Elasticity

The 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.

Part I GENERAL CONSIDERATIONS ; 1 Introduction; 1. 1 Notation for stress and displacement ; 1. 1. 1 Stress; 1. 1. 2 Index and vector notation and the summationconvention; 1. 1. 3 Vector operators in index notation; 1. 1. 4 Vectors, tensors and transformation rules; 1. 1. 5 Principal stresses and Von Mises stress ; ; 1. 1. 6 Displacement; ; 1. 2 Strains and their relation to displacements; ; 1. 2. 1 Tensile strain; ; 1. 2. 2 Rotation and shear strain; ; 1. 2. 3 Transformation of co ordinates; ; 1. 2. 4 Definition of shear strain; ; 1. 3 Stressstrain relations; ; 1. 3. 1 Lam e s content; 1. 3. 2 Dilatation and bulk modulus ; PROBLEMS; 2 Equilibrium and compatibility; 2. 1 Equilibrium equations ; 2. 2 Compatibility equations; 2. 2. 1 The significance of the compatibility equations ; 2. 3 Equilibrium equations in terms of displacements ; PROBLEMS; Part II TWODIMENSIONAL PROBLEMS ; 3 Plane strain and plane stress ; 3. 1 Plane strain ; 3. 1. 1 The corrective solution; 3. 1. 2 SaintVenant s principle ; 3. 2 Plane stress; 3. 2. 2 Relationship between plane stress and plane strain; PROBLEMS; 4 Stress function formulation ; 4. 1 The concept of a scalar stress function ; 4. 2 Choice of a suitable form ; 4. 3 The Airy stress function; 4. 3. 1 Transformation of co ordinates; 4. 3. 2 Nonzero body forces ; 4. 4 The governing equation ; 4. 4. 1 The compatibility condition ; 4. 4. 2 Method of solution ; 4. 4. 3 Reduced dependence on elastic constants ; PROBLEMS; 5 Problems in rectangular co ordinates ; 5. 1 Biharmonic polynomial functions ; 5. 1. 1 Second and third degree polynomials; 5. 2 Rectangular beam problems ; 5. 2. 1 Bending of a beam by an end load; 5. 2. 2 Higher order polynomials a general strategy; 5. 2. 3 Manual solutions symmetry considerations ; 5. 3 Fourier series and transform solutions ; 5. 3. 1 Choice of form ; 5. 3. 2 Fourier transforms; PROBLEMS; 6 End effects ; 6. 1 Decaying solutions; 6. 2 The corrective solution; 6. 2. 1 Separatedvariable solutions ; 6. 2. 2 The eigenvalue problem ; 6. 3 Other SaintVenant problems; 6. 4 Mathieu s solution; PROBLEMS; 7 Body forces ; 7. 1 Stress function formulation ; 7. 1. 1 Conservative vector fields; 7. 1. 2 The compatibility condition ; 7. 2 Particular cases ; 7. 2. 1 Gravitational loading ; 7. 2. 2 Inertia forces ; 7. 2. 3 Quasistatic problems; 7. 2. 4 Rigidbody kinematics ; 7. 3 Solution for the stress function ; 7. 3. 1 The rotating rectangular beam; 7. 3. 2 Solution of the governing equation; 7. 4 Rotational acceleration; 7. 4. 1 The circular disk ; 7. 4. 2 The rectangular bar; 7. 4. 3 Weak boundary conditions and the equation of motion; PROBLEMS; 8 Problems in polar co ordinates ; 8. 1 Expressions for stress components; 8. 2 Strain components; 8. 3 Fourier series expansion; ; 8. 3. 1 Satisfaction of boundary conditions; 8. 3. 3 Degenerate cases ; 8. 4 The Michell solution ; 8. 4. 1 Hole in a tensile field ; PROBLEMS; 9 Calculation of displacements ; 9. 1 The cantilever with an end load ; 9. 1. 1 Rigidbody displacements and end conditions ; 9. 1. 2 Deflection of the free end; 9. 2 The circular hole ; 9. 3 Displacements for the Michell solution ; 9. 3. 1 Equilibrium considerations ; 9. 3. 2 The cylindrical pressure vessel ; PROBLEMS; 10 Curved beam problems ; 10. 1 Loading at the ends; 10. 1. 1 Pure bending ; ; 10. 1. 2 Force transmission; 10. 2 Eigenvalues and eigenfunctions ; 10. 3 The inhomogeneous problem; 10. 3. 1 Beam with sinusoidal loading ; 10. 3. 2 The nearsingular problem; 10. 4 Some general considerations ; 10. 4. 1 Conclusions; PROBLEMS; 11 Wedge problems ; 11. 1 Power law tractions; 11. 1. 1 Uniform tractions ; 11. 1. 2 The rectangular body revisited; 11. 1. 3 More general uniform loading ; 11. 1. 4 Eigenvalues for the wedge angle ; 11. 2 Williams asymptotic method ; 11. 2. 1 Acceptable singularities ; 11. 2. 2 Eigenfunction expansion; 11. 2. 3 Nature of the eigenvalues; 11. 2. 4 The singular stress fields ; 11. 2. 5 Other geometries; 11. 3 General loading of the faces; PROBLEMS; 12 Plane contact problems ; 12. 1 Selfsimilarity ; 12. 2 The Flamant Solution; 12. 3 The halfplane; 12. 3. 1 The normal forceFy; 12. 3. 2 The tangential force Fx; 12. 3. 3 Summary; 12. 4 Distributed normal tractions; 12. 5 Frictionless contact problems; 12. 5. 1 Method of solution ; 12. 5. 2 The flat punch ; 12. 5. 3 The cylindrical punch (Hertz problem) ; 12. 6 Problems with two deformable bodies ; 12. 7 Uncoupled problems ; 12. 7. 1 Contact of cylinders ; 12. 8 Combined normal and tangential loading ; 12. 8. 1 Cattaneo and Mindlin s problem ; 12. 8. 2 Steady rolling: Carter s solution ; PROBLEMS; 13 Forces dislocations and cracks ; 13. 1 The Kelvin solution; 13. 1. 1 Body force problems; 13. 2 Dislocations ; 13. 2. 1 Dislocations in Materials Science ; 13. 2. 2 Similarities and differences ; 13. 2. 3 Dislocations as Green s functions ; 13. 2. 4 Stress concentrations ; 13. 3 Crack problems ; 13. 3. 1 Linear Elastic Fracture Mechanics ; 13. 3. 2 Plane crack in a tensile field ; 13. 3. 3 Energy release rate ; 13. 4 Method of images ; PROBLEMS; 14 Thermoelasticity ; 14. 1 The governing equation; 14. 2 Heat conduction; 14. 3 Steadystate problems; 14. 3. 1 Dundurs Theorem ; PROBLEMS; 15 Antiplane shear ; 15. 1 Transformation of coordinates; 15. 2 Boundary conditions; 15. 3 The rectangular bar ; 15. 4 The concentrated line force ; 15. 5 The screw dislocation ; PROBLEMS; Part III END LOADING OF THE PRISMATIC BAR ; 16 Torsion of a prismatic bar ; 16. 1 Prandtl s stress function; 16. 1. 1 Solution of the governing equation; 16. 2 The membrane analogy ; 16. 3 Thinwalled open sections ; 16. 4 The rectangular bar ; 16. 5 Multiply connected (closed) sections ; 16. 5. 1 Thinwalled closed sections ; PROBLEMS; 17 Shear of a prismatic bar ; 17. 1 The semiinverse method ; 17. 2 Stress function formulation ; 17. 3 The boundary condition; 17. 3. 1 Integrability ; 17. 3. 2 Relation to the torsion problem ; 17. 4 Methods of solution; 17. 4. 1 The circular bar; 17. 4. 2 The rectangular bar ; PROBLEMS; Part IV COMPLEX VARIABLE FORMULATION ; 18 Preliminary mathematical results ; 18. 1 Holomorphic functions ; 18. 2 Harmonic functions; 18. 3 Biharmonic functions ; 18. 4Expressing real harmonic and biharmonic functions incomplex form ; 18. 4. 1 Biharmonic functions ; 18. 5 Line integrals ; 18. 5. 1 The residue theorem; 18. 5. 2 The Cauchy integral theorem ; 18. 6 Solution of harmonic boundary value problems ; 18. 6. 1 Direct method for the interior problem for a circle ; 18. 6. 2 Direct method for the exterior problem for a circle; 18. 6. 3 The half plane ; 18. 7 Conformal mapping; PROBLEMS; 19 Application to elasticity problems ; 19. 1 Representation of vectors; 19. 1. 1 Transformation of co ordinates; 19. 2 The antiplane problem ; 19. 2. 1 Solution of antiplane boundaryvalue problems ; 19. 3 Inplane deformations; 19. 3. 1 Expressions for stresses ; 19. 3. 2 Rigidbody displacement ; 19. 4 Relation between the Airy stress function and the complexpotentials ; 19. 5 Boundary tractions ; 19. 5. 1 Equilibrium considerations ; 19. 6 Boundaryvalue problems; 19. 6. 1 Solution of the interior problem for the circle ; 19. 6. 2 Solution of the exterior problem for the circle ; 19. 7 Conformal mapping for inplane problems ; 19. 7. 1 The elliptical hole ; PROBLEMS; Part V THREE DIMENSIONAL PROBLEMS ; 20 Displacement function solutions ; 20. 1 The strain potential ; 20. 2 The Galerkin vector ; 20. 3 The PapkovichNeuber solution ; 20. 3. 1 Change of co ordinate system; 20. 4 Completeness and uniqueness; 20. 4. 1 Methods of partial integration; 20. 5 Body forces; 20. 5. 1 Conservative body force fields 20. 5. 2 Nonconservative body force fields PROBLEMS; 21 The Boussinesq potentials; 21. 1 Solution A : The strain potential; 21. 2 Solution B 21. 3; Solution E : Rotational deformation; 21. 4 Other co ordinate systems; 21. 4. 1 Cylindrical polar co ordinates; 21. 4. 2 Spherical polar co ordinates; 21. 5 Solutions obtained by superposition; 21. 5. 1 Solution F : Frictionless isothermal contact problems; 21. 5. 2 Solution G: The surface free of normal traction; 21. 6 A threedimensional complex variable solution; PROBLEMS; 22 Thermoelastic displacement potentials; 22. 1 Plane problems; 22. 1. 1 Axisymmetricproblems for the cylinder ; 22. 1. 2 Steadystate plane problems ; 22. 1. 3 Heat flow perturbed by a circular hole ; 22. 1. 4 Plane stress ; 22. 2 The method of strain suppression; 22. 3 Steadystate temperature : Solution T; 22. 3. 1 Thermoelastic plane stress; PROBLEMS; 23 Singular solutions ; 23. 1 The source solution; 23. 1. 1 The centre of dilatation; 23. 1. 2 The Kelvin solution ; 23. 2 Dimensional considerations ; 23. 2. 1 The Boussinesq solution; 23. 3 Other singular solutions ; 23. 4 Image methods; 23. 4. 1 The tractionfree half space; PROBLEMS; 24 Spherical harmonics ; 24. 1 Fourier series solution; 24. 2 Reduction to Legendre s equation; 24. 3 Axisymmetric potentials and Legendre polynomials ; 24. 3. 1 Singular spherical harmonics; 24. 3. 2 Special cases; 24. 4 Nonaxisymmetric harmonics ; 24. 5 Cartesian and cylindrical polar co ordinates; 24. 6 Harmonic potentials with logarithmic terms; 24. 6. 1 Logarithmic functions for cylinder problems ; 24. 7 Nonaxisymmetric cylindrical potentials ; 24. 8 Spherical harmonics in complex notation ; 24. 8. 1 Bounded cylindrical harmonics; 24. 8. 2 Singular cylindrical harmonics ; PROBLEMS; 25 Cylinders and circular plates ; 25. 1 Axisymmetric problems for cylinders; 25. 1. 1 The solid cylinder ; 25. 1. 2 The hollow cylinder; 25. 2 Axisymmetric circular plates; 25. 2. 1 Uniformly loaded plate on a simple support; 25. 3 Nonaxisymmetric problems ; 25. 3. 1 Cylindrical cantilever with an end load; ; PROBLEMS; 26 Problems in spherical co ordinates; 26. 1 Solid and hollow spheres ; 26. 1. 1 The solid sphere in torsion; 26. 1. 2 Spherical hole in a tensile field; 26. 2 Conical bars ; 26. 2. 1 Conical bar transmitting an axial force; 26. 2. 2 Inhomogeneous problems ; 26. 2. 3 Nonaxisymmetric problems ; PROBLEMS; 27 Axisymmetric torsion ; 27. 1 The transmitted torque ; 27. 2 The governing equation ; 27. 3 Solution of the governing equation; 27. 4 The displacement field ; 27. 5 Cylindrical and conical bars; 27. 5. 1 The centre of rotation; 27. 6 The Saint Venant problem; PROBLEMS; 28 Theprismatic bar ; 28. 1 Power series solutions; 28. 1. 1 Superposition by differentiation ; 28. 1. 2 The problems
_{P0 and P1 Properties of the solution to Pm ;}
28. 2 Solution of
_{Pm by integration ;}
28. 3 The integration process ; 28. 4 The twodimensional problem ; 28. 5. 1The corrective antiplane solution ; 28. 5. 2 The circular bar; 28. 6 The corrective inplane solution; 28. 7 Corrective solutions using real stress functions ; 28. 7. 1 Airy function ; 28. 7. 2 Prandtl function ; 28. 8 Solution procedure ; 28. 9 Example ; 28. 9. 1 Problem ; 28. 9. 3 End conditions; PROBLEMS; 29 Frictionless contact ; 29. 1 Boundary conditions; 29. 1. 1 Mixed boundaryvalue problems; 29. 2 Determining the contact area ; 29. 3 Contact problems involving adhesive forces ; 30 The boundaryvalue problem ; 30. 1 Hankel transform methods; 30. 2 Collins Method ; 30. 2. 1 Indentation by a flat punch; 30. 2. 2 Integral representation; 30. 2. 3 Basic forms and surface values ; 30. 2. 4 Reduction to an Abel equation; 30. 2. 5 Smooth contact problems; 30. 2. 6 Choice of form; 30. 3 Nonaxisymmetric problems ; 30. 3. 1 The full stress field ; PROBLEMS; 31 The pennyshaped crack ; 31. 1 The pennyshaped crack in tension ; 31. 2 Thermoelastic problems ; PROBLEMS; 32 The interface crack ; 32. 1 The uncracked interface ; 32. 2 The corrective solution; 32. 2. 1 Global conditions; 32. 2. 2 Mixed conditions ; 32. 3 The pennyshaped crack in tension ; 32. 3. 1 Reduction to a single equation ; 32. 3. 2 Oscillatory singularities ; 32. 4 The contact solution ; 32. 5 Implications for Fracture Mechanics 33 Variational methods ; 33. 1 Strain energy ; 33. 1. 1 Strain energy density ; 33. 2 Conservation of energy ; 33. 3 Potential energy of the external forces; 33. 4 Theorem of minimum total potential energy ; 33. 5 Approximate solutions the RayleighRitz method ; 33. 6 Castigliano s second theorem; 33. 7 Approximations using Castigliano s second theorem; 33. 7. 1 The torsion problem ; 33. 7. 2 The inplane problem; 33. 8 Uniqueness and existence of solution ; 33. 8. 1 Singularities ; PROBLEMS; 34 The reciprocal theorem ; 34. 1 Maxwell s Theorem ; 34. 2 Betti s Theorem; 34. 3 Use of the theorem ; 34. 3. 1 A tilted punch problem; 34. 3. 2 Indentation of a halfspace; 34. 4 Thermoelastic problems ; PROBLEMS; A Using Maple and Mathematica.