Continuum Mechanics using Mathematica®: Fundamentals, Applications and Scientific Computing

Description

This book's methodological approach familiarizes readers with the mathematical tools required to correctly define and solve problems in continuum mechanics. The book covers essential principles and fundamental applications, and provides a solid basis for a deeper study of more challenging and specialized problems related to elasticity, fluid mechanics, plasticity, materials with memory, piezoelectricity, ferroelectricity, magneto-fluid mechanics, and state changes. Key topics and features: * Concise presentation strikes a balance between fundamentals and applications * Requisite mathematical background carefully collected in two introductory chapters and two appendices * CD-ROM containing Mathematica-based exercises, examples, and programs related to the text; this user-friendly software is unique in the literature and allows readers to quickly solve complex, time-consuming problems related to the application areas covered * Recent developments highlighted through coverage of more significant applications to areas such as porous media, electromagnetic fields, and phase transitions Continuum Mechanics using Mathematica® is aimed at advanced undergraduates, graduate students, and researchers in applied mathematics, mathematical physics, and engineering. It may serve as a course textbook or self-study reference for anyone seeking a solid foundation in the field. Continuum Mechanics using Mathematica......Page 2 Contents......Page 4 Preface......Page 8 1.1 Motivation to Study Linear Algebra......Page 12 1.2 Vector Spaces and Bases......Page 13 1.3 Euclidean Vector Space......Page 16 1.4 Base Changes......Page 20 1.5 Vector Product......Page 22 1.6 Mixed Product......Page 24 1.7 Elements of Tensor Algebra......Page 25 1.8 Eigenvalues and Eigenvectors of a Euclidean Second-Order Tensor......Page 31 1.9 Orthogonal Tensors......Page 35 1.10 Cauchy’s Polar Decomposition Theorem......Page 39 1.11 Higher Order Tensors......Page 40 1.12 Euclidean Point Space......Page 41 1.13 Exercises......Page 43 Description of the Problem and Relative Algorithm......Page 47 Command Line of the Program VectorSys......Page 48 Worked Examples......Page 49 Exercises......Page 51 Worked Examples......Page 52 Exercises......Page 54 2.1 Curvilinear Coordinates......Page 55 2.2 Examples of Curvilinear Coordinates......Page 58 2.3 Differentiation of Vector Fields......Page 60 2.4 The Stokes and Gauss Theorems......Page 65 2.5 Singular Surfaces......Page 66 2.6 Useful Formulae......Page 70 2.7 Some Curvilinear Coordinates......Page 72 2.8 Exercises......Page 79 Command Line of the Program Operator......Page 80 Worked Examples......Page 82 Exercises......Page 86 3.1 Deformation Gradient......Page 87 3.2 Stretch Ratio and Angular Distortion......Page 90 3.3 Invariants of C and B......Page 93 3.4 Displacement and Displacement Gradient......Page 94 3.5 Infinitesimal Deformation Theory......Page 96 3.6 Transformation Rules for Deformation Tensors......Page 98 3.7 Some Relevant Formulae......Page 99 3.8 Compatibility Conditions......Page 102 3.9 Curvilinear Coordinates......Page 105 3.10 Exercises......Page 106 Description of the Algorithm and Instructions for Use......Page 111 Command Line of the Program Deformation......Page 112 Worked Examples......Page 113 Exercises......Page 117 4.1 Velocity and Acceleration......Page 118 4.2 Velocity Gradient......Page 121 4.3 Rigid, Irrotational, and Isochoric Motions......Page 122 4.4 Transformation Rules for a Change of Frame......Page 124 4.5 Singular Moving Surfaces......Page 125 4.6 Time Derivative of a Moving Volume......Page 128 4.7 Worked Exercises......Page 132 Aim of the Program, Input and Output......Page 135 Worked Examples......Page 136 Exercises......Page 137 5.1 General Formulation of a Balance Equation......Page 139 5.2 Mass Conservation......Page 144 5.3 Momentum Balance Equation......Page 145 5.4 Balance of Angular Momentum......Page 148 5.5 Energy Balance......Page 149 5.6 Entropy Inequality......Page 151 5.7 Lagrangian Formulation of Balance Equations......Page 154 5.8 The Principle of Virtual Displacements......Page 158 5.9 Exercises......Page 159 6.1 Constitutive Axioms......Page 162 6.2 Thermoviscoelastic Behavior......Page 167 6.3 Linear Thermoelasticity......Page 172 6.4 Exercises......Page 176 7.1 Symmetry......Page 177 7.2 Isotropic Solids......Page 180 7.3 Perfect and Viscous Fluids......Page 183 7.4 Anisotropic Solids......Page 187 7.5 Exercises......Page 189 Description of the Problem and Relative Algorithm......Page 191 Worked Examples......Page 192 Exercises......Page 193 8.1 Introduction......Page 194 8.2 Cauchy’s Problem for Second-Order PDEs......Page 195 8.3 Characteristics and Classification of PDEs......Page 199 8.4 Examples......Page 201 8.5 Cauchy’s Problem for a Quasi-Linear First-Order System......Page 204 8.6 Classification of First-Order Systems......Page 206 8.7 Examples......Page 207 8.8 Second-Order Systems......Page 211 8.9 Ordinary Waves......Page 212 8.10 Linearized Theory and Waves......Page 216 8.11 Shock Waves......Page 220 8.12 Exercises......Page 223 Command Line of the Program PdeEqClass......Page 226 Worked Examples......Page 227 Exercises......Page 230 Description of the Problem and Relative Algorithm......Page 232 Command Line of the Program PdeSysClass......Page 233 Worked Examples......Page 234 Exercises......Page 236 Description of the Problem and Relative Algorithm......Page 238 Use Instructions......Page 239 Worked Examples......Page 240 Exercises......Page 241 Description of the Problem and Relative Algorithm......Page 244 Command Line of the Program WavesII......Page 245 Worked Examples......Page 246 Exercises......Page 247 9.1 Perfect Fluid......Page 249 9.2 Stevino’s Law and Archimedes’ Principle......Page 250 9.3 Fundamental Theorems of Fluid Dynamics......Page 254 9.4 Boundary Value Problems for a Perfect Fluid......Page 258 9.5 2D Steady Flow of a Perfect Fluid......Page 260 9.6 D’Alembert’s Paradox and the Kutta–Joukowsky Theorem......Page 268 9.7 Lift and Airfoils......Page 271 9.8 Newtonian Fluids......Page 276 9.9 Applications of the Navier–Stokes Equation......Page 277 9.10 Dimensional Analysis and the Navier–Stokes Equation......Page 278 9.11 Boundary Layer......Page 280 9.12 Motion of a Viscous Liquid around an Obstacle......Page 285 9.13 Ordinary Waves in Perfect Fluids......Page 291 9.14 Shock Waves in Fluids......Page 294 9.15 Shock Waves in a Perfect Gas......Page 297 9.16 Exercises......Page 300 Command Line of the Program Potential......Page 302 Worked Examples......Page 303 Exercises......Page 308 Command Line of the Program Wing......Page 310 Worked Examples......Page 311 Exercises......Page 312 Command Line of the Program Joukowsky......Page 313 Worked Examples......Page 314 Exercises......Page 315 Command Line of the Program JoukowskyMap......Page 316 Worked Examples......Page 317 Exercises......Page 320 10.1 Basic Equations of Linear Elasticity......Page 321 10.2 Uniqueness Theorems......Page 325 10.3 Existence and Uniqueness of Equilibrium Solutions......Page 327 10.4 Examples of Deformations......Page 331 10.5 The Boussinesq–Papkovich–Neuber Solution......Page 333 10.6 Saint-Venant’s Conjecture......Page 334 10.7 The Fundamental Saint-Venant Solutions......Page 339 10.8 Ordinary Waves in Elastic Systems......Page 343 10.9 Plane Waves......Page 349 10.10 Reflection of Plane Waves in a Half-Space......Page 354 10.11 Rayleigh Waves......Page 360 10.12 Reflection and Refraction of SH Waves......Page 363 10.13 Harmonic Waves in a Layer......Page 366 10.14 Exercises......Page 369 11.1 Basic Thermodynamics......Page 371 11.2 Extended Thermodynamics......Page 374 11.3 Serrin’s Approach......Page 376 11.4 An Application to Viscous Fluids......Page 379 References......Page 383 Index......Page 386