Elasticity

Description

Content: 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 coordinates 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 coordinates 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 coordinates 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 coordinates 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 force Fy 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 18.4 Expressing 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 coordinates 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 coordinate 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 coordinate systems 21.4.1 Cylindrical polar coordinates 21.4.2 Spherical polar coordinates 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 Axisymmetric problems 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 coordinates 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 coordinates 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 The prismatic 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.