Elasticity : Theory, Applications, and Numerics

Elasticity: Theory, Applications, and Numerics, Fourth Edition , continues its market-leading tradition of concisely presenting and developing the linear theory of elasticity, moving from solution methodologies, formulations, and strategies into applications of contemporary interest, such as fracture mechanics, anisotropic and composite materials, micromechanics, nonhomogeneous graded materials, and computational methods. Developed for a one- or two-semester graduate elasticity course, this new edition has been revised with new worked examples and exercises, and new or expanded coverage of areas such as treatment of large deformations, fracture mechanics, strain gradient and surface elasticity theory, and tensor analysis. Using MATLAB software, numerical activities in the text are integrated with analytical problem solutions. Online ancillary support materials for instructors include a solutions manual, image bank, and a set of PowerPoint lecture slides. Provides a thorough yet concise introduction to linear elasticity theory and applications Offers detailed solutions to problems of nonhomogeneous/graded materials Features a comparison of elasticity solutions with elementary theory, experimental data, and numerical simulations Includes online solutions manual and downloadable MATLAB code Cover......Page 1 Elasticity: Theory, Applications, and Numerics......Page 2 Copyright......Page 3 Preface......Page 4 Contents summary......Page 5 The subject......Page 6 Feedback......Page 7 Acknowledgments......Page 8 About the Author......Page 9 Part 1: Foundations and elementary applications......Page 10 1.1 Scalar, vector, matrix, and tensor definitions......Page 11 1.2 Index notation......Page 12 1.3 Kronecker delta and alternating symbol......Page 15 Spherical coordinates......Page 549 1.5 Cartesian tensors......Page 18 1.6 Principal values and directions for symmetric second-order tensors......Page 20 1.7 Vector, matrix, and tensor algebra......Page 24 1.8 Calculus of Cartesian tensors......Page 25 1.8.3 Green's theorem in the plane......Page 28 1.9 Orthogonal curvilinear coordinates......Page 29 1 . . Exercises......Page 34 2.1 General deformations......Page 38 B......Page 586 2.3 Strain transformation......Page 47 2.4 Principal strains......Page 48 2.6 Strain compatibility......Page 49 References......Page 55 References......Page 56 2 Exercises......Page 57 3.1 Body and surface forces......Page 63 3.2 Traction vector and stress tensor......Page 64 12.3.1 Plane strain......Page 380 3.4 Principal stresses......Page 69 3.5 Spherical, deviatoric, octahedral, and von Mises stresses......Page 72 15.6 Doublet mechanics......Page 505 3.7 Equilibrium equations......Page 77 3.8 Relations in curvilinear cylindrical and spherical coordinates......Page 79 3 Exercises......Page 82 4.1 Material characterization......Page 89 7.2 Plane stress......Page 152 4.3 Physical meaning of elastic moduli......Page 94 12.3.2 Plane stress......Page 95 10.4.1 Finite simply connected domains......Page 304 7.5 Airy stress function......Page 99 5 -Formulation and solution strategies......Page 103 Cartesian coordinates......Page 545 5.2 Boundary conditions and fundamental problem classifications......Page 104 5.3 Stress formulation......Page 109 5.4 Displacement formulation......Page 110 11.5 Plane deformation problems......Page 112 16.6 Boundary element formulation......Page 113 5.7.1 Direct method......Page 115 5.7.2 Inverse method......Page 116 9.3.4 Membrane analogy......Page 117 5.7.5 Approximate solution procedures......Page 118 Boundary element method......Page 119 References......Page 120 References......Page 122 5 Exercises......Page 123 6.1 Strain energy......Page 128 14.2 Plane problem of a hollow cylindrical domain under uniform pressure......Page 133 10.4.3 Infinite domains......Page 134 8.4 Example polar coordinate solutions......Page 188 6.4.2 Betti/Rayleigh reciprocal theorem......Page 135 6.4.3 Integral formulation of elasticity—Somigliana's identity......Page 136 6.5 Principle of virtual work......Page 137 6.6 Principles of minimum potential and complementary energy......Page 139 6.7 Rayleigh–Ritz method......Page 143 9.8 Flexure formulation......Page 273 6 Exercises......Page 146 8.1 Cartesian coordinate solutions using polynomials......Page 149 7.3 Generalized plane stress......Page 155 D......Page 588 11.6 Applications to fracture mechanics......Page 160 References......Page 162 10.8 Applications to fracture mechanics......Page 321 8 -Two-dimensional problem solution......Page 167 8.2 Cartesian coordinate solutions using Fourier methods......Page 178 8.2.1 Applications involving Fourier series......Page 181 15.3 Elasticity theory with distributed cracks......Page 339 8.3.1 General Michell solution......Page 186 8.4.1 Pressurized hole in an infinite medium......Page 190 8.4.2 Stress-free hole in an infinite medium under equal biaxial loading at infinity......Page 191 8.4.3 Biaxial and shear loading cases......Page 195 8.4.4 Quarter-plane example......Page 197 8.4.6 Half-space under uniform normal stress over x≤0......Page 198 8.4.7 Half-space under concentrated surface force system (Flamant problem)......Page 199 8.4.8 Half-space under a surface concentrated moment......Page 204 8.4.9 Half-space under uniform normal loading over −a≥x≥a......Page 205 8.4.10 Notch and crack problems......Page 209 8.4.11 Pure bending example......Page 211 8.4.12 Curved cantilever under end loading......Page 212 8.5 Simple plane contact problems......Page 221 14.6 Torsion problem......Page 460 References......Page 423 9 -Extension, torsion, and flexure of elastic cylinders......Page 244 9.2 Extension formulation......Page 245 Cartesian coordinates......Page 547 9.3.1 Stress–stress function formulation......Page 247 11.2.2 Three perpendicular planes of symmetry (orthotropic material)......Page 335 11.2.3 Axis of symmetry (transversely isotropic material)......Page 252 15.4 Micropolar/couple-stress elasticity......Page 256 E......Page 589 9.6 Torsion of cylinders with hollow sections......Page 267 9.7 Torsion of circular shafts of variable diameter......Page 270 10.9 Westergaard method for crack analysis......Page 277 9 Exercises......Page 281 Part 2: Advanced applications......Page 291 A......Page 292 10.2 Complex formulation of the plane elasticity problem......Page 299 10.3 Resultant boundary conditions......Page 302 10.4.2 Finite multiply connected domains......Page 305 D.5 Thin-walled cylindrical pressure vessels......Page 583 10.6 Plane and half-plane problems......Page 311 10.7 Applications using the method of conformal mapping......Page 316 References......Page 325 10 -Exercises......Page 326 11.1 Basic concepts......Page 331 15.2 Singular stress states......Page 333 11.2.1 Plane of symmetry (monoclinic material)......Page 334 11.2.4 Cubic symmetry......Page 337 11.2.5 Complete symmetry (isotropic material)......Page 338 11.4 Torsion of a solid possessing a plane of material symmetry......Page 341 11.4.1 Stress formulation......Page 342 11.4.2 Displacement formulation......Page 344 11.4.3 General solution to the governing equation......Page 345 11.5.1 Uniform pressure loading case......Page 357 11.7 Curvilinear anisotropic problems......Page 364 11.7.1 Two-dimensional polar-orthotropic problem......Page 365 11.7.2 Three-dimensional spherical-orthotropic problem......Page 368 11 Exercises......Page 372 15 -Micromechanics applications......Page 378 Spherical components from cylindrical......Page 551 Cartesian coordinates......Page 546 13.4 Papkovich–Neuber representation......Page 383 12.5 Stress function formulation......Page 384 12.6 Polar coordinate formulation......Page 387 12.7 Radially symmetric problems......Page 388 12.8 Complex variable methods for plane problems......Page 393 References......Page 400 12 Exercises......Page 401 13.1 Helmholtz displacement vector representation......Page 405 13.2 Lamé's strain potential......Page 406 13.3 Galerkin vector representation......Page 407 13.5 Spherical coordinate formulations......Page 416 13.6 Stress functions......Page 419 13.6.1 Maxwell stress function representation......Page 422 13 Exercises......Page 424 14.1 Basic concepts......Page 431 14.3 Rotating disk problem......Page 442 14.4 Point force on the free surface of a half-space......Page 447 16.5 FEM code applications......Page 456 References......Page 467 14 Exercises......Page 469 15.1 Dislocation modeling......Page 475 15.4.1 Two-dimensional couple-stress theory......Page 493 15.5 Elasticity theory with voids......Page 500 15.7 Higher gradient elasticity theories......Page 510 15 Exercises......Page 517 16 -Numerical finite and boundary element methods......Page 520 C.1 Getting started......Page 521 16.2 Approximating functions for two-dimensional linear triangular elements......Page 522 16.3 Virtual work formulation for plane elasticity......Page 524 16 Exercises......Page 543 Spherical coordinates......Page 548 B -Transformation of field variables between Cartesian, cylindrical, and spherical components......Page 550 Displacement transformation......Page 552 Stress transformation......Page 553 C.2 Examples......Page 554 References......Page 570 D -Review of mechanics of materials......Page 571 D.1 Extensional deformation of rods and beams......Page 572 D.2 Torsion of circular rods......Page 573 D.3 Bending deformation of beams under moments and shear forces......Page 574 D.4 Curved beams......Page 581 C......Page 587 G......Page 590 I......Page 591 M......Page 592 O......Page 593 P......Page 594 S......Page 595 T......Page 597 Z......Page 598 Back Cover......Page 599