Virtual Work and Shape Change in Solid Mechanics

373 Pages·2017·4.869 MB·English

by  Michel Frémond (auth.)

Springer Series in Solid and Structural Mechanics 7
Michel Frémond
Virtual Work
and Shape
Change in Solid
Mechanics
Springer Series in Solid and Structural
Mechanics
Volume 7
Series editors
Michel Frémond, Roma, Italy
Franco Maceri, Roma, Italy
More information about this series at http://www.springer.com/series/10616
é
Michel Fr mond
Virtual Work and Shape
Change in Solid Mechanics
123
Michel Frémond
Department ofCivil Engineering
andComputer Science
University of Rome“Tor Vergata”
Rome
Italy
ISSN 2195-3511 ISSN 2195-352X (electronic)
SpringerSeries inSolid andStructural Mechanics
ISBN978-3-319-40681-7 ISBN978-3-319-40682-4 (eBook)
DOI 10.1007/978-3-319-40682-4
LibraryofCongressControlNumber:2016945146
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Preface
Mechanics is the science of motion: It predicts the motion we see with our eyes.
Motion is involved in most scientific activities and in most engineering work. The
importanceofthistopichasresultedinanaxiomatizationofmechanicsandtohave
mathematics together with experiments to be tools widely used.
Thereisalargevarietyofmotionsweseewithoureyes:themotionofthefloor
of a room, the motionof the grains of a sand pile, the motionof a debris flow, the
motionofgalaxies,etc.Theelementswhichareusedbyengineersandscientiststo
describe them may seem different.
But these theories have in common the concepts to describe the motion. The
equationsofmotionareeithersimpleorverysophisticated.Thesophisticationmay
result from the need to define and quantify precisely together the shape of the
systemwhichisconsideredandtheevolutionofthisshape,i.e.,theshapechangeor
the deformation of the system, and the velocity of deformation.
A tool to introduce the mechanical effects of the evolution of the shape of a
system is the principle of virtual work. It has a status which is perhaps too theo-
retical even if it is widely used in numerics with the so-called variational formu-
lations [1]. We show it is actually related to observation and experiments. Its
utilization is flexible and may be adapted to produce predictive theories of
numerous phenomena.
PartIisdevotedtorelatethevirtualworkprincipletowhatweseewithoureyes.
Part II shows how flexible it is. A large number of examples are given.
The principle is applied in Part III to predict the motion of solids with large
deformations. The principle requires the description of the deformations: the way
theshapeofsolidschanges.Weknowthattherearealargevarietyofpossibilities.
Thechoicehastobeassimpleaspossible,butithastocopewiththeeverydaylife
actions. It results from observations that third-order derivatives with respect to
space of the displacement are needed to have a coherent description of large
deformations.
Once the principle has defined the internal forces and given the equations of
motion,wehavetofacethederivationoftheconstitutivelawswhichdescribehow
v
vi Preface
amaterialbehaves.Equationsofmotionaregeneral.Constitutivelawsarepeculiar
to each material. Theory and observation intervene in the derivation of the con-
stitutive laws. For what concerns theory, the Clausius–Duhem inequality is the
usefultool.Forwhatconcernsobservation,experimentsguidethechoiceofthefree
energy and the choice of the pseudo-potential of dissipation.
FollowingtheexamplesofPartsIandII,weidentifyaninternalconstraintonthe
elongation matrix velocity. Following the way Lagrange takes into account an
internalconstraint,uneliaisonparfaiteinFrenchandunvincoloperfettoinItalian,
we introduce a reaction [2]. As usual in Lagrangian mechanics, this reaction is
givenbyboththeconstitutivelawsandtheequationsofmotion.Itisimpossibleto
derive entirely the value of the reaction with a constitutive law. It depends on the
whole solid and on the external actions.
These problems have been investigated at the Università degli Studi di Roma
“Tor Vergata” in the Dipartimento di Ingegneria Civile e Ingegneria Informatica
and in the framework of the Laboratorio Lagrange, bringing together Italian and
French scientists. Some of the topics have been taught in lectures given at the
Scuola di Ingegneria of the Università.
The author discussed his points of view with Profs. Franco Maceri, Olivier
Maisonneuve,andChristianLicht.Hethanksthemfortheirwatchfulviews.Hehas
appreciated the professional and kind interest of Profs. Elena Bonetti, Pierluigi
Colli,MauroFabrizio,BernardNayroles,andClaudeStolz.DoctorDanieleBianchi
was helpful to solve editing problems, and the students of the Università have
provided numerical illustrations while attending the lectures. All of them are
warmly thanked.
Rome, Italy Michel Frémond
References
1. M.Frémond,Méthodesvariationnellesencalculdesstructures,ÉcolenationaledesPontset
Chaussées,Paris(1982)
2. J.L.Lagrange,Méchaniqueanalytique,ChezLaVeuveDesaint,Libraire,Paris(1788)
Contents
Part I The Principle of Virtual Work
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 The Principle of Virtual Work. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 What We See: The Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1 The Actual Velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 The Virtual Velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 The Abstract Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 The Actions Which are Applied to the System:
The Work of the External Forces. . . . . . . . . . . . . . . . . . . . . . . . . 13
5.1 The Abstract Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6 What We See: The Velocities of Deformation . . . . . . . . . . . . . . . . 15
6.1 The Rigid System Velocities . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 The Abstract Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7 The Work to Change the Shape of the System . . . . . . . . . . . . . . . 19
7.1 The Abstract Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8 The Work to Change the Velocities of the System. . . . . . . . . . . . . 23
8.1 The Abstract Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8.1.1 The Theorem of the Kinetic Energy . . . . . . . . . . . . 25
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
vii
viii Contents
9 The Principle of Virtual Work and the Equations of Motion . . . . . 27
9.1 The Abstract Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9.2 How to Get the Principle of Virtual Work
from Experiments?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9.2.1 The Smooth Evolution . . . . . . . . . . . . . . . . . . . . . 29
9.2.2 The Non Smooth Evolution. . . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
10 Summary of the Abstract Setting to get the Equations
of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Part II Examples
11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
12 Two Points on a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
12.1 The Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
12.2 The Velocities of Deformation. . . . . . . . . . . . . . . . . . . . . . . 37
12.2.1 The Rigid System Velocities . . . . . . . . . . . . . . . . . 38
12.3 The Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
12.4 The Equations of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
13 Three Disks in a Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
13.1 The Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
13.2 The Velocities of Deformation. . . . . . . . . . . . . . . . . . . . . . . 41
13.2.1 The Rigid System Velocities . . . . . . . . . . . . . . . . . 42
13.3 The Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
13.4 The Equations of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
14 Three Balls on a Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
14.1 The System and the Velocities. . . . . . . . . . . . . . . . . . . . . . . 47
14.2 The Velocities of Deformation. . . . . . . . . . . . . . . . . . . . . . . 47
14.2.1 The Rigid System Velocities . . . . . . . . . . . . . . . . . 49
14.3 The Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
14.4 The Equations of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
14.4.1 The Theorem of the Kinetic Energy . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
15 A Deformable Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
15.1 The Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
15.2 A Sophisticated Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
15.2.1 The Velocities of Deformation. . . . . . . . . . . . . . . . 55
15.2.2 The Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
15.2.3 The Equations of Motion. . . . . . . . . . . . . . . . . . . . 56
15.2.4 The Theorem of the Kinetic Energy . . . . . . . . . . . . 57
15.2.5 The State Quantities . . . . . . . . . . . . . . . . . . . . . . . 57
Contents ix
15.2.6 The Quantities Which Describe the Evolution . . . . . 57
15.2.7 The Energy Balance . . . . . . . . . . . . . . . . . . . . . . . 57
15.2.8 The Second Law . . . . . . . . . . . . . . . . . . . . . . . . . 59
15.2.9 The Entropy Balance. . . . . . . . . . . . . . . . . . . . . . . 59
15.2.10 The Constitutive Laws. . . . . . . . . . . . . . . . . . . . . . 59
15.2.11 The Kinematic Compatibility Relationships . . . . . . . 62
15.2.12 The Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
15.2.13 The Angular Momentum Balance. . . . . . . . . . . . . . 63
15.3 A Less Sophisticated Theory. . . . . . . . . . . . . . . . . . . . . . . . 64
15.3.1 The Velocities of Deformation. . . . . . . . . . . . . . . . 64
15.3.2 The Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
15.3.3 The Equations of Motion. . . . . . . . . . . . . . . . . . . . 65
15.3.4 The State Quantity . . . . . . . . . . . . . . . . . . . . . . . . 65
15.3.5 The Quantities Which Describe the Evolution . . . . . 66
15.3.6 The Constitutive Laws. . . . . . . . . . . . . . . . . . . . . . 66
15.3.7 The Free Energy and the Actual Power
of the Internal Forces . . . . . . . . . . . . . . . . . . . . . . 66
15.4 An Upgraded Theory with the Angular Velocity . . . . . . . . . . 73
15.4.1 The Velocities of Deformation. . . . . . . . . . . . . . . . 73
15.4.2 The Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
15.4.3 The Equations of Motion. . . . . . . . . . . . . . . . . . . . 74
15.4.4 The Constitutive Laws. . . . . . . . . . . . . . . . . . . . . . 74
15.4.5 The Kinematic Relationships . . . . . . . . . . . . . . . . . 76
15.5 The Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
15.5.1 The Velocities of Deformation. . . . . . . . . . . . . . . . 77
15.5.2 The Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
15.5.3 The Equations of Motion. . . . . . . . . . . . . . . . . . . . 78
15.5.4 The Constitutive Laws. . . . . . . . . . . . . . . . . . . . . . 78
15.5.5 The Angular Momentum Balance. . . . . . . . . . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
16 Two Deformable Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
16.1 The Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
16.2 The Velocities of Deformation. . . . . . . . . . . . . . . . . . . . . . . 81
16.2.1 The Rigid System Velocities . . . . . . . . . . . . . . . . . 82
16.3 The Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
16.4 The Equations of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 83
16.5 Why the Contact Interaction of the Two Solids
Is Described by a Force?. . . . . . . . . . . . . . . . . . . . . . . . . . . 83
17 At a Distance Interactions: Continuum Reinforced by Fibers. . . . . 85
17.1 The Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
17.2 The Velocities of Deformation. . . . . . . . . . . . . . . . . . . . . . . 85
17.2.1 The Rigid System Velocities . . . . . . . . . . . . . . . . . 86