 550 600 650 650 700 790 955 1160 1575 > 2000 1660 > 2000 7.1 x 10 6.1 x 10 10 10 5.5 x 10 3.3 x 10 2.2 x 10 1.2 x 10 10 10 10 10 1.1 x 10 10 0.229 0.157 0.162 0.101 0.084 0.054 0.070 0.031 0.052 10.3 x 10 10 4.8 x 10 10 5.2 x 10 10 2.0 x 10 10 1.4 x 10 10 0.6 x 10 1.0 x 10 10 10 0.2 x 10 0.5 x 10 10 10 1.2 8 x 10 10 6 x 10 10 5.5 x 10 10 > 3 x 10 > 2 x 10 10 10 Random Random Random Random/ Polygonized Random/ Polygonized Polygonized Polygonized Polygonized Polygonized ON Table VII L a t t i c e Strains, Domain Sizes, and D i s l o c a t i o n Configurations i n Fe-Cu-Ni A l l o y (Series B) and i n Cold Rolled Ferrovac 'E' Iron Aging time at Domain Size p , cm/cm R.M.S. s t r a i n P , cm/cm Configuration 500 GC, hrs. V , A° P < EV / 2xlOO " 0 670 6.7 X i o 1 0 0.067 0. 87 X i o 1 0 Polygonized 0.16 873 3.9 X i o 1 0 0.083 1.4 X i o 1 0 Polygonized 0.30 1075 2.6 X i o 1 0 0.085 1.4 X i o 1 0 Polygonized 1.0 846 4.2 X i o 1 0 0.126 3.1 X i o 1 0 Polygonized/ i o 1 0 i o 1 0 Random 2.0 1200 2.1 X 0.074 1.1 X Polygonized 5.0 1210 2.0 X i o 1 0 0.028 0.2 X i o 1 0 Polygonized 10.0 1378 1.6 X i o 1 0 0.058 0.7 X i o 1 0 Polygonized 30.0 1535 1.3 X i o 1 0 0 Polygonized 60.0 1600 1.2 X i o 1 0 0 Polygonized 100.0 1500 1.3 X i o 1 0 Polygonized ure Fe, 1540 1.3 X i o 1 0 0.048 0.5 X i o 1 0 Polygonized cold r o l l e d 50%. LATTICE DISTANCE , A Figure 14 V a r i a t i o n of L a t t i c e S t r a i n with L a t t i c e Distance i n Medium Carbon Martensite Tempered at D i f f e r e n t Temperatures. T 1 1 r LATTICE DISTANCE , A Figure 15 V a r i a t i o n of Domain Size C o e f f i c i e n t with L a t t i c e Distance i n Medium Carbon Martensite Tempered at D i f f e r e n t Temperatures. 69 T — T • 1 r LATTICE DISTANCE , A Figure 16 V a r i a t i o n of L a t t i c e S t r a i n with L a t t i c e Distance i n Low Carbon Martensite Tempered at D i f f e r e n t Temperatures. 1.0 0.9 UJ u u. u. UJ O 0.6 •<_> UJ O Q 03 Figure 17 200 400 LATTICE DISTANCE , A -1 — 650 °C 500 °C 4C0 °C 250 °C '.JO V a r i a t i o n of Domain Size C o e f f i c i e n t with L a t t i c e Distance i n Low Carbon Martensite Tempered at D i f f e r e n t Temperatures. 70 Figure 18 200 LATTICE DISTANCE , A 400 6 0 0 V a r i a t i o n of L a t t i c e S t r a i n with L a t t i c e Distance i n Fe-1.8% Cu-1.3% Ni A l l o y (Series B) and Cold Rolled Ferrovac E Iron. 1.3 UJ U. & 0 . 7 c_> UJ N co z 0 .4 a 0.1 JL - A - a g e d 0 . 5 hr Iron , 5 0 % cold ro l l ed B - a g e d 1 0 0 hr B - a g e d 0 hr 0 2 0 0 4 0 0 600 L A T T I C E D I S T A N C E , A Figure 19 V a r i a t i o n of Domain Size C o e f f i c i e n t with L a t t i c e Distance i n Fe-1.8% Cu-1.3% Ni Al l o y (Series B) and Cold Rolled Ferrovac E Iron. 71 i n determining the domain s i z e from the computer p l o t s , and (b) s t r a i n i s being assessed at an a r b i t r a r y l a t t i c e distance of 100A° Table V also i n -cludes values for a 'true' d i s l o c a t i o n density, p . When p > p , the true s p d i s l o c a t i o n density i s e a s i l y calculated as np . When p > p , the true . ' P P s d i s l o c a t i o n density i s greater than p^ hut an exact evaluation i s not pos-s i b l e and only a lower l i m i t f o r true d i s l o c a t i o n density can be placed at Pp . In the e a r l i e r stages of tempering i n medium carbon tempered marten-s i t e , there were large f l u c t u a t i o n s i n the domain s i z e c o e f f i c i e n t p l o t (Figure 15) and thus greater confidence was placed i n d i s l o c a t i o n densities determined from r.m.s l a t t i c e s t r a i n . These l a t t e r values are inserted as true d i s l o c a t i o n density for the appropriate specimens. A f t e r tempering at 650°C, p - 2.5p for the medium carbon s t e e l . Thus the substructure was p K s highly polygonized and no r e l i a b l e estimate f o r a true d i s l o c a t i o n density could be made. True d i s l o c a t i o n densities have been plotted against tempering tempera-ture i n Figure 20. I t i s notable that for the medium carbon s t e e l , a f t e r tempering at temperatures up to 400°C, the d i s l o c a t i o n density i s comparable 82 to that reported for very heavily cold worked a l l o y s . However, Speich has shown from r e s i s t i v i t y measurements that the d i s l o c a t i o n density of , 10 3 quenched l a t h martensite varies from 30 to 90 x 10 cms/cm . In view of t h i s observation, the d i s l o c a t i o n density calculated from x-ray analysis i s not unreasonable. In the case of the low carbon tempered martensite, x-ray r e s u l t s for a l l specimens tempered at 300°C and above indicated a polygonised d i s l o c a -t i o n substructure. Only for the specimen tempered at the lowest temperature, 250°C, was a pile-up configuration i n d i c a t e . This implies that thermal re-arrangement and 'polygonisation' of the substructure d i s l o c a t i o n s which are introduced during the martensitic transformation requires much 72 Figure 20 D i s l o c a t i o n Density vs Tempering Temperature i n the Medium and Low Carbon Tempered Martensites. 73 less thermal a c t i v a t i o n i n the case of the low-carbon martensite than for the medium-carbon martensite. However, upto a tempering temperature of 450°C, the d i s l o c a t i o n densities calculated from domain s i z e and s t r a i n are within 60% of each other. This suggests that thermal rearrangement of d i s l o c a t i o n s due to tempering i n t h i s temperature range i s not extensive. The true d i s -l o cation density, though greater than p , cannot be expected to be f a r d i f -ferent from i t . Accordingly, the value of has been used as the true d i s -l o c a t i o n density i n Table VI and i n Figure 20. A f t e r tempering at 500°C and above, polygonization i n the low carbon martensite has proceeded to a much greater extent as indicated by the increasing difference i n the r e l a t i v e magnitudes of p and p . R e l i a b l e estimates of true d i s l o c a t i o n density cannot p s J be obtained under such conditions. 4.3.3 Discussion of X-Ray Results f o r Pure Iron and Fe-Cu-Ni A l l o y X-ray r e s u l t s f o r the Series B a l l o y and for pure i r o n , 50% c o l d - r o l l e d , are contained i n Table VII, and i n Figures 18, 19. No data are presented for Series A (aged only) because i n a l l cases the domain s i z e was greater than 2000A° and thus no estimates for d i s l o c a t i o n densities were possible. A t y p i c a l p l o t of domain size c o e f f i c i e n t versus l a t t i c e distance for a Series A specimen, aged for 0.5 hours, i s included i n Figure 19. Such a p l o t , which ex h i b i t s an increasing value of domain size c o e f f i c i e n t with l a t t i c e distance, i s c h a r a c t e r i s t i c of p l o t s obtained from heavily poly-gonized structures. I t was observed for the low carbon martensite tem-pered at 650°C (Figure 17), as w e l l as f o r a l l specimens i n Series A. In Series B, on the other hand, f i n i t e values for domain siz e were obtained which showed a gradual increase with aging time at 500°C. In con-t r a s t , the s t r a i n values at 100A°, which were small i n absolute magnitude, 7 4 passed through a maximum for about 1 hour aging time. Since the treatment i n Series B consisted of cold working and annealing, one might expect a continuous decrease i n s t r a i n values with increasing aging time. That the x-ray observations do not support t h i s i s a consequence of age hardening 5 8 e f f e c t s . As Hornbogen and Glenn have shown, aging i n the e a r l i e r stages involves the formation of coherent c l u s t e r s , and i t i s probably due to co-herency s t r a i n s associated with these clusters that l a t t i c e s t r a i n values increase for aging times upto 1 hour. I t should be noted that the peak y i e l d strength (0.2% o f f s e t flow stress) i n Series B also occurs a f t e r aging for 0.5 - 1 hour. The magnitude of any coherency st r a i n s i s expected to be rather small, because the atomic diameters of i r o n and copper d i f f e r only s l i g h t l y (less than 3%). However, i t i s d i f f i c u l t to separate the e f f e c t s of coherency s t r a i n s from those due to substructure d i s l o c a t i o n s , and the x-ray data for Series B cannot be u s e f u l l y interpreted. One feature to be noted i n the s t r a i n plots (Figures 14, 16, 18) is-that for some specimens the i n i t i a l peak i n l a t t i c e s t r a i n at small values of l a t t i c e distance, L, i s missed; i . e . up to some f i n i t e value of 'L', the s t r a i n i s zero. This c h a r a c t e r i s t i c can also be noticed i n C l e g g ' s ^ study of 83 Ni-ThQj and i n Sahoo's work with Al-4Cu. The magnitude of the i n i t i a l l a t -t i c e distance over which s t r a i n i s apparently zero increases as the l a t t i c e s t r a i n decreases. Furthermore, the peak i n the s t r a i n curve i s s h i f t e d to larger values of l a t t i c e distance L. This i s r e a d i l y seen from the data compiled i n Table VIII, which were obtained from Figures 14 and 18. As was e a r l i e r mentioned i n Section 3.3, i t i s not known whether t h i s e f f e c t i s a genuine one due to increased thermal rearrangement of the sub-structure d i s l o c a t i o n s or whether i t i s due to uncertainties i n the l i n e p r o f i l e analysis at small values of L. However, i t i s f e l t that when the 75 Table VIII Anomalies i n the P o s i t i o n of Peak S t r a i n i n St r a i n vs L a t t i c e Distance p l o t s . Specimen L = Distance up o to which s t r a i n equals 0 Maximum St r a i n (rms) x 10~ 2 L = L a t t i c e Max. distance at which Max. s t r a i n occurs Med. 'C Tempered 30A° 0.1463 100A° Martensite (450 QC) Med. 'C Tempered 30A° 0.1165 80A° Martensite (550°C) Series 'B' A l l o y 40A° 0.0963 200A° aged 0 b o u r s Ferrovac E Iron, 50A° 0.0618 190A° C.W. 50% Series 'B' A l l o y 110A° 0.0451 240A° aged 100 hours Specimen Table IX R e l i a b i l i t y of the Computer Analysis 3 B x 10 3 b x 10 3 8 x 10" radians radians radians Domain Domain S t r a i n Size Size 7% (Computer _0.9X ,o(Computer & 2tan6'° analysis) P gcose' analysis) Medium 'C' Tempered Martensite 250°C Low 'C Tempered Martensite 250°C Series '.B' alloyraged 0 hours Ferrovac E Iron 50% c o l d - r o l l e d 30.4 9.42 5.43 3.52 1.61 1.59 1.53 1.53 30.35 9.28 5.21 3.17 115 350 655 1077 670 1540 0.8 0.14 0.084 0.324Q at 100A° 368 650 0.25 0.2290 at 100A° 0.0666 at 100A° 0.0963 at 200A° 0.0484 at 100A° 0.0618 at 100A° 76 peak s t r a i n occurs at a much higher value of L than 100A°, then the peak value f o r s t r a i n should be used i n the Williamson and Smallman analysis rather than the lower value at L = 100A°. 4.3.4 R e l i a b i l i t y of the X-Ray Line Broadening Analysis An attempt has been made to confirm that the x-ray technique arid the. computer analysis are y i e l d i n g r e s u l t s of the ri g h t order. To th i s end, the basic Equations 14 and 15 (Section 3.1), along with the half-peak widths of of observed l i n e p r o f i l e s , have been used to calculate domain s i z e and l a t t i c e s t r a i n . These values have been compared with the r e s u l t s of the computer analysis. The approach i s best applied to those specimens which have i n -t r i n s i c broadening from one source only; small domain siz e or high l a t t i c e s t r a i n . In the present i n v e s t i g a t i o n , there i s unfortunately no specimen which exhibits such i d e a l c h a r a c t e r i s t i c s (see Tables V, VI, and VII). Inspite of t h i s drawback, Scherrer's formula (Equation 14) and Equation 15 have been used to give estimates of domain sizes and l a t t i c e s t r a i n s f o r . four specimens. These values, together with the r e s u l t s from the computer analysis are given i n Table IX. Calculations i n Table IX are based on the (220) r e f l e c t i o n since the high angle r e f l e c t i o n s are broadened to a greater extent and measurements of B are more r e l i a b l e . Comparing f i r s t the domain siz e values obtained from computer analysis with those estimated from the Scherrer formula, i t i s noticed that the l a t t e r values, f o r a l l four specimens, are of the same order of magnitude but smaller than the corresponding computer values. This i s to be expected since the Scherrer formula makes no allowance f o r the presence of l a t t i c e s t r a i n s . In the case of the Series B a l l o y specimen aged for zero hours, 77 the l a t t i c e s t r a i n i s small. Accordingly, the domain s i z e values from the two methods are found to be i n close agreement. The Ferrovac E i r o n , cold r o l l e d 50 pet, also has small nonuniform l a t t i c e s t r a i n but the domain s i z e values appear to be quite d i f f e r e n t - 1070 and 1540A°. This i s probably due to the rather small incremental broadening associated with increases i n domain siz e i n the range 1000 to 2000A°. The f i r s t two specimens; i . e . , tempered martensites, have s u b s t a n t i a l l y larger values f o r l a t t i c e s t r a i n so that there i s an appreciable difference between the two sets of domain si z e values. S i m i l a r l y i t i s noticed that the l a t t i c e s t r a i n values calcu-lated from Equation (15) are i n a l l cases larger than the corresponding values obtained from the computer analysis. Once again, t h i s can be a t t r i -buted to the simultaneous presence of domain s i z e broadening. Keeping i n mind that f o r the purpose of the rough calculations Gaussian d i s t r i b u t i o n s have been assumed for a l l the p r o f i l e s , and that Equations (14) and (15) have only l i m i t e d a p p l i c a b i l i t y , i t can be concluded that the Fourier analysis and the computer program are providing reasonable values for domain size and l a t t i c e s t r a i n . 4.4 Transmission Electron Microscopy of Tempered Iron-Carbon Martensites 4.4.1 S t r u c t u r a l Changes due to Tempering of Medium Carbon Martensite Transmission electron microscopy of tempered martensites was c a r r i e d out to confirm the conclusions of x-ray analysis; i . e . , a substructure con-s i s t i n g of a high density of random d i s l o c a t i o n s a f t e r tempering at lower temperatures i n the range studied, and a polygonized substructure a f t e r tem-pering at higher temperatures. Figure 21-25 reveal t y p i c a l microstructures of the medium carbon martensite, a f t e r tempering at various temperatures. 78 Structure a f t e r tempering at 250°C, Figure 21, i s s i m i l a r to that of as-85 quenched medium carbon martensite . I t consists of well-defined marten-s i t e laths which have a mottled appearance i n t e r n a l l y . P r e c i p i t a t i o n of carbide has occured during tempering, and the elongated p a r t i c l e s of car-bide are c l e a r l y v i s i b l e within the l a t h s , as i n the region marked A i n F i g -ure 21. I t i s d i f f i c u l t to detect carbide p a r t i c l e s along the l a t h bound-a r i e s , although T u r k a l o 8 5 and D e l i s l e e t a l 6 0 have indi c a t e d that they can also be expected to be present at such l o c a t i o n s . No attempt was made to i d e n t i f y the carbides i n the present work but on the basis of e a r l i e r i n v e s t i g a t i o n s 8 * ^ ' 8 ^ i t can s a f e l y be assumed that the observed carbide i s cementite, Fe^C, i n a l l structures tempered above 200°C. A t r a n s i t i o n carbide i s known to e x i s t when tempering i s c a r r i e d out at temperatures below 200°C. The mottled appearance of laths i n Figure 21 i s believed to be due to a high density of d i s l o c a t i o n s within the l a t h s , although the i n d i v i d u a l d i s l o c a -tions are not resolved. This i s consistent with the d i s l o c a t i o n density of 10 -3 20 x 10 cms-cms that was i n d i c a t e d by the x-ray analysis (see Table V). At the same time, the structure i s not inconsistent with the x-ray analysis of domain s i z e , since the very small domains (~350A°) indic a t e d by the anal-y s i s can e x i s t within the laths. A f t e r the medium carbon martensite has been tempered at 400°C, Figure 22, l a t h boundaries are s t i l l v i s i b l e . But w i t h i n the l a t t i c e thermal r e -arrangement and a n n i h i l a t i o n of d i s l o c a t i o n s has apparently 'coarsened' the substructure or reduced the d i s l o c a t i o n density to the extent that many i n -d i v i d u a l d i s l o c a t i o n s are readilyxesolved by transmission microscopy. The spacing and d i s t r i b u t i o n of these d i s l o c a t i o n s i s consistent with x-ray r e s u l t s , which indi c a t e d a domain s i z e of ~760A° and one d i s l o c a t i o n per do-main w a l l ; i . e . , a random d i s t r i b u t i o n of d i s l o c a t i o n s . 7 9 gure 21 Transmission Electron Micrograph of the Medium Carbon Martensite Tempered at 250°C. Figure 22 Transmission Electron Micrograph of the Medium Carbon Martensite Tempered at 400°C. Figure 25 Transmission Electron Micrograph of the Medium Carbon Martensite Tempered at 7 0 0 ° C . Figure 26 Transmission Electron Micrograph of the Low Carbon Martensite Tempered at 300°C. 82 One hour of tempering at 550°C, Figure 23, has resulted i n a c e l l u l a r substructure, and the s i z e of the c e l l s corresponds reasonably w e l l with the x-ray domain s i z e of ~1200A° for this material. A d d i t i o n a l l y i t i s noted that the cementite p a r t i c l e s (some are indicated by arrows i n Figure 23) are more spheroidal i n shape. Figures 24 and 25 represent t y p i c a l s t r u c -tures a f t e r tempering at 600°C and 700°C re s p e c t i v e l y . Well-defined f e r -r i t e . subgrains are now observed and t h e i r s i z e increases with increasing tempering temperature. These observations agree w e l l with the x-ray r e s u l t s , which indicated polygonisation a f t e r tempering at 600°C and above. The ce-mentite p a r t i c l e s are spherodized and are frequently observed t o , l i e on the sub-boundaries. No r e c r y s t a l l i z a t i o n of the matrix could be detected, even a f t e r tempering at 700°C. 4.4.2 S t r u c t u r a l Changes due to Tempering of Low Carbon Martensite The tempering of low carbon martensite involves s t r u c t u r a l changes very s i m i l a r to those which occur i n the medium carbon martensite. However, due to the lower carbon content; (a) the i n i t i a l d i s l o c a t i o n density of the mar-tensite i s lower, and (b) a f t e r tempering, the density of cementite p a r t i c l e s i s lower. As a r e s u l t , the substructure polygonises r a p i d l y at lower temper-ing temperatures. This i s c l e a r l y indicated i n Figures 26 to 29. Thus, a f t e r tempering at a low temperature of 300°C, Figure 26, many of the i n d i v i -dual d i s l o c a t i o n s are resolved i n the microscope. The di s l o c a t i o n s are quite randomly d i s t r i b u t e d , as predicted by x-ray analysis. A f t e r tempering at 400°C, there i s a noticeable trend towards polygoniztion within the o r i g i n a l laths (Fig. 27), A f t e r 500 and 550°C tempering, Figures 28(a) and 28(b), there are a c i c u l a r f e r r i t e subgrains, the shape and o r i e n t a t i o n apparently influenced by the o r i g i n a l martensite l a t h s . The presence of these subgrains also con-85 firms the x-ray results (Section 4.3.2); i . e . , a f t e r tempering at about 500°C, or higher, the substructure i s polygonized. When tempered at a s t i l l higher temperature (700°C), the structure reveals f e r r i t e subgrains with sharp boundaries (Fig. 29). Once again, the lower carbon content, and hence lower density of cementite p a r t i c l e s , i s r e f l e c t e d i n the coarser f e r r i t e subgrain siz e ,of the low carbon tempered martensite as compared to the medium carbon tempered martensite (see Figures 25 and 29; also Table X). Also, r e c r y s t -a l l i s a t i o n of the matrix was detected i n some f o i l s from the low carbon mar-tensite tempered-at 700°C. However, the f r a c t i o n of the s t e e l which had..-.' undergone r e c r y s t a l l i z a t i o n i s believed to be i n s i g n i f i c a n t on the basis of the infrequent appearance of r e c r y s t a l l i z e d regions. .The structures observed i n the present study agree reasonably w e l l with 85 82 those described i n the e a r l i e r works of Turkalo and Speich 4.4.3 Subgrain Size From the r e s u l t s of the x-ray analysis and the above de s c r i p t i o n of the s t r u c t u r a l changes taking place during tempering, i t i s evident that the sub-structure i n the e a r l i e r stages of tempering i s characterized by randomly d i s t r i b u t e d d i s l o c a t i o n s , whereas a f t e r higher temperature tempering the sub-structure i s best v i s u a l i z e d as consisting of subgrains. Having used x-ray l i n e broadening analysis to calculate the random d i s l o c a t i o n density, p, transmisssion electron microscopy was u t i l i z e d to f i n d the diameter of sub-grains, t, i n order that strength could be related to the appropriate sub-s t r u c t u r a l parameter - p or t. The values for subgrain sizes as computed from the average l i n e a r intercept (see Section 2.4.5) are compiled i n Table X. 4.4,4 Selected Area Electron D i f f r a c t i o n 86 Table X Subgrain Sizes i n Tempered Martensites Medium Carbon Martensite Low Carbon Martensite Tempered for 1 hour at Mean Linear Intercept t " , i n ym Subgrain size = 1.675t", i n ym Mean Linear Intercept t " , i n ym Subgrain s i z e t = 1.675t", i n ym 500°C 0.212 0.36 550°C 0.272 0.46 600°C 0.186 0.31 0.372 0.62 650°C 0.255 0.43 0.604 1.01 700°C 0.495 0.83 0.975 1.63 Table XI Summary of Selected Area D i f f r a c t i o n Patterns Medium Carbon Martensite Low Carbon Martensite Tempering Temperature °C No. of patterns observed No. i n Group A No. i n Group B No. of patterns observed No. i n Group A No. i n Group B 550 9 7 2 14 10 4 600 9 9 0 19 14 5 650 14 10 4 13 4 9 700 12 3 9 11 4 7 87 45 J.C.M. L i ' s model for sub-boundary strengthening predicts that the magnitude of the strengthening e f f e c t should be a function of 6, the average misorientation across the sub-boundaries (Section 1.2.5, Equation 9).. Accord-i n g l y , selected area electron d i f f r a c t i o n was employed to pursue t h i s p o s s i -b i l i t y . From 10 to 15 d i f f r a c t i o n patterns were obtained from d i f f e r e n t re-gions of each of the martensites tempered at 600°C and above for the medium carbon s t e e l , and tempered at 500°C and above for the low carbon s t e e l . Some t y p i c a l d i f f r a c t i o n patterns are included i n Figures 30 to 34. To i n -terpret these patterns a knowledge of the morphology of the as-quenched la t h martensite and of the o r i e n t a t i o n r e l a t i o n s h i p between laths i s required. 84 Krauss and Marder have shown that "the laths are generally aligned p a r a l l e l to one another i n groups that have been termed packets, blocks or sheaves". 103 These same authors had suggested that the l a t h boundaries within a packet 84 are generally low angle boundaries, but i n a more recent p u b l i c a t i o n , they modified t h i s argument and proposed that adjacent laths may be separated by low or high angle boundaries, or may be twin related. More d e f i n i t i v e work 88 89 i n t h i s regard has been done by Speich and Swann and by Chil t o n et a l , as discussed below. 88 Speich and Swann , i n t h e i r i n v e s t i g a t i o n of Fe-Ni l a t h martensites, reported that adjacent laths are frequently twin related. They went on to suggest that "twin-related laths are a r e s u l t of adjacent regions adopting d i f f e r e n t variants of the Kurdjumov - Sachs o r i e n t a t i o n r e l a t i o n s h i p to mini-mise the shape change (during the martensitic transformation) and lead to a 89 lower s t r a i n energy condition." Chil t o n et a l investigated the o r i e n t a t i o n r e l a t i o n s h i p s i n greater d e t a i l and confirmed the Speich and Swann suggestion. The Kurdjumov - Sachs r e l a t i o n s h i p s , f or adjacent f.c.c (y) and b.c.c. (a) l a t t i c e s , are given as 88 (111) / / ( O i l ) and [110] / / [ i l l ] y a y a There are 24 variants of t h i s r e l a t i o n s h i p : for a common (111)^ plane, two d i f f e r e n t <111> d i r e c t i o n s may be chosen to be p a r a l l e l to each of the three <110> d i r e c t i o n s , r e s u l t i n g i n 6 variant f or each of the {111} planes. Y Y Since there are four independent {111}^ planes, there are 24 variants of the Kurdjumov - Sachs r e l a t i o n s h i p . Keeping i n view the l i m i t e d accuracy of 89 e l e c t r o n d i f f r a c t i o n , C h i l t o n et a l divided the o r i e n t a t i o n r e l a t i o n s h i p s associated with the variants i n t o three groups. i ) where adjacent laths are o f f twin r e l a t i o n s h i p by less than 5°, i i ) where adjacent laths are o f f twin r e l a t i o n s h i p by between 5° and 20°, and i i i ) where the adjacent laths are separated by low angle boundaries, the misorientation being of the order of 10° about <110> d i r e c t i o n . a (It should be noted that i n a l l three groups the l a t t i c e s of adjacent laths can be brought into coincidence by simple r o t a t i o n about an axis of the type <110>a. This, i n turn, implies that the d i f f r a c t i o n patterns from adjacent laths should have common {110}^ poles.) Furthermore, i f a l l the variants of the K - S r e l a t i o n s h i p s are equally probable, then the frequency of experimentally observed o r i e n t a t i o n r e l a t i o n s h i p s should be such that 20% f a l l into group ( i ) , 60% into group ( i i ) , and 20% i n group ( i i i ) . C h i l t o n 89 et a l experimentally determined these frequencies to be 20%, 70% and 10%, i n d i c a t i n g strongly that adjacent laths do obey Kurdjumov - Sachs, or very s i m i l a r r e l a t i o n s h i p s . In the l i g h t of the abovementioned work, the following comments can, be made regarding some t y p i c a l d i f f r a c t i o n patterns observed i n the present i n -v e s t i g a t i o n : Figures 30(a) and 30(b): Medium Carbon Martensite, tempered at 600°C. 89 Figure 30(a) i s the d i f f r a c t i o n pattern from the selected area shown i n Figure 30(b). In the selected area, small subgrains are seen and i t i s d i f -f i c u l t to i d e n t i f y any o r i g i n a l l a t h boundaries. In the d i f f r a c t i o n pattern, a l l the spots belong to one zone only, the zone axis being <111>, i n d i c a t i n g that the selected area i s e n t i r e l y within one o r i g i n a l martensite l a t h . Each spot i s s p l i t over an angular range (- 10°) implying that the subgrains i n the selected area are bounded by low angle boundaries. On the basis of the ap-proximate number of subgrains i n the selected area, the average misorienta-ti o n between the subgrain i s estimated to be of the order of 0.5 to 1°. Figures 31(a) and 31(b): Medium Carbon Martensite, tempered at 650°C. The d i f f r a c t i o n spots can be indexed as belonging to two d i f f e r e n t zones., tjie zpne axes being <110> and <113>. Poles of {110} type are common to both 89 zones. According to Chilton's groupings , t h i s case belongs to group ( i ) . Once again, the spots belonging to the <110> zone exhibit r a d i a l spread. '• Therefore, i t i s indicated that the selected area (Figure 31b) comprises por-tions of two adjacent l a t h s , and the spread of d i f f r a c t i o n spots s i g n i f i e s the presence of subgrains with low angle boundaries. Figures 32 and 25: Medium Carbon Martensite, tempered at 700°C. The d i f f r a c t i o n pattern has been taken from an area which surrounds the sub- '. boundary marked by an arrow i n Figure 25. The pattern on indexing reveals two zones, the zone axes being <731> and <311>. There are no d i f f r a c t i o n spots common to both zones. The two l a t t i c e s can be brought into coincidence by a r o t a t i o n of - 11° about an axis whose indices have not. been determined.. Most (9 out of 12) of the patterns taken across si n g l e sub-boundaries i n medium carbon martensite tempered at 700°C had zones of high indices l i k e <731>, <531>, ets. In general, the misorientations (about u n i d e n t i f i e d and * o • probably i r r a t i o n a l axes) were of the order of 15 to 20 . Thus, the charac-90 Figure 30(b) Selected Area for the Pattern i n Figure 30(a) 91 92 Figure 3 3 Electron D i f f r a c t i o n Pattern from the Area shown i n Figure 28(a) (Low Carbon Martensite Tempered at 500°C). 9 3 Figure 34(b) Selected Area for the Pattern i n Figure 34(a). 94 t e r i s t i c s of a l a t h structure are no longer observed a f t e r tempering the martensite at 700°C. The d i f f r a c t i o n pattern of Figure 32 also shows some di f f u s e Kikuchi l i n e s (which are more r e a d i l y observed i n the o r i g i n a l negative). This i n -dicates that the l a t t i c e i s s u b s t a n t i a l l y free of any l a t t i c e s t r a i n s , which i s thus consistent with polygonization and subgrain formation i n medium car-bon martensite tempered at 700°C. Figure 33: Low Carbon Martensite, tempered at 500°C. Figure 33 i s the d i f f r a c t i o n pattern from the area shown i n Figure 28(a). Three zones that can be e a s i l y i d e n t i f i e d i n the pattern are <110>, <311> and <331>. The three have common {110} d i f f r a c t i o n spots and obey Kurdjumov - Sachs r e l a -tionships; i . e . they are due to martensite la t h s . Most of the spots belon-ing to the three zones are s p l i t , which indicates that the substructure units seen within the laths i n Figure 28(a) have low angle boundaries. Figures 34(a) and 34(b): Low Carbon Martensite, tempered at 650°C. Figure 34(a) d i f f r a c t i o n patten from an area co n s i s t i n g of three subgrains shown i n Figure 34(b). The d i s l o c a t i o n structure of the sub-boundaries i s resolved i n the selected area photomicrograph. The three zones axes were i d e n t i f i e d as <210>, <731> and <211>, the respective migorienta-tions being 8° and 17°. This case i s s i m i l a r to that of the medium carbon martensite tempered at 700°C (Figure 32), where the c h a r a c t e r i s t i c s of laths are no longer observed. The d i f f r a c t i o n patterns from the low carbon martensite tempered at 700°C were s i m i l a r i n nature to those obtained from the medium carbon martensite tempered at 700°C; i . e . across each sub-boundary, two d i f f e r e n t and high index zones were observed. However, the magnitude of the misorientation; i . e . ro-t a t i o n to bring about a coincidence of the two l a t t i c e s , was of the order of 95 only 10° as compared to 15-20° f o r the medium carbon martensite. The r e s u l t s from a l l the d i f f r a c t i o n patterns analysed i n the present study are summarized i n Table XI. Those patterns which showed one or more zones with low index zone axes and with common {110} spots (which thus i n -dicated the continuing presence of l a t h boundaries) are l i s t e d i n Group A. On the other hand, patterns with high index zones and without any common spots between the zones are included i n Group B. From the above discussion and on the basis of the contents of Table XI, some q u a l i t a t i v e conclusions can be drawn. I t i s believed that the r e s u l t s show that martensite l a t h c h a r a c t e r i s t i c s are present a f t e r tempering at temperatures up to 650°C f o r the medium carbon martensite, and upto 600°C for the low carbon martensite, even where the l a t h boundaries are not r e a d i l y discerned i n the microscope. The s l i g h t l y higher temperature l i m i t f o r the medium carbon martensite may be due to the extra pinning of the lat h bound-aries provided by the larger volume f r a c t i o n of cementite present. Further^ more, at tempering temperatures below 600-650°C, when the laths are s t i l l present, polygonization and subgrain formation take place within the lath s . Although no quantitative measurements of subgrain misoreientation are possible, the d i f f r a c t i o n patterns indicate that i n the early stages of subgrain forma-tion.*, (e.g., tempering at 500°C for the low carbon martensite) the misorienta-tl o n i s of the order of 0.5 to 2°, whereas f o r subgrains formed by tempering at 700°C, the misorientation i s of the order of 10-20°. A.5 Transmission Electron Microscopy of the Fe-Cu-Ni A l l o y and Pure Iron Transmission electron microscopy of the Fe-Cu-Ni a l l o y , a f t e r various heat treatments, was car r i e d out to investigate the nature of substructure arid to determine the copper p a r t i c l e (e phase) s i z e and spacing. 96 4.5.1 Observations of Structure The s o l u t i o n treated structure (Series A, zero aging time) consisted of equiaxed f e r r i t e grains containing some randomly d i s t r i b u t e d d i s l o c a t i o n s , A f t e r aging the as-quenched structure (Series A) for periods of upto 2 hours no s t r u c t u r a l changes were observed i n the microscope. Beyond 2 hours of aging, small copper p r e c i p i t a t e s were found i n scattered regions. In view 58 7 of the e a r l i e r studies by Hornbogen ' , and the observation that strength increases with aging time upto 1-2 hours (Table I I I ) , i t i s reasonable to conclude that clusters were being formed i n the e a r l i e r stages of aging. As the aging.time was increased further, the number and s i z e of e p a r t i c l e s progressively increased. Figure 35 reveals the structure produced by a 100 hour aging treatment. Many of the copper p a r t i c l e s are observed to be strung along d i s l o c a t i o n s . In Series B, the solu t i o n treated specimens were cold r o l l e d p r i o r to any aging. Figure 36 i s a transmission electron micrograph of the a s - r o l l e d structure. For comparison, the corresponding structure of pure iron .(Ferro-r vac EJ) , a f t e r cold r o l l i n g 50%, i s shown i n Figure 37. The sub s t r u c t u r e ' c e l l s i n the pure i r o n have di s l o c a t i o n s heavily concentrated i n the c e l l walls. By contrast, the deformed iron-copper a l l o y shows a more uniform d i s t r i b u t i o n of di s l o c a t i o n s . Electron d i f f r a c t i o n patterns from the areas seen i n Figures 36 and 37 are reproduced i n Figures 38 and 39. In each case, d i f f r a c t i o n ppots from only one zone are observed (<100> for Fe-Cu-Ni and <110> for pure iron) and a l l the spots are s p l i t or spread over an angular range. This, i n -dicates that the regions seen i n Figures 36 and 37 are each within a s i n g l e o r i g i n a l f e r r i t e grain and that the c e l l walls separate regions of l a t t i c e which, are only s l i g h t l y misoriented with "respect to each other (approximately 1 to 4 ) . Figure 35 Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged for 100 hours (Series A). Figure 36 Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Solution-Treated and Cold-Rolled 50 percent. Figure 37 Transmission Electron Micrograph of Ferrovac E Iron, Quenched and Cold-Rolled 50 percent. 99 Figure 39 Electron D i f f r a c t i o n Pattern from the Area Shown in Figure 37 (Cold Rolled Ferrovac E Iron). 100 The x-ray line broadening analysis of these two cold rolled specimens (Fe-Cu-Ni alloy and pure iron) revealed domain sizes of 670 and 1540A°, and nonuniform lattice strains of 0.0963 and 0.0618, respectively. In both cases, 80 the Williamson and Smallman analysis indicates a polygonized configuration of dislocations. In the micrographs (Figures 36 and 37) i t i$ d i f f i c u l t to identify domains of the size indicated by x-ray analysis. However, the spac^ ing of dislocations in c e l l walls is expected to be very small, and the size indicated by the analysis could be an average of the spacing of dislocations in the substructure over a large area, including both the walls and interiors of cells. Another possible explanation for the difference between x-ray do-main size and the observed c e l l size is associated with long-range stresses due to c e l l walls, as a result of which the lattice spacing would continu-ously change as one moves from c e l l wall to c e l l interior. This continuous change in lattice parameter can result in adjacent regions diffracting in-coherently . and thus yielding a smaller x-ray domain size value as compared to the c e l l size. The presence of a polygonized substructure can be rationalized from the 90 point of view of slip pclygonization ; i.e., when a large number of d i s l q c a -tions of the same ri.gn accumulate in a c e l l wall, the strain energy associated with the wall I.s quite high. A more stable configuration is achieved when the dislocations run parallel and on top of each other, as is observed in classical polygonization. The only difference is that the distance between the dislocations along the wall is not constant in the case of slip polygoniza-tion, Comparing the x-ray results for the two cold rolled specimens, the dislocation density is larger for the iron-copper-nickel alloy. This is qualitatively consistent with the structures shown in the relevant micrographs, and is readily explained by the inhibiting effect of solute atoms (copper) 101 on cross s l i p ^ . On aging the c o l d - r o l l e d a l l o y (Series B), recovery of the substructure was observed to take place. Again the f i r s t E-phase p a r t i c l e s were sighted, only a f t e r aging for >_ 2 hours. T y p i c a l p r e c i p i t a t e s a f t e r aging for 10 and 1Q0 hours are shpwn i n Figures AO and 41 res p e c t i v e l y . In Figure 40 most of the copper p r e c i p i t a t e s are seen to l i e along d i s l o c a t i o n s , which i s unexpected i n view of the small nucleation b a r r i e r associated with the p r e c i p i t a t i o n of 5 8 copper . A f t e r aging f o r 100 hours (Figure 41), the copper p a r t i p l e s ap-peared to be more uniformly d i s t r i b u t e d . Photqmicrographs were enlarged 2.5 times and then used to evaluate the average copper p a r t i c l e s i z e . Figures 42(a), 42(b), 43(a) are more micrographs from the same serie s of specimens but at lower magnification to show the e f f e c t s of aging on the c e l l s produced by cold working. As a r e s u l t of the aging treatment, d i s l o c a t i o n rearrange-ment has taken place i p the c e l l w alls, r e s u l t i n g i n low energy d i s l o c a t i o n networks. Several of these networks are seen at 'A' i n Figures 42(a) and 43(a), and they constitute the boundaries of the c e l l s or subgrains. These micrographs were used to determine subgrain s i z e , using the l i n e a r intercept method as described e a r l i e r i n Section 2.4.5. Figure 43(b) shows a d i f f r a c -t i o n pattern from the area i n F i g . 43(a). The pattern once again shows only one zone, with spots spread over an angular range. This means that during the recovery process, the misorientation across the c e l l or subgrain bound-aries has not increased to a s i g n i f i c a n t degree. 4.5.2 Analysis of Structure Tables XII and XIII summarize the res u l t s of various quantitative measure-ments on electron micrographs. The true diameter of copper p a r t i c l e s , d^, was calculated by d i r e c t measurements on at least three enlarged micrographs 1 0 2 O-2/i. i _ i Figure 40 Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged for 10 hours, (Series B). • ft (• i & I'M J Figure 41 Transmission Electron Micrograph of the Fe-Cu-Nl A l l o y , Aged for 100 hours (Series B). Figure 42(a) Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged for 10 hours (Series B). Figure 42(b) Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged for 10 hours (Series B). Figure 43(a) Transmission Electron Micrograph of the Fe-Cu-Ni A l l o y , Aged for 100 hours (Series B). Figure 43(b) Electron D i f f r a c t i o n Pattern from the Area Shown i n Figure 4 3(a). Table XII e Phase P a r t i c l e Size and Extent of P r e c i p i t a t i o n i n the Fe-Cu-Ni.Alloy. Seri .c s A Series B Aging time at 500°C d , V A n x 10 f l x 10 4 % p r e c i p i t -ation f2 = 0.016-f 1 • d , A V n x 10 f l x 10 4 % p r e c i p i t -ation = 0. f2 016-f 2 hrs. - - - -0 -160 x I O - 4 — — — = 0 -160 X IO" 4 5 hrs. 80 17.0 4.55 2.8 155 x 10~ 4 80 47 12.5 7.7 147. 5 x 10" 10 hrs. 102 27.2 15 10 145 x 10" 4 99 72 38 26.5 122 X i o " 4 30 hrs. 108 68.6 47.7 29.8 112. 3 x 10" "60 hrs. 129 67 79 49.4 81 X i o " 4 LOO hrs. 105 52 92.6 58 67.4 x I O - 4 148 56.7 100 62.0 60 X i o " 4 4 4 Table XIII Dispersion Parameters and Subgrain Sizes in the Fe-Cu-Ni Alloy. Aging Time Series A Series B D, A° D/d s D, A° D/d s t", y t = t" x 1.675, y 2 hrs. 0.32 0.53 5 hrs. 3062 47 1847 27 0.34 0.56 10 hrs. 2143 27 1311 16.5 0.34 0.56 30 hrs. 1276 14.4 0.36 0.60 60 hrs. 1177 11.1 0.35 0.54 100 hrs. 1272 10.4 1212 10.0 0.37 0.63 Ferrovac E Iron 0.75 1.26 50% cold rolled 107 for each aged material. Hornbogen'' has pointed out that since contrast i s due only to d i f f r a c t i o n , there i s bound to be some inaccuracy i n the d i r e c t measurements of p a r t i c l e diameter. This inaccuracy would be further increased i n the early stages of aging, when some coherency s t r a i n s might be present. The number of p a r t i c l e s per unit volume x^ as obtained by counting the number of dark-appearing p a r t i c l e s i n the enlarged micrographs. This number was m u l t i p l i e d by a factor of two since, according to contrast theory, there should be equal numbers of 'black' and 'white' p a r t i c l e s i f the o r i e n t a t i o n i s random. To calculate the volume of the f o i l , an average thickness of 1500A° was assumed. In Table XII ,n denotes the number of p a r t i c l e s per cubic centimeter, f^/2 i s the volume f r a c t i o n of v i s i b l e p r e c i p i t a t e s i n the t h i n f o i l and i s the volume f r a c t i o n present as c l u s t e r s which i s taken as = (0.016 - f ^ ) . Other parameters have been defined e a r l i e r . Copper p a r t i c l e s i z e and f r a c -t i o n of copper p r e c i p i t a t e d have been pl o t t e d against the aging time for both Series A and Series B i n Figures 44 and 45 r e s p e c t i v e l y . Cold working p r i o r to aging has apparently not affected the size of e-phase p a r t i c l e s attained a f t e r a given aging time, but has s i g n i f i c a n t l y increased the t o t a l amount of copper which has p r e c i p i t a t e d as e. This suggests that cold working i n -creases the i n i t i a l nucleation rate but has l i t t l e e f f e c t on growth rate of 5 8 the p r e c i p i t a t e s . Hornbogen has speculated that nucleation of clusters might take place at vacancies. The extra vacancies generated by cold work would thus aid the nuclpation but, i f they anneal out i n a short time at 500°C, the vacancies would not influence the volume d i f f u s i o n of copper atoms and hence the copper p a r t i c l e s i z e . In Series B the subgrain s i z e was observed to increase from 0.53u to 0.63y as the aging time was increased from 2 hours to 100 hours, T h i s slow increase i n subgrain s i z e may be a t t r i b u t e d to the presence of some p r e c i p i -108 AGING TIME , hr Figure 4 4 Plot of log ( P a r t i c l e Diameter) vs Aging Time fo r the Fe-Cu-Ni A l l o y s . AGING TIME , hr Figure 4 5 Plot of Percent P r e c i p i t a t i o n vs Aging Time for the Fe-Cu-Ni A l l o y s . 109 tates i n the c e l l walls (see Figure 42b f o r example). F i n a l l y , i t should be noted that even a f t e r aging for 100 hours at 500°C, p r e c i p i t a t i o n i s i n -complete i n both Series A and Series B. 4.6 Strengthening Mechanisms i n Tempered Martensites 4.6.1 Contribution from Dispersed Cementite P a r t i c l e s Strengthening due to noncoherent second phase p a r t i c l e s i s usually evaluated on the basis of the Orowan mechanism, i r . which d i s l o c a t i o n s by-pass the p a r t i c l e s by bowing between them. The Orowan contribution to the i n i -21 t,ial y i e l d s t r e s s , based on K e l l y and Nicholson's treatment, i s given by: a = Jib . 1 ( 1 + I O J D ( 4 ) or TT 2 1-v D 2b Ashby's modification, which has been discussed i n Section 1.2.4, may be stated: °or = f~ ' 1 (1 + l ^ D 1 * cos<Hn { i(l + ( f - l)sin$} (5) o s d To compute the value of a from Equation 5, the expression cos< j)£n{— o (1 +(^j l ) s i n ( f > ) needs to be maximized with respect to ( j > , where 2<f> i s the . s angle subtended by the bowed-out segments of a d i s l o c a t i o n at the p a r t i c l e . In the present i n v e s t i g a t i o n , for the range of D and d g values contained i n Table IV for the tempered carbon martensites, the values of <)> were deter-mined by t r i a l and error to vary between 11° and 19°. This l i e s within the o 28 approximate l i m i t s of zero to 30 suggested by Ashby . Using the afore-mentioned values of < f ) , Equation (5) predicts an Orowan stress which i s smaller by 12 to 15% than that predicted from K e l l y and Nicholson's state-ment, Equation (4). 110 However, i n the e a r l i e r stages of tempering of martensite, when the edge-to-edge p a r t i c l e spacings are small and Orowan stresses are expected to be r e l a t i v e l y large, the cementite p a r t i c l e s have an elongated shape (see Figures 7,8,21). The consequence of such p a r t i c l e shape, as shown by Bush 91 and K e l l y i s that for the same value of mean free path A, the e f f e c t i v e D value would be smaller than that calculated from the Edelson-Baldwin formula (Section 2.4.4) which assumes a s p h e r i c a l p a r t i c l e shape. Although i t i s d i f f i c u l t to evaluate the exact magnitude of a shape correction, i t should be recognised that the correction, when applied to Equation 4, i s i n the opposite d i r e c t i o n to that due to Ashb,y's modification of the bowing-out model. In addition, there are some short and long-range in t e r a c t i o n s be-tween p a r t i c l e s and di s l o c a t i o n s which are neglected i n the theories of d i s -persion strengthening. I t i s therefore possible that data which appear to be i n better agreement with Equation (4) would be i n at least as good agree-ment with Equation (5) i f corrections were made for the various factors l i s t e above. Ashby's statement of the Orowan theory i s d i f f i c u l t to apply i n i t s pre-sent form. For t h i s reason, and because the Kelly-Nicholson statement has repeatedly been shown i n previous reported work to give exc e l l e n t predictions of dispersed-phase strengthening, Equation (4) has been used i n the present study. Results of applying Equation (4) to the tempered martensite data are shown i n Table XIV. 4.6.2 Contribution from Random D i s l o c a t i o n Arrays The substructure a f t e r tempering at the lower temperatures was charac-t e r i z e d by a random array of dislocations(see Section 4.3.2). Examination of the corresponding electron micrographs (Figures 21 and 26) also reveals Table XIV Dispersion Strengthening Tempered Martensites. i n Medium Carbon Martensite (0.42%C, l.l%Mn) Low Carbon (0.11%C Martensite , 0.9% Mn) Tempering Temperature °C D, ym -2 a , MNm or' D, ym -2 a , MNm or 250 0.06 585 0.10 390 300 0.065 555 0.11 375 350 0.07 535 0.12 345 400 0.075 505 0.125 330 450 0.08 480 0.15 295 500 0.09 430 0.22 205 550 0.10 395 0.315 160 600 0.115 350 0.46 110 650 0.19 235 0.57 95 700 0.30 160 1.06 55 112 the presence of l a t h boundaries, which have not been treated as independent b a r r i e r s to deformation. The structures are equivalent to those of cold worked metals i n which there i s a high density of disl o c a t i o n s within each 8 10 grain. Under these conditions, Keh and Weissman and Dingley and McLean have shown for b.c.c metals that the flow stress i s a unique function of d i s l o c a t i o n density independent of the grain s i z e . Thus the substructure i s completely described as a high density array of random d i s l o c a t i o n s . The contribution a , of substructure d i s l o c a t i o n s to the strength has been calculated from Equation (1) a = 2 a y b p 1 / 2 (1) P g where a was taken as 0.4 from the work of Keh and Weissmann . An approximate but independent check on the value of a can be obtained from the t e n s i l e test and x-ray r e s u l t s of the present work f o r cold worked Ferrovac E i r o n . The domain s i z e and nonuniform l a t t i c e s t r a i n were found to be 1540A° and 0.0618% resp e c t i v e l y (Table V I I ) . The corresponding d i s -l o c a t i o n densities were calculated to be 1.3 x 10"^ and 0.75 x 10'^ cm-cm ^ . A 'polygonized' configuration i s indicated by the analysis. However, the two estimates of di s l o c a t i o n s d e n s i t i e s d i f f e r by less than 50 percent, and the true d i s l o c a t i o n density can be expected to be only s l i g h t l y larger than 1 0 - 3 1.3 x 10 cm-cm . The y i e l d strengths (0.2% o f f s e t flow stresses) f o r pure i r o n i n the as-quenched and i n the quenched and 50%-rolled conditions -2 were found to be 160 and 310 MNm respectively. In the as-quenched i r o n , the grain boundaries have a strengthening e f f e c t , whereas i n the cold r o l l e d 8 10 9 2 state, they do not. ' ' The grain s i z e of the quenched i r o n was approxi-mately lOOu and the magnitude of the associated strengthening e f f e c t i s e s t i -9 3 - 2 mated to be 28 MNm . Therefore the dis l o c a t i o n s generated during cold 10 -3 r o l l i n g (1.3 x 10 cm*cm ) must account for a strengthening increment of 113 _2 {310 - (160-28)} = 178 MNm . S u b s t i t u t i n g the appropriate values i n Equa-t i o n (1) gives a = 0.42. Considering that the true d i s l o c a t i o n density may 30 2 be s l i g h t l y larger than 1.3 x 10 ' lines/cm i . e . , the value of a would be accordingly decreased, excellent agreement i s observed with the value derived g by Keh and Weissmann from transmission e l e c t r o n microscopy work. A s i m i l a r c a l c u l a t i o n from t e n s i l e and x-ray r e s u l t s from the Fe-Cu-Ni a l l o y specimens i n the as-quenched, and the 50% cold r o l l e d condition y i e l d s a value of 0.46 for a. Table XV contains the r e s u l t s of computations of o p f o r those tempered martensites where the substructure had e a r l i e r been shown to consist of ran-dom d i s l o c a t i o n s . 4.6.3 Contribution from Subgrain Boundaries A f t e r tempering at higher temperatures i n the range, the substructure was shown to consist of subgrains (see Sections 4.4). The strengthening e f f e c t of subgrains can be evaluated e i t h e r by means of the modified H a l l -45 55 Petch type of r e l a t i o n s h i p due to L i or on the basis of the Langford-Cohen model as discussed i n Section 1.2.5. The two r e l a t i o n s h i p s are expressed: r r - N u.b / 8 9 N 1 / 2 ^ - l / 2 , Q v ( L l ) °, - 0 o + 2 Tr(l-v) (7rb ) t ( 9 ) (Langford-Cohen) a = a + 3.0ubt - 1 (10) o Equation (10) was derived by Langford and Cohen for the case of elongated c e l l s (with preferred orientation) i n cold drawn i r o n wire. A s i m i l a r expres-sion can be established for s p h e r i c a l subgrains of random o r i e n t a t i o n . Let a be the applied t e n s i l e stress and a the l a t t i c e f r i c t i o n s tress so that the average e f f e c t i v e shear s t r e s s acting on a d i s l o c a t i o n i n i t s s l i p plane r-c T . a-a i s equal to ( — 5 — ) . I t can be shown that i f t i s the true subgrain diameter, Table XV Strengthening Contribution of Random D i s l o c a t i o n Arrays i n Tempered Martensites Tempering Temperature °C Medium (0. Carbon 42%C, Martensite l.l%Mn) Low Carbon ( o . l l % C , Martensite 0.9%Mn) P , cm/ 3 cm -2 a , MNm P P , cms/cm -2 a , MNm P 250 20 X i o 1 0 685 8 i n 1 0 x 10 440 300 17 X i o 1 0 630 6 i n 1 0 x 10 375 350 13 X i o 1 0 545 5 .5 i n 1 0 x 10 355 400 8 X i o 1 0 440 > 3 i n 1 0 x 10 > 275 450 4 X i o 1 0 310 > 2 i n 1 0 x 10 > 225 500 3 X i o 1 0 275 550 2.3 X i o 1 0 230 600 1.5 X i o 1 0 185 115 (2 then the average subgrain diameter in a random (slip) plane is ^ - j t. Now the work done on a dislocation by the applied stress in sweeping out the area of the slip plane within a subgrain is , O s , TT , 2 x 2 The energy required to create a dislocation length equal to the peri-meter of the slip plane 2 where T is the line tension of the dislocation. Using a value of yb /2 for T, and equating the above two energies, we obtain: , a" go, v TT 2 2 1,2 /2 2 ~4'"3 = ~2 ,1T/"3 Z or a = a + 4.9^- (31) o t On theoretical grounds alone, i t is d i f f i c u l t to reject or support either Li's model or the Langford-Cohen model. Possibly transmission electron micro-scopy on slightly deformed metals containing a subgrain substructure would shed some light on the exact mechanism of subgrain strengthening. However, in the absence of such evidence, i t was decided to plot the results accord-ing to both of the above relations and to find out which provided the better f i t . Accordingly, for those specimens in which subgrain sizes could be mea-sured, (Table X), a -.a was evaluated from Tables II and XIV, and plotted ' y-s or -1/2 " -1 against t and t as shown in Figures 46 and 47 respectively. The experi-mental point for medium carbon martensite tempered at 600°C x^ as ignored in drawing the least squares f i t line on both plots because i t was in such poor agreement with a l l other points. Equally good linear plots are observed in both cases and can be expressed by the following equations: 116 480 -420 360 300 3 5 Figure 47 Plot of (a -a ) vs t f o r the Medium 6 y.s. or and Low Carbon Tempered Martensites, 117 (o - a ) = 155 + 0.2t 1/2MNm 2 y.s or and (a - a ) = 270 + 81 x 10 _ 6t _ 1MNm - 2 y.s or = 270 + 4.2ubt~J]MNm where a i s the 0.2% of f s e t flow stress and t i s i n meters, y • s -3/2 In the Hall-Petch p l o t (Fig. 46) the slope i s 0.2 MNm . In e a r l i e r 41 42 -3/2 works by Embury et a l and Warrington , values of 0.32 MNm and 0.35 MNm were obtained for cold drawn ir o n and f o r subgrains i n commercially pure i r o n (obtained by cold working and annealing) r e s p e c t i v l e y . I f the above three values of the Petch slope, i n the order stated, are inserted i n Equation (9), the values f o r 8 can be calculated to be 12°, 33° and 37° r e s p e c t i v e l y . In contradiction to t h i s r e s u l t , a l l three of the above materials can be shown to have low angle boundaries. Thus the experimental slopes appear to be much too large to be explained by L i ' s model. Furthermore, as discussed l a t e r , the intercept on the stress axis i s much smaller f o r the Hall-Petch case than would be expected. The intercept obtained from the Langford-rCohen p l o t , on the other hand, i s of the r i g h t magnitude. The observed slope of 4.2pb i s also i n good agreement with the t h e o r e t i c a l value of 4.9yb. 4.6.4 A d d i t i v i t y of Strengthening Mechanisms Having estimated the magnitude of the strengthening e f f e c t s from various obstacles and compiled them i n Tables XVI and XVII, an attempt can be made to account f o r the t o t a l y i e l d strength of tempered martensites. However, one question that needs p r i o r c l a r i f i c a t i o n concerns the a d d i t i v i t y of v a r i -ous strengthening e f f e c t s . Generally, i t i s accepted that i f a d i s l o c a t i o n , while moving i n i t s s l i p plane, encounters two types of b a r r i e r s simultane-ously, then the e f f e c t s are additive. On the other hand, i f the b a r r i e r s 118 are to be overcome successively, the stronger of the two should contrpl the flow stress. Two examples of simultaneous b a r r i e r s a,re those i m p l i c i t i n Hall-Petch Equation ( 7 ) and i n Equation ( 4 ) written to account for strength-ening by p a r t i c l e s of a dispersed incoherent second phase. In these cases a mobile d i s l o c a t i o n i s seen to overcome i n t e r a c t i o n with.either a grain bound-ary or a p a r t i c l e while s t i l l moving i n the l a t t i c e , so that l a t t i c e f r i c -t i o n and s o l i d s o l u t i o n strengthening can be, added to the grain s i z e term or to the Orowan s t r e s s , r espectively. In a case where both a dispersed phase and subgrains are present, t h e i r a d d i t i v i t y i s not immediately evident. However, the p h y s i c a l picture behind 4 5 5 5 L i ' s model or Langford-Cohen model f o r subgrain strengthening does not require the sub-boundary or c e l l w a l l to be a p h y s i c a l b a r r i e r i n i t s e l f . Instead, the subgrain s i z e influences the motion of a d i s l o c a t i o n through the presence of other d i s l o c a t i o n s i n L i ' s model, or through the increased length of the d i s l o c a t i o n loop created i n the Langford-Cohen model., Consequently, every d i s l o c a t i o n segment experiences the subgrain s i z e e f f e c t at a l l times during i t s motion and the subgrain strengthening e f f e c t i s akin to l a t t i c e f r i c t i o n and s o l i d solution e f f e c t s . I t i s thus concluded that strengthening contributions due to l a t t i c e f r i c t i o n , solutes, subgrains and an incoherent p a r t i c l e dispersion, should a l l be additive. In a s i m i l a r fashion, i t can be shown tphat the strengthening e f f e c t of randomly d i s t r i b u t e d d i s l o c a t i o n s can be added to dispersed phase strengthening. In Tables XVI and XVII for medium carbon tempered martensites and low carbon tempered martensites r e s p e c t i v e l y , the following quantities have been l i s t e d ; i ) y i e l d strength of the tempered martensites, a y s-i i ) predicted strengthening contribution of the cementite p a r t i c l e s , Table XVI Strengthening Contributions i n the Medium Carbon Tempered Martensite -2 a , MNm t ' V MNm 2 Tempering a Temperature ^" °C -2 -2 , MNm a , MNm s • or -2 a , MNm P Hall-Petch Relation at = 0 . 2 t - 1 / 2 Langford-Cohen a = 81 x 1 0 " ^ t _ 1 250 1600 585 685 330 300 1530 555 630 345 350 1450 535 545 370 400 1300 505 440 355 450 1230 480 310 440 500 1040 430 275 335 550 900 395 239 275 600 840 350 185 355 260 305 * 135 ** 230 650 690 235 - 305 190 * 150 265 700 540 160 220 90 * 160 ** 290 * a i f Hall-Petch 0 type r e l a t i o n used f o r substructure s trengthening. ** o" o i f Langf ord-Cohen model used for substructure strengthening Table XVII Strengthening Contributions i n the Low Carbon Tempered Martensite. -2 a , MNm -2 ff , MNm o Tempering Temperature —2 a , MNm y.s. -2 a , MNm or -2 a , MNm P JHall-Petch Relation JLangford-1 Cohen a = °C at = 0 . 2 t " 1 / 2 1 t 81 x 1 0 - 6 t _ 1 256 1040 390 440 --210 300 1010 375 375 260 350 900 345 355 200 400 830 330 >275 <225 450 780 295 >225 <260 500 700 205 335 230 160* ** 265 550 600 160 295 175 * 145 A* 265 600 520 110 250 130 160 ** 280 650 440 95 200 80 * 145 ** 265 700 360 55 155 50 * -150 250 O q i f Hall-T Jetch type r e l a t i o n used for ^ substructure strengthening. ~o i f Langf ord-Cohen model used f o r substructure s teng-th ening. 121 a or i i i ) predicted strengthening e f f e c t of randomly d i s t r i b u t e d disloca-tion s , a , P iv) predicted strengthening e f f e c t of subgrains, a , for both H a l l -Petch type of r e l a t i o n s h i p and the Langford-Cohen model. v) a^, which i s simply a f i x e d component of the t o t a l observed y i e l d strength which i s not accounted fjor by P Q R » » ° r o^. The assumption i s made (and l a t e r supported) that i s comprised of those strengthening c o n t r i b u r tions which are independent of tempering temperature. For the heavily tem-pered martensites where substructure has been i d e n t i f i e d as consisting of subgrains, two estimates of O q are given for two estimates of the subgrain strengthening. In fa c t O q includes contributions from l a t t i c e f r i c t i o n (Peierl-Nabarro s t r e s s ) , from elements i n s o l i d s o l u t i o n (both i n t e r s t i t i a l and s u b s t i t u t i o n a l ) and from the small increment of s t r a i n hardening a s s o c i -ated with deforming the specimen i n tension to 0.002 s t r a i n . Knowing the i d e n t i t y of the various contributions to o , a lower l i m i t can be place on i t s magnitude. Since the equilibrium s o l u b i l i t y of carbon i n a-riron at room 19 20 temperature i s approximately 80 ppm, i t should contribute approximately ' _2 145 MNm to a (Equation 2). The concentrations of s u b s t i t u t i o n a l a l l o y i n g o elements are comparable i n both the s t e e l s and the y i e l d strength contribu-12 13 2 88 t i o n expected therefrom ' i s approximately 60-70 MNm . The P e i e r l s stress _2 i s u n l i k e l y to exceed 35-40 MNm , so that o^ should have a minimum value of _2 approximately 240 MNm . From Tables XVI and XVII, i t i s clear that the value of the intercept, o , obtained by using the Hall-Petch type of r e l a t i o n s h i p •r 2 for subgrain strengthening i s roughly only 150 MNmr . Furthermore, for the 122 medium carbon martensite tempered at 600°C, substructure strengthening was -2 185 MNm when calculated on the basis of random d i s l o c a t i o n density, and -2 355 MNm when treated as subgrains by the Hall-Petch approach. The two values d i f f e r by almost a factor of two. If the Langford-Cohen model i s used to evaluate subgrain strengthening, a i s found to be 270 MNm which i s of the r i g h t magnitude; i . e . , s l i g h t l y higher than the minimum expected value. Furthermore, i f substructure strength-ening i s evaluated from both the random d i s l o c a t i o n density concept and the subgrain concept for medium carbon martensite tempered at 600°C, the two _2 values are 185 and 260 MNm . The discrepency i s much smaller than when the Hall-Petch r e l a t i o n i s used, and may be a t t r i b u t e d to some extent to an under estimation of the subgrain s i z e . This i s also r e f l e c t e d i n the r e l a -_2 t i v e l y lower absolute value of a (230 MNm ) f o r t h i s specimen. J o c The above discussion indicates that Hall-Petch type of r e l a t i o n s h i p i s not v a l i d f o r subgrains i n tempered martensite. A number of previously pub-li s h e d data i n support of the r e l a t i o n have been re-examined to see i f they can also be f i t t e d to the Langford-Cohen p r e d i c t i o n . The data from Warring-ton's work"''"' on ~lmm wire specimens unfortunately had to be excluded, because the grain s i z e i n the wires ranged from 0.2 to 1.5mm, and the specimens could 49 not be considered t y p i c a l l y p o l y c r y s t a l l i n e . Embury et a l expressed the flow stress of cold drawn commercially pure i r o n as a = 105 + 0.32t~ 1 / / 2MNm~ 2(t i n meters) Their data, as taken from p l o t s of the flow stress and c e l l s i z e versus s t r a i n have been tabulated i n Appendix A (Table A l ). I t i s found that the data also f i t w e l l to the r e l a t i o n a = 215 + 4.47ubt - 1MNnf 2 _2 The expected value for a i n t h i s case would be about 175 MNm , and the 123 slope compares favourably with the t h e o r e t i c a l value of 3.0ub arri v e d at by Langford and Cohen"'"' for cold drawn structures. 94 83 S i m i l a r l y , least squares f i t s were obtained from B a l l ' s and Sahoo's data (Appendix A, Tables A2 and A3 respectively) on subgrain s i z e and strength for pure aluminum. The appropriate equations are given below: -1/2 -2 B a l l ' s data a = -4.5 + 0.076t MNm (t i n meters) Replotted accord- a = 9.5 + 14ybt "'"MNm 2 ing to Langford-Cohen model -1/2 -2 Sahoo's data a = -54 + 0.137t MNm (t i n meters) Replotted accord- a = 4.2 + 10.5ybt "Wm 2 ing to Langford-Cohen model The Langford-Cohen plo t s y i e l d e d p o s i t i v e values of reasonable magni-tude for the "matrix strength" of aluminum, whereas the Hall-Petch r e l a t i o n -ship gave negative values. However, the slopes for the t ^ r e l a t i o n s h i p are high compared to the t h e o r e t i c a l l y expected value of 4.9ybfor s p h e r i c a l sub-83 grains. This anomaly might be r e l a t e d to the observation that the Petch -312 slope for high angle boundaries (0.04 MNm ) i n r e c r y s t a l l i z e d aluminum i s smaller then that for subgrain boundaries by a factor of 2 to 4. Moreover, -1/2 -1 the slope determined from t or t p l o t s depends c r i t i c a l l y on the method by which subgrain si z e has been determined, whereas the intercept i s much less dependent on i t . Since the Langford-Cohen model predicts better values f o r the intercept i n a l l cases, i t i s reasonable to conclude that the Langford-Cohen model i s more consistent i n p r e d i c t i n g subgrain strengthening e f f e c t s i n aluminum as well as i n i r o n . Again examining the O q values i n Table XVI for medium carbon tempered _2 martensite', a l l the values are greater than 240 MNm , the lower l i m i t expected for a . Ip the e a r l i e r stages of tempering, however, O q has much higher -2 values, of the order of 350 MNm . This i s r e a d i l y explained i f the assump-124 ti o n i s made that the f e r r i t e matrix has more than the equilibrium amount of carbon i n s o l i d s o l u t i o n a f t e r tempering at low temperatures i n the range. 20 According to Wert (Equation (2)), approximately 0.02% carbon i n so l u t i o n would y i e l d a value of the observed magnitude. In j:he low carbon tempered martensite (Table XVII), i t i s d i f f i c u l t to arr i v e at a value for O q since l i t t l e confidence i s placed i n y i e l d strength values f o r the specimens tempered at 250-350°C. Inspite of t h i s , an average -2 value of approximately 225MNm can be assigned for the f i r s t f i v e specimens. _2 This value, although smaller than the lower l i m i t of 240MNm expected, i s at least of the ri g h t magnitude. At higher tempering temperatures, where substructure strengthening i s due to subgrains, the average value of i s 270MNm~2. The above discussion indicates that the carbide dispersion and the sub-structure (described e i t h e r as random d i s l o c a t i o n s or subgrains) are the only features of the structures of tempered steels which undergo s i g n i f i c a n t change as the tempering temperature i s increased from 250 to 700°C. In medium carbon tempered martensite, they account c o l l e c t i v e l y f o r 80 pet of the strength a f t e r tempering at 250°C, and approximately 50 pet of the strength a f t e r tem-pering at 700°C. The corresponding f r a c t i o n s f or low carbon martensite are 80 pet and 30%. The rest of the strength i n each case i s accounted for by s o l i d s o l u t i o n strengthening and l a t t i c e f r i c t i o n . 4.6.5 Comparison with Previous Work The present conclusions are i n marked disagreement with those of Cox^ 4 1 and Tyson . Cox at t r i b u t e s 60-70 pet of the strength of tempered 0,30 to 0.38 pet C martensites to i n t e r s t i t a l strengthening over the whole range of tempering temperatures. The values assigned by Cox"'" to Orowan hardening 125 were about one quarter of what would be predicted from Equation (4) f o r h i s own reported p a r t i c l e spacings. Moreover, Cox used a = 0.2 i n Equation (1), thus assigning a small strengthening r o l e to d i s l o c a t i o n s . His estimates of d i s l o c a t i o n density, based on microscopy, were much lower than those of the present work. His conclusion that r e l a t i v e l y large concentrations of carbon remained i n s o l u t i o n a f t e r tempering even at high temperatures was i n d i r e c t l y based on estimates of the volume f r a c t i o n of carbide i n specimens observed by transmission microscopy. 4 Tyson argued that the v a r i a t i o n i n carbide spacing, and accordingly i n the Orowan s t r e s s , accounted for a l l the strength differences i n s t e e l s tempered by Turkalo and Low9"* i n the temperature range 426° to 675°C. How-ever, i n applying Equation (4) , Tyson^ e f f e c t i v e l y kept "~^ n^" a t a constant l e v e l of 2.8. This i s approximately v a l i d f o r D - 1.0 um, but cj>/Trx£n^-decreases to 2 for D ~ O.lym, which was wi t h i n the range of D values being considered. Thus Tyson greatly overestimated the contribution of Orowan strengthening at the p a r t i c l e spacing of i n t e r e s t . 2 Smith and Hehemann tempered S.A.E. 4340 martensite and b a i n i t e i n the temperature range of 315°C to 540°C. They a t t r i b u t e d the v a r i a b l e part of the observed y i e l d strength to a combination of dispersion strengthening and hardening due to l a t h boundaries. Smith and Hehemann also assigned a constant contribution (independent of tempering temperature) to y i e l d strength from d i s l o c a t i o n s within the martensite laths • In f a c t , as e a r l i e r pointed out by Cox"'", the martensite l a t h s i z e does not change appreciably with i n -creasing tempering temperature, but the d i s l o c a t i o n density w i t h i n the laths does decrease markedly, as confirmed by x-ray r e s u l t s i n the present study. 81 88 Smith and Hehemann c i t e d the work of Kunitake and of Speich and Swann 81 i n support of t h e i r approach. However, Kunitake worked with low carbon martensites which were tempered at temperatures of 550°C and higher. The 126 structures observed by Kunitake showed r e l a t i v e l y well-defined f e r r i t e grains, and the s t r u c t u r a l parameter he used was "the mean free f e r r i t e path as de-termined by f e r r i t e boundaries", not the l a t h s i z e implied by Smith and Hehe-2 88 mann . Speich and Swann defined the substructure contribution to strength as being due to "increased d i s l o c a t i o n density plus martensite/martensite boundaries and the c e l l walls or i n t e r n a l twins within the transformed region." 2 They did not, as suggested by Smith and Hehemann , ascribe a separate strength-ening contribution to d i s l o c a t i o n s within the martensite l a t h s . 96 Kossowsky and Brown have suggested that dislocation-cementite networks cause strengthening i n a manner analogous to grain boundary strengthening. However, such networks were rarely, observed.and the authors themselves did not present any microscopic evidence f o r such networks. 4.7 Strengthening Mechanisms i n the Copper Bearing Steel 4.7.1 Strength i n the Underaged Condition Peak strengths were observed i n both Series A and B a f t e r aging for ap-proximately one hour. The f i r s t few p r e c i p i t a t e s were seen i n specimens that had been aged for approximately 2 hours, but were so sparse that the volume f r a c t i o n could be considered e s s e n t i a l l y zero. Thus the increase i n y i e l d strength for aging times upto one hour can be a t t r i b u t e d to a progressive increase i n the volume f r a c t i o n of c l u s t e r s present. In Series B ? some re-covery of the substructure introduced by cold working occurred simultaneously with c l u s t e r formation. However, i n the absence of quantitative information on the progress of c l u s t e r i n g , no attempt has been made to account i n d e t a i l for the strength of these 'underaged' a l l o y s . 4.7.2 Contribution of Incoherent e P r e c i p i t a t e s 127 With increase i n aging time beyond 2 hours, the y i e l d strength de-creased (Table III) , and i n the conventional sense the a l l o y would be des-cribed as 'overaged'. However, the p r e c i p i t a t i o n of copper as e was f a r from complete a f t e r 2 hours at 500°C. Even a f t e r 100 hours of aging, quantitative electron microscopy revealed that -38 pet of the copper i n the s t e e l could not be accounted for as incoherent e p r e c i p i t a t e (Table XII). Thus with i n -creasing aging time there was a steady increase i n the amount of incoherent p r e c i p i t a t e and a corresponding decrease i n the amount of coherent c l u s t e r s . The strengthening e f f e c t of non-coherent p a r t i c l e s i s generally evalu-97 ated on the basis of the Orowan mechanism. Recently, R u s s e l l and Brown have advanced another theory f o r the s p e c i f i c case where the p r e c i p i t a t i n g phase has a lower shear modulus than the matrix, which i s the case for i r o n -copper a l l o y s . The authors of this theory suggest that the p r e c i p i t a t e d phase, whether coherent or not and whatever i t s p a r t i c l e s i z e , i s always sheared by the passage of d i s l o c a t i o n s through i t . However, there i s con-siderable transmission electron microscopic evidence to contradict t h i s 5 8 98 suggestion for iron-copper alloys. Hornbogen and Pattanaik et a l found evidence of the shearing of incoherent p r e c i p i t a t e s only a f t e r 8-10 pet s t r a i n . F u j i i et a l ^ during t h e i r i n v e s t i g a t i o n of a heavily overaged Fe-6 at. pet. Cu a l l o y , observed almost semicircular d i s l o c a t i o n loops between the copper p a r t i c l e s ; i . e . , the configuration required to by-pass the p a r t i c l e s by bowing between them. Moreover, the s l i p planes i n the i r o n matrix and i n the incoherent copper p r e c i p i t a t e need not always be p a r a l l e l . In f a c t , p a r a l l e l s l i p planes would occur s t a t i s t i c a l l y f o r only one i n s i x p r e c i p i t a t e s , so that i n most instances shearing would also require the formation of a jog and the p u l l i n g of a dipole. E n e r g e t i c a l l y , t h i s would favour by-passing rather than shearing. The absence of evidence of d i s l o c a t i o n loops around 5 p a r t i c l e s was a t t r i b u t e d by F u j i i et a l to the lower shear modulus of copper^ 128 which would cause loops to be attracted into the m a t r i x - p a r t i c l e i n t e r f a c e . The v a l i d i t y of the Orowan mechanism i n the deformation of the overaged Fe-6%Cu a l l o y was further confirmed when F u j i i ' s r e s u l t s on heavily aged s p e c i -mens provided excellent agreement with Equation (4). The bulk of evidence suggests, therefore, that the contribution of noncoherent p a r t i c l e s should be evaluated on the basis of the Orowan model. The magnitude of the Orowan stress can be estimated from the e-phase p a r t i c l e spacing, the l a t t e r being calculated on the basis of the volume f r a c t i o n of e p r e c i p i t a t e d . However, i t i s also observed (Table XII) that the stress concentration factor at the p a r t i c l e , D/d g, has very large values for the low copper a l l o y used i n the present work, the lowest value of the 33 24 f a c t o r being 10. Consequently, according to Ashby and A n s e l l , the by-passing of the p a r t i c l e s should be accomplished by cross s l i p i n preference to the Orowan mechanism. I f y i e l d i n g i s c o n t r o l l e d by cross s l i p , one would expect a strong temperature dependence of the y i e l d s t r e s s . The fact that 29 34 35 t h i s temperature dependence has not been observed by others ' ' has been 38 explained by Hirsch as follows: i f we consider an edge d i s l o c a t i o n approach-ing a p a r t i c l e i n i t s s l i p plane, then i n order that the d i s l o c a t i o n may cross s l i p , i t must be bent to an extent that a segment of the o r i g i n a l d i s l o c a t i o n acquires a screw component before i t can undergo cross s l i p . The stress re-quired to accomplish t h i s bending i s i n form the same as the Orowan s t r e s s , so that even i f cross s l i p takes place the y i e l d stress i s c o n t r o l l e d by the Orowan st r e s s . The explanation seems adequate enough for the above mentioned 29 34 35 work ' ' where the stress concentration factor varied from approximately 3 (for an overaged Al-4Cu a l l o y ) to 6 (for i n t e r n a l l y oxidised Cu-Si and Cu-Be single c r y s t a l s ) . However, when the stress concentration f a c t o r i s much larger, as i t i s i n the present study, i t i s apparent that the edge d i s l o c a t i o n i n the above example can a t t a i n a screw component at a stress lower than the 129 Orowan s t r e s s . In addition , i n the present study the matrix strength i t -s e l f (with contributions from s o l i d s o l u t i o n hardening and coherent c l u s t e r hardening) i s expected to be large, so that the absolute magnitude of the stress on the d i s l o c a t i o n segment adjacent to the p a r t i c l e would be extremely high. Therefore, i t i s suggested that when the copper bearing s t e e l i s de-formed a f t e r the heat treatments used i n the present i n v e s t i g a t i o n , d i s l o c a -tions can overcome the incoherent p r e c i p i t a t e s by cross s l i p at a stress much lower than the Orowan s t r e s s , and that the contribution of noncoherent p r e c i p i t a t e s can therefore be neglected i n explaining the strength. This explanation seems to be i n contradiction with the aforementioned work qf F u j i i et al~* who found good agreement between t h e i r data for a 6 at.% Cu a l l o y and the Orowan theory. The 6% copper a l l o y , a f t e r the heat-treat-ment used by F u j i i et a l contained a large volume f r a c t i o n of e phase, and probably no c l u s t e r s . Thus, the stress concentration f a c t o r could be ex-pected to be as low as 3, compared with at l e a s t 10 f o r the lower copper a l l o y heat-treated as i n the present work. F u j i i et a l also studied a low-copper s t e e l , which was aged at 500°C as i n the present work. Whereas they at t r i b u t e d strengthening again to the Orowan mechanism, t h e i r Orowan plo t had a slope which was only 2/3 of the value predicted from Equation (4). In the l i g h t of the present discussion, i t i s l i k e l y that the low-copper a l l o y was aged by F u j i i et a l such as to produce a high density of c l u s t e r s , and that the observed strengthening was due mainly to coherent p r e c i p i t a t e s . A s i m i l a r explanation would account for the low Orowan slope obtained by Pattanaik with aged Fe-5% Cu a l l o y s . 4.7.3 Contribution of Coherent Clusters and Substructure Since the magnitude of coherency s t r a i n s i n t h i s system i s expected to 130 be small, most of the hardening due to coherent c l u s t e r s should be due to the requirement that the di s l o c a t i o n s shear c l u s t e r s . This hardening has 21 24 been treated by K e l l y and Nicholson and A n s e l l , as discussed more f u l l y i n Section 1.2.3. The amount of strengthening, °~ from t h i s source i s given by o 1/2 ° c o h = f - f2 <3> A r e l i a b l e value for y, the surface energy of the new in t e r f a c e pro-duced during the shearing process, i s not av a i l a b l e . Accordingly the y i e l d strength (0.2% o f f s e t flow stress) a f t e r various aging times for Series A 1/2 specimens has been p l o t t e d against (Tables III and XII) and y has been evaluated from the slope of the st r a i g h t l i n e drawn to f i t the data. The i n -tercept on the stress axis, a , would be expected to include contributions from P e i e r l s s t r e s s , s o l i d s o l u t i o n strengthening and grain boundary strength-ening. Figure 48 shows such a p l o t , and the st r a i g h t l i n e f i t obtained by the method of least squares can be expressed 3 1 / 2 -2 (Series A) a = 120 + 3.1 x 10 f , MNm y. S I The slope, 3.1 x 103MNm 2 , when equated to y i e l d s a value of 380 ergs/cm 2 for y> the surface energy. In Series B, an add i t i o n a l and va r i a b l e strengthening contribution i s expected from substructure. The substructure i s i n the form of subgrains which are the r e s u l t of allowing recovery of the cold r o l l e d matrix to occur during aging. This contribution cap be estimated on the basis of the Langford-Cohen model, which was found to be applicable i n the case of subgrain strength-ening i n tempered martensites (Section 4.6.3 and 4.6.4). The subgrain s i z e varies from 0.53y to 0.63y as the aging time i s increased from 2 hours to 100 hours, and the corresponding strengthening contribution (a = 4.2ybt ^) _2 decreases from 155 po 130MNm . From the Series B data, the value for y,» 1 3 1 132 the surface energy, can now be computed from the slope of a (a - a^) vers-1/2 us f ^ p l o t , Figure 49. The s t r a i g h t l i n e shown i n F i g . 49 was obtained by the method of least squares and can be expressed as 3 1 / 2 -2 (a - a ) = 205 + 2.6 x 10 f . MNm y-s t 2 2 This slope correspnds to a y value of 330 ergs/cm , and the intercept on -2 the stress axis i s 205MNm . Because there are a large number of data points available i n the p l o t f o r Series B, greater confidence Is placed i n the value of y obtained from Series B data. In deriving Equation (3), i t i s assumed that the only new i n t e r f a c e created by the shearing of a c l u s t e r i s that between the i r o n and the copper. No account i s taken of the possible i n t e r f a c e created within the c l u s t e r since the Burger's vector of a d i s l o c a t i o n i s d i f f e r e n t i n i r o n and i n copper. The assumption i s w e l l j u s t i f i e d i n the present case since the Burger's vec-tor i n i r o n and copper are i n f a c t quite s i m i l a r and thus only a m i s f i t d i s -21 l o c a t i o n might be created around the c l u s t e r . K e l l y and Nicholson have estimated the energy of such a d i s l o c a t i o n to be = 0 . 6 y b ( ^ ) 2 where Ab i s the difference i n the magnitudes of the Burger's vectors i n the matrix and the p r e c i p i t a t e . Substituting appropriate values for the Fe-Cu system; -19 11 5 63 5c 10 Energy of the m i s f i t d i s l o c a t i o n = 0 . 6 x 7 . 6 x 1 0 x — - • 2.48 x 10 2 - 10 ergs/cm Thus the energy of the m i s f i t d i s l o c a t i o n i s r e l a t i v e l y i n s i g n i f i c a n t , and the value of y found i n the present i n v e s t i g a t i o n can be assumed to be associated with the iron-copper i n t e r f a c e . 133 No estimates could be found i n the l i t e r a t u r e for the i n t e r f a c i a l en-ergy between s o l i d i r o n and s o l i d copper at room temperature, against which 2 the present value of -330 ergs/cm could be compared. However, the high 99 angle grain boundary energies for i r o n and copper are reported to be 520 2 and 650 ergs/cm r e s p e c t i v e l y , both of which values are within a factor of 21 two of the value of y obtained here. K e l l y and Nicholson have suggested that an approximate value of y m&Y be obtained on the basis of van der Merwe's"^^ treatment of the energies of t i l t and twist boundaries. The 21 rough estimate derived by K e l l y and Nicholson was 0.03yb. Using appropri-ate values for i r o n and copper, the respective numbers are calculated to be 2 565 and 370 ergs/cm . 97 In the theory recently advanced by R u s s e l l and Brown to explain strength-ening i n a l l o y systems where the dispersed phase has a lower shear modulus than the matrix, the p r e c i p i t a t e i s sheared, and a segment of d i s l o c a t i o n tends to lag behind i n the dispersed phase so that the applied stress has to bend the d i s l o c a t i o n before i t can leave the dispersed phase. The extent of this bending depends on the r e l a t i v e magnitudes of the shear modulii of the dispersed phase and the matrix, u-^  and y^. The amount of strengthening i s , i n turn, a function of the extent of bending. Thus, i f voids are dispersed P l i n the matrix, then — = 0 and the included angle between the arms of a d i s -' v2 6 l o c a t i o n would be almost zero. On the other hand, when the dispersed phase has the same shear modulus as the matrix, then y^/y 2 = 1» t n e included angle (2<j)) i s found to be 180°, and no strengthening i s expected. When th e i r theory i s applied to the iron-copper a l l o y containing co-herent c l u s t e r s , R u s s e l l and Brown predict that the strengthening e f f e c t i s propprtional to the square root of the volume f r a c t i o n of coherent c l u s t e r s . 21 This r e l a t i o n s h i p i s the same as that arrived at by K e l l y and Nicholson on the basis of the creation of a new i n t e r f a c e . Experimentally i t i s there-134 fore d i f f i c u l t to d i s t i n g u i s h between the two theories. R u s s e l l and Brown describe t h e i r own experimental data according to the equation a , = -,71 + 3.57 x 10 3f 1 / 2MNm~ 2 coh 1/2 3 -2 The c o e f f i c i e n t of f , i . e . , 3.57 x 10 MNm , i s claimed to be i n a l -most exa^ct agreement with the predictions of t h e i r theory. In the present 1/2 3 work, the slope of the (a -a ) versus f„ p l o t (Figure 47) was 2.6 x 10 r y . s t ' 2 b -2 MNm , which i s i n reasonable accord with the theory. However, cer t a i n objections can be raised against the Ru s s e l l and Brown theory. F i r s t , i n s i m p l i f y i n g the statement of the theory, i t was assumed that a l l clusters were of the same s i z e and that the diameter d of a l l clusters was approximately 25A°, i r r e s p e c t i v e of the aging conditions under which the clusters were produced. In p r a c t i c e , a range of sizes of clusters would be expected, and 25A° i s a low value for the c l u s t e r s i z e . The shear modulus of a c l u s t e r i s shown to be a function of i t s s i z e , and according to E _ u . ' j • J u >-u c l u s t e r c l u s t e r n c , o r . o equations derived by the authors, • = — — = 0.96 for d = 25A , &. y. v ir o n i r o n where E denotes the energy of the d i s l o c a t i o n i n a p a r t i c u l a r plane and y i s the shear modulus. This r a t i o 0.96 would correspond to an included angle of 147° between the arms of a d i s l o c a t i o n at the c l u s t e r . Thus, according to the theory, strengthening i s caused because a s t r a i g h t d i s l o c a t i o n ( i n -cluded angle 180°) i s to be bent to an included angle of 147°. However, when Russ e l l and Brown needed a value for i n t e r c l u s t e r spacing, they chose to use the centre-to-centre spacing of nearest-neighbour c l u s t e r s , instead of the mean free path between cl u s t e r s which would be required for a s t r a i g h t d i s l o c a t i o n . I f a correction i s applied f or t h i s discrepancy, the predicted strengthening e f f e c t i s reduced by a factor of ~4 for the composition i n -vestigated i n the present study, which would leave the experimental r e s u l t s in poor agreement with theory. 135 Ruspell and Brown also suggest that, i n p r i n c i p l e , t h e i r theory should be applicable even when the shear modulus of the matrix i s less than that of 101 102 the p r e c i p i t a t e . However, K e l l y and Dash and Fine have investigated age hardening i n the Al-Ag and Al-Zn systems, r e s p e c t i v e l y , and i n both cases the strength could be adequately explained on the basis of Equation (3). The value of y obtained by K e l l y f o r the Al-Ag system was consistent with the value obtained from the heat of reversion. Also, i s approximately 10% higher than so that when a correction for the small coherent p r e c i p i -tate i s applied,.the two u values turn out to be within 2% of each other. Accordingly, the angle to which the. d i s l o c a t i o n i s to be bent i s approximately 160°, and n e g l i g i b l e hardening would be expected therefrom. It i s concluded therefore that f o r the heat treatments studied i n the present i n v e s t i g a t i o n , the v a r i a b l e part of the strength of the iron-copper n i c k e l a l l o y i s adequately explained on the basis of the shearing of clusters and the creation of a new in t e r f a c e . An a d d i t i o n a l and s i g n i f i c a n t strength-ening increment due to subgrain boundariesj i s provided when the solu t i o n treated a l l o y i s cold worked before aging. 4.7.4 Matrix Strength In Tables XVIII and XIX, the various strengthening contributions are compiled for Series A and B, resp e c t i v e l y . The values of a C Q ^ quoted f o r Series A are the r e s u l t of calcu l a t i o n s based on the estimate of y obtained from Series B. -2 ' The average 0 values i n Series A and B are 175 and 200MNm resp e c t i v e l y . An approximate estimate of 'matrix strength' can also be made'on the basis of the known composition of the a l l o y . Thus the 60 ppm of i n t e r s t i t i a l s pre-20 -2 sent should contribute about 125MNm to the y^Leld strength of ir o n . Another 136 Table XVIII Strengthening Contributions i n Series A, Fe-Cu-Ni A l l o y . Aging time at 500°C, hours -2 a , MNm y.s. f 1/2 2 °coh = 2.6 x 103f^/2, MNm"2 o = 0 y.s. coh MNm-2 2 525 0.1265 330 195 5 505 0.1247 320 185 10 475 0.1204 310 165 100 375 0.0821 210 165 Table XIX Strengthening Contributions i n Series B, Fe-Cu-Ni A l l o y . Aging time a at 500°C, y - S ' hours MNm 1/2 coh 3.1/2 t, um 2.6 x 10 f MNm 4.2 ubt "2 - MNm a = o a - a - a y.s. t coh. MNm~2 2 5 10 30 60 100 685 665 635 585 565 540 0.1265 0.1214 0.1105 0.1060 0.09 0.0775 330 315 285 275 230 200 0.533 0.56 0.56 0.604 0.593 0.625 150 140 140 130 135 130 205 21Q 210 180 200 210 137 -2 -2 35-40MNm may be added for P e i e r l s s t r e s s , and 55-70MNm for n i c k e l i n 12 13 s o l i d s o l u t i o n ' , brings the expected matrix strength f o r Series B to -2 215-235MNm . In Series A, an a d d i t i o n a l contribution to a can be expected o r 93 from grain boundaries. For a grain s i z e of -50u, t h i s contribution i s ex-_2 pected to be approximately 40MNm , so that the estimated value of matrix -2 strength r i s e s to >255MNm . In the two s e r i e s , therefore, the experimental a values are lower than expected, o r I t i s perhaps worth noting that i f the subgrain strengthening f o r Series B i s evaluated by means of the Hall-Petch r e l a t i o n (Section 4.5.3), i t becomes -2 larger, and O q reduces to the s t i l l lower value of 75MNm . Again, therefore, the choice of the Langford-Cohen model to describe strengthening for a poly-gonized substructure i s indicated by the r e s u l t s . 138 5. SUMMARY AND CONCLUSIONS (1) Strengthening mechanisms i n three heat-treated low a l l o y s t e e l s have been investigated. The three materials were: a medium carbon (0.42% C, .1.1% Mn) martensite tempered i n the range 250-700°C, a low carbon (0.11% C, 0.9% Mn) martensite tempered i n the same temperature range, and an age-hardening copper-bearing steel(1.8% Cu) which was aged at 500°C e i t h e r d i r e c t l y (Series A), or with 50% cold reduction by r o l l i n g a f t e r s o l u t i o n treatment (Series B). The techniques employed were electron microscopy (both r e p l i c a and trans-mission) and x-ray l i n e broadening a n a l y s i s . (2) In the present study, the X-ray l i n e broadening technique has been used e f f e c t i v e l y for the quantitative analysis of transformation-induced d i s l o c a -t i o n substructures. (3) On the basis of electron d i f f r a c t i o n work, i t has been concluded that i n the tempered martensites, l a t h c h a r a c t e r i s t i c s are retained even a f t e r the l:>.th boundaries can no longer be delineated by transmission microscopy. However, a f t e r tempering at 700°C f o r 1 hour, the substructure consists of subgrains, the boundaries of which do not appear to be r e l a t e d to the o r i g i -n a l l a t h boundaries. (4) In the l i g h t of current theories for substructure strengthening, i t has been concluded that the contributions of substructure and dispersed non-coherent p a r t i c l e s to y i e l d strength are additive. This should be true 139 whether the substructure consists of randomly d i s t r i b u t e d d i s l o c a t i o n s or of subgrains. (5) The observed dependence of y i e l d strength on subgrain s i z e was i n a l l cases consistent with the predictions of the Langford-Cohen model; i . e . , sub-grain strengthening was inversely proportional to the subgrain s i z e . By con-t r a s t , the experimental data were not consistent with a Hall-Petch type of r e l a t i o n s h i p . (6) For the iron-carbon martensites of the present work that were tempered at r e l a t i v e l y low temperatures, that part of the y i e l d strength which va r i e d with tempering temperature could be accounted f o r e n t i r e l y by contributions from randomly d i s t r i b u t e d d i s l o c a t i o n s ( o r i g i n a l l y introduced during the martensitic transformation) and from dispersed cementite p a r t i c l e s (Orowan mechanism). (7) A f t e r tempering at higher temperatures i n the range studied, the sub-structure of tempered martensites had transformed into subgrains. Once again, subgrain strengthening and dispersion strengthening adequately ac-counted for a l l the v a r i a b l e part of the y i e l d strength. (8) The f i n d i n g of the present work that d i s l o c a t i o n substructure accounts f o r a major part of the y i e l d strength of tempered medium carbon martensites i s i n marked contrast with a l l previous studies of strengthening mechanisms i n these materials. (9) In the peak-aged condition for the Fe-1.8% Cu-1.3% Ni a l l o y studied, almost a l l the copper was present as coherent c l u s t e r s . Even a f t e r aging 140 o f o r 100 hours at 500 C. The p r e c i p i t a t i o n of copper as incoherent p r e c i p i -tates was found to be incomplete. (10) In marked contrast with the conclusions of previous workers, i t has been concluded that the strengthening contribution of incoherent p r e c i p i -tates i n age-hardening copper-bearing s t e e l s containing <2% Cu i s i n s i g n i f i -cant i n the optimally aged condition due to the large r a t i o of p a r t i c l e spacing to p a r t i c l e diameter. Instead, strengthening -is associated with shearing of coherent c l u s t e r s . (11) In the peak-aged and over-aged conditions of the copper-bearing s t e e l , subgrains (introduced by cold working and subsequent recovery during aging-Series B) make a contribution to strength which i s e s s e n t i a l l y independent of aging time. 141 APPENDIX A Table A l Flow Stress and C e l l Size Data f o r 49 Cold Drawn Iron Wire (Embury et a l ) Flow Stress, C e l l Size t \ m 1 MNm -2 t,um 322.0 0.970 1.03 x 10 6 372.5 0.600 1.67 x 10 6 400.0 0.450 2.22 x 10 6 407.0 0.317 3.15 x 10 6 483.0 0.300 3.33 x 10 6 669.5 0.215 4.65 x 10 6 795.0 0.144 6.95 x 10 6 142 Table A2 Y i e l d Stress and Corresponding Subgrain Sizes 94 i n Pure Aluminum ( B a l l ) Y i e l d Strength Subgrain Size t \ m ^ ._T -2 t, um MNm ' 10.0 40.0 2.5 x i o 4 10.0 25.0 4.0 X i o 4 13.9 18.0 5.6 X i o 4 13.5 14.0 7.1 X i o 4 13.6 20.5 4.9 X i o 4 16.5 16.5 6.1 X i o 4 24.7 12.0 8.3 X i o 4 18.5 16.0 6.25 x 10 4 21.7 11.5 8.7 X i o 4 26.4 5.2 19 X i o 4 Table A3 Room Temperature Y i e l d Strengths and the Corresponding Subgrain Diameters f o r Pure Aluminum a f t e r Various Thermomechanical Treat-83 ments. (Sahoo ) -1 -1 Material Condition Subgrain t , m Y i e l d Strength Size, t, pm -2 MNm Cold r o l l e d 60% 0 .82 1 .22 X 10° 89 Cold Rolled 70% 0 .76 1 .32 X i o 6 101 Cold Rolled 80% o .70 1 .43 X i o 6 108 Cold Rolled annealed at 30 min. 70% and 300°C for 1 .33 0 .75 X i o 6 76 Cold Rolled annealed at 1 hr. 70% and 300°C for 1 .58 0 .63 X i o 6 52 Cold Rolled annealed at 2 hrs. 70% and 300°C for 2 .28 0 .44 X i o 6 39 Cold Rolled annealed at 6 hrs. 70% and 300°C for 3 .15 0 .32 X i o 6 16 Cold Rolled annealed at 1 hr. 70% and 100°C for 0 .85 1 .18 X i o 6 97 Cold Rolled annealed at 1 hr. 70% and 250°C for 1 .00 1 .00 X i o 6 87 144 REFERENCES 1. Cox, A.R., Strength of Metals and A l l o y s , Suppl Jap. Inst, of Metals, 1968, v o l . 9, 118-125. 2. Smith, D.W., and Hehemann, R.F., J . Iron and S t e e l Inst., 1971, v o l . 209, 476-481. 3. Hyam, E.D., and Nutting, J., J . Iron and Steel Inst., 1956> v o l . 184, 148-165. 4. Tyson, W.R., Acta Met., 1963, v o l . 11, 61-62. 5. 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