Elasticity

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.