出版日期:2015-1-1
ISBN:9787510084539
作者:[俄] Leonid P Lebedev,[美] Michael J Cloud
页数:363页
书籍目录
Foreword
Preface
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
3.6 Eigenvalues and Eigenvectors of a Second—Order Symmel ricTensor
3.7 The Cayley—Hamilton Theorem
3.8 Other Properties of Second—Order Tensors
3.9 Extending the Dyad Idea
3.10 Tensors of the Fourth and Higher Orders
3.11 Functions of Tensorial Arguments
3.12 Norms for Tensors， and Some Spaces
3.13 Differentiation of Tensorial Functions
3.14 Problems
4.Tensor Fields
4.1 Vector Fields
4.2 Differentials and the Nabla Operator
4.3 Differentiation of a Vector Function
4.4 Derivatives of the Frame Vectors
4.5 Christoffel Coefficients and their Properties
4.6 Covariant Differentiation
4.7 Covariant Derivative of a Second—Order Tensor
4.8 Differential Operations
4.9 Orthogonal Coordinate Systems
4.10 Some Formulas oflntegration
4.11 Problems
5.Elements of Differential Geometry
5.1 Elementary Facts from the Theory of Curves
5.2 The Torsion of a Curve
5.3 Frenet—Serret Equations
5.4 Elements of the Theory of Surfaces
5.5 The Second Fundamental Form of a Surface
5.6 Derivation Formulas
5.7 Implicit R，epresentation of a Curve； Contact of Curves
5.8 Osculating Paraboloid
5.9 The Principal Curvatures of a Surface
5.10 Surfaces of Revolution
5.11 Natural Equations of a Curve
5.12 A Word About Rigor
5.13 Conclusion
5.14 Problems
Applications in Mechanics
6.Linear Elasticity
6.1 Stress Tensor
6.2 StrainTensor
6.3 Equation of Motion
6.4 Hooke's Law
6.5 Eqrulibrium 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 Muumum Total Energy Principle
6.11 Ritz's Method
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 ∑
7.3 Shell Eqrulibrium 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 PlateTheory
7.9 On Non—Classical Theories of Plates and Shells
Appendix A Formulary
Appendix B Hints and Answers
Bibliography
Index
作者简介
这是一部简明清晰，并且自称体系的有关张量、张量域及其应用的教材。书中囊括了学习有关张量应用的必需的所有内容，以及阐明了这些材料如何应用在力学中，也包括弹性线性理论基础和弹性壳体。前四章表述了所有的主体结果，剩余的部分教我们如何将这些结果应用于微分几何和连续力学中的各种情况中，如弹性体、弹性板和弹性壳体。在这个版本中，每章末都提供了练习和问题解答提示和答案。扩展附录部分以手册的形式综述了书中包括的所有重要公式。