Continuum Mechanics

The Birthplace of Mathematical Models

Inbunden, Engelska, 2015

AvMyron B. Allen,Myron B. (University of Wyoming) Allen,Myron B Allen

1 539 kr

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Presents a self-contained introduction to continuum mechanics that illustrates how many of the important partial differential equations of applied mathematics arise from continuum modeling principlesWritten as an accessible introduction, Continuum Mechanics: The Birthplace of Mathematical Models provides a comprehensive foundation for mathematical models used in fluid mechanics, solid mechanics, and heat transfer. The book features derivations of commonly used differential equations based on the fundamental continuum mechanical concepts encountered in various fields, such as engineering, physics, and geophysics.The book begins with geometric, algebraic, and analytical foundations before introducing topics in kinematics. The book then addresses balance laws, constitutive relations, and constitutive theory. Finally, the book presents an approach to multiconstituent continua based on mixture theory to illustrate how phenomena, such as diffusion and porous-media flow, obey continuum-mechanical principles.Continuum Mechanics: The Birthplace of Mathematical Models features: Direct vector and tensor notation to minimize the reliance on particular coordinate systems when presenting the theoryTerminology that is aligned with standard courses in vector calculus and linear algebraThe use of Cartesian coordinates in the examples and problems to provide readers with a familiar settingOver 200 exercises and problems with hints and solutions in an appendixIntroductions to constitutive theory and multiconstituent continua, which are distinctive for books at this levelContinuum Mechanics: The Birthplace of Mathematical Models is an ideal textbook for courses on continuum mechanics for upper-undergraduate mathematics majors and graduate students in applied mathematics, mechanical engineering, civil engineering, physics, and geophysics. The book is also an excellent reference for professional mathematicians, physical scientists, and engineers.

Produktinformation

Utgivningsdatum2015-08-11
Mått165 x 241 x 25 mm
Vikt649 g
FormatInbunden
SpråkEngelska
Antal sidor288
FörlagJohn Wiley & Sons Inc
ISBN9781118909379

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Preface v1 Geometric Setting 11.1 Vectors and Euclidean Point Space 21.1.1 Vectors 21.1.2 Euclidean Point Space 61.1.3 Summary 81.2 Tensors 81.2.1 First-Order Tensors and Vectors 81.2.2 Second-Order Tensors 111.2.3 Cross Products, Triple Products, and Determinants 151.2.4 Orthogonal Tensors 201.2.5 Invariants of a Tensor 211.2.6 Derivatives of Tensor-Valued Functions 241.2.7 Summary 272 Kinematics I: The Calculus of Motion 292.1 Bodies, Motions, and Deformations 292.1.1 Deformation 322.1.2 Examples of Motions 332.1.3 Summary 362.2 Derivatives of Motion 362.2.1 Time Derivatives 372.2.2 Derivatives with Respect to Position 382.2.3 The Deformation Gradient 402.2.4 Summary 422.3 Pathlines, Streamlines, and Streaklines 432.3.1 Three Types of Arc 432.3.2 An Example 452.3.3 Summary 492.4 Integrals Under Motion 492.4.1 Arc, Surface, and Volume Integrals 492.4.2 Reynolds Transport Theorem 552.4.3 Summary 573 Kinematics II: Strain and its Rates 593.1 Strain 593.1.1 Symmetric Tensors 603.1.2 Polar Decomposition and the Deformation Gradient 643.1.3 Examples 663.1.4 Cauchy–Green and Strain Tensors 683.1.5 Strain Invariants 703.1.6 Summary 713.2 Infinitesimal Strain 723.2.1 The Infinitesimal Strain Tensor 723.2.2 Summary 753.3 Strain Rates 753.3.1 Stretching and Spin Tensors 763.3.2 Skew Tensors, Spin, and Vorticity 793.3.3 Summary 843.4 Vorticity and Circulation 843.4.1 Circulation 843.4.2 Summary 883.5 Observer Transformations 893.5.1 Changes in Frame of Reference 893.5.2 Summary 954 Balance Laws 974.1 Mass Balance 984.1.1 Local Forms of Mass Balance 994.1.2 Summary 1024.2 Momentum Balance 1024.2.1 Analysis of Stress 1044.2.2 Inertial Frames of Reference 1104.2.3 Momentum Balance in Referential Coordinates 1134.2.4 Summary 1144.3 Angular Momentum Balance 1154.3.1 Symmetry of the Stress Tensor 1174.3.2 Summary 1184.4 Energy Balance 1194.4.1 Thermal Energy Balance 1224.4.2 Summary 1244.5 Entropy Inequality 1244.5.1 Motivation 1254.5.2 Clausius–Duhem Inequality 1264.5.3 Summary 1274.6 Jump Conditions 1274.6.1 Singular Surfaces 1294.6.2 Localization 1324.6.3 Summary 1355 Constitutive Relations: Examples of Mathematical Models 1375.1 Heat Transfer 1385.1.1 Properties of the Heat Equation 1405.1.2 Summary 1425.2 Potential Theory 1435.2.1 Motivation 1435.2.2 Boundary Conditions 1445.2.3 Uniqueness of Solutions to the Poisson Equation 1465.2.4 Maximum Principle 1475.2.5 Mean Value Property 1505.2.6 Summary 1515.3 Fluid Mechanics 1525.3.1 Ideal Fluids 1525.3.2 An Ideal Fluid in a Rotating Frame of Reference 1545.3.3 Acoustics 1555.3.4 Incompressible Newtonian Fluids 1585.3.5 Stokes Flow 1595.3.6 Summary 1635.4 Solid Mechanics 1645.4.1 Static Displacements 1645.4.2 Elastic Waves 1675.4.3 Summary 1706 Constitutive Theory 1736.1 Conceptual Setting 1746.1.1 The Need to Close the System 1746.1.2 Summary 1766.2 Determinism and Equipresence 1776.2.1 Determinism 1776.2.2 Equipresence 1776.2.3 Summary 1786.3 Objectivity 1796.3.1 Reducing Functional Dependencies 1806.3.2 Summary 1826.4 SYMMETRY 1836.4.1 Changes in Reference Configuration 1836.4.2 Symmetry Groups 1866.4.3 Classification of Materials 1896.4.4 Implications for Thermoviscous Fluids 1936.4.5 Summary 1936.5 Admissibility 1946.5.1 Implications of the Entropy Inequality 1956.5.2 Analysis of Equilibrium 1976.5.3 Linear, Isotropic, Thermoelastic Solids 1996.5.4 Summary 2027 Multiconstituent Continua 2037.1 Constituents 2047.1.1 Configurations and Motions 2047.1.2 Volume Fractions and Densities 2067.1.3 Summary 2087.2 Multiconstituent Balance Laws 2097.2.1 Multiconstituent Mass Balance 2107.2.2 Multiconstituent Momentum Balance 2127.2.3 Multiconstituent Angular Momentum Balance 2147.2.4 Multiconstituent Energy Balance 2157.2.5 Multiconstituent Entropy Inequality 2167.2.6 Isothermal, Nonreacting Multiphase Mixtures 2177.2.7 Summary 2197.3 Fluid Flow in a Porous Solid 2207.3.1 Modeling Assumptions for Porous Media 2217.3.2 Balance Laws for the Fluid and Solid Phases 2237.3.3 Equilibrium Constraints 2257.3.4 Linear Extensions From Equilibrium 2267.3.5 Commentary 2287.3.6 Potential Formulation of Darcy’s Law 2297.3.7 Summary 2337.4 Diffusion in a Binary Fluid Mixture 2347.4.1 Modeling Assumptions for Binary Diffusion 2357.4.2 Balance Laws for the Two Species 2357.4.3 Constitutive Relationships for Diffusion 2367.4.4 Modeling Solute Transport 2397.4.5 Summary 242A Guide to Notation 243A.1 General Conventions 243A.2 Letters Reserved for Dedicated Uses 244A.3 Special Symbols 245B Vector Integral Theorems 247B.1 Stokes’s Theorem 248B.2 The Divergence Theorem 249B.3 The Change-of-variables Theorem 252C Hints and Solutions to Exercises 253References 265Index 269