Treatise on Classical Elasticity: Theory and Related Problems

805 Pages·2013·11.327 MB·English

by  Petre P. Teodorescu (auth.)

Mathematical and Analytical Techniques with
Applications to Engineering
Petre P. Teodorescu
Treatise on
Classical Elasticity
Theory and Related Problems
Mathematical and Analytical Techniques
with Applications to Engineering
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Petre P. Teodorescu
Treatise on Classical
Elasticity
Theory and Related Problems
123
Petre P.Teodorescu
Faculty ofMathematics
Universityof Bucharest
Bucharest
Romania
ISSN 1559-7458 ISSN 1559-7466 (electronic)
ISBN 978-94-007-2615-4 ISBN 978-94-007-2616-1 (eBook)
DOI 10.1007/978-94-007-2616-1
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Preface
Themechanicsofdeformablesolidsbringsitscontributionbothtotheknowledge
of the phenomena of the surrounding physical reality, from a theoretical point of
view,andtothedeterminationofthestateofstrainandstressinvariouselements
of construction, from practical considerations. The deformable solids have a
particularly complex character; a mathematical modelling of them is not simple
andoftenleadstoinextricabledifficultiesofcomputation.Oneofthemostsimple
mathematicalmodelsand,atthesametime,themostusedone,isthemodelofthe
elastic body—especially the linear elastic model; despite its simplicity, even this
model of real body may lead to great difficulties of calculation.
Ingeneral,theengineeringconstructionshavebeenbased,duringthecenturies,
onempiricalmethods;beginningwiththeseventeenthcentury,onehasobtaineda
lot of results, which form what now is called ‘‘Strength of Materials’’, where
simplifying supplementary hypotheses have been introduced. As a matter offact,
thisdenominationisnotaproperone,becauseitcorrespondsonlytoamechanical
phenomenon modelled by the so-called ‘‘strength theory’’; we maintained this
denomination, being still currently used. The theory of elasticity, chapter of the
mechanics of deformable solids with a theoretical character, succeeds to express
better the physical phenomenon, giving results closer to the reality, in certain
limits;itbecameascienceonlyatthemiddleofninthcentury,beingincontinuous
development even today.
Thepracticalimportanceofabookonthetheoryofelasticity,whichis—atthe
same time—an introduction to the mechanics of deformable solids, consists in
putting in evidence points of view and scientific methods of computation in a
domain in which simplified methods or with a non-accurate limit of validity are
still used. The actual technical progress and the necessity to use a minimum of
materials in various constructions ask for a better determination of the state of
strain and stress which takes place in a civil or mechanical construction; the
engineering design may be thus improved.
Thefirsteightchaptersdealwiththeconstructionofthemathematicalmodelof
a deformable solid, giving special attention to the linear elastic bodies; the for-
mulation of the fundamental problems is followed by their solution in
v
vi Preface
displacements end stresses. The importance of the concentrated loads is put into
evidence, as well in the case of Cosserat-type bodies. Another group of four
chapters contains static and dynamic spatial problems, treated systematically by
the same method of potential functions.
The following two chapters deal with some special problems: particular cases,
treated in the same systematical manner and the case of anisotropic and non-
homogeneous bodies.
The last two chapters contain introductions to thermoelasticity and linear vis-
coelasticity. Special accent is put on the solving methodology as well as on the
mathematical tool used: vectors, tensors and notions of the field theory. Contin-
uous and discontinuous phenomena and various mechanical quantities are pre-
sented in unitary form by means of the theory of distributions. Some appendices
give the book an autonomy with respect to other works, a special mathematical
knowledge being not necessary.
Concerning the first six chapters, I must mention the kind co-operation of
ProfessorVasileIlle,TechnicalUniversityofCluj-Napoca,whounfortunatelyhas
passed away. I am grateful to Mariana Gheorghi(cid:2)ta˘ for her valuable help in the
presentation of this book. The excellent cooperation of the team of Springer,
Dordrecht, is gratefully acknowledged.
Thebookcoversawidenumberofproblems(classicalornewones)asonecan
seefromitscontents.Itusedtheknownliterature,aswellastheoriginalresultsof
the author and his more than 50 years’ experience as Professor of Mechanics and
ElasticityattheUniversityofBucharest.Itisaddressedtoalargecircleofreaders:
mathematicians (especially those involved in applied mathematics), physicists
(particularly those interested in mechanics and its connections), engineers of
various specialities (civil, mechanical engineers, etc., who are scientific
researchers or designers), students in various domains etc.
Bucharest, Romania, January 2013 P. P. Teodorescu
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Aim of Mechanics of Deformable Solids. . . . . . . . . . . . . . . . 1
1.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Models in the Mechanics of Deformable Solids. . . . . 4
1.2 Fundamental Computation Hypotheses.
Short Historical Account. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Fundamental Computation Hypotheses . . . . . . . . . . . 9
1.2.2 Short Historical Account. Development Trends . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Geometry and Kinematics of Deformation. . . . . . . . . . . . . . . . . . 33
2.1 Finite Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Material and Space Co-ordinates. Strains. . . . . . . . . . 35
2.2 Infinitesimal Deformations. . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.1 Displacement Gradient . . . . . . . . . . . . . . . . . . . . . . 44
2.2.2 Continuity Equations. Computation of Displacements
and Rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.3 Elementary States of Deformation . . . . . . . . . . . . . . 60
2.2.4 Displacements and Strains in Curvilinear
Co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2.5 Kinematics of Deformation . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3 Mechanics of Stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1 Stress Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . 73
3.1.2 Variation of Stresses Around a Point . . . . . . . . . . . . 78
3.2 Stress Tensor. Equations of Equilibrium and Motion . . . . . . . 84
3.2.1 Stress Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
vii
viii Contents
3.2.2 Equations of Equilibrium and Motion. . . . . . . . . . . . 95
3.2.3 Elementary States of Stress . . . . . . . . . . . . . . . . . . . 98
3.2.4 Finite Deformations . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2.5 Stresses in Curvilinear Co-ordinates . . . . . . . . . . . . . 108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 Mathematical Models in Mechanics of Deformable Solids . . . . . . 115
4.1 Elastic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1.1 Construction of the Elastic Model . . . . . . . . . . . . . . 116
4.1.2 Elastic Potential. Green’s Theory. . . . . . . . . . . . . . . 122
4.1.3 Hooke’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.2 Inelastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.1 Bodies with Plastic Properties . . . . . . . . . . . . . . . . . 155
4.2.2 Bodies with Viscous Properties . . . . . . . . . . . . . . . . 169
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5 General Equations of the Theory of Elasticity. Formulation
of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.1 General Equations of the Theory of Elasticity . . . . . . . . . . . . 191
5.1.1 Statical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.1.2 Dynamical Problems. . . . . . . . . . . . . . . . . . . . . . . . 196
5.2 Statical Problem. Potential Functions . . . . . . . . . . . . . . . . . . 199
5.2.1 Formulations in Displacements. . . . . . . . . . . . . . . . . 199
5.2.2 Formulations in Stresses . . . . . . . . . . . . . . . . . . . . . 205
5.3 Dynamical Problem. Potential Functions. . . . . . . . . . . . . . . . 219
5.3.1 Formulations in Displacements. . . . . . . . . . . . . . . . . 219
5.3.2 Formulations in Stresses . . . . . . . . . . . . . . . . . . . . . 227
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
6 Principles and General Theorems of the Theory of Elasticity.
Computation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.1 Principles and General Theorems of the Theory
of Elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.1.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.1.2 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . 248
6.1.3 Other Considerations. . . . . . . . . . . . . . . . . . . . . . . . 256
6.1.4 Simply Connected Domains.
Multiply Connected Domains. . . . . . . . . . . . . . . . . . 262
6.2 Computation Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
6.2.1 Particular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 266
6.2.2 General Methods of Computation. . . . . . . . . . . . . . . 279
6.2.3 Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . 281
6.2.4 Method of Fundamental Solutions . . . . . . . . . . . . . . 288
6.2.5 Other Computation Methods . . . . . . . . . . . . . . . . . . 301
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Contents ix
7 Introduction to the Theory of Cosserat Type Bodies . . . . . . . . . . 307
7.1 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
7.1.1 Introduction to Cosserat Type Bodies. . . . . . . . . . . . 307
7.1.2 State of Deformation. . . . . . . . . . . . . . . . . . . . . . . . 312
7.1.3 State of Stress and Couple-Stress. . . . . . . . . . . . . . . 322
7.1.4 Constitutive Laws. . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.2 Formulations of the Static and Dynamic Problems.
General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
7.2.1 Formulation of the Static Problem . . . . . . . . . . . . . . 332
7.2.2 Formulation of the Dynamic Problem. . . . . . . . . . . . 336
7.2.3 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 345
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
8 Theory of Concentrated Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 357
8.1 Case of Linearly Elastic Bodies. . . . . . . . . . . . . . . . . . . . . . 357
8.1.1 Construction of Concentrated Loads. . . . . . . . . . . . . 357
8.1.2 Tensor Properties of Concentrated Loads. . . . . . . . . . 372
8.1.3 Solutions for Concentrated Loads. . . . . . . . . . . . . . . 376
8.2 Case of Linearly Elastic Cosserat Type Bodies . . . . . . . . . . . 383
8.2.1 Solutions for Concentrated Loads. . . . . . . . . . . . . . . 384
8.2.2 Centres of Dilatation. Centre of Rotation. . . . . . . . . . 386
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
9 Elastic Space. Elastic Half-Space. . . . . . . . . . . . . . . . . . . . . . . . . 393
9.1 Elastic Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
9.1.1 Volume Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
9.1.2 Concentrated Loads . . . . . . . . . . . . . . . . . . . . . . . . 397
9.2 Elastic Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
9.2.1 Action of a Periodic Load. . . . . . . . . . . . . . . . . . . . 400
9.2.2 Action of a Local Load. . . . . . . . . . . . . . . . . . . . . . 404
9.2.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
9.2.4 Methods of the Theory of Distributions. . . . . . . . . . . 412
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
10 Elastic Eighth-Space. Elastic Quarter-Space . . . . . . . . . . . . . . . . 427
10.1 Elastic Eighth-Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
10.1.1 Action of a Periodic Normal Load . . . . . . . . . . . . . . 428
10.1.2 Action of a Local Normal Load. . . . . . . . . . . . . . . . 430
10.1.3 Action of a Local Tangential Load. . . . . . . . . . . . . . 446
10.1.4 Particular Cases. Application. . . . . . . . . . . . . . . . . . 455
10.2 Elastic Quarter-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
10.2.1 Action of a Local Normal Load. . . . . . . . . . . . . . . . 462
10.2.2 Action of a Local Tangential Load. . . . . . . . . . . . . . 467
10.2.3 Particular Cases. Applications . . . . . . . . . . . . . . . . . 473
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478