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E-raamat: Introduction to the Variational Formulation in Mechanics: Fundamentals and Applications

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  • ISBN-13: 9781119600954
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Introduction to the Variational Formulation in Mechanics: Fundamentals and ApplicationsSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 09-Dec-2019
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119600954
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Introduces readers to the fundamentals and applications of variational formulations in mechanics

Nearly 40 years in the making, this book provides students with the foundation material of mechanics using a variational tapestry. It is centered around the variational structure underlying the Method of Virtual Power (MVP). The variational approach to the modeling of physical systems is the preferred approach to address complex mathematical modeling of both continuum and discrete media. This book provides a unified theoretical framework for the construction of a wide range of multiscale models.

Introduction to the Variational Formulation in Mechanics: Fundamentals and Applications enables readers to develop, on top of solid mathematical (variational) bases, and following clear and precise systematic steps, several models of physical systems, including problems involving multiple scales. It covers: Vector and Tensor Algebra; Vector and Tensor Analysis; Mechanics of Continua; Hyperelastic Materials; Materials Exhibiting Creep; Materials Exhibiting Plasticity; Bending of Beams; Torsion of Bars; Plates and Shells; Heat Transfer; Incompressible Fluid Flow; Multiscale Modeling; and more.

  • A self-contained reader-friendly approach to the variational formulation in the mechanics
  • Examines development of advanced variational formulations in different areas within the field of mechanics using rather simple arguments and explanations
  • Illustrates application of the variational modeling to address hot topics such as the multiscale modeling of complex material behavior
  • Presentation of the Method of Virtual Power as a systematic tool to construct mathematical models of physical systems gives readers a fundamental asset towards the architecture of even more complex (or open) problems

Introduction to the Variational Formulation in Mechanics: Fundamentals and Applications is a ideal book for advanced courses in engineering and mathematics, and an excellent resource for researchers in engineering, computational modeling, and scientific computing.

Preface xv
Part I Vector and Tensor Algebra and Analysis 1(46)
1 Vector and Tensor Algebra
3(20)
1.1 Points and Vectors
3(3)
1.2 Second-Order Tensors
6(11)
1.3 Third-Order Tensors
17(5)
1.4 Complementary Reading
22(1)
2 Vector and Tensor Analysis
23(24)
2.1 Differentiation
23(5)
2.2 Gradient
28(2)
2.3 Divergence
30(2)
2.4 Curl
32(2)
2.5 Laplacian
34(1)
2.6 Integration
35(3)
2.7 Coordinates
38(7)
2.8 Complementary Reading
45(2)
Part II Variational Formulations in Mechanics 47(234)
3 Method of Virtual Power
49(60)
3.1 Introduction
49(1)
3.2 Kinematics
50(16)
3.2.1 Body and Deformations
50(5)
3.2.2 Motion: Deformation Rate
55(6)
3.2.3 Motion Actions: Kinematical Constraints
61(5)
3.3 Duality and Virtual Power
66(8)
3.3.1 Motion Actions and Forces
67(2)
3.3.2 Deformation Actions and Internal Stresses
69(2)
3.3.3 Mechanical Models and the Equilibrium Operator
71(3)
3.4 Bodies without Constraints
74(7)
3.4.1 Principle of Virtual Power
75(5)
3.4.2 Principle of Complementary Virtual Power
80(1)
3.5 Bodies with Bilateral Constraints
81(6)
3.5.1 Principle of Virtual Power
81(5)
3.5.2 Principle of Complementary Virtual Power
86(1)
3.6 Bodies with Unilateral Constraints
87(7)
3.6.1 Principle of Virtual Power
89(3)
3.6.2 Principle of Complementary Virtual Power
92(2)
3.7 Lagrangian Description of the Principle of Virtual Power
94(3)
3.8 Configurations with Preload and Residual Stresses
97(3)
3.9 Linearization of the Principle of Virtual Power
100(3)
3.9.1 Preliminary Results
101(1)
3.9.2 Known Spatial Configuration
102(1)
3.9.3 Known Material Configuration
102(1)
3.10 Infinitesimal Deformations and Small Displacements
103(3)
3.10.1 Bilateral Constraints
104(1)
3.10.2 Unilateral Constraints
105(1)
3.11 Final Remarks
106(1)
3.12 Complementary Reading
107(2)
4 Hyperelastic Materials at Infinitesimal Strains
109(56)
4.1 Introduction
109(1)
4.2 Uniaxial Hyperelastic Behavior
109(4)
4.3 Three-Dimensional Hyperelastic Constitutive Laws
113(3)
4.4 Equilibrium in Bodies without Constraints
116(7)
4.4.1 Principle of Virtual Work
117(1)
4.4.2 Principle of Minimum Total Potential Energy
117(1)
4.4.3 Local Equations and Boundary Conditions
118(2)
4.4.4 Principle of Complementary Virtual Work
120(1)
4.4.5 Principle of Minimum Complementary Energy
121(1)
4.4.6 Additional Remarks
122(1)
4.5 Equilibrium in Bodies with Bilateral Constraints
123(5)
4.5.1 Principle of Virtual Work
125(1)
4.5.2 Principle of Minimum Total Potential Energy
125(1)
4.5.3 Principle of Complementary Virtual Work
126(1)
4.5.4 Principle of Minimum Complementary Energy
127(1)
4.6 Equilibrium in Bodies with Unilateral Constraints
128(3)
4.6.1 Principle of Virtual Work
128(1)
4.6.2 Principle of Minimum Total Potential Energy
128(1)
4.6.3 Principle of Complementary Virtual Work
129(1)
4.6.4 Principle of Minimum Complementary Energy
130(1)
4.7 Min-Max Principle
131(3)
4.7.1 Hellinger-Reissner Functional
131(2)
4.7.2 Hellinger-Reissner Principle
133(1)
4.8 Three-Field Functional
134(2)
4.9 Castigliano Theorems
136(8)
4.9.1 First and Second Theorems
136(3)
4.9.2 Bounds for Displacements and Generalized Loads
139(5)
4.10 Elastodynamics Problem
144(4)
4.11 Approximate Solution to Variational Problems
148(14)
4.11.1 Elastostatics Problem
148(6)
4.11.2 Hellinger-Reissner Principle
154(2)
4.11.3 Generalized Variational Principle
156(2)
4.11.4 Contact Problems in Elastostatics
158(4)
4.12 Complementary Reading
162(3)
5 Materials Exhibiting Creep
165(64)
5.1 Introduction
165(1)
5.2 Phenomenological Aspects of Creep in Metals
165(3)
5.3 Influence of Temperature
168(2)
5.4 Recovery, Relaxation, Cyclic Loading, and Fatigue
170(3)
5.5 Uniaxial Constitutive Equations
173(9)
5.6 Three-Dimensional Constitutive Equations
182(6)
5.7 Generalization of the Constitutive Law
188(3)
5.8 Constitutive Equations for Structural Components
191(8)
5.8.1 Bending of Beams
192(3)
5.8.2 Bending, Extension, and Compression of Beams
195(4)
5.9 Equilibrium Problem for Steady-State Creep
199(10)
5.9.1 Mechanical Equilibrium
199(2)
5.9.2 Variational Formulation
201(4)
5.9.3 Variational Principles of Minimum
205(4)
5.10 Castigliano Theorems
209(5)
5.10.1 First and Second Theorems
209(2)
5.10.2 Bounds for Velocities and Generalized Loads
211(3)
5.11 Examples of Application
214(5)
5.11.1 Disk Rotating with Constant Angular Velocity
214(3)
5.11.2 Cantilevered Beam with Uniform Load
217(2)
5.12 Approximate Solution to Steady-State Creep Problems
219(6)
5.13 Unsteady Creep Problem
225(2)
5.14 Approximate Solutions to Unsteady Creep Formulations
227(1)
5.15 Complementary Reading
228(1)
6 Materials Exhibiting Plasticity
229(52)
6.1 Introduction
229(1)
6.2 Elasto-Plastic Materials
229(6)
6.3 Uniaxial Elasto-Plastic Model
235(8)
6.3.1 Elastic Relation
235(1)
6.3.2 Yield Criterion
236(2)
6.3.3 Hardening Law
238(2)
6.3.4 Plastic Flow Rule
240(3)
6.4 Three-Dimensional Elasto-Plastic Model
243(10)
6.4.1 Elastic Relation
244(2)
6.4.2 Yield Criterion and Hardening Law
246(3)
6.4.3 Potential Plastic Flow
249(4)
6.5 Drucker and Hill Postulates
253(2)
6.6 Convexity, Normality, and Plastic Potential
255(3)
6.6.1 Normality Law and a Rationale for the Potential Law
255(2)
6.6.2 Convexity of the Admissible Region
257(1)
6.7 Plastic Flow Rule
258(2)
6.8 Internal Dissipation
260(2)
6.9 Common Yield Functions
262(4)
6.9.1 The von Mises Criterion
263(1)
6.9.2 The Tresca Criterion
264(2)
6.10 Common Hardening Laws
266(1)
6.11 Incremental Variational Principles
267(5)
6.11.1 Principle of Minimum for the Velocity
268(1)
6.11.2 Principle of Minimum for the Stress Rate
269(1)
6.11.3 Uniqueness of the Stress Field
270(1)
6.11.4 Variational Inequality for the Stress
270(1)
6.11.5 Principle of Minimum with Two Fields
271(1)
6.12 Incremental Constitutive Equations
272(7)
6.12.1 Constitutive Equations for Rates
273(2)
6.12.2 Constitutive Equations for Increments
275(3)
6.12.3 Variational Principle in Finite Increments
278(1)
6.13 Complementary Reading
279(2)
Part III Modeling of Structural Components 281(96)
7 Bending of Beams
285(16)
7.1 Introduction
285(1)
7.2 Kinematics
285(4)
7.3 Generalized Forces
289(1)
7.4 Mechanical Equilibrium
290(4)
7.5 Timoshenko Beam Model
294(4)
7.6 Final Remarks
298(3)
8 Torsion of Bars
301(14)
8.1 Introduction
301(1)
8.2 Kinematics
301(3)
8.3 Generalized Forces
304(1)
8.4 Mechanical Equilibrium
305(4)
8.5 Dual Formulation
309(6)
9 Plates and Shells
315(62)
9.1 Introduction
315(1)
9.2 Geometric Description
316(4)
9.3 Differentiation and Integration
320(3)
9.4 Principle of Virtual Power
323(3)
9.5 Unified Framework for Shell Models
326(6)
9.6 Classical Shell Models
332(15)
9.6.1 Naghdi Model
332(3)
9.6.2 Kirchhoff-Love Model
335(5)
9.6.3 Love Model
340(2)
9.6.4 Koiter Model
342(2)
9.6.5 Sanders Model
344(2)
9.6.6 Donnell-Mushtari-Vlasov Model
346(1)
9.7 Constitutive Equations and Internal Constraints
347(13)
9.7.1 Preliminary Concepts
348(2)
9.7.2 Model with Naghdi Hypothesis
350(7)
9.7.3 Model with Kirchhoff-Love Hypothesis
357(3)
9.8 Characteristics of Shell Models
360(9)
9.8.1 Relation Between Generalized Stresses
360(1)
9.8.2 Equilibrium Around the Normal
361(3)
9.8.2.1 Kirchhoff-Love Model
361(1)
9.8.2.2 Love Model
362(1)
9.8.2.3 Koiter Model
363(1)
9.8.2.4 Sanders Model
363(1)
9.8.3 Reactive Generalized Stresses
364(5)
9.8.3.1 Reactions in the Naghdi Model
364(2)
9.8.3.2 Reactions in the Kirchhoff-Love Model
366(3)
9.9 Basics Notions of Surfaces
369(10)
9.9.1 Preliminaries
369(1)
9.9.2 First Fundamental Form
370(2)
9.9.3 Second Fundamental Form
372(3)
9.9.4 Third Fundamental Form
375(1)
9.9.5 Complementary Properties
375(2)
Part IV Other Problems in Physics 377(58)
10 Heat Transfer
379(14)
10.1 Introduction
379(1)
10.2 Kinematics
379(2)
10.3 Principle of Thermal Virtual Power
381(5)
10.4 Principle of Complementary Thermal Virtual Power
386(2)
10.5 Constitutive Equations
388(2)
10.6 Principle of Minimum Total Thermal Energy
390(1)
10.7 Poisson and Laplace Equations
390(3)
11 Incompressible Fluid Flow
393(18)
11.1 Introduction
393(1)
11.2 Kinematics
394(2)
11.3 Principle of Virtual Power
396(7)
11.4 Navier-Stokes Equations
403(2)
11.5 Stokes Flow
405(2)
11.6 Irrotational Flow
407(4)
12 High-Order Continua
411(24)
12.1 Introduction
411(1)
12.2 Kinematics
412(6)
12.3 Principle of Virtual Power
418(7)
12.4 Dynamics
425(2)
12.5 Micropolar Media
427(2)
12.6 Second Gradient Theory
429(6)
Part V Multiscale Modeling 435(66)
13 Method of Multiscale Virtual Power
439(36)
13.1 Introduction
439(1)
13.2 Method of Virtual Power
439(8)
13.2.1 Kinematics
439(3)
13.2.2 Duality
442(3)
13.2.3 Principle of Virtual Power
445(1)
13.2.4 Equilibrium Problem
446(1)
13.3 Fundamentals of the Multiscale Theory
447(2)
13.4 Kinematical Admissibility between Scales
449(13)
13.4.1 Macroscale Kinematics
449(2)
13.4.2 Microscale Kinematics
451(2)
13.4.3 Insertion Operators
453(3)
13.4.4 Homogenization Operators
456(2)
13.4.5 Kinematical Admissibility
458(4)
13.5 Duality in Multiscale Modeling
462(5)
13.5.1 Macroscale Virtual Power
462(2)
13.5.2 Microscale Virtual Power
464(3)
13.6 Principle of Multiscale Virtual Power
467(1)
13.7 Dual Operators
468(5)
13.7.1 Microscale Equilibrium
468(2)
13.7.2 Homogenization of Generalized Stresses
470(2)
13.7.3 Homogenization of Generalized Forces
472(1)
13.8 Final Remarks
473(2)
14 Applications of Multiscale Modeling
475(26)
14.1 Introduction
475(1)
14.2 Solid Mechanics with External Forces
475(15)
14.2.1 Multiscale Kinematics
476(3)
14.2.2 Characterization of Virtual Power
479(1)
14.2.3 Principle of Multiscale Virtual Power
480(2)
14.2.4 Equilibrium Problem and Homogenization
482(5)
14.2.5 Tangent Operators
487(3)
14.3 Mechanics of Incompressible Solid Media
490(10)
14.3.1 Principle of Virtual Power
491(2)
14.3.2 Multiscale Kinematics
493(2)
14.3.3 Principle of Multiscale Virtual Power
495(2)
14.3.4 Incompressibility and Material Configuration
497(3)
14.4 Final Remarks
500(1)
Part VI Appendices 501(58)
A Definitions and Notations
503(36)
A.1 Introduction
503(1)
A.2 Sets
503(1)
A.3 Functions and Transformations
504(3)
A.4 Groups
507(2)
A.5 Morphisms
509(1)
A.6 Vector Spaces
509(3)
A.7 Sets and Dependence in Vector Spaces
512(1)
A.8 Bases and Dimension
513(1)
A.9 Components
514(2)
A.10 Sum of Sets and Subspaces
516(1)
A.11 Linear Manifolds
516(1)
A.12 Convex Sets and Cones
516(1)
A.13 Direct Sum of Subspaces
517(1)
A.14 Linear Transformations
517(5)
A.15 Canonical Isomorphism
522(1)
A.16 Algebraic Dual Space
523(3)
A.16.1 Orthogonal Complement
524(1)
A.16.2 Positive and Negative Conjugate Cones
525(1)
A.17 Algebra in V
526(2)
A.18 Adjoint Operators
528(1)
A.19 Transposition and Bilinear Functions
529(3)
A.20 Inner Product Spaces
532(7)
B Elements of Real and Functional Analysis
539(16)
B.1 Introduction
539(2)
B.2 Sequences
541(1)
B.3 Limit and Continuity of Functions
542(2)
B.4 Metric Spaces
544(2)
B.5 Normed Spaces
546(3)
B.6 Quotient Space
549(1)
B.7 Linear Transformations in Normed Spaces
550(2)
B.8 Topological Dual Space
552(1)
B.9 Weak and Strong Convergence
553(2)
C Functionals and the Gateaux Derivative
555
C.1 Introduction
555(1)
C.2 Properties of Operator X
555(1)
C.3 Convexity and Semi-Continuity
556(1)
C.4 Gateaux Differential
557(1)
C.5 Minimization of Convex Functionals
557(2)
References 559(16)
Index 575
9781946160577
Dedication xi
Foreword xv
Jed Horne
Preface xxxv
1 Introduction 1(29)
The Controversy
Early New Orleans History
Vieux Carre Architecture
Tout Ensemble
Creoles Versus Americans
Growth Of American Section
Postwar Growth And Decline
Twentieth Century
French Quarter Waterfront
Tradition Of Conflict
2 The Controversy Begins 30(9)
Moses Report
Morrison And Transportation
Bartholomew Report
Bridge And Causeway Construction
Central Area Committee Role
Prospectus
Riverfront Expressway On Major Street Plan
3 Reports And Recommendations 39(6)
Downs Report
Preliminary Engineering Report
Continued Expressway Opposition
Proposal By Dock And Levee Boards
Unanimous Preservation Opinion
4 Preservationists Mobilize As Expressway Plans Develop 45(8)
Approval Of City Bonds
Nomats Traffic Study
Domblatt Engineering Report
The Tunnel
Expressway Becomes Part Of Interstate System
Plans For Public Hearing
Intensified Expressway Battle
5 Highway Department Hearing Leads To Further Study 53(8)
Highway Department Hearing
Fromherz Report
Guidelines Growth
6 Expressway Opponents Take The Initiative 61(6)
Domblatt Tunnel Study
Freeway Protest Meeting
AIA Opposition
Secretary Udall Enters The Controversy
Challenge In Washington Post
Saturday Evening Post Skirmish
7 Preservationists Suggest Alternative Routes-landmark Question Tabled 67(8)
Katz Plan
Tulane Report
Tulane Report Attacked
National Historic Landmark Designation
8 Route Approved But Alternate Proposals Continue 75(10)
Schiro-Levy Plan
Federal Approval Of Elevated Expressway
National Trust Involved
Attacks On Federal Decision
Support From Help
Borah-Baumbach Report
Engineers Support Planning Process
New Schiro Efforts
9 Highway Opponents Seek Objective Study 85(5)
Requests For Study Of Alternate Routes
Continued Controversy
10 Hearing On Moreau Resolution Held 90(10)
Favorable Testimony
The Opposition
Closing Arguments
The Vote
11 National Campaign Brings Support 100(9)
Hope For Federal Study
National Campaign To Save The Vieux Carre
National Trust Resolution
Federal Legislation
12 New Proposals Meet Continued Opposition 109(9)
Compromise Proposals
Reaffirmation Of Opposition
Local Reaction To National Opposition
Impact Study Announced
Preliminary Draft Of Impact Study
Highway Department Moves Ahead
13 Impact Study Released 118(11)
Bureau Opposition To Elevated Highway
Reaction Of Expressway Opponents
Opposition From Freeway Advocates
14 Expressway Controversy Escalates 129(6)
Highway Department Grade-Level Study
Pontalba Suit
Proposal For Multiarched Expressway
New Bureau Position
15 A National Issue Develops 135(9)
Arthur D. Little Report
Bridwell Hearing
Pontalba Suit
16 Freeway Opponents Gain New Allies 144(7)
Calls For Comprehensive Planning
Rise Of Crescent Council
Schiro Request For HUD Study
Expanding Opposition
Governor Appoints Freeman Committee
17 Bridwell Backs Grade-Level Expressway 151(8)
Bridwell In New Orleans
Federal Grade-Level Plan
Expressway Opponents' Growing Strength
The Howard Bill
Discussion Of Grade-Level Solution
Disclosure Of Bridwell's Intentions
18 New Studies Underscore Need For Freeway 159(5)
Freeman Committee Report
Voorhies Report
19 Grade-Level Expressway Accepted 164(9)
Bridwell Buckling?
Rader Report Released
Planning Commission Accepts Surface Roadway
Highway Department Supports Grade-Level Facility
City Council Hearing
Bridwell Approval Of Roadway
20 Nixon Administration Reexamines Expressway Issue 173(11)
Withdrawal Of Approval
Advisory Council Review
Reaction To Advisory Council Recommendations
Metropolitan Area Committee Joins The Battle
Appeal To New Federal Highway Administrator
Bureau Attacks Advisory Council Recommendations
Preservationists Attack
Increased Central Area Council Effort
21 Braman Hearing Held 184(9)
Meeting With Supporters
Meeting With Opponents
22 Riverfront Expressway Canceled 193(13)
Reaction Of Opponents
Reaction Of Proponents
Newspaper React
The Tunnel
Volpe Decision Formally Announced
National Reaction
Request To Remove I-310 From Interstate System
23 Calls For Comprehensive Planning Issued 206(8)
Call For Unity
Revival Of Riverfront Expressway?
Boggs Announces Washington Meetings
Requests For Delay
Freeway Opponents Decline To Attend Meetings
Boggs And Expressway Opponents Reach Accord
24 Washington Meetings Produce More Controversy 214(10)
The Meetings
Local Reaction To Washington Meetings
Borah-Baumbach Counterattack
Brown Criticizes Volpe And Boggs
Request For Federal Funding Halt
Revivalists Go Underground
25 Studies Affecting The Vieux Carre Riverfront 224(6)
Quarterfront
Quarterfront Questioned
Centroport
26 Conradt Report Issued 230(6)
Traffic Analysis
Disagreement With Voorhies
Evaluation Of Metropolitan Expressway System
Importance Of Report
Reaction
27 Roadway Revival Possible? 236(5)
New Roadway Proposal
Roadway Or Expressway?
Belt Theory
28 Changing The Process 241(3)
Afterword By Diane Donley 244(16)
Notes 260(49)
Additional Images 309(36)
Plaque Dedication 345(2)
In Memoriam: Richard
0. Baumbach		 347(2)
In Memoriam: William E. Borah 349(4)
Select Bibliography 353(8)
Chronology Of The Vieux Carre
Riverfront Expressway Controversy 361(8)
Acknowledgments 369(2)
Index 371
EDGARDO OMAR TAROCO, PHD, was a Full Researcher at the National Laboratory for Scientific Computing (LNCC/MCTIC) and at the National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC), Petrópolis, Brazil.

PABLO JAVIER BLANCO, PHD, is a Full Researcher at the National Laboratory for Scientific Computing (LNCC/MCTIC) and at the National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC), Petrópolis, Brazil, and Associate Professor at the Catholic University of Petrópolis, Brazil.

RAÚL ANTONINO FEIJÓO, is a Full Researcher at the National Laboratory for Scientific Computing (LNCC/MCTIC) and at the National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC), Petrópolis, Brazil.
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