Holdings: Handbook of continuum mechanics

Handbook of continuum mechanics: general concepts, thermoelasticity

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Bibliographic Details
Main Author:	Salençon, Jean 1940- (Author)
Format:	Book
Language:	English French
Published:	Berlin [u.a.] Springer 2001
Series:	Physics and astronomy online library
Subjects:	Continuum mechanics Kontinuumsmechanik Lehrbuch
Online Access:	Inhaltsverzeichnis
Physical Description:	XIX, 803 S. Ill., graph. Darst. 2 Beil.
ISBN:	3540414436

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Record in the Search Index

_version_	1804128509701914624
adam_text	CONTENTS I. MODELLING THE CONTINUUM ............................... 1 1 SCALE, MODEL, AND VALIDATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 CONCEPTS AND FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 THE MAIN IDEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 FRAME OF REFERENCE AND COORDINATE SYSTEM . . . . . . . . . 9 2.3 CONFIGURATIONS OF THE SYSTEM . . . . . . . . . . . . . . . . . . . . . 10 2.4 MATERIAL FRAME INDIFFERENCE . . . . . . . . . . . . . . . . . . . . . . . 11 3 LAGRANGIAN DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 CONTINUITY HYPOTHESES . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 WEAKENING OF CONTINUITY HYPOTHESES. . . . . . . . . . . . . . . 15 3.4 PHYSICAL INTERPRETATION OF THE LAGRANGIAN DESCRIPTION: PATHLINES . . . . . . . . . . . 16 3.5 STREAKLINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 PARTICLE VELOCITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.7 GENERALISED REFERENCE CONFIGURATION . . . . . . . . . . . . . . . 17 4 EULERIAN DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 DETERMINING PATHLINES . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 STREAMLINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 STEADY MOTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.5 SEMI-STEADY MOTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.6 NOTATION FOR VELOCITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 II. DEFORMATION ............................................. 31 1 TRANSPORT, TRANSFORMATION, AND DEFORMATION . . . . . . . . . . . . . . 37 2 CONVECTIVE TRANSPORT IN A HOMOGENEOUS TRANSFORMATION . . . . 38 2.1 HOMOGENEOUS TRANSFORMATION . . . . . . . . . . . . . . . . . . . . . 38 2.2 MATERIAL VECTOR AND CONVECTIVE TRANSPORT . . . . . . . . . . 39 2.3 TRANSPORT AND EXPANSION OF A VOLUME . . . . . . . . . . . . . 40 2.4 TRANSPORT OF AN ORIENTED SURFACE . . . . . . . . . . . . . . . . . . 42 VIII CONTENTS 3 DEFORMATION IN A HOMOGENEOUS TRANSFORMATION . . . . . . . . . . . . 43 3.1 EXPANSION TENSOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 USING THE EXPANSION TENSOR . . . . . . . . . . . . . . . . . . . . . . 45 3.3 GREENLAGRANGE STRAIN TENSOR . . . . . . . . . . . . . . . . . . . . 48 3.4 POLAR FACTORISATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 DEFORMATION OF THE CONTINUUM: GENERAL CASE . . . . . . . . . . . . . 51 4.1 BASIC PRINCIPLE: THE HOMOGENEOUS TANGENT MAP . . . . 51 4.2 TRANSPORT EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 DEFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 DISPLACEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.5 POLAR FACTORISATION AND RIGID BODY TRANSFORMATION. . 55 4.6 FRAME INDIFFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 INFINITESIMAL TRANSFORMATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 LINEARISED STRAIN TENSOR . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3 GRADIENT OF A TENSOR FIELD ON THE CURRENT CONFIGURATION . . . . . . . . . . . . . . . . . . . . . 58 6 GEOMETRICAL COMPATIBILITY OF A LINEARISED STRAIN FIELD . . . . . 59 6.1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 COMPATIBILITY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.4 APPLICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7 FINAL COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.1 TRANSFORMATION AND DEFORMATION . . . . . . . . . . . . . . . . . . 65 7.2 LAGRANGIAN PARAMETRISATION RELATIVE TO A GENERALISED CONFIGURATION . . . . . . . . . . . . 65 7.3 PRACTICAL INVESTIGATION OF STRAINS . . . . . . . . . . . . . . . . . . 66 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 III. KINEMATICS .............................................. 81 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2 LAGRANGIAN KINEMATICS OF THE CONTINUUM . . . . . . . . . . . . . . . . . 87 2.1 CONVECTIVE TRANSPORT AND MATERIAL DERIVATIVE. . . . . . . 87 2.2 LAGRANGIAN STRAIN RATE . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3 EULERIAN KINEMATICS OF THE CONTINUUM . . . . . . . . . . . . . . . . . . . 90 3.1 MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.2 MATERIAL DERIVATIVE OF A VECTOR . . . . . . . . . . . . . . . . . . . . 90 3.3 EULERIAN STRAIN RATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4 USE OF THE STRAIN RATE TENSOR . . . . . . . . . . . . . . . . . . . . . 92 3.5 SPIN TENSOR. RATE OF VOLUME DILATATION . . . . . . . . . . . . 95 3.6 COMPARISON WITH LINEARISED STRAIN FOR INFINITESIMAL TRANSFORMATIONS . . . . . . . . . . . . . . . . . . 98 3.7 GEOMETRICAL COMPATIBILITY OF A STRAIN RATE FIELD . . . . 99 3.8 RIGID BODY MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 CONTENTS IX 3.9 WEAKFORMULATION OF GEOMETRICAL COMPATIBILITY . . . . 100 3.10 INFINITESIMAL TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . 101 3.11 FRAME INDIFFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4 MATERIAL DERIVATIVES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1 MATERIAL DERIVATIVES IN THE LAGRANGIAN DESCRIPTION . . 102 4.2 MATERIAL DERIVATIVES IN THE EULERIAN DESCRIPTION . . . . . 103 4.3 MATERIAL DERIVATIVE OF A POINT FUNCTION . . . . . . . . . . . . 104 4.4 MATERIAL DERIVATIVE OF A VOLUME INTEGRAL . . . . . . . . . . . 105 4.5 MATERIAL DERIVATIVE OF A CIRCULATION. . . . . . . . . . . . . . . . 115 4.6 MATERIAL DERIVATIVE OF A FLUX . . . . . . . . . . . . . . . . . . . . . 116 5 CONSERVATION OF MASS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1 EQUATION OF CONTINUITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 INTEGRAL FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 MATERIAL DERIVATIVE OF A MASS INTEGRAL IN THE EULERIAN DESCRIPTION . . . . . . . . . . . . . . . . . . . . . . . 120 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 IV. THE VIRTUAL WORK APPROACH TO THE MODELLING OF FORCES ............................... 133 1 THE PROBLEM OF MODELLING FORCES . . . . . . . . . . . . . . . . . . . . . . . . 139 1.1 MODELLING FORCES IN A SYSTEM OF MATERIAL POINTS . . . . . 139 1.2 THE VIRTUAL WORKMETHOD . . . . . . . . . . . . . . . . . . . . . . . . 144 2 DUALISATION AND VIRTUAL WORK FOR A SYSTEM OF MATERIAL POINTS . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.1 SYSTEM COMPRISING A SINGLE MATERIAL POINT . . . . . . . . . 144 2.2 SYSTEM COMPRISING SEVERAL MATERIAL POINTS . . . . . . . . . 145 2.3 VIRTUAL VELOCITY, VIRTUAL MOTION AND VIRTUAL WORK. . 147 2.4 STATEMENT OF THE PRINCIPLE OF VIRTUAL WORK . . . . . . . . . 149 2.5 VIRTUAL MOTIONS IN RELATION TO THE MODELLING OF FORCES . . . . . . . . . . . . . . . . . . . . . . . 149 3 VIRTUAL WORKMETHOD FOR A SYSTEM OF MATERIAL POINTS . . . . . . 150 3.1 PRESENTATION OF THE VIRTUAL WORKMETHOD . . . . . . . . . . 150 3.2 EXAMPLE OF AN APPLICATION . . . . . . . . . . . . . . . . . . . . . . . 150 3.3 REMARKS ON THIS APPLICATION OF THE VIRTUAL WORKMETHOD . . . . . . . . . . . . . . . . . . . . . . 154 3.4 GEOMETRICAL COMPATIBILITY OF * * IJ . SYSTEMS OF HINGED RODS . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4 THE VIRTUAL WORKMETHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.1 GENERAL PRESENTATION OF THE METHOD . . . . . . . . . . . . . . . 156 4.2 SUMMARY OF THE VIRTUAL WORKMETHOD . . . . . . . . . . . . . 158 4.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.4 CHANGE OF FRAME. FRAME INDIFFERENCE . . . . . . . . . . . . . . 159 5 RIGID BODY MOTIONS. DISTRIBUTORS AND WRENCHES . . . . . . . . . . . 160 5.1 DISTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 X CONTENTS 5.2 WRENCHES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.3 RESTRICTION OF A LINEAR FORM DEFINED ON A SPACE OF VIRTUAL MOTIONS TO THE RIGID BODY VIRTUAL MOTIONS 162 5.4 WRENCH OF A FORCE SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . 162 5.5 FIELDS OF DISTRIBUTORS AND WRENCHES. DIFFERENTIATION . 163 6 GENERAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.1 DEFINING THE SYSTEM AND ITS MOTIONS . . . . . . . . . . . . . . . 165 6.2 VIRTUAL WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.3 LAW OF MUTUAL ACTIONS AND FUNDAMENTAL LAW OF DYNAMICS . . . . . . . . . . . . . . . . 166 6.4 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7 MOMENTUM THEOREM. KINETIC ENERGY THEOREM . . . . . . . . . . . . 168 7.1 DEFINITION OF THE SYSTEM AND ITS MOTIONS . . . . . . . . . . . 168 7.2 WRENCH OF THE QUANTITIES OF ACCELERATION. MOMENTUM WRENCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.3 CONSERVATION OF MOMENTUM. . . . . . . . . . . . . . . . . . . . . . . 170 7.4 EULERS THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.5 KINETIC ENERGY THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.6 DISCONTINUOUS REAL VELOCITY FIELD. SHOCKWAVES . . . . . 173 8 LOOKING AHEAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 V. MODELLING FORCES IN CONTINUUM MECHANICS .............. 183 1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 2 SCALAR FIELD MODEL OF INTERNAL FORCES. PRESSURE . . . . . . . . . . . . 190 2.1 VIRTUAL MOTIONS. VIRTUAL RATE OF WORKBY QUANTITIES OF ACCELERATION . . 190 2.2 VIRTUAL RATE OF WORKBY EXTERNAL FORCES . . . . . . . . . . . 191 2.3 VIRTUAL RATE OF WORKBY INTERNAL FORCES . . . . . . . . . . . . 194 2.4 EQUATIONS OF MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 2.5 RELEVANCE OF THE MODEL. PRESSURE FIELD . . . . . . . . . . . . . 198 2.6 THE FUNDAMENTAL LAW OF DYNAMICS. . . . . . . . . . . . . . . . 199 2.7 DISCONTINUOUS VIRTUAL VELOCITY FIELD . . . . . . . . . . . . . . . 199 3 MODELLING INTERNAL FORCES BY A TENSOR FIELD. STRESSES . . . . . . . 202 3.1 VIRTUAL MOTIONS. VIRTUAL RATES OF WORKBY QUANTITIES OF ACCELERATION AND EXTERNAL FORCES. . . . . . . . . . . . . . . . 202 3.2 VIRTUAL RATE OF WORKBY INTERNAL FORCES . . . . . . . . . . . . 203 3.3 EQUATIONS OF MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 3.4 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 3.5 CAUCHY STRESS TENSOR. STRESS VECTOR . . . . . . . . . . . . . . . 208 3.6 MODELLING INTERNAL FORCES WITHIN THE CONTINUUM USING THE STRESS VECTOR . . . . . . . . . . . . . . . . . . . . . . . . . . 211 3.7 EXPLICIT FORMS OF THE EQUATIONS OF MOTION . . . . . . . . . . 214 3.8 DISCONTINUOUS VIRTUAL VELOCITY FIELD . . . . . . . . . . . . . . . 215 CONTENTS XI 3.9 DISCONTINUITIES IN THE STRESS FIELD . . . . . . . . . . . . . . . . . 218 3.10 EULERS THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 3.11 KINETIC ENERGY THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . 223 3.12 RATE OF WORKBY DEFORMATION . . . . . . . . . . . . . . . . . . . . . 224 3.13 GEOMETRICAL COMPATIBILITY REVISITED. A WEAKFORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 3.14 WEAKFORMULATION OF THE EQUATIONS OF MOTION . . . . . . 225 3.15 FRAME INDIFFERENCE OF THE CAUCHY STRESS TENSOR . . . . . 226 4 LAGRANGIAN FORMULATION OF STRESSES . . . . . . . . . . . . . . . . . . . . . . 226 4.1 PIOLAKIRCHHOFF STRESS TENSOR . . . . . . . . . . . . . . . . . . . . . 226 4.2 PIOLALAGRANGE STRESS TENSOR. EQUATIONS OF MOTION . . 230 5 REVIEW AND PROSPECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.1 MECHANICIST AND PHYSICIST . . . . . . . . . . . . . . . . . . . . . . . . 232 5.2 SUMMARY OF RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.3 MICROPOLAR MEDIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 VI. LOCAL ANALYSIS OF STRESSES ............................... 249 1 IMPLEMENTING THE NOTION OF STRESS . . . . . . . . . . . . . . . . . . . . . . . 255 2 SOME PRACTICAL DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 2.1 DIMENSIONS AND UNITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 2.2 NORMAL STRESS AND SHEAR STRESS . . . . . . . . . . . . . . . . . . . 256 2.3 SIGN CONVENTION. PHYSICAL INTERPRETATION . . . . . . . . . . . 258 2.4 CAUCHY RECIPROCAL THEOREM . . . . . . . . . . . . . . . . . . . . . . 258 2.5 CHANGE OF BASIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 2.6 PRINCIPAL AXES. PRINCIPAL STRESSES . . . . . . . . . . . . . . . . . 262 2.7 INVARIANTS OF THE STRESS TENSOR. REPRESENTATION THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . 263 2.8 DEVIATORIC STRESS TENSOR. . . . . . . . . . . . . . . . . . . . . . . . . . 265 3 MOHR REPRESENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.1 MOHR REPRESENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.2 MOHR CIRCLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 3.3 DESCRIPTION OF PRINCIPAL CIRCLES . . . . . . . . . . . . . . . . . . . 269 3.4 PRACTICAL CONSEQUENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 271 3.5 EXAMPLES OF STRESS STATES . . . . . . . . . . . . . . . . . . . . . . . . 271 4 YIELD CONDITIONS FOR ISOTROPIC MATERIALS . . . . . . . . . . . . . . . . . . 274 4.1 PRESENTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 4.2 GENERAL PRINCIPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 4.3 TRESCA YIELD CRITERION . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 4.4 VON MISES YIELD CRITERION . . . . . . . . . . . . . . . . . . . . . . . . 277 5 MATERIAL DERIVATIVE OF THE STRESS TENSOR . . . . . . . . . . . . . . . . . . 278 5.1 MATERIAL DERIVATIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 5.2 INTRINSIC DERIVATIVE (TRUESDELL RATE) . . . . . . . . . . . . . . . 279 5.3 COROTATIONAL DERIVATIVE (JAUMANN RATE) . . . . . . . . . . . 280 XII CONTENTS SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 VII. THERMOELASTICITY ........................................ 295 1 FROM EXPERIENCE TO A CONSTITUTIVE LAW . . . . . . . . . . . . . . . . . . . 301 2 EXPERIMENTAL OBSERVATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 2.1 GENERALITIES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 2.2 SIMPLE TENSION TEST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 2.3 FURTHER EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . 304 2.4 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 3 THERMODYNAMICS OF CONTINUOUS MEDIA . . . . . . . . . . . . . . . . . . . 306 3.1 FIRST LAW: ENERGY EQUATION . . . . . . . . . . . . . . . . . . . . . . 306 3.2 SECOND LAW: FUNDAMENTAL INEQUALITY . . . . . . . . . . . . . . 311 3.3 LAGRANGIAN EXPRESSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 313 4 THERMOELASTIC CONSTITUTIVE LAWS . . . . . . . . . . . . . . . . . . . . . . . . 315 4.1 ELASTICITY HYPOTHESIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 4.2 UNCONSTRAINED THERMOELASTICITY . . . . . . . . . . . . . . . . . . . 316 4.3 THERMOELASTICITY WITH INTERNAL CONSTRAINTS . . . . . . . . . 321 4.4 MATERIAL SYMMETRIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 4.5 ISOTROPIC THERMOELASTIC MATERIAL IN THE REFERENCE CONFIGURATION . . . . . . . . . . . . . . . . . . . . 325 4.6 INTERNAL CONSTRAINTS FROM THE EULERIAN STANDPOINT . . . 327 5 UNCONSTRAINED LINEAR THERMOELASTICITY . . . . . . . . . . . . . . . . . . . 328 5.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 5.2 PHYSICAL LINEARISATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 5.3 ISOTROPIC LINEAR THERMOELASTIC MATERIAL . . . . . . . . . . . . 332 5.4 INFINITESIMAL TRANSFORMATION AND GEOMETRICAL LINEARISATION . . . . . . . . . . . . . . . . . . . . 334 5.5 STABILITY OF THERMOELASTIC MATERIALS . . . . . . . . . . . . . . . 340 5.6 TYPICAL VALUES FOR MATERIALS IN COMMON USE . . . . . . . 343 5.7 EXAMPLES OF ANISOTROPIC THERMOELASTIC MATERIALS. . . . 343 6 HISTORICAL PERSPECTIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 VIII. THERMOELASTIC PROCESSES AND EQUILIBRIUM ............... 361 1 QUASI-STATIC THERMOELASTIC PROCESSES. . . . . . . . . . . . . . . . . . . . . 367 1.1 THE NEED FOR A CONSTITUTIVE LAW . . . . . . . . . . . . . . . . . . 367 1.2 FORMULATING THE QUASI-STATIC THERMOELASTIC PROBLEM 369 1.3 SOLVING THE QUASI-STATIC THERMOELASTIC PROBLEM. . . . . 372 1.4 SOME EXAMPLES OF BOUNDARY CONDITIONS . . . . . . . . . . . 374 2 LINEARISING THE QUASI-STATIC THERMOELASTIC PROCESS. . . . . . . . . 376 2.1 LINEARISATION HYPOTHESES . . . . . . . . . . . . . . . . . . . . . . . . . 376 2.2 THE PRINCIPLES OF LINEARISATION . . . . . . . . . . . . . . . . . . . . 377 CONTENTS XIII 2.3 USUAL EQUATIONS FOR LINEARISED THERMOELASTIC PROCESSES . . . . . . . . . . . . . 380 3 LINEARISED QUASI-STATIC THERMOELASTIC PROCESSES . . . . . . . . . . . 381 3.1 DECOUPLING THE HEAT PROBLEM . . . . . . . . . . . . . . . . . . . . . 381 3.2 THERMOELASTIC EQUILIBRIUM . . . . . . . . . . . . . . . . . . . . . . . 381 3.3 UNIQUENESS THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 3.4 NATURAL INITIAL STATE AND PRINCIPLE OF SUPERPOSITION . . 382 3.5 NONZERO INITIAL SELF-EQUILIBRATING STRESS STATE . . . . . . . 383 3.6 PRELOADED INITIAL STATE . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 4 SOLUTION TO THE THERMOELASTIC EQUILIBRIUM PROBLEM . . . . . . . . 385 4.1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 385 4.2 STATICALLY ADMISSIBLE STRESS FIELDS. KINEMATICALLY ADMISSIBLE DISPLACEMENT FIELDS . . . . . . 386 4.3 THERMOELASTIC EQUILIBRIUM AND ASSOCIATED ISOTHERMAL ELASTIC EQUILIBRIUM . . . . . . 389 4.4 SOLUTION METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 5 DISPLACEMENT METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 5.1 BASIC PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 5.2 ISOTHERMAL EQUILIBRIUM AND HOMOGENEOUS ISOTROPIC MATERIALS. NAVIER EQUATION . . . . . . . . . . . . . . . 393 5.3 STRONGLY HETEROGENEOUS MATERIALS AND THERMAL SHOCKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 6 STRESS METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 6.1 BASIC PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 6.2 ISOTHERMAL EQUILIBRIUM AND HOMOGENEOUS ISOTROPIC MATERIALS. BELTRAMIMICHELL EQUATIONS . . . . . 398 6.3 STRONGLY HETEROGENEOUS MATERIALS AND THERMAL SHOCKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 7 TORSION OF A CYLINDRICAL ROD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 7.1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 402 7.2 SOLUTION: DISPLACEMENT METHOD . . . . . . . . . . . . . . . . . . . 403 7.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 7.4 INVARIANCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 7.5 ROD WITH CIRCULAR CROSS SECTION . . . . . . . . . . . . . . . . . . 409 7.6 SOLUTION BY THE STRESS METHOD . . . . . . . . . . . . . . . . . . . . 412 7.7 YIELD POINT OF THE ROD IN TORSION . . . . . . . . . . . . . . . . . 413 8 SAINT VENANT PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 IX. CLASSIC TOPICS IN THREE-DIMENSIONAL ELASTICITY ........... 429 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 2 TENSIONCOMPRESSION OF A CYLINDRICAL ROD . . . . . . . . . . . . . . . . 436 2.1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 436 2.2 FORM OF THE SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 XIV CONTENTS 2.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 2.4 YIELD POINT OF THE ROD IN SIMPLE TENSION OR COMPRESSION . . . . . . . . . . . . . . . . . 440 2.5 TENSIONTORSION OF A CYLINDRICAL ROD . . . . . . . . . . . . . . 440 3 NORMAL BENDING OF A CYLINDRICAL ROD . . . . . . . . . . . . . . . . . . . . 442 3.1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 442 3.2 FORM OF THE SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 3.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 3.4 YIELD POINT OF THE ROD IN NORMAL BENDING. . . . . . . . . . 450 4 OFF-AXIS BENDING OF A CYLINDRICAL ROD . . . . . . . . . . . . . . . . . . . . 451 4.1 BENDING NORMAL TO THE OY AXIS . . . . . . . . . . . . . . . . . . . 451 4.2 OFF-AXIS BENDING. STATEMENT OF THE PROBLEM . . . . . . . . 452 4.3 FORM OF THE SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 4.4 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 5 BENDING OF A CYLINDRICAL ROD WITH AXIAL LOADING . . . . . . . . . . 454 5.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 5.2 SOLUTION AND REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 5.3 SAINT VENANT PROBLEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 6 ELASTIC EQUILIBRIUM OF A HOLLOW SPHERE UNDER PRESSURE . . . . . 456 6.1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 456 6.2 FORM OF THE SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 6.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 6.4 YIELD POINT OF A HOLLOW SPHERE UNDER PRESSURE . . . . . . 460 7 ELASTIC EQUILIBRIUM OF A CYLINDRICAL TUBE UNDER PRESSURE . . . 461 7.1 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 461 7.2 FORM OF THE SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 7.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 X. VARIATIONAL METHODS IN LINEARISED THERMOELASTICITY ..... 483 1 DIRECT METHODS AND VARIATIONAL METHODS . . . . . . . . . . . . . . . . . 489 1.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 1.2 DIRECT METHODS OF SOLUTION . . . . . . . . . . . . . . . . . . . . . . . 491 1.3 PRESENTATION OF VARIATIONAL METHODS . . . . . . . . . . . . . . . 492 1.4 THE VIRTUAL WORKTHEOREM . . . . . . . . . . . . . . . . . . . . . . . 493 1.5 NOTIONS OF CONVEXITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 1.6 EXPRESSING THE LINEARISED THERMOELASTIC CONSTITUTIVE LAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 2 MINIMUM OF THE POTENTIAL ENERGY . . . . . . . . . . . . . . . . . . . . . . . . 499 2.1 CONVEXITY OF C ( S I ,* D I ) . . . . . . . . . . . . . . . . . . . . . . . . . . 499 2.2 MINIMUM PRINCIPLE FOR DISPLACEMENTS . . . . . . . . . . . . . . 499 2.3 EXPLICIT EXPRESSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 2.4 UNIQUENESS OF THE SOLUTION . . . . . . . . . . . . . . . . . . . . . . . 502 2.5 THERMOELASTIC MATERIAL WITH INTERNAL CONSTRAINTS . . . . 503 CONTENTS XV 3 MINIMUM OF THE COMPLEMENTARY ENERGY . . . . . . . . . . . . . . . . . . 504 3.1 CONVEXITY OF S ( F ,S T I ,T D I ) . . . . . . . . . . . . . . . . . . . . . . . 504 3.2 MINIMUM PRINCIPLE FOR STRESSES . . . . . . . . . . . . . . . . . . . 505 3.3 EXPLICIT EXPRESSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 3.4 UNIQUENESS OF THE SOLUTION . . . . . . . . . . . . . . . . . . . . . . . 508 3.5 COMBINING MINIMUM PRINCIPLES FOR DISPLACEMENTS AND STRESSES . . . . . . . . . . . . . . . . . . . . 509 3.6 THERMOELASTIC MATERIAL WITH INTERNAL CONSTRAINTS . . . . 510 3.7 PRESTRESSED AND PRELOADED INITIAL REFERENCE STATE. . . . 511 4 VARIATIONAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 4.1 CONVERSES TO THE MINIMUM PRINCIPLES . . . . . . . . . . . . . . 512 4.2 VARIATIONAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 5 NATURAL INITIAL STATE. ISOTHERMAL EQUILIBRIUM . . . . . . . . . . . . . . 519 5.1 EXPRESSIONS FOR THE ELASTIC ENERGY . . . . . . . . . . . . . . . . . 519 5.2 CLAPEYRON EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 5.3 EXAMPLE: APPARENT MODULUS OF A HETEROGENEOUS CYLINDER . . . . . . . . . . . . . . . . . . . . . . 522 5.4 THE MAXWELLBETTI RECIPROCITY THEOREM . . . . . . . . . . . 525 6 SELF-EQUILIBRATING STRESS FIELDS. MINIMUM POTENTIAL THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 6.1 SELF-EQUILIBRATING STRESS FIELDS FOR THE PROBLEM . . . . . 526 6.2 MINIMUM COMPLEMENTARY ENERGY AND MINIMUM POTENTIAL THEOREM. . . . . . . . . . . . . . . . . . 529 6.3 REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 7 PARAMETRIC PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 7.1 AIM OF THE STUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 7.2 STATICALLY ADMISSIBLE AND KINEMATICALLY ADMISSIBLE FIELDS FOR THE PARAMETRIC PROBLEM . . . . . . . . . . . . . . . . . 532 7.3 SELF-EQUILIBRATING STRESS FIELDS FOR THE PARAMETRIC PROBLEM . . . . . . . . . . . . . . . . . . . . . . . 535 7.4 EXAMPLES OF PARAMETRIC PROBLEMS . . . . . . . . . . . . . . . . . 536 8 ENERGY THEOREMS FOR PARAMETRIC THERMOELASTIC EQUILIBRIUM 541 8.1 CASTIGLIANOS THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 8.2 MINIMUM POTENTIAL THEOREM . . . . . . . . . . . . . . . . . . . . . 545 8.3 ISOTHERMAL EQUILIBRIUM FROM THE NATURAL INITIAL STATE IN LINEAR ELASTICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 9 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 XI. STATICS OF ONE-DIMENSIONAL MEDIA ....................... 569 1 THE PROBLEM OF ONE-DIMENSIONAL MODELLING. . . . . . . . . . . . . . . 577 2 STATICS OF WIRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 2.1 GEOMETRICAL MODEL. REAL MOTIONS . . . . . . . . . . . . . . . . . 579 2.2 VECTOR SPACE OF VIRTUAL MOTIONS . . . . . . . . . . . . . . . . . . . 580 XVI CONTENTS 2.3 VIRTUAL RATE OF WORKBY EXTERNAL FORCES . . . . . . . . . . . 580 2.4 VIRTUAL RATE OF WORKBY INTERNAL FORCES . . . . . . . . . . . . 582 2.5 EQUILIBRIUM EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 2.6 CONSISTENCY OF THE MODEL. PHYSICAL INTERPRETATION . . . 584 2.7 DISCONTINUITIES IN THE INTERNAL FORCE FIELD . . . . . . . . . . 586 2.8 INTEGRATING THE EQUILIBRIUM EQUATIONS . . . . . . . . . . . . . 587 2.9 DISCONTINUITIES IN THE VIRTUAL VELOCITY FIELD . . . . . . . . 588 2.10 RELEVANCE OF THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 589 3 STATICS OF BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 3.1 GUIDING IDEAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 3.2 GEOMETRICAL MODEL. REAL MOTIONS . . . . . . . . . . . . . . . . . 593 3.3 VECTOR SPACE OF VIRTUAL MOTIONS . . . . . . . . . . . . . . . . . . . 594 3.4 VIRTUAL RATE OF WORKBY EXTERNAL FORCES . . . . . . . . . . . 594 3.5 VIRTUAL RATE OF WORKBY INTERNAL FORCES . . . . . . . . . . . . 596 3.6 EQUILIBRIUM EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 3.7 CONSISTENCY OF THE MODEL. PHYSICAL INTERPRETATION . . . 600 3.8 DISCONTINUITIES IN THE INTERNAL FORCE FIELD . . . . . . . . . . 601 3.9 INTEGRATING THE EQUILIBRIUM EQUATIONS . . . . . . . . . . . . . 602 3.10 DISCONTINUITIES IN THE VIRTUAL MOTION . . . . . . . . . . . . . . 603 3.11 RELEVANCE OF THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 604 3.12 COMPARING ONE-DIMENSIONAL AND THREE-DIMENSIONAL MODELS . . . . . . . . . . . . . . . . . . . . 607 3.13 THE NAVIERBERNOULLI CONDITION . . . . . . . . . . . . . . . . . . . 610 4 STRUCTURES COMPOSED OF ONE-DIMENSIONAL MEMBERS . . . . . . . . 613 4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 4.2 ENDPOINT AND SUPPORT BOUNDARY CONDITIONS . . . . . . . . 614 4.3 BOUNDARY CONDITIONS AT ASSEMBLY JOINTS . . . . . . . . . . . 618 4.4 CONNECTIONS AND SUPPORTS . . . . . . . . . . . . . . . . . . . . . . . . 621 4.5 STATIC ANALYSIS OF STRUCTURES . . . . . . . . . . . . . . . . . . . . . . 621 4.6 KINEMATIC ANALYSIS OF STRUCTURES . . . . . . . . . . . . . . . . . . 623 4.7 PLANE STRUCTURES LOADED IN-PLANE . . . . . . . . . . . . . . . . . 625 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 XII. THERMOELASTIC STRUCTURAL ANALYSIS ....................... 645 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 2 THERMOELASTIC BEHAVIOUR OF THE ONE-DIMENSIONAL MEDIUM . . 652 2.1 FORMULATING A CONSTITUTIVE LAW . . . . . . . . . . . . . . . . . . . 652 2.2 INFINITESIMAL TRANSFORMATION. DISPLACEMENT AND STRAIN DISTRIBUTORS . . . . . . . . . . . . . . 653 2.3 VIRTUAL WORKTHEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 2.4 GUIDING IDEAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 2.5 ISOTHERMAL ELASTIC BEHAVIOUR OF A STRAIGHT BEAM ELEMENT FROM THE NATURAL INITIAL STATE . . . . . . . . . . . . . 656 CONTENTS XVII 2.6 THERMOELASTIC BEHAVIOUR OF A STRAIGHT BEAM ELEMENT IN A PRESTRESSED INITIAL STATE . . . . . . . . . . . . . . . . . . . . . . 663 2.7 THERMOELASTIC BEHAVIOUR OF THE ONE-DIMENSIONAL MEDIUM IN STRUCTURAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 2.8 EXTENSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 3 LINEARISED THERMOELASTIC EQUILIBRIUM OF ONE-DIMENSIONAL STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . 667 3.1 SMALL PERTURBATION HYPOTHESIS . . . . . . . . . . . . . . . . . . . . 667 3.2 STATICALLY DETERMINATE PROBLEMS . . . . . . . . . . . . . . . . . . 668 3.3 STATICALLY INDETERMINATE PROBLEMS . . . . . . . . . . . . . . . . . 668 3.4 MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 3.5 PLANE STRUCTURES LOADED IN-PLANE . . . . . . . . . . . . . . . . . 670 3.6 STRAIGHT BEAMS LOADED IN-PLANE . . . . . . . . . . . . . . . . . . 672 4 EXAMPLE APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 4.1 STATICALLY DETERMINATE PROBLEMS . . . . . . . . . . . . . . . . . . 674 4.2 STATICALLY INDETERMINATE PROBLEMS . . . . . . . . . . . . . . . . . 676 5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 APPENDICES I. ELEMENTS OF TENSOR CALCULUS ............................. 693 1 TENSORS ON A VECTOR SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 1.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 1.2 FIRST RANKTENSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 1.3 SECOND RANKTENSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 2 TENSOR PRODUCT OF TENSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 2.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 2.2 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 2.3 PRODUCT TENSORS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 3 TENSOR COMPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 3.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 3.2 CHANGE OF BASIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 3.3 MIXED SECOND RANKTENSORS . . . . . . . . . . . . . . . . . . . . . . 705 3.4 TWICE CONTRAVARIANT OR TWICE COVARIANT SECOND RANKTENSORS . . . . . . . . . . . 707 3.5 COMPONENTS OF A TENSOR PRODUCT . . . . . . . . . . . . . . . . . . 708 4 CONTRACTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 4.1 DEFINITION OF THE CONTRACTION OF A TENSOR . . . . . . . . . . . 708 4.2 CONTRACTED MULTIPLICATION . . . . . . . . . . . . . . . . . . . . . . . . 709 4.3 DOUBLY CONTRACTED PRODUCT OF TWO TENSORS . . . . . . . . 711 4.4 TOTAL CONTRACTION OF A TENSOR PRODUCT . . . . . . . . . . . . . 713 4.5 DEFINING TENSORS BY DUALITY . . . . . . . . . . . . . . . . . . . . . . 713 4.6 INVARIANTS OF A MIXED SECOND RANKTENSOR . . . . . . . . . . 714 XVIII CONTENTS 5 TENSORS ON A EUCLIDEAN VECTOR SPACE . . . . . . . . . . . . . . . . . . . . . 714 5.1 DEFINITION OF A EUCLIDEAN SPACE . . . . . . . . . . . . . . . . . . . 714 5.2 APPLICATION: DEFORMATION IN A LINEAR MAPPING . . . . . . 715 5.3 ISOMORPHISM BETWEEN E AND E . . . . . . . . . . . . . . . . . . 715 5.4 COVARIANT FORM OF VECTORS IN E . . . . . . . . . . . . . . . . . . . 717 5.5 FIRST RANKEUCLIDEAN TENSORS AND THE CONTRACTED PRODUCT . . . . . . . . . . . . . . . . . . . . . . 718 5.6 SECOND RANKEUCLIDEAN TENSORS OF SIMPLE PRODUCT FORM AND THEIR CONTRACTED PRODUCTS . . . . . . . 719 5.7 SECOND RANKEUCLIDEAN TENSORS . . . . . . . . . . . . . . . . . . . 721 5.8 RANK N EUCLIDEAN TENSORS . . . . . . . . . . . . . . . . . . . . . . . . 726 5.9 CHOICE OF ORTHONORMAL BASIS IN E . . . . . . . . . . . . . . . . . 726 5.10 PRINCIPAL AXES AND PRINCIPAL VALUES OF A REAL SYMMETRIC SECOND RANKEUCLIDEAN TENSOR . 727 6 TENSOR FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 6.1 DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 6.2 DERIVATIVE AND GRADIENT OF A TENSOR FIELD . . . . . . . . . . 729 6.3 DIVERGENCE OF A TENSOR FIELD . . . . . . . . . . . . . . . . . . . . . . 731 6.4 CURVILINEAR COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . . 732 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 II. DIFFERENTIAL OPERATORS: BASIC FORMULAS .................. 741 1 ORTHONORMAL CARTESIAN COORDINATES . . . . . . . . . . . . . . . . . . . . . . 743 1.1 COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 1.2 VECTOR FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 1.3 SCALAR FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 1.4 SECOND RANKTENSOR FIELD . . . . . . . . . . . . . . . . . . . . . . . . 744 2 GENERAL CARTESIAN COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . . 744 2.1 COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 2.2 VECTOR FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 2.3 SCALAR FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 2.4 SECOND RANKTENSOR FIELD . . . . . . . . . . . . . . . . . . . . . . . . 745 3 CYLINDRICAL COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 3.1 PARAMETRISATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 3.2 VECTOR FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 3.3 SCALAR FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 3.4 SYMMETRIC SECOND RANKTENSOR FIELD . . . . . . . . . . . . . . 747 4 SPHERICAL COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 4.1 PARAMETRISATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 4.2 VECTOR FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 4.3 SCALAR FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 4.4 SYMMETRIC SECOND RANKTENSOR FIELD . . . . . . . . . . . . . . 749 CONTENTS XIX III. ELEMENTS OF PLANE ELASTICITY ............................. 751 1 PLANE PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 2 PLANE STRAIN THERMOELASTIC EQUILIBRIUM . . . . . . . . . . . . . . . . . . 757 2.1 PLANE LINEARISED STRAIN TENSOR . . . . . . . . . . . . . . . . . . . . 757 2.2 PLANE STRAIN DISPLACEMENT FIELD. . . . . . . . . . . . . . . . . . . 758 2.3 PLANE STRAIN THERMOELASTIC EQUILIBRIUM IN A HOMOGENEOUS AND ISOTROPIC MATERIAL . . . . . . . . . . . 758 2.4 SOLUTION BY THE DISPLACEMENT METHOD . . . . . . . . . . . . . . 760 2.5 SOLUTION BY THE STRESS METHOD . . . . . . . . . . . . . . . . . . . . 764 2.6 REMARKS ON THE PLANE STRAIN TWO-DIMENSIONAL PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . 767 2.7 TWO-DIMENSIONAL BELTRAMI*MICHELL EQUATION . . . . . . . 767 2.8 BODY FORCES DERIVING FROM A POTENTIAL. AIRY FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 2.9 CYLINDRICAL TUBE UNDER PRESSURE . . . . . . . . . . . . . . . . . . 770 3 PLANE STRESS THERMOELASTIC EQUILIBRIUM . . . . . . . . . . . . . . . . . . 771 3.1 PLANE STRESS TENSOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 3.2 PLANE STRESS FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 3.3 PLANE STRESS THERMOELASTIC EQUILIBRIUM IN A HOMOGENEOUS AND ISOTROPIC MATERIAL . . . . . . . . . . . 771 3.4 SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 3.5 CYLINDRICAL TUBE UNDER PRESSURE . . . . . . . . . . . . . . . . . . 778 SUMMARY OF MAIN FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 BIBLIOGRAPHY .................................................. 783 SUBJECT INDEX ................................................ 789
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spelling	Salençon, Jean 1940- Verfasser (DE-588)122726022 aut Mécanique des milieux continus Handbook of continuum mechanics general concepts, thermoelasticity Jean Salençon Berlin [u.a.] Springer 2001 XIX, 803 S. Ill., graph. Darst. 2 Beil. txt rdacontent n rdamedia nc rdacarrier Physics and astronomy online library Continuum mechanics Kontinuumsmechanik (DE-588)4032296-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Kontinuumsmechanik (DE-588)4032296-8 s DE-604 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009358613&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Salençon, Jean 1940- Handbook of continuum mechanics general concepts, thermoelasticity Continuum mechanics Kontinuumsmechanik (DE-588)4032296-8 gnd
subject_GND	(DE-588)4032296-8 (DE-588)4123623-3
title	Handbook of continuum mechanics general concepts, thermoelasticity
title_alt	Mécanique des milieux continus
title_auth	Handbook of continuum mechanics general concepts, thermoelasticity
title_exact_search	Handbook of continuum mechanics general concepts, thermoelasticity
title_full	Handbook of continuum mechanics general concepts, thermoelasticity Jean Salençon
title_fullStr	Handbook of continuum mechanics general concepts, thermoelasticity Jean Salençon
title_full_unstemmed	Handbook of continuum mechanics general concepts, thermoelasticity Jean Salençon
title_short	Handbook of continuum mechanics
title_sort	handbook of continuum mechanics general concepts thermoelasticity
title_sub	general concepts, thermoelasticity
topic	Continuum mechanics Kontinuumsmechanik (DE-588)4032296-8 gnd
topic_facet	Continuum mechanics Kontinuumsmechanik Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009358613&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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