Engineering viscoelasticity

Engineering Viscoelasticity covers all aspects of the thermo- mechanical response of viscoelastic substances that a practitioner in the field of viscoelasticity would need to design experiments, interpret test data, develop stress-strain models, perform stress analyses, design structural components, and carry out research work. The material in each chapter is developed from the elementary to the esoteric, providing the background in mathematics and mechanics that are central to understanding the subject matter being presented. This book also examines how viscoelastic materials respond to the application of loads, and provides practical guidelines to use them in the design of commercial, military and industrial applications. Preface 6 Acknowledgments 8 Contents 9 1 Fundamental Aspects of Viscoelastic Response 14 Abstract 14 1.1...Introduction 14 1.2...The Nature of AmorphousAmorphous PolymersPolymers 16 1.3...Mechanical Response of ViscoelasticViscoelastic Materials 17 1.3.1 Material Response to Step-StrainStrain Loading 18 1.3.2 Material Response to Step-StressStress Loading 20 1.3.3 Material Response to Cyclic StrainStrain Loading 23 1.3.4 Material Response to Constant StrainStrain Rate Loading 24 1.4...EnergyEnergy Storage and Dissipation 25 1.5...GlassGlass TransitionTransition and Regions of ViscoelasticViscoelastic Behavior 29 1.6...AgingAging of ViscoelasticViscoelastic Materials 33 References 34 2 Constitutive Equations in Hereditary Integral Form 35 Abstract 35 2.1...Introduction 36 2.2...Boltzmann’s Superposition Principle 37 2.3...PrinciplePrinciple of FadingFading MemoryMemory 42 2.4...Closed-CycleCycle Condition 46 2.5...Relationship Between Modulus and Compliance 49 2.5.1 Elastic Relationships 50 2.5.2 ConvolutionConvolution Integral Relationships 51 2.5.3 LaplaceLaplace-Transformed Relationships 53 2.6...Alternate Integral Forms 55 2.7...Work and Energy 58 2.8...Problems 60 References 64 3 Constitutive Equations in Differential Operator Form 65 Abstract 65 3.1...Introduction 66 3.2...Fundamental Rheological ModelsModels 66 3.2.1 Linear Elastic SpringSpring 67 3.2.1.1 Constitutive Equation 67 3.2.1.2 Response to Strain Loading 68 3.2.1.3 Response to StressStress Loading 68 3.2.2 Linear Viscous DashpotDashpot 69 3.2.2.1 Constitutive Equation 69 3.2.2.2 Response to Strain Loading 70 3.2.2.3 Response to StressStress Loading: 70 3.3...Rheological Operators 71 3.3.1 Fundamental Rheological Operators 72 3.3.2 General Rheological Operators 73 3.3.3 Rheological Equations in LaplaceLaplace Transformed Space 75 3.3.4 Initial Conditions for Rheological ModelsModels 75 3.4...Construction of Rheological ModelsModels 79 3.5...Simple Rheological ModelsModels 81 3.5.1 KelvinKelvin--Voigt SolidSolid 81 3.5.1.1 Constitutive Equation 82 3.5.1.2 Response to StrainStrain Loading 83 3.5.1.3 Response to StressStress Loading 83 3.5.2 MaxwellMaxwell--Wiechert FluidFluid 86 3.5.2.1 Constitutive Equation 87 3.5.2.2 Response to StrainStrain Loading 87 3.5.2.3 Response to Stress Loading 89 3.6...Generalized ModelsModels 91 3.6.1 Generalized KelvinKelvin ModelModel 91 3.6.2 Generalized MaxwellMaxwell ModelModel 93 3.7...Composite ModelsModels 95 3.7.1 Standard Linear SolidSolid 96 3.7.1.1 Response to StrainStrain Loading 97 3.7.1.2 Response to StressStress Loading 99 3.7.2 Three-Parameter FluidFluid 100 3.8...Problems 101 References 103 4 Constitutive Equations for Steady-State Oscillations 104 Abstract 104 4.1...Introduction 104 4.2...Steady-State Constitutive Equations from Integral Constitutive Equations 105 4.3...Steady-StateSteady-State ConstitutiveConstitutive Equations from DifferentialDifferential Constitutive Equations 113 4.4...Limiting Behavior of ComplexComplex Property Functions 116 4.5...EnergyEnergy Dissipation 120 4.6...Problems 121 References 123 5 Structural Mechanics 124 Abstract 124 5.1...Introduction 124 5.2...BendingBending 126 5.2.1 HereditaryHereditary IntegralIntegral Models 128 5.2.2 DifferentialDifferential Operator Models 129 5.2.3 Models for Steady-StateSteady-State Oscillations 130 5.3...TorsionTorsion 131 5.3.1 HereditaryHereditary IntegralIntegral Models 132 5.3.2 DifferentialDifferential Operator Models 133 5.3.3 Models for Steady-StateSteady-State Oscillations 134 5.4...Column BucklingBuckling 135 5.4.1 HereditaryHereditary IntegralIntegral Models 136 5.4.2 DifferentialDifferential Operator Models 138 5.5...Viscoelastic SpringsSprings 139 5.5.1 Axial SpringSpring 140 5.5.2 Shear SpringSpring 142 5.5.3 BendingBending SpringSpring 143 5.5.4 TorsionTorsion SpringSpring 145 5.6...Elastic--Viscoelastic CorrespondenceCorrespondence 147 5.7...Mechanical VibrationsVibrations 149 5.7.1 ForcedForced VibrationsVibrations 150 5.7.2 FreeFree VibrationsVibrations 155 5.8...Problems 156 References 159 6 Temperature Effects 160 Abstract 160 6.1...Introduction 160 6.2...TimeTime TemperatureTemperature Superposition 161 6.3...Phenomenology of the Glass TransitionTransition 168 6.4...Effect of PressurePressure on the Glass TransitionTransition TemperatureTemperature 170 6.5...HygrothermalHygrothermal Strains 171 6.6...Problems 172 ReferencesReferences 174 7 Material Property Functions and Their Characterization 175 Abstract 175 7.1...Introduction 176 7.2...Experimental CharacterizationCharacterization 176 7.2.1 Constant Strain TestTest 176 7.2.2 Constant Stress TestTest 179 7.2.3 Constant Rate TestTest 179 7.2.4 Dynamic TestsTests 180 7.3...AnalyticalAnalytical Forms of ConstitutiveConstitutive Functions 180 7.3.1 Material Property FunctionsProperty Functions in PronyProny Series Form 180 7.3.2 Material Property FunctionsProperty Functions in PowerPower-LawLaw Form 182 7.4...InversionInversion of Material Property FunctionsProperty Functions 182 7.4.1 Approximate InversionInversion of Material Property FunctionsProperty Functions 183 7.4.2 Exact InversionInversion of Material Property FunctionsProperty Functions 185 7.5...NumericalNumerical CharacterizationCharacterization of Material Property FunctionsProperty Functions 189 7.5.1 NumericalNumerical CharacterizationCharacterization of the Shift FunctionShift Function 190 7.5.2 NumericalNumerical CharacterizationCharacterization of Modulus and ComplianceCompliance 192 7.6...Source Data Files and Computer Application 197 7.6.1 Source Data Files 197 7.6.2 Computer Application 197 7.7...Problems 198 References 201 8 Three-Dimensional Constitutive Equations 202 Abstract 202 8.1...Introduction 202 8.2...Background and Notation 203 8.3...Constitutive Equations for AnisotropicAnisotropic Materials 205 8.4...Constitutive Equations for OrthotropicOrthotropic Materials 208 8.5...Constitutive Equations in Integral TransformTransform Space 211 8.6...Integral Constitutive Equations for Isotropic Materials 213 8.7...Constitutive Equations for IsotropicIsotropic IncompressibleIncompressible Materials 219 8.8...Differential Constitutive Equations for IsotropicIsotropic Materials 220 8.9...Problems 222 ReferencesReferences 226 9 Isothermal Boundary-Value Problems 227 Abstract 227 9.1...Introduction 227 9.2...Differential Equations of MotionMotion 228 9.2.1 Balance of Linear MomentumMomentum 228 9.2.2 BalanceBalance of Angular MomentumMomentum 229 9.3...General Boundary-Value Problem 229 9.4...Quasi-StaticQuasi-Static Approximation 231 9.5...Classification of Boundary-Value Problems 232 9.5.1 Traction Boundary-ValueBoundary-Value Problem 232 9.5.2 Displacement Boundary-ValueBoundary-Value Problem 232 9.5.3 Mixed Boundary-ValueBoundary-Value Problem 233 9.6...Incompressible Materials 234 9.7...Materials with SynchronousSynchronous Moduli 235 9.8...Separation of Variables in the Time Domain 236 9.9...Integral-Transform Correspondence Principles 239 9.10...The Poisson’s Ratio of Isotropic Viscoelastic Solids 242 9.11...Problems 243 References 246 10 Wave Propagation 247 Abstract 247 10.1...Introduction 247 10.2...HarmonicHarmonic WavesWaves 248 10.2.1 Materials of Integral Type 249 10.2.2 Materials of Differential Type 250 10.3...ShockShock Waves 254 10.3.1 Materials of Integral Type 257 10.3.2 Materials of Differential Type 262 10.4...Problems 265 References 267 11 Variational Principles and Energy Theorems 268 Abstract 268 11.1...Introduction 268 11.2...VariationVariation of a FunctionalFunctional 269 11.3...Variational Principles of Instantaneous Type 270 11.3.1 First CastiglianoCastigliano-Type Principle 271 11.3.2 Second CastiglianoCastigliano-Type Principle 275 11.3.3 Unit Load Theorem 280 11.4...ReciprocalReciprocal Theorems 286 11.4.1 Static Conditions 286 11.4.2 Dynamic Conditions 287 11.5...Problems 289 ReferencesReferences 289 Appendix A Mathematical Background 290 Appendix B Elements of Solid Mechanics 331 Index 357 Engineering Viscoelasticity covers all aspects of the thermo- mechanical response of viscoelastic substances that a practitioner in the field of viscoelasticity would need to design experiments, interpret test data, develop stress-strain models, perform stress analyses, design structural components, and carry out research work. The material in each chapter is developed from the elementary to the advanced, providing the background in mathematics and mechanics that are central to understanding the subject matter being presented. The book examines how viscoelastic materials respond to the application of loads, and provides practical guidelines to use them in the design of commercial, military and industrial applications. This book also: · Facilitates conceptual understanding by progressing in each chapter from elementary to challenging material · Examines in detail both differential and integral constitutive equations, devoting full chapters to each type and using both forms in applications throughout the book · Presents variational principles and energy theorems from a practical point of view, using these to develop computational tools such as the viscoelastic equivalent of the Unit-Load and the Unit-Displacement theorems · Provides more than 50 examples, solved in detail, and complete solution guides (hints) to all assigned problems. · Features code segments to curve-fit WLF time-temperature shift and Dirichlet-Prony functions, and a computer application to obtain the convolution inverse of Prony series. Engineering Viscoelasticity is as ideal book for first graduate-level courses on viscoelasticity, upper-level undergraduate level courses in engineering, and practicing professionals Front Matter....Pages i-xv Fundamental Aspects of Viscoelastic Response....Pages 1-21 Constitutive Equations in Hereditary Integral Form....Pages 23-52 Constitutive Equations in Differential Operator Form....Pages 53-91 Constitutive Equations for Steady-State Oscillations....Pages 93-112 Structural Mechanics....Pages 113-148 Temperature Effects....Pages 149-163 Material Property Functions and Their Characterization....Pages 165-191 Three-Dimensional Constitutive Equations....Pages 193-217 Isothermal Boundary-Value Problems....Pages 219-238 Wave Propagation....Pages 239-259 Variational Principles and Energy Theorems....Pages 261-282 Errata to: Engineering Viscoelasticity....Pages E1-E20 Back Matter....Pages 283-353