Asymptopia September 1999

Thanks to a donation from Dr Gordon Moore and his wife Betty, the University has given the go-ahead for the construction of a new 7.5 million pound library for the physical Sciences, technology and mathematics adjacent to the Centre for Mathematical Sciences. Dr Moore is the Chairman Emeritus of Intel Corporation, which he co-founded in 1968. In the mid 1960s, while director of development at Fairchild Semiconductor, he made astonishingly accurate predictions about the growth of computing power, which together became known as `Moore's Law' which, combined with the microprocessor, first introduced by Intel in 1971, is the foundation for today's microcomputer revolution. Artist's impression of the new Betty and Gordon Moore Library   The Betty and Gordon Moore Library will bring together collections of scientific resources currently scattered across Cambridge and will also provide high quality links to electronic data sources and electronic retrieval systems, which will help make it a new intellectual focus for Cambridge science and technology. The new library will be part of Cambridge University Library, which is a legal deposit library and so entitled to claim a copy of every book and journal published in the UK. The building that will house the new Library has already been designed as part of the overall development of the Clarkson Road site. The Library will be circular in form, unique in Cambridge but with precedents in other times, such as the library designed by Thomas Jefferson at the University of Virginia in 1817. The new Library will provide a home for Professor Stephen Hawking's papers and electronic archive, which he has offered to donate to the University. Initially, these papers will include Professor Hawking's hand-written material dating from before 1973 and an early draft of A Brief History of Time. In the longer term, the Library will provide a digitised archive for more recent material stored in electronic media by Professor Hawking. The archive will continue a great Cambridge tradition of preserving the papers of famous scientists for future study.

The buildings that will form the new Centre for Mathematical Sciences are being constructed principally from reinforced concrete. The structure of the pavilions and the central core are highly co-ordinated with the building services and environ-mental engineering. The concrete structure is designed to exploit its thermal capacity and this is reflected in the architecture of the buildings. From the earliest concept designs, it was recognised that something special should be created for the structure of roof which will cover the central core.The architect's design concept produced a central sunken core area, topped by a grass roof. The ground to the sides of the core is shaped, so that natural daylight can be brought into the lower level. At the upper ground level, the roof covers the large meeting and circulation area spanning a maximum of 21 metres. For a building, the geometry is complex and the loading from the grass roof is high. For ease of waterproofing, the roof slab was to be constructed in concrete, but a number of options were considered for the support structure. The choice lay between an arch or a beam structure. An arch form resists the applied loads by a combination of axial compression and bending moment. This is in contrast to a beam which only resists the applied forces by flexing and developing bending moment. Clearly, the arch form will deflect very much less than a beam for a particular applied load. A common problem with using arch structures for building roofs is that minimum headroom is needed at the edge. This means that the springing point needs to be raised which creates a problem in carrying the arch thrust down to the floor level. A number of structural schemes were considered, including a balanced cantilever in structural steel and a portal frame in reinforced concrete. However, after weighing up the advantages and disadvantages a buttressed arch structure was chosen. Buro Happold are the civil and structural engineering consultants for the Centre for Mathematical Sciences. This information is issued to the contractor (John Laing Construction Ltd) who, through the appointment of specialist sub-contractors, constructs the building.   The geometry of the central core has not been easy to construct. The space tapers and the arches are arcs of circles but each is of a different radius. The problem of the high point loads and the provision of tolerance was solved by using high grade cast iron castings at the arch springing point with a pin connection enabling the loads from the arches to be accurately analysed. Iron castings are commonly used in mechanical engineering but their use in construction represented a particular challenge. So that they can carry the intense loads at the end of the arches, the cast iron pin assemblies were made to an extremely low tolerance. By using a steel frame as a skeleton in the buttresses, the attachment points for the castings could be very accurately positioned. The castings were made by BAS Castings Ltd. All the Steel Arches were successfully erected by the steelwork sub-contractor (Hawk Engineers) just before Christmas 1998. The concrete roof was cast in situ and the waterproofing and planting system will be a specialist system provided by Euroroof Ltd.

Solid Mechanics is a small but vibrant activity within the Department of Applied Mathematics and Theoretical Physics. Its basic concern is the analysis of the internal forces in solid materials and structures, and assessment of their consequences. The most serious consequence is failure of the structure, which is at best inconvenient and at worst can be life-threatening. The subject is massive and this note can discuss only a couple of examples.

Photograph courtesy of EQE International

It may not be a great comfort to know that virtually all structures -- including the aeroplanes in which we all fly -- contain cracks! There is, however, a well-defined subject called fracture mechanics. A crack does no harm so long as it does not grow catastrophically. To explain, it is necessary to talk about stress. This is force divided by area. Its significance may be related to common experience by considering whether you would prefer to have your foot trodden on by a heavy man wearing big shoes, or by a diminutive lady wearing stiletto heels: the former applies the greater force but it is spread over a bigger area and so is less damaging. The paradox with a crack is that very high stresses are unavoidably generated near its edge. The simplest theory actually predicts that these stresses become infinite. Why, therefore, is the material not pulled apart? More refined theory (which involves the analysis of a set of nonlinear partial differential equations) shows why the stresses near the crack edge in fact remain finite. Thus, it explains why the crack need not extend, until the applied loading is sufficiently intense. More important, it shows the precise way in which the simplest theory becomes modified by the nonlinearity, and provides the engineer with an easily-- parameter -- the `stress intensity factor' -- which should not be allowed to exceed a critical value. Even theory at this level suffices for much of engineering design and analysis, and is employed, for instance, in conjunction with programmes of inPservice inspection of airframes and aero engines to decide when a component should be removed from service. Modern materials are often so tough that the assumptions underlying the simple theory do not apply: they withstand much larger stresses and more sophisticated theory is needed. Research is still progressing in some areas but much understanding has already been acquired and is exploited in practice. A problem under active study at present is the dynamic propagation of a crack. The equations describing propagation are exceptionally difficult to solve, but noteworthy progress has recently been made, and the resolution of questions such as why cracks tend to branch when they run fast is in prospect.

Different examples of failure are provided by microelectronic components. A silicon chip is often sealed within a plastic block and it is important that this plastic package should remain bonded to the chip. The package is nearly always under stress, introduced upon cooling from its fabrication temperature, so decohesion is a possibility. Such devices are inspected after manufacture, and those with cracks are discarded. The problem here is the development of efficient and economic methods of inspection. Ultrasound is used and the problem is to identify those characteristics of the returned signal that diagnose the state at the chip-package interface. Theory underpinning a novel use (at the National Institute of Standards and Technology, Washington, DC) of an Tacoustic microscopeU, in which amplitude rather than travel time is mapped, is under development in the Department. There are other semiconductor devices in which stress is `designed in', to achieve a desired electronic performance. A thin layer consisting of a mixture of silicon and germanium may be deposited on a substrate of pure silicon, for example. The atoms tend to line up, even though the germanium atoms would prefer more space, so the layer is under stress. Failure in this context means relaxation of the stress by some means: the device remains intact but its performance alters. Mechanisms for stress relaxation (involving the introduction of dislocations), and means for its avoidance, have been clarified through work within the Department. Another development was the first satisfactory explanation of the relation between elastic mismatch and electronic performance of `quantum wire' structures.

Almost all of this work is inter-disciplinary in nature: the theory can be done in the Department but it has to address the right problems! Often this involves collaboration with some outside organisation. There are, however, also strong interdepartmental links within Cambridge, and one quite recent development has been the establishment of the Cambridge Centre for Micromechanics, whose participants come mainly from DAMTP, and the Departments of Engineering and Materials Science. Present concerns of the Centre include analysis of the shock-absorbing capacity of metal foams and (avoidance of) the failure by micro-buckling of fibre-reinforced composites. The Centre provides, internally, a forum for collaboration and, externally, a strong interdisciplinary group with the potential to tackle challenging problems in materials technology.

The Dill Faulkes Educational Trust has made an exceptionally generous donation of one million pounds to the Isaac Newton Institute for Mathematical Sciences, to finance construction of the Gatehouse on the Clarkson Road Site.

The Gatehouse is strategically placed at the east end of the site, between the Newton Institute and the new library. It will provide the main access to the site for all pedestrians and cyclists approaching from the Grange Road end of Clarkson Road. It is on the main east-west axis of the site, and is of central importance from an architectural point of view. The Architects have responded to the challenge with a building of exciting design. It contains a semi-circular Seminar Room, accommodating 50, at first floor level, and three `pi-by-three' offices at ground floor level. The building is completed by staircase and lift leading to a landing over the archway entrance to the site.

The General Board has agreed to assign the Gatehouse to the Newton Institute, with the proviso that the Seminar Room may, when circumstances permit, occasionally be made available for Departmental use. The Newton Institute is the national institute for the mathematical sciences. It was established in 1992, and was this year awarded one of the prestigious Queen's Anniversary Prizes for Higher and Further Education, in recognition of its achievements over the last seven years. The Institute runs visitor research programmes on selected themes which attract leading specialists world-wide, for periods of up to six months, for intense interactive research. The current programmes are on Complexity and Information Theory, on Structure Formation in the Universe, and on Solid Mechanics and Materials Science. These programmes are all deliberately interdisciplinary in character, and demonstrate the wide applicability of mathematics to a very great range of problems of public interest and concern.

Our newsletter now has a new format and a new name. Mathematicians (including those who progressed no further than `O' levels!) will recall the term asymptote as describing a curve that gets ever closer to a straight line. The point at which they finally meet is asymptopia and we are hoping that the plan for a new home for mathematics in Cambridge is reaching its own Asymptopia.