The impact of F.R.N. Nabarro on the LEDS theory of workhardening

Progress in Materials Science 54 (2009) 707–739

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The impact of F.R.N. Nabarro on the LEDS theory of workhardening Doris Kuhlmann-Wilsdorf * Applied Science Emerita, Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, United States

a r t i c l e

i n f o

a b s t r a c t F.R.N. Nabarro has had a profound impact on the development of workhardening theory. Albeit, on account of his opposition to the LEDS/LES theory, that may be recognized as an emerging new paradigm, this has not always been positive. The LEDS/LES paradigm is ultimately based on Taylor’s 1934 workhardening theory. According to it, all plastic deformation of all solids under virtually any conditions, whether or not dislocations play a role in it, produces structures that closely approach minimum free energy, subject to Newton’s third law, i.e. force equilibrium. In due course, the LES/ LEDS theory is expected to replace the previous SODS (‘‘self-organizing dislocation structures’’ approach) that was supported by Nabarro. Namely, in contrast to the SODS approach, the LEDS/LES theory has long since quantitatively accounted for the bulk of all important aspects of crystalline plasticity and their correlated dislocation structures. It is argued that the protracted resistance against paradigm change, as also by Nabarro, is in line with Kuhn’s pertinent discussion in ‘‘The Structure of Scientific Revolutions’’. Ó 2009 Elsevier Ltd. All rights reserved.

Contents 1.

2.

Part 1: personal reminiscences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Germany at the end of WWII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The background of post WWII research on plastic deformation . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. First personal and professional interactions with F.R.N. Nabarro. . . . . . . . . . . . . . . . . . . . . . . . . Part II: development of workhardening theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The beginnings of post WWII workhardening theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* Address: 2600 Barracks Road, Apt. 278, Charlottesville, VA 22901, United States. Tel./fax: +1 434 295 4920. E-mail address: [email protected] 0079-6425/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2009.03.007

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2.2. 2.3. 2.4.

3.

The Peierls–Nabarro force, dislocation uncertainty and melting . . . . . . . . . . . . . . . . . . . . . . . . The mesh-length theory and the ICSMA conference series . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of the LEDS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Determining the low-energy form of dislocation cells via computer calculations . . . . 2.4.2. Dependence of cell morphology on pressure and strain rate . . . . . . . . . . . . . . . . . . . . 2.4.3. LEDS formed in fatigue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. The Stage I to Stage II LEDS transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5. Understanding cells and cell blocks via the LEDS hypothesis . . . . . . . . . . . . . . . . . . . . 2.4.6. Worksoftening and grain boundary substance studied with H.G.F. Wilsdorf . . . . . . . . 2.4.7. Deformation band studies with E. Aernoudt, E. A. Starke, Jr. and K. Winey . . . . . . . . . 2.4.8. Dislocations/lattice vacancy interactions: ‘‘mushrooming’’ and planar vs. wavy glide . . . 2.4.9. The onset of Stage IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.10. The speed dependence of the flow stress and link length distribution in cell walls . . . . 2.4.11. Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.12. Alloy hardening – solid solutions and precipitates . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.13. Grain boundary hardening – two versions of the Hall–Petch relationship . . . . . . . . . 2.4.14. Conferences and final collaboration with Frank Nabarro. . . . . . . . . . . . . . . . . . . . . . . Part 3: workhardening theory in light of Kuhn’s ‘‘Structure of Scientific Revolutions’’ . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Part 1: personal reminiscences 1.1. Germany at the end of WWII Frank Nabarro entered my life at the first post WW II annual conference of the German Society of Materials Science, then still named Deutsche Gesellschaft für Metallkunde (DGM), in Stuttgart, in mid-September of 1948, i.e. quite soon after the currency reform that ushered in German post WW II reconstruction. I then was a lowly post doc of my thesis supervisor Prof. Georg Masing at Göttingen University, having in December 1947 completed a Ph.D. thesis on micro-strains of hard-drawn copper wires [1]. A critical turning point herein had been my April 1945 decision to interpret my results in terms of dislocations rather than, for example, metal amorphization through slip lines. As it happened, while I pondered this decision over several days in my home town Bremen, where I had retreated in early April of 1945 to be with my parents at the end of WWII when the whole civil order had effectively broken down and Göttingen University had closed, Bremen was subjected to almost incessant crashing artillery fire, namely by British troops who slowly advanced on Bremen. When finally the first British soldiers arrived, all streamed out of shelters and bunkers to greet them as liberators, not only from the bombardment but no less from the Nazi dictatorship that to the very end had threatened death to anyone who would show signs of wanting to surrender. Alas the occupying soldiers drove us from the street with pointed guns, and after an initial total curfew, there followed a long-lasting order against ‘‘fraternization’’ between occupying troops and Germans. All this was rather different from the Allied radio broadcasts to which in the preceding war years we had listened under threat of the death penalty. Even so, out of sight of officers, fraternization proceeded apace. It took only a few days to form friendly bonds between Germans and Allied occupation troops. From May to early August 1945, while Göttingen University remained closed, I worked as a photocopyist at the US occupation administration of Bremen, where my GI coworkers tried to ‘‘fraternize’’ more than I was prepared, and presented occasional chocolate bars which were very gratefully received since we were hungry most of the time. Namely, in those early days after the end of WWII, the daily German food ration amounted to round about 1400 cal. From this already inadequate level, food rations slowly deteriorated both in quantity and quality, finally in mid-1948 to bottom out at 800 cal daily.

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Similarly there was a painful shortage of heating material to make us shiver through the winters, and on account of inhibitory rules and almost worthless money, shops were bare and there was no significant rebuilding until the 1948 currency reform. Instead, still remaining factories were dismantled and shipped to allied countries for reparations. It was the time when Germans were ‘‘punished for the concentration camps’’, never mind that the only concentration camps of which we had known, had been the tool for our subjugation under the Nazi dictatorship, and when following the Morgenthau plan, Germany was going to be transformed into an agricultural country. In August 1945, friends in Göttingen sent word, via two women whom I had not known before, that the University had re-opened, and together with them I bicycled back through the intervening 130 miles. There was no other option since railroad traffic had not yet been re-established. As part of the appalling breakdown of proper facilities, also the post office did not function. Meanwhile, the need for communications had never been more urgent, as perhaps as many as one third of the German people was displaced and/or tried to locate displaced family members and friends. My August 1945 bicycle trip from Bremen to Göttingen was one of the eeriest experiences of my life: except for troop traffic, and except for a sprinkling of pedestrians and cyclists, highways and streets were deserted as there were virtually no private cars or motor cycles nor gas stations. The countryside was lovely but many fields were untended. Most impressively, the one large city on our way, Hannover, was even more thoroughly destroyed by bombing than was Bremen. As we approached on a big road, as we traveled for miles through Hannover, and as we left it, the typical sight was ruins as far as the eye could see. The continuing impoverishment of Germany changed abruptly when it became clear that Europe could not recover, and was in growing danger of falling under communist rule, as long as at its center, Germany festered in a state of progressive decay. In the summer of 1948, this then prompted (i) the currency reform that permitted the re-ignition of economic activity, and (ii) the infusion of capital through the Marshall Plan. That the first annual meeting of the DGM took place soon after the currency reform was no accident, of course, and similarly it was no accident that a small number of selected foreigners could now for the first time visit Germany, whereas until then it had been all but impossible for foreign civilians to cross the German borders, and similarly for Germans to get out. One of the earliest foreign visitors was Frank Nabarro as THE honored guest at the 1948 conference of the DGM in Stuttgart. Of course, I was most eager to participate, but while previously we had had money (albeit not enough for the black market) but shops were empty, now shops were full and we had no money. Now railroads, that had not been operating through most of 1945, and thereafter had tended to be so overcrowded that, on account of choked doors, it was common for passengers to scramble through windows, finally offered decent service, but we could not afford tickets. I therefore decided to hitchhike to the Stuttgart Conference and my later husband, Heinz, fearing for my safety, insisted on coming along for my protection. Thus over the span of about 1 week Heinz and I hitchhiked from Bremen to Stuttgart. 1.2. The background of post WWII research on plastic deformation At the time, my April 1945 decision in Bremen to base my thesis on dislocation theory had been far from obvious: dislocations were still entirely hypothetical and were in competition especially with the hypothesis that glide destroyed the crystalline order and left behind hard amorphous zones. Anyway, by deciding that dislocations must be real, and by interpreting my results accordingly, my research had become highly compatible with Frank Nabarro’s interests. Indeed, by the time of our first meeting at the Stuttgart conference, I had read Nabarro’s relevant papers with Mott [2,3] and had admiringly decided that Mott and Nabarro were the most important scientists in my research area. As it was, my thesis topic had been suggested to Prof. Masing by my mentor and fatherly friend Professor Richard Becker who was eager to refine and extend pre-WWII thesis work by Boas [4] and Orowan [5] under his direction, then at the University of Berlin. In my early student days, Becker had been fairly recently displaced to Göttingen University, from his earlier prestigious position in Berlin, namely on account of his suspected anti-Nazi opinions, aggravated no doubt by his past association with his two Jewish graduate students, Boas and Orowan. At Göttingen Becker enjoyed good relationships with his colleagues all around, few if any of whom were Nazi’s as far as I could tell;

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but under threat of getting into a concentration camp, everyone conformed to the appearance of being a loyal Nazi and one could discover the true political opinions only of one’s closest friends. Anyway, in Berlin, Becker had evidently not conformed well enough so as to fall into disgrace even though he was one of the top German theoretical physicists. In fact, as recently reported by Avery [6], while still a student at New College in Oxford before WW II, Frank Nabarro had had to abandon his very first research project, namely on the theory of magnetic coercivity, when after several months of effort it was discovered that the problem had already been solved by Becker. In Göttingen, Becker received only a tiny research budget and his office, located just outside of the main entrance of the large Department of Physics, had been the doorkeeper’s cubbyhole. Fortunately, the German strong tradition of tenure for professors prevented worse, as indeed it also protected the Jewish father of my Roman Catholic study friend Maria Ehrenberg: as a previously highly regarded physiologist at Göttingen’s medical school, Prof. Ehrenberg continued drawing his salary throughout the Nazi dictatorship, and to do a modicum of research, perhaps on account of Maria’s Roman-Catholic mother. The Ehrenberg’s continued to live in their lovely cultured home in which I was an occasional guest, and Maria freely studied at the University. However, throughout the Hitler dictatorship, Prof. Ehrenberg was not permitted to lecture and he took pains to enter and leave his university office and laboratory through a side door and generally to be as inconspicuous as possible. 1.3. First personal and professional interactions with F.R.N. Nabarro Stuttgart was among the most heavily bombed German cities and there, too, had as yet been next to no repair or reconstruction except for the clearing of loose debris. As a result, along with most conference attendees, I stayed in the ‘‘Bahnhof’s Hotel’’ which was the erstwhile air raid bunker underneath the Stuttgart main rail road station. It consisted of a warren of small cubicles with two-tier bunk beds plus a small metal locker for each guest, separated by concrete walls that reached to only about one foot below the ceiling so as to permit ventilation. Communal basic toilets and showers served the guests and there was a rudimentary cafeteria. I trust that Frank Nabarro, as the single, highly honored foreign guest, was more luxuriously housed. Normally, as a lowly post doc I should have remained well in the background. However, few of the dignitaries of the DGM could speak English well enough to carry on even a halting conversation, and only a handful of people in all of Germany could intelligently speak about dislocations. Thus, quite unexpectedly, I was catapulted from a lowly post doc to Frank Nabarro’s translator, guide and companion for the duration of the conference. So at meals I sat by his side at the head of the table, surrounded by Germany’s most important metallurgists from academia and industry, during lectures I translated for him, and at entertainments I was there with him in the best places, such as at a charming chamber concert in a palais outside of Stuttgart. It was a wonderful way of becoming acquainted, and Frank Nabarro and I quickly formed a personal friendship that has endured for most of my life. Of course we talked about the war events and what happened to German Jews. At that point I was still unaware that Frank was Jewish but he was evidently satisfied with my answer (that has not ever changed) that ‘‘the Germans’’ were not only innocent in the Holocaust but had known nothing about it, and that this needed no other proof than that whatever happened to the Jews in the East, occurred outside of the German borders and had been a state secret. We might compare the situation with alleged torture of US prisoners overseas, except with still less opportunity for information since there was no free press and we were greatly intimidated by the ever-present threat of being consigned to a concentration camp, if not to be sentenced to death, for perhaps knowing the ‘‘wrong’’ things. And in this line, ‘‘Feindfunk’’, i.e. foreign radio broadcasts (to which my mother who was a native New Yorker had listened regularly under threat of the death penalty), never gave any relevant information. Consequently, only a tiny minority of Germans knew anything whatever about what we now call the Holocaust, and even fewer had had any part in it. And more importantly yet, the very fact of its secrecy proves that the Holocaust would have been most unpopular, – for which dictatorship will ever hide popular activities? But I also confessed that we, ‘‘the Germans’’, had witnessed abysmal discrimination against German Jews, albeit not physical abuse, without having had the courage to do anything about it. I told

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Frank Nabarro that from 1933 until her 1936 emigration to Philadelphia, my best friend, Liesel Stern, was Jewish, so that I had had the opportunity to observe the official Nazi anti-Semitism much more directly than the great majority of Germans. And finally, as I told him, because this shameful treatment of Jews had been meted out in our name, it was my duty, as it was the duty of all Germans, to make personal amends as best we could for the rest of our lives. In return Frank confided had he had done secret intelligence work in the war. As I found nothing bad about that, he asked me what I thought of ‘‘black’’ intelligence, but at the time I had no idea what that was and he dropped the subject. In retrospect, perhaps Nabarro had been involved in ‘‘black’’ intelligence that I might have found very objectionable, perhaps directly under Sir Basil Schonland who by then had gone to South Africa to be the president of the South African CSIR (Council for Scientific and Industrial Research). Anyway, Frank’s wartime service must have been highly meritorious as he was awarded the honor of MBE, ‘‘Member of the Order of the British Empire’’. At the time of the Stuttgart conference, Frank had recently married Schonland’s personal secretary, Margaret Dalziel, a wonderful woman whose later friendship I have greatly valued for many years. But besides lots of political, philosophical and personal topics, we talked and talked about dislocations. For me it was a glorious conference. And evidently Frank had also enjoyed our meeting, even though, as he said, he did not understand anything of my own lecture on corrosion [7], as at the time I was being groomed to become a university Professor and Masing endeavored to broaden my education. Anyway, on return to his position in the Department of Physics of Bristol University under N.F. Mott, Frank recommended that I be invited for a brief spell of research. Further, the Nabarro’s offered, and I gladly accepted, that while in Bristol I could sub-let their vacant second bedroom and share their meals at fair cost, in their second floor apartment in a row house in Great George Street, conveniently close to the Royal Fort, the site of the Physics department. And so it was that on January 2nd 1949, when air traffic did not yet exist and there still were no international rail connections into or out of Germany, I left Göttingen for Hook van Holland in a British troop train (in a sealed compartment so as an ‘‘enemy alien’’ not to endanger the soldiers?). I crossed the channel per ferry in a stormy night, and went on to Bristol by passenger train. In mid-afternoon of January 3rd 1949, then, Frank and Margaret met me at the Bristol railroad station. Within an hour or two they took me to the Cabot Tower, a lovely Bristol outlook point, and to the Bristol Suspension Bridge that spans the Bristol gorge. Thus my travel to England and arrival in Bristol was somewhat physically stressed. There soon was some emotional stress, too, as Frank was not always as totally charming and polite as he had been in Stuttgart. Thus on the second day, I believe, when Frank had asked me about the Peierls (now known as the Peierls–Nabarro) force and I responded that I was not very strong in mathematics and had not studied this topic, he burst out: ‘‘Oh my God, and to think that I had you come because I thought you were competent!’’. In fact, I was exposed to frequent somewhat uncomplimentary language from various colleagues. A kind of ‘‘rough and tumble’’ language was common in Mott’s department, and perhaps among students at British universities generally, and it did not help that I was a German-born woman. I accepted this as part of a different culture to which I had to get used, and in spite of an occasionally rough superficial style, Frank and I got on very well. By contrast, Professor Mott was unfailingly polite, kind and helpful. Quite a few times I was invited to parties that Prof and Mrs. Mott tended to give for faculty, staff and students, and a few times I was their single personal guest. As a result I became very fond of them and their two young daughters. Also Prof. Mott’s secretaries, Alice Terry and Mrs. Langdon, were wonderful cultured people. Among others, noticing my threadbare clothing, they gave me a few discarded dresses that were very welcome; and as a guest in Mr. and Mrs. Terry’s home, I saw television for the first time in my life. In fact, while before coming to England I had been prepared to dislike the British, who in my mind had been responsible for the WWII destruction of Germany and the post WWII misery, it took not long at all that I became an admirer of the British people, as I have remained ever since. I found them more genuinely kind and Christian than the Germans, and time and time again I was addressed by strangers who somehow noticed that I was German, simply to speak to me most kindly and offer support if that should have been needed. Initially, I had been invited to Bristol for only 6 weeks. However, I had had enough of dictatorship, war, deprivation and occupation. Therefore I was delighted that Frank supported my wish to stay, and

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that very soon Prof. Mott offered me a permanent position as a post doc, albeit at a much lower than customary salary (a reason why second-hand dresses were gratefully accepted). Over the work week, Margaret was away studying music in Birmingham (her father was a professional violinist in an important Scottish orchestra), to be home only on weekends when she cleaned house and prepared meals for the coming week. Indeed, Margaret then was and always remained to be an excellent wife and model housekeeper. As a result, Frank and I spent an inordinate amount of time together, and this was overwhelmingly devoted to discussing dislocations. Dislocations were our topic from breakfast to 11 am tea time in the tea room of the Department, at which Prof. Mott liked to talk with students, post docs and faculty. Thereafter we might work at our respective desks in different rooms until we met for lunch (usually with Jan and Minnie van der Merwe and one or two other friends, often in the cafeteria of the nearby Bristol museum). After lunch there might again be a few hours of work at our desks or in discussion with other colleagues until afternoon teatime, thereafter to leave for home and our common dinner at about 7 pm. And, what did we do on the way home, during and after dinner? Discussed dislocations and other defects such as vacancies and twins, of course, often until near mid-night. For the remainder, as Prof. Mott did not assign me any tasks but suggested that I do as I pleased, I spent most of my desk time on studying the dislocation literature, in which I had been very deficient because international journals had not reached German libraries since 1939. In those days, Frank started writing his outstandingly successful book on dislocations [8]. In this connection he asked me to do a literature search on slip lines and to write it up as a chapter in his book. This I gladly accepted as an excellent task. However, when in a few months I was ready, his book was still very far from completion, and Frank suggested that I go ahead and publish my survey as an independent paper, as I did [9]. All in all I learned a very great deal from Frank and felt a great debt of gratitude, for (i) having me brought out of Germany and (ii) taught me dislocation theory. I vowed to myself that I would ever be his friend, and always would try to benefit him, no matter what. My stay with the Nabarro’s ended in March 1949 when Frank seized an opportunity to work with A. H. Cottrell at the University of Birmingham From then on I lived in a rented room at Arley Hill Road. We kept in touch, though, and saw each other at various informal conferences in Bristol or in Cambridge. Additionally, later in the year I visited Frank and Margaret as their house guest in Birmingham, when they already had their fist-born son, David. Even though Prof. Mott offered me a permanent position, much to my regret I had to leave Bristol because of my marriage to Heinz Wilsdorf on January 4th, 1950. Namely, international travel to and from Germany had finally been resumed in the spring of 1949, and Heinz had come to England in hopes that we would marry, but since I was still undecided he stayed in the German YMCA in London. His move had become possible because Heinz’s aunt in Milwaukee had sent the necessary funds and he had been personally invited to England by my greatly admired mentor and dearly loved friend Dr. Elsie Briggs. Dr. Briggs was the Bristol University ‘‘Appointments Officer’’ to whom I had been introduced through Professor Mott and who over the preceding months had comforted me in many a stressful situation. Initially Heinz and I had planned to emigrate to the USA, which at that time was possible from England but not from Germany, but literally within days of receiving the expected visas, issuance of visas for Germans was suspended. Still, we did not want to return to Germany and as I was very happy in Bristol, we decided to try to stay in England, preferably in or near Bristol. Albeit in spite of sincere efforts by Prof. Mott, Heinz was unable to find a position in England, – most likely because a young German who had been a soldier was mistrusted. But, then, love took over, we threw caution to the wind and married anyway, in the German Lutheran church in London, with Elsie Briggs as one of our witnesses. At that point, with Heinz unemployed and we living precariously on my small stipend, Jan van der Merwe secured Dr. Schonland’s reluctant consent that Heinz would operate the CSIR’s new Phillips electron microscope in South Africa. Namely, after completing his Ph.D. in the fall of 1949, Jan with his dear wife Minnie had returned to his position at the CSIR in Pretoria, and meanwhile the CSIR had had acquired a Phillips electron microscope. At that time, electron microscopes had become the research instruments of choice but of which only few existed and only very few persons knew how to operate. However Heinz had since 1945 worked under Prof. H. König with the Göttingen Siemens electron microscope (that after the war had been hidden in a garage to protect it from

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requisitioning for reparations) and was an expert electron microscopist. Now the CSIR had one of the precious electron microscopes but no one to operate it, while South African born Schonland, being strongly anti-German, gave in to hire Heinz only after dedicated persuasion by Jan and others. And thus Heinz left for South Africa in early February of 1950 where Jan and Minnie van der Merwe most graciously accepted him into their household. This was an early but decisive chapter in our lifelong friendship that had started in January of 1949 in Bristol and still endures between Jan and myself, nearly 60 years later, after our mates have passed on, Heinz in April 2000 and Minnie in March 2006. In fact, Jan, at 86, spent 3 months, from August to November 2008, on a research project with Prof. Gary Shiflet in the Department of Materials Science and Engineering at the University of Virginia. I followed Heinz to South Africa in June 1950, after having fulfilled my obligations to Prof. Mott. I was fortunate to soon be appointed lecturer (i.e. Assistant Professor) in the Department of Physics at the University of the Witwatersrand (or ‘‘Wits’’ for short) in Johannesburg, that had only minimal research activity. We found a modern apartment in the then quite small town Kempton Park, next to the international airport of South Africa that then was under construction to be opened in 1953, I believe. From there, Heinz and I daily commuted in opposite directions by train, each for about 40 min. At the CSIR in Pretoria, among others Heinz developed new techniques for the investigation of metal surfaces by electron microscopy, and I at Wits was very busy with a heavy teaching load, besides in due course bearing and caring for our two children, Gabriele born November 2nd, 1953 and Michael born March 2nd, 1956. From the outset, our life in South Africa was conflicted by the split of the white population into two major camps, namely English and ‘‘Boers’’. That split dated from the 1899 to 1902 Boer War that had caused many deaths, especially among Boer women and children in British concentration camps, and had left deep bitterness. In WWII, the Boers had sided with Germany, and the English of course with the Allies. Further, the current government was Afrikaans and had been elected on a platform of ‘‘apartheid’’, the official policy of race separation that was meant to create mutually cooperative but independent white and black South African states. Herein, whites were supposed to have few rights in black states and vice versa, which policy was broadly supported by Afrikaners and Germans, but it was strongly opposed by the English part of the population. Now Wits was very much on the English side, and as a result quite anti-German. In fact I believe the chairman of the Department of German and I were the only Germans at Wits, while a significant fraction of the faculty and students were Jewish, far above their respective overall share of the white population. And in line with Wits anti-government stance, we accepted a number of ‘‘colored’’ (i.e. mixed blood) and black students, in defiance of government policy. In fact we favored them academically, as we recognized the extra difficulties that they faced in daily life. By contrast, the CSIR meanwhile was headed by the highly respected and cultured Afrikaner Prof. Meiring Naude, and there the Afrikaans influence was dominant. Heinz and I were therefore steeped in opposing cultures and had equally many friends on opposite sides of the political spectrum. Sensitized by the Nazi dictatorship, I opposed discrimination on account of race and felt politically at home at Wits. In fact, I believe it was in 1952, I became one of ‘‘15 courageous Wits faculty members’’ (as it was put on a front-page head line by the Johannesburg Star) who signed an open letter to protest the so-called ‘‘packing of the supreme court’’ that in due course permitted to legally take property rights from Blacks in towns and from Coloreds in the Cape as part of the apartheid laws. Also, together with my physics colleague and friend G. Wiles, whom I had already met in Bristol, I once or twice participated in interracial functions at the residence of the Archbishop of Johannesburg, the predecessor of Desmond Tutu. But yet I had no doubt of the sincerity and admirable character of the Afrikaners who strongly supported apartheid. On the practical level, Heinz and I could see no way out: I could not get myself to treat blacks and coloreds any different than whites, and as a result was very popular with our servants, but also was not much respected and widely taken advantage of. Our good friends, Wits Professor of economics Ludwig Lachmann and his wife Dr. Margot Lachmann, who as Jews had under Hitler emigrated from Berlin, were in very much the same frustrating position. Meanwhile I was loath to disappoint Jan and Minnie van der Merwe and their family members and friends who had the highest ideals and were ever fair and kind to their black servants but also knew how to maintain respect and prevent pilfering.

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Anyway, Heinz and I concluded that we would never fit in properly and decided to leave for the United States. Fatefully, in 1951 our greatly beloved chairman of Physics, Prof. Evans, died unexpectedly, and in 1953 Frank Nabarro was appointed as his successor, backed by Schonland’s strong recommendation. As Frank claimed, our presence in South Africa and my position at Wits had been a strong inducement for his acceptance of the offer; albeit we had already tentatively decided not to stay permanently, as already indicated. While negotiations with Nabarro proceeded at a level high above me, Alan Cottrell, who had visited us in South Africa and knew of our intention to leave, urged me to tell Nabarro of our tentative plans. However, I did not want to unduly influence Frank, believing that the chairmanship at Wits would be a great opportunity for him, and at any rate I could not seriously believe that my presence or absence should make much of a difference in his career decision. And so Frank arrived, and with his typical great energy and outstanding intelligence began to develop the Wits physics department into one of, if not the best in South Africa. Frank and Margaret bought a home within a few miles of the Wits campus that initially consisted of three interconnected rondavels with thatched roofs. Margaret was by then the mother of two, David and Ruth, but two more boys and one girl arrived after we had already left for the United States. Throughout, I closely cooperated with Frank and single-mindedly endeavored to help him in whatever he wanted to do. Albeit, this was not always easy since I had had, and endeavored to maintain, warm personal relationships with my Wits colleagues, as in fact succeeded. Alas unhappiness developed between Nabarro and my physics colleagues, partly no doubt because they were not strong researchers, and in due course there was a complete turnover of the Wits physics faculty except for myself. As already indicated, even though both Heinz and I loved South Africa and could not think of how better the country could have been governed under the circumstances, we yet could not get used to Apartheid. We therefore continued our efforts to after all emigrate to the United States. This opportunity arose when Bob Maddin offered me a position in the Metallurgical Engineering Department of the University of Pennsylvania in Philadelphia of which he had recently become the Chairman. Bob Maddin also located a position for Heinz in Philadelphia, namely at the Franklin Research Laboratories, where within 5 years Heinz rose to become one of the three Technical Directors. Alas, high-level mismanagement of the Franklin Institute Research Laboratory first threatened its health, and not long after brought about its demise. Therefore Heinz accepted the position of Chairman of the newly instituted Department of Materials Science at the then strictly male University of Virginia (UVA). Since Heinz would not go to UVA without me, I was offered a professorship in ‘‘engineering physics’’, as the first ever female full professor in any school of UVA except nursing. Alas, to my dismay I was for quite a few years subjected to some (now long past) painful discrimination for having the wrong sex. Intermittently, Heinz and I saw the Nabarro’s and their five children on visits, partly as their houseguests. In return Frank as also Jan van der Merwe (sometimes with his family) visited UVA on numerous occasions, as over the years we maintained close relationships with South Africa. We welcomed quite a few graduate students and post docs from Pretoria University, Wits and the University of Port Elizabeth. In return we encouraged graduating Ph.D. students from UVA to go for post doctoral studies to South Africa. The first of my Ph.D. students to spend a year in South Africa, as post doc under Nabarro and thereby beginning a lifelong close association with South Africa, was William A. (Bill) Jesser. After his return to America he became my departmental colleague and later my Chairman. Happily, Bill and his wife Barbara are still my good friends, and at time of writing, Jan van der Merwe is once again at UVA for a 3-month research period on epitaxy, as already mentioned. The best student in my very first physics undergraduate class at Wits, i.e. of 1950/1951, had been John Matthews. After graduating, going on to a Ph.D. and then as a post doc under Frank at Wits, he had become an expert electron microscopist and had begun devising highly original methods to experimentally test Jan van der Merwe’s theory of epitaxy. Soon after my arrival at UVA in 1963, John Matthews joined me as a post doc (from January 1964 to January 1966) while largely supervising Bill Jesser’s Ph.D. thesis and writing several publications with him. Thereafter, John Matthews briefly returned to Wits but then came back to the States where, as a member of IBM’s Yorktown Heights laboratory, he continued to make foundational contributions to the experimental investigation of epitaxy. Throughout, these studies were based on Jan van der Merwe’s epochal theory that, beginning with his

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Bristol Ph.D. thesis and still continuing, he continued to develop so as, historically, to become the father of what we now call ‘‘nanotechnology’’. But, of course, enormous contributions were made by many others, also. John Matthews passed on much too young, but in terms of experimental investigations of epitaxy, historically, he occupies the equivalent place to that of Jan van der Merwe in theory. I am grateful that I can still count various members of John’s family, including his widow, his brother and some members of his brother’s family among my dear friends, although I barely know John’s most famous son, the band leader David Matthews, and do not know his other children, Jane and Peter as well as I would wish. Other South African scientists who have intermittently come to the United States on account of our and our departmental colleagues’ close connection with the country, and thereby have enriched international science and incidentally Heinz’s and my life, include Conrad Ball, Louis Bredell, Max Braun, John Crawford, J.S. (Koos) Fourie, Horace Gaigher, Paul Jackson, I.A. (Sakkie) Kotze, A. Leach, Robert MacCrone, Johan Malherbe, Johan Prins, Pauline Stoop, and J.D. (Koos) Vermaak, and in return from here the following went for research time to South Africa: Ted Duncan, Robert A. Johnson, Greg Olsen, George Proto, C. Tom Schamp and Gary Shiflet.

2. Part II: development of workhardening theory 2.1. The beginnings of post WWII workhardening theory While in Bristol, I refined my theory of workhardening [10] that I had started as part of my Ph.D. thesis. It was based on two major assumptions: (i) that crystals always contained easily acting sources of dislocations that, in principle, could emit unlimited sequences of one type and sign of dislocation, and (ii) that the dislocations would form pile-ups against plentiful obstacles. Both points had already been central in my Ph.D. thesis, wherein the concept of pile-ups, although not by that name, had been taken from Kochendörfer [11]. Albeit in Kochendörfer’s book [11] and in my thesis [1], it had not yet been recognized that the dislocation spacing in pile-ups would be denser at the head than towards the tail. Cottrell introduced that insight [12] and also that the stress at the leading dislocation would be proportional to the number of piled-up dislocations. Even so, the mathematics of the dislocation distribution in pile-ups had as yet not been solved. Rather, following unpublished work by Heilbronn, Nabarro found the maximum number of dislocations that could be piled-up in an interval of specified length under a specified stress [13], but the full solution had to wait for the work of Eshelby, Frank and Nabarro [14]. In much greater doubt yet was the nature of the dislocation sources whose presence was strongly suggested by my thesis research. While Taylor had assumed that mosaic block walls would provide the requisite sources [15], Mott still adhered to the short-lived hypothesis by Frank that dislocations would reach high speeds and multiply through reflection at obstacles and free surfaces [16]. At Bristol, in protracted somewhat exasperating discussions, I had repeatedly tried to convince F.C. Frank that the fact of very slow non-jerky deformations, as documented in my thesis, disproved his multiplication by reflection mechanism, and that there must be easily acting dislocation sources. Alas, typically Charles Frank cut me off in the end, saying ‘‘So what!’’ and turning his back. Thus it was somewhat against my better insight when Prof. Mott inserted into the first part of Ref. [10] that my theory might be applicable after the reflection of fast dislocations had run its course. Later, when Charles Frank with Read had discovered the Frank–Read dislocation multiplication mechanisms [17] (that I had failed to discover over many, many hours of trying because I never ventured into three dimensions) Charles wrote me a note saying: ‘‘Doris, I have found your sources of dislocations’’ or similar. Alas in the publication [17] he neglected to mention my work or our discussions. Yet at all odds, for me the most important new knowledge I obtained at Bristol was the theory of epitaxy that Jan van der Merwe developed in his Ph.D. thesis under F.C. Frank. To my understanding, it ultimately made modern computers possible. Prof. Becker who, as already indicated, had suggested my thesis research, had been deeply disappointed by my Ph.D. results. Already the theses of Boas and Orowan had been designed by him to test his own, then widely accepted theory [18] that crystal plasticity resulted from quantum-like elementary

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glide steps. These would be triggered when, through thermal activation, the resolved shear stress in small volume elements reached a critical value. Alas, both Boas’ and Orowan’s theses had turned out to be inconclusive. Now I had found [1] that the strains investigated in all three theses were of the nature of ‘‘aftereffects’’ (‘‘Nachwirkungen’’) with total strains always remaining below the elastic strain and therefore of no relevance for plastic deformation. As a bonus, in the process I had found the explanation for the logarithmic time law of aftereffects that had previously been unsuccessfully tackled by quite a few eminent physicists [19]. More pertinently, I found the dependence of the activation energy of the microcreep studied on stress and temperature to be incompatible with Becker’s theory. Alas, Becker did not want to accept this result but reacted with saying: ‘‘High time that someone competent do this work’’ or similar, and proceeded to commission his young coworkers Leibfried and Haasen to repeat the measurements with a different experimental set-up. In due course, these very able colleagues found closely the same results as I had, but seemingly more politically astute than I, Leibfried and Haasen did not specifically relate their results to Becker’s theory and neglected to make any mention of my work [20]. I was, of course, deeply disappointed in return, – one of many similar disappointments. Did Becker take no notice? I do not know. Even though we all remained on very friendly terms, we never discussed this issue. Having developed a greatly improved technique for studying surface profiles of solids by means of the electron microscope, with then unheard-of resolutions as small as 2 nm, Heinz followed my suggestion to apply the technique to the study of slip lines. As one of the first results, he discovered the ‘‘elementary structure’’ on deformed aluminum [21]. Its ultra-fine scale reconfirmed that there must be multitudes of easily acting dislocation sources, at least in aluminum. While I had been sidelined in South Africa, Seeger, the brilliant pupil of Kochendörfer and Dehlinger, began to develop the pile-up theory of workhardening. In the course of years he and his group made immense contributions to our knowledge of crystal plasticity, among others establishing that there were three workhardening stages, probing the effects of workhardening on a variety of physical properties, prominently among them magnetism, and most importantly providing a wealth of information on dislocation structures [22,23]. Parts of Seeger’s dislocation pile-up workhardening theory are still lingering in the current ‘‘SODS’’ paradigm. According to it, plastic deformation causes the generation of pile-ups with an on average constant number of dislocations. The pile-ups were thought to repel each other through their long-range stresses and thereby limit the dislocation density and average path to cause Stage II with a constant slope. Stage III was postulated to start when the applied stress becomes so high as to force cross slip at the heads of the pile-ups and thereby extend the dislocation path. Peter Haasen contributed especially to the ideas on Stage III, while Hirsch and Mitchell [24] modified the theory by postulating that the pile-up stresses would cause tangles of ‘‘unpredicted glide’’ about the heads of pile-ups. In addition, a wealth of experimental evidence on dislocations and dislocation structures was developed by P.B. Hirsch and coworkers, prominently among them P. Hazzledine, A. Howie, T. Mitchell, V. Vitek and M.J. Whelan. As already indicated, after having left South Africa and after some period as fulltime housewife and mother, I accepted a part-time faculty position under Bob Maddin at the University of Pennsylvania. As all women who have been in a similar situation will know, a return to one’s profession under such circumstances is difficult. As a first task Bob Maddin wanted me to theoretically interpret his Ph.D. student Hiroshi Kimura’s thesis research results, namely on point defects in quenched metals. In this pursuit Hiroshi came for discussions two or three times weekly to our apartment after Gabriele and Michael had been put to bed. Hiroshi was an outstandingly able scientist who later became Professor at Sendai University and we achieved some significant results [25,26]. One year later, expanding the scope of my part-time associate professorship at U. Pennsylvania, I had to begin teaching a graduate course on crystal defects and plasticity – again at night. This became a challenge, among others because I had to confront Seeger’s dislocation pile-up theory. Alas, there was no choice but to dismiss it for two reasons. Firstly, as I had already realized earlier without publishing it, and as was later much more persuasively shown by Hirsch and Mitchell [24], Seeger’s derivation of the Stage II workhardening parameter, based on long-range interactions among pile-ups, was flawed. Secondly, and decisively, based on an analysis of the ‘‘elementary structure’’ [21], Jan van der Merwe had long since shown that interacting parallel pile-ups would invariably convert into low-energy dipolar edge dislocation mats, that are essentially free of long-range stresses and may be regarded

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as building blocks of Taylor lattices [27]. As a result I had reluctantly given up my own workhardening theory [10] as it was based on pile-ups. Accordingly I was somewhat bemused when Seeger’s theory found wide acceptance, indicating that few if any theoreticians in the crystal plasticity area ever read our paper [27]. 2.2. The Peierls–Nabarro force, dislocation uncertainty and melting Still more vexing was the need to teach the Peierls–Nabarro force theory. Herein I quickly realized that, try as I might, for lack of mathematical sophistication I could not understand Nabarro’s famous papers ([28,29] see also [8] p.175–176). Therefore, almost in desperation when time began to run out, I decided to derive the frictional stress on a moving dislocation in an otherwise perfect crystal in my own way, employing a minimum of mathematics. To my relief I soon found two simple models [30] suitable for teaching in my course, but to my initial dismay both yielded the friction stress at least two orders of magnitude higher than had been found by Nabarro, while subsequent authors had had obtained still smaller values. Fortunately my result was in rather good agreement with an earlier value by Huntington [31] who had already noted that Nabarro’s model comprised an unnatural symmetry, namely so as to make the energies of the two symmetrical dislocation configurations in a simple cubic lattice to be exactly alike, whereas simple logic expects them to be the two opposite extremes in a cyclic variation of dislocation core energy. On reflection, I surmised that approximations in Nabarro’s highly elegant but complex mathematics must have canceled out the whole effect, and that subsequent workers, following Nabarro’s method, had reduced the remaining errors so that, if they had managed to eliminate the errors entirely, they would have found the friction stress to be zero [30]. That left the problem why, then, measurements of the Peierls–Nabarro force in pure fcc and hcp metals had yielded values still below those of Nabarro, and thus orders of magnitude lower than Huntington’s and mine. Contemplating this puzzle I realized that dislocations in an idealized crystal lattice of stationary atoms, would be subject to higher frictional stresses than dislocations in real lattices of vibrating atoms. Namely, even at absolute zero the Heisenberg uncertainty of atoms, and much more so thermal vibrations at increasing temperatures, will smear out the dislocation axis position from a mathematical line into an extended volume of ‘‘dislocation uncertainty’’, thereby reducing the amplitude of the periodic changes of dislocation core energy when a dislocation moves through a crystal lattice. In fact, it is readily seen that the Peierls–Nabarro stress vanishes when the uncertainty area of the axis straddles one whole periodicity interval in the average crystal plane normal to the dislocation. This effect is entirely separate from thermal activation, which I found to be negligibly small under almost all circumstances. Also, significant dislocation uncertainty effectively rules out dislocation ‘‘jogs’’. Analyzing the softening effect resulting from dislocation axis uncertainty, I found it to strongly increase with dislocation core diameter, that in theory is roughly inversely proportional to the theoretical critical resolved shear stress of dislocation-free lattices. Accordingly, on account of a large dislocation axis uncertainty, close-packed planes in fcc and hcp have almost no Peierls–Nabarro stress even at absolute zero temperature, whereas among the pure diamond lattice materials, such as germanium and silicon, and of course diamond, the dislocation core is very contracted. In those, therefore, dislocation uncertainty is minimal. This, then, explains the great hardness of diamond-type crystals up to high temperatures. The width of the dislocation axis uncertainty has intermediate values in other lattices, and as a result, the level of the Peierls–Nabarro stress in other crystal types, e.g. in bcc and rock salt-type crystals, is intermediate [30]. In criticism of the dislocation axis uncertainty theory, Nabarro ([8] p. 177–178) linked it to the prior work of Dietze [32] who in evaluating the Peierls–Nabarro stress had modeled the interactive forces of thermally vibrating atoms facing each other across the slip plane as regions of Gaussian stress distributions. As would be expected, with rising temperature such a ‘‘diffuseness’’ in the atomic positions reduces the Peierls force. Albeit the ‘‘diffuseness’’ of the dislocation axis position in accordance with dislocation uncertainty is a different effect and Nabarro conceded that dislocation axis uncertainty reduces the Peierls stress much more strongly than does atomic diffuseness. Nonetheless he proceeded to criticize [30] by the argument that dislocation motion ‘‘requires the correlated motion of a group of

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atoms on one side of the glide plane in one direction and of a group on the other side in opposite direction. The calculation (i.e. in [30]) fails to recognize that the expected amplitude of this correlated motion is much less than that of the uncorrelated motions of a single pair of atoms’’ BUT by the above wording Nabarro reveals a misunderstanding of the essential point of dislocation uncertainty, namely that it limits the precision with which the dislocation axis is defined and thereby imparts to it an energy that is averaged over the range of its uncertainty; THAT is the effect of dislocation core uncertainty. Also, Nabarro continued to believe that the energy in the two symmetrical positions is the same, which remained a matter of disagreement between us. Pursuing the properties of dislocation cores still further, I found that their free energy decreases with rising temperature so as to vanish at some point. And numerical calculations of the temperature of vanishing dislocation core energy always happened to fall into the region of the melting points of the various substances that I considered. It is a tribute to Nabarro’s cited monumental book ‘‘Theory of Crystal Dislocations’’ [8] that it also comprises a chapter on the dislocation theory of melting (chapter 11.1.2). Regrettably, when I developed the dislocation theory of melting and of the liquid state, Nabarro’s book had not yet appeared so that I did not have the benefit of his treatment, while conversely our papers were too late to be included in his book. Anyway, Nabarro correctly cites several predecessors of the theory, some of which I missed. Much the same is true for the evolution of my workhardening theory, initially under the name ‘‘mesh-length theory’’. Does melting really happen by the filling of a material with dislocation cores, in accordance with my theory of melting? [33] I was led to conclude that the answer is ‘‘Yes’’. For one, the expected changes of a host of physical properties agree with observations [33]. Further, (i) the value of the surface energy between crystals and their melts is explained by the theory [34], as are (ii) the radial correlation functions of all pure metal melts that were studied [35], and (iii) the electron diffraction patterns of aluminum while being melted in the electron microscope [36], as well as (iv) X-ray patterns of metals that are amorphized through severe plastic deformation [37,38]. In fact, a recent literature search suggests that the dislocation core melting theory is about to find broad acceptance [39]. 2.3. The mesh-length theory and the ICSMA conference series I returned to my early interest in workhardening theory in 1962. Based on Ref. [27] together with very extensive studies of strain-induced surface markings by Heinz [40–44], and my best reading of the literature, I developed ‘‘A New Theory of Workhardening’’ [45] which became a ‘‘Citation Classic’’ [46] and became known as the ‘‘mesh-length theory’’. Ref. [45] treats only Stage II, assumed to be strictly linear, without proposing any specific models. It simply claimed that plastic straining required continual generation and movement of new dislocations that were formed by supercritical bowing and required bypassing of parallel dislocations, both at stresses that are inversely proportional to the average dislocation link length or spacing, as the case may be. Consequently the flow stress, s, must rise in proportion with the root of dislocation density, q, as

pffiffiffiffi ðs  so Þ ¼ aGb q

ð1Þ

where so is the friction stress, G is the shear modulus and a is a proportionality constant. Eq. (1) is in fact one of the most persistent relationships in all of workhardening but at the time it was at variance with the preponderance of experimental evidence, except for measurements by Bailey and Hirsch [47]. Another important prediction of [45] that has stood the test of time is ‘‘similitude’’, i.e. that with increasing strain and flow stress, dislocation structures would reduce in scale in accordance with Eq. (1) but remaining similar to themselves. Another conclusion of [45] was that dislocation intersections (often dubbed ‘‘forest cutting’’) contributed little if anything to the flow stress of wavy-glide materials, in opposition to Basinski’s hypothesis [48,49] that forest cutting is the predominant hardening mechanism in metals that, however, was never reduced to a quantitative theory. In fact, in Ref. [45] it is argued that, at least in pure fcc metals, forest cutting cannot be an important hardening mechanism because in these, the onset of double glide is not accompanied by any discontinuous increase of flow stress (see Fig. 1).

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Fig. 1. The idealized four-stage workhardening curve of wavy-glide single crystals, e.g. pure fcc and bcc metals. In actual fact, Stage IV can never be observed in conjunction with Stages I, II and III because is starts at larger strains than can be applied to single crystals in either tension or shear. (Fig. 26 of [123]).

Albeit, Nabarro, Basinski and Holt [50] neglected to include the mesh-length theory in their widely read review article on scientific studies of plastic deformation. Perhaps Nabarro had felt offended by my work on the Peierls stress? In fact, Frank and I continued our friendly relationships and mutual visits at least until 1996, but our professional interactions were impaired as increasingly Nabarro claimed that he could not understand the gradually evolving LEDS theory, i.e. that the defect structures in deforming solids would always be close to minimum free energy while needing to be compatible with the material’s structure and flow stress. Meanwhile Peter Haasen had succeeded Masing as department head in Göttingen, and soon instituted the bi-annual series of ICSMA (International Conferences on the Strength of Metals and Alloys). In it, Nabarro was a frequently invited speaker but the mesh-length theory and later the LEDS and LES theories of plastic deformation evolving therefrom were systematically ignored. As a result, until 1988, Ref. [52] of 1979 remained to be my one and only contribution to any ICSMA conference. It theoretically explains the shape evolution of fatigue hysteresis loops in pure fcc metals through similitude as reproduced in Fig. 2, with excellent agreement that speaks for itself. The evolving stalemate between three competing workhardening/flow stress theories that was revealed in the literature survey by Nabarro, Basinski and Holt [50] with their neglect of the meshlength theory, prompted John Hirth and Hans Weertman to organize an international conference designed to bring all parties together, i.e. proponents of pile-ups [22,23,53], critical dislocation bowing [45] and ‘‘forest cutting’’ [48]. As a result, Hirsch and Mitchell refined their incisive criticism of Seeger’s pile-up theory and expanded Hirsch’s stress relieved pile-up theory [24], and I contributed the ‘‘unified’’ theory of Stages II and III [54]. Herein it was for the first time clearly enunciated ([54], p. 99–100) that actually realized dislocation configurations: (i) must be in equilibrium, in the sense that the sum of the externally applied stress and the stress fields of all dislocations in the crystal must yield a resolved shear stress equal to or smaller than so, the frictional stress, at the position of every dislocation. (ii) The actual configuration must represent the configuration of minimum free energy among all configurations accessible to the system under the given conditions of straining, such as speed of straining, temperature, stacking fault energy, etc. Therefore the same configuration, but on a scale inversely proportional to the difference between the applied stress and so, is also in equilibrium at any other stress...

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Fig. 2. Left: Hysteresis loops of copper single crystals at different values of the cumulative strain, (cpl)cum, as indicated, according to Mughrabi [51]. Also shown, in connection with the loop at a strain of 2.0, is the method of determining sB and sF. Right: The experimental hysteresis loop for the cumulative strain of 2.0 from left, together with theoretically derived loops, obtained by transforming both stress and strain in proportion with the root of the cumulative strain, i.e. ðcpl Þ1=2 cum . (Fig. 1 of [52]).

(This is the principle of similitude). Among a number of additional still valid insights, [54] explains that changes in friction stress, so, and of dislocation mobility will cause transitions from one ‘‘stage’’ to another. G. Langford and M. Cohen’s classic study of the composite workhardening curve of drawn iron wires and correlated dislocation structures (all of cell type) [55] provided an opportunity for a critical test of [54]. By determining the values of the different numerical parameters and deriving from these the workhardening curve, excellent agreement was obtained (see Fig. 3). 2.4. Development of the LEDS theory 2.4.1. Determining the low-energy form of dislocation cells via computer calculations Fig. 3 may be viewed as proof of the mesh-length theory, at least for drawn ion wire. Here, in line with similitude, dislocation cells shrink inversely proportional to the flow stress that in turn is determined by critical bowing of always the longest link lengths From this basis, the development of the LEDS (Low-Energy Dislocation Structures) theory proceeded, based on the ‘‘LEDS hypothesis’’, i.e. that at any given flow stress, the actual dislocation arrangements have the lowest free energy that the existing dislocation content could achieve. The first step herein were computer calculations of the stresses of model dislocation walls, dislocation cells and dislocation cell aggregates by the very talented Ph.D. students Moore (compare [57]) and Bassim (see [58]). As in those days computers were still in their infancy, this project involved much innovative research in devising novel methods for graphically presenting stress distributions about dislocations, namely in order to overcome the severe limitations posed by the, from the present standpoint ridiculously low, 64 kb memory of UVA’s central computer. These computations were designed to identify the dislocation structures that at given dislocation content would have the lowest stresses and strain energies, and thus would develop in deformed crystalline materials. Not unexpectedly, dislocation cells representing rigidly rotated volume elements were determined to have by far lower free energies than isolated dislocation boundaries. Less obviously, but in line with

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Fig. 3. Composite workhardening curve of severely drawn iron wire (reproduced from Ref. [55]), compared with calculated values according to the mesh-length theory [54] (small circles). Besides so, the fit involves only one adjustable parameter, namely n, the ratio of the longest link lengths (whose supercritical bowing determines the flow stress) to the average link length (determined from micrographs). The above fit represents n ffi 3.3, while theoretically it is n ffi 3. The curvature in the graph is due to the log term in the Frank–Read stress and proves that the flow stress is indeed determined by supercritical dislocation bowing and not by dislocation bypassing that would yield linearity Similarly, the slight curvature of Stage II in wavy glide had already been shown in [54] to be precisely accounted for by critical dislocation bowing. (Reproduced from [56]).

long-standing X-ray determinations of crystallographers, the free energy of a ‘‘mosaic block structure’’ of joined-together dislocation cells was found to be by far smaller than that of assemblies of isolated dislocation cells, i.e. morphologically like pebbles in cement. However, somewhat unexpectedly, the dislocation cell structure of lowest free energy was/is ‘‘the checkerboard pattern’’, i.e. an assembly of parallelepipidal (i.e. in the simplest case cubic) dislocation cells whose interiors are rigidly rotated relative to the undisturbed lattice, about the same rotation axis and by equal angles but in alternating directions. Alas, also these foundational additions to workhardening theory were ignored in ICSMA conferences and mostly in the literature at large. Indeed, detractors doubted the results of the outlined computer calculations declaring, as Basinski had once put it publicly and thereafter was frequently repeated: ‘‘garbage in, garbage out’’. Initially that attack could not be decisively countered since checkerboard patterns had not yet been observed, but they have been widely documented since, in a variety of materials, and resulting from a large range of different straining geometries and stress levels. Subsequently, too, Jan van der Merwe was able to show analytically that the checkerboard dislocation cell pattern has indeed minimum free energy [59]. The further evolution of the LEDS theory greatly benefited from collaborations with top experimentalists who provided a wide range of experimental data and electron micrographs. Roughly in historical order, as follows: 2.4.2. Dependence of cell morphology on pressure and strain rate Murr and Kuhlmann-Wilsdorf [60] re-examined electron micrographs, that Murr had accumulated over nearly a decade, of dislocation cell structures formed under room temperature, short-duration (2 ls), high-stress (80–460 kbar) shock pressure pulses. The mean dislocation cell diameter was found to be inversely proportional to the shock pressure, in agreement with similitude as well as the results of Refs. [55] and [56]. Next, at 250 kbar, in nickel the cell size was 0.5 lm, independent of pulse duration between 0.5 and 6.0 ls shock duration, except that at the shortest pulse duration, i.e. 0.5 ls, only

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the beginnings of dislocation cells could be discerned. It follows that in room temperature nickel the dislocations move into their low-energy configurations at a speed of

v disl ffi 5  107 m=5  107 s ffi 1 m=s

ð2Þ

This, then, strongly supported the LEDS principle up to very high but not unlimited strain rates. 2.4.3. LEDS formed in fatigue Next, the interaction with Campbell Laird, who is widely recognized for his many fundamental studies on fatigue, was most fruitful and extended over several years [61–66]. Those studies went into considerable detail and quantitative explanations were developed for a host of observations on wavyglide single crystals under constant amplitude fatigue that to my knowledge have never been seriously criticized. Specifically, the ‘‘loop patches’’ formed at low cumulative strains arise from mutual random trapping of glide dislocations that of course are LEDS’s. In the course of fatiguing, the loops in the patches flip cyclically and gradually develop towards Taylor lattices, a LEDS of still lower free energy, that are penetrated by channels or veins in which screw-oriented glide dislocations move cyclically. The associated back (sB) and friction stresses (sF) were analyzed and explained in detail in terms of loop flipping as well as glide dislocation motion (see Fig. 2). Further considered was the transformation from the loop patch and vein structure into persistent slip bands with their ladder structure. Herein, the loop patches that approximate Taylor lattices transform into a series of parallel dipolar walls of still lower free energy than loop patches (see Fig. 4). Eventually, as first shown by Scoble and Weissmann ([67] (Fig. 4)), continued fatigue cycling causes multiple glide and thereby introduces additional Burgers vectors. These facilitate the formation of dislocation cells, preferentially with the checkerboard pattern and thus yet lower free energy. Indeed, throughout, dislocation structure transformations in fatigue were found to be driven by reductions of free energy. Among several outstanding supporting micrographical investigations is one by Charsley and Kuhlmann-Wilsdorf [68]. It illuminates still another dislocation structure type, namely so-called ‘‘maze structures’’ formed in multiple glide fatigue that superficially look like cell structures. These are formed of dipolar or multipolar dislocation walls that have no net Burgers vector content and therefore are not associated with lattice rotations. Specifically, the walls studied in Ref. [68] are parallel to {1 0 0} planes and thus are in mutually perpendicular orientation, and each comprise two mutually perpendicular Burgers vectors. The peculiar morphology of maze structures, at least in this case, arises because wall intersection are sites of heightened energy and therefore are

Fig. 4. Microstructure in the persistent slip band of a fatigued Ge crystal that is in the process of converting from a ‘‘ladder structure’’ of dipolar edge dislocations (extending from top left to bottom right), into a ‘‘checkerboard pattern’’ of dislocation cells parallel to the primary {1 1 1} plane. The cell rotations are by ±0.25°, and the rotation axis is normal to the primary Burgers vector in the primary slip plane. (Copy of Fig. 13 of [67]).

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avoided in favor of ‘‘L’’ joints and ‘‘T’’ joints. Many other examples of maze structures with different morphologies have been reported in the literature, specifically by Wang, Laird and Chai [69] and Lepistö, Kuokkala and Kettunen [70] of which micrographs were included in the survey article [71], edited by Nabarro and Duesbery. And, again, as far as it is known, all maze structures approach minimum free energy. 2.4.4. The Stage I to Stage II LEDS transformation Returning to unidirectional plastic deformation, Jackson and Kuhlmann-Wilsdorf [72] and Kuhlmann-Wilsdorf and Comins [73] clarified the transition from Stage I to Stage II of the workhardening curve. Namely, Stage I is dominated by ‘‘dipolar mats’’ that are formed of parallel rows of opposing positive and negative edge dislocations parallel to the primary slip plane. No matter how they come about, through the mutual trapping of parallel sequences of primary edge glide dislocations, or through cross slip or perhaps still other mechanisms and a combination of them, they densify via similitude as the applied stress rises in Stage I. Thereby, effective tensile and compressive strains in the slabs between the opposing dislocations build up that are due to the extra atomic planes on one side and missing planes on the other. Once they reach a critical level, these trigger the corresponding tensile and compressive glide, namely on the most highly stressed ‘‘unpredicted’’ slip system (i.e. not predictable from the macroscopically applied stress) that necessarily is inclined against the primary slip plane. The reactions between the primary edge dislocations and the ‘‘unpredicted’’ dislocations, transform the initial dipolar edge dislocation mats into parallel networks with a high concentration of ‘‘Lomer–Cottrell locks’’. Somewhat counter-intuitively these are tilt boundaries with their tilt axis lying in the primary slip plane and with alternating sense of rotation, as in Fig. 5. Foxall, Duesbery and Hirsch [74]. This, then, is the structure of Stage II, wherein further straining increases the dislocation density in the networks and decreases the link length in them. That the described ‘‘unpredicted’’ slip triggers the Stage I to Stage II transition, and its origin, had already been found by others [75–77], but [72] and [73] placed it into context with the LEDS hypothesis. In fact, details are quite complicated and have been unraveled for fcc Stage I in connection with the LEDS hypothesis, especially by Takamura and coworkers in an extraordinarily meticulous series of systematic observations extending over many years [78]. Moreover, we do not yet know how the dislocation

Fig. 5. A Stage II ‘‘carpet structure’’ of the kind typical for wavy-glide materials, in an extended niobium single crystal according to Foxall, Duesbery and Hirsch [74]. The ‘‘hedges’’ seen at lower right are believed to be the beginnings of the cell structure that develops in Stage III.

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structure develops further. The difficulty will be clear from Fig. 6 that illustrates the overall geometry that leads to the ‘‘carpet structure’’ of Fig. 5. It shows that, in essence, from Stage I to Stage II ‘‘unpredicted slip’’ reduces the free energy of a Taylor lattice by reducing the tensile-compressive stresses between rows, and in the process rotates the lattice. But there will now be rotated tensile/compressive stresses, and in the course of further straining these are bound to cause domains of mutually misoriented Taylor lattices whose geometry we do not yet know. In Fig. 6b, the indicated rotated Taylor lattice is idealized because (i) crystallographically, the reacting ‘‘unpredicted’’ dislocations cannot be parallel to the initial edge dislocations so that instead of parallel edge dislocations, networks are formed that include a screw component. (ii) The rows of the rotated Taylor-lattice form tilt walls of alternating sense of rotation (i.e. a ‘‘carpet structure’’ such as in Fig. 5), i.e. tilted slabs of alternating misorientation, within which there must be remnants of the same kind of tensile/compressive stresses that causes the Stage I to II transformation. The detailed geometry of that configuration and how it densifies in Stage II are not yet known. Even while the computer calculations of Moore and Kuhlmann-Wilsdorf [57] and Bassim and Kuhlmann-Wilsdorf [58] had revealed the basic principles of dislocation cell morphology, there were still

Fig. 6. (a) Idealized geometry of the Stage I structure of dipolar mats that at the end of Stage I have densified to form a Taylor lattice. (b): ‘‘Unpredicted’’ intersecting glide relieves the tensile/compressive stresses between the rows but in the process, trapped ‘‘unpredicted’’ dislocations react with the Stage I edge dislocations to form a Taylor lattice that is rotated by 90° as sketched (reproduced from [73]).

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huge gaps in our understanding of what types of cells would form under any particular set of conditions, with what shapes and sizes and what rotation angles, and comprising what dislocations. To provide the answers to many of those questions is the merit of Niels Hansen, the group leader, and the absolutely first-class electron microscopists Darcy Hughes and Bent Bay [79–84] with Darcey also having excellent gifts as a theorist. In studies of ‘‘wavy-glide’’ pure aluminum and nickel, and of ‘‘planar glide’’ 5.5at% Al–Mg alloy, rolled up to 90% or deformed in torsion, their observations and theoretical analysis confirmed the LEDS hypothesis and yielded additional insights. Specifically, in the wavy-glide materials studied, all defect structures were cell walls, i.e. dislocation rotation walls, with undetectably low longer-range stresses and typical rotations of up to several degrees except for ‘‘dense dislocation walls’’ (DDW’s) with typically stronger rotations. The DDW’s delineated ‘‘cell blocks’’ (CB’s) that comprised several to many ordinary dislocation cells. Isolated cells, like pebbles in concrete, were not found, in line with their relatively high energy as found in the already discussed computer calculations by Moore and Bassim [57,58]. Also in line with those computations, within any one cell block, alternating cell wall rotations of similar size in opposite direction, as in ‘‘checkerboard patterns’’, were common. 2.4.5. Understanding cells and cell blocks via the LEDS hypothesis Within CB’s, similitude appeared to be obeyed, as the sizes of ordinary cells shrink with increasing deformation. Herein new cell walls form in cell interiors and, by splitting into two, these open volume elements of relatively rotated material between them. Similarly, but more rapidly, CB’s shrink, as DDW’s split in the course of straining and thereby open up relatively rotated volume elements, but in this case those rotated volume elements are not more or less equiaxed but form plate-like ‘‘microbands’’ (MB’s). The same phenomenon of CB’s and the manner of their size reduction in the course of straining, that was observed in the pure wavy-glide metals, was also found in the planar glide Al–5.5%Mg alloy, but in that case the interiors of the CB’s were filled with Taylor-lattice structure, instead of dislocation cells. This is in line with the earlier results in Refs. [73–77] that cells have a lower free energy than Taylor lattices but require for their formation (i) extra Burgers vectors and (ii) a modicum of threedimensional dislocation mobility, neither of which is available in planar glide. Evidently most important in actual structures were the DDW-bounded ‘‘cell blocks’’, each of which were found to be deformed through the operation of one or more slip systems but always short of the five needed for homogeneous deformation. This was shown by the fact that the combination of simultaneously operating slip systems abruptly changed at the DDW’s. Thus it was realized that the cell blocks would be of fundamental importance in the macroscopic deformation process, and especially in texture formation, but their true nature remained mysterious until through the interaction with Ed Starke and his coworkers the immense importance of deformation bands and the rules of their generation were discovered (see ‘‘Deformation Band Studies’’ below). 2.4.6. Worksoftening and grain boundary substance studied with H.G.F. Wilsdorf H.G.F. Wilsdorf suffered a stroke in 1990 that affected his left side so that for his remaining years, until April 2000, he was confined to a wheelchair. With his science career thus suddenly interrupted, Heinz first turned to his love of books and in particular his extensive collection of German American imprints, i.e. German language books printed in America from the early eighteenth to the mid-nineteenth century. Over the years Heinz had assembled an impressive collection of such books, mostly hymnals and volumes of sermons, many illustrated with lovely wood cuts. Now, in spite of his severe handicap, Heinz learned the rudiments of word processing and with admirable perseverance, able to use only one hand, he wrote a scholarly book on his collection that was published and even went into a second edition [86]. Following his wish, this collection has since been donated to the library of his alma mater ‘‘Göttingen University’’ in Germany (see http://www.heinz-wilsdorf.com/collectionE.html [87]). Thereafter, Heinz returned to some still unsolved problems in his research on ultra-high strength MA (mechanically alloyed) alloys that had been cut short by his stroke. Based on his and his coworkers’ large collection of electron micrographs and data, we thus tackled the question of why such alloys tend to exhibit strong worksoftening [88–90]. In this pursuit, the LEDS theory was found to apply also

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to these extreme materials, and the worksoftening to be largely due to a decrease of the frictional stress, so of Eq. (1), that in those alloys is relatively much larger than in typical pure metals and alloys. Most interestingly, careful examination of electron micrographs of those MA alloys revealed exquisitely thin nonmetallic films along grain boundaries [90]; i.e. we rediscovered ‘‘Tammann’s grain boundary substance’’. This had been described nearly a century ago by Gustav Tammann (who many consider to be the father of materials science and who had been Professor Masing’s Ph.D. thesis advisor in Göttingen). But the grain boundary substance had long since vanished from observation, and not having been found in modern metals it had become a historical curiosity with no relevance for modern technological metals. Now we established that Tammann’s grain boundary substance was present in at least some ultra-strong MA alloys, and possibly significantly contributed to their strength. Further, Hall Petch hardening in MA alloys together with their worksoftening was treated in another paper [91] (see also below). 2.4.7. Deformation band studies with E. Aernoudt, E. A. Starke, Jr. and K. Winey In 1998, inspite of 55 years of research on crystalline plasticity, I had only very rudimentary knowledge of deformation bands, mainly gained from general textbooks as a student, and from an isolated study by Kuhlmann-Wilsdorf and Aernoudt [85]. In the latter we had shown that the morphology of, and slip system selection in, parallel, longitudinally extended, evenly spaced deformation bands observed in rolled and cross rolled iron, with alternating operation of two different slip systems, conformed to the LEDS hypothesis. Based on this interesting but certainly not typical case, deformation bands seemed to be infrequent artifacts of plastic deformation without fundamental significance. This opinion changed abruptly and dramatically when my colleague E.A. Starke Jr. appointed me to the Ph.D examination committee of his student S.S. Kulkarni, namely on compressed aluminum alloy samples that incidentally exhibited some impressive deformation bands (DB’s). From these, I recognized deformation bands to be the common result of multiple glide and to be the likely source of the cell blocks (CB’s) described in my work with Hansen, Bay and Hughes. Therefore I proceeded to more closely consider DB’s, derive their theoretical minimum free energy in terms of the sum of their surface energy and the strain energy in the surrounding material, and to compare the results with observations and data in the literature [92–95] in order to assess how closely DB’s conform with the LEDS hypothesis. By way of explanation: The strain energy in the surroundings of DB’s arises because DB’s delineate volume elements with any one particular selection of simultaneously acting slip systems that falls short of the five that would permit deformation homologous with the externally imposed shape change, and which selection abruptly changes across DB boundaries. In fact, only up to three independent slip systems appear ever to operate simultaneously within any one volume element, so that the local micro-strain tends to be significantly different from the imposed macro-strain. With increasing strain, the volume elements underlying individual DB’s shrink in size by subdivision and the generation of secondary DB’s in them, and on to tertiary DB’s in secondary DB’s. In retrospect, it became clear that this development of DB structures had been observed in the micrographs of Bay and Hughes [81,83,84], but on a too small scale as if it could have been recognized. On this understanding, the first-order theoretical expectation from the LEDS hypothesis turned out to be very simple indeed, namely that at minimum, the free energy of a deformation band yields (compare [123])

L=W 2 ffi sDB =ð0:035 GbÞ

ð3Þ

where L and W are the length and width of the DB, sDB is the flow stress, and G and b are the material’s shear modulus and Burgers vector magnitude, respectively. Comparing Eq. (3) with the own experimental results as well as with all useable data that could be located in the literature, not a single violation of Eq. (3), beyond reasonable error of the data, was found [93]. Even more persuasive was a comparison with a case of deformation bands in a diblock copolymer heavily fatigue cycled by Winey [96]. Herein, also, L/W2 was found to be constant, in fact within a factor of less than two. In a still more persuasive confirmation of the minimum stored free energy hypothesis also in plastics, in another diblock copolymer as a function of cycling frequency and temperature, always that lamella orientation had been established that had the lowest shear modulus in

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the shearing plane and direction [97]. Finally, on a very large scale, as a qualitative confirmation, it may be noted that the walls of some Arizona, Colorado and Utah canyons show cross sections of what appear to be intricate DB’s in geologically tilted and deformed sedimentation rock layers, including secondary and perhaps even tertiary DB’s. By implication, then, also in the rock formations exposed in those canyon walls there are planes and directions of lower than average shear strength, namely presumably parallel to sedimentation planes and old fluid flow directions that give rise to DB structures. Thus deformation bands also form (i) in non-crystalline materials, (ii) up to quite large scales, and (iii) at least where relevant measurements have been made, they have been found to conform to minimum free energy. Correspondingly the LEDS theory has been expanded into the generalized form of the LES (Low Energy Structure) theory of solid plasticity, without restriction as to the detailed deformation mechanisms involved. Based on the already mentioned survey of DB data available for metals in the literature, it was possible to determine the relative decrease of free energy due to the formation of DB’s. It was found therefrom that the energy stored in DB’s amounts to only 10% of the elastic energy at stress sDB, and that the energy in them could heat the material through only about 0.01 °C [103]. Hence, in a wide range of deformation conditions, 0.01 °C is the characteristic, and amazingly close, approach to free energy minimum, i.e. to thermo-dynamical equilibrium, of the microstructures established by plastic deformation. 2.4.8. Dislocations/lattice vacancy interactions: ‘‘mushrooming’’ and planar vs. wavy glide Observationally, the difference between planar and wavy glide is that planar glide materials continue deforming with Taylor-lattice type structures in Stage II, while wavy-glide materials experience a Stage II/Stage III transition and from then on deform with dislocation cells, i.e. with mosaic block structures. Conventionally, as also in the SODS approach, this difference tends to be explained through the suppression of cross slip and climb in planar glide, namely on account of low stacking fault energies. These cause dislocations to extend into partials and thereby inhibit three-dimensional dislocation motion that is necessary for the formation of dislocation cell walls. While this argument poses no problem for the LEDS theory, it is evidently incomplete since Al–Mg alloys exhibit planar glide with Taylor-lattice type structures (see Fig. 7) and yet have high stacking fault energies. The resolution of this paradox is found from ‘‘mushrooming’’ [98], the complex interactions between dislocations and supersaturated vacant lattice sites, that in turn are largely generated by gliding dislocations. Namely, besides a modicum of three-dimensional dislocation mobility, the formation of cell boundaries, and thus Stage III glide, requires the easy availability of dislocations with other than primary Burgers vectors. In wavy-glide materials, mushrooming provides these by means of dislocation loop nucleation via supersaturated glide-generated vacancies in all slip plane orientations. However, on account of the large size of the solute Mg atoms, the equilibrium thermal vacancy concentration in Al–Mg alloy is so high [99] that the glide-generated vacancies do not constitute a sufficient supersaturation for loop nucleation [99]. This, then, explains the suppression of Stage III both by low stacking fault energies and in Al–Mg alloys. A critical distinction between Stage II in planar and wavy glide is that it is straight in planar glide but mildly curved (like Stage IV of Fig. 5) in wavy glide. Such a curvature proves that supercritical dislocation bowing (namely out of cell walls) controls the flow stress in both wavy-glide Stages II and IV. By contrast, the linearity of the stress–strain curves in planar glide, in both Stages I and II, shows that in these the flow stress is determined by dislocation bypassing, i.e. between parallel bipolar edge dislocation arrays, and in Taylor-lattice like structures as in Fig. 7. Wavy-glide Stage III, on the other hand, is represented by Voce curves. In these, the workhardening coefficient decreases linearly with the flow stress as well as with the testing temperature, as first shown by Mecking and Lücke [100,101]. In the LEDS theory, this functional dependence is readily explained through the realization that parallel dislocations of opposite Burgers vectors will annihilate each other if they approach to within a critical annihilation distance, and that the annihilation can be assisted by thermal activation. Thereby the LEDS theory readily accounts for both the shape and the temperature dependence of wavy-glide Stage III [102].

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Fig. 7. Taylor-lattice type microstructure parallel to the rolling plane of Al+5.5%Mg alloy after 10% reduction in thickness. The absence of cell walls is shown by the uniform background intensity across what otherwise might appear to be cell walls. Also note the quasi-uniform dislocation distribution, and boundaries across which the Burgers vector selection changes as indicated by abrupt background contrast changes and that these are parallel to {1 1 1} slip plane directions as shown by the broken lines at top left. The double arrow indicates the rolling direction. The still unknown morphology of this structure remains to be clarified. (Courtesy of D.A. Hughes).

2.4.9. The onset of Stage IV According to present best knowledge, Stage IV arises when dislocation cells have become so small that glide-generated vacant lattice sites diffuse to the cell walls too rapidly as to build up the requisite supersaturation for mushrooming. While there remain sufficient Burgers vectors in the walls so that cell morphology is not affected, the dislocation links in the walls become much straighter. Therefore, arriving glide dislocations and cell wall links mutually annihilate already at larger critical distances, and the workhardening coefficient is lowered without change of cell structure. 2.4.10. The speed dependence of the flow stress and link length distribution in cell walls In Ref. [103] the distribution of link lengths in cell walls was theoretically deduced to have the form of Fig. 8 (left), wherein the average link length is n = 3 times larger than the longest link length that would at any time bow out super-critically under an applied flow stress. Namely, an intersecting glide dislocation impinging on a link in a cell wall, will inevitably react with it to form a new link. In the process, the previous one link between its anchoring nodes is replaced by three links. Further, the link formed by the reaction of the pre-existing and impinging dislocation could be situated anywhere between the two nodes, and in turn the nodes would somewhat reposition. Intuitively, therefore, the shape of the left curve of Fig. 8 was deduced to well describe the dislocation link length distribution generated by the manifold repetition of the above process. At the time of writing [103] there was no supporting experimental evidence for that conclusion. Since then, Lin, Lee and Ardell [104] have made excellent determinations of the link length distribution in high temperature, steady state compression creep of aluminum close to its melting point (at 647 °C, top right of Fig. 8, at the indicated stresses), and in primary creep of rock salt (bottom right of Fig. 8) at the indicated temperatures and stresses. The close similarity among the three distributions, that strongly supports the corresponding part of the LEDS theory, is evident. The initial intent of Ref. [103] had been to explain the experimentally observed mild speed dependence of the flow stress of ductile metals that did not seem to be thermally activated. It was concluded

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Fig. 8. Predicted (top left, mildly re-scaled Fig. 48 of Ref. [103]) and measured (right, Fig. 53 of [104]) link length distributions in cell walls. The most important aspect of this Figure, besides the excellent qualitative verification of prediction through experimental data, is that the experimental link length distributions (at right) are evidently nearly independent of material and straining conditions, and this in a wide range of stresses and in a significant range of temperatures, as had been predicted based on the LEDS principle.

that the discussed speed dependence arises because higher strain rates require the simultaneous activation of an increased number of links with the correspondingly larger critical bowing stress, as indicated by the lettering at top left of Fig. 8. At least semi-quantitatively, Fig. 8 top left fully explains the observed strain rate dependence of the flow stress in wavy glide. 2.4.11. Recovery For this writer, metal recovery has been the earliest effect to be explained by means of dislocation movements [105], albeit at that time, i.e. in 1949, incorrectly. Even so, the then derived logarithmic time dependence of recovery,

Rr ¼ ðs  so Þ=ðsmax  so Þ ¼ 1  A ln ð1 þ BtÞ

ð4Þ

with Rr the relative flow stress after recovery time s, and with the constant A proportional to the absolute temperature, has ever since been widely observed. Now the same approach as in Ref. [105] but with realistic dislocation models, for both planar and wavy glide, yielded equations that were gratifyingly close to observation for the cases of pure aluminum and aluminum alloy [106]. Specifically, the theory developed in Ref [106] predicts recovery in accord with Eq. (4) for planar glide, and recovery largely following Eq. (4) but including a mild dish-shape distortion for wavy glide. Figs. 9 and 10 (extracted from [107], and from [108] using data from Refs. [109] and [110]), document that the

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Fig. 9. Flow stress of pure Al (wavy glide) as a function of log (t) in accordance with Eq. (4), after pre-strains from 5% to 25%, recovered at (a) 120 °C, (b) 160 °C and (c) 200 °C. Note the mild dish-shape curvature in deviation from Eq. (4). This is expected for wavy glide but is absent in planar glide, as seen in Fig. 10 (Fig. 4 of [107]).

prediction of [106] is fulfilled for AlMg alloys as planar glide materials, and for wavy-glide Al. Several additional examples in the literature support the same result also for other metals [105]. 2.4.12. Alloy hardening – solid solutions and precipitates Alloy hardening is such a very wide-ranging, diverse and technologically vastly important subject that it could not very well be ignored by an encompassing theory of the mechanical properties of solids. It is therefore highly significant that all of its various aspects have been successfully explained by means of the LEDS theory. Specifically, solid solution hardening in fcc metals, in particular, depends on the degree to which the solute decreases the stacking fault energy, and thereby transforms a pure wavy-glide metal into an a-brass alloy, partly or completely, assuming no drastic change in vacancy energy. If stacking fault energy is not an issue, solid solution alloying acts mostly to change (and typically to increase) the friction stress so, which has the effect of simply shifting the stress– strain curve upward. Precipitates, by contrast, limit the mean free dislocation path to, roughly, the distance between them, i.e. the ‘‘Orowan distance’’. This essentially raises the initial yield stress to the Orowan stress, sOr, since up to sOr the mean free dislocation-free path is so small that the resulting plastic strain compares to the elastic deformation. Consequently, the stress strain curve appears to start with elastic deformation up to sOr, after which it continues as for the unalloyed material. (See Fig. 11).

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Fig. 10. As Fig. 9 but at room temperature for different Al–Mg alloys (planar glide) and pre-strains as indicated. Note that Figs. 9 and 10 are in accord with the LEDS theory. Note also the amazing persistence of the log–time law of Eq. (4) for 17 years(!). Fig. 12 of Nes [108], based on Refs. [109] and [110]).

Fig. 11. Idealized workhardening curves of a pure metal (curve 0) and precipitation hardened alloys of the same metal but with an increasing percentage of precipitates (curves 1, 2 and 3). The curves appear to begin with an elastic part up to the Orowan stress, beyond which they have the same shape as the matrix material However, on account of dislocation multiplication by critical bowing between the precipitates, the initial slopes correspond to only about one half of the elastic modulus. For details see [71] (Fig. 2 of [88]).

2.4.13. Grain boundary hardening – two versions of the Hall–Petch relationship In relation to the various aspects of grain boundary hardening, much the same applies as to alloy hardening, as above. It, too, is fully explained by the LEDS theory. The Hall–Petch relationship 1=2

ðs  so Þ ¼ K HP =DG

ð5Þ

with DG the grain size, results if the shear strength of the grain boundaries is proportional to the flow stress. If that strength is independent of the flow stress, the exponent is 1/3. (See [88]).

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2.4.14. Conferences and final collaboration with Frank Nabarro For the sake of completeness, herewith a brief listing of important conferences: The 1966 ‘‘Workhardening’’ [111] conference in Evanston, organized by Hirth and Weertman for the express purpose of stimulating constructive dialogs among proponents of the major competing workhardening models, and the 1977 conference ‘‘Work Hardening in Tension and Fatigue’’ [112] organized by Tony Thompson with a similar aim spawned the papers ‘‘Unified Theory of Stages II and III of Workhardening in Pure FCC Metal Crystals’’ [54] and ‘‘Recent Progress in Understanding of Pure Metal and Alloy Hardening’’ [113]. Also happily important have been various Riso conferences under the leadership of Niels Hansen and similarly a conference led by E.A. Starke, Jr. (see [106]) the participation in which yielded a fruitful collaboration on deformation bands. A paper initially meant for the ‘‘Dislocations 2000’’ conference at NIST in Gaithersburg, MD, June 17–22, 2000 was not accepted and appeared separately in Materials Science and Engineering [114]. Further, most important were the following conferences, to which also Nabarro contributed: ‘‘Low-Energy Dislocation Structures’’, Proc. Intl. Conf. on Low-Energy Dislocation Structures, UVA, School of Engr. and Appl. Science, Charlottesville, VA, Aug. 10-14, 1986 (Eds. M.N. Bassim, W.A. Jesser, D. Kuhlmann-Wilsdorf and H.G.F. Wilsdorf, Elsevier Sequoia, Lausanne, 1986); see also Mater. Sci. Engrg., 81 (1986), pp. 1–574 and 86 (1987) pp.19–92. [115]; ‘‘Low-Energy Dislocation Structures II’’, Proc. Intl. Conf. on Low-Energy Dislocation Structures, UVA, School of Engr. and Appl. Science, Charlottesville, VA, Aug. 14-17, 1989 (Eds. M.N. Bassim, W.A. Jesser, D. Kuhlmann-Wilsdorf and G.J. Shiflet, Elsevier Sequoia, Lausanne, 1989). 458 pages; see also Mater. Sci. Engg., A113 (1989) pp. 1–454. [116]; ‘‘Low-Energy Dislocation Structures IV’’, Proc. Intl. Conf. on Low-Energy Dislocation Structures IV, Univ. of Manitoba, Winnipeg, Canada, June 5–8, 1995 (Eds. M.N. Bassim, W.A. Jesser and D. Kuhlmann-Wilsdorf, Akademie Verlag, Berlin, Germany, 1995), 443 pages; see also physica status solidi (a)149 (#1) 1995 pp. 1–443. [117]. These conferences have been highly successful; and the specially bound conference volumes contain a wealth of data and micrographs regarding a great variety of LEDS’s. Among others, my paper ‘‘Modelling of Plastic Deformation via Segmented Voce Curves Linked to Characteristic LEDS’s which are Generated by LEDS Transformations between Workhardening Stages’’ [102], that presents the theory of the shape of Stage III, appeared in [117], while Ref. [70] by Lepistö et al. formed part of LEDS I ([115]). Further, I contributed ‘‘Strengthening Through LEDS’’ [118] to the ICSMA Conference edited by Kettunen, Lepistö and Lehtonen, Tampere, Finland, Aug. 22-26, Pergamon Press, Oxford, NY, 1988), Vol. I, pp. 221-226 and ‘‘Why Do Dislocations Assemble into Interfaces in Epitaxy as Well as in Crystal Plasticity? To Minimize Free Energy’’, [119] to the Oct. 10–12, 2001 Symposium in honor of Jan van der Merwe. In retrospect, I am much indebted to Frank Nabarro for again and again acting as the Devil’s Advocate regarding the LEDS theory and thereby to cause deeper thinking and greater effort to dig into the literature than otherwise would have been the case. His strongest argument in his Devil’s Advocate role was that the LEDS theory could not possibly account for the apparently a-thermal increase of the flow stress with rising strain rate. Fortunately this objection could be removed by means of Ref. [103] (see Fig. 8) already discussed. An opportunity to publish with Frank came about with a paper on the nucleation of small angle boundaries [120] written during one of Nabarro’s many stays as visiting professor in our Department at UVA. At issue in that common paper was a then recent addition to the LEDS theory, namely the nucleation of new cell walls in the process of shrinking average cell size (and thus increasing number of cells) in Stages III and IV, so as to account for similitude. Frank proceeded to mathematically show that the envisaged wall nucleation would be a first-order transition. Even so he felt unable to accept the LEDS theory as evidenced in his paper ‘‘Complementary Models of Dislocation Patterning’’ [121], Herein he considered two alternative types of ‘‘patterning’’, namely the formation of Taylor lattices and of mosaic block structures, with the aim of deciding whether such ‘‘patterning’’ was driven energetically (i.e. as in the LEDS theory) or purely kinematically (a la SODS). Nabarro concluded agnostically saying that:

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‘‘even in models of rectilinear edge dislocations which cannot cross slip, dislocation patterning can be driven by two quite distinct mechanisms. The implication is that a unified theory for the real threedimensional case is hardly possible. One can only hope for detailed studies of individual mechanisms, and of the transitions from one mechanism to another.’’ And yet, as shown in a long review article in Philosophical Magazine [122] as well as in the 2002 Campbell Memorial lecture ‘‘Advancing Towards Constitutive Equations for the Metal Industry Via the LEDS Theory’’, (Metall. Trans B, 35, pp. 5–54, 2004) [123], an enormous range of problems in the metal industry could and should now be tackled by means of the LEDS theory, optimally bolstered by the last remaining open issue, namely creep. To complete the LEDS theory of creep should in fact pose no major problem, namely by using Orowan’s proposed balance between hardening through creep deformation and softening through recovery, since both parts of the theory are already available. And for further surveys of the LEDS theory [124–130,122] may be consulted. 3. Part 3: workhardening theory in light of Kuhn’s ‘‘Structure of Scientific Revolutions’’ As undergraduates we tend to be taught that the international community of scientists is collectively wresting its secrets from nature, and that each scientist rejoices in the advances of any other. The personal experiences of this writer, related above, have not exactly borne out this idealistic view point. What, if anything, went wrong? Should anyone be blamed, and if so who? This question may perhaps best be pursued in relation to Kuhn’s foundational book ‘‘The Structure of Scientific Revolutions’’ [131]. According to it, within its different disciplines science progresses in repeated phases, roughly as follows: (i) As the result of a ‘‘scientific revolution’’, a new ‘‘paradigm’’ of theoretical interpretation is accepted. (ii) In the following, often protracted, period of ‘‘normal science’’, the details and consequences of the paradigm are studied and new data are accumulated, whereby results that contradict the paradigm tend to be discarded as failed efforts. (iii) As results accumulate that contradict a paradigm, in response to increasing doubt, a ‘‘revolution’’ may occur wherein a new paradigm is adopted that is in better harmony with the data. Kuhn realized that a failing paradigm tends to be fiercely defended by its adherents. In this connection Kuhn quotes Max Planck who wrote [132]: ‘‘A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.’’ The perhaps saddest case of the collective resistance of scientists to new paradigms, of all, is that of the Hungarian physician Ignaz Philipp Semmelweis (1818–1865). He discovered that numerous women died of puerperal fever after childbirth in hospitals because doctors and nurses, with unwashed hands and blood and pus spattered clothes, spread the infection from one young mother to another. Semmelweis watched helplessly and increasingly frantically. He pleaded for asepsis to no avail, and eventually, being ridiculed and harassed to boot, became insane in desperation. Pertinently, in our own field of solid plasticity, R. Hill in a brief passage in Ref. [133] wrote regarding G.I. Taylor, (p.145/145): ‘‘Taylor. . . did not find it easy, however, to convince contemporary metallurgists of the intrinsic superiority of his general approach. In later expositions he observed with some asperity (rare indeed) that other methods depended ‘on knowing the form which the answer will take before starting to solve the problem’ or on ‘verifying particular hypotheses by special measurements’’. In fact, after several brilliant years of research, Taylor left the area of solid plasticity rather abruptly to become highly acclaimed in hydrodynamics, in which he received numerous honors including seven honorary doctorates. It seems, then, that G.I. Taylor abandoned plastic deformation because he found colleagues either not willing or unable to develop theory by strict logic without bias, – and the preceding discussion seems to indicate that our research area has continued to labor under this same difficulty. In retrospect, we may partly blame this problem on the conflict between two alternative paradigms in solid plasticity since 1934, namely (i) that dislocations generate high-energy obstacles against glide,

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especially in the form of pile-ups, and (ii) that dislocations assemble into low-energy structures, initially recognized by Taylor in the form of Taylor lattices. The obstacle option, first introduced by Kochendörfer [11], prevailed as it was somewhat adopted by Orowan under the guidance of Becker [5], next provided the decisive hypothesis in this writer’s Ph.D. thesis [9,10], thereafter was adopted from Kochendörfer by his pupil Seeger [22,23], and on to Cottrell, Mott, Hirsch and the majority of materials scientists, to eventually morph into the SODS approach. By contrast the competing paradigm of low-energy dislocation structures, i.e. LEDS, is based on Taylor’s brilliant 1934 workhardening theory [15] but was not appreciated, was neglected and fell into oblivion when Taylor left the area. Much later, this writer adopted this paradigm after having contemplated a large number of dislocation structures, generated in many different materials in a wide range of strains and via very different modes of straining, and finding that they clearly were not pile-ups but LEDS’s. Yet, the difficulty of displacing the old paradigm proved to be enormous, and the new LEDS paradigm, that explains all of the evidence and so clearly is in line with physics, has been fiercely fought, in line with historical precedence as in the case of Semmelweis. In fact, solid state plasticity is right now without a paradigm, as pile-up theories have no adherents left and still popular models such as Kocks’ penetration of glide dislocations through assemblies of ‘‘forest dislocations’’ and Mughrabi’s (also two-dimensional model [134,135]) cannot form the basis of a new paradigm as they lack generality. From this perspective, it seems, then, (1) that for a few decades now we have been witnessing an episode of scientific stagnation of the kind that has repeatedly occurred in the past, notoriously including the delay of medical asepsis in the 1800’s with the loss of many lives (including that of my grandmother); (2) that following Kuhn, such episodes are brought about by the lack of a widely accepted new paradigm, and (3) that, as they result from very widespread human traits, those episodes should not be blamed on specific ‘‘villains’’. BUT: Clearly, right now, in the area of solid plasticity a paradigm change is urgently needed to clear the way towards purposeful, collaborative efforts to develop useful results for society, such as constitutive equations that will help in optimizing industrial processes [123]. Some of the human psychology underlying the inertia that makes paradigm shifts so difficult but also so badly needed, seem to be revealed in a remarkable result of a 1975 study. This addressed suspected political influence in the NSF ‘‘Materials Research Division’’ [145] that at the time was responsible for funding of US material science departments, as shown in Fig. 12. Plotted herein from left to right, i.e. along the horizontal axis, is the number of times the average faculty member in any US materials science department had been cited in the international literature. As a first approximation, therefore, the left to right placement of the different departments, whether indicated by dots, crosses or small circles, is a measure of the quantity and quality of the combined research success of each department, namely increasing from lowest at left to highest at right. By contrast, the vertical axis indicates the NSF research funding received per faculty member in multiples of the national average. Specifically, the top dark dot, near the middle in horizontal direction, indicates the then highest NSF research funding per faculty member of any US materials science department, namely about 2.3 times the national average, inspite of its mediocre research success. By contrast, the three departments that received the lowest funding per person among the departments in the black dots grouping, are found at the extreme right and the extreme left. Thus the scientifically best and lowest departments in the group indicated by black dots, had the same funding success, but each still above the national average. Namely, in Fig. 12, departments are separated into three groups according to funding success. The black dots represent departments that appeared to be favored for non-scientific reasons, the crosses represent departments that received funding near the national average, and those indicated by open circles were evidently disadvantaged, again apparently for non-scientific reasons (that prompted the hearings). While the three-tiered grouping of funding could be shown to arise from biases within the NSF Division, the average departmental quality is very much the same in all three groups. How, then, could the peculiar funding distribution among the groups arise, since funding decisions had to be buttressed by ‘‘peer review’’ assessments? Answer: the similar bell-shapes of the three curves, in lieu of the expected general rise from lower left to upper right, strongly suggests the corresponding bias among the ‘‘peer reviewers’’. If so, the conclusion emerges that average peer reviewers generated the counter-intuitive decrease of funding from average to superior ability, through a bias against their strongest competitors. If true, this would be very sobering: Contrary to our ideals, Fig. 12 seems to show that, shielded by anonymity, ‘‘peer reviewers’’ tend to discriminate against their most able competitors. We may intuit

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Fig. 12. 1975 statistics of relative funding (vertical axis) received by US materials science departments from the NSF Materials Division, as a function of relative (bottom) and absolute (top) citation rates of the average faculty member in each department, according to a statement made before the US House Subcommittee on Science Research and Technology. Departments have been grouped according to received funding per faculty member, i.e. well funded (dark dots, top curve), average funding (crosses, middle curve) and poorly funded (bottom curve). It was alleged that this funding spread was due to extraneous effects that could not be objectively justified. The peaking in the middle of each of the three curves appears to have been due to reviewer prejudices against top performers in favor of faculty members of average ability. Anyway, departments with a poor research record received low funding (left) as also did departments with high citation rates, meaning with top research performance (right). The curvature thus may indicate that the average reviewer favored faculty with average research ability and discriminated against those of higher ability than he/she. [136].

that this situation is favored by lack of scientific direction when opinion instead of established fact is allowed to guide decisions. If so, such counterproductive biases will greatly diminish, when to our great advantage we shall finally agree on a widely shared paradigm so as to enter Kuhn’s phase (ii), which in Kuhn’s words is the ‘‘often protracted period of ‘normal science’, wherein the details and consequences of the paradigm are studied and new data are accumulated’’, i.e. is the by far most fruitful period in a cycle. Acknowledgements Gratitude is due to the organizers of the present symposium for permitting, indeed encouraging me, to freely express my somewhat unconventional opinions and conclusions. References [1] Kuhlmann D, Masing G. Investigations on the plastic deformation of copper wire. Zeitsch Metall 1948;9:361–75. [2] Mott NF, Nabarro FRN. An attempt to estimate the degree of precipitation hardening, with a simple model. Proc Phys Soc (London) 1940;52:86–9.

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