MATHEMATICA AND MAPLE

PROBLEMS

SOLUTION MANUAL

WILLIAMS' ASYMPTOTIC METHOD

PREFACE TO THE THIRD EDITION

The first edition of this book, published in 1992, was based on a one semester graduate course on Linear Elasticity that I have taught at the University of Michigan since 1983. In two subsequent revisions, the amount of material has almost doubled and the character of the book has necessarily changed, but I remain committed to my original objective of writing for those who wish to find the solution of specific practical engineering problems. With this in mind, I have endeavoured to keep to a minimum any dependence on previous knowledge of Solid Mechanics, Continuum Mechanics or Mathematics. Most of the text should be readily intelligible to a reader with an undergraduate background of one or two courses in elementary Mechanics of Materials and a rudimentary knowledge of partial differentiation. Cartesian tensor notation and the index convention are used in a few places to shorten the derivation of some general results, but these sections are carefully explained, so as to be self-explanatory.

Modern practitioners of Elasticity are necessarily influenced by developments in numerical methods, which promise to solve all problems with no more information about the subject than is needed to formulate the description of a representative element of material in a relatively simple state of stress. As a researcher in Solid Mechanics, with a primary interest in the physical behaviour of the systems I am investigating, rather than in the mathematical structure of the solutions, I have frequently had recourse to numerical methods of all types and have tended to adopt the pragmatic criterion that the best method is that which gives the most convincing and accurate result in the shortest time. In this context, `convincing' means that the solution should be capable of being checked against reliable closed-form solutions in suitable limiting cases and that it is demonstrably stable and in some sense convergent. Measured against these criteria, the `best' solution to many practical problems is often not a direct numerical method, such as the finite element method, but rather one involving some significant analytical steps before the final numerical evaluation. This is particularly true in three-dimensional problems, where direct numerical methods are extremely computer-intensive if any reasonably accuracy is required, and in problems involving infinite or semi-infinite domains, discontinuities, bonded or contacting material interfaces or theoretically singular stress fields. By contrast, I would immediately opt for a finite element solution of any two-dimensional problem involving finite bodies with relatively smooth contours, unless it happened to fall into the (surprisingly wide) class of problems to which the solution can be written down in closed form. The reader will therefore find my choice of topics significantly biassed towards those fields identified above where analytical methods are most useful.

As in the second edition, I encourage the reader to become familiar with the use of symbolic mathematical languages such as Maple and Mathematica, since these tools open up the possibility of solving considerably more complex and hence interesting and realistic elasticity problems. They also enable the student to focus on the formulation of the problem (e.g. the appropriate governing equations and boundary conditions) rather than on the algebraic manipulations, with a consequent improvement in insight into the subject and in motivation. Finally, they each posess post-processing graphics facilities that enable the user to explore important features of the resulting stress state. The reader can access numerous files for this purpose at the website for this volume on www.springer.com or at my University of Michigan homepage http://www-personal.umich.edu/~jbarber/elasticity/book.html, including the solution of sample problems, electronic versions of the tables in Chapters 21,22, and algorithms for the generation of spherical harmonic potentials. Some hints about the use of this material are contained in Appendix A, and more detailed tips about programming are included at the above websites. Those who have never used Maple or Mathematica will find that it takes only a few hours of trial and error to learn how to write programs to solve boundary-value problems in elasticity.

This new edition contains four additional chapters, including two concerned with the use of complex-variable methods in two-dimensional elasticity. In keeping with the style of the rest of the book, I have endeavoured to present this material in a such a way as to be usable by a reader with minimal previous experience of complex analysis who wishes to solve specific elasticity problems. This necessarily involves glossing over some of the finer points of the underlying mathematics. The reader wishing for a more complete and rigorous treatment will need to refer to more specialized works on this topic. I have emphasised the relation between the complex and real (Airy and Prandtl) stress functions, including algorithms for obtaining the complex function for a stress field for which the real stress function is already known. The complex variable methods and notation developed in Chapter 19 are also used in the development of a hierarchical treatment of three-dimensional problems for prismatic bars of fairly general cross-section in Chapter 28. The other major addition is a new chapter on variational methods, including the use of the Rayleigh-Ritz method and Castigliano's second theorem in developing approximate solutions to elasticity problems. The new edition contains numerous additional end-of-chapter problems. As with previous editions, a full set of solutions to these problems is available to bona fide instructors on request to the author. Some of these problems are quite challenging, indeed several were the subject of substantial technical papers within the not too distant past, but they can all be solved in a few hours using Maple or Mathematica. Many texts on Elasticity contain problems which offer a candidate stress function and invite the student to `verify' that it defines the solution to a given problem. Students invariably raise the question `How would we know to choose that form if we were not given it in advance?' I have tried wherever possible to avoid this by expressing the problems in the form they would arise in Engineering --- i.e. as a body of a given geometry subjected to prescribed loading. This in turn has required me to write the text in such a way that the student can approach problems deductively. I have also generally opted for explaining difficulties that might arise in an `obvious' approach to the problem, rather than steering the reader around them in the interests of brevity. I have taken this opportunity to correct the numerous typographical errors in the second edition, but no doubt despite my best efforts, the new material will contain more. Please communicate any errors to me. As in previous editions, I would like to thank my graduate students and more generally scientific correspondents worldwide whose questions continue to force me to re-examine my knowledge of the subject. I am also grateful to Professor John Dundurs for permission to use Table 9.1 and to the Royal Society of London for permission to reproduce Figures 13.2, 13.3. Vikram Gavini, David Hills and Alexander Korsunsky were kind enough to read drafts of the complex-variable chapters and made useful suggestions on presentation, but they are in no way responsible for my rather idiosyncratic approach to this subject.

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