## ElasticityThe subject of Elasticity can be approached from several points of view, - pending on whether the practitioner is principally interested in the mat- matical structure of the subject or in its use in engineering applications and, in the latter case, whether essentially numerical or analytical methods are envisaged as the solution method. My ?rst introduction to the subject was in response to a need for information about a speci?c problem in Tribology. As a practising Engineer with a background only in elementary Mechanics of - terials, I approached that problem initially using the concepts of concentrated forces and superposition. Today, with a rather more extensive knowledge of analytical techniques in Elasticity, I still ?nd it helpful to go back to these roots in the elementary theory and think through a problem physically as well as mathematically, whenever some new and unexpected feature presents di?culties in research. This way of thinking will be found to permeate this book. My engineering background will also reveal itself in a tendency to work examples through to ?nal expressions for stresses and displacements, rather than leave the derivation at a point where the remaining manipulations would be mathematically routine. The ?rst edition of this book, published in 1992, was based on a one semester graduate course on Linear Elasticity that I have taught at the U- versity of Michigan since 1983. |

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### Contents

2 | |

3 | |

2 EQUILIBRIUM AND COMPATIBILITY | 25 |

Part II | 34 |

3 PLANE STRAIN AND PLANE STRESS | 37 |

4 STRESS FUNCTION FORMULATION | 45 |

5 PROBLEMS IN RECTANGULARCOORDINATES | 54 |

6 END EFFECTS | 77 |

18 PRELIMINARY MATHEMATICALRESULTS | 273 |

19 APPLICATION TO ELASTICITYPROBLEMS | 293 |

Part V | 318 |

20 DISPLACEMENT FUNCTION SOLUTIONS | 319 |

21 THE BOUSSINESQ POTENTIALS | 333 |

22 THERMOELASTIC DISPLACEMENTPOTENTIALS | 347 |

23SINGULAR SOLUTIONS | 362 |

24 SPHERICAL HARMONICS | 377 |

7 BODY FORCES | 91 |

8 PROBLEMS IN POLAR COORDINATES | 109 |

9 CALCULATION OF DISPLACEMENTS | 123 |

10 CURVED BEAM PROBLEMS | 135 |

11 WEDGE PROBLEMS | 149 |

12 PLANE CONTACT PROBLEMS1 | 171 |

13 FORCES DISLOCATIONS AND CRACKS | 199 |

14 THERMOELASTICITY | 219 |

15 ANTIPLANE SHEAR | 226 |

Part III | 238 |

16 TORSION OF A PRISMATIC BAR | 241 |

17 SHEAR OF A PRISMATIC BAR1 | 259 |

Part IV | 270 |

25 CYLINDERS AND CIRCULAR PLATES | 391 |

26 PROBLEMS IN SPHERICALCOORDINATES | 405 |

27 AXISYMMETRIC TORSION | 419 |

28 THE PRISMATIC BAR | 429 |

29FRICTIONLESS CONTACT | 449 |

30 THE BOUNDARYVALUE PROBLEM | 459 |

31 THE PENNYSHAPED CRACK | 479 |

32 THE INTERFACE CRACK | 487 |

33 VARIATIONAL METHODS | 499 |

34 THE RECIPROCAL THEOREM | 517 |

A USING MAPLE AND MATHEMATICA | 529 |

531 | |

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### Common terms and phrases

antiplane applied axisymmetric beam biharmonic biharmonic equation body force boundary conditions boundary-value problem Chapter coefficients complete stress field complex contact area contact problems coördinate system corrective solution corresponding crack cross-section curved cylinder defined determine dislocation displacement field distribution edges eigenvalue elastic equilibrium equations example expressions Figure Find the stress force F Fourier Fourier series half-space harmonic function hence hole holomorphic function identically in-plane integral J.R. Barber ln(r loaded Mechanics of Materials method non-zero normal traction obtain particular solution perturbed plane strain plane stress polar coördinates polynomial punch radius rectangular region result rigid rotation satisfy Science+Business Media B.V. shear stress shear traction singular solve spherical Springer Science+Business Media stress and displacement stress components Substituting superposing surface symmetric temperature tensile tensile stress theorem torque torsion traction-free two-dimensional uniform vector weak boundary conditions wedge zero