published in Fracture Research in Retrospect (A.A. Balkema) by
K.B. BROBERG

ABSTRACT: To have walked a mile with a scientist of Waloddi Weibull's stature is a privilege, treasurable for a lifetime. Here some personal experiences from encounters with Professor Weibull are related. He is perhaps best known for his statistical distribution function, but his interests ranged from statistical aspects on fracture and fatigue across several other fields, particularly oceanography. He was a man with fine humour combined with sharp logic. Several anecdotes are connected with him, a few of which are related here.

KEYWORDS: Weibull distribution, statistical fracture mechanics, Weibull anecdotes, oceanography.

INTRODUCTION

This essay is not intended to discuss Weibull's scientific achievements, but only to relate some personal experiences.

SOME BIOGRAPHICAL DATA

Professor Ernst Hjalmar Waloddi Weibull was born on the 18th of June 1887. He started his career as a navy officer, which gave him the opportunity to follow a two year course for navy officers at the Royal Institute of Technology (RIT) in Stockholm, 1909‑1911, whereupon be studied at the Royal Naval Staff College in Stockholm, 1911‑1913. He received a Licentiate degree in 1924 and was appointed Professor of Machine Elements at RIT the same year. He chaired the Standardisation Commission of the Swedish Machine Industry from 1926 to 1931. During the years 1931‑1940 be was Prorector of RIT, and he was appointed Honorary Doctor at Uppsala University in 1932. In 1941 he became scientific advisor to the company AB Bofors in Karlskoga, but kept his association with RIT as Professor of Engineering Physics 1941‑1953, a personal research position paid by AB Bofors. He gave a short course each year for about a dozen students, and it was as one of these that I first met him, 1948.

Weibull became a member of the Royal Swedish Institute for Engineering Research (now the Royal Swedish Academy of Engineering Sciences) in 1927, of the Royal Society of Sciences in Uppsala in 1941, of the Royal Swedish Academy of Sciences in 1943 and of the Swedish Academy of Military Sciences in 1946. In 1978 he was awarded the Great Gold Medal from the Swedish Academy of Engineering Sciences, its highest distinction. When King Carl XVI Gustav presented him with the medal, Waloddi Weibull could say: `Seventy‑one years ago I got my officer's commission from the hand of Your Majesty's grandfather's grandfather'.

Weibull's scientific interests were very varied. Best known, of course, is his interest in fracture mechanics. In one early paper (Weibull 1938) he described how the loading velocity and the presence of `bounding media' (distilled water, nitrit acid, copper sulphate and sodium hydrate) influenced cracking in glass plates, loaded by the contact pressure of a steel ball. In the Introduction he mentions a preliminary report `submitted to the Royal Swedish Institute of Scientific Industrial Research in February 1925' about these experiments, but the paper from 1938 contained additional material. He concluded that 'the ultimate strength to a high degree depends on the rate of application of load', that 'if the glass plate is contiguous to certain electrolytes, the ultimate strength may in some cases increase more than 100%' and that 'the ultimate strength in the vase of two‑dimensional state of stress in glass and bakelite plates shows that the ultimate strength of these materials is in all probability determined by the maximum tensile stress, and not by the shearing stress or the shear strain energy'. He attributed the action of `certain electrolytes' to their influence on the surface energy.

The statistical theory of fracture was published in two issues of the Proceedings of the Royal Swedish Institute for Engineering Sciences (Weibull 1939a, b). Since these original papers appeared when World War Il broke out, and in proceedings with comparatively small distribution, he synthesised results of these in another paper twelve years later (Weibull 1951). This became his probably best known and most cited paper. The importance of the 1939 papers for the field of statistical fracture mechanics, however, has been recognised by leading scientists. G.R. Irwin's presentation to the 32nd Annual Meeting of ASTM in Atlantic City, N.J., June 26‑July 1, 1960 referred to the first of the 1939 papers (Weibull 1939a). Irwin was obviously particularly attracted to the `relative mathematical simplicity' of the Weibull analysis (Irwin 1960).

In the first of the 1939 papers (Weibull 1939a) Weibull discusses the volume dependence on statistical strength parameters and also polydimensional stress states, non-uniform stresses, torsional stresses and bending stresses. In the second one (Weibull 1939b) he discusses regular and irregular materials. Regular materials (isotropic or anisotropic) are those `in which formation of cracks continues as long as the edge of the crack is subjected to tensile stress', and irregular materials are those `in which formation of cracks ... ceases before it has spread over the whole crosssectional area'. He takes a rope as an example of incoherent irregularity (rupture of one strand influences the stressel on the other strands along the whole rope) and wood as an example of coherent irregularity (rupture of one fibre will only lead to an increase of stress in the immediate vicinity).

Weibull's interest in fracture mechanics later turned towards issues of fatigue. He published one book on fatigue testing (Weibull 1961). He was actively pursuing his scientific interests until he died, peacefully, at the age of 92, on the 12th of October 1979.

Weibull took part in two large oceanographic expeditions, one in the Mediterranean with the research vessel Skagerack in 1946, and the other round the world in 1947-1948 with the yacht Albatross. Among other things he studied the thickness of the sediment layer at the sea bottom as well as underwater sounds produced by marine animals.

As scientific advisor to AB Bofors he worked with explosives from different points of view and produced a paper on waves in compressible media (Weibull 1948). In this paper he used an elegant mathematical formulation of nonlinear wave propagation. His interest in mathematics is evident from a speech he gave in 1951 at the annual meeting of the Swedish Academy of Engineering Sciences, and I cannot resist quoting some of the passages (my translations are within quotation marks). In this speech he claims that 'the advent of complicated and immensely efficient computers opens the way for something which could be called experimental mathematics'. He also discusses operational analysis and game theory, mentioning von Neumann and Morgenstern, 'who through their theory for strategic games created a new mathematical discipline, originally intended to substitute the classical differential approach in national economical calculations'. His comments on computers are very interesting. He mentions the planhed Swedish machine BESK, which was almost completed and would produce 100,000 additions in 8 seconds and as many multiplications in 48 seconds, which would be 2 1/z times faster than the famous American machine ENIAC. He mentions 'solution of symbolic (hon-numerical) equations' and translation of languages as possible future applications. He thinks that `an electronic machine would be particularly well suited to build a Bach-fugue on a suggested theme, carefully avoiding forbidden octave- and quintparallels and other prohibitions, perhaps so perfect that a suitable statistical disturbance must be introduced to obtain a human touch on the product'. He allo compares a computer with a human brain and concludes that the computer can exceed the brain as regards speed but otherwise is 'at about the same level as an earthworm'. He concludes his speech with a discussion of Norbert Wiener's cybernetics.

WALKING WITH A MAESTRO

Professor Weibull suggested that I made my exam work at RIT during the spring of 1949 in Karlskoga, using equipment and material at AB Bofors. He introduced me to an excellent and quite recent work by R.M. Davies in Aberystwyth on a Hopkinson pressure bar with capacitive measurement of the motion of the end opposite to the impacted end (Davies 1948). He was very interested in the experiments, and be gave me all the support I needed. Sometimes he shared experiences from his work at AB Bofors, for instance when he investigated the cause of an accident in the manufacture of dynamne. At that time he had a temporary young helger, and they were going to detonate a sizeable dynamite charge. The charge was some hundred metres away, it was to be detonated electrically by the quick turning of a handle (a safety precaution). Nothing happened, however, and Weibull said to the boy: `Stay here and I'11 see what is wrong'. He went to the charge, assuming that there was some shortcut in the connection of the cord to the detonating cap, but everything seemed to be in order. He went back to the boy and said that the error was probably somewhere along the cord. 'Yes', said the boy, `it must be. I have been turning the handle here all the time'. Weibull found that very amusing.

He used to tell me about his experiences with the Albatross expedition. Somewhere the crew found a big turtle, which the cook turned into turtle soup. `He needed a lot of sherry', said Weibull, `he came back to the supply, asking for more several times'. Anyway, everybody agreed that the result was excellent, but afterwards the zoologist onboard found a remark in a zoology book: 'This species is inedible'. `It must have been the sherry', said Weibull. During the expedition he got news by cable that he had become father to a baby girl. The whole crew offered to be godfathers on condition that the girl was named Moana (first a pronounced as a in `father') after the most beautiful girl they bad seen in Polynesia. And so it was. Weibull told me how he liked to listen to Moana's playmates shouting her name with their typical dialect from the province of Scania in snuthem Sweden, where he had the family's summer house, Bockamöllan.

In 1952 Weibull telephoned me to ask if I would be interested in accompanying him on a small expedition to the Mediterranean, something that I immediately accepted. The purpose was to determine the thickness of the sediment layer by an acoustical method, and I should handle the recording equipment. The plan was to place small detonating charges at the bottom and then record the different waves, direct and reflected. For preparation he had arranged a smaller expedition to the Baltic on the ketch Agneta, owned by his friend, chief-engineer Oscar Wiberg. The Baltic is not very deep, so the charges could be attached to an electric cord and detonated from the ship. A microphone was placed just below the water surface, and the signals were recorded via an amplifier that I bad built. Now and then we got signals even before the detonation, and Weibull, with his experience from earlier expeditions interpreted the signals to me: 'This isa haddock' or 'This is a crab'. Everything worked well. We bad a narrow escape once, when Weibull, who handled the charges, was just about to detonate one when he discovered that the cord had got caught in something and the charge was just at the ship's hull, instead of at the bottom. He never told Wiberg about this incident.

Several years later Waloddi Weibull told me that Oscar Wiberg had died in Switzerland at the age of well over 80. `He went there to improve his technique in slalom', Said Weibull, `and then be died of a heart attack right on the slope'. I could notice a gentle smile: It was obvious that Weibull did not begrudge his old friend such a dignified and stately departure from this world.
Some new problems arose in connection with the expedition to the Mediterranean and Weibull solved these with characteristic ingenuity. One was to let the charge detonate when it had reached the bottom, even though a depth of 1-3 km did not allow use of an electric cord. Weibull put the small charge and an electrical battery in a can and used a thin membrane as lid. At sufficiently high water pressure the lid collapsed, making electric contact between the battery and the detonating cap. But how to avoid detonation before the charge bad reached the bottom? This was arranged with a PMMA lid above the membrane, thick enough to withstand even the pressure at the bottom. Acetone was poured in the space between the membrane and the PMMA plate. Weibull had made experiments at home to find the appropriate proportions, so that the PMMA lid was weakened by the acetone at the proper speed. There was never a failure. In the Tyrrhenian Sea between Corsica-Sardinia and Italy the depth is up to 3 km, and Weibull had calculated that the charge would reach the bottom after about 6 minutes, and then it was only to wait a few minutes more, until the acetone had done its job.

The ship Elie Monnier

Next problem: What recording system could be used when the exact time of the event could not be known better than to within a few minutes? Weibull used the simplest way possible: A tape recorder. One of his friends - he had friends and useful contacts everywhere - had a position at the Swedish Radio that allowed him to lend us a tape recorder that had an upper frequency of 40 kHz, which was fully sufficient. I picked it up in Stockholm and travelled by train to Toulon, the main harbour town for the French Navy in the Mediterranean. There the 600 ton ship Elie Monnier was put at our disposal. It was equipped for diving and belonged to the French Navy. I don't remember exactly, but I think some friend of Weibull in the French Government or the French Academy of Sciences had arranged that. Weibull also bad company with a Swedish oceanographer, Professor Hans Pettersson from Göteborg, who took bottom samples at some places.

The sediment layer is very thick in the Mediterranean. There are deposits from rivers such as the Po, the Rhone and the Nile, and there is sand from the Sahara. The increase of sediment thickness was regarded to be about 7 mm per century, much higher than in the oceans. In a few million years there will be a lot of sediment, but the increase is still slow enough that several iron meteors were found in the core samples collected by Professor Pettersson.

We started from Toulon, cruised through the Tyrrhenian Sea and landed after a week or so in Alger. At several positions we made a stop, Weibull filled the charge device with acetone and dropped it into the sea. Then be placed the microphone at the side of the ship just below the surface, and I started the tape recorder. Everything went fine, bot we noticed that the microphone was hardly needed. Each wave impinging on the hull of Elie Monnier could be heard clearly, even though the charge was only a few dozen grams at 3 km depth. First, of course, came the direct wave from the charge, then came the wave that bad gone down through the sediment layer to the crystalline bottom and turned back. Then the first wave came back after re flections at the free surface and the bottom, and so on. Several wave reflections, gradually becoming weaker, could be heard. In short, the result was that the sediment layer was generally very thick, more than 1 km, and rather stratified, which created a number of weak reflections.

In Alger Professors Weibull and Pettersson attended a geology conference, and I did some sightseeing. One night I walked through the casbah, completely unaware of the tensions between different ethnic groups at that time. During lunch and dinner I joined Weibull and his company. I remember how someone told us that he had asked Professor de Geer what he thought about Wegener's theory of the Continental Drift and got the answer that this theory was one of the most stupid things he ever had heard of. `I do not understand why he reacted so angrily', said the man (whose identity I unfortunately cannot recall), `because I think the theory might make sense'. Weibull agreed. When the conference was finished, we took a plane to Nice, both of us using the discount for students. Weibull had a house in Alassio on the Italian Riviera, and I spent a few days there with him and his family before returning to Stockholm.

At a time when exploratory travel was an art, Waloddi Weibull was a connoisseur. He kept a card register over places he had visited, and when going to revisit them he simply brought along the appropriate cards. There he had, for instance, written about hotels, their specialities of food and wines, etc. He also kept card registers on other things, such as literature references.

My next encounter with Waloddi Weibull was at the IUTAM Symposium in Stresa in 1960. One day there was an excursion to the Borromeo Islands in Lago Maggiore. We went through the beautiful Palazzo Borromeo on Isola Bella, and Weibull told me about the algebra he was constructing for stochastic variables. The conversation went like this: 'You see, the rules for stochastic variables are not the same as for ordinary variables. So, for instance, if A = B + C, it does not follow that B = A - C. This is a nice floor, and doesn't seem to be too difficult to make. Perhaps I could do something like that on our veranda in Bockamöllan. Likewise, the derivative of a constant is not automatically zero. Doesn't that guide resemble Mussolini? Maybe it had been better if Mussolini had been a guide instead'. And so it went on, and when the tour around the castle was finished, his description about his theory was also finished, and I am sure that he did not miss anything of what the guide had shown us.

Weibull had planned to leave Stresa before the symposium was finished, but I felt very flattered that he stayed to listen to my lecture on 'The propagation of a brittle crack'. After that it took several years until I met him again. I had become Professor of Solid Mechanics at the Lund Institute of Technology in 1961, and had therefore moved to Lund. Some years later I visited him a few times in Bockamöllan. I think it was during one of these visits that he told me some stories about his distribution function. A zoologist, who was interested in a particular variety of seashells, bad tried to use the Gaussian distribution for the shell sizes and had got some puzzling results. Weibull applied his distribution function. As is well known, if plotted in a special way, the result should be a straight line, but here an almost piecewise straight line emerged, with two straight parts. This was somewhat disturbing, but the result prompted the zoologist to look carefully at the shells, and he found that what he had believed was a homogeneous sample was in fact shells from two different species. Actually, in this way a new species was discovered.

Another story about his distribution function has to do with the strength of welded joints. The Swedish company ASEA had some problems with certain welded details and decided to make a series of laboratory tests. Weibull was asked to help with the evaluation. He used his distribution function and found a straight line, and was content with that. However, when be reported the results at a meeting at ASEA, the chief-engineer asked him about the minimum strength. The Weibull distribution is cut off so that, for instance, the probability of finding an 18 year old male human being with the height 1 cm is exactly zero, and not finite as in other distributions. He told me that be felt very embarrassed when be suddenly noticed that his results did not indicate a minimum strength. The chief-engineer then said: `Do you imply, Professor, that our welds are so bad that they cannot stand any load at all?' Weibull did not know what to answer, but 'the head of the laboratory came to my rescue', he said, with the message: `Yesterday there was one sample that we could not test, because it broke before we could mount it in the testing machine'.

In 1966 he visited the Lund Institute of Technology and gave a lecture on his distribution function (Fig. 1).

Figure 1. Professor Weibull presenting his distribution function during a lecture at the Lund Institute of Technology on the 3rd ofNovember 1966.

In the mid seventies, when he was close to 90 years, I was invited by one of Waloddi Weibull's relatives to a dinner in Malmö, where he was the guest of honour. After the dinner we took a walk through the garden, and he described his new ideas about fatigue. If I remember correctly, it was at that time he told me about possibilities of reducing the number of fatigue tests needed for a statistical evaluation of fatigue strength, by using a slightly tapered test piece. Each test would then give two results: The number of cycles to failure and the position of the fracture. I was really impressed by his enthusiasm, his energy and his eagerness to communicate his ideas.

In 1976 he donated a sizeable collection of scientific works to my department at the Lund Institute of Technology. This treasure is still very much appreciated and is on permanent display at the department.

Waloddi Weibull was a great scientist and a very good friend. In my mind I can still hear him talking and I can remember his fine humour and his sharp logic. And each time I find his distribution function or his statistical theory of fracture mentioned, I recall Mediterranean sediment, Isola Bella and Bockamöllan - happy to have known a man like him.

REFERENCES

Davies, R.M. 1948. A critical study of the Hopkinson pressure bar. Phil. Trans. Roy. Soc. London A 240:375-457.
Irwin, GR. 1960. Size and shape effects on fracture of solids. In ASTM STP 283.
Weibull, W. 1938. Investigations into strength properties of brittle materials. Proceedings of the Royal Swedish Institute for Engineering Research No. 149.
Weibull, W. 1939a. A statistical theory of the strength of materials. Proceedings of the Royal Swedish Institute for Engineering Research No. 151.
Weibull, W. 1939b. The phenomenon of rupture in solids. Proceedings of the Royal Swedish Institute for Engineering Research No. 153.
Weibull, W. 1948. Waves in compressible media. Trans. Roy. Inst. Technology, Stockholm No. 18.
Weibull, W. 1951. A statistical distribution function of wide applicability. J. Appl. Mech. 18: 293297.
Weibull, W. 1961. Fatigue Testing and the Analysis of Results. New York: Pergamon Press.