Lectures on Viscoelasticity Theory This book contains notes for a one semester course on viscoelasticity given in the Division of Applied Mathematics at Brown University. The course serves as an introduction to viscoelasticity and as a workout in the use of various standard mathematical methods. The reader will soon find that he needs to do some work on the side to fill in details that are omitted from the text. These are notes not a completely de­ tailed explanation. Furthermore much of the content of the course is in the prob­ lems assigned for solution by the student. The reader who does not at least try to solve a good many of the problems is likely to miss most of the point. Much that is known about viscoelasticity is not discussed in these notes and references to original sources are usually not given so it will be difficult or impossible to use this book as a reference for looking things up. Readers wanting something more like a treatise should see Ferrys Viscoelastic Properties of Polymers Lodges Elastic Liquids the volumes edited by Eirich on Rheology or any issue of the Transactions of the Society of Rheology. These works emphasize physical aspects of the subject. On the mathematical side Gurtin and Sternbergs long paper On the Linear Theory of Viscoelasticity (ARMA~ 291(1962)) remains the best reference for proofs of theorems. I. Viscoelastic Response in Shear 1. Stress Relaxation 2. Creep 3. Response Functions 4. Models 5. Superposition 6. Tensile Response 7. Relation between Modulus and Compliance 8. Sinusoidal Shearing 9. Nomenclature 10. Energy Storage and Loss 11. Oscillation with Increasing Amplitude II. Fourier and Laplace Transforms 1. Fourier Transforms 2. Two sided Laplace Transforms 3. Laplace Transforms 4. Elementary Formulas 5. Convolutions III. Relations Between Modulus and Compliance. 1. Limits and Moments: Fluids 2. Limits and Moments: Solids 3. Solids and Fluids 4. Scale Invariant Response 5. Approximate Transform Inversion 6. Direct Approximation 7. Approximate Relation between Modulus and Complex Modulus 8. Graphs of Moduli and Compliances IV. Some One Dimensional Dynamical Problems 1. Torsional Oscillations 2. Plane Shear Waves V. Stress Analysis 1. Quasi static Approximation 2. Stress strain Relations 3. Simplest Deformations of Isotropic Materials 4. Simple Tension: Basic Approximation Methods 5. The Correspondence Principle 6. Example: Flat headed Punch 7. Example: Tube under Internal Pressure 8. Incompressible Materials VI. Thermal Effects 1. The Time Temperature Shift Factor 2. Example: Runaway 3. Variable Temperature Histories 4. Example: Simple Tension of a Cooling Hod 5. Thermal Expansion VII. Large Deformations with Small Strains 1. Example: Simple Rotation 2. Example: Torsion 3. Small Distortions with Large Rotations 4. Relative Strain 5. Isotropic Materials 6. Fluids 7. Example: Steady Simple Shearing 8. Example: Oscillatory Shearing 9. Motions with Uniform Velocity Gradient 10. Anisotropic Fluids VIII. Slow Viscoelastic Flow 1. Viscoelastic Flow 2. Flow Diagnosis 3. Slow Viscoelastic Flow: Asymptotic Approximations 4. Slow Viscoelastic Flow: Three dimensional Equations 5. Orders of Approximation in Slow Motion 6. The Rivlin Ericksen Tensors 7. Solution of Problems 8. Ordinary Perturbations 9. A Useful Identity 10. Plane Flow 11. Flow in Tubes IX. Viscometric Flow 1. Stress 2. Example: Channel Flow 3. Slip Surfaces Shear Axes and Shear Rate 4. Dynamical Problems 5. Flow In Tubes 6. Viscometric Equation in Terms of Rivlin Ericksen Tensors 7. Centripetal Effects 8. Boundary Layers.