Mechanics of Viscoelastic Solids Aleksey D. Drozdov Institute for Industrial Mathematics, Beersheba, Israel JOHN WILEY & SONS Chichester New York Weinheim Brisbane Toronto Singapore
Contents Preface xi 1 Constitutive Models in Linear Viscoelasticity 1 1.1 THE BOLTZMANN SUPERPOSITION PRINCIPLE 2 1.2 THE SHORT-TERM RESPONSE IN PHYSICALLY AGED POLYMERS 8 1.2.1 A MODEL OF ADAPTIVE LINKS 9 1.2.2 A BALANCE EQUATION FOR THE NUMBER OF LINKS 10 1.2.3 COMPARISON WITH EXPERIMENTS 16 1.2.4 TIME-VARYING LOADING 18 1.3 THE LONG-TERM RESPONSE IN PHYSICALLY AGED POLYMERS 23 1.3.1 A MODEL OF ADAPTIVE LINKS 25 1.3.2 A FREE VOLUME CONCEPT 29 1.3.3 CONSTITUTIVE EQUATIONS 41 1.3.4 COMPARISON WITH EXPERIMENTS 47 1.4 CONSTITUTIVE RELATIONS FOR THREE-DIMENSIONAL LOADING 56 1.5 THERMODYNAMIC POTENTIALS OF A TRANSIENT NETWORK 63 1.5.1 STRAIN ENERGY DENSITY 63 1.5.2 THERMODYNAMIC POTENTIALS AND CONSTI TUTIVE EQUATIONS 69 1.5.3 PROPERTIES OF RELAXATION FUNCTIONS... 73 1.6 VARIATIONAL PRINCIPLES IN VISCOELASTICITY... 83 1.6.1 VARIATIONAL PRINCIPLES AND GOVERNING EQUATIONS 83 1.6.2 GIBBS' PRINCIPLE AND THE SECOND LAW OF THERMODYNAMICS 88 1.6.3 DEBONDING OF A STRIP LINKED TO A RIGID FOUNDATION BY A VISCOELASTIC ADHESIVE LAYER 90
viii CONTENTS 1.7 PHASE TRANSITIONS IN VISCOELASTIC MEDIA 106 1.7.1 FORMULATION OF THE PROBLEM 106 1.7.2 AN INTEGRAL OVER A VARYING DOMAIN... 110 1.7.3 INTERFACIAL EQUILIBRIUM EQUATIONS 114 1.7.4 LOCAL MELTING OF A VISCOELASTIC MEDIUM. 120 References 125 2 Constitutive Models in Nonlinear Viscoelasticity... 135 2.1 A TRANSIENT NETWORK WITH NONLINEAR LINKS.. 136 2.1.1 CONSTITUTIVE EQUATIONS 140 2.1.2 COMPARISON WITH EXPERIMENTS 145 2.1.3 THREE-DIMENSIONAL LOADING 148 2.1.4 TORSION OF A CONICAL PIPE 149 2.1.5 THE CORRESPONDENCE PRINCIPLES 155 2.2 A NETWORK WITH AN EYRING-TYPE CLOCK 157 2.2.1 STRESS-STRAIN RELATIONS 159 2.2.2 THE GENERALIZED ARRHENIUS EQUATION... 161 2.2.3 COMPARISON WITH EXPERIMENTS 163 2.3 A NETWORK WITH AN ENTROPY-DRIVEN CLOCK... 172 2.3.1 THERMODYNAMIC POTENTIALS 174 2.3.2 A DISSIPATION-DRIVEN INTERNAL CLOCK... 177 2.3.3 COMPARISON WITH EXPERIMENTS 180 2.4 CONSTITUTIVE RELATIONS IN VISCOELASTOPLAS- TICITY 189 2.4.1 A MODEL OF ADAPTIVE LINKS 189 2.4.2 COMPARISON WITH EXPERIMENTS 196 References 203 3 Constitutive Models in Finite Viscoelasticity 209 3.1 OPERATOR LINEAR CONSTITUTIVE RELATIONS...210 3.1.1 KINETIC EQUATIONS 210 3.1.2 STRAIN ENERGY DENSITY 211 3.1.3 THERMODYNAMIC POTENTIALS 217 3.1.4 COMPARISON WITH EXPERIMENTS 221 3.1.5 THE PRINCIPLE OF MINIMUM FREE ENERGY.. 225 3.1.6 THERMODYNAMIC STABILITY 233 3.2 THE MOONEY-RIVLIN MODEL 237 3.2.1 DIFFERENTIAL CONSTITUTIVE EQUATIONS... 237 3.2.2 UNIAXIAL EXTENSION 239 3.2.3 SIMPLE SHEAR 245 3.2.4 TORSION OF A CIRCULAR CYLINDER 248 3.3 STRAIN ENERGY DENSITY OF A NETWORK 254 3.3.1 UNIAXIAL EXTENSION 256
CONTENTS ix 3.3.2 BIAXIAL EXTENSION 259 3.4 CONSTITUTIVE EQUATIONS BASED ON THE FREE VOLUME CONCEPT 271 3.4.1 THE TIME-STRAIN SUPERPOSITION PRINCIPLE 273 3.4.2 A NEO-HOOKEAN VISCOELASTIC MEDIUM... 275 3.4.3 UNIAXIAL EXTENSION 276 3.4.4 SIMPLE SHEAR 284 3.4.5 RADIAL DEFORMATION OF A SPHERICAL SHELL 289 3.5 A NETWORK WITH AN ENTROPY-DRIVEN CLOCK... 299 3.5.1 A CONCEPT OF TRANSIENT NETWORKS 300 3.5.2 THERMODYNAMIC POTENTIALS 313 3.5.3 SIMPLE SHEAR 317 References 323 4 Constitutive Models in Thermoviscoelasticity 329 4.1 THERMORHEOLOGICALLY SIMPLE MEDIA 330 4.1.1 TIME-TEMPERATURE SUPERPOSITION PRINCI PLE '. 331 4.1.2 THERMAL SHIFT FACTOR. 332 4.1.3 NONISOTHERMAL LOADING 337 4.2 THERMORHEOLOGICALLY COMPLEX MEDIA 340 4.2.1 BASIC HYPOTHESES 340 4.2.2 A CONCEPT OF TRANSIENT NETWORKS (MODEL I) 343 4.2.3 A CONCEPT OF TRANSIENT NETWORKS (MODEL I) 351 4.2.4 NONISOTHERMAL LOADING 358 4.3 THE NONISOTHERMAL RESPONSE IN NONAGING POLYMERS 364 4.3.1 THREE-DIMENSIONAL LOADING 371 4.3.2 THE STANDARD THERMOVISCOELASTIC SOLID. 372 4.3.3 COOLING OF A RECTILINEAR BAR 378 4.3.4 COOLING OF A CYLINDRICAL SHELL 382 4.4 LINEAR MEDIA SUBJECTED TO PHYSICAL AGING... 400 4.4.1 CONSTITUTIVE EQUATIONS 401 4.4.2 COOLING OF A SPHERICAL PRESSURE VESSEL. 404 4.5 NONLINEAR MEDIA SUBJECTED TO PHYSICAL AGING 414 4.5.1 STRAIN ENERGY DENSITY OF A NETWORK (ISOTHERMAL LOADING) 415 4.5.2 STRAIN ENERGY DENSITY OF A NETWORK (NONISOTHERMAL LOADING) 417 4.5.3 THERMODYNAMIC POTENTIALS 419 4.5.4 CONSTITUTIVE RELATIONS (FINITE STRAINS). 421 4.5.5 CONSTITUTIVE RELATIONS (SMALL STRAINS). 425
x CONTENTS 4.5.6 CONSTITUTIVE RELATIONS FOR PHYSICALLY AGED POLYMERS 427 4.5.7 THE AGING SHIFT FACTOR 434 4.5.8 UNIAXIAL EXTENSION 439 4.5.9 COOLING OF A PLATE 446 References 459 Index 465