# Mechanics of Deformable Bodies

### Lectures on Theoretical Physics

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Mechanics of Deformable Bodies: Lectures on Theoretical Physics, Volume II covers topics on the mechanics of deformable bodies. The book discusses the kinematics, statics, and dynamics of deformable bodies; the vortex theory; as well as the theory of waves. The text also describes the flow with given boundaries. Supplementary notes on selected hydrodynamic problems and supplements to the theory of elasticity are provided. Physicists, mathematicians, and students taking related courses will find the book useful.

## Table of Contents

Chapter I. Kinematics of Deformable Bodies

1. A Fundamental Theorem of Kinematics

2. Review of Vector Analysis

3. The Theorems of Gauss, Stokes, and Green

4. Some Remarks on Tensor Analysis

Chapter II. Statics of Deformable Bodies

5. Concept of Stress; General Classification of Deformable Bodies

6. Equilibrium of Incompressible Fluids (Hydrostatics)

7. Statics of Compressible Fluids

8. The State of Stress of an Elastic Solid

9. Strain-Stress Relations, Elastic Constants, Elastic Potential

10. Viscous Pressures and Dissipation, Particularly in Incompressible Fluids . 73

Chapter III. Dynamics of Deformable Bodies

11. Euler's Equations for a Perfect Incompressible Fluid

12. Derivation of Euler's Equations from Hamilton's Principle. The Pressure, a Lagrange Multiplier

13. Euler's Equations for the Perfect Compressible Fluid and Their Application to Acoustics

Appendix. Comparison of Compressible and Incompressible Flows

14. Dynamics of the Elastic Body

15. The Quasi-Elastic Body as Model of the Ether

16. Dynamics of Viscous Fluids. Hydrodynamics and Hydraulics. Reynolds' Criterion of Turbulence

17. Some Remarks on Capillarity

Chapter IV. Vortex Theory

18. Helmholtz's Vortex Theorems

1. The Differential Form of the Conservation Theorem

2. The Integral Form of the Conservation Theorem

3. The Spatial Distribution of the Vorticity

19. Two- and Three-dimensional Potential Flow

20. A Fundamental Theorem of Vector Analysis

21. Straight and Parallel Vortex Filaments

1. The Single Vortex Filament

2. Two Vortex Filaments of Equal Strength and Opposite or Equal Sense

3. A Theorem Concerning the "Center of Mass" of Two or More Vortices

4. The Law of Areas for a System of Vortex Filaments

5. General Remarks on the Dynamics of Vortices

6. Atmospheric Vortices

22. Circular Vortex Rings

Chapter V. Theory of Waves

23. Plane Gravity Waves in Deep Water

24. Plane Gravity Waves in Shallow and Moderately Deep Water

25. Plane Capillary Waves and Combined Capillary-Gravity Waves

26. The Concept of Group Velocity

27. Circular Waves

1. The Periodic Case. Introduction of Bessel Functions

2. Single Disturbance. The Fourier-Bessel Integral

3. Integration with Respect to k. The Method of the Stationary Phase

4. Integration with Respect to a. Discussion of a Limiting Case

28. Ship Waves (Kelvin's Limit Angle and Mach's Angle)

Chapter VI. Flow With Given Boundaries

29. Flow Past a Plate

30. The Problem of the Wake; Surfaces of Discontinuity

31. The Problem of the Free Jet Solved by Conformal Mapping

32. Karman's Vortex Street

Appendix. The Drag Problem

33. Prandtl's Boundary Layer

Chapter VII. Supplementary Notes On Selected Hydrodynamic Problems

34. Lagrange's Equations of Motion

35. Stokes' Resistance Law

36. The Hydrodynamic Theory of Lubrication

37. Riemann's Shock Waves. General Integration of Euler's Equations for a Compressible Fluid in One-dimensional Flow

38. On Turbulence

A. Some Properties of Turbulent Flow

B. Older Attempts at a Mathematical Theory of Reynolds' Turbulence Criterion

C. Reformulation of the Turbulence Problem; the Origin of the Turbulence

D. The Limiting Case of Isotropic Turbulence

E. A Mathematical Model to Illustrate the Turbulence Problem

Chapter VIII. Supplements To The Theory Op Elasticity

39. Elastic Limit, Proportional Limit, Yield Point, Plasticity, and Strength

40. Crystal Elasticity

41. The Bending of Beams

42. Torsion

43. Torsion and Bending of a Helical Spring

44. The Elastic Energy Content of a Rectangular Parallelepiped

45. The Surface Waves of the Elastic Half-Space

A. Reflection of a Plane Transverse Spatial Wave

B. Elastic Surface Waves

Problems

Chapter I

Chapter II

Chapter III

Chapter IV

Chapters V and VI

Chapter VII

Chapter VIII

Answers and Comments

Appendix I

Appendix II

Appendix III

Appendix IV

Index

## Product details

- No. of pages: 408
- Language: English
- Copyright: © Academic Press 1950
- Published: January 1, 1950
- Imprint: Academic Press
- eBook ISBN: 9781483220277

## About the Author

### Arnold Sommerfeld

#### Affiliations and Expertise

University of Munich

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