Philippe A. CHARlEZ Mining Engineer from Facult Palytechnique de Mans Ph.D from Institut de Physique du Glabe de Pars Rack Mechanics Expert at Total Compagnie Fran~aise des Plroles

1991

ROCK ECHANICS

volume 1 IHEOREIICAL FUNDAMENIALS

Foreword by Vincent MAURV

Chairman of Comit Franvais de Mcanique des Raches

Rack Mechancs Expert al Elf Aquitane

t EDITIONS TECHNIP 27 RUE GINOUX 75737 PARIS ceOE)( 15

Table of contents

Foreword

Preface

Nomenclature

INTRODUCTION. Some hasic concepts of solid mechanics

1 MECHANICS OF CONTINUOUS BASIC CONCEPTS

1 STATE OF STRAIN

1.1

1.2 1.3

lA

1.5

1.1.1 1.1.2 1.1.3 1.1.4

description of the strain of a solid Affine transformation. VonCiept of displacement Convective transport of a vector . ...... . . . . . .. . ..... . Convective transport of a volume ............. ...... . .. . Convective transport of an oriented surface ............... .

leCOITapC,sltlon of the transformation. Rigidity condition .......... . Eulerian description of the strain of a body ...................... . 1.3.1 Affine Eulerian transformation ........................... . 1.3.2 Convective transport oC a vector ................. ....... . 1.3.3 Norm of a vector. Decomposition of K .................... . 1.3.4 Convective transport of a volume ......................... . 1.3.5 of tensor [} as a fundion of velocities .......... . 1.3.6 of the acceleration in au Eulerian Summary table of the Lagrangian and Eulerian formulae in the case of transformations . . . . . . . . . . . . . . . . .. . .. . . State of strain under the hypothesis of small

VII

IX

XXI

1

9

9 9

10 11 11 12 13 13 14 14 14 15 15

16 16

XII Table of contents

1.6 Geometrical of the strain tensor ....................... 18 1.6.1 Diagonal atraina ... ... .. . ..... ...... ................. 19 1.6.2 Non strains ...... ............................... 19 1.6.3 Volume variations. Firat of the tensor f .......... 20 1.6.4 Elongation of the vector Invariant of the second arder ... 20

1. 7 Plane state of strain 21 1.8 State of strain in cylindrical coordinates ... ....................... 21

1.8.1 Curvilinear coordinates and natural reference frame ........ 21 1.8.2 Specific case of coordinates ....... ............. 22

1.9 Equations of compatibility ....... . ... .......................... 24 Bbliography ....................... . ..... .. ............. 25

2 STATE OF STRESS 27 27 28 30 31 32 33 33

2.1 Internal forces and stress vector ......... . ........ . 2.2 Equilibrium of the tetrahedron ........... . 2.3 2.4 2.5 2.6

2.7 2.8 2.9

Concept of boundary condition ................. . Momentum balance equilibrium eql11at,lOllS Kinetic energy theorem . . ..... , ............ , .................... . Theorem of kinetic momentum. of the stress tensor ..... 2.6.1 Invariant quadratic form ... ... .... . . . . . . . . . . .. . ..... 2.6.2 Diagonalization of the stress tensor with """'1"\""or.

to its principal dircctions . . . . . . . . . . . . .. ........ . ...... . Change of cartesian reference frame .............................. . Equilibrium equations in coordinate .............. , .... . Stress tensor in Lagrangian variables ......... .

2.10 Plane state of stress. Mohr's cirde .... .. .. ..... .. . ......... .

3 THERMODYNAMICS OF CONTINUOUS MEDIA

3.1 3,2 3.3 3.4 3.5 3.6 3.7 3.8

A. REVIEW OF

Internal energy of a system ... First of thermodynamics ............ . .................. . Second state fundon: entropy of a system ....................... . Second of thermodynamic." ...... , . . .. . . . . . .. . ...... , .. Free energy , ................................. _ ................... .

and free enthalpy of a fluid .. ,........... . .. . state functions ...................................... .

variable and state equation

34 35 35 36 38 41

43

43 44 44 45 46 46 47 47

Ta.bie of contents XIII

3.9 Total differentiation o state unction .............................. 48 3.9.1 Calorimetric coefficients ................................... 48 3.9.2 Thermoelastic coefficients o a fluid ........................ 49 3.9.3 Further equalities between partal derivatives .............. 50

3.10 Expression of a fluid entropy ...................................... 51

B. CONSTITUTIVE EQUATIONS OF SOLIDS

3.11 The fundamental inequality o Clausius-Duhem .............. .... . . 51 3.11.1 Mass balance.............................................. 52 3.11.2 Momentllm conservation ................................... 52 3.11.3 First principie of thermodynamics ......................... 52 3.11.4 Second principIe of thermodynamics ....................... 53 3.11.5 Fundamental inequality of Clausius-Dllhem ................ 53

3.12 Choice of state variables .......................................... 54 3.12.1 The memory of a material................................. 54 3.12.2 Observable state variables ................................. 54 3.12.3 Concealed or internal state variables ....................... 54

3.13 Thermodynamic potential ......................................... 55 3.14 Case of reversible behaviour elastici ty ............................. 56 3.15 Hooke's law ....................................................... 57 3.16 Case of irreversible behaviour ..................................... 57 3.17 Dissipation potential .............................................. 58 3.18 Yield locus and plastic behaviour .................................. 59 3.19 Plastic flow rule and continuity condition .......................... 62 3.20 Specific case of standard laws ..................................... 65

3.20.1 Hill's principIe of maximum plastic work ................... 65 3.20.2 Uniqueness of the solution (or Hill's theorem) .............. 66

3.21 Conclusion........................................................ 68 Bibliography ............................................................ 68

11 MECHANISM OF MATERIAL STRAIN

4 LINEAR ELASTICITY. GENERAL THEORY 73 4.1 Hooke's ]aw ....................................................... 73 4.2 Thermodynamic considerations. Symmetry of the rigidity matrix .. 74 4.3 Case of isotropic materials ........................................ 74

4.3.1 Generalzation to any Cartesian system of coordinates ...... 76 4.3.2 Physical interpretation of isotropy ......................... 76

XIV

4.4

4.5 4.6 4.7 4.8 4.9 4.10 4.11

Ta.ble al cantents

The common elastic constants .................................... . 4.4.1 Young's modulus and Poisson's ra.tio .................... . 4.4.2 Hydrostatic hulk modulus ...................... . ....... . 4.4.3 Shear modulus ........................................... . Further of Hooke's equations .......................... . The Beltrami-Mitehell differential equations ................. . ... .

of the elastic solution of a boundary problem ........ .

theorem ....................................... . in cyIndrical coordinates ................ . .... .

77 77 78 78 79 79 81 82 83 83 84 84

5 PLANE THEORY OF ELASTICITY 85

85 86

5.1 Basie of state of strain 5.2 Stress harmonic ,."""t'lrm potential ........................ . 5.3 Plane coordinates ............................... 87 5.4 Application to the calculation of stresses in infinite pi ates .......... 87

5.4.1 Determination of function for an infinite plate ....... 87 5.4.2 Effect of a circular disturbance. Kirsch'g problem 89 5.4.3 Effect of a pressure on the borehole ............ 92

5.5 The finite elastic solid: a.pproximate solution ............... 92 5.6 The method of of Muskhelishvili ............... 98

5.5.1 Analytical functions and Cauchy-Riemann conditions (CRC) 98 5.6.2 Application to the biharmonic equation .................... 100 5.6.3 Expression of stresses and . . . . . .. . . . . . . . . . . . . 101

5.7 Transformation of the basie formula.......... 102 5.8 conditions in the image plane .......... 103 5.9 Determination of by integrais ......... 105 5.10 Applieation to the case of an infinite containing an elliptical

cavity ..................... .. ..................................... 106 5.11 Conclusion................... .................................... 109 Bibliography ..... ................... . .. . . ......... .... ........... 110

6 BEHAVIOUR OF A MATERIAL CONTAINING CAVITIES 111

6.1 6.2

6.3

Phenomenological Strain energy associated with a Definition of effective bulk modulus ........................................................ . Specific types o cavities: pares and microcracks '" .............. .

111

111 113

TabIe oE contents xv

6.4 Evolution of the effective modulus with loading ................ . . . . 115 6.5 Determination of the cracking spectrum using Morlier's method .... 116 6.6 Closure of a crack population under a compressive stress field ...... 119 6.7 Additional observations concerning the closure of the microcracks .. 121 6.8 Conclusion. Concept of porosity ................................... 122 Bibliography ............................. .............................. 122

7 THERMODYNAMICS OF SATURATED POROUS MEDIA 123

7.1 Basic hypothesis of thermoporomechanics ......................... 124 7.2 The importance ofthe Lagrangian description for writing conservative

laws .............................................................. 124 7.3 Mass conservation ................................................. 125 7.4 Conservation of linear momentum and mechanical energy balance. . 126 7.5 First principle of thermodynamics ................................. 127 7.6 Second principIe of thermodynamics inequality of Clausius-Duhem. 128 7.7 Choice of state variables (intrinsic dissipation) ..................... 129 7.8 Constitutive state law and thermodynamic potential ............... 130 7.9 Case of reversible behaviour. Laws of thermoporoelasticity ....... ,. 131 7.10 Case of irreversible behaviour ..................................... 131 7.11 Diffusion laws of thermoporomechanics ............................ 131

7.11.1 First diffusion law: hydraulic diffusion law or Darcy's law .. 132 7.11.2 Second diffusion law: heat diffusion law or Fourier's law .... 132 7.11.3 Hydraulic and thermal diffusivity laws ..................... 132

Bibliography ............................................................ 133

8 INFINITESIMAL THERMOPOROELASTICITY 135

8.1 Hooke's law in thermoporoelasticity. Concept of elastc etIective stress 135 8.1.1 Decomposition of the state of stress. Hooke's law of a porous

medium ................................................... 136 8.1.2 Biot's coefficient and elastic effective stress. . . . . . . . . . . . . . . . . 137

8.2 Volume variations accompanying the deformation of a saturated porous medium ................................................... 138 8.2.1 Bulk volume variations .................................... 138 8.2.2 Variation in pore volume .................................. 138 8.2.3 Relative porosity variation ................................. 140

8.3 Mass variations accompanying the deformationof a saturated porous medium ........................................................... 141

8.4 Undrained behaviour. Skempton's coefficient and undrained elastic constants ......................................................... 141

8.5 Thermal effeds ....................................... ........... 144

XVI Table of contents

8.6 Entropy variation accompanying a transformation ........ ........ 145 8.6.1 (m = O) isothermal (T:;;;;; T) test.... .. 146 8.6.2 Undrained (m = O) isochoric (e:u = O) test... ....... 146 8.6.3 Isochoric (eu = O) isothermal (T = To) test............ 146

8.7 Variation in fluid free enthalpy during a transformation '" .. _..... 147 8.8 potential ......................................... 148 8.9 Relation between thermal expansion coefficients ................... 150 8.10 of hydraulic diffusivity ..... . . . . . . . . . . . . . .. ......... 151 8.11 Particular cases .................................. ................ 151 8.12 o thermal diffusivity .............. _ ... . . .. ............ 152 8.13 Resolution of a thermoporoelastie boundary

Beltram-Mitchell and consolidation eQllatlOrlS

9 THE TRIAXIAL TEST AND THE MEASUREMENT OF THERMOPOROELASTIC PROPERTIES

9.1 of the test and of the experimental

153 156 156

159

159 9.2 cireuts ......... _. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . .. ... 161 9.3 Strains measurement ......................................... .... 162 9.4 Friction ......... , ........... _ ................ _ . . . . . . .. . . 163

9.4.1 Friction o the piston ................................ . .. 163 9.4.2 Fl'iction of movng piston ...................... ..... ..... 164

9.5 and installation of the sample ........... ............. 164 9.6 saturation of the sample ....................... _ ... _ . _ . . 165 9.7 Calculation of from the consolidation time.... ........ 167 9.8 Undrained hydrostatic compression measurement of B and 168

9.8.1 The measuring circuit of pore presEure _.................... 170 9.8.2 The heterogeneity of the stress field ........... .... ....... 173

9.9 Second of consolidation ................... ................. 173 9.10 Measurement of drained elastic parameters ....................... . 9.11 Measurement of undraned elastic . . . . .. .. ., ....... ., 9.12 Measurement of Biot's coefficient and matrix bulk modulus ...... . 9.13 Measurement of the coeffic.ents of thermal " .......... .

9.13.1 Thermal expansion coefficient of fluid ................ . 9.13.2 Measurement of Q:u and O'B ....... '" .... .

9.14 Thermal conductivity ...................... . 9.15 heat ................ _ .......... .

173 174 175 176 177 177 178 180

181

Tabie oE contents XVII

10 THERMOPOROELASTOPLASTICITY. GENERAL THEORY AND APPLICATION 183

A. GENERAL CONCEPTS

10.1 Constitutive laws in ideal thermoporoelastoplasticity .............. 183 10.1.1 Variationsin pressure associated with a TPEP transformation 183 10.1.2 Constitutive law in TPEP ..... ........................... 184 10.1.3 Variation in entropy associated with a TPEP

transformation ............................................ 185 10.1.4 Variation in fluid free enthalpy ............................. 186 10.1.5 Thermodynamic potential in TPEP ........................ 186

10.2 InequalityofClausius-Duhem and concept ofplasticeffective stresses 187 10.3 Physical concept of hardening. Calculation of hardening modulus

and of plastic multiplier ........................................... 188 10.4 Incrementallaw in the case of an associated plastic flow rule ....... 191 10.5 Generalization of elastoplasticity: concept of tensorial zone ........ 192 10.6 Laws wi th more than two tensorial zones: theory of mul timechanisms . . 193 10.7 Laws with an infinity of tensorial zones ............................ 193

B. THE CAMBRIDGE MODEL

10.8 Space of parameters ............................................... 194 10.9 Phenomenological study: normally consolidated clay under

hydrostatic compression ........................................... 195 10.9.1 Behaviour in the elastic domain .... . . . . .. . . . . . . . . . . . . . . . . . . 196 10.9.2 Behaviour in the plastic domain ........................... 197

10.10 Behaviour of a clay under deviatoric stress. Critical state concept .. 198 10.11 Expression of the plastic work ..................................... 200 10.12 Determination of the yield locus ................................... 200 10.13 Hardening law .................................................... 201 10.14 Plastic flow rule and hardening modulus .... . . . . . . . . . . . . . . . . . . . . . . . 202 10.15 Application of the Cambridge model to sorne specific stress paths .. 204

10.15.1 Isotropic consolidation ..................................... 204 10.15.2 Anisotropic consolidation .................................. 204 10.15.3 Oedometric consolidation .................................. 205 10.15.4 Undrained triaxial test .................................... 206

10.16 Diffusivity equations associated with the Cam-Clay ................ 207 10.17 The concep~ of overconsolidation application to triaxial tests ....... 208

10.17.1 Undrained overconsolidated test........................... 208 10.17.2 Drained overconsolidated test .............................. 212

XVIII Table o( contellts

C. THE CONCEPT OF INTERNAL FRICT/ON THE MOHR-COULOMB CR/TER/ON

10.18 The ,.."", .. ,,..,, 214 10.19 The line ............................. .... 215 10.20 Yield locus in the space of principal stresses ...................... . 10.21 Special case of triaxial test ... . ................................. . 10.22 Special case o biaxialloading .................................... . 10.23 Tension cutoffs .... 10.24 Generalization of Mohr-Coulomb criterion: concept of intrinsic curve 10.25 Tbe non-assoeiativeness of tbe plastic fiow rule ................... . 10.26 The Rudnicki and Rice model .................................... . 10.27 Correlation between 'f'Y\1'\l'!I'Hrrt> and Mohr-Coulomb models

D. APPLlCATION OF THE LADE MODEL TO THE BEHAVIOUR OF CHALK

216 218 218 219 220 221 222 224

10.28 10.29

under bydrostatic loading . .. ..... ..... 226 under deviatoric loading ................ 227

Lade model ........ . . . . . . . . . . . . . . .. .. .... 228 10.30.1 Elastic behaviour. modulus ...... 228 10.30.2 behaviour under deviatoric loading .... 228 10.30.3 behaviour undel hydrostatic loading ... 232

10.31 Shao and simplified model ............................. 233 10.32 Taking into account resistan ce to traction ............... .... .... 235 10.33 Lade's model and of effective stresses ........... .. ...... 236 Bibliography .......... :... ............................................ 237

OF 11 FISSURING

11.1

MECHANISMS COHESION LOSS

241 241

11.2 Basle of brittle ...... ........................... 241 11.3 Stress field assciated with a. crack concept of stress intensity factor 243 11.4 Generalization of the o stress intensity factor ............. 245 11.5 Physical o the stress factors ................. 247 11.6 Calculation o the stress fa.ctor .......................... 248

11.6.1 11.6.2

with a rectilinear crack in a uniaxial stress field 248 with rectilinear crack in any far stress field 249

TabIe o[ contents

11.6.3 Infinite plate with a concentrated force on the crack 11.6.4 Infinite with rectilinear crack and continuous LVU,UU',r.

11.7 Condition for crack initlation. Griffith criterion ................... . 11. 7.1 Writing the first .. .. . ... . ....................... . 11.7.2 Kinetic energy associated with the of a crack .. 11.7.3 Griffith criterion .......................................... .

11.8 Growth of an initiated crack. "lU=JI:5L

xx TabIe ol contents

12.7 Expression of * in the case of l. crack ....... , ..................... 289 12.8 Introduction of the "damage" variable ............................. 290 12.9 State law. Expression of the thermodynamic potential ............. 290 12.10 Inequality of Clausius-Duhem. Associated thermodynamic Corees. . . 293 12.11 No damage. Open cra.ck ........................................... 294 12.12 No damage. Closed crack ............................ ,............ 295 12.13 Damage ........ ..... .... ......... ....... ...... ............ ........ 298

12.13.1 Specific case in which the crack is open .................... 298 12.13.2 Specific case in which the crack is closed ............... , .. , 299

Bibliography ............................................................ 302

13 APPEARANCE OF SHEARING BANDS IN GEOMATERIALS 303

A. INTRODUCTION. BASIC CONTRADICTION

B. THE MOHR-COULOMB CRITERION. THE CONVENTIONAL MACROSCOPIC A PPROACH

C. THE MICROSTRUCTURAL APPROACH OF TJIE SHEARING BAND

13.1 The rock considered as a material with a population of cracks 306 13.2 Rupture probability of a. single crack under biaxialloading ......... 307 13.3 Collapse of a sample concept. Concept of reference volume. . . . . . . . . 309 13.4 Case of heterogeneous state of stress ............................... 313 13.5 Pseudo-three-dimensional extension and shape of the failure envelope 313 13.6 Nemat Nasser's micromechanical model ...................... ,.... 315

D. APPEARANCE OF A SJIEARING BAND SEEN AS A BIFURCATION

13.7 Existence of the phenomenon Desrues's experimental approach ..... 317 13.8 Mathematical formulation of localizaban ...................... ,... 318

13.8.1 Kinematic condition ....................................... 320 13.8.2 Static condition ........................................... 320 13.8.3 Rheological condition ...... ,............................... 320

13.9 Elasticity and .bifurcation ................ ,........................ 321 13.10 Case of Rudniki and Rice's elastoplastic model .................... 321 13,11 Bifurcation and associativeness .............................. ,..... 326 13.12 Discontinuous bifurcation ......................................... 326 13.13 Conclusion and recommended research ............................ 326 Bibliography ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

INDEX..... ....... .......................... .................. ......... 329

N omenclature

MATRICIAL NOTATION

,v

6,lj tr(1) = AH = AH

k

1= i1 V Aij = UiVj

1= \l i1 A .. _ OU, 1) -OXj

2: oAs u. - __ J - OX' j J - 2: OU, \JU= -

ox' i

e ==

vectors.

vector Nabla.

tensor of the second order.

trace of a tensor.

contraded product of two tensors.

matricial product of two vectors.

transposition of a tensor.

symmetric part of a. tensor.

gradient of a vector.

divergence of a tensor.

di vergence of a vector.

tensor of the fourth order.

product of two tensors.

scalar product of two vectors.

XXII

MAIN SYMBOLS

vector.

f., strain tensor.

E

[{

v

."

B

elastic strain.

plastic strain.

stress tensor.

deviatoric stress tensor.

first Lame constant.

second Lame constant, vis-"""'un" .. .., friction coef-

ficient.

Young's modulus.

bulk modulus, kinetc energy.

Poisson's ratio.

shear modulus.

drained modulus.

drained bulk modulus.

drained Posson's ra.tio.

undraned modulus.

undrained bulk modulus.

undrained Poisson's ratio.

matrix bulk modulus.

Biot's coefficient.

Biot's modulus .

coefficent.

drained coefficient.

expanSlOn

undrained thermal expansion coefficient.

(XI

L

k

Nomenc1ature

fluid thermal expansion coef-ficient.

matrix thermal expansion co-efficient.

latent heat.

isotropic ... '!''''''",,,

thermal conductvity, Cam-Clay swelling coefficient.

heat capacity at constant vo-lumic deformation.

u, U, Um interna! energy.

W, '1/;, 'l/;m free energy.

S, S, s".. entropy.

h, hm enthalpy.

9, gm free enthalpy.

>. multiplier, compressibility coefficient.

M critical state line.

e

4>(z ),

void ratio.

porosity.

bulk volllme.

pore volume.

matrix volume.

fil:st complex potential of M llscbelish vili.

W(z), 'I/;(z} second complex potential of M llschelishvili.

d .... "".u,,'I'>" variable.

Nomenclature

H

9

isothermal consolidaton coef-ficient.

hardening modulus.

energy release rate.

r

le

XXIII

surface energy. shape coefficient.

stress factors.

fracture

INTRODUCTION

Sorne basic concepts of solid rnechanics

Very gene rally speaking, two categories of physical parameters can be distin-guished in mechanics:

(a) The dynamic quantites which give rise to motion. These are mainly forces or force couples.

(b) The kinematic qua.ntities which describe motion geometrically. These are mainly displacements, velocities a.nd accelerations.

Before getting down to the founda.tions of continuum mechanics, there are certa.in general concepts that need to be recalled. These will be a. good starting-p'oillt for a proper understanding of rock mechanics.

REPRESENTATION OF THE MOVEMENT OF A POINT IN SPACE AND TIME

To describe the movement of a. moving object, an observer requires a reference frame and a dock. A' reference frame is defined by a.n,origin (which we will assume to be identical with the observer) and a basis which, depending on the case, can be orthogonal and unit vedors. We shall assume it to be Galilean, i.e linked to the earth. At a given moment, the moving object will be localized in space. Its position will be represented mathematically by a vector linking the origin to the moving object that is

3

OM = Xi i (1) i=l

2 lntroduction

where Xi are the coordinates of the point, C the vectors o the basis. If this latter is orthogonal and unit vectors

(2) in which Dij is the Kronecker symbol. Given a moving object initially situated at point X defining the "initial configuration". At the instant t, the moving object has a velocity v(t) and is situated at point i(t). These parameters define the "present configuration". There are two separate methods of representing the movement that we shall describe succinctly below.

Eulerian configuration

The movement is described by evaluating the present velocity of the moving object on the basis of its present position x(t) and of time

v(t) =

Introduction 3

appearing in x(t) , the particulate derivatives a.re total derivativ~. We may note lastly that the kinematic variables X, X, V, and the acceleration f do not depend on the representation mode and are equal irrespective of the configuration chosen since, in both cases, they are compared with the same reference frame. One must then avoid any confusion between representation mode and change of reference frame.

INTRINSIC QUANTITIES AND PRINCIPLE OF OBJECTIVITY

The kinematic quantities decribed below (x, V, f) are not intrinsic for they depend on the reference frame. It is known, for example, that an observer will give a different description of a moving object depending on whether or not he himself is moving.

On the other hand, certain quantities, such as stresses, cannot depend on the choice of the reference frame. They are said to be objective, that i8 invariant in any change of reference frame. Now, such is not always the case if one does not proceed carefully. For instance, let us consider a solid in rotation subjeded to a tension load (Fg. 1). In a fixed reference frame Ro the vector can be written

(5)

----F

e

Fg. 1. PrincipIe o objcctivily.

while in a "mobile" reference frame Rw linked to the solid, this same vector wiU be

(6)

4 Intreducton

If the rotational velocity w is constant CB = wt), the rate of the tension load in RQ and Rw is such that

and

dF (' _ _) dt = Fow - sm Wtcl + cos wte2

dF =0 dt in Rw

(7)

(8)

The force rate depends then on the chosen referimce frame. Clearly in this example the physica.l reality ls the second oue (wth respect to Rw) snce is assumed to be constant. If one wishes however to use the fixed reference frame, it is necessary, for the rate to be objective, to eliminate the rigid movement of Rw with respect to Ro.

There are several methods of eliminating the movement of the observer. The best known are the convective derivative (calculation of the variation rates with respect to the material itself) and the Jauman derivative (material derivative with respect to the corotational reference fra.me). In the framework of this study, we shall consider that the partieulate derivatives are objective and always expressed with respect to a fixed observer. This lS a perfectly realistic hypothesis fOl rock strain where movements are aIways sma.ll and sIow.

FUNDAMENTAL LAW OF DYNAMICS: THEOREM OF LINEAR MOMENTUM

The linear momentum of a mass point m with velocity v is the vector quantity mv. The linear momentum theorem is expressed as follows: "When the velocity of a mass point m varies because of the influence of forces applied to it, the resulting variation in the linear momentum is su eh that"

(9)

If this reasoning is extended to a solid o density p, of volume V and of external surface S subjeded both to surface forces o resultant F and to body forces of resultant

f~ the theorem can be written

(10)

Intraductin 5

MOMENTUM

definition, the knetic momentum with resped to a point 0, of a mass point o masa m driven by a velocity i! ia the vector ; such that

l OM Ami! This characterizes the circular motion of a masa point around an axis

(Fig. 2). Indeed, if a F, this point will can create a. movement point will be driven

is linked to an axis z, on the action of any force rotate around the z axis since ony its tangential force ifIt

is taken up the rigid bond). The rotating mass i! su eh that

(12) in which w is the ~UI~~"~' o tbe considered point. With resped to the z axis,

"'11),Ul'" momentum ; such that

A 2 (- A -) 2 -= r mw Un Uf = r mwuz (13)

z y

""---'---I----iIlllllllB... X

Fig. 2. Theorem of kinetc mornentum.

Extending the CaoVUJlHj

6 lntrodudioll

Ir one rep(aces (14) in (11) and derives wjtb respcd Lo time one obt.ains

(16)

or by applying (9)

(17)

since iOMjdt = r (d .. jdt) = rwiit = V. "A solid begins lo rotate around ao axis ir the resulting: momentum oC the Corces

aeting 0 0 this poio!. is not nil."

CONCLUSIONS

From Eqs (10) and (17). one can then state the conditions under which a salid will not move (dvjdt;;; dwjdt;: O).

JI. is necessary and sufficient that: (a) The resultant of ihe (orces applied 1.0 ihis salid be zeto. (b) Tbe resulting moment.um oC these forces also be zero. In this case, tbe salid i8 said Lo be in static equilibrium. This i5 the general

Cramework oC rack mechanics.

BmLIOGRAPHY

COUARRAZE, G" snd GROSSIORD, J .L. , Initiation d la rhlologie, Technique et docu-mentation Lavoisier, 1983. FRANEAU, J ., Physique glnirale, Academy Press, Bruxelles, 1970. GERMAIN , P ., Mcanique, tome I, Ed. Ellipse, Ecole polytecbnique, 1986. STUOT2, P., Lois de comportement : pnnc1pes gnirata, manuel de Rbologie des gomatriaux, Presses de I'ENPC, 1987, pp. 103-127.

Part 1 Mechanics

of continuous media Basic concepts

CHAPTER 1

State of strain

1.1 LAGRANGIAN DESCRIPTION OF THE STRAIN OF A SOLID

Given a solid S located with tesped to a rcference (rame of fixcd axes Ro and given X the initial coordioate of any paint of this sol id.

Consider a motian of this point which at time t is in a position i(t). We explained in the inlroduction that it was possible to describe the salid mo-

tian referring to its initial position and time; it is expressed using the Lagrangian transformation

x= 4)L(X ,t) where X a~d t are known as Lagrange variables.

1.1.1 Affine Lagrangian transformation Concept of displacement

( 1.1)

The displacement vector , of the considered point is the difference betweeo the initial configuratian and the present configuration relative to the initiaJ oonfiguration so that (1.1) can be written

ii~.i(+;(X,t) (1.2) where (X,t) is the displacement vector.

Tbe Lagrangian transformation is known as "affine" ir the displacement vector varies linearly with X

(X,t) = lf(t) ,i( (13)

10 P.rs l. MechaniCIJ of cont;nll"WI mli". B .. ic coneept"

Deriving this expression witb respect to Ji one abtains

8Ui where H'j = ax.

,

f!(t) Is the "displacemenl gradient" associaled with ~he affine transformation . Replacing (1.3) in (1.2) one obtains

where L is tbe unit tensor. Writing

J'(t) = l HI(O

=>; = J'(t).X thell deriving again witb resped to X, one obtains

8; J'(t) = uX ux where F. . . = --'

'1 8X. ,

f(t) is thus the "transforma.tion gradient" matrix.

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

The Lagrangian "afline" description can be generalized to tbe case of any hans--facmalion. The affine lransformation becomes then of the incremental type and one can still "read" (1.7) by saying that in the vidnity of a.ny point M, the function c$ can he approximaled by a linear fundian known as a "linear tangent lransformation" such that

di= J'(X,')dX (1.9)

1.1.2 Convective tran.sport o a vector

Ir we apply the pteceding formula to a vector Po at zetO time transforming itself into P, one obtains

(1.10) so that the vedorial variation P will be equal to

(1. 11)

Gbaptcr 1. Statc 01 slrain 11

1.1.3 Convective transpbrt of a volume (Fig. 1.1)

[f one has three initia.l orthogonal unit vectors Po, Qo, Ro tbe assodated volume is such that

v, = (p, t"~,) . R, (1.12)

Fig. 1.1. Convective transport of a volume.

Indeed, Po and Qo beiDg perpendicular, Po" 00:::: IPoIIQolit;

Similarly, if P, Q, ii represent these same vectors alter transformation, v = (PI";) R (1.13)

Taking account of (1.10), one obtains

V = (fp, A fa,) IR. (1.14) which can also be written alter development

v = J Va (1.15) in which J:::: del 1!1 is the Jacobian ofthe transformation.

1.1.4 Convective transport of an oriented surface (Fig. 1.2) Using tbe same reasoning, one can write

Qa " Po = Soiio (1.16)

12 Part l. Me.:halliC$ o( continuoWl media. Buic concepts

-

- -0 0 O

'0

rig. 1.2. Conveclive t rAJlsporl ot 811 orienled surhce.

( 1.17) (1.15) can also be written

(1.18) that is, ta.king account of (1.10)

(1.19)

1.2 DECOMPOSITION OF THE TRANSFORMATION RIGIDITY CONDITION

A material becomes strained when the melric properties (distances and angles) of tbe respective body are modified, On the contrary ir the moticns affecting the salid do oot modify the metric properties, one can speak oC "rigid moticn" . Let us translale these definitioos into mathematical formulae by expressing tlle norm of any vector P

, , ,- - ,( -) ( -) --IP 1= P.? = [Po. fPo = PogPo (1.20) with

q= I[.! being a symmetric tensor (1.21) One conclucles from (1.20) that the motian is rigid provided that

9=1 ( 1.22) 1 GeneralIy, tramposition is omitted for the scaru product except where the resuhs lLre affecteJ

(when a ma~rix is introduced lor example).

Clu.pter 1. S(ate of , Ir";" 13

Relation (1.20) can then be decomposed ioto a rigid part and a st raining part

, - [' 1 - - [' l -IPI = Po' 2'- D . Po+ Po , 2('+ D . Po (1.23) Equation (1.23) shows that the second term i5 characteristic of a purely rigid

motion and the litst of apure strain. Therefore, one defines the straio GN!t:n LfJgrangt: tensor 5uch that

,,= ! (C- 1) - 2 -

In lhe 5ame way as g ~ is a symmetric tensor.

1.3 EULERIAN DESCRIPTION OF THE STRAIN OF A BODY

(1.24)

Tbe Eulerian description, contrarily to the Lagrangian description defines moLion ooly on the basis of the coordinates of lhe present configuration i and of time.

The Eulerian t ransformation exprcsscs the veloeity of a point of the salid in tht! pre5ent configuration as a funetion of z and time t whi ch are consequcntly known as "Eulerian variables".

The transformation is thus expressed by the equation

v= t$E(i, t) (1.25) ; being tite present velocity of the considered poiut .

1.3.1 Affine Euleran transformation

This assumes that the velocity is a linear fundion of the presenl coordina tes. that

"

(1.25) We m ay observe that i is no looger (as was the ioit ial coordinate X) a constant .

As we shall see litis maJ.:es the derivation with respect to time much more complexo If one takes the Lagtangian t ransformaLion again , and derives ii with respecl to

time, one obtains

d% . --=v=FX =KF.Y dt - - -

:::} K=t".r l [being tbe derivative of fwith resped to timc.

(1. 27)

( 1.28)

14 Pa.rt l. Mechanics of conljnuous media. Buje coneepts

1.3.2 Convective transport of a vector

If one applies Eq. (1.26) to a vector P, it may be deduced that

(1.29)

1.3.3 Norm oC a vector. Decomposition of !S Let us decompose first [( into its syrnrnctrie an:l its skew symmetric par~s such

that

[S=[J+g (1.30) wit.h

1 D= - (K+ 'K) - 2 - - l= ~ (K- 'K) - 2 - -

We can study the evolution oftbe norm of the vector P by derivil!g it witb respect to time, that is

d (_ _) di p. P

= pfSP+lS; p PIS p+p'IS ,P pWp

(1.31)

In the Eulerian description of motion, the movement is rigid (i.e. the norm of P does not change with time) ifand only ir Q= O. So t,he symmetric part of!f. characterizes the late o strain in the salid wbile Q, skew symmetric part, represents tbe rigid motiaD (tbe cate of ratation).

1.3.4 Convective trnnsport of a volume

As for a Lagrangian transformation, one can ca\culate the variation in volume associated witb an Euleria.n transformation.

Considering P, Q and Ras three vectors, the volume associated i8 suro that (1.32)

The deri~tive of V with resped to time is such that

( 1.33)

Ch .. p ter l . Stllte or strain 15

( 1.34) from which one can easily show (by taking vectors parallel Lo tbe reference axes)

,

V = L J( V i=l

Ol what. is exactly t.he S

1.4 SUMMARY TABLE OF THE LAGRANGIAN AND EULERIAN FORMULA E IN THE CASE OF HOMOGENEOUS TRANSFORMATIONS

LAGRANGF. EULER

'1tansformation z = X + If(')X ti = ~(t) . % z=[(') -X K~f!1

Convect.ive transport. of el vector P=f Po P=/SP

Convedive transport oC volume V = J Vo V = "W)V J = d,t([)

Convective transport oC a surface f'S:;: Jn~ Sg

Candtion oC rigidity Q:;: l with p=O Q= '[ -E

Strain tensor 24= fl'- f-[ p= ~ [" eH '(" e0J

Accelcration aiJ

'i = &t _8ii( ;_ "'(= al + "V~v V

1.5 STATE OF STRAIN UNDER THE HYPOTHESIS OF SMALL PERTURBATIONS (SPH)

Let U8 tale the Lagrangian description again and assume "hat the variation oC norru associated wi th any vector j5 is sufficiently small 1.0 neglect the infinitely small of the second arder.

Given Cl.P the variation associated with the vector P between nitial and present configurations, thaL is

p = Po + Cl.P (1.43) Tbc nOIm of this vector 18 therefore

(1.44)

Negled ing 6.p2 (SPlI ) and taking account or the fad that 6.P = It Po, where!! 18 Lhe displacement grl\dient, (see Eq. (1 .11 ) one obtains

22 --- -2-(')-P = Po + lj . Po . Pu + Po . .ti . Po = Po + Po' Fj + IJ . Po (1.45) that 18 by writing

( 1.46) one obtains

(1.47) In othcr words, p 2 = P; (rigid mot ion) if and only if f = O. The vector 6.P cal!

now be written by decomposin g !Jin its symmetric part and its skew symmetric part:

/lP = H P,= -(H+H)+ - H - 'H) po - - [1 , 1 ( 1-- 2--2- - ( 1A8)

or writing 2 = 1j - '{! ( 1.49)

Since in the case of a rigid moton, ~ is zera, n, skew symmetric par' or the tensor H(d isplacemcnt gradient) eepresents tbe rigid motion whi le" represents the strain . Foe that reason, IS called "sLrain tensor~ in the hypothesis of small perturbat ions. lt Is represented by the symmetric part oC the displacement gradient !.enSQr. In the case of a purely strai ni ng motion, one will have

(1.50) A fi rsL conseqncnce of Lhe small perturbatious hypothesis is the identification of

the Green Lagrange tensor - wi th the tensor f. lndeed

~ = !(C-I)=!('F.P- I ) 2-- 2 ---1 (1.5 1) = :; [([+ 'in ([ + if) - 1[

'"

1 :; [('1!+if)[ =[

18 Pllrt , . Medul.lIk ... o( continuow media. B4.Sic conupu

A further oonsequence oC thc spa i8 the identity of Lagrangian and EuJeran configu rations. Indeed ir ';(X , t) and ~ s(i, t) represellt the same qUAlltity

(1.52) , displacement vector being small, we will have

( 1.53) This identity shows us tllat ene can (in SPll) derive ind iscrimillately witb respect

to X 01 i and tbaL Lbe particulat.e derivative becomes a partal derivative witb re-sped to time. One can &ka undcrstand why it ill preferahle to speak about "Sma.ll perturbalions" rather than "SmaH deformations". In raet, ene has lo take into con-sideration thaL botb the displacerncnt and the displa.ccment gradien t llhavc to be smaU. Finally {rom (1.6), (1.7) sud ( 1.26) one deduces

( 1.51) and replacing (1.54) in (1.30)

D= - H+ fH = i: 1 [ . . J - 2 - - -

(1.55)

smce the particulate der ivative c.oincides with a par t ia.! deri vative witb respect 1.0 time.

Ta sum up, one should nole a1l the following basie formulae in the SPH

1.6 GEOMETRlCAL SIGNIFICAN CE OF THE STRAIN TENSOR

(1.56)

The state of atrain at any ptu't oC the solid ls therefore rep resented by a symmetric tensor ~ auch that

( 1.57)

The diagonal componenls are known as normal strains and the non diagonal ones as shear alrains . Their geometric significante can be underst.ood as Collow$. If one considers Eq . (1.50) taking account oC (1.57), one obtains

CI,apter J. S tate o ( s train

1.6.1 Diagonal shains

c:u P .. ~ + e:r;v P"v + E~. P", Cr~ Po.:: + CYII POlI + EVI PN

19

(1.58)

Ir the initia.l vector is pa.rallel to O:r;, Po, ~ Poe = O and ir one assigns B.n inerease in length 6 P", , thc Eqs (1.58) are reduced to

ll.P:r; = En Po", (1.59) Cn rep tesen1s then the relat.ive variation of a vector parallel lo the axis %. The reasolli ng ..... ould be idcnti cal ror y and z .

1 .6.2 Non diagonal strains

Given the initial vect.or merged witb O. whose coordinales are PD'" =- Pa., =- O allcl Paz.

Let. us apply to it a displacemenl ll.P" (Fig. 1.3)

Fi. 1.3. Non d ia,:ollal sl'3ins.

The Eqs (1.58) are titen redueed t.o

.P" being smal1 Olle can write

I"P,I 8 = e". = [PDII

( 1.60)

(1.61)

20 Part l. Mccl>llnjcs Q( c:ont;mwus meroa, Buje conc"'pts

e ll O i:, tbcn ehllrllderi:ltic of the ~lip of 11 plane perpendicular to .t and parllllel lo y. This slip creatcs a distortion oC thc medium and is characLeristic oC ts change in sha.pe. It IS called shear slrain.

1.6.3 Volume variations. First invariant of the t ensor

We have shown ill tbe case of ao Eulerian configuration [Eq. (1.36)] dV - ~ t, (D)V dt -

( 1.62) TakiD.g &Ccount oC (1.55) fOI SPB we will have

( 1.63) which ;5 the firsl invariant oC {. dV IV is known as "cuhic cxpansioD" .

As a condusian, the normal strains charac.t.erizc the relative changcs in length and eventually in voLume while l he shear strains charac1erize the changes of form oC a continuou!I mediurn.

1 .6.4 Elongation of the vector j5. Invariant of the second arder The elongation of a vector P represeots its relative variation in length. If Po

becomes after sLrain P =. p~ + -;;P \\le define the elongation! (scalar quantity ) such that

IPI -IP.I IPI

Uuder the hypothesis ofsmall perturbations, one has

- IPI -I P.I _ _ _ _ !PI

IPI' -11'.1' 11'1 ' -11'.1' IPI( IPI + IP.I) '" 2d'

where d is tbe norm of P"(P,, == d) and a unit vector parallel to P" . Taking accou nL of ( 1.50) one obtains in Lile SPH

!PI' ~ !P.I' + 2P,. M ~ IPI' -11'.1' ~ 2 P, . ~ . p. By substituting (1.66) in (1.65) one ootains finally

(1.6')

(1.65)

( 1.66)

( 1.67) is the invariant oC the second order oC tbc tensor { and js t lterefore independent of Lile reference frame .

C"'plfll' 1. Slate uf str",n 21

1.7 PLANE STATE OF STRAIN

A solid is in a state of pla.ne strain paralJel to 11. plane Oxy if the displacement component w (Le. perpendicular to Oxy) is zero and if the components Iinked to tbis plane (i.c. tl and v) depelld only on x and y hui not on z.

Consequently, this definition induces that

0' f:yy = f}y

1 [8u 8W] F:rz ="2 f}z + ox :::; O 1 [o, Ow] Cy ="2 f}z + ay = o

oW CH ::: OZ :::; o

The st.ate of strain is thell expressed by the tensor

and , the elongation (. Ln a. direction 8 (with resped to Or) by

e:::; f:rr cos 2 8 + f:YII sin 2 8 + 2.!,..y sin O cos 8

1.8 STATE OF STRAIN IN CYLINDRICAL COORDINATES

(1.68)

( 1.69)

(1.70)

In certain specific problems such as wellbore stabili ty, it is often useful to refer to other types of reference feame than the conventional Cactesian coardinates system.

One uses curvilinear coardinates defining a "'local reference (rame" associated with the specific point where the state of stra.in is calculated.

1.8.1 Curvilinear coordinates and natural reference frame

Given a system of Cartesiall arthogonal and llnit vectors coordina.tes Xl, X2 , X3 and given l, e2, ej the vectors associated with this basis (~ ~:::; 6'J' le. 1 :::; 1).

Let us cnvisage a change of variable sucil that a point M previously localizcd by the coordinates Xl, X Z, X3 will after change of referente Crame be

(1.71)

22 PM( l. Mechanics Q( continuous m~; . Basic concepts

In arder that thc sequencc UI, U2, tia should make it possible to achicve an llU-ambiguous acatian of point Al, it is neccssary that there should be ane to one cor-respondence between u; and Xi . It i5 therefore necessary tho.t there exists a unique inver:>c of (l.71). Furtherrnore we sho.11 assume tho.1 the ti; are continuous derivable functions with respect to the Xi (and conversely). A point M so dcf1ned, one can

~ effed an infinitesimal displacemenl dOM while only vruying tI and monitoriog U2 and U3. Ooe describes thus a curve known as "coordinate tine" associated with tll. ln tlle same way ooe could describe starting from M two other "coordinatc lines", ene associated with U2 , the other with u;:! (Fig. 1.4).

Fig. 1.4. Natural eference IIlCes associa.te

ChapUr J . S'Al", 01 "lr"in

tlHl.t

z = pcosO y psin8 , ,

23

(1.73)

In the Eucl idian space, any point M can then equally well be located ei ther by a. sequence of values x, v, Z or by the ~uence p, O, z called "cylindrical coordinate of pomt M ". A set of coordinale tines (Fig. 1.5) is dcscri bcd , consisting of straightlines lile O/M (fJ and:: constant, p variable), PH (p and O cOllstant , z variable) and drclcs of radiliS O/M (p and z constants, 9 variable). In each point of the space it is tben possible to define a local reference frRlIle of basis 9" , g" g, such that [~(gq. 1.72)]

9, c060r + s iu O, ff. -psin O~ + p cos 9Fy (1.71) g, ~ i,

z

'z

-

O' 'U

"

' ;; O

fi; . 1.5. Cylindrit'1I1 coord inalc syslcm.

Arno ng these three vect.ors, 5, is nol unit vector (its norm is equal to p) .

Iff,1 ~ 1 Iff.1 ~ p Iff,1 ~ 1 (1.75)

24 Pare l . "f~chan;e~ o( c(meimrou. Ined ia , BlI.! ic C(J II Cl:'p~

'fhe :; ~I ain ten:r.or a.:>1IOciated with tiJe 1I0n P,ud idian bMis gp, g/l, 9. is called "tensor oC na tural strains" . 11. does nol correspond to t he physical one sinct th e lJ(ulS is not unil vedeTS. Qfie can deduce the components of Lhe physical si raio tensor ("'1 from those oC l ile natural slrain tern;or ':, by lhe relationship

(1.76)

In t he case of an ort hogonaJ and u ni ~ vedors local referente frame, the n a~ural components are equal to t he physical components.

111 Lhe case oC cylind rical coordillates, one can tbererorc define a sLrain tensor (associated wiLh l he local basis c;,. " ez) suth lhat

[

Ch ... pter 1. Stat", of stra.ill

one obtains the equality

One could show similarl)'

82,'I + 02~z = 2 Oy OZ2 0112 oy8z

Three other compatibility equations can be obtained

that is,

82e:&:& = .!!.... [~ 8(:,. + {)f;,,~ + /J'ry] 8y/Jz 82: 8x 8y oz

One oould show similar!y

25

(1.80)

(1.81)

(1.82)

(1.83)

(1.84 )

(1.85)

(1.86)

The compatibility equations show that the strain field must be continuous (since derivable two timesq across the medium. Their physical significance is clear: the state ofstrain in a point (or in a smal l volume e/ernent) must be compatible with the strains of its ncighbours; the compatibility equations characlerize conlinuity oC the matter.

BIBLIOGRAPHY

DRAGON , A., Plasticit et endommll'gement, cours de ttoisieme cyde, universit de Poitiers - UER Sciences. ENSMA 1988. JA EGER, J.C., and COOK, N.W.G., Fundamentafs 01 rock mechanlcs, third edition, Chapman and Hall , London , 1979.

26 Part 1. Mech/Ul;es o( cont;nu

CHAPTER 2

State of stress

2.1 INTERNAL FORCES AND STRESS VECTOR

Let us consider asolid in equilibrium under the Retioo oC external force5 distri buted on its external surfa.te. We neglect presently body forces , forces of inertia and mo-ment! of forces. Let us cut this solid by a surface (Fig. 2.1) and el us take as p08itive

~hc external side of the section and negative Lba.t situated towards the remaining material. The sold can only remain in equiJibrium ir one applies 10 th!! positive side of the section, Corces wbose resultant ir is equal to that exerted previously on the removed part of ihe body. We define the stress vector fin aL point M oC the section, such tbat

(2.\ )

The elementary force dP" acting on the infinitesimal surface dS whose externru normal is is therefore such that

With resped to a Cartesian reference frame x , y, z the stress vec~or ca.n be rcprescnted by its three components u.,,, , (J'l/'" (J'zn t.ha.t i5,

p .. = L (J"nei (2.3) in wbich ii are the vector5 of.the Cartesian basis.

28 Part l . Mechan;u of cOtltiuuo ... rncili . Bas,c (;oncep"

O"zn -', '.

+ + dS

y

(J . ......... . ......... ::;.:

,

rilo 2. J. Definilion o lhe slreu ,,tor .

2.2 EQUILffiRIUM OF THE ELEMENTARY TETRAHEDRON1

s

The stress vector is flot representalive o lhe mechanical state in a. point of tbe material. We , hall show that the com plete st ress state is represented by a. set o f tbrce tres, vectors on three perpendicular planes.

Far l his purpose let. llS consider (Fig. 2.2a) a coordtnatc syst.em x , y , z. On each coordinate plane tbere acts a stress vector that is

p ... = u"re" + ullzi', + qu~ Py = uryez + Uyye; + (T~!I~

p~ = uu~ + uy,ell + uui. (2.4)

We may now write lbe statie equilibrium of a.n infinitely small tetrahedron OABC (Fig. 2.2b) . This will be writLen in 1\ vectorial form

t. A foz + m A P, +11 A P. ::::- A Pn (2 .5)

IThi. demonl~T.~ion neAIKL5 Ih~ praence of body fOf'CetI ..nd fOf'C6 of illuti ... Ooe c..n "how lhal in a f,eoeroJ CNe the resulu aTe Wlmodilied. lile V(!lIlne of Ihe ~tlrahedrOJlle"dio, more rapidly lo,..ard. ero lh..n iu exluooJ surf&ee.

Ch.pter 2. S t.a te of.!tre.l'

:

,

,

..... ,

(J

I : i/ UyZ ! t ~ (J ..................... .

"

........

.. P" .. ~_ ......

-l'y

fl: Z.Za. CompollefllS o r SU-CS>l ,",cc: lO l"S on coordlnal t pllllltll .

y

where t, m, 11 are the di rector cosines of aJld A the area of triangle ABe. By identify ing in (2.5) tbe coefficiente of~ , e~ and e~ one obtaine the three

equations " .. (Tn: t + (T",,~m + O'"r . n a,o ~ (l"OJr t + (TuI/m + (T~ .. n a ,o ~ (Tu: l + (T~~m + O"un

wh ich can also be written

Pn =e (2.6) with

( a n a., a n ) 2 ~ a,. a" a" ",. "., a H

(2 .7)

2

-

"

o r----*c--I~ }'

,

FiC Z.2b. Equihbrium of lhe element!lry tclrnhedroll.

Tbe stress sta~ in any point of the sold is then represented by a tensor oC tbe secand arder known as "stress tensor". The quantities appearing in the tensor are called stress compoMnts" . The dillgonal compone.nts are called "normal slresses"; they are pOIIitive ir tradions. The non diagonal componenLs (0"" with -#J) are on t,he aLher haDd called sbear slresses. The koowledge oC this tensor makes it possible Lo calculate t he stress vedar on any face! passing lhrough this point..

2.3 CONCEPT OF BOUNDARY CONDITlON

In each point of the external surCace oC the solid the equilibrium cand itioD must be respected . Thus. ir F is the surface force apptied on the houndary of the solid and ir repreaents the cxternal normal to this BUrrare, in any point of this external surfa.ce one has

F = e . M E S (2.8) This equat.ion is known as boundary condition. Eq. (2.8) SboWB that ~ is a linear

application betwecn F and .

Chapler ,. SCate 01 s'reu

2.4 MOMENTUM BALANCE EQUILIBRIUM E Q UAT IONS

From thc moment.um balance one can deduce a first. property oC tbe stress tensor. This fundl\rnental principIe (see lnt.roduction) can be writLen

(2.9)

Ir one assurnes tbat the external (orces are both sur(ace forces F applied on .. he e:tlernaJ boundary and bod)' (orces l (gravity for uample) , Eq. (2.9) can be written

(2.10)

ABsuming mass conservation (i .e. djdt Iv pdV = O), taking acoount o( (2.8) and app lying the divergence t lleorem to the surface integral one obtains

1. di! 1. 1. -p-dV = V' !ZdV+ /dV v dt v v (2.11) or loeally (in other words at any point of the salid) i( one extends the above formula l o any volume V

_ d "\l.!?: +f=P dt (U,)

When tbe body Corees CM be neglected and when l he nertia effecls are negligible (:U 2) is reduced to

'V. !Z = 0 (2.13) o,

{}du +

8tr 3:, +

lJus~ O & Tu & lJ tr,,,,

+ lJ(T 1/"

+ lJtr 11 % O (2.14) & Tu &

lJ(T u: +

lJtr . , +

{}tru O & Tu a, = These equations known as "cquilibrium equations" must be vcrified in any point

oC tbc solid.

2.5 KINETI C ENERGY THEOREM

Assuming j = O in Eq. ('2. 12) Bnd multiplying eacll mernbcr by , (displacement vee.t.or a' Lbe considercd poinL) we ob Lain

_(~ ) dv _ u , v !! :;Pdt - u

Observing tba'

i] . (V 2") == 'V . C,!!) - 2" ; (\7 ) aCter int.egration on the volu me V of Lhe solid, one obtains

Usiag lhe dassical properly

l !! : (V' l u)"' dV 1 (2 .:)dV

(2. 15)

(1.16)

(1. 17)

and applying the divergente theorem to the first !.erm oC Lhe left hand member one i5 led to

fs (2)iidS- 1(2f)dV ~ 1p(~~)dV (2.18) e l takin g account oC (2 .8)

(2. 19)

The fi rst term of the leCt hand member rep rcsents Lbe acLion of the exlerna! forces through Lhe displacemenLS oC the ex ternaJ sur(ace, Lhe second Lhe acUen ofthe internal forees, the tl'lird on the rigilt hand member, lhe action of the forces of ncrLia. Thi!! is a. diroct oonsequence of the rnomentum balance. (2.19) can also be written in tcrms of energy tate. For this , one derives it with respect to time (the rorces are supposed to remain constant )

fs F ;dS- l

ChapCer~. S'aCt o( . Irea 33

Tben, one finally obt&ins

(2.22) where P~Z I and P;nJ are respectivel)' tbe ene.rgy late of external and internal forces.

2.6 THEOREM OF KINETIC MOMENTUM SYMMETRY OF THE STRESS TENSOR

Up to now we. have only vcrified the momentum balance. It is also necessary tbat the resulting moment around each coordinate axis, in &Il)'

point af t he solid, be nil. Lel us cansider an infinitesimal cu be of m aterial and observe thal on1y the COIn-

panenla acting in ea.clJ of the planes (shear components) are able ta aeate a ratatian. Around z axis, the couple u wz tends to create an anticlockwise rotation while the couple U zw creates a clackwise retatlan. The resulting momentum around axis z can then be written (Fig. 2.3)

(2 .23) The momentu m equilibrium can tben be wri tten

(2 .24) T he two otber equilibria (around z and V) would similarly Icad ta

(2.25) Tbe momentaequilibrium shows that 2' is a sy mmctrc tensor. Two basic canse-

quences can be deduced fram this proper ty.

2.6.1 Invariant quadratic form

lf!! ia the stress tensor expressed in reference frame xyz and ji (P. , P,I , P.) any

vedar expressed with resped to thiB Btlmtl re{eren ee fl'ame, the qu1l.d.ro.t.i.c fo rm

( ) 2 2 :. 112' Pz,P,JtP. = un:P~ +O'noP, +uup. +20'zy p. f1~ + 'luz. PzP~ + 'luy.P,P.

is invariant (in othcr words independent or the coordinate system). put in a matricial form

(2.26)

Eq. (2.26)

34

o

;(! u. .. ---_. -- ------_.

--------;f---dy

.,)"---------------"'

Fil"_ 2 .J. Theorem oC the kinetic momenlum (symmeu')' of the stress tensor).

2.6.2 Diagonalization of the stress tensor with respect to its principal directions

y

2" being symmetric, t here exlst three values 0'1,0"2 . (13 diagonalizing the tensor when expressed in a specifi c set ofaxes named principal directions. Tbey are roots of t.he characteristic equation

where 11, I'J, 13 are three invariant8 such that.

l = l7u :+U .... +(Ju

12 = -(Q'y~O'"u + l7u tT:u + O"rrUyy) + U~, + (T~., + O';~ 13 = U.,rO'n(1u +20"y.orzU,.. .. - 0" ..... 17;.

, , -O'"y~cr~r - Uuu.,y

0"10 (7 ~, (13 are called prin,jpal strcsses. The tensor ~ can theo be expressed

(2.28)

(2 29)

35

(

U,

~;; ~ (2 .30) In a plane perpendicula r to & principal direction, the shear components of thc

stress tensor ate thus nil.

2.7 CHANGE OF C ARTESIAN REFERENCE FRAME

Givcn a coordinate system z, y , z wi t ll respect to which Lile statc oC stress al poinl O is representcd by a ten80r !! . Let us consider anothet orlhogonal unit vectors coordinate systern %', 11 ,r wiLh a new basis j, ~, i':;, such tba1.

-, " n-e, = ~ rijej ,

where ~ (; = 1, 2,3) are lhe buis vedors of lbe first coordinale system . l he P" are tbe components of vector i: in lbe previous basis.

(2.31)

In (2.31)

(2.32) 'fhe change oC Cartcsian ,eferencc frame is lben defi ned by a malrix f (nKessa.rily

orthogonal). Following lhe defin ition oC a malrix

T=T,;(,,')) (2.33) one can easily show that tbe relation;hip between Lhe stress tensor !'!' (with respect to x' , rI, z') an d ~ is such thaL

~ . = e . ~ .' f (2.34)

2.8 EQUlLffiRIUM EQ U ATIONS IN CYLINDRlC AL C OORDINATE

As for the strain tensor (see C hapter 1, Eq. ( 1.74)], in cylindrical coordinates, one introd uces the orthogonal (but not neusslltily un it vectors) 10caJ refelence f,ame g"

g, , ~ . The sute of stress is then represented by the physical tensor

(2.35)

36 Part l. Mechllnics o( cont;nuous medi". Ba.sic c

Chapter 2. State of stress 37

that is

(2.44)

U is known as the mixed "Piola Lagrange" stress tensor. Its physical meaning clearly appears in Fig.2.4. Whereas the "Cauchy" stress tensor represents the action of a current force f on a current area increment da (normal to ), the "Piola Lagrange" stress tensor represents the action of the same current force f on the initial area increment dao (whose normal is o). This fact appears clearly in Eq. (2.44): U.t.f is (like l!), purely Eulerian whereas tE is a mixed Eulerian Lagrangian tensor so that TI has to be mixed to balance fe. This is the reason why U is nol a symmetric tensor.

-T

--.

a)lnilial un9lrained configur alion

-n

-::--__ .~ T

b )Currenl str ained configuration

Fig 2.4. Geometrical represenlalion of lhe Piola-Lagrange and Cauchy stress tensors.

The energy rate o the internal forces can be expressed then in Lagrangian con-figuration that is

(2.45)

Equation (2.41) has the disadvantage of using the velocity gradient. Furthermore U is a mixed non symmetric tensor. One can obtain a more homogeneous formulation by defining the "Piola-Kirchoff" stress tensor. Indeed

Chapter 2. State of stress

l

y

(Jyx (Jxx

x

Fig. 2.5. Plane slale of stress.

y

n

Fig. 2.6. Equilibrium of lhe lelrahedron.

39

.. x

This latter equation makes it possible to calculate on AB the norma.l component tr and the shear component T.

Chapter 2. State of stress 41

Two diametrically opposed points on the circle are then representative of the state of stress on two perpendicular facets. We may note that on these perpendicular facets, the shears are opposed (but of the same sign). We may further note that for a hydrostatic plane loading (0'"1 = 0'"2), Mohr's circle is reduced to a point.

BIBLIOGRAPHY

GERMAIN, P., 1986, Mcanique, Vol. I and TI. Ed. Ellipse, Ecole polytechnique, Paris. JAEGER, J.C., and COOK, N.W.G, 1979, Fundamentals ofrock mechanics, 3,d edition, Chapman & Hall, London. LEMAITRE, J., and CHABOCHE, J .L., 1988, Mcanique des matriaux solides, Dunod, Paris. MUSKHELISHVILI, N.I., 1977, Some basic problems of the mathematical theory of elas-ticity, reprint of the 2nd English edition, Noordhoff international publishing.

44 Part l. Mechanics o( continuous media. Basic concepts

3.2 FIRST PRINCIPLE OF THERMODYNAMICS

Let us consider a transCormation during which one subjects the system to a vari-ation oC internal energy AU by modiCying the kinetic energy oC its partic!es. Experi-ence shows that only a part of this variation of interna] energy carries out mechanieal work. In order to satisfy the fundamental principie oC energy eonservation, one has to imagine another form of energy: heat. The physical interpretation is simple: if one nereases the velocity of the partic!es, one also inereases the frietions between parti-c!es from which results an energy loss by heat dissipation. Dnder these eonditions the energy balance can be written

AU = AW+AQ (3.1) A W representing mechanical work and AQ the quantity oC heat furnished to the external medium. If, in addition, in the eourse of transCormation, the kinetie maero-seopic energy oC the system is modified (this energy being brought into the system is to be put in the leH-hand si de) Eq. (3.1) can be written

AU + Al( = AW + AQ (3.2) in which Al( is Lhe variation in kinetic energy of the system during transformation. One has to differentiate c!early the microscopic kinetie energy of partic!es which define internal energy from kinetic energy which results from a maeroscopic motion of the solid. The incremental form of Eq. (3.2) can be written

d .. dt (U + K) = We.:t + Q

in whieh vV .:! is the power of the external forces and Q the heat fiow.

3.3 SECOND STATE FUNCTION: ENTROPY OF A SYSTEM

(3.3)

The first principie of thermodynamics alone is noL sufficient to explain the trans-formation of a system. Indeed, experience shows Lhat in the absence of any exchange of heat or work with the exterior an isolaLed system is able to evolve.

The most classical experiment is that of a container of volume 2V shared between two compartments A and B of Lhe same volume V, separated by a screen pierced with a pinhole. If initially, A contains a gas and B is empty, the system evolves and the gas spreads spontaneously throughout the pinhole until the pressure becomes uniform or, which is equivalent, until the number of particles is equal in A and B. This so-called equilibrium state is irreversible, since the system never reverts spontaneously to its initial state (by spontaneously is meant without additional energy supply).

Chapter 3. Thermodynamics of contnuous media 45

The problem is therefore equivalent to that of the distribution of N particles between two compartments A and B.

Now, the number of possible combinations for placing n particles in one compart-ment and N-n in the other is such that

N! W71 = ---:-;-:-:-----:-: n!(N - n)!

Jt can be se en that \lVn , known as "complexion number", is maximum for n = N /2. In other words the system always evolves towards a state in which the complexion number is maximum.

The equilibrium state of a system is therefore a state of maximum probability. The entropy of a system containing N particles in any state (i.e. not necessarily

in equilibrium) is by definition s = k en W" (3.4)

in which Wn is the complexion number of the system and k a constant known as Bo!tzman's constant.

In the equilibrium state, the complexion number will assume its maximum value Wo so that the equilibrium entropy will be equal to

s = k en Wo (3.5)

This shows us that the evolution of a system towards an equilibrium state is always accompanied by an increase in entropy. On the contrary, the entropy of a closed system in equilibrium is stationary.

3.4 SECOND PRINCIPLE OF THERMODYNAMICS

These considerations enable us to state the second principie of thermodynamics by postulating the existen ce of a second state fundion S known as entropy such that

dS> dQ - T

in which T is caBed the absolute temperature of the system. The second principie thus enables one to define two fundamental processes:

(3.6)

(a) Irreversible transformations: these satisfy Eq. (3.6) with the supersc.ript sign meaning that they cannot be reversed. This will be the case for plastiClty.

(b) Reversible transformations: these represent the ideal Jimit case for which Eq. (3.6) is an equality. They can be reversed which means that the sys-tem can revert to its initial state by returning all the energy it has received during transformation. This will be the case with isothermal elasticity.

46 Part J. Mechanics ol continuous media. Basic concepts

3.5 FREE ENERGY

It is possible from U and S to define other state functions. If we imagine an isothermal reversible transformation (T = Const) without variation of macroscopic kinetic energy, the two principIes can be written respectively

or by substtuton of dQ

dU dQ

dWext + dQ TdS

diJI = dWext wth

iJI = U - TS

(3.7)

(3.8) (3.9)

known as free energy of the system. This s a state functon in the same way as U and S.

3.6 ENTHALPY AND FREE ENTHALPY OF A FLUID

Three fundamental concepts define mechanically a perfect fluid: (a) Zero tension resistan ce. (b) Zero shear resistance. (c) Normal stress equal in al! directions.

These properties show us that a fluid possesses a stress tensor such that

( -p O O) ~ = O -p O O O -p

in which p is the fluid pressure (the minus sign shows that this relates to compression). Consequently the force exerted on an external element of surface dS of the system

will be equal to d] = -pdS (3.10)

--> If one displaces this force by an element of length di (parallel to ) one reduces the volume of fluid, ana the work carried out (at constant pressure)

--> dWext = -pdS di = -pdV (3.11)

If one again assumes the macroscopic kinetic energy to be zero the first principIe can be written for this isobaric transformation

dH =dQ (3.12)

Chapter 3. Thermodynamics of continuous media 47

with

H = U+pV (3.13) H is known as the enthalpy of the system. It is also a state function. In the same

way one can define the free enthalpy of a system by defining the state function

G= H-TS (3.14)

3.7 SPECIFIC STATE FUNCTIONS

All the state functions can be reduced to the mass of the system. One spea.ks then of specific quantities written u, s, h, "p, g. If p is the density of the constituent of the system (assumed to be unique), one will then have for example

u = PUdV

We should note that the specific enthalpy of a fluid can be written

(3.16)

3.8 STATE VARIABLE AND STATE EQUATION

Experience shows very clearly that three variables influence the behaviour of a fluid (except chemical phenomena): volume, pressure and temperature. It would seem judicious then to choose these variables when describing the state of a fluid. In fact, these variables are not independent: it is known for exa.mple that if one heats a fluid while maintaining its volume constant, its pressure increases uncontrollably. One therefore has to admit that these three va.riables are linked by an equation: the state equation, which is generally written

f(p, V, T) = O (3.17)

The function f must be determined experimentally: it defines the behaviour of the system. One therefore speaks of a "constitutive law": it must obey the thermo-dynamics principIes, but only experimentation makes its determination possible. The subjective choice of state variables determines then decisively the expected results of a constitutive law.

3.9 TOTAL DIFFERENTIATION OF STATE FUNCTION

St.ate runchons are generally Ilot ac.cessiblc t o experimentation . Ho weve r con51-der ing their total diffcrentials one can deduce ccrtain irn portant propert.ies from Lhcm and define uperimentally measurahl(" coefficients .

3.9.1 Calorimetric coefficients

Ir o lle caU5eS the temp

For obvious rea.qons, Cv is known as specific heal at eonstant volume and Cp speciftc heal. al. constaul pr cssu rc . ThcorcticaIly thcy dcpcnd only Oll thc fl L d con-sidered.

It is also possible to connect specific heat al. constant pressu re and entropy. Indel., in lhc case of a reversible transformation

dQ ~ TdS (3.14) or taking aceounl of(3. 18)

meclr:::. TriS (325) ~1 orcvcr comjrlering T and p as fltate variables, lhe lotal diffcrcntial of S can be

writtcn

dS= (~;)p dT+ (~~')T dp Ir olle assurnes an isoLari r transfor m atioll (dp = O), e = Cp

(3.2&) and (3.26), one obta;ns

~ (DS) _ C, m ar p T

3 .9 .2 T h erm oelast ic coeffic ients oC a fluid

(3.'6)

and , by identi fyi:Lg

(327)

Depending on whether one chooses the pair of state variables p - T , p - V or V - T , ane can define three different transformations and one associat.es with each of lhem a differell t lhermoelastic coefficLent:

(a) Isothc l'nla l com p ressio n (p and Vare state variables) In tiL is case Oll': defi nes the bu lk modulus of thc fluid /{J such that

(3.28)

lhe minus sign indicating t.hat an increase in pressure creat.es a reduct.iolL ilL volume .

(b) b obaric h eating (T and Vare st.ate variables) In t.his case one defines the volume expansion coefficient cr J of lhe fluid sll-ch that

d\l = VajdT (c) Isoch o ri c h ea t ing (T and pare state variables)

One defines tiLe coeffi(".ient X SUCIl that

dp ~ PXdT

(3.'9)

(330)

50 P.rt l. Mecha.niu 01 con,inuc>us mema. B ... je conc" p t,.

We may note that each of the Eqs (3.28) (3.29) and (3.30) can be expressed in tbe form of partial derivative!!I since they correspoll d fOI each special case to a well-defined Lransformat ion

_1 = _~ (av) KI V 8p T

01 = ~ (W) V liT p

x= ~(ap) p liT v (3.3 1)

3.9.3 Further equalities between partial derivatives

At this stage it is useful to deri .... e a few remarkable equalities by exploiting the Cauchy Riemann condition fol' a funetian F(z, y), that is

dF ~ Pd

Chapter 3. Thermodynamia of continuus media 51

3.10 EXPRESSION OF A FLUID ENTROPY

The definition of the various Ihermoelastic and calorime~ric coefficients (accessibJe to measuremellt) makcs it possible to express state funct.ions such as entropy , interna! energy or free energy. If we choose p and T as statc variables the pcrfect differential of S c.an be wrltten as

dS = (85) dT (8S) d 8T +{) P , P T

(3 .39)

takin,ll; account of(3.27) (3.31) and (3.37) one is \ed t.o the expression

Cm dS= ~ dT-aVdp (3.10)

()f hy divirling the t.wo memhers hy m

e al ds= -----.EdT- - dp T p (3.41 )

in wLich, oS is lile specific eIltropy and p lile fluid demity

B. CONSTITUTIVE EQUATIONS OF SOLIDS

'rhe behaviour of solids should be consistent with thermodynamic rest.rictions and the balance equations, but, as we shall see, thermodynamics can only provide "cl ues" regarding constitutive equations. Jt is up to thc experirncntcr to choose judiciously, l.he state variables for ol.fH

52 Pan l . MedJ ... n;a 01 co:mt;nuow media. Basic -COl>CepLt"

3.11.1 Mass balance

Ir pis the solid density and V its volu me, it can be written

(3.42)

3.11.2 Momentum conservation

This hllS been expounded in various forms in the InLtoduction and in Chapter 2. 1t expresses the conscrvation of mechanical power in the form (Eqs (2.20) and (2.22) - in the absence of body forces)

h F iJdS - [ (rz : { }dll = i

ptl~U .t +r-'V q in wh.,ich is the deriva\.ive of u with tcspect 1.0 t ime.

3.11.4 Second princ ipie of thermodynamics

53

(3.48)

Its genera! expresslou (:.U:i) can be clarifled by tak lng account of(:i .45) 10 the form

- > - dV- -dS dS J.' 1. q dt-vT sT (3.49) or by ntroJudog spedlk. entrop)' and hy applyi ng Lhe di\'ergence theorem to tbc surfacc nt.cgrat of (3.49)

ds ri r Pdi+ 'V ' T - T~O

3.11.5 F\mdamental inequality of Clausius-Duhem

Lel us extracL r from Eq. (3 .18) and replace il in (3.50). One obtai ns

Observing thaL

d, p- + \l dI

f_-.!. [pu- u { +'V . q1 >O T T - - ' J -

V. (f) = T'V q-if 'V T T T'

o ne can write (3.5 1) in the fOlln

P [T dS _ dh] + 2: : t dl dt

- 'VT O -, .- > l' -

Let us introduce the specili c free energ.)'

one fina lly obtaillS

.p = u - Ts

d lb ::: du _ T ds _ s dT dt di di dt

( . .) VT (2: ; f )-P t/J+sT -iy?' O (3 .53) is known as tll e ineq uali t,y of Clauslus. Duhem .

(3.50)

(3.51 )

(3 .52)

(3.53)

54 Part J. Mechanics of continuous media. B.uic concepu

3.12 CHOICE OF STATE VARIABLES

The inequality oC CJausius-Dubem defines the f,hcrmodynamic admissibil ity of thc system. Al every moment in its evolu tion this has to be satisfied . The lhermodynamic potentiaJ depends, a.~ we saw in the previous paragraph , on a certain number of variables known a8 :>tate variables . Th~ variables can be "measurable" but also internaJ Ol "hidden" onc&. The choice is based on phenomcllologlcal obSo!rvations. It results then partally rrom the subjectivity of thc experimenter.

3.12.1 The m emory of a material

Any material call have a precise memory of the past, in particular of the irre-vcrsibilities it may h'\\'e experienc.ed. This is apparcnt in t lle dassic dio.gram repte-sented in Fig. 3.1. A material wiU bchave differcntly depending on whether it has bel!:n loaded up to point A (no memory) or up to point 8 .

In this (:asc, during e. future loading the irreversibilities will appear in B and not in A as previously. In thermody namic formalism one will therefore have te define a certAin oumber of "memo ry" variables also known as internal hardening varial;.les. As suggested above , these oons ideralion., lcad us lo envisage two types of slate variables : the measurable variables and the internal variables.

3.12.2 Observable state variables

The state variables truly accessible lo expedmentation are those deduc.ed conven-tionally from me

dccoupling of the reversible 3ncl irre\'ersible processes: lile "pa rtitiolli ng rule" which is writ~cn under the h)'pothesis of sma ll perturbatiolls

(3.54) Wc ma)' note that in the case of a purcly reversible process (e.lasticitr) f.,e becomes atl observable variable .

(J

u

(J

o ~ __________ ~ ____ ~ ___ f

" f' ~. 3.1 Cnncept of hard~n\llg.

2. Lastly, lile va ri ablf'_

6 Pan l. Mcchanics of conlmuo~ ,"edia. Basic COllccpls

or taking i\ ccoun~ of Lhe s l. rain part it ion;ng hypothesis (3 .54)

1!J :: tb k ,T.{" ,V.I (3 .56) The function 1/1 being a sti\t,e funaion Dne can caku late its to t.al differential thaL

=

a" 8. + 8T dT + tJ Vt dVt

(3.57)

a" D" a" 8(f _ {f') : d({ - e) + 8TdT + av. dV.I: 8Tj Jp EN> Oct : df.t + fJrdT+ {jVi dVt

=

Thc partilioning hypot hcsis enables one theterarc to wrile the deri vative (with respect lO tinll!) of '" in lhe form

: _ EN . . ~ 8"' 1' fJTj ir, 1p - Q{e - ... + iJT + 8V. k

repla.d ng (3 .58) in (3 .53) ;"nd takiug aecou nt of (3.54) olle ohtains

3. 14 C ASE OF REVERSIBLE BEIIAVIOUR ELASTICITY

(358)

(3 .59)

Let us consider a reversible transformation (no irreversible strain i;.P and no evo-lu tion oC lhe hard ening variables) al. constant and un iform temperature ('r = 0, 'V. T:;:: O). In this case Eq . (3.59) becomes equality and yields

8~ ~ = p u,-' (3.60) One may also consi der the eMe of a reversible t ransformation consisting of a

uniform heating of the soli d. In this case (3.59) implics that a~

,= - 8T (3 .61)

\

/

60 Pllrt l. Mechllnics of continuou$ medill. Basic concepts

fundion of order zero which means that /{J is homogeneous of order 1 (since !Z is the derivative of /(J). These properties have fundamental consequences on the formalism of dissipative phenomena (independent of time scale). 1st consequence

If /{J is homogeneous of order 1 then /{Jo conjugate of /{J is homogeneous of infinite order.

Indeed, given k and m, the respective homogeneity orders of /{Jo and /{J. /(J and I{)-being conjugate, one can write [see Eq. (3.70)].

i;.P :!Z - /(J 01{) 8u :!Z-/(J

considering the Euler identi ty l

taking account of (3.69)

which can also be written (3.73)

(3.74) Similarly one would prove I{)* :::::; 1{) ( m - 1) which leads by eliminating I{)- from the

last two equations to

k=~ m-l

The order of 1{) being 1 (m == 1) the order of I{)" is ~hen inn.nite. 2nd consequence

(3.75)

If there is dissipation, then!Z belongs to a hypersurface I(q: , Ak) == O knowD as yield locus.

If there is dissipatioD,!Z = o/{J/oi;.P =!l. (e, Ak) in which Q. is a bomogeneous fundion of order zero that is

(3.76) which can also be written

(3.77) ,\ being arbitrary; one can choose ,\ such that '\if == l. The elimination of the '\i~ ... '\i~ from the 6 equations of the type (3.77) leads to a scalar equation of the type

(3.78) lOne of the fundamental properties of homogeneous functions is the Euler identity: if 0(x) is

homogeneous of arder m, then

x 80 = m0 8x

66 Part l. Mechamcs oE continuous media. Basic concepts

yield locus O~~~=-__ ~~ ________ ~-__ ___

P' Fig. 3.5. Principie of maximum plasUe wOl"k.

![' being any plastically admissible field (f(!!:') ~ O]. Eq. (3.97) alone expresses there-fore normality and convexity. It shows that for an associated plastic law, the material "works" plastically to the maximum limit of its possibilities. Plastic dissipation is thus maximum.

3.20.2 Uniqueness of the solution (or Hill's theorem)

A further important consequence of the associativeness of the plastic law is the uniqueness of the solution of a boundary-value problem. Let us consider a certain volume of material, V. In certain portions of this volume V, the loadings paths correspond to plastic loading, in another portion they correspond to elastic unloading posterior to a primary plastic state and, in the remainder to a purely elastic state. We assume that the present state of stress![ at all points of the solid is known as well as the yield locus J(t ).

We assume an associated plastic flow rule U == F). Let us suppose that a loading increment dF (or an increment of displacement dil) is applied to the surface of the solido The stress increment and the associated strain increment must verify three conditions:

1. Firstly, the components of the incremental strain field must be kinematically admissible, that is

(3.98)

in which dil is the displacement increment associated with df .

ChapLer 3. Thermodynamics of contnuOU5 media 67

2. Subsequently, the components of the incremental stress field must be statically admissible, that is

or \1. d~ dq: Ti

o dF

in V on S (3.99)

3. Lastly, the components of the two fields must be plastically admissble, in other words (the condition is written here in terms of increments instead of particulate derivatives) remembering Eqs (3.89) and (3.90) in the case of normality, one obtains

with

1 df = A tl : da + - (da : n ) . n H > O - '" - H---

a a~

in which -4 el is the elastic tensor.

(3.100)

Let us now assume two sets of increments (d~, df. ) and (d~', df.') that satisfy the three condtions (3.98), (3.99), (3.100). They are at the same time kinematically, statically and plastically admissible, and let us consider the volume integral 1 such that

1= 1 {[d~'-dQ"]: [df'-df]}dV Condition (3.98) enables one to write

1 = 1 {( dQ"' - dQ") : [\1 0 (d' - d)} dV or agam

(3.101)

(3.102)

l = 1 \1. [(dQ"' - dQ") . (d' - d)] dV -1 (d' - d) [\1. (dz' - dQ")] dV (3.103) The second term is zero [Eq. (3.99)]. By applying to the first right-hand ter m the

divergence theorem, one obtains

1 = [(dz' - dq:) . (d' - d)] . dS (3.104) in which Ti is the externa! normal to the solid surface. As on the exterl1al surface, the loading increment dF (andJor displacement increment) is prescribed

dF = d~ = dQ"' dil = d'

on bF on bu

which means that (3.104) has to be zero, that is coming back to (3.101)

1 [( dQ"' - d~ ) : (df.' - df )] d V = O (3.105)

(3.106)

68 Part l. Mechanics oE continllous media. Basic concepts

To come back now to the constitutive relation (3.100), let us calculate the inte-grand of Eq. (3.106)

1 (d[' - d[) : (df.' - df) = H(d[' - d[): [(c/d[': 2 - exd[ : 2)2] (3.107) +4 el : (d[' - d[) : (dll' - d[)

with

ex 1 if f([ ) = O and du : n ~O O'. O ir f(1l ) < O or f([ ) = O and d[ :n

Chapter 3. Thermodynamics oi contjnuous medja 69

LABETHER, 1985, Mesures thermiques et flux, Masson. LEMAITRE, J., and CHABOCHE, J .L., 1988, Mcanique des matriaux solides, Dunod, Paris. NOWACKI, W., 1986, Thermoelasiicity, Pergamon Press, Polish scientific publishers. DUZIAUX, R., and PERRIER, J., 1978, Mcanique des fluides applique, Dunod, Paris. REIF, F., 1965, Statistical physics, Mac Graw Hill, Berkeley Physics course. ROCARD, Y., 1967, Thermodynamique, Masson.

Part 11

Mechanis of materIal strain

CHAPTER4

Linear elasticity General theory

Many sedimentary rocks display elastic behaviour, in other words, instantaneous and aboye aH reversible. It seems worthwhile therefore to review certain fundamental concepts of contnuum media elasticity with a view to extending them subsequently to saturated porous media. There are many stout volumes that deal exhaustively with linear elasticity: we would menton in particular the works of Musckelishvili, Timoshenko and Ooodier, and Oreen and Zerna.

4.1 HOOKE'S LAW

In its original conception we have seen [Eq. (3.63)] that the theory of linear elas-ticity is based on the foHowing hypothesis:

"The components of the stress tensor at a given point of a solid are linear and homogeneous functions of the components of the strain tensor at the same pont."

This definition induces automatically the small perturbations hypothesis

rJ=A:e - ::: ....

( 4.1)

in which A known as "elastic tensor" is a tensor of the fourth order containing 81 compon~nts in the general case. Since!?: and f are symmetric tensors, A only contains 36 independent components. Because of linearity A is then an intrinsi~ char-acteristic of the considered material independent of stress ~nd strain components.

, )

Ghapter 4. Linear elasticity. General theory 75

coordinate z. In configuratiol1 Hooke's first can now written, isotropy)

(4.6)

z

r+--+~3 ,+-+ __ ~2

2

x b

Fig. 4.1. Isotropy with respecl 1.0 pl'incipal axes.

dentifying and (4.6) one

12 - A13) 13

so that (4.6) can be expressed in the form

us write

12 = A All -

Observing that the cubical expansion is such that 1 + z + 3

(Einstein convention) one finally obtains

lTl

similar rotations around y account the symmetry of .Q.)

one would

AH + 2J12

+ 2J13

obtain

(4.7)

(4.8)

(4.9)

(4.10) into

(4.11)

76 Part TI. Mechanism 01 material strain

or in a tensorial form

(4.12) In linear and isotropic elasticity, the nllmber of elastic constants is reduced to two:

Lame 's constants A and J.

4.3.1 Generalization to any Cartesian system of coordinates

Equation (4.12) can be generalized to any Cartesian reference frame by using the quadratic invariant form of tensors !! and f..

Let us consider a vector P with respect to a reference frame linked to the principal directions 1, 2, 3.

The quadratic invariant is then written

( 4.13) or taking account of Eq. (4.12)

2 - - 2 nO" = .AulPI + 2JP & . P = .\tu!?1 + 2Jn& - - -

(4.14)

Given now any reference frame x, y, z with respect to which the components of P are~, "1,(. The invariance of (4.14) makes it possible to write it with respect tO!! and f. expressed in the new basis, that is after development

O"xxe + O"yy"12 + O"zze + 20"yz"1( + 20"zx(e + 20"xy~"1 = Akl; (e + r2 + (2) + 2J (xxe + &yy"12 + &.zz(2

+2&yz"1( + 2:zx(e + 2xyer) (4.15)

By identifying the variolls coefficients of the two members one obtains a relation-ship identical to (4.12) i.e.

4.3.2 Physical interpretation of isotropy

Hooke's law (4.16) shows that: (a) Normal stresses generate only normal strains. (b) Shear stresses generate only shear strains

or, which is identical. (e) Normal stresses are responsible for changes in volume. (d) Shear stresses are responsable for changes in formo

( 4.16)

Chilpter 4. Linear elasticty. General theory 77

4.4 TIIE COMl\10N ELASTIC CONSTANTS

Lame's constants >. and J.l are rarely used in practice. Other elastic constants can indeed defined [roIn loading

4.4.1 Young's rnodulus and Poisson's ratio

If one a.

78 Part II. Mechanism o( material strain

On the other hand, by eliminating in (4.17) .\ea between one of the first two equations and the last, one obtains

U zz = -2J.LE.yy + 2J.LE.zz ( 4.20)

U zz 2 eyy 2 -=- J.L-+ J.L ezz E. zz

(4.21)

that is by taking account of (4.19) one obtains

(4.22) in which v = .\ / 2(.\ + J.l) is known as "Poisson's ratio".

Figure 4.2 shows the physical significance of E and v: E represents the rigidity of the material under uniaxal loading while v represents the capability of the material to transfer its deformability perpendicularly to th