Mechanics of solids and shells : theories and approximations

521 Pages·2003·5.68 MB·English

by  Gerald Wempner

MECHANICS
OF
SOLIDS
AND
SHELLS
Theories and Approximations
© 2003 by CRC Press LLC
MECHANICS
OF
SOLIDS
AND
SHELLS
Theories and Approximations
Gerald Wempner
Professor emeritus
Georgia Institute of Technology
Atlanta, Georgia
U.S.A.
Demosthenes Talaslidis
Professor
Aristotle University Thessaloniki
Thessaloniki
Greece
CRC PR ESS
Boca Raton London New York Washington, D.C.
© 2003 by CRC Press LLC
9654 disclaimer Page 1 Monday, September 16, 2002 1:06 PM
Library of Congress Cataloging-in-Publication Data
Wempner, Gerald, 1928-
Mechanics of solids and shells : theories and approximations / Gerald Wempner, G.
Talaslidis Demosthenes.
p. cm. — (Mechanical engineering series)
Includes bibliographical references and index.
ISBN 0-8493-9654-9 (alk. paper)
1. Mechanics, Applied. 2. Solids. 3. Shells (Engineering) I. Demosthenes, G.
Talaslidis. II. Title. III. Mechanical engineering series (Boca Raton, Fla.)
TA350 .W525 2002
620.1¢.05—dc21 2002073733

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Preface
The mechanics of solids and the mechanics of shells have long histories;
countless articles and books have been written by eminent scholars. Most
books address specific aspects which are necessarily restricted. Many are
limited by kinematical assumptions, e.g., small strains and/or small rota-
tions. Others are confined to specific behaviors, e.g., elasticity. This book
is also necessarily restricted but in a different way:
This book is intended as a reference for scholars, researchers, and prac-
titioners, to provide a reliable source for the mathematical tools of anal-
ysis and approximation. As such, those aspects which are common to all
continuous solids are presented in generality. These are the kinematics
(Chapter 3), the kinetics (Chapter 4), and the energetics (Chapter 6); the
only common limitations are the continuity and cohesion of the medium.
The basic quantities, e.g., strains, stresses, energies, and the mathemat-
ical relations are developed precisely. To achieve the requisite precision
and to reveal the invariant properties of physical entities, the foundations
are expressed via the language of vectorial and tensorial analysis (Chap-
ter 2). Additionally, geometrical and physical interpretations are empha-
sized throughout. Since large rotations play a central role in structural
responses, e.g., instabilities, the decomposition of rotations and strains is
givenspecialattentionanduniquegeometricalinterpretations. Wherespe-
cial restrictive conditions are invoked, they are clearly noted.
With a view toward practical applications, the authors have noted the
physical implications of various approximations. In the same spirit, only a
few specific results are presented; these are the ”exact” solutions of much
significanceinengineeringpractice(simplebendingandtorsion,andexam-
ples of actual stress concentrations).
Theutilityofthisbookisenhancedbytheunificationofthevarioustop-
ics: The theories of shells (Chapters 9 and 10) and finite elements (Chap-
ter 11) are couched in the general concepts of the three-dimensional con-
tinuous solid. Each of the variational principles and theorems of three di-
mensions(Chapter6)hastheanalogouscounterpartinthetwodimensions
oftheshell(Chapter9).
© 2003 by CRC Press LLC
This book does provide original presentations and interpretations: Prin-
ciplesofworkandenergy(Chapter6)includethebasicconceptsofKoiter’s
monumentalworkonstabilityatthecriticalload. Additionally,thevarious
complementaryfunctionalsareexpressedintermsofthealternativestrains
and stresses; the various versions are fully correlated and applicable to fi-
nite deformations. The presentation of the Kirchhoff-Love shell (Chapter
9) includes original treatments for the plastic behavior of shells.
A final chapter views the finite element as a device for the approxima-
tion of the continuous solution. The mathematical and physical attributes
are described from that viewpoint. As such, the presentation provides a
meaningful bridge between the continuum and the discrete assembly.
Our rationale for the content and the structure of this book is best ex-
hibited by the following sequential decomposition:
• The foundations of all theories of continuous cohesive solids are set
forth in generality in the initial Chapters 3 and 4.
• Theestablishedtheoriesofelasticity,plasticity,andlinearviscoelas-
ticity are presented in the subsequent Chapter 5. These are cast in
the context of classical thermodynamics. The specific mathemati-
cal descriptions of materials are limited to those which have proven
effective in practice and are supported by physical evidence.
• TheprinciplesofworkandenergyarepresentedinthenextChap-
ter6;thebasicformsaregivenwithoutkinematicallimitations.
Only the Castigliano theorem is restricted to small deformations.
• Theformulationsoflinearelasticityandviscoelasticityarecontained
inChapter7. Thisfacetofoursubjecthasevolvedtotheextentthat
it is vital to a complete understanding of the mechanics of solids.
Here, the basic formulations are couched in the broader context of
the general theory. These linear theories are included so that our
book provides a more complete source of reference.
• The differential geometry of surfaces is essential to the general the-
ories of shells. Chapter 8 presents the geometric quantities and
notationswhichareemployedinthesubsequenttreatmentofshells.
• The mechanics of shells is presented in two parts: Chapter 9 sets
forth a theory which is limited only by one kinematic hypothesis;
a normal is presumed to remain straight. No restrictions are im-
poseduponthemagnitudeofdeformations. Assuch,themechanics
ofChapter 9encompassestheKirchhoff-Lovetheoryofthesubse-
quent Chapter 10. The latter includes the traditional formulations
of elastic shells, but also presents original theories of elastic-plastic
shells.
© 2003 by CRC Press LLC
• The final Chapter 11 places the basic notions of finite elements in
the context of the mechanics of solids and shells. The most funda-
mentalaspectsaredescribedfromthemathematicalandmechanical
perspectives.
We trust that the preceding preview serves to reveal our intended unifi-
cation ofthegeneralfoundations,thevarioustheoriesandapproximations.
The authors’ perspectives have been influenced by many experiences, by
interactionswithcolleaguesandbytheeffortsofmanypredecessors. Afew
works are most notable: The classic of A. E. H. Love, the insightful works
of S. P. Timoshenko, the lucid monograph by V. V. Novozhilov, and the
text by A. E. Green and W. Zerna are but a few which have shaped our
views. Ourgraspofenergeticformulationsandinstabilitycriteriaaretrace-
able to the important contributions by E. Trefftz, B. Fraeijs de Veubeke,
and W. T. Koiter. Throughout this text, the authors have endeavored to
acknowledge the origins of concepts and advances. Inevitably, some are
overlooked; othersareflawedbyhistoricalaccounts. Theauthorsapologize
to persons who were inadvertently slighted by such mistakes.
On a personal note, Gerald Wempner must acknowledge his most influ-
ential teacher: His father, Paul Wempner, was a person with little formal
education, but one who demonstrated the value of keen observation and
concerted intellectual effort. Demosthenes Talaslidis would like to express
his gratitude to his wife Vasso for her tolerance and support throughout
preparation of this book. Both authors owe a debt of gratitude to Vasso
Talaslidis and Euthalia Papademetriou; their forbearance and hospitality
have enabled our collaboration. The authors are also indebted to Profes-
sor Walter Wunderlich who encouraged their earlier research. Finally, the
authors are obliged to Mrs. Feye Kazantzidou for her careful attention to
the illustrations.
Gerald Wempner and Demosthenes Talaslidis
Atlanta and Thessaloniki, July 2002
© 2003 by CRC Press LLC
Contents
1I ntroduction
1.1PurposeandScope
1.2MechanicalConceptsandMathematicalRepresentations
1.3IndexNotation
1.4Systems
1.5SummationConvention
1.6PositionofIndices
1.7VectorNotation
1.8KroneckerDelta
1.9PermutationSymbol
1.10SymmetricalandAntisymmetricalSystems
1.11AbbreviationforPartialDerivatives
1.12Terminology
1.13SpecificNotations
2 Vectors,Tensors,andCurvilinearCoordinates
2.1Introduction
2.2 Curvilinear Coordinates, Base Vectors, and Metric
Tensor
2.3ProductsofBaseVectors
2.4ComponentsofVectors
2.5SurfaceandVolumeElements
2.6DerivativesofVectors
2.7TensorsandInvariance
2.8AssociatedTensors
2.9CovariantDerivative
2.10 Transformation from Cartesian to Curvilinear
Coordinates
2.11IntegralTransformations
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3 Deformation
3.1Conceptof aContinuousMedium
3.2GeometryoftheDeformedMedium
3.3DilationofVolumeandSurface
3.4 Vectors and Tensors Associated with the Deformed
System
3.5NatureofMotioninSmallRegions
3.6Strain
3.7TransformationofStrainComponents
3.8PrincipalStrains
3.9MaximumShearStrain
3.10 Determination of Principal Strains and Principal
Directions
3.11DeterminationofExtremalShearStrain
3.12EngineeringStrainTensor
3.13StrainInvariantsandVolumetricStrain
3.14DecompositionofMotionintoRotationandDeformation
3.14.1RotationFollowedbyDeformation
3.14.2DeformationFollowedbyRotation
3.14.3IncrementsofRotation
3.15PhysicalComponentsoftheEngineeringStrain
3.16Strain-DisplacementRelations
3.17CompatibilityofStrainComponents
3.18RatesandIncrementsofStrainandRotation
3.19EulerianStrainRate
3.20StrainDeviator
3.21ApproximationofSmallStrain
3.22ApproximationsofSmallStrainandModerateRotation
3.23ApproximationsofSmallStrainandSmallRotation
4 Stress
4.1StressVector
4.2CoupleStress
4.3ActionsUponanInfinitesimalElement
4.4EquationsofMotion
4.5TensorialandInvariantFormsofStressandInternalWork
4.6TransformationofStress—PhysicalBasis
4.7Propertiesof aStressedState
4.8HydrostaticStress
4.9StressDeviator
4.10AlternativeFormsoftheEquationsofMotion
4.11SignificanceofSmallStrain
4.12ApproximationofModerateRotations
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4.13 ApproximationsofSmallStrainsandSmallRotations;Lin-
earTheory
4.14Example:Bucklingof aBeam
5 BehaviorofMaterials
5.1Introduction
5.2GeneralConsiderations
5.3ThermodynamicPrinciples
5.4ExcessiveEntropy
5.5HeatFlow
5.6Entropy,EntropyFlux,andEntropyProduction
5.7WorkofInternalForces(Stresses)
5.8AlternativeFormsoftheFirstandSecondLaws
5.9Saint-Venant’sPrinciple
5.10ObservationsofSimpleTests
5.11Elasticity
5.12Inelasticity
5.13LinearlyElastic(Hookean)Material
5.14MonotropicHookeanMaterial
5.15OrthotropicHookeanMaterial
5.16 Transversely Isotropic (Hexagonally Symmetric)
HookeanMaterial
5.17IsotropicHookeanMaterial
5.18HeatConduction
5.19HeatConductionintheHookeanMaterial
5.20CoefficientsofIsotropicElasticity
5.20.1SimpleTension
5.20.2SimpleShear
5.20.3UniformHydrostaticPressure
5.21AlternativeFormsoftheEnergyPotentials
5.22 Hookean Behavior in Plane-Stress and
Plane-Strain
5.23JustificationofSaint-Venant’sPrinciple
5.24YieldCondition
5.25YieldConditionforIsotropicMaterials
5.26TrescaYieldCondition
5.27vonMisesYieldCriterion
5.28PlasticBehavior
5.29IncrementalStress-StrainRelations
5.30GeometricalInterpretationoftheFlowCondition
5.31ThermodynamicInterpretation
5.32TangentModulusofElasto-plasticDeformations
5.33TheEquationsofSaint-Venant,L´ evy,Prandtl,andReuss
5.34HenckyStress-StrainRelations
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5.35 Plasticity without a Yield Condition;
EndochronicTheory
5.36AnEndochronicFormofIdealPlasticity
5.37ViscousBehavior
5.38NewtonianFluid
5.39LinearViscoelasticity
5.40IsotropicLinearViscoelasticity
5.40.1DifferentialFormsoftheStress-StrainRelations
5.40.2IntegralFormsoftheStress-StrainRelations
5.40.3RelationsbetweenComplianceandModulus
6 PrinciplesofWorkandEnergy
6.1Introduction
6.2HistoricalRemarks
6.3Terminology
6.4Work,KineticEnergy,andFourier’sInequality
6.5ThePrincipleofVirtualWork
6.6ConservativeForcesandPotentialEnergy
6.7PrincipleofStationaryPotentialEnergy
6.8ComplementaryEnergy
6.9PrincipleofMinimumPotentialEnergy
6.10StructuralStability
6.11StabilityattheCriticalLoad
6.12EquilibriumStatesNeartheCriticalLoad
6.13EffectofSmallImperfectionsupontheBucklingLoad
6.14PrincipleofVirtualWorkAppliedto aContinuousBody
6.15 Principle of Stationary Potential Applied to a Continuous
Body
6.16GeneralizationofthePrincipleofStationaryPotential
6.17GeneralFunctionalandComplementaryParts
6.18PrincipleofStationaryComplementaryPotential
6.19ExtremalPropertiesoftheComplementaryPotentials
6.20FunctionalsandStationaryTheoremofHellinger-Reissner
6.21 Functionals and Stationary Criteria for the
ContinuousBody;Summary
6.22GeneralizationofCastigliano’sTheoremonDisplacement
6.23VariationalFormulationsofInelasticity
7 LinearTheoriesofIsotropicElasticityandViscoelasticity
7.1Introduction
7.2UsesandLimitationsoftheLinearTheories
7.3KinematicEquationsof aLinearTheory
7.4LinearEquationsofMotion
7.5LinearElasticity
© 2003 by CRC Press LLC