张量分析及其在力学中的应用

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

《张量分析及其在力学中的应用》张量分析是研究连续介质力学的重要数学工具。张量分析及其在连续介质力学中的应用紧密结合工程力学来介绍张量分析的基本理论和实用计算。《张量分析及其在力学中的应用》共分七章，内容包括：矢量与张量，笛卡尔张量，张量场论，张量场函数的导数，张量分析在线弹性理论中的应用，张量分析在流体力学中的应用。