2000_Book_MechanicsOfDeformableSolids.pdf

Issam Doghri
Mechanics of Deformable Solids

Springer-Verlag Berlin Heidelberg GmbH
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Engineering
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Issam Doghri
Mechanics of
Deformable Solids
Linear, Nonlinear, Analytical and Computational Aspects
with 233 Figures
" Springer

Dr. Issam Doghri
Associate Professor of Applied Mechanics
Universite catholique de Louvain
CESAME
Euler Building
4 Avenue G. Lemaitre
B - 1348 Louvain-la-Neuve
Belgium
EMAIL: doghri@mema.ucl.ac.be
Cataloging-in-Publication Date applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Doghri, Issam:
Mechanics of deformabIe solids : linear, nonlinear, analytical and
computational aspects 1 Issam Doghri.
(Engineering online library)
ISBN 978-3-642-08629-8 ISBN 978-3-662-04168-0 (eBook)
DOI 10.1007/978-3-662-04168-0
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Preface
This book is based on lllY experience in teaching. research, computer software
developlllcnt and consulting.
The audience for this book includes students, engineers or researchers in me-
chanical engineering, civil engineering, applied mathelllatics, materials sci-
ence, and other people who wish or rlPcd to have a good introduction to the
mechanics of deforlllable solids and structures.
SOllle chapters of the book are usually taught at the undergraduate level (e.g.,
beallls and plates in linear elasticity) while others (e.g., large deformations,
numel'ic,l! algoritluns, shells) are usually studied Cît the graduate level. Some
topies are even at the fringe of research (e.g.. computational algorithllls for
nonlinear lllechanics, damage mec:hanics. lllicro-mechanics).
1 belieye however t1wt my presentation of advanced subjects is dear enough
to be accessible to undel'graduate students and prac:ticing engineers if they
are willing to invest SOllle effort. whic:h 1 tried to minimize. ;Iost mec:hanical
ami civil engineers will be faced in their professiemal life with at least some
problems ]wlonging to the advanced materiaL and even if they use commer-
eial software to solve thell1, a correct understanding of the basic ideas will
help thelll to proc:eed properly when setting up a problem, running a code
<Inel analyzing the results.
1 shall now enumerate the reaSOllS whic:h make llle think that this book is
original anei worth publishing.
• Containeel in one textbook are three S'ubjects of major interest in solid
lllechanies:
(1) Theory of linear ebstic:ity (three-elimensional theory, yariational for-
llluiations, two-elilllensionai problellls, torsion. therll1o-elasticity, etc:.)
(2) TIlf'ory of structures (beall1s, plates alld shells) in linear isotropic
elasticity.
- (3) Nonlinear lllechanics (plasticity. viscoplasticity, finite strains, etc.)
induding c:olllputationai algorithllls.
Although numerous text.books exist on each Olle of the threc subjects, and
eyen an individual parts of each subject (e.g., books on beam theory -often
callecl strength of materials). 1 am not mvare of il book which contains a
comprehensive treatment of ali three subjeets.

VI
• Of course -otherwise the book would be too lengthy- 1 had to make some
choices. Several subjects are omitted completely (e.g., dynamics, vibra-
tions, waves). AIso, each chapter is actually an introduction (or a primer)
to the relevant theme. However, the presentation is given with enough gen-
emlity and depth to achieve two goals:
- (1) Allow the readers to grasp the fundamentals of each subject and
solve basic or most common problems.
- (2) Permit them to read and study more advanced or detailed texts on
the subject if they wish to do so.
• There is a good balance between engineering and mathematics. 1 always
try to introduce a subject via an intuitive approach and then tackle its
formulation and analysis in mathematical terms. Many textbooks use a
mathematical equipment which is either too limited or too sophisticated:
I tried to strike the right compromise. I view mathematics as a tool, but
a wonderful one. On the one hand, it should not be so heavy as to render
the purpose obscure, and on the other hand, when used properly, it offers
insight into the engineering problem at hand. AIso, since there is usually
more than one approach to a given subject, 1 have always chosen one which
is simple, but not simplistic. As mentioned above, the presentation always
has the appropriate depth aud generality.
• I use seveml notation systems: tensor (symbolic) notation (e.g., 0"), in-
dex (component) notation (l7ij), matrices ([17ij]) and arrays ({171 }). I often
present formulations in at least two different notation systems. The read-
ers can use the notation they feell110re comfortable with: however they are
encouraged to try to understaud and use aU of them. Each notation has
its advantages. Tensor notation provides a neat qualitative understanding
of basic principles and results, index notation is usually what the students
tind easiest to deal with, and matrices and arrays are most useful for com-
puter implementation. However, each notation system has its shortcomings
and cases where it becol11es too cumbersome or should be used with care.
• There is an emphasis on nonlinear material models, including sophisti-
cated ones (e.g., nonlinear kinematic hardening, ductile damage, micro-
mechanically-based models). In each case, three aspects are examined: basic
experimental facts, mathel11atical formulation and numerical implementa-
tion.
• Computational methods oc:c:upy a good portion of the book. This is es-
pecially true in the chapters dealing with nonlinear mechanics, where the
emphasis is put on numerical methods, but the correc:t framework is al-
ready introduced in the chapter dealing with variational formulations in
linear elasticity. Usually, analytical and numerical methods in mechanics
are viewed by the students as completely different worlds: they are taught
in separate courses, by different teachers using separate approaches aud
notations. lIy aim is to show the readers that everything stems from basic
principles; there are SOl11e problems which can be solved in closed form

VII
(usually after making several reasonable assumptions), and some problems
for which we seek approximate solutions by numerical means.
• In each chapter, there are severai problems which are completely worked
out. Some of them cover fundamental issues of each subject and allow the
readers to put the basic theory into practice. Other problems are rather
involved and lengthy, but shed more light on the subject or cover some
aspects which have not been treated in the basic theory. AII of them have
been proposed as examination subjects.
• There are several appendices which contain useful results and formulae.
• The table of contents is given elsewhere: the logic behind the chapter fiow
is as folIows.
Chapter 1 gives the minimal background information which is needed in
order to start. and the following chapters are arranged according to what 1
found from my teaching experience to be an increasing degree of difficulty.
This has three advantages for the students:
- (1) They can start solving problems very quickly and therefore their
interest remains intact.
- (2) They c:an grasp fundamental notions such as stress, strain, tension,
compression. bending, torsion, energy. etc., rapielly.
- (3) They can better appreciate the simplifying assumptions and sub-
tleties of elifferent theories (e.g., solving a beam problem first by el-
ementary beam theory -strength of materials- and then by the two-
elimensional theory of elasticity).
Acknowledgments
During my stuelies at College Saeliki (Tunis), Ecolc ."{ationale eI·Ingenieurs
ele Tunis, Universite Pierre et l'darie Curie (Paris) anei Ecole Normale
Superieure ele Cachan (France), 1 was very luc:ky to have some truly re-
markable anel inspiring teachers who made 111e see and pursuc the beauty in
Science. Also, there are sorue classmates from those years whose frieuelship 1
stiU c:herish today.
In my professional career at the University of California-Santa Barbara,
Centric Engineering Systems (California), Universite catholique ele Louvain
(Belgium) and as a consultant for various companies. 1 hael the privilegc of
working with many really talented people with whom 1 had very interesting
discussions anei interactions.
1 typeel the book rnyself using l5IEX, but 1 benefiteel from the precious
help of two of my graeluate stuelents: Serge ],vlunhoven anei Svetoslav Nikolov.
Serge integrateel alI figures in the source files, formatteel the whole book anei
helpeel with numerous worel-processing problems. Svetoslav prepareel most of
the figures from my hanel-elrawn .. graffiti"' anel enelureel my enelless c:hanges
with patience. Vithout the help of Serge anei Svetoslav, the book woulcl have
taken much longer to be reaely. 1 am very grateful to them.
When 1 starteel this project, 1 thought that it woulel take me at most
one year of moelerate work in oreler to put together 80me of my papers,

VIII
lecture notes and other handwritten notes. The endeavor ended up taking
alI my free time during two and a half years, including vacations, weekends,
evenings and a four-month sabbaticalleave. 1 am deeply thankful to my wife
for her unwavering support and encouragement, and 1 hope to make up to
her and our two daughters for alI the time that 1 did not spend with them.
Last, but not least, 1 am very grateful to my parents for giving me a
nurturing home and a good education, and to my Creator for blessing me
with good health and overalI luck in life.
Louvain-Ia-Neuve, December 1999.

Table of Contents
Preface....................................................... V
1. Basic mechanics ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 On tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Stress................................................. 3
1.3 Strain................................................. 5
1.4 Principal invariants and eigenvalues of stress and strain ..... 6
1.5 lIohr·s stress circles .................................... 7
1.6 Equilibrium............................................ 8
1.7 Local formulation of static problems . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Continuity equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
1.9 COl11patibility equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
1.10 Strength criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
1.11 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16
1.12 Strain energy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
1.13 Navier equations ....................................... 23
1.14 Beltrami-Mitchell cOl11patibility equations ................. 24
1.15 Saint-Venant·s principle, Uniqueness, Superposition, Special
theories .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25
1.16 Solved problem: composite cylinder under axialload . . . . . . .. 26
2. Variational formulations, work and energy theorems . . . . .. 29
2.1 Local formulation of static problems ...... . . . . . . . . . . . . . . .. 29
2.2 Virtual work theorel11 (VWT) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30
2.3 Displacel11ent-based variational forl11ulation . . . . . . . . . . . . . . .. 33
2.4 Potential energy theorel11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
2.5 Stress-based variational formulation. . . . . . . . . . . . . . . . . . . . . .. 37
2.6 Complementary energy theorel11 . . . . . . . . . . . . . . . . . . . . . . . . .. 38
2.7 Energy bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
2.8 Stored energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40
2.9 Maxwell-Betti reciproc:ity theorem . . . . . . . . . . . . . . . . . . . . . . .. 41
2.10 Castigliano's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
2.11 Introduction to nUl11eric:all11ethods ....................... 45
2.11.1 Method of Ritz .................................. 45

X Table of Contents
2.11.2 Finite element method (FEM) ..................... 45
2.12 Solved problem: volume change of an axially compressed body 47
3. Theory of beams (strength of materials) . . . . . . . . . . . . . . . . .. 49
3.1 Definitions and geometric properties . . . . . . . . . . . . . . . . . . . . .. 49
3.2 Pure bending of a straight beam: 3D elasticity solution . . . . .. 51
3.3 Basic assumptions of beam theory . . . . . . . . . . . . . . . . . . . . . . .. 55
3.3.1 Externalloads................................... 56
3.3.2 Internalloads (stress resultants) . . . . . . . . . . . . . . . . . . .. 57
3.3.3 Equilibrium equations. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
3.3.4 Navier-Bernoulli assumption . . . . . . . . . . . . . . . . . . . . . .. 59
3.3.5 Constitutive equations .. . . . . . . . . . . . . . . . . . . . . . . . . .. 59
3.4 Displacement boundary and continuity conditions .......... 60
3.5 Computation of internalloads (stress resultants) ........... 62
3.6 Computation of normal and shear stresses . . . . . . . . . . . . . . . .. 64
3.7 Statically determinate or indeterminate problems .......... 66
3.8 Strain energy ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
3.9 Work and energy theorems .............................. 68
3.10 Influence lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
3.11 Solved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
3.11.1 Shear reduced area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
3.11.2 Statically determinate problems . . . . . . . . . . . . . . . . . . .. 71
3.11.3 Statically indeterminate problems .. . . . . . . . . . . . . . . .. 77
3.11.4 llaxwell-Betti reciprocity theorem . . . . . . . . . . . . . . . . .. 80
3.11.5 Castigliano's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
3.11.6 Virtual work theorem (VWT) . . . . . . . . .. .. . . . . . . . . .. 90
4. Torsion of beams ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
4.1 Formulation with a warping function . . . . . . . . . . . . . . . . . . . . .. 95
4.2 Formulation with a conjugate function .... . . . . . . . . . . . . . . .. 98
4.3 Formulation with Prandtl's stress function . . . . . . . . . . . . . . . .. 98
4.4 Contour lines of the stress function ....................... 100
4.5 Maximum tangential stress .............................. 101
4.6 Membrane analogy ..................................... 102
4.7 Multi-connected sections- Particular case .................. 104
4.8 Multi-connected sections- General case .................... 104
4.9 Thin tubes with variable thickness ........................ 107
4.10 Potential energy ........................................ 108
4.11 Cylindrical coordinates .................................. 109
4.12 Solved problems ........................................ 110
4.12.1 Elliptic section ................................... 110
4.12.2 Triangular section ................................ 113
4.12.3 Notched circular section ........................... 114
4.12.4 Thin rectangular section .......................... 116
4.12.5 Ritz's method- Square section ...................... 117

Table of Contents XI
4.12.6 Hollow elliptic section- Special method .............. 118
4.12.7 Hollow elliptic section- General method ............. 119
4.12.8 Thin circular tube ................................ 121
4.12.9 Thin-walled section with multiple voids ............. 121
5. Theory of thin plates ..................................... 123
5.1 Definitions and notation ................................. 123
5.2 Internalloads (stress resultants) .......................... 123
5.3 Equilibrium equations ................................... 126
5.4 Displacements .......................................... 127
5.5 Strains ................................................ 129
5.6 Constitutive equations .................................. 129
5.7 Summary: two un-coupled problems ...................... 130
5.8 Fundamental P.D.E. for bending problem .................. 131
5.9 Boundary conditions .................................... 133
5.10 Contradictions in Kirchhoff-Love theory ................... 135
5.11 Plates with two simply supported opposite edges - Levy's
method ............................................... 136
5.13 Influence function ...................................... 140
5.14 Solved problems ........................................ 140
5.14.1 Uniformly loaded rectangular plate with two simply
supported opposite edges and two built-in eelges ...... 140
5.14.2 Uniformly loaeleel rectaugular plate with two simply
supporteel opposite eelges anei two free eelges ......... 142
6. Bending of thin plates in polar coordinates ............... 143
6.1 Change of coordinates ................................... 143
6.2 Axisymmetric problems ................................. 146
6.4 Solveel problems ........................................ 149
6.4.1 Uniformly loaeleel plate ............................ 149
6.4.2 Uniform loael along a concentric circle ............... 150
6.4.3 Uniform pressure on a concentric elisk ............... 153
6.4.4 Plate simply supporteel on a number of points ....... 155
6.4.5 Ritz's method .................................... 160
7. Two-dimensional problems in Cartesian coordinates ...... 163
7.1 Plane strain ........................................... 163
7.2 Plane stress .......................................... 164
7.3 Summary: plane strain versus plaue stress ................. 165
7.4 Airy stress function ..................................... 167
7.5 Polynomial solutions ................................... 167
7.6 Solution by Fourier series ................................ 169
7.7 Generalized plane stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

XII Table of Contents
7.8 Solved problems ........................................ 173
7.8.1 Concentrated load at the end of a cantilever beam .... 173
7.8.2 Uniform loads on the upper and lower surfaces of a
simply supported beam ........................... 177
7.8.3 Uniform load on a cantilever beam ................. 182
7.8.4 Uniform load on a beam with two clamped ends ...... 182
7.8.5 Compression of a beam in the height-direction ....... 182
7.8.6 Body forces- Beam under its own weight ............ 187
8. Two-dimensional problems in polar coordinates ........... 193
8.1 Change of coordinates ................................. 193
8.2 Summary: plane strain versus plane stress ................ 195
8.3 Airy stress function ..................................... 196
8.4 Axisymmetric plalle problems ............................ 198
8.5 Periodic Airy stress functions ............................ 200
8.6 Generalized plane strain ................................. 201
8.7 Solved problems ........................................ 201
8.7.1 Hollow circular cylinder under inner and outer pressures201
8.7.2 Composite hollow cylinder under inner and outer pres-
8.7.3
8.7.4
8.7.5
8.7.6
8.7.7
8.7.8
8.7.9
8.7.10
8.7.11
8.7.12
sures ............................................ 205
Coil ,vinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Bending of a curved beam . . . . . . . . . . . . . . . . . . . . . . . . . 209
Traction of a circular arch ......................... 214
Rotating disk of uniform thic:kness .................. 215
Rotating disk of variable thickness .................. 218
Stress concentration in a plate with a small circular hole222
Force 011 the straight edge of a semi-infinite plate ..... 224
Pressure on the straight edge of a semi-infinite plate .. 228
Compression of a disk along a diameter ............. 230
Compression of a disk over two opposing arcs ........ 231
9. Thermo-elasticity ......................................... 233
9.2 Heat equation .......................................... 234
9.3 Thermo-mechanical problem ............................. 235
9.3.1 Thermal problem ................................. 236
9.3.2 l'dec:hanic:al problem .............................. 237
9.4 Thermal stresses: some remarks .......................... 237
9.5 Solved problems ........................................ 239
9.5.1 Axisymmetric thermal stresses in a hollow cylinder ... 239
9.5.2 Thermal stresses in a composite cylinder ............ 242
9.5.3 Transient thermal stresses in a thin plate ............ 245

Table of Contents XIII
10. Elastic stability ........................................... 249
10.1 Introduction ........................................... 249
10.2 Direct and energy methods .............................. 250
10.3 Euler's method for axially compressed columns ............. 252
10.3.1 Critical buckling load ............................. 252
10.3.2 Critical buckling stress ............................ 254
10.3.3 Rell1arks ........................................ 256
10.4 Energy-based approximate rnethod ....................... 256
10.5 Non-conservative loads .................................. 258
10.6 Solved problems ........................................ 259
10.6.1 Two rigid bars connec:ted with a spring ............. 259
10.6.2 Colurnn clamped at one end ....................... 260
10.6.3 Column clamped at one end and simply supported at
the other ........................................ 261
10.6.4 Colull1n clall1ped at both ends ..................... 263
10.6.5 Column elastically built-in at one end and sill1ply sup-
ported <It the other ............................... 264
10.6.6 Column with non-uniform properties ................ 266
10.6.7 Ecc:entric: compressive load ........................ 267
10.6.8 Beam-column under compressive and bending forces .. 268
10.6.9 Energy method .................................. 269
11. Theory of thin shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11.1 Geoll1etry of the mid-surface ............................. 273
11.2 First fundamental fOrIn ................................. 274
11.3 Second fundamental form ............................... 278
11.4 Compatibility conditions of Codazzi and Gauss ............ 280
11.5 Surface of revolution ................................... 280
11.5.1 General case .................................... 280
11.5.2 Conic surfaces ................................... 283
11.6 Gradient of a vector field in curvilinear coordinates ......... 283
11.7 Kinematics of the mid-surface ........................... 285
11.8 Displacements anei strains outside the mid-surface .......... 287
11.8.1 General theory .................................. 287
11.8.2 Application: plates in rectangular coordinates ....... 289
11.8.3 Application: plates in polar coordinates ............ 290
11.9 Internal loads (stress resultants) ......................... 290
11.10 Equilibrium equations .................................. 292
11.10.1 General theory .................................. 292
11.10.2 Application: plates in rec:tangular coordinates ....... 295
11.10.3 Application: plates in polar coordinates ............ 295
11.121'dembrane theory ...................................... 298
11.13 Further reading ........................................ 298

XIV Table of Contents
12. Elasto-plasticity .......................................... 301
12.1 One-dimensional model ................................. 301
12.2 Three-dimensional model ................................ 304
12.3 Linear elasticity ........................................ 305
12.4 Equivalent stress ....................................... 306
12.5 Hardening ............................................. 306
12.6 Flow rules ............................................. 307
12.7 Tangent operator, loading/unloading, hardening/softening ... 308
12.8 Elementary examples ................................... 312
12.8.1 Uniaxial tension-compression ...................... 312
12.8.2 Simple shear .................................... 313
12.9 Boundary-value problem ................................ 313
12.10Numerical algorithms ................................... 313
12.10.1 Finite element method (F.E.M.) ................... 313
12.10.2 Return mapping algorithm ........................ 315
12.10.3 Consistent tangent operator ....................... 318
12.11 A general framework for material models .................. 320
12.11.1 State variables .................................. 320
12.11.2 Equations of state ............................... 321
12.11.3 Flow rules ...................................... 322
12.11.4 Rate-independent plasticity ....................... 323
12.11.5 Heat equation ................................... 324
12.12 A class of non-associative plasticity models ................ 325
12.13 Further reading ........................................ 327
13. Elasto-viscoplasticity ..................................... 329
13.2 Three-dimensional model. ............................... 331
13.3 Numerical algorithms ................................... 332
13.3.1 Return mapping algorithm ........................ 333
13.3.2 Consistent tangent operator ....................... 333
13.4 Further reading ........................................ 335
14. Nonlinear continuum mechanics .......................... 337
14.1 Kinematics ............................................ 337
14.1.1 Description of motion ............................ 338
14.1.2 Material time derivative .......................... 338
14.2 Deformation ........................................... 339
14.2.1 Deformation gradient ............................ 339
14.2.2 Polar decomposition ............................. 340
14.2.3 Spectral decompositions .......................... 341
14.2.4 Length variation ................................. 343
14.3 Strain measures ........................................ 344
14.3.1 One-dimensional case ............................ 344
14.3.2 Three-dimensional case ........................... 344

Table of Contents XV
14.4 Strain rates ........................................... 346
14.5 Balance laws .......................................... 348
14.5.1 Transport formula ............................... 348
14.5.2 Conservation of mass ............................. 349
14.5.3 Conservation of linear momentum ................. 350
14.5.4 Conservation of rotational momentum .............. 350
14.5.5 Cauchy stress tensor ............................. 351
14.5.6 Eulerian strong formulation ....................... 351
14.5.7 Eulerian weak formulation ........................ 352
14.5.8 Balance of work aud energy rates .................. 353
14.5.9 Nominal stress .................................. 354
14.5.10 Lagrangian weak formulation ...................... 354
14.5.11 Lagrangian strong formulation .................... 355
14.6 Conjugate stress and strain measures ..................... 356
14.6.1 Definition and examples .......................... 356
14.6.2 Interpretation of the second Piola-Kirchhoff stress ... 358
14.6.3 Uniaxial tension/compression ..................... 358
14.7 Objectivity ............................................ 359
14.7.1 Definition ...................................... 360
14.7.2 Examples ....................................... 361
14.8 Objective stress rates ................................... 363
14.8.1 Examples ....................................... 363
14.8.2 A family of objective rates ........................ 365
14.9 Laws of thermodynamics ................................ 366
14.9.1 First law ....................................... 366
14.9.2 Second law ..................................... 366
14.9.3 Clausius-Duhem inequality ....................... 367
14.10 Further reading ........................................ 367
15. Nonlinear elasticity ....................................... 369
15.1 Hyperelasticity and hypoelasticity ........................ 369
15.1.1 Definitions ...................................... 369
15.1.2 Hyperelasticity and material objectivity ............ 369
15.1.3 Elasticity tensors ................................ 371
15.1.4 Incompressibility constraint ....................... 373
15.2 Principal invariants and principal stretches ................ 374
15.3 Isotropic hyperelasticity in principal invariants ............. 375
15.3.1 Formulation .................................... 375
15.3.2 l'lodified neo-Hookean model ...................... 377
15.3.3 Modified Mooney-Rivlin model .................... 378
15.4 Isotropic hyperelasticity in principal stretches .............. 379
15.4.1 Formulation .................................... 379
15.4.2 Modified Ogden's model .......................... 380
15.5 Examples of homogeneous deformations ................... 381
15.5.1 Homogeneous simple shear ........................ 381

XVI Table of Contents
15.5.2 Uniform extension ............................... 383
15.5.3 Pure dilatation .................................. 384
15.6 Linearization .......................................... 385
15.6.1 Linearization of the deformation ................... 386
15.6.2 Linearization of constitutive equations ............. 387
15.6.3 Linearization of the equations of elasto-statics ....... 387
15.6.4 Variational formulations .......................... 388
15.6.5 Linearization of the weak formulation .............. 389
15.7 Mixed variational formulation ........................... 390
15.7.1 Formulation .................................... 390
15.7.2 Incompressibility constraint ....................... 392
15.8 Appendices ............................................ 393
15.8.1 The Piola identity ............................... 393
15.8.2 Linearization of a pressure B.C.................... 393
15.8.3 Differentiation of an isotropic function of a second-
order symmetric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
15.8.4 Elasticity tensors for principal stretch formulation ... 395
16. Finite-strain elasto-plasticity .............................. 397
16.1 First theory ........................................... 397
16.1.1 lIultiplicative decomposition of the deformation gra-
dient ........................................... 397
16.1.2 Hyperelastic-plastic constitutive equations .......... 399
16.1.3 Stress-strain relations ............................ 401
16.1.4 Flow rules ...................................... 402
16.1.5 Elastic predictor ................................. 404
16.1.6 Time discretization of the plastic flow rule .......... 404
16.1.7 Return mapping algorithm in principal stresses and
strains ......................................... 406
16.1.8 Algorithmic tangent moduli ....................... 407
16.1.9 Summary of the algorithm ........................ 408
16.1.10 Application: Quadratic logarithmic free energy and
h flow ..................... , ................... 410
16.2 Second theory ......................................... 413
16.2.1 Additive decomposition of the rate of deformation
and hypoelasticity ............................... 413
16.2.2 Computation of the strain increment ............... 415
16.2.3 Polar decomposition algorithm .................... 417
16.2.4 A time-integration algorithm for the rotation matrix . 419
16.2.5 Application: the Jaumann objective stress rate ...... 420
16.2.6 Summary ....................................... 421
16.2.7 Incremental objectivity ........................... 422

Table of Contents XVII
17. Cyclic plasticity .......................................... 423
17.2 Three-dimensional model. ............................... 425
17.3 Dissipation inequality ................................... 428
17.4 Plastic multiplier ...................................... 428
17.5 Tangent operator ...................................... 429
17.6 Hardening modulus .................................... 429
17.7 Retum mapping algorithm .............................. 430
17.8 Consistent tangent operator ............................. 433
17.9 Numerical simulation ................................... 435
18. Damage mechanics ....................................... 439
18.1 Damage variable ....................................... 439
18.2 Three-dimensional constitutive model ..................... 442
18.3 Dissipation inequality ................................... 445
18.4 Plastic multiplier ...................................... 445
18.5 Tangent operator ...................................... 445
18.6 Hardening modulus .................................... 446
18.7 Closed-form solutions for loadings with constant triaxiality .. 447
18.8 Return mapping algorithm .............................. 450
18.8.1 Corrections over the elastic predictor ............... 458
18.8.2 Summary of the algorithm ........................ 458
18.8.3 Non-damaged case ............................... 458
18.9 Consistent tangent operator ............................. 459
18.9.1 Non-damaged case ............................... 461
18.10 Numerical simulations .................................. 461
18.10.1 Ductile failure under uniaxial tension .............. 462
18.10.2 Ductile failure under simple shear ................. 462
18.10.3 A post-processor for crack initiation ............... 462
18.11 Further reading ........................................ 465
19. Strain localization ........................................ 469
19.1 lIotivation: a one-dimensional example ................... 469
19.2 Uniqueness and ellipticity ............................... 472
19.3 Strain localization ...................................... 474
19.3.1 Continuous bifurcation ........................... 475
19.3.2 Discontinuous bifurcation ......................... 475
19.3.3 Summary ....................................... 477
19.4 Analytical results for initially homogeneous plane problems .. 477
19.4.1 General strain-softening models ................... 477
19.4.2 A ductile damage model .......................... 481
19.5 Numerical results for a ductile damage model .............. 481
19.5.1 Biaxial loadings in plane stress .................... 481
19.5.2 Notched plate with a macro-defect ................. 486
19.6 Nonlocalor intemal-length models ....................... 495

XVIII Table of Contents
19.7 A two-scale homogenization procedure.................... 497
19.8 Numerical algorithms ................................... 502
19.9 Elasticity with damage- Model without threshold .......... 504
19.9.1 Local constitutive equations ...................... 504
19.9.2 Nonlocal macroscopic formulation ................. 505
19.9.3 Numerical simulations ............................ 505
19.10 Elasticity with damage- Model with threshold ............. 508
19.10.1 Local constitutive equations ...................... 508
19.10.2 Nonlocal macroscopic formulation ................. 508
19.10.3 Numerical simulations ............................ 509
19.11 Appendices ............................................ 511
19.11.1 Strain localization criterion in 2D .................. 511
19.11.2 Macroscopic free-energy potential .................. 513
19.11.3 Macroscopic dissipation potential .................. 516
20. Micro-mechanics of materials ............................. 517
20.1 lVIicro/macro approach .................................. 517
20.2 Homogenization schemes ................................ 519
20.2.1 Average strains and stresses ....................... 519
20.2.2 Voigt model .................................... 521
20.2.3 Reuss model .................................... 522
20.2.4 Self-Consistent model ............................ 522
20.2.5 Mori-Tanaka model .............................. 524
20.3 lficro/macro constitutive model for semi-crystalline polymers 525
20.3.1 Crystalline phase ................................ 526
20.3.2 Al11orphous phase ............................... 529
20.3.3 Interl11ediate phase .............................. 534
20.3.4 Single inclusion .................................. 536
20.3.5 Overall behavior ................................. 536
20.3.6 Numerical sil11ulations ............................ 536
20.4 Further reading ........................................ 541
A. Cylindrical coordinates ................................... 543
B. Cardan's formulae ........................................ 549
C. Matrices for the representation of second- and fourth-order
tensors ................................................... 551
C.1 Storage ............................................... 551
C.2 Change of coordinates .................................. 554
D. Zero-stress constraints .................................... 557
D.1 Small-strain h elasto-plasticity .......................... 557
D.2 General small-strain models ............................. 559
D.3 General finite-strain models ............................. 560

1. Basic mechanics
In this chapter, we recall some basic equations and definitions for stress,
strain, equilibrium, compatibility, strength criteria, Hooke's law, etc., which
are needed in subsequent chapters. We tried to separate results which are
independent of any constitutive model from those which are specific to lin-
ear elasticity. The reader is assumed to have some knowledge of continuum
mechanics, tensor analysis, linear algebra, etc.
1.1 On tensors
Throughout the book, boldface symbols denote tensors, the order of which
is indicated by the context. Einstein's summation convention over repeated
indices is used unless otherwise indicated:
3
aikbkj == L aikbkj
k=l
Dots and colons are used to indicate tensor products contracted over one and
two indices, respectively:
U· v = UiVi;
(a· b)ij = aikbkj;
(C : a)ij = Cijklalk;
(a· U)i =aijUj;
a : b = aikbki;
(C: D)ijkl = CijmnDnmkl
Tensor products are designated by 0, e.g.,
(U0V)ij =UiVj; (a0b)ijkl =aijbkl
There are numerous references on tensors; for this book an introduction such
as the one given in (Ogden, 1984) or (Segel, 1977) is sufficient. What we need
to remember is that in order for a "mathematical object with indices" to
represent tensor components, it has to obey precise transformation rules. For
example, in a change of coordinates from one orthonormal basis (el,e2,e3)
to another (el, e2, e3), a vector (first-order tensor),

2 1. Basic mechanics
transforms according to:
where Q is a 3 x 3 proper-orthogonal matrix:
QikQjk = QkiQkj = Oij, det Q = 1,
with Oij denoting Kronecker's symbol:
Oij = 1 if i = j; Oij = O if i =1 j
Row (i) of matrix Q contains the components of vector ei in basis (el, e2, e3).
A second-order tensor,
transforms according to:
aij = Qikak/Qjl, aij = QkiaklQlj,
The second- and fourth-order identity tensors (1 and 1) are defined by:
1
l ij = Oij, I ijkl = "2 (OikOjl + OilOjk)
The inverse of a second-order tensor a is denoted by a-l so that:
-1 -1 1· ( -1) ( -1) ~
a . a = a . a = , l.e. aik a kj = a ilalj = Uij
The inverse of a fourth-order tensor C is denoted by C-1 so that:
C: C-1 = C-1 : C = 1, i.e. Cijmn(C-l)nmkl = (C-l)ijpqCqpkl = I ijkl
In several chapters, two lemmas will be useful:
Lemma:
1
Let: C = 1 - Da (8) b and De = a : b,
then: C invertible {::::::} D =1 De,
and:
-1 1
C = 1 + D _ De a (8) b, (1.1)
with a and b second-order tensors and C a fourth-order tensor. The proof
for this lemma is straightforward. There is a similar lemma for the case when
a and bare vectors and C a second-order tensor:
Lemma:
1
Let: C = 1- Da (8) b and De = a· b,
then:
and:
C invertible {::::::} D =1 De,
1
C-l = 1 + D _ De a (8) b (1.2)

1.2 Stress 3
1.2 Stress
Consider an elementary uniaxial tension test, Le. a bar (axis ez ) with uniform
cross section is subjected to forces ±Fez applied at its end sections and along
its axis (Fig. 1.1). In this test, a (uniaxial) stress, i.e. an internal force per
unit area, develops and is defined by
F
O"u = A'
where A is the area value of the deformed cross section.
r-===1
ca
(1.3)
Fig. 1.1. Uniaxial tension test on a cylindrical specimen: initial (a) and deformed
(b) states.
This notion is generalized to multi-axial loadings by defining a Cauchy
stress tensor iT which is a second-order symmetric tensor (see Chap. 14 and
references therein for details). Since iT is symmetric, it can be stored as a
3 x 3 symmetric matrix in a given orthonormal basis:
(1.4)
It is essential to remember that in a change of coordinates from an orthonor-
mal basis (8(1» to another (8(2», the stress matrices transform according
to:
(1.5)
where the change of coordinates matrix [Q] is proper-orthogonal:
Q .QT =QT . Q =1; det Q =1 (1.6)
For an application of Eqs. (1.5,6), see Appendix A.
Similarly to the uniaxial case, the stress components O"ij are interpreted
as internal forces per unit area. If we isolate an elementary parallelepiped
of dimensions dx x dy x dz in a stressed body, then on each face of the
parallelepiped acts an internal force (a vector) whose three components per
unit area in the Cartesian basis (ez , ey, ez ) are plotted in Fig. 1.2.
It is important to notice the convention used in Fig. 1.2. On the facet of
outside normal ez for instance, acts the following force per unit area:

(-)
a yy
--
1
(-)1
aYXff
'" 1 (+
__( I axz
,
(-)
a yz
(+)
azz
I ( _
laxz
a(+) ,
xy
Fig. 1.2. Stresses acting on the facets of an elementary parallelepiped of dimensions
dx x dy x dz
(1.7)
Upper scripts (+) and (-) in Fig. 1.2 have the following meaning:
(±) _ oaxx (dx)
axx - axx ± âx 2'
idem for other stress components. Equation (1.7) is generalized as follows:
the force per unit area acting on a facet with outside unit normal n is:
(1.8)
In component form, this can be written as:
(1.9)
Note that the so-called "stress vector" (o-T . n) is not in general collinear
with n. The normal stress an is defined by:
an = (o-T . n) . n = ajinjni
The sign convention adopted in this book is that an is positive if it is tensile,
and negative if it compresses the facet (in civil engineering, an opposite sign
convention is usually used).
We shall see later on that it is useful to define a deviatoric stress tensor
8 which is, by design, traceless (tr 8 =O):
1
8 == o- - -(tr 0-)1,
3
i.e. (1.10)

1.3 Strain 5
1.3 Strain
Consider the uniaxial tension test of Fig. 1.1 again. If the initial and final
values of the bar length are 10 and 1, respectively, then a strain along the axis
of the bar is defined as:
(1 - 1
0 )
f:u =
10
(1.11)
This notion is generalized to multi-axial loadings by defining a strain ten-
sor, which is defined in the injinitesimal case as the following second-order
symmetric tensor (see Chap. 14 and references therein for details):
(1.12)
where u(x) is the displacement field and " the gradient operator. In Carte-
sian coordinates, Eq. (1.12) can be written under the following form:
f:i. = ~(âUi + âUj)
3 2 âXj âXi
(1.13)
Since e is symmetric, it can be stored as a 3 x 3 symmetric matrix in a given
orthonormal basis:
(1.14)
We now give a geometrical interpretation of the strain components in Carte-
sian coordinates (for details, see (Love, 1927)). Consider material line vectors
e x and e y which transform onto e~ and e~ after deformation. The diagonal
(or direct) strain components f:xx and f:yy have the same interpretation as
in the uniaxial stress case (Eq. (1.11)); they represent relative length varia-
tion in the (x) and (y) directions, respectively. As for the shear strain f:xy , it
represents angle variation:
1 I I
f:xy = 2ex· e y
It is essential to remember that in a change of coordinates from an orthonor-
mal basis to another, the strain matrices obey the same transformation rules
as the stresses:
(1.15)
It can be shown (e.g., Chap. 14) that for infinitesimal strains, (tr e) measures
the variation of volume:
dV
- ~ l+tre
dVo '
i.e.
dV -dVo
tr e ~ dVo '

where dVo and dV are elementary volumes before and after deformation,
respectively. It is useful to define the deviatoric part of E as follows:
e == E - ~(tr E)l, i.e. eij = tij - ~(tmm)Oij (1.16)
It is obvious that -by design- e is traceless (tr e =O), therefore e represents
the isochoric part of the strain tensor.
1.4 Principal invariants and eigenvalues of stress and
strain
Let 7] be a second-order symmetric tensor (rlij = 1/ji); typical examples are
the stress (u) and strain (E) tensors. There are three scalar quantities which
do not change from one orthonormal basis to another. They are called the
principal invariants of 7] and can be expressed as follows:
It = tr 7] =1/11 +1/22 +'/]33;
12 ~ [Ii - tr (7]2)] = 11/22 1/23
1+1
1/11 1/13
1+1
1/11 1/12 1;
1/23 1/33 1/13 1/33 1/12 1/22
1/11 1/12 1/13
13 det7] = 1/12 1/22 1/23 (1.17)
1/13 1/23 1/33
When changing from an orthonormal basis 8(1) to another 8(2), a necessary
"sanity check" consists in verifying that It, 12 and 13 remain unchanged:
There exists an orthonormal basis (e(1), e(2), e(3») in which 7] is represented
by a diagonal matrix:
~ l'
1/3
(1.18)
where 1/i are the eigenvalues (or principal values) of 7] and e(i) the principal
directions. By definition, 1/i and e(i) are such that:
7]. e(i) =1/ie(i); i =1, 2,3; no sum over i (1.19)
Equation (1.19) shows that the eigenvalues are solution of the following equa-
tion:
det(7]- 0:1) =O, (1.20)

1.5 Mohr's stress circles 7
which can be rewritten as follows:
_0:3 +Il 0:2 - I20: + h = O, (1.21)
where It, I2 and I3 are the principal invariants of TI defined by (1.17). Equa-
tion (1.21) can be solved in closed form using Cardan's formulae (see Ap-
pendix B). Equations (1.17) show that the principal invariants and values
are related by:
(1.22)
Principal invariants and values play a central role in modeling isotropic ma-
terials, examples are given in Sect. 1.10 and chapters 12, 15 and 16. For
isotropic and elastic materials, it can be shown that the principal directions
of stress and strain coincide (see Sect. 15.8.3).
Finally, there are cases where TI has the following matrix representation:
'1712 O 1
'1722 O ,
O '1733
(1.23)
Examples are stress or strain matrices for plane problems (Chaps. 7 and 8)
and bending moment matrices for plates (Chaps. 5 and 6). In those cases,
solving Eq. (1.20) becomes an easy task: simple algebra gives the principal
values as follows:
"',' ="n;'In ± [ ("n;*')'+'Ii,j"', *="" (1.24)
1.5 Mohr's stress circles
A useful graphical representation of normal and shear stresses is due to the
German engineer Otto Mohr (1835-1918). Consider a facet of outward unit
normal n, the "stress vector" u T . n acting on the facet can be decomposed
into normal and shear components as follows:
(1.25)
Designating by an and as the scalar measures of normal and shear stresses:
(1.26)
it is shown -e.g., (Mase, 1970)- that points (an, as) are necessarily situated
within the shaded area of Fig. 1.3. The three Mohr's stress circles of Fig. 1.3
are determined by the principal stresses which are assumed to be distinct and
ordered according to: al < a2 < a3. The radii of the circles are: (a2 - ad/2,
(a3 - a2)/2 and (a3 - ad/2. The latter being the radius of the largest circle,
Fig. 1.3 shows that it is equal to the maximum shear stress. This result is
used for instance in Tresca's yield criterion (Sect. 1.10).

Fig. 1.3. Mohr's stress circles. Normal (O"n) and shear (0".) stresses are necessarily
within the shaded area, with 0"1 < 0"2 < 0"3 being the principal stresses.
1.6 Equilibrium
For a deformed body in static equilibrium, the following vector equation must
be satisfied in each material point:
div (TT +f = O, (1.27)
where li [N/m3J are forces per unit volume and (div) designates the di-
vergence operator. In Cartesian coordinates, Eq. (1.27) is equivalent to the
following system of three scalar equations:
oau oa21 oa31 f
--+--+--+ 1
OXI OX2 OX3
oa12 00"22 00"32 f
--+--+--+ 2
OXI OX2 OX3
oa13 oa23 oa33 f
--+--+--+ 3
OXI OX2 OX3
o,
= 0,
o (1.28)
Using the convention on repeated (or dummy) indices, this system can be
rewritten in the following compact form:
Oaji f O
--+ i=
OXj
(1.29)
Equilibrium equations are found from the balance of linear momentum (see
Chap. 14). Alternatively, they can be derived directly by writing the equilib-
rium conditions for the elementary parallelepiped of Fig. 1.2, as in (Filonenko-
Borodich, 1958) for instance (or as in Secs. 5.3 for plates and 11.10 for shells).
Indeed, it can be easily shown that equilibrium of forces gives Eqs. (1.28)
while equilibrium of moments gives three scalar equations:
i.e., (T is symmetric (aji = aij).

1.7 Local formulation of static problems 9
1.7 Local formulation of static problems
Consider a solid body which before deformation occupies an open set il of
]R3. The body is subjected to forces per unit volume f in il and to the
following boundary conditions (B.Cs.): forces per unit area F on a part FF
of its boundary and imposed displacements U on a part Fu of the boundary
(Fig. 1.4). We assume that Fu ::j:. 0, FF n Fu = 0, and FF U Fu = F, where
F designates the boundary of the domain. The position vector of a material
partide in a fixed global frame is designated by x.
U
n
Fig. 1.4. Loads and boundary conditions
The problem is to find the fields of displacements u(x), strains €(x) and
stresses lT(x) which satisfy the following equations:
u = U on Fu
div lTT + f = O in il
lTT . n =F on FF
(displacement B.Cs.)
(equilibrium)
(force or "traction" B.Cs.)
1
€ = 2(V'u + V'Tu) (infinitesimal strains)
"Constitutive equations"
In Cartesian coordinates, Eqs. (1.30) become:
Ui = Ui on Fu
âaji f O' n
--+ i= lllu
âXj
ajinj =Fi on FF
fij = ~ (âUi + âUj)
2 âXj âXi
(1.30)
(1.31)

As a reminder of the notions of open and closed sets, consider the unit
interval, Le., {x E IRjO ~ x ~ 1}, then n =]0, 1[ is the open set, n = [0,1] is
the closed set, and r = {0,1}.
We have not written specific constitutive equations in (1.30e) to empha-
size the point that Eqs. (1.30a-d) are valid for any constitutive model. The
simplest material model is linear elasticity for which Eqs. (1.30e) become (see
Sect. 1.11):
O' =c : €, Le. (Tij =Cijkl€lk,
where the elasticity operator c is named after the British physicist Robert
Hooke (1635-1703).
For simplicity, we have Iimited ourselves to the B.Cs. presented above,
but other B.Cs. are possible (see references in Chap. 2), provided that the
problem is well posed. For mathematicai details, see (Parton and Perline,
1984a), but for the purpose of this book, it suffices to follow the following
"common sense" rules in order to understand this important notion:
- (1) The body must be in static equilibrium, Le. the resultant of alI forces
must vanish, and the resultant moment w.r.t. a fixed point must be zero.
- (2) We cannot impose in a given point of the boundary r and in the same
direction a force and a displacement at the same time. In other words,
we cannot impose Fi and Ui in the same point (we either impose Fi and
compute the corresponding displacement in the i-direction, or impose Ui
and compute the reaction force in the i-direction). What can be imposed
at a given point of rare: three displacement components or three force
components or a displacement component and two force components in the
other two orthogonal directions or a force component and two displacement
components in the other two orthogonal directions.
An important class of problems corresponds to the case where only force
B.Cs. are applied to the boundary r (Le., ru = 0). First, the body must in
static equilibrium (condition (1)) and then the stress solution will be unique
but the displacement field will be defined up to a rigid body displacement.
In several chapters -including this one- we make the small-perturbation
hypothesis (SPH), Le. we assume that strains, displacements and rotations
are "small". Therefore, infinitesimal strains (1.30d) are used as strain mea-
sures. Displacements are assumed to be small compared to a representative
dimension of the body (e.g., for a beam, its length, for a circular plate, its
diameter). An important consequence of SPH is that we write (and solve
for) equilibrium and boundary condition equations on the initial, undeformed
(thus known) configuration of a body. Exceptions to SPH are the search for
possible buckling modes in Chap. 10 and the study of finite-strain problems
in Chaps. 14 to 16.
Finally, problem (1.30) is formulated in a formal way, Le. one which is
not mathematically rigorous. For a mathematical presentation in the case of
linear elasticity, see (Parton and Perline, 1984a-b, 1983) for example.

1.8 Continuity equations 11
1.8 Continuity equations
Consider a body under imposed forces and displacements as described in Sect.
1.7 and let n be a surface inside the body. Two regions are thus defined,
one on each side of n and are designated by (1) and (2). Their outward
unit normals on n are n(1) and n(2) =_n(l), respectivelYj see Fig. 1.5. The
surface forces acting an side (1) of n are, by definition, uel) T ·n(1) and those
acting on side (2) of n are: U(2) T . n(2). Equilibrium of forces on n requires
that:
uel) T . n(l) +u(2)T . n(2) = O on n
Since n(2) = -n(1), this becomes:
u(l)T. n = u(2)T. n on n, (1.32)
with n = n(l) ar n(2). Equation (1.32) requires the continuity of the "stress
vector" u T ·n. Note that continuum mechanics does not require the continuity
of the entire stress tensor u. The field of "stress vectors" can be thought of
as that of cohesive forces holding the body together.
There is a second continuity requirement: the displacement vector must
be continuous across rI :
(1.33)
IT this condition is violated, then the material breaks along rl (e.g., a crack
may appear).
IT the surface n is an interface between two different materials, then
continuity conditions (1.32-33) must hold if the two materials are perfectly
"glued" together at n, otherwise the two materials may separate (e.g., de-
lamination in laminated composites).
There are many interesting cases where two solids are in contact but
tangential (sliding) ar normal separation can occur in some areas. In those
cases, contact conditions other than (1.32-33) must be writtenj see (Johnson,
1987), (Doghri et al., 1998) and references therein.
u
F
Fig. 1.5. Continuity conditions across a surface n

1.9 Compatibility equations
Some problems are such that one can guess the form fij(Z) = fji(Z) of the
strain field. The stresses can be computed from the constitutive equations and
equilibrium equations (1.30b) and force B.Cs. (1.3Oc) can then be checked.
However, there remains a question: is it possible to find a displacement field
Ui(Z) such that the strain-displacement relations (1.30d) are satisfied? The
answer to this question is yes if compatibility equations are satisfied (the six
components of a symmetric strain tensor cannot be specified arbitrarily since
a displacement vector has only three components). In Cartesian coordinates,
these equations read (e.g., (Love, 1927)):
82 .
8 i' = O; i, j, k, 1 = 1, 2, 3
Xi Xk
(1.34)
It can be shown that these equations are reduced to a linear combination of
the following six equations:
821011 + 821023 821012 821013 =O
8X28X3 8x18xl 8x18x3 8x18x2
82t22 + 82t13 _ 82t21 _ 82t23 =O
8x18x3 8x28x2 8X28X3 8x28xl
82t33 + 82t12 82t31 82t32 =O
8x18x2 8x38x3 8x38x2 8x38xl
2 â2f12 _ â2f11 _ â2f22 = O
âx1âx2 âx2âx2 âx1âxl
2 82t13 _ 82t11 _ 82t33 = O
âx18x3 8x3âx3 8x18xl
2 â2f23 _ â2f22 _ â2f33 =O
âx2âx3 âx3âx3 âx2âx2
(1.35)
These conditions are necessary and sufficient for a simply connected domain,
otherwise additional conditions must be written in order to ensure that a
single-valued displacement field is found (examples are given in Chaps. 4 and
8). We recall the definition of a simply connected domain: it is such that any
closed curve contained in the domain can be continuously shrunk to a point.
As a counter example, a tube is not simply connected, it is multi-connected.
1.10 Strength criteria
Figure 1.6 shows an idealized stress-strain response of a metallic specimen
under uniaxial tension. It is seen that if the stress exceeds a certain level
u(A) =Uy, then u"'''' is no longer proportional to t",,,,. Also, if the specimen

1.10 Strength criteria 13
is unloaded at any point along (AB), then to a zero stress (O"(C) = O) cor-
responds a non-zero strain (€(C) "1 O), i.e. an irreversible or permanent or
plastic deformation takes place. Elasto-plasticity theory is studied in chapters
12 and 16.
O"(B) ~----=_'9'--
o
€(B)
Fig. 1.6. Uniaxial stress-strain response of a metallic specimen
For a uniaxial tension or compression test in the (x) direction, it is seen
that the response is linear elastic as long as the following yield criterion is
satisfied:
(1.36)
where O"y is a material parameter known as the initial yield stress. For multi-
axialloadings, criterion (1.36) was generalized by von Mises as follows:
(1.37)
where J2 (u) is the von Mises equivalent stress defined by:
3 3
J2(u) =(_S:S)I/2, ie J(u) (s S )1/2
2 .. 2 = 2" ij ji , (1.38)
where sis the deviatoric part ofthe Cauchy stress. Since J2 (u) is the second
invariant of the deviatoric stress, the yield criterion is both isotropic and
pressure insensitive, this last property being generally well verified for metals.
It will also be shown in Sect. 1.12 that the square of J2 (u) is -up to a factor-
equal to the elastic distortion energy. By developing the expression of J2 (u):
h(u) = {~[(0"11 - 0"22)2 + (0"22 - 0"33)2 + (0"33 - 0"11)2]
+3 (0"~2 + 0"~3 +0"~1)}1/2, (1.39)

we can double-check that it is independent of hydrostatic pressure. It is easy
to check that for uniaxial tension in the (1) direction, we have J2 (u) = lanl,
and this explains the presence of the factor 3/2 in the definition of the van
Mises stress. For simple shear in the (1, 2) plane, we have h(u) = laI2h/3.
The van Mises stress can be "visualized" as being equal -up ta factar-
ta the so-called octahedral shear stress. Assume that the principal stresses
ai and directions e(i), i =1, 2, 3, are knawn. The principal stress directians
form an orthonormal basis B == (e(l), e(2), etaj). The so-called octahedral
plane makes equal angles with thase directians (Fig. 1.7). The auter unit
normal n ta this plane has the follawing camponents in B:
(1.40)
The stress matrix in B is diaganal and has the follawing expression:
[a] = [~ ~2 ~ 1
O O aa
(1.41)
The surface force acting on the octahedral plane has the following components
in B:
(1.42)
It can be decomposed inta normal and tangential (or shear) components as
follows:
(1.43)
The normal stress component is given by:
T I I
an =(u . n) . n =-(al +a2 +aa) =-tr u
3 3
(1.44)
From Eqs. (1.42-44), the tangential surface force is found to have the following
components in B:
(1.45)
Recalling the definition of the deviatoric stress 8 -Eq. (1.10)- it is found that:

1.10 Strength criteria 15
1 [SI 1
r = Vă :: (1.46)
The norm of r is the so-called octahedral shear stress:
1 1
IIrli = J3(s~ + s~ + S~)1/2 = J3(8 : 8)1/2 (1.47)
It is seen that the von Mises equivalent stress J2 (u) is related to IIrli by:
(1.48)
Fig. 1.7. Orthonormal hasis of principal stress directions (e(l), e(2}, e(3}) and oc-
tahedral plane with outer normal n
An isotropic yield criterion should be either a function of the principal
stress invariants (e.g., von Mises criterion) or a symmetric function of the
principal stresses (e.g., 7resca criterion). The latter criterion is defined as
follows:
(maximum shear stress) ~ a;
IT we know the principal stresses (al, a2, a3), the maximum shear stress is
simply equal to the radius of the largest Mohr's circle (Sect. 1.5); thus the
Tresca yield criterion becomes:
(1.49)
As an application of the von Mises and Tresca yield criteria, consider a
bi-axial stress state where the stress matrix in a Cartesian basis is given by:
[
al
[a]= ~ (1.50)

Using Eq. (1.39), it is found that the von Mises yield criterion takes the
following form:
(1.51)
Introducing the following change of variables:
0"1 0"2 1 0"1 0"2 1
El == (O"Y + O"y) "j'i' E 2 == (- O"y + O"y) "j'i' (1.52)
it is seen that inequality (1.51) defines an elliptical sur/ace:
(1.53)
The ellipse is plotted in Fig. 1.8, which can be interpreted easily: if the stress
point (O"dO"y, 0"2/O"y) is inside the elliptical surface, then the state is linear
elastic. II the stress point is outside the ellipse, then irreversible (plastic)
deformation occurs.
The Tresca criterion gives a hexagonal surface which is also plotted in Fig.
1.8; its interpretation is identical to that of the von Mises elliptical surface.
The Tresca surface is easy to obtain. The stress matrix (1.50) is diagonal and
the principal stresses are simply 0"1, 0"2 and 0"3 = O. Mohr circles for three
different stress regions are plotted in Fig. 1.8.
Figure 1.8 shows that the von Mises and Tresca criteria mostly give the
same predictions but that in general the latter criterion is more conservative.
In practice however, this is not important, because engineers generally use a
safety factor Ks > 1, Le. the yield (or "strength") criteria (1.37, 49) are used
with a right-hand-side equal to O"y / Ks.
The von Mises and Tresca yield criteria are applicable for ductile isotropic
materials. For brittle materials such as ceramics or glass, it is observed that
they break if the tensile stress exceeds a certain material parameter O"J or the
tensile strain exceeds a parameter EJ. Such materials remain linear elastic as
long as:
(1.54)
otherwise they break.. In the strength criteria (1.54), 0"1 designates the largest
principal tensile stress and EI the largest principal tensile strain.
1.11 Linear elasticity
The results which were presented so far are independent of any specific ma-
terial behavior. We now restrict our attention to the important case of linear

1.11 Linear elasticity 17
o
von Mises
Fig. 1.8. For a bi-axial stress state, the von Mises and Tresca yield criteria give
elliptical and hexagonal surfaces, respectively
elasticity (note that nonlinear elastic models do exist, see Chap. 15). By "elas-
tic" we mean "reversible", Le. if the external solicitation which is applied to
the body is removed, the latter retrieves its initial shape. For instance, in the
uniaxial tension example of Fig. 1.1, if the tensile loading is brought to zero,
the bar retrieves its initial geometry (length lo and cross section area Ao).
For the same test, a "linear elastic" material is such that the (uniaxial) stress
(Txx is proportional to the axial strain Exx:
(1.55)
where E > Ois a material modulus named after the British scientist Thomas
Young (1773-1829). Typical values of E are: 200 GPa (Le., 200 x 109 N/m2)
for steel, 70 GPa for aluminum and 0.1 GPa for rubber. For multi-axial
loadings, Eq. (1.55) is generalized as follows:
(T =e : €, Le. (Tij =CijklElk, (1.56)
where e is called Hooke's opemtor. It is a fourth-order tensor which is
positive-definite (see Chap. 2) and has the following symmetries (e.g., (Du-
vaut, 1990)):
Cijkl =Cjikl =Cijlk =Cklij (1.57)
On a first glance, e has 34 =81 components, but using the symmetries of e,
€ and (T, it can be shown that e has only 21 independent components. An
interesting consequence for computer implementation is that e can be stored

as a symmetric 6 x 6 matrix while € and lT can be stored as 6 x 1 arrays; see
Appendix C.
An important particular case is that of isotropic materials, for which
mechanical properties are independent of the loading direction. For those
materials, it can be shown -e.g., (Duvaut, 1990)- that Cijkl has the following
expression:
(1.58)
where the material parameters A and J.I. are named after the French engineer
Gabriel Lame (1795-1870) and are the only material properties needed to de-
scribe isotropic linear elastic behavior (instead of 21 in the general anisotropic
case). Using Eqs. (1.57,58), the stress-strain reIation (1.56b) becomes:
(1.59)
where (tr) is the trace operator. We now wish to invert Eqs. (1.59), Le. express
strains in terms of stresses. The "trick" is to relate the trace of lT to that of
€. Computing the trace on each side of (1.59b), it is found that:
tr lT = (3A + 2J.1.)tr € (1.60)
Substitution of this result into Eq. (1.59b) gives:
lT A
€ = - - 2 (A 2) (tr lT)l,
. Uij A r
l.e. f.ij = -2J.1.- - 2J.1.(3A + 2J.1.) UmmUij
2J.1. J.I. 3 + J.I.
(1.61)
As a first application, consider the same uniaxial tension test as before; the
stress matrix in the Cartesian basis is:
[uJ = [T ~ ~1 (1.62)
The corresponding strain components are given by Eqs. (1.61b) as:
A+J.I. A
f.xx = J.I.(3A + 2J.1.) U xx ; f.yy =f.zz = - 2J.1.(3A + 2J.1.) U xx ;
f.xy =f.yz = f.xz =O (1.63)
Usually, the material parameters which are measured experimentally are not
A and J.I. but E and v which can be defined from a uniaxial tension test as
follows:
E = uxx ; v = _~ = _f.zz ,
f.xx f.xx f.xx
(1.64)

1.11 Linear elasticity 19
Le. Young's modulus E measures the stiffness in the tension (rudal) direction
and v measures the lateral contraction. This ratio v is named after the French
engineer S.D. Poisson (1781-1840). Comparing Eqs. (1.63-64), the following
identities are found:
(1.65)
The Lame coefficients are then easily deduced:
A _ Ev . _ E
- (1- 2v)(1 + v)' f-L - 2(1 +v)
(1.66)
Using identities (1.65), it is easy to check that the strain-stress relations (1.61)
can be rewritten under the following simpler form:
l+v v . l+v v
€ = ~u - E(tr u)1, l.e. Eij = ~Uij - EUmmbij (1.67)
As a second application, consider a state of pure shear in the (x, y) plane
(Fig. 1.9). The stress matrix in the Cartesian basis is given by:
The strains are given by Eqs. (1.67) as:
l+v
Exy = ~Uxy; Exx = Eyy = Ezz = Eyz = Exz = O
Using Eq. (1.66b), we have:
Uxy
f-L=-
2Exy
(1.68)
This shows that the so-called shear modulus f-L has a similar role to that of E
in a uniaxial tension test, it is the constant slope of the straight line: shear
stress (uxy ) versus engineering shear strain (2Exy ); Fig. 1.9.
As a third application, consider a state of hydrostatic pressure. The stress
matrix in the Cartesian basis is given by:
[uj = [-: ~p ~ l'
O O -p
(1.69)
where p > Ois the applied pressure. The strains are given by Eqs. (1.67) as:
(1 - 2v)
(xx = (yy = Ezz = E (-p); (xy = Eyz = Exz = O

Fig. 1.9. A state of pure shear in the (x, y) plane
A so-called bulk modulus r;, is defined by:
E
3r;,::::: --2- =3A+2J.t,
1- v
(1.70)
using Eqs. (1.65). Thus we have €xx =-p/3r;,. Actually, using Eq. (1.60), we
have the very general result:
tr (T
tr €=--
3r;,
Equations (1.70, 71) show that:
1
(v-t'2) =? (r;,-too) =? (tr€-tO)
(1.71)
(1.72)
Since for infinitesimal strains, (tr €) measures the variation of volume, Eq.
(1.72) shows that for an incompressible material: v = 1/2 or r;, -t 00 (this
means in practice: r;,/J.t ~ 103 ). The larger the value of r;, (w.r.t. J.t), the less
compressible the material. That's why r;, is also called "compressibility mod-
ulus" .
Exercise: show that the stress-strain relations can be written under the
following format using r;, and G ::::: J.t:
(T = 2G€ + (r;, - ~G)(tr €)1, Le. aij = 2G€ij + (r;, - ~G)€mm8ij (1.73)
In summary, for a linear isotropic material, we need two independent material
parameters which are the Lame coefficients A and J.t or Young's modulus E
and Poisson's ratio v or the bulk and shear moduli r;, and G::::: J.t. Those three
pairs are related together by Eqs. (1.65, 66, 70). Parameters E, A, J.t =G and
r;, have the dimension of a stiffness [N/m2], while vis dimensionless. One of
several equivalent stress-strain relations (1.59,67,73) canbe used, depending
on the particular problem at hand.
Exercise: find the strain versus stress relations (1.67) by direct application
of lemma (1.1).

1.12 Strain energy 21
1.12 Strain energy
Consider a linear spring under a tensile force F (Fig. 1.10). The tension
T = F in the spring is related to the displacement U by T = kU, where
k [N/mJ is the stiffness ofthe spring. This relation can be written as follows:
(1.74)
where W(U) is the energy of deformation of the spring.
Fig. 1.10. Linear spring: initial (a) and deformed (b) states.
In linear elasticity, the analogous relation to T = kU is: (T =C : €, which
can be written as:
d 1
(T--(-€·c·€)
- d€ 2 . . ,
~
W(€)
(1.75)
where, by analogy with Eq. (1.74), W(€) is the strain energy per unit volume
(a better justification will be given in Chap. 2). The strain energy can be
rewritten as follows:
1 1
W(€) = "2€ :(T = "2{ij(Jji (1.76)
For a uniaxial tension test in the (x) direction, W(€) has the simple expres-
sion:
(1.77)
which has a simple interpretation: in uniaxial tension, W(€) is simply the area
under the stress-strain line (the area of the shaded triangle in Fig. 1.11).

W
Fig. 1.11. In uniaxial tension, the strain energy per unit volume (W) is simply
equal to the area of the shaded triangle
For generalloadings, using the isotropic linear elastic relations (1.59), the
following expression is found:
(1.78)
It is useful to rewrite this expression a couple of times. Using the deviatoric
strain e, Eq. (1.16), simple algebra gives:
W(e) = (~ + ~)(tmm)2 + p.eijeji,
and this becomes -using definition (1.70) of the bulk modulus ""-
W(e) = ~(tmm)2 +~
' - - " wd;'(e)
wvol(f",,,,)
(1.79)
It appears that W(e) is the sum of two terms: Wtlol(tmm) which represents
the part of the energy due to the change of volume, and WdiB(e) which
corresponds to the change of shape. The latter is called the distortion energy.
Note that the interpretations of the bulk and shear moduli given in Sect. 1.11
are consistent with Eq. (1.79): "" is attached to the volume variat ionenergy,
while p. appears in the distortion part of the energy.
We now rewrite Eq. (1.79) one more time using the stress tensor. Re-
calling the definition of the deviatoric stress tensor s, Eq. (1.10), and using
Eqs. (1.59, 60), it is found that the deviatoric stress and strain tensors are
proportional:
s =2p.e (1.80)
Finally, substituting into Eq. (1.79) and recalling (1.70),3"" =3A + 2p., it is
found that the strain energy is the following stress function:
W(e) =W(u) = (umm)2 + SijSji (1.81)
18"" 4p.
--.....-..- '-v-"
Wvol(u",,,,) Wd;.(S)

1.13 Navier equations 23
Recalling Eq. (1.38), it is seen that the (elastic) distortion energy W di8 (s) is
proportional to the square of the von Mises equivalent stress:
(1.82)
Consequently, the yield criterion (1.37) can be written in terms ofthe (elastic)
distortion energy as follows:
2
WdiB(s) < O'y
- 6ţt'
where O'y is the initial yield stress.
1.13 Navier equations
(1.83)
Some problems are such that the form of the displacement field u(x) can
be guessed. The strains are then computed from the strain-displacement re-
lations (1.30d) and the stresses from the constitutive equations of isotropic
linear elasticity (1.59). Equilibrium equations (1.30b) and force B.Cs. (1.30c)
can then be checked. It is desirable to "automate" this process once and for
alI, Le. to express the equilibrium equations in terms of the displacements.
In Cartesian coordinates, using Eqs. (1.31d, 59), Eqs. (1.31b) become:
8 [ 8um 8Ui 8uj )]
- >'~ij -- +ţt(- + - + fi = O
8xj 8xm 8xj 8Xi
We now assume that the Lame coefficients are uniform, i.e. independent of
the position vector x. The previous equations then become:
8 (8um ) 8 (8ui 8uj ) j O
A - - - +ţt- -+- + i=
8Xi 8xm 8xj 8xj 8Xi
After renaming dummy indices, the equations can be rewritten as:
8 8um 82Ui
(>. +ţt)-8 (-8) +ţt 8 8 + fi =O
Xi Xm Xj Xj
(1.84)
The three scalar equations thus obtained are named after the French engineer
Navier (1785-1836). They can be rewritten under the following tensor form
which can be used in other coordinate systems (e.g., cylindrical or spherical):
(>. +ţt)V(div u) + ţtLlu +f =O, (1.85)
where Ll designates the Laplacian operator. Actually, we shall use this pro-
cedure quite often: when we need to differentiate w.r.t. position, we first con-
sider Cartesian coordinates, where the computations are the easiest, then we
try to find a tensor or "intrinsic" form which can be used for other coordinate
systems.

1.14 Beltrami-Mitchell compatibility equations
Some problems are such that one can guess the form Uij(m) = Uji(m) of the
stress solution (several examples are given for plane problems in chapters 7
and 8). This (trial) stress field must satisfy equilibrium equations (1.30b) and
force B.Cs. (1.30c). Strains can be computed from the constitutive equations.
Those strains must satisfy the compatibility equations of Sect. 1.9 in order
to ensure that displacements verifying (1.31d) can be found.
In a stress-based approach, it is desirable to express the compatibility
equations in terms of stresses, so one can check the suitability of a guess
from the beginning. Assuming isotropic linear elasticity, we can substitute
Eqs. (1.61) into (1.35), but we arrive to a simpler representation if we use
the method of Beltrami and Mitchell which proceeds as follows. Define:
This is known (up to a sign) as the hydrostatic stress. Now assume that the
material properties E and v are uniform in space. Substituting Eqs. (1.61)
into (1.34) leads to:
_ 3v (O" EPuH + Okl 82uH _ O'k 82uH _ 0'1 82uH )
E '38xk8Xl 8Xi8Xj '8Xj8Xl 3 8xi8xk
+__ '3 + ___
'_ _ 3 = O
1 +v ( EPu" 82Ukl 82u'k 82U'I)
E 8xk8Xl 8xi8Xj 8Xj8Xl 8Xi8xk
(1.86)
Setting k =l, summing over the repeated index k =l, and rearranging terms,
we obtain:
..dUij +3 82uH _ ~(8Uik) _ ~(8Ujk)
8Xi8xj 8xj 8Xk 8Xi 8Xk
-~ (Oii..dUH + EPuH ) = O,
1 +v 8Xi8xj
(1.87)
where ..d designates the Laplacian operator. If tr satisfies equilibrium equa-
tions (1.31b), then Eqs. (1.87) take a simpler form:
(1.88)
These are six independent compatibility equations. Further simplification can
be obtained by setting i = k and j = lin Eqs. (1.84). After summation over
repeated indices and rearrangement of terms, we obtain:
~(8Uij) _ 3 (1 - v) ..dUH =O
8Xi 8xj 1 +v
(1.89)

l.15 Saint-Venant's principle, Uniqueness, Superposition, Special theories 25
Assuming that u(x) satisfies equilibrium equations (1.31b), Eq. (1.89) be-
comes:
l+v
3(1 _ v) div i, (1.90)
where (div) designates the divergence operator. Substitution into Eq. (1.88)
gives:
(1.91)
An important case is when the external forces per unit volume i are uniform
in space (e.g., forces due to gravity, assuming a uniform density). In this case,
Eqs. (1.91) become much simpler:
3 82uH
LlUij + -1-- ~ = O, with LlUH = O
+ v UXiUXj
Taking the Laplacian again on (1.92a), we obtain:
(1.92)
(1.93)
Le. the stress field must be bi-harmonic. Note that the same result applies
for the strains: LlLltij =O.
1.15 Saint-Venant's principle, Uniqueness,
Superposition, Special theories
There exists a principle due to the French engineer Barre de Saint-Venant
(1797-1886) which is very useful and many interesting problems could not
be solved without it. This principle states that if a system of external loads
applied on a part rA of the surface of a body is replaced by another, statically
equivalent system, then at a sufficient distance from rA, the stresses due
to the two systems will be practically the same. In some simple cases, the
principle can even be demonstrated, e.g. (Parton and Perline, 1984a). The
principle is better understood by considering its applications; several of them
are given throughout the book, e.g. chapters 3, 4, 7 and 8.
For linear elasticity, a fundamental result which is proven in Chap. 2
is that if the basic problem of Sect. 1.7 is well posed, then its solution is
unique. This is a powerful tool for solving problems in linear elasticity: if by
experience or intuition, a solution is found which satisfies aU the equations
(Le., (1.30) or (1.31)), then it is the solution to the problem.
Another powerful result in linear elasticity is the so-called superposition
principle which can be stated as follows.

Consider two different loading systems (i), i = 1,2,
I(i} in il, F(i} on r F and U(i} on rUj solution: u(i}(x), O'(i)(x) and
€(i}(x). Consider now a third loading system (O):
[al(I} + {31(2}] in il, [aF(I} +(3F(2)] on rF and [aUei} +{3U(2}] on ru ,
where a and (3 are given scalars. The "principle" states that the solution of
problem (O) is:
[au(1} (x) +{3U(2} (x)], [aO'(1} (x) +{30'(2} (x)] and [a€(I}(x) +(3€(2) (x)].
Numerous applications of this principle are given throughout the book,
e.g. chapters 3 to 8. One has to remember however that this "principle" stems
from the fact that the basic problem formulated in Sect. 1.7 is linear when
the constitutive model is linear elastic; when the problem becomes nonlinear
because of material or geometric nonlinearities, the superposition "principle"
does not apply.
Finally, in many interesting engineering applications, we do not solve the
basic problem of Sect. 1.7, but modijied and simplijied versions of it. Such
cases arise for so-called structures (beams, plates and shells). Beams are solids
for which one dimension (the length) is much longer than the other two di-
mensions. Plates and shells are solids with one dimension (the thickness)
much smaller than the other two dimensions; when the mid-thickness surface
is planar, the structure is called a plate, otherwise it is a shell. Based on
kinematic assumptions, simplified or special theories are developed for struc-
tures and new variables are defined in order to describe their deformed state.
For beams, internal loads, Le. stresses integrated through the cross section
are defined instead of stresses. For plates and shells, internal loads per unit
length, Le. stresses integrated over the thickness, are used. For bending of
beams, plates and shells, the curvature of the deformed middle fiber or sur-
face is used instead of the usual strain tensor. Special theories offer dramatic
simplifications of the original three-dimensional (3D) problem by reducing it
to a one-dimensional (lD) problem along the middle fiber for beams, and to
a two-dimensional (2D) problem on the middle surface for plates and shells.
For details, see chapters 3, 5, 6 and 11.
1.16 Solved problem: composite cylinder under axial
load
A composite solid is made up of two concentric circular cylinders of length
l: a fiber of radius Rj, Young's modulus Ej and Poisson's ratio Vj, and a
matrix of radii Rj and Rm and elastic properties Em and Vm (Fig. 1.12).
Axial and uniform displacements ±(U/2)ez are applied to faces z = ±l/2.
Lateral surface r = Rm is stress-free and perfect adherence at the interface
r = Rj is assumed. We introduce the following notation for convenience:
R U
C == (~)2, e == l' p== -arr(Rj), (1.94)

1.16 Solved problem: composite cylinder under axialload 27
where C represents the fiber volume fraction, e the axial (uniform) strain and
p the (continuous) interface pressure.
matrix : Em, lIm
Fig. 1.12. Composite cylinder under axial tension/compression.
Working with cylindrical coordinates (Appendix A), one can prove that
the stress solution is given as follows:
• Fiber: Urr = UOO = -p, Uzz = constant == uzI,
• Matrix: urr,OO = =t= [ ( ~mr=t= 1]1 ~CP'
Uzz =constant == Uzm , (1.95)
and that the shear stresses vanish everywhere:
UrO =UO z = Urz = O (1.96)
Indeed, the stress expressions satisfy the stress B.Cs., continuity of the stress
vector at the interface, the only non-trivial equilibrium equation:
dUrr Urr - UOO - O
-- + --'-'----'-'-
dr r -,
and the compatibility equations. In order to find the stresses, one can look
for the displacement field under the form:
u(r, 0, z) =u(r)er + ezez
Actually, we shall see in Chap. 8 that the stresses in the matrix are those of a
Lame's hollow cylinder under internal pressure P and zero external pressure.
Isotropic linear elasticity gives the strain field as follows:
. (1 - vI - 2vJ)
• Flber: (rr = (00 = -vIe - EI p,
C
x -
CP'
1-
[
c;r=t= 1 ± 2vm]
(1.97)

the other strains being uniform:
€zz = e, €r() =€()z = €rz = O (1.98)
Condition €zz = e ("generalized plane strain") allows to compute the axial
stresses azf and a zm :
The radial displacement field u(r) is simply found from the relations:
du
u = r€()(), dr = €rr
(1.99)
The interface pressure is computed from displacement continuity at the in-
terface (r = Rf). The following expression is found:
v = (1 +vf )(1 - 2vf) (1 + vm) (~ 1 _ 2 ) ~
- Ef + Em C + Vm 1 - C
(1.100)
Note that if Vm = vf then p = O, azf = eEf' azm = eEm and arr = a()() = O
everywhere: under axial loading, the composite behaves as a system of two
parallel bars, because there is no mismatch in the lateral contraction.
The equivalent axial stifJness E is defined as follows:
E = < azz >
- ,
e
(1.101)
where < azz > is the stress average:
_1[2 22]
< azz >= R2 7rRfazf + 7r(Rm - Rf)azm = Cazf + (1- C)azm
7r m
(1.102)
Substituting the expression of p into those of the axial stresses, the axial
stiffness is found to be:
E =pEf + (1 - C)Em,. +2C (vm ; Vf )2
.. (1.103)
The term under brace is the one given by a simple mixture rule, the additional
term is due to the multi-axial nature of the stress state.

2. Variationa1 formulations, work and energy
theorems
In the previous chapter, we presented the local or "strong" formulation, which
allows to solve various (but rather simple) problems in statics. Often, as we
shall see in subsequent chapters, the formulation is used under approximate
forms (e.g., theories of beams, plates and shells). However, the most powedul
numerical methods (e.g., finite elements) which are used to find approximate
solutions to problems that cannot be solved in closed form (Le., most in-
dustrial problems) are not based on the local formulation but on the global
formulations which will be developed in Secs. 2.2 to 2.6; this constitutes one
of the major interests of this chapter.
2.1 Local formulation of static problems
For simplicity, we consider an orthonormal Cartesian coordinate system
(O, Xl, X2, X3), but the results which we shall find in this chapter can be
easily rewritten for other coordinate systems.
Consider a solid body which before deformation occupies an open set il
of]R.3. The body is subjected to forces per unit volume fi [N/m3 ] in il and
to the following boundary conditions (B.Cs.): forces per unit area Fi [N/m2 ]
on a part FF of its boundary and imposed displacements Ui [m] on a part
Fu of the boundary (Fig. 2.1). We assume that Fu =1 0, FF n Fu = 0, and
FF UFu =F, where F designates the boundary of the domain.
We now recall from Sec. 1.7 the local formulation (Po) of the problem in
statics (or quasi-statics).
le Formulation (Po):
Find the fields of displacements Ui(X), strains €ij(X) and stresses O"ij(X) which
satisfy the following equations:

30 2. Variational formulations, work and energy theorems
u
Xa
n
Fig. 2.1. Loads and boundary conditions
Ui =Ui on r u
aUij f o· n
--+ i= fiu
aXj
Uijnj =Fi on rF
fij = ~ (~:; + ~i) == U(i,j)
(displacement B.C.)
(equilibrium)
(force or "traction" B.C.)
(infinitesimal strains)
(2.1)
For now, we do not consider any particular constitutive model, because we
shall see that the global formulations of Sec. 2.2 do not depend on any ma-
terial model.
For simplicity, we have limited ourselves to the B.Cs. presented above,
but of course, other B.Cs. are possible in practice. For slightly more general
B.Cs., see (Hughes, 1987) and for a much more general setting, see (Duvaut,
1990). What we should keep in mind is that the problem under study must
be well posed, as explained in Sec. 1.7.
2.2 Virtual work theorem (VWT)
A multiplication of the equilibrium equations with (a smooth vector field) Wi
followed by an integration over il give:
r(aUij + /i)wi dil =O
ln aXj
An integration by parts leads to:
(2.2)

We have:
2.2 Virtual work theorem (VWT) 31
~ (aij +aji) Wi,j (aij being symmetric)
1
'2aij (Wi,j +Wj,i) (permutation of "dummy" indices)
aijw(i,j) (W(i,j): infinitesimal strain associated with Wi )
We also have aijnj =Fi on rF ; Eq. (2.3) can then be rewritten as:
We now introduce the following two sets:
Y {v = {vd I v "sufficiently smooth" and Vi = Ui on ru}
Y* {v* = {vn Iv* "sufficiently smooth" and vt = Oon ru} (2.5)
In order to avoid discussing mathematical technicalities, we use the vague
condition "sufficiently smooth"; roughly speaking it means that the displace-
ment fields must be such that aH mathematical operations are legitimate
(e.g., Vi, Vi,j, V;' vt,j E L2(n), the space of square-integrable functions).
If v E Y, then v is said to be "kinematically admissible" (K.A.) For
instance, the displacement solution u of problem (Po) is K.A. (u E Y). An
element v* of Y* can be viewed as the difference of two K.A. fields. For
instance, we can take v* = v - u, where u is the displacement solution
(u E Y) and v is any K.A. field (v E Y).
If we take w = v (with v E Y) in (2.4), we can reformulate (Po) in the
following way:
"Formulation (Pt}:
Find the displacement field u which satisfies:
uEY
r aijv(i,j) dn = r fiVi dn + r FiVi dr + r aijnjUi dr, "Iv E Y
ln ln lrF lru
"Constitutive equations" (2.6)
If we take w =v* (with v* E Y*) in Eq. (2.4), we can reformulate (Po) in
the following way:
" Formulation (P2 ):
Find the displacement field u which satisfies:

uEY
rO"ijVei,j} dJl = r/ivi dJl +! Fivi dF, Vv* E Y*
la la ~
"Constitutive equations" (2.7)
In the literature, either one of formulations (Pt) or (P2) is known as global
or weak formulation of a static problemj they are also known under the name
virtual work theorem (VWT)
le We shall now prove that formulations (Po) and (P2) are equivalent. We
have already proven that (Po) => (P2), we only need to show that (P2) =>
(Po). We have:
Using the divergence theorem, Eq. (2.7b) becomes (we assume that u is a
solution of (P2 )):
-!O"ijvinj dF + rO"ij,jvi dJl + r/ivi dJl + r Fivi dF = O, Vv* E Y*
r la la lrF
(2.8)
Since vi = Oover Fu, this is equivalent to:
We now need to show that O"ij,j +fi = Oin Jl and O"ijnj = Fi on FF' Following
(Hughes, 1987), we first choose vi = (O"ij,j + fi) rP where:
(i) rP > O in Jl' (ii) rP = O on F and (iii) rP is "sufficiently smooth" (the
last two conditions guarantee that v* E Y*). Replacing in Eq. (2.9), we find:
1,(O"ij,j + fi)",(O"ij,j + /i)J~ dJl = O, (2.10)
2':0 >0
which implies that O"ij,j + fi =Oin Jl.
We now choose vi = (O"ijnj - Fi) 'ljJ, where:
(i) 'ljJ > Oon FF, (ii) 'ljJ = Oon Fu and (iii) 'ljJ is "sufficiently smooth" (we
do have v* E Y*). Replacing in Eq. (2.9), we find (using the result that we
obtained in the first step: O"ij,j + /i = O):
(2.11)

2.3 Displacement-based variational formulation 33
which implies that (aijnj - Fi) =Oon rF.
To conclude, we have proven that if u is solution of (P2 ), it is also a
solution of (Po) .
.. Remarks:
Equation (2.4), at the basis of the VWT is independent of the constitu-
tive model. Therefore, the VWT is written in identical fashion for plasticity,
viscoplasticity, etc., under the small perturbation hypothesis (SPH). Even in
large deformations, it has a similar form (see Chap. 14). It is helpful to read
formulation (P2 ) of the VWT in the following manner:
(The work of the internalloads in the deformations due to the virtual
displacement field v*) = (The work of the externalloads in the virtual
displacement field v*), Vv* E T*.
In reality, the work of the internalloads is defined with an opposite sign,
so the VWT actually reads:
(The work of the internalloads ...) + (The work of the externalloads...) =O.
We feeI however that for applications, the first reading is easier.
The VWT is the foundation of powerful numerical methods, in particular the
finite element method (FEM).
The notion of work allows to take into account without difficulties cases of
forces or moments which are concentrated or per unit length, while the local
formulation (Po) is not suited for that. Actually, in some textbooks, the VWT
is used as point of departure, and (Po) is found as a consequence (e.g., see
Salenc;on (1988a)).
In formulation (P2 ), we say that (aij and Vii,j»)' (Ii and vi) and (Fi and vi)
are dual or conjugate variables, because the scalar -inner- product of each
pair gives a work.
With an appropriate choice of dual variables, it is possible to construct special
theories (e.g., beams, plates and shells) directly from the VWT (e.g., see
Salenc;on (1988b)). For instance, for the beams studied in Chap. 3, we do not
take (stress and strain) but: (bending moment and curvature), (normal force
and axial strain), (shear force and shear strain).
2.3 Displacement-based variational formulation
From now on, we shall restrict our attention to linear elasticity:
(2.12)
In this chapter, we allow for heterogeneous (Cijkl =Cijkl(Z)) and anisotropic
materials. Hooke's operator Cijkl is:

-symmetric: Cijkl = Cjikl = Cijlk = Cklij,
-positive definite: '<Ix E fl and '<111 second-order symmetric tensor ("7ij = "7ji),
Cijkl"7ij"7kl 2: O and Cijkl"7ij"7kl =O=> "7ij =O
In order to study the "weak" formulation (P2 ) , we will rewrite it with the
help of the following notations:
a(u,w) fa CijkIU(k,I}W(i,j) dfl;
< j,w > = r J;Wi dfl; < F,w >rF= r FiWi dr
la lrF
(2.13)
It is easy to verify that a(., .), < .,. > and < .,. >rF are symmetric and
bilinear forms. Symmetry means that:
a(v*,u);
a(u,v*)
< j,v* > < v*,j >; < F,v* >rF=< v*,F >rF
Bilinearity means that we have linearity w.r.t. each argument:
a(o:u + f3v, v*)
a(u, o:v* + f3w*)
o:a(u,v*) + f3a(v,v*),
o:a(u,v*) + f3a(u,w*),
(2.14)
(2.15)
idem for < .,. > and < .,. >rF' Due to the positiveness of e, a(.,.) is a
positive form: a(w, w) 2: O.
We now present a new formulation (P3 ) of the original problem.
.. Formulation (P3):
Find the displacement field u(x) solution of the following problem:
uEY
a(u,v*) =< j,v* > + < F,v* >rF' '<Iv* E Y* (2.16)
.. We have shown in Sec. 2.2 that formulations (Po) and (P2 ) are equiv-
alent, and therefore (Po) and (P3 ) are equivalent. We shall now prove that
if (P3 ) admits a displacement solution, then this solution is unique. Let us
assume that there exist two displacement fields u and v which are both so-
lutions of (P3 ). We then have:
a(v,v*) - a(u,v*) =O, '<Iv* E Y* (2.17)
And since a(.,.) is bilinear, this is equivalent to:

2.4 Potential energy theorem 35
a(v - u,v*) =O, lv* E Y* (2.18)
Let us choose v* = v - u (recall that since u and v E Y, then v* E Y*).
We then have: a(v - u, v - u) = O. Since c is positive definite, we obtain:
V(i,j) - U(i,j) =O, and therefore:
v - u = "rigid body displacement" , (2.19)
Le., there exist two vectors a and b such that in each point M of the solid
body we have:
~
v-u= b+OM xa (2.20)
The rest ofthe prooffollows (Duvaut, 1990): if ru contains three non-aligned
points M l , M2 and M3 , then since u =U =v on ru, we obtain the equalities:
~ ~ ~
O=b+OMl xa=b+OM2 xa=b+OM3 xa, (2.21)
from which we deduce that:
~ ~ ~
Ml M2 xa = O; M2M3 xa = O; M3 Ml xa = O (2.22)
Since MI, M2 and M3 are not aligned, the 3 equalities imply that a =O, and
consequently b =O.
We have thus proved the uniqueness of the displacement field solution of
(P3 ) (and since (Po) and (P3 ) are equivalent, we also have uniqueness for
(Po)). Note that we have not demonstrated the existence of a solution; for a
proof see (Duvaut, 1990).
2.4 Potential energy theorem
The potential energy I(w) associated with a displacement field w is defined
as:
1
I(w) == 2a(w,w)- < j,w > - < F,w >rF (2.23)
We now introduce a new formulation (P4 ), which is the potential energy
theorem.
,. Formulation (P4):
Find the displacement field u(a:) solution of the following problem:
uEY
I(u) ::; I(v), Iv E Y (2.24)

We are going to prove that problems (P3 ) and (P4 ) are equivalent. Our proofs
of the theorems of potential energy, complementary energy and energy bounds
follow those of Duvaut(1990); for other proofs see, e.g. (Lanczos, 1970), (Lipp-
mann, 1972), (Dym and Shames, 1973), (Oden and Reddy, 1976), (Mason,
1980), (Parton and Perline, 1984b). First, we show that (P3 ) ::::} (P4 ). We
have:
I(v) - I(u)
1 1
2a(v,v)- < j,v > - < F,v >rF -2a(u,u)+ < j,u >
+ < F,u >rF
1
2a(v - u,v - u)
, ,
....
:;:::0
+a(u,v-u)- < j,v -u > - < F,v -u >rF
" ,
..
= O, using (P3 ) with v* =v - u
We have used the properties of bilinearity and symmetry of the forms in order
to rearrange the terms as shown.
We now show that (P4 ) ::::} (P3). Let v be any element of Y and a a real
number. Let w = u + a(v - u); we have w E Y. Using (P4 ) with w along
with the bilinearity and symmetry properties, we obtain:
a 2
2"f(v - u,v - u),+a[a(u,v - u)- < j,v - u >
....
:;:::0
- < F,v - u >rFl2:: 0, Va E lR
The discriminant must be :S O, which implies that:
a(u,v - u)- < j,v - u > - < F,v - u >rF = 0,
Le., (P3 ) (recall that (v - u) E Y*) .
.. Mechanical interpretation:
For a K.A. displacement field v, we have:
~a(v,v): strain energy (see Chap. 1 and Sec. 2.8);
< j,v > + < F,v >rF : work of externalloads;
I(v) = ~a(v,v)- < j,v > - < F,v >rF: potential energy.
(2.25)
(2.26)
Theorem (P4) states that among all K.A. displacement fields v, the so-
lution u is the one which minimizes the potential energy. This theorem has
important applications; it allows for instance to find approximate solutions
to problems which cannot be solved with the local or "strong" formulation
(Po). Ritz's method (Sec. 2.11.1) is based on (P4 ).

2.5 Stress-based variational formulation 37
2.5 Stress-based variational formulation
Let us write the constitutive equations as follows:
(2.27)
(D: compliance tensor, c: stiffness tensor). Let '11 be a second-order symmetric
tensor ('T}ij ='T}ji). We multiply the constitutive equations by 'T}ij and integrate
over n:
(2.28)
Since '11 is symmetric, we have:
(2.29)
Integration by parts gives then:
InDijkto'kl'T/ij dn = ru(r/ijnjdr - rUi ââ% dn
lr ln Xj
r Ui'T}ijnjdr + r Ui'T}ijnjdr
lru lrF
_r Ui â'T}ij dn
ln âXj
(2.30)
We now introduce the following two sets:
E
E*
{7' = (Tij) ITij = Tji, ~~j + li = Oin n, Tijnj = Fi on rF}
{7'* = (Ti}) ITi} = Tji' ~~; = Oin n, Ti}nj = Oon r F } (2.31)
If 7' E E, then 7' is said to be "statically admissible" (S.A.) For example,
the stress field u solution of (Po) is S.A. (u EE). Elements of E* are called
self-equilibrated stress fields, i.e. in equilibrium without body or surface forces
(an example from another context: residual or initial stress fields are in self-
equilibrium). AIso, an element 7'* E E* can be viewed as being the difference
of two S.A. fields. For instance, we may consider: 7'* = 7' - u, where u is the
stress solution (u E E) and 7' is any S.A. field (7' EE).
Let us write (2.30) for '11 =7'*, where 7'* E E*:
(2.32)
We are going to rewrite this result using the following notations:

A(O", '11) = laDijklUkl'f/ij dn
< U,'11 >ru = ( Ui'f/ijnjdr
iru
(2.33)
It can be easily checked that A(.,.) and < .,. >ru are bilinear and that A(.,.)
is symmetric positive definite.
We now introduce a new formulation (Ps) of the original problem.
It Formulation (Ps):
Find the stress field O"(a:) solution of the following problem:
O"EE
A(O",T*) =< U,T* >ru' VT* E E* (2.34)
We have already shown that (Po) ~ (Ps)j it can be shown that problems
(Po) and (Ps) are equivalent.
It We are now going to show that the stress field solution of (Ps) is unique.
Let us assume that there exist two stress fields O" and T which are both
solutions of (Ps). We then have:
A(O",T*) - A(T,T*) =O, VT* E E* (2.35)
Linearity of A(.,.) w.r.t. the first argument implies that:
(2.36)
Let us choose T* =O" - T (recall that T* E E*)j we have:
A(O" - T, O" - T) = O~ O" - T = 0, (2.37)
due to the positiveness of A(., .).
2.6 Complementary energy theorem
First, we introduce the following notation:
(2.38)
We now introduce a new formulation (P6 ), which is the complementary energy
theorem.

2.7 Energy bounds 39
,. Formulation (P6):
Find the stress field u solution of the following problem:
uEI)
J(U) ~ J(T), "IT E I) (2.39)
,. We are going to show that problems (P5 ) and (P6 ) are equivalent. First,
we show that (P5 ) => (P6). Using the properties of A(.,.) and < .,. >ru ' the
following result is easily established:
1
J(T) - J(u) = 2A(T - U,T - u) +
... ,
>0
A(U,T - u)- < U,T -u >ru
... "
'"
= O, using (P5 ) with T* = T - u
(2.40)
We now prove that (P6 ) => (P5 ). Let T be any element of E and a a real
number. Let TI = u + a(T - u); we have TI E E. Using (P6) with TI, and
taking into account the properties of A(., .) and < ., . >ru, we obtain:
a 2
2 ;1(T - U, T - u), +a [A(U,T - u)- < U,T - u >rul2:: O, Va E IR
'"
~o
(2.41)
The discriminant must be ~ O, which implies that:
A(U,T - u)- < U,T - u >ru= O, (2.42)
Le., (P5) (recall that (T - u) E E*).
,. Mechanical interpretation:
(-J(T)) is called the complementary energy of the S.A. stress field T. The-
orem (P6 ) states that among alI S.A. stress fields T, the solution u is the
one which maximizes the complementary energy. We shall see in the next
section that the displacement and stress fields (u, u) solutions of problem
(Po) satisfy: I(u) = -J(u).
2.7 Energy bounds
If u and u are the displacement and stress fields solutions of problem (Po),
we have:

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