
At a Glance
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| Preface | p. xv |
| Tensors | p. 1 |
| First Steps with Tensors | p. 1 |
| Multilinear forms | p. 1 |
| Linear mapping | p. 2 |
| Multilinear form | p. 2 |
| Dual space, vectors and covectors | p. 3 |
| Dual space | p. 3 |
| Expression of a covector | p. 3 |
| Einstein summation convention | p. 5 |
| Change of basis and cobasis | p. 7 |
| Tensors and tensor product | p. 10 |
| Tensor product of multilinear forms | p. 10 |
| Tensor of type (0/1) | p. 11 |
| Tensor of type (1/0) | p. 12 |
| Tensor of type (0/2) | p. 14 |
| Tensor of type (2/0) | p. 16 |
| Tensor of type (1/1) | p. 18 |
| Tensor of type (q/p) | p. 20 |
| Symmetric and antisymmetric tensors | p. 22 |
| Operations on Tensors | p. 25 |
| Tensor algebra | p. 25 |
| Addition of tensors | p. 25 |
| Multiplication of a tensor by a scalar | p. 25 |
| Tensor multiplication | p. 26 |
| Contraction and tensor criteria | p. 27 |
| Contraction | p. 27 |
| Tensor criteria | p. 31 |
| Euclidean Vector Space | p. 33 |
| Pre-Euclidean vector space | p. 33 |
| Scalar multiplication and pre-Euclidean space | p. 33 |
| Fundamental tensor | p. 34 |
| Canonical isomorphism and conjugate tensor | p. 35 |
| Canonical isomorphism | p. 35 |
| Conjugate tensor and reciprocal basis | p. 37 |
| Covariant and contravariant representations of vectors | p. 40 |
| Representation of tensors of order 2 and contracted products | p. 42 |
| Euclidean vector spaces | p. 45 |
| Exterior Algebra | p. 49 |
| p-forms | p. 49 |
| Definition of a p-form | p. 49 |
| Exterior product of 1-forms | p. 50 |
| Expression of a p-form | p. 52 |
| Exterior product of p-forms | p. 56 |
| Exterior algebra | p. 57 |
| q-vectors | p. 59 |
| Point Spaces | p. 63 |
| Point space and natural frame | p. 63 |
| Point space | p. 63 |
| Coordinate system and frame of reference | p. 64 |
| Natural frame | p. 66 |
| Tensor fields and metric element | p. 69 |
| Transformations of curvilinear coordinates | p. 69 |
| Tensor fields | p. 73 |
| Metric element | p. 74 |
| Christoffel symbols | p. 75 |
| Definition of Christoffel symbols | p. 75 |
| Ricci identities and Christoffel formulae | p. 78 |
| Absolute differential, Covariant derivative, Geodesic | p. 80 |
| Absolute differential of a vector, covariant derivatives | p. 80 |
| Absolute differential of a tensor, covariant derivatives | p. 83 |
| Geodesic and Euler's equations | p. 85 |
| Absolute derivative of a vector (along a curve) | p. 86 |
| Volume form and adjoint | p. 88 |
| Volume form | p. 88 |
| Adjoint | p. 91 |
| Differential operators | p. 92 |
| Gradient | p. 92 |
| Divergence | p. 99 |
| Curl | p. 101 |
| Laplacian | p. 103 |
| Exercises | p. 106 |
| Lagrangian and Eulerian Descriptions | p. 147 |
| Lagrangian Description | p. 147 |
| Configuration | p. 147 |
| Deformation and Lagrangian Description | p. 148 |
| Flow and hypotheses of continuity | p. 152 |
| Trajectories | p. 153 |
| Streakline | p. 154 |
| Velocity and acceleration of a particle | p. 155 |
| Abstract configuration | p. 156 |
| Eulerian Description | p. 157 |
| Definition; Comparison between L- and E-descriptions | p. 157 |
| Trajectory and velocity | p. 158 |
| Streamline | p. 161 |
| Steady motion | p. 163 |
| Exercises | p. 165 |
| Deformations | p. 171 |
| Homogeneous Transformation | p. 172 |
| Definition of homogeneous transformations | p. 172 |
| Convective transport | p. 174 |
| Convective transport of a vector | p. 174 |
| Convective transport of a volume | p. 174 |
| Simple shear | p. 176 |
| Cauchy-Green deformation tensor and stretch | p. 177 |
| (Right) Cauchy-Green deformation tensor | p. 177 |
| Stretch | p. 180 |
| Shear angle | p. 181 |
| Principal stretches | p. 182 |
| Finite strain tensor | p. 184 |
| Polar decomposition | p. 186 |
| Pure stretch and rotation | p. 186 |
| Euler-Almansi strain tensor | p. 190 |
| Rigid body transformation | p. 191 |
| Tangential Homogeneous Transformation | p. 193 |
| Deformation gradient | p. 193 |
| Homogeneous transformations of elements | p. 196 |
| Transport of vectors, volume deformation, and area deformation | p. 196 |
| Stretches | p. 199 |
| Strain | p. 201 |
| Displacement and gradient | p. 203 |
| Material displacement gradient | p. 204 |
| Spatial displacement gradient | p. 206 |
| Curvilinear coordinate system | p. 208 |
| Infinitesimal Transformation | p. 210 |
| Tensor notions relating to infinitesimal transformations | p. 211 |
| Compatibility conditions | p. 216 |
| Rigid body transformation | p. 220 |
| Exercises | p. 222 |
| Kinematics of Continua | p. 263 |
| Lagrangian Kinematics | p. 263 |
| Homogeneous transformation motion | p. 264 |
| General motion and gradient | p. 266 |
| Eulerian Kinematics | p. 268 |
| Homogeneous transformation motion | p. 268 |
| Velocity field | p. 268 |
| Material derivative of a vector | p. 269 |
| Material derivative of a volume | p. 269 |
| Eulerian rates | p. 271 |
| General motion and velocity gradient | p. 274 |
| Velocity gradient tensor and Eulerian rates | p. 274 |
| Lagrangian and Eulerian strain tensors | p. 279 |
| Rate of rotation | p. 280 |
| Decomposition of motion | p. 285 |
| Rigid body motion | p. 286 |
| Material Derivatives of Circulation, Flux, and Volume | p. 287 |
| About the particle derivative | p. 287 |
| Physical quantity | p. 287 |
| Vector field | p. 290 |
| Tensor field | p. 292 |
| Material derivative of circulation | p. 293 |
| Material derivative of flux | p. 296 |
| Material derivative of volume integral | p. 299 |
| Lagrangian and Eulerian approaches | p. 299 |
| Proper motion case | p. 303 |
| Exercises | p. 305 |
| Fundamental Laws; Principle of Virtual Work | p. 315 |
| Conservation of Mass and Continuity Equation | p. 315 |
| Axiom of mass conservation | p. 315 |
| Continuity equation | p. 316 |
| Continuity equation in the Lagrangian description | p. 316 |
| Continuity equation in the Eulerian description | p. 317 |
| Mass flow rate | p. 319 |
| Material derivative of integral of mass density | p. 320 |
| Isochoric motion, steady and irrotational flows | p. 322 |
| Isochoric motion | p. 322 |
| Steady flow | p. 326 |
| Steady isochoric flow | p. 327 |
| Irrotational flow | p. 328 |
| Isochoric irrotational flow | p. 329 |
| Fundamental Laws of Dynamics | p. 330 |
| Body forces and surface forces | p. 330 |
| Principles of linear momentum and moment of momentum | p. 333 |
| Cauchy's stress tensor | p. 337 |
| Cauchy's stress tensor and principles of dynamics | p. 342 |
| Linear momentum principle and equilibrium equations | p. 342 |
| Moment of momentum principle | p. 344 |
| The generalized Cauchy's theorem | p. 346 |
| Poisson's theorem | p. 347 |
| Theorem of Kinetic Energy | p. 348 |
| Theorem of kinetic energy in the Eulerian description | p. 348 |
| Theorem of kinetic energy in the Lagrangian description | p. 350 |
| Study of Stresses | p. 356 |
| Reciprocity of stresses | p. 356 |
| Principal stresses | p. 358 |
| Stress invariants; deviator | p. 364 |
| Stress quadric of Cauchy and Lame stress ellipsoid | p. 370 |
| Geometrical constructions and Mohr's circles | p. 374 |
| (Mohr's) stress plane | p. 374 |
| Stress vector and plane of Mohr | p. 375 |
| Description of Mohr's circles | p. 378 |
| Particular stresses | p. 380 |
| Principle of Virtual Work | p. 384 |
| Preliminary recalls | p. 384 |
| Rigid body motion | p. 386 |
| Expressions of virtual power (and virtual work) | p. 389 |
| Principle of virtual work | p. 391 |
| Thermomechanics and Balance Equations | p. 393 |
| Balance equation | p. 393 |
| Proper motion | p. 393 |
| Material domain | p. 398 |
| Fixed domain | p. 401 |
| First principle of thermodynamics | p. 402 |
| Principle | p. 402 |
| Balance equations and local forms | p. 404 |
| Potential energy of body forces | p. 406 |
| Internal energy and balance equation | p. 407 |
| Second principle of thermodynamics | p. 409 |
| Principle | p. 409 |
| Clausius-Duhem inequality | p. 411 |
| Dissipation and reversibility | p. 412 |
| Conclusion and constitutive equations | p. 414 |
| Exercises | p. 417 |
| Linear Elasticity | p. 455 |
| Elasticity and Tests | p. 455 |
| Generalized Hooke's Law in Linear Elasticity | p. 458 |
| Generalized Hooke's law | p. 458 |
| Quadratic forms and strain energy function | p. 459 |
| Isotropic material and Lame coefficients | p. 463 |
| Constitutive equations | p. 463 |
| Young's modulus and Poisson's ratio | p. 466 |
| Bulk modulus | p. 469 |
| Shear modulus | p. 471 |
| Hooke's law's expression in a general coordinate system | p. 472 |
| Navier's equations of motion | p. 475 |
| Equations and Principles in Elastostatics | p. 477 |
| Navier's equation; the Beltrami equations of compatibility | p. 478 |
| Principle of superposition | p. 481 |
| Saint-Venant's principle | p. 484 |
| Classical Problems | p. 485 |
| Plane problems | p. 485 |
| Plane stress problems | p. 485 |
| Plane strain problems | p. 488 |
| Classical problems in elastostatics | p. 491 |
| Uniaxial stresses | p. 491 |
| Torsion of a circular cylinder body | p. 493 |
| Torsion of cylindrical shafts | p. 496 |
| Exercises | p. 501 |
| Summary of Formulae | p. 541 |
| Bibliography | p. 575 |
| Glossary of Symbols | p. 577 |
| Index | p. 585 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9781402010552
ISBN-10: 1402010559
Published: 31st January 2003
Format: Hardcover
Language: English
Number of Pages: 612
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: US
Dimensions (cm): 34.29 x 16.51 x 3.18
Weight (kg): 1.02
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