# The Nonlinear Theory of Elastic Shells

## 1st Edition

### One Spatial Dimension

**Authors:**A. Libai

**eBook ISBN:**9780323150811

**Imprint:**Academic Press

**Published Date:**28th January 1988

**Page Count:**428

## Description

The Nonlinear Theory of Elastic Shells: One Spatial Dimension presents the foundation for the nonlinear theory of thermoelastic shells undergoing large strains and large rotations. This book discusses several relatively simple equations for practical application.

Organized into six chapters, this book starts with an overview of the description of nonlinear elastic shell. This text then discusses the foundation of three-dimensional continuum mechanics that are relevant to the shell theory approach. Other chapters cover several topics, including birods, beamshells, and axishells that begins with a derivation of the equations of motion by a descent from the equations of balance of linear and rotational momentum of a three-dimensional material continuum. This book discusses as well the approach to deriving complete field equations for one- or two-dimensional continua from the integral equations of motion and thermodynamics of a three-dimensional continuum. The final chapter deals with the analysis of unishells.
This book is a valuable resource for physicists, mathematicians, and scientists.

## Table of Contents

Preface

Chapter I: Introduction

A. What is a Shell?

B. Elastic Shells and Nonlinear Behavior

C. Approaches to Shell Theory

D. The Approach of this Book to Shell Theory

E. Outline of the Book

References

Chapter II: The Generic Equations of 3-Dimensional Continuum Mechanics

A. The Integral Equations of Motion

B. Stress Vectors

C. Heat

D. The Clausius-Duhem (-Truesdell-Toupin) Inequality

E. The First Piola-Kirchhoff Stress Tensor

F. Gross Equations of Motion

References

Chapter III: Longitudinal Motion of Straight Rods with Bi-Symmetric Cross-Sections (BIRODS)

A. Geometry of the Undeformed Rod

B. Integral Equation of Motion

C Differential Equation of Motion

D. Jump Condition and Propagation of Singularities

E. The Weak Form of the Equation of Motion

F. The Mechanical Work Identity

G. Mechanical Boundary Conditions

H. The Principle of Virtual Work

I. The Mechanical Theory of Birods

J. The Mechanical Theory of Elastic Birods

1. Restrictions on the Strain-Energy Density

2. The Equation of Motion in Displacement and Intrinsic Forms

K. Variational Principles

1. Hamilton's Principle

2. Variational Principles for Elastostatics

L. Thermal Equations

M. The First Law of Thermodynamics

N. Thermoelastic Birods

1. Linearized Constitutive Equations for N, h, and ι

References

Chapter IV: Cylindrical Motion of Infinite Cylindrical Shells (Beamshells)

A. Geometry of the Undeformed Shell and Planar Motion

B. Integral Equations of Cylindrical Motion

C. Initial and Spin Bases

D. Jump Conditions and Propagation of Singularities

E. Differential Equations of Cylindrical Motion

F. The Weak Form of the Equations of Motion

G. The Mechanical Work Identity

H. Mechanical Boundary Conditions

1. General Boundary Conditions for Nonholonomic Constraints

2. Classical Boundary Conditions

3. Typical Boundary Conditions

Appendix: The Principle of Mechanical Boundary Conditions (by Dawn Fisher)

I. The Principle of Virtual Work

J. Potential (Conservative) Loads

1. Dead Loading (and a Torsional Spring)

2. Centrifugal Loading

3. Pressure Loading (Constant or Hydrostatic)

4. General Discussion and Examples

K. Strains

L. Alternative Strains and Stresses

M. The Mechanical Theory of Beamshells

N. Elastic Beamshells and Strain-Energy Densities

1. Quadratic Strain-Energy Densities

2. General Strain-Energy Densities

3. Strain-Energy Density by Descent from 3-Dimensions

O. Elastostatics

1. Inextensional Beamshells

2. Pressure Laded Beamshells

P. Elastodynamics

1. Displacement-Shear Strain Form

2. Stress Resultant-Rotation Form

3. A System of 1st-Order Equations

4. Classical Flexural Motion

Q. Variational Principles for Beamshells

1. Hamilton's Principle

2. Variational Principles for Elastostatics

R. The Mechanical Theory of Stability

1. Buckling Equations

2. Shallow Beamshells

S. Some Remarks on Failure Criteria and Stress Calculations

1. Some Large-Strain Refinements

2. Determination of the Deformed Configuration

T. Thermodynamics

U. The Thermodynamic Theory of Stability of Equilibrium

References

Chapter V: Torsionless, Axisymmetric Motion of Shells of Revolution (Axishells)

A. Geometry of the Undeformed Shell

B. Integral Equations of Motion

C. Differential Equations of Motion

D. Differential Equations of Torsionless, Axisymmetric Motion

E. Initial and Spin Bases

F. Jump Conditions and Propagation of Singularities

G. The Weak Form of the Equations of Motion

H. The Mechanical Work Indentity

I. Mechanical Boundary Conditions

J. The Principle of Virtual Work

K. Load Potentials

1. Self-Weight (Gravity Loading)

2. Centrifugal Loading

3. Arbitrary Normal Pressure

L. Strains

M. Compatibility Conditions

N. The Mechanical Theory of Axishells

O. Elastic Axishells and Strain-Energy Densities

1. Quadratic Strain-Energy Densities

2. General Strain-Energy Densities

3. Elastic Isotropy

4. Approximations to the Strain-Energy Density

5. Strain-Energy Density by Descent from 3-Dimensions

P. Alternative Strains and Stresses

Q. Elastostatics

1. General Field Equations Using the Modified Strain-Energy Density Y=Φ-gQ

2. General Field Equations Using a Mixedenergy Density Ψ

R. The Simplified Reissner Equations for Small Static Strains

1. Membrane Theory

2. Moderate Rotation Theory

3. Nonlinear Shallow Shell Theory

S. Special Cases of the Simplified Reissner Equations

1. Cylindrical Shells and Membranes

2. Circular Plates and Membranes

3. Conical Shells

4. Spherical Shells and Membranes

5. Toroidal Shells and Membranes of General Cross Section

T. Nonlinear, Large-Strain Membrane Theory (Including Wrinkling)

1. What is a Membrane?

2. Constitutive Relations

3. Field Equations and Boundary Conditions

4. Some Special Large-Strain Problems

5. Asymptotic Approximations

6. Wrinkled Membranes

U. Elastodynamics

1. Displacement-Shear Strain Form

2. Rotation-Stress Resultant Form

3. A System of 1st-Order Equations

4. Intrinsic Form

5. Some Special Topics

V. Variational Principles for Axishells

1. Hamilton's Principle

2. Variational Principles for Elastostatics

3. Remarks

4. Examples

W. The Mechanical Theory of Stability of Axishells

1. Buckling Equations

2. Effects of Other Loads

3. Simplification of the Buckling Equations

4. Buckling of Axisymmetric Plates

5. The Postbuckling of Axishells: A Short Survey

X. Thermodynamics

References

Chapter VI: Shells Suffering 1-Dimensional Strains (Unishells)

A. In-Plane Bending of Pressurized Curved Tubes

1. Introduction

2. Geometry

3. External Loads

4. Semi-Inverse Approach

5. Kinematics of Deformation

6. Equilibrium Equations

7. Constitutive Relations

8. Boundary and End Conditions

9. Reductions and Approximations

10. The Collapse of Tubes in Bending

B. Variational Principles for Curved Tubes

1. General Remarks

2. Principles of Virtual Work and Stationary Total Potential

3. Extended Variational Principles

C. Helicoidal Shells

1. Deformation of an Arbitrary Shell

2. Dependence of the Metric and Curvature Components on the Same, Single, Surface Coordinate Implies a General Helicoid

3. The Geometry of a General Helicoid

D. Force and Moment Equilibrium

E. Virtual Work and Strains

F. The Rotation Vector for 1-Dimensional Strains

G. Strain Compatibility

H. Component Form of the Field Equations

1. Compatibility Conditions

2. Force Equilibrium

3. Moment Equilibrium

4. Constitutive Relations

I. Special Cases

1. Axishells

2. Pure Bending of Pressurized Curved Tubes

3. Torsion, Inflation, and Extension of a Tube

4. Extension and Twist of a Right Helicoidal Shell

5. Inextensional Deformation

References

Appendices

A. Notation

1. General Scheme of Notation

2. Global Notations

B. Guide to 3-Dimensional Strain-Energy Densities

Index

## Details

- No. of pages:
- 428

- Language:
- English

- Copyright:
- © Academic Press 1988

- Published:
- 28th January 1988

- Imprint:
- Academic Press

- eBook ISBN:
- 9780323150811