The Nonlinear Theory of Elastic Shells

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ISBN: 9780521472364 | 0521472369
Cover: Hardcover
Copyright: 2/13/1998

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Elastic shells are pervasive in everyday life. Examples of these thin-walled structures range from automobile hoods to basketballs, veins, arteries and soft drink cans. This book explains shell theory, with numerous examples and applications. As a second edition, it not only brings all the material of the first edition entirely up to date, it also adds two entirely new chapters on general shell theory and general membrane theory. Aerospace, mechanical and civil engineers, as well as applied mathematicians, will find this book a clearly written and thorough information source on shell theory.

PREFACE

xiii

(2)

PREFACE TO THE FIRST EDITION

CHAPTER I: INTRODUCTION

(10)

A. What is a Shell?

(1)

B. Elastic Shells and Nonlinear Behavior

(1)

C. Approaches to Shell Theory

(1)

D. The Approach of this Book to Shell Theory

(1)

E. Outline of the Book

(2)

References

(5)

CHAPTER II: THE GENERIC EQUATIONS OF THREE-DIMENSIONAL CONTINUUM MECHANICS

(10)

A. The Integral Equations of Motion

(2)

B. Stress Vectors

(1)

C. Heat

(1)

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

(1)

E. The First Piola-Kirchhoff Stress Tensor

(1)

*F. Gross Equations of Motion

(4)

References

(1)

CHAPTER III: LONGITUDINAL MOTION OF STRAIGHT RODS WITH BISYMMETRIC CROSS SECTIONS (BIRODS)

(30)

A. Geometry of the Undeformed Rod

(1)

B. Integral Equation of Motion

(4)

C. Differential Equation of Motion

(1)

*D. Jump Condition and Propagation of Singularities

(3)

E. The Weak Form of the Equation of Motion

(2)

F. The Mechanical Work Identity

(1)

G. Mechanical Boundary Conditions

(2)

H. The Principle of Virtual Work

(3)

1. The Principle of Virtual Work and boundary conditions in statics

(1)

I. The Mechanical Theory of Birods

(1)

J. The Mechanical Theory of Elastic Birods

(2)

1. Restrictions on the strain-energy density

(1)

2. The equation of motion in displacement and intrinsic forms

(1)

K. Variational Principles

(7)

1. Hamilton's Principle

(1)

2. Variational principles for elastostatics

(4)

L. Thermal Equations

(1)

M. The First Law of Thermodynamics

(1)

N. Thermoelastic Birods

(2)

1. Linearized constitutive equations for N, h, and t

(1)

References

(3)

CHAPTER IV: CYLINDRICAL MOTION OF INFINITE CYCLINDRICAL SHELLS (BEAMSHELLS)

(108)

A. Geometry of the Undeformed Shell and Planar Motion

(2)

B. Integral Equations of Cylindrical Motion

(4)

1. Natural initial and boundary conditions

(1)

C. Initial and Spin Bases

(1)

*D. Jump Conditions and Propagation of Singularities

(1)

E. Differential Equations of Cylindrical Motion

(1)

F. The Weak Form of the Equations of Motion

(1)

G. The Mechanical Work Identity

(1)

H. Mechanical Boundary Conditions

(6)

1. General boundary conditions for nonholonomic constraints

(2)

2. Classical boundary conditions

(1)

3. Typical boundary conditions

(3)

*Appendix: The Principle of Mechanical Boundary Conditions

(1)

Dawn Fisher

I. The Principle of Virtual Work

(2)

J. Potential (Conservative) Loads

(7)

1. Dead loading (and a torsional spring)

(1)

2. Centrifugal loading

(1)

3. Pressure loading (constant or hydrostatic)

(1)

*4. General discussion and examples

(5)

K. Strains

(1)

L. Alternative Strains and Stresses

(3)

M. The Mechanical Theory of Beamshells

(1)

N. Elastic Beamshells and Strain-Energy Densities

(13)

1. Quadratic strain-energy densities

(3)

*2. General strain-energy densities

(4)

*3. Strain-energy density by descent from three dimensions

(5)

O. Elastostatics

(11)

1. Inextensional beamshells

(4)

2. Pressure loaded beamshells

100

(5)

P. Elastodynamics

105

(9)

1. Displacement-shear strain form

106

(3)

2. Stress resultant-rotation form

109

(1)

3. A system of first-order equations

109

(1)

4. Classical flexural motion

110

(1)

5. Cartesian frames

111

(3)

Q. Variational Principles for Beamshells

114

(8)

1. Hamilton's Principle

114

(1)

2. Variational principles for elastostatics

115

(7)

R. The Mechanical Theory of Stability

122

(14)

1. Buckling equations

127

(6)

2. Shallow beamshells

133

(3)

S. Some Remarks on Failure Criteria and Stress Calculations

136

(4)

1. Some large-strain refinements

138

(1)

2. Determination of the deformed configuration

139

(1)

T. Thermodynamics

140

(7)

U. The Thermodynamic Theory of Stability of Equilibrium

147

(2)

References

149

(10)

CHAPTER V: TORSIONLESS, AXISYMMETRIC MOTION OF SHELLS OF REVOLUTION (AXISHELLS)

159

(184)

A. Geometry of the Undeformed Shell

160

(1)

B. Integral Equations of Motion

161

(5)

C. Differential Equations of Motion

166

(1)

D. Differential and Integral Equations of Torsionless, Axisymmetric Motion

166

(2)

E. Initial and Spin Bases

168

(1)

*F. Jump Conditions and Propagation of Singularities

168

(1)

G. The Weak Form of the Equations of Motion

169

(1)

H. The Mechanical Work Identity

169

(1)

I. Mechanical Boundary Conditions

170

(2)

J. The Principle of Virtual Work

172

(1)

K. Load Potentials

173

(2)

1. Self-weight (gravity loading)

173

(1)

2. Centrifugal loading

174

(1)

3. Arbitrary normal pressure

174

(1)

L. Strains

175

(2)

M. Compatibility Conditions

177

(1)

N. The Mechanical Theory of Axishells

178

(1)

O. Elastic Axishells and Strain-Energy Densities

179

(13)

1. Quadratic strain-energy densities

180

(2)

*2. General strain-energy densities

182

(2)

*3. Elastic isotropy

184

(2)

*4. Approximations to the strain-energy density

186

(1)

*5. Strain-energy density by descent from three dimensions

187

(5)

P. Alternative Strains and Stresses

192

(4)

Q. Elastostatics

196

(4)

1. General field equations using the modified strain-energy density Y = XXX -- gQ

197

(2)

2. General field equations using a mixed-energy density XXX

199

(1)

R. The Simplified Reissner Equations for Small Static Strains

200

(6)

1. Membrane theory

204

(1)

2. Moderate rotation theory

205

(1)

3. Nonlinear shallow shell theory

205

(1)

S. Special Cases of the Simplified Reissner Equations

206

(53)

1. Cylindrical shells and membranes

206

(9)

2. Circular plates and membranes

215

(6)

3. Conical shells

221

(8)

4. Spherical shells and membranes

229

(21)

5. Toroidal shells and membranes of general cross section

250

(9)

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

259

(21)

1. What is a membrane?

259

(3)

2. Constitutive relations

262

(3)

3. Field equations and boundary conditions

265

(3)

4. Some special large-strain problems

268

(7)

5. Asymptotic approximations

275

(1)

6. Wrinkled membranes

276

(4)

U. Elastodynamics

280

(10)

1. Displacement-shear strain form

280

(2)

2. Rotation-stress resultant form

282

(1)

3. A system of first-order equations

283

(1)

4. Intrinsic form

284

(3)

5. Some special topics

287

(1)

6. Twisted axishells

288

(2)

V. Variational Principles for Axishells

290

(11)

1. Hamilton's Principle

290

(1)

2. Variational principles for elastostatics

291

(5)

3. Remarks

296

(2)

4. Examples

298

(3)

W. The Mechanical Theory of Stability of Axishells

301

(21)

1. Buckling equations

303

(5)

2. Effects of other loads

308

(1)

3. Simplification of the buckling equations

309

(4)

4. Buckling of axisymmetric plates

313

(4)

5. The postbuckling of shells of revolution: a short survey

317

(5)

X. Thermodynamics

322

(6)

References

328

(15)

CHAPTER VI: SHELLS SUFFERING ONE-DIMENSIONAL STRAINS (UNISHELLS)

343

(46)

A. In-Plane Bending of Pressurized Curved Tubes

344

(13)

1. Introduction

344

(1)

2. Geometry

345

(1)

3. External loads

345

(1)

4. Semi-inverse approach

345

(1)

5. Kinematics of deformation

346

(2)

6. Equilibrium equations

348

(1)

7. Constitutive relations

349

(1)

8. Boundary and end conditions

349

(3)

9. Reductions and approximations

352

(2)

10. The collapse of tubes in bending

354

(3)

B. Variational Principles for Curved Tubes

357

(6)

1. General remarks

357

(1)

2. Principles of Virtual Work and Stationary Total Potential

358

(2)

3. Extended variational principles

360

(3)

C. Helicoidal Shells

363

(2)

1. Deformation of an arbitrary shell

363

(1)

2. Dependence of the metric and curvature components on the same, single surface coordinate implies a general helicoid

364

(1)

3. The geometry of a general helicoid

365

(1)

D. Force and Moment Equilibrium

365

(3)

E. Virtual Work and Strains

368

(2)

F. The Rotation Vector for One-Dimensional Strains

370

(2)

G. Strain Compatibility

372

(1)

H. Component Form of the Field Equations

372

(6)

1. Compatibility conditions

372

(3)

2. Force equilibrium

375

(1)

3. Moment equilibrium

375

(1)

4. Constitutive relations

376

(2)

I. Special Cases

378

(7)

1. Axishells

378

(1)

2. Pure bending of pressurized curved tubes

378

(1)

3. Torsion, inflation, and extension of an infinite tube

378

(2)

4. Extension and twist of a right helicoidal shell

380

(3)

5. Inextensional deformation

383

(2)

References

385

(4)

CHAPTER VII: GENERAL NONLINEAR MEMBRANE THEORY (INCLUDING WRINKLING)

389

(64)

A. The Equations of Motion

389

(2)

B. Natural Initial and Force Boundary Conditions

391

(1)

C. Shocks

392

(1)

D. The Membrane Stress Resultant Tensor

393

(2)

E. Differential Equations of Motion

395

(1)

F. Hybrid and Component Forms of the Equations of Motion

396

(4)

1. A useful hybrid notation

396

(1)

2. Alternative hybrid forms

397

(1)

3. Component forms

398

(2)

G. Weak Form of the Equations of Motion

400

(1)

*1. Direct derivation of the weak form from the integral form

401

(1)

H. The Mechanical Work Identity and the Lagrangian Strain Tensor

401

(2)

I. Strain Compatibility

403

(3)

J. Boundary Conditions and the Principle of Virtual Work

406

(5)

1. Classical boundary conditions for static membrane problems

410

(1)

K. Load Potentials

411

(3)

1. Pressure a function of total volume only

412

(1)

2. Position dependent pressure

413

(1)

L. Constitutive Relations for Elastic Isothermal Membranes

414

(7)

1. Quadratic strain-energy densities

416

(1)

2. Complementary-energy densities

417

(1)

3. Inextensional true membranes

418

(1)

4. Large [O(1)] strain membrane problems

419

(1)

5. Reduction from three dimensions

419

(2)

M. Elastostatics

421

(6)

1. Angular discontinuities

422

(1)

2. The nature of the membrane problem

423

(2)

3. Small-strain, large-rotation theory

425

(1)

4. Small-strain, linearized theory

425

(2)

N. Circular Cylindrical Membranes

427

(4)

1. Example

429

(2)

O. Wrinkling of Membranes

431

(6)

P. The Bending and Wrinkling of Inflated Membrane Tubes

437

(2)

Q. Plane Wrinkled Membranes

439

(2)

R. Isotropic States of Stress and Soap Films

441

(2)

S. Dynamics of Membranes

443

(5)

Appendix: Membrane States

448

(1)

References

448

(5)

CHAPTER VIII: GENERAL SHELLS

453

(58)

A. Geometry of the Undeformed Shell

453

(2)

B. Integral Equations of Motion

455

(2)

C. Natural Initial and Boundary Conditions

457

(1)

D. Shocks

458

(1)

E. Stress Resultant and Stress Couple Tensors

459

(1)

F. Differential (Local) Equations of Motion

460

(1)

G. Hybrid Form of the Equations of Motion

461

(1)

H. Weak Form of the Equations of Motion

461

(2)

I. The Mechanical Work Identity

463

(1)

J. Strains

464

(2)

1. Local or objective rates

464

(2)

K. Strain Compatibility Conditions

466

(2)

L. Finite Rotation Vectors

468

(2)

1. An alternative, coordinate-free representation of XXX

469

(1)

M. Algebraic and Differential Mechanical Boundary Conditions; Virtual Work

470

(2)

N. Load Potentials

472

(1)

O. The Mechanical Theory of Shells

473

(1)

P. Elastic Shells and Strain-Energy Densities

474

(2)

1. Mixed-energy density

475

(1)

Q. The Kirchhoff Hypothesis

476

(18)

1. Form S (for spin basis)

476

(3)

2. An alternative approach to the Kirchhoff Hypothesis: form D (for deformed basis)

479

(6)

3. Some typical strain-and mixed-energy densities

485

(4)

4. Limiting forms as XXX = H/L XXX 0

489

(1)

5. The effects of transverse shearing strains on XXX

490

(1)

6. The approximate two-dimensional strain-energy density

490

(2)

7. Example: the neo-Hookean strain-energy density

492

(1)

8. Note on the external power of the edge loads under the Kirchhoff Hypothesis

493

(1)

R. Component Form of the Equations of Motion

494

(4)

1. Alternative displacement components

496

(1)

2. Component form under the Kirchhoff Hypothesis (D-form)

496

(2)

S. Elastostatics

498

(4)

1. Satisfying force equilibrium with a stress function

498

(1)

2. Satisfying moment equilibrium with another stress function

498

(1)

3. Field equations for stress function and finite rotation vectors: general theory

499

(1)

4. Field equations for stress function and finite rotation vectors: the Kirchhoff Hypothesis

499

(3)

5. Intrinsic static field equations

502

(1)

T. Simplifications of the Shell Equations

502

(5)

1. Dynamic, quasi-shallow shell theory

503

(2)

2. Shell membranes

505

(2)

References

507

(4)

APPENDICES

511

(6)

A. Guide to Notation

511

(2)

1. General scheme of notation

511

(1)

2. Global notations

511

(2)

B. Some Isotropic, Three-Dimensional Strain-Energy Densities

513

(1)

References

514

(3)

INDEX

517

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The Nonlinear Theory of Elastic Shells

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