The Nonlinear Theory of Elastic Shells

by A. Libai , J. G. Simmonds

• ISBN13: 9780521472364
• ISBN10: 0521472369
• Edition: 2nd
• Format: Hardcover
• Copyright: 1998-02-13
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

PREFACE	xiii	(2)
PREFACE TO THE FIRST EDITION	xv
CHAPTER I: INTRODUCTION	1	(10)
A. What is a Shell?	1	(1)
B. Elastic Shells and Nonlinear Behavior	1	(1)
C. Approaches to Shell Theory	2	(1)
D. The Approach of this Book to Shell Theory	3	(1)
E. Outline of the Book	4	(2)
References	6	(5)
CHAPTER II: THE GENERIC EQUATIONS OF THREE-DIMENSIONAL CONTINUUM MECHANICS	11	(10)
A. The Integral Equations of Motion	11	(2)
B. Stress Vectors	13	(1)
C. Heat	14	(1)
D. The Clausius-Duhem (-Truesdell-Toupin) Inequality	15	(1)
E. The First Piola-Kirchhoff Stress Tensor	15	(1)
*F. Gross Equations of Motion	16	(4)
References	20	(1)
CHAPTER III: LONGITUDINAL MOTION OF STRAIGHT RODS WITH BISYMMETRIC CROSS SECTIONS (BIRODS)	21	(30)
A. Geometry of the Undeformed Rod	21	(1)
B. Integral Equation of Motion	21	(4)
C. Differential Equation of Motion	25	(1)
*D. Jump Condition and Propagation of Singularities	25	(3)
E. The Weak Form of the Equation of Motion	28	(2)
F. The Mechanical Work Identity	30	(1)
G. Mechanical Boundary Conditions	31	(2)
H. The Principle of Virtual Work	33	(3)
1. The Principle of Virtual Work and boundary conditions in statics	36	(1)
I. The Mechanical Theory of Birods	36	(1)
J. The Mechanical Theory of Elastic Birods	36	(2)
1. Restrictions on the strain-energy density	37	(1)
2. The equation of motion in displacement and intrinsic forms	37	(1)
K. Variational Principles	38	(7)
1. Hamilton's Principle	40	(1)
2. Variational principles for elastostatics	41	(4)
L. Thermal Equations	45	(1)
M. The First Law of Thermodynamics	46	(1)
N. Thermoelastic Birods	46	(2)
1. Linearized constitutive equations for N, h, and t	48	(1)
References	48	(3)
CHAPTER IV: CYLINDRICAL MOTION OF INFINITE CYCLINDRICAL SHELLS (BEAMSHELLS)	51	(108)
A. Geometry of the Undeformed Shell and Planar Motion	51	(2)
B. Integral Equations of Cylindrical Motion	53	(4)
1. Natural initial and boundary conditions	56	(1)
C. Initial and Spin Bases	57	(1)
*D. Jump Conditions and Propagation of Singularities	58	(1)
E. Differential Equations of Cylindrical Motion	59	(1)
F. The Weak Form of the Equations of Motion	60	(1)
G. The Mechanical Work Identity	60	(1)
H. Mechanical Boundary Conditions	61	(6)
1. General boundary conditions for nonholonomic constraints	61	(2)
2. Classical boundary conditions	63	(1)
3. Typical boundary conditions	64	(3)
*Appendix: The Principle of Mechanical Boundary Conditions	67	(1)
Dawn Fisher
I. The Principle of Virtual Work	67	(2)
J. Potential (Conservative) Loads	69	(7)
1. Dead loading (and a torsional spring)	69	(1)
2. Centrifugal loading	69	(1)
3. Pressure loading (constant or hydrostatic)	70	(1)
*4. General discussion and examples	71	(5)
K. Strains	76	(1)
L. Alternative Strains and Stresses	77	(3)
M. The Mechanical Theory of Beamshells	80	(1)
N. Elastic Beamshells and Strain-Energy Densities	81	(13)
1. Quadratic strain-energy densities	82	(3)
*2. General strain-energy densities	85	(4)
*3. Strain-energy density by descent from three dimensions	89	(5)
O. Elastostatics	94	(11)
1. Inextensional beamshells	96	(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
Asterisk (*) indicates a section that may be omitted without loss of continuity