Tensor Analysis with Applications in Mechanics.
Başlık:
Tensor Analysis with Applications in Mechanics.
Yazar:
Lebedev, Leonid P.
ISBN:
9789814313995
Yazar Ek Girişi:
Fiziksel Tanımlama:
1 online resource (378 pages)
İçerik:
Contents -- Foreword -- Preface -- Preface to the First Edition -- Tensor Analysis -- 1. Preliminaries -- 1.1 The Vector Concept Revisited -- 1.2 A First Look at Tensors -- 1.3 Assumed Background -- 1.4 More on the Notion of a Vector -- 1.5 Problems -- 2. Transformations and Vectors -- 2.1 Change of Basis -- 2.2 Dual Bases -- 2.3 Transformation to the Reciprocal Frame -- 2.4 Transformation Between General Frames -- 2.5 Covariant and Contravariant Components -- 2.6 The Cross Product in Index Notation -- 2.7 Norms on the Space of Vectors -- 2.8 Closing Remarks -- 2.9 Problems -- 3. Tensors -- 3.1 Dyadic Quantities and Tensors -- 3.2 Tensors From an Operator Viewpoint -- 3.3 Dyadic Components Under Transformation -- 3.4 More Dyadic Operations -- 3.5 Properties of Second-Order Tensors -- The tensor transpose -- Tensors raised to powers -- Symmetric and antisymmetric tensors -- 3.6 Eigenvalues and Eigenvectors of a Second-Order Symmetric Tensor -- 3.7 The Cayley-Hamilton Theorem -- 3.8 Other Properties of Second-Order Tensors -- Tensors of rotation -- Polar decomposition -- Deviator and ball tensor representation -- 3.9 Extending the Dyad Idea -- 3.10 Tensors of the Fourth and Higher Orders -- Isotropic tensors -- 3.11 Functions of Tensorial Arguments -- Linear functions -- Isotropic scalar-valued functions -- Isotropic tensor-valued functions -- 3.12 Norms for Tensors, and Some Spaces -- Some elements of calculus -- Some normed spaces -- 3.13 Differentiation of Tensorial Functions -- On symmetric tensor functions -- 3.14 Problems -- 4. Tensor Fields -- 4.1 Vector Fields -- Cylindrical coordinates -- Spherical coordinates -- 4.2 Differentials and the Nabla Operator -- 4.3 Differentiation of a Vector Function -- 4.4 Derivatives of the Frame Vectors -- 4.5 Christoffel Coeffcients and their Properties -- Euclidean vs. non-Euclidean spaces.
4.6 Covariant Differentiation -- 4.7 Covariant Derivative of a Second-Order Tensor -- 4.8 Differential Operations -- 4.9 Orthogonal Coordinate Systems -- Differentiation in the orthogonal basis -- 4.10 Some Formulas of Integration -- 4.11 Problems -- 5. Elements of Differential Geometry -- 5.1 Elementary Facts from the Theory of Curves -- Curvature -- Moving trihedron -- Curves in the plane -- 5.2 The Torsion of a Curve -- 5.3 Frenet-Serret Equations -- 5.4 Elements of the Theory of Surfaces -- First fundamental form -- Geodesics -- 5.5 The Second Fundamental Form of a Surface -- Normal curvature of the surface -- 5.6 Derivation Formulas -- Some useful formulas -- 5.7 Implicit Representation of a Curve -- Contact of Curves -- Contact of curves -- Contact of a curve with a circle -- evolutes -- Contact of nth order between a curve and a surface -- 5.8 Osculating Paraboloid -- 5.9 The Principal Curvatures of a Surface -- 5.10 Surfaces of Revolution -- 5.11 Natural Equations of a Curve -- Natural equation of a curve in the plane -- 5.12 A Word About Rigor -- 5.13 Conclusion -- 5.14 Problems -- Applications in Mechanics -- 6. Linear Elasticity -- 6.1 Stress Tensor -- Forces -- Equilibrium equations of a continuum medium -- Stress tensor -- Principal stresses and principal area elements -- 6.2 Strain Tensor -- 6.3 Equation of Motion -- 6.4 Hooke's Law -- 6.5 Equilibrium Equations in Displacements -- 6.6 Boundary Conditions and Boundary Value Problems -- 6.7 Equilibrium Equations in Stresses -- 6.8 Uniqueness of Solution for the Boundary Value Problems of Elasticity -- 6.9 Betti's Reciprocity Theorem -- 6.10 Minimum Total Energy Principle -- 6.11 Ritz'sMethod -- Korn's inequality -- 6.12 Rayleigh's Variational Principle -- 6.13 Plane Waves -- 6.14 Plane Problems of Elasticity -- 6.15 Problems -- 7. Linear Elastic Shells.
7.1 Some Useful Formulas of Surface Theory -- 7.2 Kinematics in a Neighborhood of Σ -- 7.3 Shell Equilibrium Equations -- 7.4 Shell Deformation and Strains -- Kirchhoff's Hypotheses -- 7.5 Shell Energy -- 7.6 Boundary Conditions -- 7.7 A Few Remarks on the Kirchhoff-Love Theory -- 7.8 Plate Theory -- Resultant forces and moments in a plate -- equilibrium equations -- Boundary conditions -- Strain energy for the plate -- Lagrange variational principle in plate theory -- Uniqueness of solution -- Remarks -- 7.9 On Non-Classical Theories of Plates and Shells -- Reissner's approach to plate and shell theory -- Plate and shell theories of higher order -- Micropolar shells or 6th-parametrical shell theory -- Appendix A Formulary -- Chapter 1 -- Chapter 2 -- Chapter 3 -- Chapter 4 -- Chapter 5 -- Chapter 6 -- Chapter 7 -- Appendix B Hints and Answers -- Chapter 1 Exercises -- Chapter 1 Problems -- Chapter 2 Exercises -- Chapter 2 Problems -- Chapter 3 Exercises -- Chapter 3 Problems -- Chapter 4 Exercises -- Chapter 4 Problems -- Chapter 5 Exercises -- Chapter 5 Problems -- Chapter 6 Exercises -- Chapter 6 Problems -- Chapter 7 Exercises -- Bibliography -- Index.
Özet:
The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies. In this case general curvilinear coordinates become necessary. The principal basis of a curvilinear system is constructed as a set of vectors tangent to the coordinate lines. Another basis, called the dual basis, is also constructed in a special manner. The existence of these two bases is responsible for the mysterious covariant and contravariant terminology encountered in tensor discussions. A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems. This book is a clear, concise, and self-contained treatment of tensors, tensor fields, and their applications.The book contains practically all the material on tensors needed for applications. It shows how this material is applied in mechanics, covering the foundations of the linear theories of elasticity and elastic shells. The main results are all presented in the first four chapters. The remainder of the book shows how one can apply these results to differential geometry and the study of various types of objects in continuum mechanics such as elastic bodies, plates, and shells. Each chapter of this new edition is supplied with exercises and problems - most with solutions, hints, or answers to help the reader progress. An extended appendix serves as a handbook-style summary of all
important formulas contained in the book.
Notlar:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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