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BL
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| Record Number
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860054
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| Main Entry
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Irgens, Fridtjov,1935-
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| Title & Author
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Tensor analysis /\ Fridtjov Irgens.
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| Publication Statement
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Cham, Switzerland :: Springer,, [2019]
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| Page. NO
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1 online resource
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| ISBN
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3030034127
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: 3030034135
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: 9783030034122
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: 9783030034139
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3030034119
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9783030034115
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| Notes
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6 General Coordinates in Euclidean Space E3
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Includes index.
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| Contents
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Intro; Preface; A Short Presentation of the Contents of This Book; References; Contents; Symbols; 1 Mathematical Foundation; 1.1 Matrices and Determinants; 1.2 Cartesian Coordinate Systems. Scalars and Vectors; 1.2.1 Displacement Vectors; 1.2.2 Vector Algebra; 1.3 Cartesian Coordinate Transformations; 1.4 Curves in Space; 1.5 Dynamics of a Mass Particle; 1.6 Scalar Fields and Vector Fields; 2 Dynamics. The Cauchy Stress Tensor; 2.1 Kinematics; 2.1.1 Lagrangian Coordinates and Eulerian Coordinates; 2.1.2 Material Derivative of Intensive Quantities
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2.1.3 Material Derivative of Extensive Quantities2.2 Equations of Motion; 2.2.1 Euler's Axioms; 2.2.2 Newton's Third Law of Action and Reaction; 2.2.3 Coordinate Stresses; 2.2.4 Cauchy's Stress Theorem and Cauchy's Stress Tensor; 2.2.5 Cauchy's Equations of Motion; 2.3 Stress Analysis; 2.3.1 Principal Stresses and Principal Stress Directions; 2.3.2 Stress Deviator and Stress Isotrop; 2.3.3 Extreme Values of Normal Stress; 2.3.4 Maximum Shear Stress; 2.3.5 State of Plane Stress; 2.3.6 Mohr-Diagram for State of Plane Stress; Reference; 3 Tensors; 3.1 Definition of Tensors; 3.2 Tensor Algebra
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3.2.1 Isotropic Tensors of Fourth Order3.2.2 Tensors as Polyadics; 3.3 Tensors of Second Order; 3.3.1 Symmetric Tensors of Second Order; 3.3.2 Alternative Invariants of Second Order Tensors; 3.3.3 Deviator and Isotrop of Second Order Tensors; 3.4 Tensor Fields; 3.4.1 Gradient, Divergence, and Rotation of Tensor Fields; 3.4.2 Del-Operators; 3.4.3 Directional Derivative of Tensor Fields; 3.4.4 Material Derivative of Tensor Fields; 3.5 Rigid-Body Dynamics. Kinematics; 3.5.1 Pure Rotation About a Fixed Axis; 3.5.2 Pure Rotation About a Fixed Point; 3.5.3 Kinematics of General Rigid-Body Motion
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3.6 Rigid-Body Dynamics. Kinetics3.6.1 Rotation About a Fixed Point. The Inertia Tensor; 3.6.2 General Rigid-Body Motion; 3.7 Q-Rotation of Vectors and Tensors of Second Order; 3.8 Polar Decomposition; 3.9 Isotropic Functions of Tensors; References; 4 Deformation Analysis; 4.1 Strain Measures; 4.2 Green's Strain Tensor; 4.3 Small Strains and Small Deformations; 4.3.1 Small Strains; 4.3.2 Small Deformations; 4.3.3 Principal Strains and Principal Directions for Small Deformations; 4.3.4 Strain Deviator and Strain Isotrop for Small Deformations; 4.3.5 Rotation Tensor for Small Deformations
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4.3.6 Small Deformations in a Material Surface4.3.7 Mohr-Diagram for Small Deformations in a Surface; 4.4 Rates of Deformation, Strain, and Rotation; 4.5 Large Deformations; 5 Constitutive Equations; 5.1 Introduction; 5.2 Linearly Elastic Materials; 5.2.1 Generalized Hooke's Law; 5.2.2 Some Basic Equations in Linear Elasticity. Navier's Equations; 5.2.3 Stress Waves in Elastic Materials; 5.3 Linearly Viscous Fluids; 5.3.1 Definition of Fluids; 5.3.2 The Continuity Equation; 5.3.3 Constitutive Equations for Linearly Viscous Fluids; 5.3.4 The Navier-Stokes Equations; 5.3.5 Film Flow; References
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| Abstract
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This book presents tensors and tensor analysis as primary mathematical tools for engineering and engineering science students and researchers. The discussion is based on the concepts of vectors and vector analysis in three-dimensional Euclidean space, and although it takes the subject matter to an advanced level, the book starts with elementary geometrical vector algebra so that it is suitable as a first introduction to tensors and tensor analysis. Each chapter includes a number of problems for readers to solve, and solutions are provided in an Appendix at the end of the text. Chapter 1 introduces the necessary mathematical foundations for the chapters that follow, while Chapter 2 presents the equations of motions for bodies of continuous material. Chapter 3 offers a general definition of tensors and tensor fields in three-dimensional Euclidean space. Chapter 4 discusses a new family of tensors related to the deformation of continuous material. Chapter 5 then addresses constitutive equations for elastic materials and viscous fluids, which are presented as tensor equations relating the tensor concept of stress to the tensors describing deformation, rate of deformation and rotation. Chapter 6 investigates general coordinate systems in three-dimensional Euclidean space and Chapter 7 shows how the tensor equations discussed in chapters 4 and 5 are presented in general coordinates. Chapter 8 describes surface geometry in three-dimensional Euclidean space, Chapter 9 includes the most common integral theorems in two- and three-dimensional Euclidean space applied in continuum mechanics and mathematical physics.
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| Subject
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Calculus of tensors.
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| Subject
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Calculus of tensors.
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| Dewey Classification
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515/.63
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| LC Classification
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QA433.I74 2019
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