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Foundations Of The Non Linear Theory Of Elasticity by V. V. Novozhilov

This book is based on a course of lectures given by the author in 1947 in the Mathematical-Mechanical Department of Leningrad National University. It is devoted to the exposition of the theory of elasticity without any assumptions restricting the magnitude of elongations, displacements, or angles of rotation. It also examines, in a general formulation, the connection between stresses and strains in an isotropic elastic body.

Translated from the First (1948) Russian Edition by F. Bagemihl, H. Komm, W. Seidel

You can get the book here and here

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Contents

Preface

Chapter I. The Geometry of Strain
§ 1. Coordinates
§ 2. The Angles Determining the Directions of the Coordinate Lines
§ 3. Strain Components
§ 4. Transformation of Strain Components Under Change of Axes
§ 5. Principal Axes of Strain
§ 6. Transformation of the Parameters e_r and u Under Change of Coordinate Axes
§ 7. Geometrical Meaning of the Parameters
§ 8. Fibers Preserving Direction Under Deformation
§ 9. Invariants of Strain and Rotation
§ 10. The General Picture of the Deformation in the Neighborhood of an Arbitrary Point of the Body
§ 11. Change in Volume
§ 12. On the Magnitude of Elongations and Shears
§ 13. The Theory of Small Deformations
§ 14. The Case of Small Deformations and Small Angles of Rotation
§ 15. The Transition to the Equations of the Classical Theory
§ 16. On the Transition to Curvilinear Coordinates

Chapter II. The Equilibrium of an Element of Volume of a Body
§ 17. Stresses
§ 18. Formulas for Transformation of Stress Components Under Change of Coordinate System
§ 19. Conditions for Equilibrium of an Elementary Parallelepiped Isolated From a Deformed Body
§ 20. Transformation of the Equations of Equilibrium of an Element of Volume to the Cartesian Coordinates of the Points of the Body Before Its Deformation
§ 21. Simplification of the Equations of Equilibrium in the Case of Small Elongations and Shears
§ 22. Simplification of the Equations of Equilibrium for Small Rotations
§ 23. Transition to the Classical Equations of Equilibrium
§ 24. Transition to Curvilinear Coordinates

Chapter III. Strain Energy, Boundary Conditions, Stress-Strain Law
§ 25. Strain Energy
§ 26. The Principle of Virtual Displacements
§ 27. Derivation of the Differential Equations of Equilibrium of a Deformed Isotropic Body from the Principle of Virtual Displacements
§ 28. The Relation Between Stress and Strain Components
§ 29. Boundary Conditions
§ 30. The Simplification of the Derived Equations in the Case of a Small Deformation
§ 31. Hooke’s Law
§ 32. On the Applicability of Equations (III.38) to Elastic-Plastic Deformations
§ 33. On the Simplest Variants of Nonlinear Stress-Strain Relations
§ 34. Conclusion

Chapter IV. Formulation of Elastic Problems in Terms of Stresses
§ 35. Two Further Forms for the Equations of Equilibrium of a Volume Element
§ 36. Simplification of Equations (IV.7) and (IV.8) for Small Deformations
§ 37. Still Another Form of the Boundary Conditions
§ 38. Simplification of Equations (IV.7) and (IV.8) for Small Angles of Rotation
§ 39. The Generalization of Saint-Venant’s Relations to the Case of Large Rotations and Strains
§ 40. Simplification of the Equations (IV.26) for Small Deformations
§ 41. On the Formulation of the Problems of the Theory of Elasticity in Terms of Stresses and Strains

Chapter V. The Problem of Elastic Stability
§ 42. Nonuniqueness of Solutions in the Theory of Elasticity
§ 43. The Differential Equations Which Determine the Critical Loads
§ 44. Boundary Conditions of the Problem of Elastic Stability
§ 45. Energy Criterion for the Determination of Critical Loads

Chapter VI. On the Deformation of Flexible Bodies
§ 46. Deformation of Plates
§ 47. Two-Dimensional Deformation of an Infinitely Long Strip
§ 48. Deformation of Shells
§ 49. On the Nature of Kirchhoff’s Assumptions
§ 50. Deformation of Rods (First Approximation)
§ 51. Deformation of Rods (Second Approximation)
§ 52. Pure Torsion
§ 53. The Final Expressions for the Strain Components of a Thin Rod
§ 54. Conclusion

Bibliography

Foundations Of The Non Linear Theory Of Elasticity by V. V. Novozhilov

This book is based on a course of lectures given by the author in 1947 in the Mathematical-Mechanical Department of Leningrad National University. It is devoted to the exposition of the theory of elasticity without any assumptions restricting the magnitude of elongations, displacements, or angles of rotation. It also examines, in a general formulation, the connection between stresses and strains in an isotropic elastic body.

Translated from the First (1948) Russian Edition by F. Bagemihl, H. Komm, W. Seidel

You can get the book here and here

Follow us on

Twitter https://x.com/MirTitles

Mastadon https://mastodon.social/@mirtitles

Bluesky https://bsky.app/profile/mirtitles.bsky.social

Tumblr https://www.tumblr.com/mirtitles

Internet Archive https://archive.org/details/mir-titles

Fork us on gitlab https://gitlab.com/mirtitles

Contents

Preface

Chapter I. The Geometry of Strain
§ 1. Coordinates
§ 2. The Angles Determining the Directions of the Coordinate Lines
§ 3. Strain Components
§ 4. Transformation of Strain Components Under Change of Axes
§ 5. Principal Axes of Strain
§ 6. Transformation of the Parameters e_r and u Under Change of Coordinate Axes
§ 7. Geometrical Meaning of the Parameters
§ 8. Fibers Preserving Direction Under Deformation
§ 9. Invariants of Strain and Rotation
§ 10. The General Picture of the Deformation in the Neighborhood of an Arbitrary Point of the Body
§ 11. Change in Volume
§ 12. On the Magnitude of Elongations and Shears
§ 13. The Theory of Small Deformations
§ 14. The Case of Small Deformations and Small Angles of Rotation
§ 15. The Transition to the Equations of the Classical Theory
§ 16. On the Transition to Curvilinear Coordinates

Chapter II. The Equilibrium of an Element of Volume of a Body
§ 17. Stresses
§ 18. Formulas for Transformation of Stress Components Under Change of Coordinate System
§ 19. Conditions for Equilibrium of an Elementary Parallelepiped Isolated From a Deformed Body
§ 20. Transformation of the Equations of Equilibrium of an Element of Volume to the Cartesian Coordinates of the Points of the Body Before Its Deformation
§ 21. Simplification of the Equations of Equilibrium in the Case of Small Elongations and Shears
§ 22. Simplification of the Equations of Equilibrium for Small Rotations
§ 23. Transition to the Classical Equations of Equilibrium
§ 24. Transition to Curvilinear Coordinates

Chapter III. Strain Energy, Boundary Conditions, Stress-Strain Law
§ 25. Strain Energy
§ 26. The Principle of Virtual Displacements
§ 27. Derivation of the Differential Equations of Equilibrium of a Deformed Isotropic Body from the Principle of Virtual Displacements
§ 28. The Relation Between Stress and Strain Components
§ 29. Boundary Conditions
§ 30. The Simplification of the Derived Equations in the Case of a Small Deformation
§ 31. Hooke’s Law
§ 32. On the Applicability of Equations (III.38) to Elastic-Plastic Deformations
§ 33. On the Simplest Variants of Nonlinear Stress-Strain Relations
§ 34. Conclusion

Chapter IV. Formulation of Elastic Problems in Terms of Stresses
§ 35. Two Further Forms for the Equations of Equilibrium of a Volume Element
§ 36. Simplification of Equations (IV.7) and (IV.8) for Small Deformations
§ 37. Still Another Form of the Boundary Conditions
§ 38. Simplification of Equations (IV.7) and (IV.8) for Small Angles of Rotation
§ 39. The Generalization of Saint-Venant’s Relations to the Case of Large Rotations and Strains
§ 40. Simplification of the Equations (IV.26) for Small Deformations
§ 41. On the Formulation of the Problems of the Theory of Elasticity in Terms of Stresses and Strains

Chapter V. The Problem of Elastic Stability
§ 42. Nonuniqueness of Solutions in the Theory of Elasticity
§ 43. The Differential Equations Which Determine the Critical Loads
§ 44. Boundary Conditions of the Problem of Elastic Stability
§ 45. Energy Criterion for the Determination of Critical Loads

Chapter VI. On the Deformation of Flexible Bodies
§ 46. Deformation of Plates
§ 47. Two-Dimensional Deformation of an Infinitely Long Strip
§ 48. Deformation of Shells
§ 49. On the Nature of Kirchhoff’s Assumptions
§ 50. Deformation of Rods (First Approximation)
§ 51. Deformation of Rods (Second Approximation)
§ 52. Pure Torsion
§ 53. The Final Expressions for the Strain Components of a Thin Rod
§ 54. Conclusion

Bibliography

Foundations Of The Non Linear Theory Of Elasticity by V. V. Novozhilov

This book is based on a course of lectures given by the author in 1947 in the Mathematical-Mechanical Department of Leningrad National University. It is devoted to the exposition of the theory of elasticity without any assumptions restricting the magnitude of elongations, displacements, or angles of rotation. It also examines, in a general formulation, the connection between stresses and strains in an isotropic elastic body.

Translated from the First (1948) Russian Edition by F. Bagemihl, H. Komm, W. Seidel

You can get the book here and here

Follow us on

Twitter https://x.com/MirTitles

Mastadon https://mastodon.social/@mirtitles

Bluesky https://bsky.app/profile/mirtitles.bsky.social

Tumblr https://www.tumblr.com/mirtitles

Internet Archive https://archive.org/details/mir-titles

Fork us on gitlab https://gitlab.com/mirtitles

Contents

Preface

Chapter I. The Geometry of Strain
§ 1. Coordinates
§ 2. The Angles Determining the Directions of the Coordinate Lines
§ 3. Strain Components
§ 4. Transformation of Strain Components Under Change of Axes
§ 5. Principal Axes of Strain
§ 6. Transformation of the Parameters e_r and u Under Change of Coordinate Axes
§ 7. Geometrical Meaning of the Parameters
§ 8. Fibers Preserving Direction Under Deformation
§ 9. Invariants of Strain and Rotation
§ 10. The General Picture of the Deformation in the Neighborhood of an Arbitrary Point of the Body
§ 11. Change in Volume
§ 12. On the Magnitude of Elongations and Shears
§ 13. The Theory of Small Deformations
§ 14. The Case of Small Deformations and Small Angles of Rotation
§ 15. The Transition to the Equations of the Classical Theory
§ 16. On the Transition to Curvilinear Coordinates

Chapter II. The Equilibrium of an Element of Volume of a Body
§ 17. Stresses
§ 18. Formulas for Transformation of Stress Components Under Change of Coordinate System
§ 19. Conditions for Equilibrium of an Elementary Parallelepiped Isolated From a Deformed Body
§ 20. Transformation of the Equations of Equilibrium of an Element of Volume to the Cartesian Coordinates of the Points of the Body Before Its Deformation
§ 21. Simplification of the Equations of Equilibrium in the Case of Small Elongations and Shears
§ 22. Simplification of the Equations of Equilibrium for Small Rotations
§ 23. Transition to the Classical Equations of Equilibrium
§ 24. Transition to Curvilinear Coordinates

Chapter III. Strain Energy, Boundary Conditions, Stress-Strain Law
§ 25. Strain Energy
§ 26. The Principle of Virtual Displacements
§ 27. Derivation of the Differential Equations of Equilibrium of a Deformed Isotropic Body from the Principle of Virtual Displacements
§ 28. The Relation Between Stress and Strain Components
§ 29. Boundary Conditions
§ 30. The Simplification of the Derived Equations in the Case of a Small Deformation
§ 31. Hooke’s Law
§ 32. On the Applicability of Equations (III.38) to Elastic-Plastic Deformations
§ 33. On the Simplest Variants of Nonlinear Stress-Strain Relations
§ 34. Conclusion

Chapter IV. Formulation of Elastic Problems in Terms of Stresses
§ 35. Two Further Forms for the Equations of Equilibrium of a Volume Element
§ 36. Simplification of Equations (IV.7) and (IV.8) for Small Deformations
§ 37. Still Another Form of the Boundary Conditions
§ 38. Simplification of Equations (IV.7) and (IV.8) for Small Angles of Rotation
§ 39. The Generalization of Saint-Venant’s Relations to the Case of Large Rotations and Strains
§ 40. Simplification of the Equations (IV.26) for Small Deformations
§ 41. On the Formulation of the Problems of the Theory of Elasticity in Terms of Stresses and Strains

Chapter V. The Problem of Elastic Stability
§ 42. Nonuniqueness of Solutions in the Theory of Elasticity
§ 43. The Differential Equations Which Determine the Critical Loads
§ 44. Boundary Conditions of the Problem of Elastic Stability
§ 45. Energy Criterion for the Determination of Critical Loads

Chapter VI. On the Deformation of Flexible Bodies
§ 46. Deformation of Plates
§ 47. Two-Dimensional Deformation of an Infinitely Long Strip
§ 48. Deformation of Shells
§ 49. On the Nature of Kirchhoff’s Assumptions
§ 50. Deformation of Rods (First Approximation)
§ 51. Deformation of Rods (Second Approximation)
§ 52. Pure Torsion
§ 53. The Final Expressions for the Strain Components of a Thin Rod
§ 54. Conclusion

Bibliography

Foundations Of The Non Linear Theory Of Elasticity by V. V. Novozhilov

This book is based on a course of lectures given by the author in 1947 in the Mathematical-Mechanical Department of Leningrad National University. It is devoted to the exposition of the theory of elasticity without any assumptions restricting the magnitude of elongations, displacements, or angles of rotation. It also examines, in a general formulation, the connection between stresses and strains in an isotropic elastic body.

Translated from the First (1948) Russian Edition by F. Bagemihl, H. Komm, W. Seidel

You can get the book here and here

Follow us on

Twitter https://x.com/MirTitles

Mastadon https://mastodon.social/@mirtitles

Bluesky https://bsky.app/profile/mirtitles.bsky.social

Tumblr https://www.tumblr.com/mirtitles

Internet Archive https://archive.org/details/mir-titles

Fork us on gitlab https://gitlab.com/mirtitles

Contents

Preface

Chapter I. The Geometry of Strain
§ 1. Coordinates
§ 2. The Angles Determining the Directions of the Coordinate Lines
§ 3. Strain Components
§ 4. Transformation of Strain Components Under Change of Axes
§ 5. Principal Axes of Strain
§ 6. Transformation of the Parameters e_r and u Under Change of Coordinate Axes
§ 7. Geometrical Meaning of the Parameters
§ 8. Fibers Preserving Direction Under Deformation
§ 9. Invariants of Strain and Rotation
§ 10. The General Picture of the Deformation in the Neighborhood of an Arbitrary Point of the Body
§ 11. Change in Volume
§ 12. On the Magnitude of Elongations and Shears
§ 13. The Theory of Small Deformations
§ 14. The Case of Small Deformations and Small Angles of Rotation
§ 15. The Transition to the Equations of the Classical Theory
§ 16. On the Transition to Curvilinear Coordinates

Chapter II. The Equilibrium of an Element of Volume of a Body
§ 17. Stresses
§ 18. Formulas for Transformation of Stress Components Under Change of Coordinate System
§ 19. Conditions for Equilibrium of an Elementary Parallelepiped Isolated From a Deformed Body
§ 20. Transformation of the Equations of Equilibrium of an Element of Volume to the Cartesian Coordinates of the Points of the Body Before Its Deformation
§ 21. Simplification of the Equations of Equilibrium in the Case of Small Elongations and Shears
§ 22. Simplification of the Equations of Equilibrium for Small Rotations
§ 23. Transition to the Classical Equations of Equilibrium
§ 24. Transition to Curvilinear Coordinates

Chapter III. Strain Energy, Boundary Conditions, Stress-Strain Law
§ 25. Strain Energy
§ 26. The Principle of Virtual Displacements
§ 27. Derivation of the Differential Equations of Equilibrium of a Deformed Isotropic Body from the Principle of Virtual Displacements
§ 28. The Relation Between Stress and Strain Components
§ 29. Boundary Conditions
§ 30. The Simplification of the Derived Equations in the Case of a Small Deformation
§ 31. Hooke’s Law
§ 32. On the Applicability of Equations (III.38) to Elastic-Plastic Deformations
§ 33. On the Simplest Variants of Nonlinear Stress-Strain Relations
§ 34. Conclusion

Chapter IV. Formulation of Elastic Problems in Terms of Stresses
§ 35. Two Further Forms for the Equations of Equilibrium of a Volume Element
§ 36. Simplification of Equations (IV.7) and (IV.8) for Small Deformations
§ 37. Still Another Form of the Boundary Conditions
§ 38. Simplification of Equations (IV.7) and (IV.8) for Small Angles of Rotation
§ 39. The Generalization of Saint-Venant’s Relations to the Case of Large Rotations and Strains
§ 40. Simplification of the Equations (IV.26) for Small Deformations
§ 41. On the Formulation of the Problems of the Theory of Elasticity in Terms of Stresses and Strains

Chapter V. The Problem of Elastic Stability
§ 42. Nonuniqueness of Solutions in the Theory of Elasticity
§ 43. The Differential Equations Which Determine the Critical Loads
§ 44. Boundary Conditions of the Problem of Elastic Stability
§ 45. Energy Criterion for the Determination of Critical Loads

Chapter VI. On the Deformation of Flexible Bodies
§ 46. Deformation of Plates
§ 47. Two-Dimensional Deformation of an Infinitely Long Strip
§ 48. Deformation of Shells
§ 49. On the Nature of Kirchhoff’s Assumptions
§ 50. Deformation of Rods (First Approximation)
§ 51. Deformation of Rods (Second Approximation)
§ 52. Pure Torsion
§ 53. The Final Expressions for the Strain Components of a Thin Rod
§ 54. Conclusion

Bibliography

Foundations Of The Non Linear Theory Of Elasticity by V. V. Novozhilov

This book is based on a course of lectures given by the author in 1947 in the Mathematical-Mechanical Department of Leningrad National University. It is devoted to the exposition of the theory of elasticity without any assumptions restricting the magnitude of elongations, displacements, or angles of rotation. It also examines, in a general formulation, the connection between stresses and strains in an isotropic elastic body.

Translated from the First (1948) Russian Edition by F. Bagemihl, H. Komm, W. Seidel

You can get the book here and here

Follow us on

Twitter https://x.com/MirTitles

Mastadon https://mastodon.social/@mirtitles

Bluesky https://bsky.app/profile/mirtitles.bsky.social

Tumblr https://www.tumblr.com/mirtitles

Internet Archive https://archive.org/details/mir-titles

Fork us on gitlab https://gitlab.com/mirtitles

Contents

Preface

Chapter I. The Geometry of Strain
§ 1. Coordinates
§ 2. The Angles Determining the Directions of the Coordinate Lines
§ 3. Strain Components
§ 4. Transformation of Strain Components Under Change of Axes
§ 5. Principal Axes of Strain
§ 6. Transformation of the Parameters e_r and u Under Change of Coordinate Axes
§ 7. Geometrical Meaning of the Parameters
§ 8. Fibers Preserving Direction Under Deformation
§ 9. Invariants of Strain and Rotation
§ 10. The General Picture of the Deformation in the Neighborhood of an Arbitrary Point of the Body
§ 11. Change in Volume
§ 12. On the Magnitude of Elongations and Shears
§ 13. The Theory of Small Deformations
§ 14. The Case of Small Deformations and Small Angles of Rotation
§ 15. The Transition to the Equations of the Classical Theory
§ 16. On the Transition to Curvilinear Coordinates

Chapter II. The Equilibrium of an Element of Volume of a Body
§ 17. Stresses
§ 18. Formulas for Transformation of Stress Components Under Change of Coordinate System
§ 19. Conditions for Equilibrium of an Elementary Parallelepiped Isolated From a Deformed Body
§ 20. Transformation of the Equations of Equilibrium of an Element of Volume to the Cartesian Coordinates of the Points of the Body Before Its Deformation
§ 21. Simplification of the Equations of Equilibrium in the Case of Small Elongations and Shears
§ 22. Simplification of the Equations of Equilibrium for Small Rotations
§ 23. Transition to the Classical Equations of Equilibrium
§ 24. Transition to Curvilinear Coordinates

Chapter III. Strain Energy, Boundary Conditions, Stress-Strain Law
§ 25. Strain Energy
§ 26. The Principle of Virtual Displacements
§ 27. Derivation of the Differential Equations of Equilibrium of a Deformed Isotropic Body from the Principle of Virtual Displacements
§ 28. The Relation Between Stress and Strain Components
§ 29. Boundary Conditions
§ 30. The Simplification of the Derived Equations in the Case of a Small Deformation
§ 31. Hooke’s Law
§ 32. On the Applicability of Equations (III.38) to Elastic-Plastic Deformations
§ 33. On the Simplest Variants of Nonlinear Stress-Strain Relations
§ 34. Conclusion

Chapter IV. Formulation of Elastic Problems in Terms of Stresses
§ 35. Two Further Forms for the Equations of Equilibrium of a Volume Element
§ 36. Simplification of Equations (IV.7) and (IV.8) for Small Deformations
§ 37. Still Another Form of the Boundary Conditions
§ 38. Simplification of Equations (IV.7) and (IV.8) for Small Angles of Rotation
§ 39. The Generalization of Saint-Venant’s Relations to the Case of Large Rotations and Strains
§ 40. Simplification of the Equations (IV.26) for Small Deformations
§ 41. On the Formulation of the Problems of the Theory of Elasticity in Terms of Stresses and Strains

Chapter V. The Problem of Elastic Stability
§ 42. Nonuniqueness of Solutions in the Theory of Elasticity
§ 43. The Differential Equations Which Determine the Critical Loads
§ 44. Boundary Conditions of the Problem of Elastic Stability
§ 45. Energy Criterion for the Determination of Critical Loads

Chapter VI. On the Deformation of Flexible Bodies
§ 46. Deformation of Plates
§ 47. Two-Dimensional Deformation of an Infinitely Long Strip
§ 48. Deformation of Shells
§ 49. On the Nature of Kirchhoff’s Assumptions
§ 50. Deformation of Rods (First Approximation)
§ 51. Deformation of Rods (Second Approximation)
§ 52. Pure Torsion
§ 53. The Final Expressions for the Strain Components of a Thin Rod
§ 54. Conclusion

Bibliography