Nonlinear Effects in Fluids and Solids

This volume of scientific papers is dedicated with gratitude and esteem to Ronald Rivlin and is offered as a token of appreciation by former students, col­ laborators, and friends. Ronald Rivlin's name is synonymous with modem developments in contin­ uum mechanics. His outstanding pioneering theoretical and experimental re­ ·search in finite elasticity is a landmark. From his work there has followed a spate of developments in which he played the leading role-the theory of fiber-rein­ forced materials, the developments of the theory of constitutive equations, the theory of materials with memory, the theory of the fracture of elastomers, the theory of viscoelastic fluids and solids, the development of nonlinear crystal physics, the theory of small deformations superimposed on large, and the effect of large initial strain on wave propagation. It is in Rivlin's work that universal relations were first recognized. Here also are to be found lucid explanations of physical phenomena such as the Poynting effect for elastic rods in torsion. Addi­ tionally, he and his co-workers predicted the presence of secondary flows for viscoelastic fluids in straight pipes of noncircular cross section under a uniform pressure head. While some others may have displayed a cavalier lack of concern for physical reality and an intoxication with mathematical idiom, Rivlin has al­ ways been concerned with genuine mathematical and physical content. All of his papers contain interesting and illuminating material-and may be read with profit by anyone interested in continuum mechanics.

Table of Contents:
1. On Structural Changes in Two-Phase Materials.- 1. Introduction.- 2. Stochastic State-Space and Operators in Z.- 3. Evolution of States during the Transient Period.- 4. Structural Changes of the Aluminum - Resin Composite.- References.- 2. Introduction to Nonlinear Elasticity.- 1. Introduction.- 2. Kinematics of Finite Deformation.- 3. Cauchy Stress Principle and Equations of Motion.- 4. Mechanical Energy Principle and Hyperelasticity.- 5. Change of Frame and Material Frame Indifference.- 6. Material Symmetry Transformations.- 7. Isotropic Hyperelastic Materials.- 8. Blatz - Ko Constitutive Equation.- 9. Incompressible Materials.- 10. Rivlin - Saunders Strain Energy Function.- 11. Inflation Response of a Balloon.- 12. Remarks on Other Kinds of Internal Constraints.- 13. Boundary-Value Problems and Nonuniqueness in Elastostatics.- 14. Universal Inverse Solutions.- 15. Nonuniversal Inverse Solutions: Example.- 16. Class of Universal Relations.- 17. Truesdell’s Problem: Restrictions on Constitutive Equations.- 18. Elastic Stability and Nonuniqueness.- 19. Concluding Remarks.- References.- 3. Propagating and Static Exponential Solutions in a Deformed Mooney - Rivlin Material.- 1. Introduction.- 2. Mooney - Rivlin Materials.- 3. Small Deformations Superimposed on Large.- 4. Propagating and Static Exponential Solutions.- 5. Propagation Condition - Secular Equation.- 6. Homogeneous Waves.- 7. Propagating Evanescent Solutions.- 8. Static Exponential Solutions.- References.- 4. Circularly Polarized Waves of Finite Amplitude in Elastic Dielectrics.- 1. Introduction.- 2. Basic Equations.- 3. Simple Shearing and Polarization.- 4. Circularly Polarized Progressive Waves.- 5. Circularly Polarized Standing Waves.- 6. Approximations.- References.- 5. Static Theory of Point Defectsin Nematic Liquid Crystals.- 1. Introduction.- 2. Basic Theory.- 3. Dirichlet Problems.- 4. Stable Point Defects.- Acknowledgments.- References.- 6. Function Spaces and Fading Memory.- 1. Introduction.- 2. Stability of the Solution of a Volterra Integral Equation.- 3. Cauchy Problem for the Laplace Equation.- 4. Concept of Fading Memory for an Elastic Solid.- References.- 7. On Thermomechanical Formulation of Theories of Continuous Media.- 1. Introduction.- 2. General Thermomechanical Theory.- 3. Thermomechanical Theory of Shells and Plates.- 4. Thermomechanical Theory of Rods.- Acknowledgment.- References.- Supplementary Reference.- 8. Steady Flow of Slightly Viscoelastic Fluids.- 1. Introduction.- 2. Theorem Concerning the Flow of Slightly Viscoelastic Fluids.- 3. Some General Remarks on Flows with Axial Symmetry.- 4. Convergent Flow in a Right-Circular Cone.- References.- Supplementary References.- 9. Some Anomalies in the Structure of Elastic-Plastic Theory at Finite Strain.- 1. Introduction.- 2. Elastic - Plastic Coupling.- 3. Objectivity.- 4. Elastic - Plastic Constitutive Relations.- Acknowledgment.- References.- Supplementary Reference.- 10. Nonlinear Thermoelastic Constitutive Equations for Transversely Isotropic and Orthotropic Materials.- 1. Introduction.- 2. Notation and General Theory.- 3. Constitutive Equations for Transversely Isotropic Materials.- 4. Transverse Isotropy — Simple Shearing Deformations.- 5. Constitutive Equations for Orthotropic Materials.- 6. Orthotropy — Simple Shearing Deformations.- Acknowledgment.- References.- 11. Energy Minimization for Membranes.- 1. Introduction.- 2. Inextensible Networks.- 3. Networks with Shearing Stiffness.- 4. Rubber Membranes.- 5. Stress.- Acknowledgment.- References.- 12. On Boundary Layer-LikeStructures in Finite Thermoelasticity.- 1. Introduction.- 2. Governing Equations.- References.- 13. On the Thickness Limitation for Euler Buckling.- 1. Introduction.- 2. Potential Energy.- 3. Neutral Equilibrium of Critical States.- 4. Further Developments for the General Material.- 5. Asymptotic Analysis for a Thin Plate.- 6. Results for a Neo-Hookean Plate.- 7. Bifurcation Conditions for a Thick Plate.- References.- 14. Irreducible Constitutive Expressions.- 1. Introduction.- 2. Decomposition Procedure.- 3. Generating Functions.- 4. Procedure for Generating Irreducible Expressions.- References.- 15. Determination of the Strain Energy Density Function for Compressible Isotropic Nonlinear Elastic Solids by Torsion-Normal Force Experiments.- 1. Introduction.- 2. Governing Equations.- 3. Change of Variables.- 4. Axial Force, Twisting Moment.- 5. Material Identification by Torsion Experiments.- 6. Summary.- References.