Topics in Nonlinear Physics

InhaltsangabeNonlinear Problems in Physics.- Nonlinear Problems in Physics.- The Nonlinear Field Theories in Mechanics.- 1. Thermodynamics of Homogeneous Processes.- Temperature and Heat.- Homogeneous Processes.- The Thermodynamic State.- Thermodynamic Processes.- Histories.- Thermodynamic Constitutive Equations.- Example 1. The Classical Caloric Equations of State.- 2. Kinematics. Changes of Frame.- Geometry.- Primitive Elements of Thermomechanics.- Bodies, Configurations, Motions.- Mass Density.- Reference Configuration.- Deformation Gradient.- Change of Reference Configuration.- Current Configuration as Reference.- Stretch and Rotation.- Stretching and Spin.- Changes of Frame.- 3. Force and Work, Laws of Motion.- Physical Principles of Mechanics.- Forces and Moments in Continuum Mechanics.- Momentum and Moment of Momentum.- The Euler-Cauchy Stress Principle.- Energy and Heat Work.- Equations of Balance.- Differential Equations of Continuum Mechanics.- 4. Constitutive Equations. Simple Materials.- Nature of Constitutive Equations.- Equivalent Processes.- Constitutive Equations.- Axioms for Constitutive Equations.- Simple Materials.- Exercise 4.1 (Zaremba-Noll Theorem).- Reduction for Material Frame-Indifference.- Internal Constraints.- 5. The Isotropy Group.- Isotropy Group.- Orthogonal Part of the Isotropy Group.- Change of Reference Configuration.- 6. Solids, Isotropic Materials, Fluids, Fluid Crystals.- Main Properties of the Isotropy Group.- Isotropic Materials.- Solids.- Fluids.- Fluid Crystals.- 7. Motions with Constant Stretch History.- Definition.- Noll's Theorem.- Determination of the Deformation History from the First Three Rivlin-Ericksen Tensors.- Equivalence of Simple Materials with Rivlin-Ericksen Materials in Motions with Constant Stretch History.- Classification of Motions with Constant Stretch History.- 8. The Stress System in Viscometric Flows of Incompressible Fluids.- Recapitulation.- Functional Equations for the Response Function.- Illustration by Means of Shearing Flow.- Consequences of Invariance Under Reflections.- The Viscometric Functions. Normal-Stress Effects.- Position of the Classical Theory.- 9. Dynamical Conditions in Viscometric Flows.- Recapitulation.- Dynamic Compatibility.- Shearing Flow.- Channel Flow.- Helical Flows.- Flow Between Rotating Cylinders.- Flow in a Circular Pipe.- Other Viscometric Flows.- 10. Impossibility of Rectilinear Plow in Pipes.- Problem.- Explicit Constitutive Equation.- Dynamical Equation.- Compatibility.- 11. Elastic Materials.- Statics of Simple Materials.- Reduction for Frame-Indifference.- Elastic Fluids.- Natural States.- Isotropic Elastic Materials.- Unconstrained Bodies.- Incompressible Bodies.- Remarks.- 12. Non-Homogeneous Universal Solutions for Incompressible Elastic Bodies.- Universal Solutions.- List of the Universal Solutions.- Inflation or Eversion, Torsion, and Extension of a Cylinder.- Torsion and Tension of a Solid Cylinder.- Eversion.- Remarks on Method.- 13. Hyperelastic Materials.- The Piola-Kirchhoff Stress Tensor.- Definition of a Hyperelastic Material.- The Two Isotropy Groups.- Minimum of the Stored-Energy Function.- Need for an A Priori Inequality.- Coleman and Noll's Inequality.- 14. Work Theorems in Hyperelasticity.- Nonsense About Perpetual Motion.- Virtual Work of the Traction on the Boundary.- Homogeneous Processes in Homogeneous Bodies.- The First Work Theorem.- The Second Work Theorem.- The Third Work Theorem.- 15. Thermomechanics. Equipresence.- Fading Memory.- Thermodynamics and Continuum Mechanics Reviewed.- General Principles of the Thermomechanics of Continua.- Thermodynamic Process.- Constitutive Equations. Equipresence.- The Clausius-Duhem Inequality.- The Reduced Dissipation Inequality.- Simple Thermodynamic Materials.- 16. Thermodynamics of Simple Materials with Fading Memory.- Quasi-Elastic Response. Fading Memory.- Dissipation and Quasi-Elastic Response.- Applications.- A Final Remark on the Stored-Energy Function in Elasticity.- 17.