Viscoelastic Structures Mechanics of Growth and Aging

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1 Viscoelastic Structures Mechanics of Growth and Aging Aleksey D. Drozdov Institute for Industrial Mathematics Ben-Gurion University of the Negev Be'ersheba, Israel ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto

2 Kinematics of Continua 1.1 Basic Definitions and Formulas Description of Motion Tangent Vectors The Nabla Operator Deformation Gradient Deformation Tensors and Strain Tensors Stretch Tensors Relative Deformation Tensors Rigid Motion Generalized Strain Tensors Volume Deformation Deformation of the Surface Element Objective Tensors Velocity Vector and Its Gradient Corotational Derivatives The Rivlin-Ericksen tensors Bibliography Constitutive Models in Linear Viscoelasticity 2.1 Differential Constitutive Models Differential Constitutive Models Fractional Differential Models 2.2 Integral Constitutive Models Boltzmann's Superposition Principle Connections Between Creep and Relaxation Measures A Model of Adaptive Links Spectral Presentation of the Function X(t, т) Three-Dimensional Loading 2.3 Creep and Relaxation Kernels Creep and Relaxation Kernels for Nonaging Media Creep and Relaxation Kernels for Aging Media Properties of Creep and Relaxation Measures

3 Xll Contents 2.4 Thermodynamic Potentials and Variational Principles in Linear Viscoelasticity Thermodynamic Potentials of Aging Viscoelastic Media Variational Principles in Viscoelasticity Gibbs' Principle and the Second Law of Thermodynamics Thermodynamic Inequalities in Linear Viscoelasticity A Model of Adaptive Links for Aging Viscoelastic Media A Model of Adaptive Links Validation of the Model Prediction of Stress-Strain Curves for Time-Varying Loads 93 Bibliography 97 Nonlinear Constitutive Models with Small Strains Nonlinear Differential Models Nonlinear Integral Models Uniaxial Loading Three-Dimensional Loading A Model for Crosslinked Polymers A Model of Adaptive Links Determination of Adjustable Parameters Constitutive Equations for Three-Dimensional Loading Correspondence Principles in Nonlinear Viscoelasticity A Model for Non-Crosslinked Polymers A Model of Adaptive Links A Generalized Model of Adaptive Links Validation of the Model 153 Bibliography 161 Nonlinear Constitutive Models with Finite Strains Differential Constitutive Models The Rivlin-Ericksen Model The Kelvin-Voigt Model The Maxwell Model The Standard Viscoelastic Solid Fractional Differential Models, Fractional Differential Operators with Finite Strains Fractional Differential Models Uniaxial Extension of an Incompressible Bar Radial Deformation of a Spherical Shell Uniaxial Extension of a Compressible Bar Simple Shear of a Compressible Medium Integral Constitutive Models Linear Constitutive Equations Constitutive Equations in the Form of Taylor Series 205

4 Contents хш BKZ-Type Constitutive Equations Semilinear Constitutive Equations A Model of Adaptive Links A Model of Adaptive Links The Lagrange Variational Principle Thermodynamic Stability of a Viscoelastic Medium Constitutive Equations for Incompressible Media Extension of a Viscoelastic Bar A Constitutive Model in Finite Viscoelasticity A Model of Adaptive Links Uniaxial Extension of a Viscoelastic Bar Biaxial Extension of a Viscoelastic Sheet Torsion of a Viscoelastic Cylinder 248 Bibliography 255 Constitutive Relations for Thermoviscoelastic Media Constitutive Models in Thermoviscoelasticity Thermorheologically Simple Media The Proportionality Hypothesis The McCrum Model A Model of Adaptive Links in Thermoviscoelasticity Governing Equations A Refined Model of Adaptive Links Constitutive Models for the Nonisothermal Behavior Constitutive Equations for Isothermal Loading Constitutive Equations for Nonisothermal Loading Three-Dimensional Loading The Standard Thermoviscoelastic Solid Cooling of a Cylindrical Pressure Vessel 313 Bibliography 328 Accretion of Aging Viscoelastic Media with Finite Strains Continuous Accretion of Aging Viscoelastic Media A Model for Continuous Accretion Continuous Accretion of a Viscoelastic Cylinder Continuous Accretion of an Elastoplastic Bar Winding of a Cylindrical Pressure Vessel The Lame Problem for an Accreted Cylinder Winding of a Composite Cylinder with Account for Resin Flow Kinematics of Deformation Governing Equations Accretion on a Rigid Mandrel Accretion with Small Strains 406

5 Contents 6.4 Volumetrie Growth of a Viscoelastic Tissue A Brief Historical Survey Constitutive Equations Compression of a Growing Bar The Lame Problem for a Growing Cylinder 430 Bibliography 436 Accretion of Viscoelastic Media with Small Strains Accretion of a Viscoelastic Conic Pipe Formulation of the Problem Kinematics of Accretion Constitutive Equations Governing Equations (Model 1) Governing Equations (Model 2) Numerical Analysis Accretion of a Viscoelastic Spherical Dome Formulation of the Problem Governing Equations Determination of Preload Displacements in an Accreted Dome Numerical Analysis Debonding of Accreted Viscoelastic Beams Accretion of a Two-Layered Beam Accretion of an Elastic Beam on a Nonlinear Winkler Foundation Torsion of an Accreted Elastoplastic Cylinder Formulation of the Problem Stresses and Strains in a Growing Cylinder Accretion of an Elastic Cylinder An Elastoplastic Cylinder with One Plastic Region An Elastoplastic Cylinder with Two Plastic Regions 509 Bibliography 510 Optimization Problems for Growing Viscoelastic Media An Optimal Rate of Accretion for Viscoelastic Solids Torsion of an Accreted Viscoelastic Cylinder With Small Strains Extension of an Accreted Elastic Bar with Finite Strains Optimal Accretion of an Elastic Column Formulation of the Problem and Governing Equations Optimal Regime of Loading Optimal Regime of Accretion Preload Optimization for a Wound Cylindrical Pressure Vessel Formulation of the Problem and Governing Equations 542

6 Contents Winding of a Nonaging Cylindrical Pressure Vessel Winding of an Aging Cylindrical Pressure Vessel Optimal Design of Growing Beams Formulation of the Problem and Governing Equations Optimal Thickness of a Nonaging Elastic Beam Optimal Thickness of an Aging Elastic Beam Optimal Solidification of a Spherical Pressure Vessel Formulation of the Problem Temperature Distribution Stresses and Displacements Stresses in a Pressure Vessel after Cooling Numerical Analysis 582 Bibliography 589 Index 593

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