Mechanical Response of Polymers: An Introduction

by Alan S. Wineman , K. R. Rajagopal

• ISBN13: 9780521643375
• ISBN10: 0521643376
• Format: Hardcover
• Copyright: 2000-06-05
• Publisher: Cambridge University Press

Summary

Automobiles, household appliances, and electronic devices all make use of polymeric materials. As polymers are used increasingly in sophisticated industrial applications, it has become essential that mechanical engineers, who have traditionally focused on the behavior of metals, become as capable and adept with polymers. This text provides a thorough introduction to the subject of polymers from a mechanical engineering perspective, treating stresses and deformations in structural components made of polymers. Three themes are developed. First, the authors discuss the time-dependent response of polymers and its implications for mechanical response. Secondly, descriptions of mechanical response are presented for both time-dependent and frequency-dependent material properties. Finally, the stress-strain-time relation is applied to determine stresses and deformations in structures. With numerous examples and extensive illustrations, this book will help advanced undergraduate and graduate students, as well as practising mechanical engineers, make optimal and effective use of polymeric materials.

Author Biography

Alan S. Wineman is a Professor in the Department of Mechanical Engineering and Applied Mechanics and Member of the Macromolecular Science and Engineering Center, University of Michigan.

Table of Contents

Preface	ix
Discussion of Response of a Viscoelastic Material	1	(27)
Comparison with the Response of Classical Elastic and Classical Viscous Materials	1	(1)
Response of a Classical Elastic Solid	1	(2)
Response of a Classical Viscous Fluid	3	(3)
Comments on Material Microstructure	6	(1)
Response of a Viscoelastic Material	7	(3)
Typical Experimental Results	10	(5)
Material Properties	15	(2)
Linearity of Response	17	(4)
Aging Materials	21	(7)
Problems	24	(4)
Constitutive Equations for One-Dimensional Response of Viscoelastic Materials: Mechanical Analogs	28	(26)
Maxwell Model	28	(7)
Kelvin--Voigt Model	35	(5)
Three-Parameter Solid or Standard Linear Solid	40	(5)
N Maxwell Elements in Parallel	45	(5)
N Kelvin--Voigt Elements in Series	50	(1)
Relaxation and Creep Spectra	51	(3)
Problems	52	(2)
Constitutive Equations for One-Dimensional Linear Response of a Viscoelastic Material	54	(13)
General Restrictions on the Constitutive Equation	54	(4)
Linearity of Response: Superposition of Step Increments	58	(4)
Linearity of Response: Superposition of Pulses	62	(2)
Creep Forms of the Constitutive Equation	64	(1)
Summary of Forms of the Constitutive Equation	64	(3)
Problems	65	(2)
Some Features of the Linear Response of Viscoelastic Materials	67	(21)
Relation Between Relaxation and Creep Functions	67	(5)
Characteristic Creep and Relaxation Times	72	(3)
Characteristic Relaxation, Creep, and Process Times	75	(5)
Some Examples Illustrating Implications of Fading Memory	80	(8)
Problems	83	(5)
Histories with Constant Strain or Stress Rates	88	(27)
Constant Strain Rate Deformation	88	(4)
Constant Strain Rate Deformation and Recovery	92	(5)
Influence of Rise Time T* or Strain Rate α	97	(1)
Work Done in a Constant Strain Rate Deformation and Recovery Test	98	(2)
Repeated Cycles	100	(1)
Step Strain and Recovery	100	(3)
Ramp Strain Approximation to a Step Strain History	103	(2)
Constant Stress Rate Loading and Unloading History	105	(10)
Problems	109	(6)
Sinusoidal Oscillations	115	(33)
Sinusoidal Strain Histories	115	(5)
Sinusoidal Stress Histories	120	(3)
Relation Between G* (ω) and J* (ω)	123	(1)
Work per Cycle During Sinusoidal Oscillations	124	(1)
Complex Viscosity	125	(1)
Examples of Calculation of G* (ω) and J* (ω)	125	(5)
Low and High Frequency Limits of G* (ω) and J* (ω)	130	(5)
Fourier Integral Theorem, Fourier Transform	135	(2)
Expressions for G(t) and J(t) in Terms of G* (ω) and J* (ω)	137	(3)
Work Done During a General Deformation History	140	(8)
Problems	142	(6)
Constitutive Equation for Three-Dimensional Response of Linear Isotropic Viscoelastic Materials	148	(24)
Introduction	148	(1)
Linearity	149	(1)
Uniaxial Extension, Poisson's Ratio, Isotropy	150	(3)
Uniaxial Extension Along the x2 and x3 Directions	153	(1)
Shear Response	153	(1)
Constitutive Equation for Three-Dimensional Response	154	(1)
A Relation Between Poisson's Ratio and the Extensional and Shear Material Properties	155	(2)
Volumetric and Pure Shear Response	157	(3)
Stress in Terms of Strain History	160	(1)
Sinusoidal Oscillations	160	(3)
Laplace Transformation of the Constitutive Equations	163	(1)
Effect of Viscoelasticity on Principal Directions of Stress and Strain	164	(2)
Summary of Constitutive Relations	166	(2)
Relations for Special Cases of Volumetric Response	168	(4)
Problems	168	(4)
Axial Load, Bending, and Torsion	172	(47)
Introduction	172	(1)
Structural Components Under Axial Load	172	(4)
Pure Bending of Viscoelastic Beams	176	(1)
Kinematics of Deformation	177	(2)
Constitutive Equation	179	(1)
Force Analysis	180	(1)
Stress, Bending Moment, and Curvature Relations	181	(1)
Deformation of Beams Subjected to Transverse Loads	182	(3)
Beams on Hard Supports, Correspondence Principle	185	(5)
Delayed Contact, Direct Method of Solution	190	(5)
Interaction of Polymeric Structural Components, a Viscoelastic Beam on a Viscoelastic Support	195	(4)
Extrusion of a Bar, Tracking the History of a Material Element	199	(3)
Traveling Concentrated Load on a Beam	202	(5)
Torsion of Circular Bars	207	(5)
Analysis of Viscoelastic Structures	212	(7)
Problems	212	(7)
Dynamics of Bodies with Viscoelastic Support	219	(13)
Introduction	219	(1)
Comparison of Spring--Damper and Viscoelastic Supports	219	(2)
Forced Oscillations	221	(7)
Free Oscillations	228	(4)
Problem	231	(1)
Boundary Value Problems for Linear Isotropic Viscoelastic Materials	232	(15)
Introduction	232	(1)
Governing Equations	232	(2)
Correspondence Theorem for Quasi-Static Motion	234	(3)
Breakdown of the Correspondence Principle	237	(1)
Application of the Correspondence Principle: Pressure Loading of a Viscoelastic Cylinder	238	(2)
Application of the Correspondence Principle: Torsion of Bars of Non-Circular Cross-Section	240	(3)
Direct Solution Methods	243	(4)
Problem	246	(1)
Influence of Temperature	247	(46)
Introduction	247	(1)
Thermally Induced Dimensional changes	247	(1)
Mechanical Response at Different Temperatures	248	(3)
Time--Temperature Superposition	251	(2)
Experimental Support for Time--Temperature Superposition	253	(1)
General Comments	254	(2)
Effect of Temperature on Characteristic Stress Relaxation Time	256	(1)
Other Material Property Functions	257	(1)
Implications of Time--Temperature Superposition for Processes	258	(1)
Rate of Work	259	(1)
An Experimental Study	260	(2)
Extension to Time-Varying Temperature Histories	262	(6)
Constitutive Equation for Time-Varying Temperature Histories	268	(1)
Thermo-Viscoelastic Response of a Three Bar Structure: Formulation	269	(2)
Thermo-Viscoelastic Response of a Three Bar Structure: Development of Frozen-in Deformation	271	(5)
Thermo-Viscoelastic Response of a Three Bar Structure: Frozen-in Forces	276	(9)
Thermo-Viscoelastic Response of a Three Bar Structure: Cooling Induced Warping	285	(7)
Thermo-Viscoelastic Response of a Three Bar Structure: Comments	292	(1)
Appendix A Operator Notation for Time Derivatives	293	(2)
Appendix B Laplace Transform	295	(4)
Appendix C Volterra Integral Equations	299	(6)
Appendix D Formal Manipulation Methods	305	(3)
Appendix E Field Equations in Cartesian and Cylindrical Coordinates	308	(3)
References	311	(2)
Index	313