Tensor Analysis and Continuum Mechanics

This book is designed for students in engineering, physics and mathematics. The material can be taught from the beginning of the third academic year. It could also be used for self­ study, given its pedagogical structure and the numerous solved problems which prepare for modem physics and technology. One of the original aspects of this work is the development together of the basic theory of tensors and the foundations of continuum mechanics. Why two books in one? Firstly, Tensor Analysis provides a thorough introduction of intrinsic mathematical entities, called tensors, which is essential for continuum mechanics. This way of proceeding greatly unifies the various subjects. Only some basic knowledge of linear algebra is necessary to start out on the topic of tensors. The essence of the mathematical foundations is introduced in a practical way. Tensor developments are often too abstract, since they are either aimed at algebraists only, or too quickly applied to physicists and engineers. Here a good balance has been found which allows these extremes to be brought closer together. Though the exposition of tensor theory forms a subject in itself, it is viewed not only as an autonomous mathematical discipline, but as a preparation for theories of physics and engineering. More specifically, because this part of the work deals with tensors in general coordinates and not solely in Cartesian coordinates, it will greatly help with many different disciplines such as differential geometry, analytical mechanics, continuum mechanics, special relativity, general relativity, cosmology, electromagnetism, quantum mechanics, etc.. Contents Preface Chapter 1. Tensors 1. First Steps with Tensors 1.1 Multilinear Forms 1.1.1 Linear Mapping 1.1.2 Multilinear Form 1.2 Dual Space, Vectors and Covectors 1.2.1 Dual Space 1.2.2 Expression of a Covector 1.2.3 Einstein Summation Convention 1.2.4 Change of Basis and Cobasis 1.3 Tensors and Tensor Product 1.3.1 Tensor Product of Multilinear Forms 1.3.2 Tensor of Type \binom{0}{1} 1.3.3 Tensor of Type \binom{1}{0} 1.3.4 Tensor of Type \binom{0}{2} 1.3.5 Tensor of Type \binom{2}{0} 1.3.6 Tensor of Type \binom{1}{1} 1.3.7 Tensor of Type \binom{q}{p} 1.3.8 Symmetric and Antisymmetric Tensors 2. Operations on Tensors 2.1 Tensor Algebra 2.1.1 Addition of Tensors 2.1.2 Multiplication of a Tensor by a Scalar 2.1.3 Tensor Multiplication 2.2 Contraction and Tensor Criteria 2.2.1 Contraction 2.2.2 Tensor Criteria 3. Euclidean Vector Space 3.1 Pre-Euclidean Vector Space 3.1.1 Scalar Multiplication and Pre-Euclidean Vector Space 3.1.2 Fundamental Tensor 3.2 Canonical Isomorphism and Conjugate Tensor 3.2.1 Canonical Isomorphism 3.2.2 Conjugate Tensor and Reciprocal Basis 3.2.3 Covariant and Contravariant Representations of Vectors 3.2.4 Representations of Tensors of Order 2 and Contracted Products 3.3 Euclidean Vector Spaces 4. Exterior Algebra 4.1 p-FORMS 4.1.1 Definition of a p-Form 4.1.2 Exterior Product of I-Forms 4.1.3 Expression of a p-Form 4.1.4 Exterior Product of p-Forms 4.1.5 Exterior Algebra 4.2 q-VECTORS 5. Point Spaces 5.1 Point Space and Natural Frame 5.1.1 Point Space 5.1.2 Coordinate System and Frame of Reference Change of coordinate system Coordinate systems 5.1.3 Natural Frame 5.2 Tensor Fields and Metric Element 5.2.1 Transformations of Curvilinear Coordinates Transformation of tensor components 5.2.2 Tensor Fields 5.2.3 Metric Element 5.3 Christoffel Symbols 5.3.1 Definition of Christoffel Symbols 5.3.2 Ricci Identities and Christoffel Formulae 5.4 Absolute Differential, Covariant Derivatives, Geodesic 5.4.1 Absolute Differential of a Vector, Covariant Derivatives 5.4.2 Absolute Differential of a Tensor, Covariant Derivatives 5.4.3 Geodesic and Euler's Equations 5.4.4 Absolute Derivative of a Vector (Along a Curve) 5.5 Volume Form and Adjoint 5.5.1 Volume Form 5.5.2 Adjoint 5.6 Differential Operators 5.6.1 Gradient 5.6.1a Gradient of a Function 5.6.1b Gradient of a Tensor 5.6.2 Divergence 5.6.2a Divergence of a Vector Field 5.6.2b Divergence of a Tensor Field 5.6.3 Curl 5.6.3a Curl of a Covector Field 5.6.3b Curl of a Tensor Field 5.6.4 Laplacian 5.6.4a Laplacian of a Function 5.6.4b Laplacian of a Vector Field 6. Exercises Chapter 2. Lagrangian and Eulerian Descriptions 1. Lagrangian Description 1.1 Configuration 1.2 Deformation and Lagrangian Description 1.3 Flow and Hypotheses of Continuity 1.4 Trajectories 1.5 Streakline 1.6 Velocity and Acceleration of a Particle 1.7 Abstract Configuration 2. Eulerian Description 2.1 Definition; Comparison between Lagrangian and Eulerian Descriptions 2.2 Trajectory and Velocity 2.3 Streamline 2.4 Steady Motion Exercises Chapter 3. Deformations 1. Homogeneous Transformation 1.1 Definition of Homogeneous Transformations 1.2 Convective Transport 1.2.1 Convective Transport of a Vector 1.2.2 Convective Transport of a Volume 1.2.3 Simple Shear 1.3 Cauchy-Green Deformation Tensor and Stretch 1.3.1 (Right) Cauchy-Green Deformation Tensor 1.3.2 Stretch 1.3.3 Shear Angle 1.3.4 Principal Stretches 1.4 Finite Strain Tensor 1.5 Polar Decomposition 1.5.1 Pure Stretch and Rotation 1.5.2 Euler-Almansi Strain Tensor 1.6 Rigid Body Transformation 2. Tangential Homogeneous Transformation 2.1 Deformation Gradient 2.2 Homogeneous Transformations of Elements 2.2.1 Transport of Vectors, Volume Deformation, and Area Deformation 2.2.2 Stretches 2.2.3 Strain 2.3 Displacement and Gradient 2.3.1 Material Displacement Gradient 2.3.2 Spatial Displacement Gradient 2.3.3 Curvilinear Coordinate System 3. Infinitesimal Transformation 3.1 Tensor Notions Relating to Infinitesimal Transformations 3.2 Compatibility Conditions 3.3 Rigid Body Transformation Exercises Chapter 4. Kinematics of Continua 1. Lagrangian Kinematics 1.1 Homogeneous Transformation Motion 1.2 General Motion and Gradient 2. Eulerian Kinematics 2.1 Homogeneous Transformation Motion 2.1.1 Velocity Field 2.1.2 Material Derivative of a Vector 2.1.3 Material Derivative of a Volume 2.1.4 Eulerian Rates 2.2 General Motion and Velocity Gradient 2.2.1 Velocity Gradient Tensor and Eulerian Rates 2.2.2 Lagrangian and Eulerian Strain tensors 2.2.3 Rate of Rotation 2.2.4 Decomposition of Motion 2.3 Rigid Body Motion 3. Material Derivatives of Circulation, Flux, and Volume 3.1 About the Particle Derivative 3.1.1 Physical Quantity 3.1.2 Vector Field 3.1.3 Tensor field 3.2 Material Derivative of Circulation 3.3 Material Derivative of Flux 3.4 Material Derivative of Volume Integral 3.4.1 Lagrangian and Eulerian Approaches 3.4.2 Proper Motion Case Exercises Chapter 5. Fundamental Laws; The Principle of Virtual Work 1. Conservation of Mass and Continuity Equation 1.1 Axiom of Mass Conservation 1.2 Continuity Equation 1.2.1 Continuity Equation in the Lagrangian Description 1.2.2 Continuity Equations in the Eulerian Description 1.2.3 Mass Flow Rate 1.3 The Material Derivative of the Integral of Mass Density 1.4 Isochoric Motion; Steady and Irrotational Flows 1.4.1 Isochoric Motion 1.4.2 Steady Flow 1.4.3 Steady Isochoric Flow 1.4.4 Irrotational Flow 1.4.5 Isochoric Irrotational Flow 2. Fundamental Laws of Dynamics 2.1 Body Forces and Surface Forces 2.2 Principles of Linear Momentum and Moment of Momentum 2.3 Cauchy's Stress Tensor 2.4 Cauchy's Stress Tensor and the Principles of Dynamics 2.4.1 Linear Momentum Principle and Equilibrium Equations 2.4.2 Moment of Momentum Principle 2.4.3 The Generalized Cauchy's Theorem 2.4.4 Poisson's Theorem 3. Theorem of Kinetic Energy 3.1 Theorem of Kinetic Energy in the Eulerian Description 3.2 Theorem of Kinetic Energy in the Lagrangian Description 4. Study of Stresses 4.1 Reciprocity of Stresses 4.2 Principal Stresses 4.3 Stress Invariants; Deviator 4.4 Stress Quadric of Cauchy and Lamé Stress Ellipsoid 4.5 Geometrical Constructions and Mohr's Circles 4.5.1 (Mohr's) Stress Plane 4.5.2 Stress Vector and Plane of Mohr 4.5.3 Description of Mohr's Circles 4.5.4 Particular Stresses 5. Principle of Virtual Work 5.1 Preliminary Recalls 5.2 Rigid Body Motion 5.3 Expressions of Virtual Power (and Virtual Work) 5.4 Principle of Virtual Work 6. Thermomechanics and Balance Equations 6.1 Balance Equation 6.1.1 Proper Motion 6.1.2 Material Domain 6.1.3 Fixed Domain 6.2 First Principle of Thermodynamics 6.2.1 Principle 6.2.2 Balance Equations and Local Forms 6.2.3 Potential Energy of Body Forces 6.2.4 Internal Energy and Balance Equation 6.3 Second Principle of Thermodynamics 6.3.1 Principle 6.3.2 Clausius-Duhem Inequality 6.3.3 Dissipation and Reversibility 6.4 Conclusions and Constitutive Equations Exercises Chapter 6. Linear Elasticity 1. Elasticity and Tests 2. The Generalized Hooke's Law in Linear Elasticity 2.1 The Generalized Hooke's Law 2.2 Quadratic Forms and Strain Energy Function 2.3 Isotropic Material and Lame Coefficients 2.3.1 Constitutive Equations 2.3.2 Young's Modulus and Poisson's Ratio 2.3.3 Bulk Modulus 2.3.4 Shear Modulus 2.3.5 Hooke's Law's Expression in a General Coordinate System 2.3.6 Navier's Equations of Motion 3. Equations and Principles in Elastostatics 3.1 Navier's Equation; The Beltrami Equation of Compatibility 3.2 Principle of Superposition Reciprocity Theorem 3.3 Saint-Venant's Principle 4. Classical Problems 4.1 Plane Problems 4.1.1 Plane Stress Problems 4.1.2 Plane Strain Problems 4.2 Classical Problems in Elastostatics 4.2.1 Uniaxial Stresses (or Simple Extension) 4.2.2 Torsion of a Circular Cylindrical Body 4.2.3 Torsion of Cylindrical Shafts Exercises Summary of Formulae Chapter 1 Section 1 Section 2 Section 3 Section 4 Section 5 Chapter 2 Section 1 Section 2 Chapter 3 Section 1 Section 2 Section 3 Chapter 4 Section 1 Section 2 Section 3 Chapter 5 Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Chapter 6 Section 2 Section 3 Section 4 Bibliography Glossary of Symbols Latin Letters Greek Letters Other Symbols Index

this Volume Reviews The Theory Of Tensors And The Foundations Of Continuum Mechanics As Preparation For Areas Of General Relativity, Fluid Dynamics, Engineering, And Differential Geometry. Talpaert (faculty Of Science And Engineering At Several African And European Universities) Covers Concepts Including Canonical Isomorphism, Euclidean Vector Spaces, Exterior Algebra, Point Spaces, Metric Element, And Christoffel Symbols. The Book, Which May Be Used As A Text For Third-year Students Of Physics, Engineering, Or Mathematics, Lays Groundwork For More Technical Subjects Including Strength Of Materials, Plasticity, And Viscoelasticity. Annotation ©2003 Book News, Inc., Portland, Or

"This work lays the groundwork for more technical subjects as strength of materials, plasticity, vicoelasticity, and nonlinear continuum mechanics. The material is presented with great pedagogical care, a summary of formulae and a glossary of symbols are provided, as well as ninety-five solved problems. The book is suitable as a text for third-year students of mathematics, physics and engineering, and for anyone wishing to acquire insight into the mathematics of mechanics, the mathematics of physics, the mathematics of engineering, continuum mechanics, elasticity and viscoelasticity, linear and multilinear algebra, or matrix theory."--BOOK JACKET The idea of tensor took form at the end of the 19th century when it became necessary to express pressure forces in continua.