J.R.BARBER: ELASTICITY

 This web page relates to the second edition ISBN 1-4020-0964-X (Hardback) and 1-4020-0966-6 (Paperback). A third edition has just been published by Springer.

MATHEMATICA AND MAPLE

 In connection with the second edition, I am making available a set of files in Maple and Mathematica that facilitate the solution of boundary value problems in Elasticity. These can be accessed at the URL http://www-personal.umich.edu/~jbarber/elasticity/maple-and-mathematica2.html

If you want to explore this resource, I suggest you start by clicking on either Programming in Maple or Programming in Mathematica and then on to `Catalogue of Maple files' or `Catalogue of Mathematica files'. If you have never used these methods to solve problems, you will surprised how effective they are. You will however need to have Mathematica or Maple installed on your computer system. Additional resources for Mathematica solutions of some elasticity problems can be found at http://documents.wolfram.com/applications/structural/. In particular, Chapters 3 and 4 of this resource apply to problems from Chapters 17 and 16 respectively of `Elasticity'.

ERRATA

In the first printing of the second edition, pages 30 and 300 were incorrectly left blank. The correct text for these pages, which contains problems 2:3 to 2:8 and problems 21.4, 21.5 respectively, can be downloaded at `Page 30' and`Page 300' . If you find any other errors in the book or the electronic files, please let me know at jbarber@umich.edu. You can download my most recent list of errata at `Errata'.

PROBLEMS

 The second edition contains 223 end-of-chapter problems. These range from routine applications of the methods described in the chapter to quite challenging problems suitable for student projects. Most three dimensional problems are only really practicable when using Mathematica or Maple. Problems 24.8 and 24.9 are more difficult than they look! This is because the tractions on the spherical surface due to the point force(s) cannot be written in terms of a finite Fourier series in beta. This contrasts with the corresponding cylindrical problems 12.1, 12.2, 12.3. In fact, these tractions are still weakly (logarithmically) singular, though the dominant singularity associated with the point force has been removed. They can be removed by an infinite series of spherical harmonics, but the series will be rather slowly convergent. A more rapidly convergent solution can be obtained by first removing the logarithmic singularity.

SOLUTION MANUAL

 A solution manual is available, containing detailed solutions to all the problems, in some cases involving further discussion of the material and contour plots of the stresses etc. Bona fide instructors should contact me at jbarber@umich.edu if they need the manual and I will send it out as zipped .pdf files. Please tell me which edition of the book you have.

WILLIAMS' ASYMPTOTIC METHOD

 An analytical tool using MatLab has been developed for determining the nature of the stress and displacement fields near a fairly general singular point in linear elasticity. This is based on the method outlined in Section 11.2. For more information, click here.

TABLE OF CONTENTS

I: GENERAL CONSIDERATIONS

CHAPTER 1 INTRODUCTION

CHAPTER 2 EQUILIBRIUM AND COMPATIBILITY

II: TWO-DIMENSIONAL PROBLEMS

CHAPTER 3 PLANE STRAIN AND PLANE STRESS

CHAPTER 4 STRESS FUNCTION FORMULATION

CHAPTER 5 PROBLEMS IN RECTANGULAR COORDINATES

CHAPTER 6 END EFFECTS

CHAPTER 7 BODY FORCES

CHAPTER 8 PROBLEMS IN POLAR COORDINATES

CHAPTER 9 CALCULATION OF DISPLACEMENTS

CHAPTER 10 CURVED BEAM PROBLEMS

CHAPTER 11 WEDGE PROBLEMS

CHAPTER 12 PLANE CONTACT PROBLEMS

CHAPTER 13 FORCES, DISLOCATIONS AND CRACKS

CHAPTER 14 THERMOELASTICITY

CHAPTER 15 ANTIPLANE SHEAR

III: END LOADING OF THE PRISMATIC BAR

CHAPTER 16 TORSION OF A PRISMATIC BAR

CHAPTER 17 SHEAR OF A PRISMATIC BAR

IV: THREE-DIMENSIONAL PROBLEMS

CHAPTER 18 DISPLACEMENT FUNCTION SOLUTIONS

CHAPTER 19 THE BOUSSINESQ POTENTIALS

CHAPTER 20 THERMOELASTIC DISPLACEMENT POTENTIALS

CHAPTER 21 SINGULAR SOLUTIONS

CHAPTER 22 SPHERICAL HARMONICS

CHAPTER 23 CYLINDERS AND CIRCULAR PLATES

CHAPTER 24 PROBLEMS IN SPHERICAL COORDINATES

CHAPTER 25 AXISYMMETRIC TORSION

CHAPTER 26 FRICTIONLESS CONTACT

CHAPTER 27 THE BOUNDARY-VALUE PROBLEM

CHAPTER 28 THE PENNY-SHAPED CRACK

CHAPTER 29 THE INTERFACE CRACK

CHAPTER 30 THE RECIPROCAL THEOREM

PREFACE TO THE SECOND EDITION

 Since the first edition of this book was published, there have been major improvements in symbolic mathematical languages such as Maple and Mathematica and this has opened up the possibility of solving considerably more complex and hence interesting and realistic elasticity problems as classroom examples. It also enables 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. During the past 10 years I have developed files in Maple and Mathematica to facilitate this process, notably electronic versions of the Tables in the present Chapters 19 and 20 and of the recurrence relations for generating spherical harmonics. One purpose of this new edition is to make this electronic material available to the reader through the Kluwer website www.elasticity.org. I hope that readers will make use of this resource and report back to me any aspects of the electronic material that could benefit from improvement or extension. Some hints about the use of this material are contained in Appendix A. 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.

I have also taken the opportunity to include substantially more material in the second edition - notably three chapters on antiplane stress systems, including Saint-Venant torsion and bending and an expanded section on three-dimensional problems in spherical and cylindrical coordinate systems, including axisymmetric torsion of bars of non-uniform circular cross-section.

Finally, I have greatly expanded the number of end-of-chapter problems. 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. A full set of solutions to these problems is in preparation and will be made available to bona fide instructors on request.

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