The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction

One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity. J. Willard Gibbs This book is an outgrowth of lectures I have given, on and off over some sixteen years, in graduate courses at the California Institute of Technology, and, in abbreviated form, elsewhere. It is, nevertheless, not meant to be a textbook. I have aimed at a full exposition of the phenomenological theory of linear viscoelastic behavior for the use of the practicing scientist or engineer as well as the academic teacher or student. The book is thus primarily a reference work. In accord with the motto above, I have chosen to describe the theory of linear viscoelastic behavior through the use of the Laplace transformation. The treatment oflinear time-dependent systems in terms of the Laplace transforms of the relations between the excitation add response variables has by now become commonplace in other fields. With some notable exceptions, it has not been widely used in viscoelasticity. I hope that the reader will find this approach useful.

Table of Contents:
1. Introductory Concepts.- 1.0 Introduction.- 1.1 Constitutive Equations.- 1.2 Stress.- 1.3 Strain and Rate of Strain.- 1.4 Purely Elastic Linear Response.- 1.4.1 Single-Plane Symmetry.- 1.4.2 Three-Plane Symmetry or Orthotropy.- 1.4.3 Axisymmetry, or Transverse Isotropy.- 1.4.4 Isotropy.- 1.4.4.1 The Generalized Hooke’s Law and the Elastic Moduli..- 1.4.4.2 The Generalized Hooke’s Law and the Elastic Compliances.- 1.5 Purely Viscous Linear Response.- 1.5.1 The Generalized Newton’s Laws.- 1.6 Problems.- References.- 2. Linear Viscoelastic Response.- 2.0 Introduction.- 2.1 Linear Time-dependent Behavior.- 2.1.1 Differential Representations: The Operator Equation.- 2.1.2 Integral Representation. The Boltzmann Superposition Integrals.- 2.1.3 Excitation and Response in the Transform Plane.- 2.2 The Impulse Response Functions.- 2.3 The Step Response Functions.- 2.4 The Slope Response Functions.- 2.5 The Harmonic Response Functions.- 2.6 Excitation and Response in the Time Domain.- 2.7 Problems.- References.- 3. Representation of Linear Viscoelastic Behavior by Series-Parallel Models.- 3.0 Introduction.- 3.1 The Theory of Model Representation.- 3.1.1 The Elements of Electric Circuit Analysis.- 3.1.2 The Elements of Mechanical Model Analysis.- 3.2 Electromechanical Analogies.- 3.3 The Elementary Rheological Models.- 3.3.1 The Spring and Dashpot Elements.- 3.3.2 The Maxwell and Voigt Units.- 3.4 Models with the Minimum Number of Elements.- 3.4.1 The Standard Three- and Four-Parameter Series-Parallel Models.- 3.4.1.1 The Models of the Standard Linear Solid.- 3.4.1.2 The Models of the Standard Linear Liquid.- 3.4.2 The Non-Standard Three- and Four-Parameter Series-Parallel Models.- 3.4.2.1 The Non-Standard Three-Parameter Models.- 3.4.2.2 The Non-Standard Four-Parameter Models.- 3.4.3 Other Four-Parameter Models.- 3.4.4 Behavior of the Standard Models in the Complex Plane. Initial and Final Values.- 3.4.5 Response of the Standard Models to the Standard Excitations.- 3.5 Models with Large Numbers of Elements.- 3.5.1 The Generalized Series-Parallel Models.- 3.5.1.1 The Wiechert and Kelvin Models.- 3.5.1.2 Spectral Strength and Time Dependence in Wiechert and Kelvin Models.- 3.5.1.3 The Canonical Models.- 3.5.2 Non-Standard Series-Parallel Models.- 3.5.3 Non-Series-Parallel Models with Large Numbers of Elements..- 3.6 Model Fitting.- 3.6.1 Procedure X.- 3.6.2 Collocation Method.- 3.6.3 Multidata Method.- 3.6.4 Algorithm of Emri and Tschoegl.- 3.7 Series-Parallel Models and the Operator Equation.- 3.8 Problems.- References.- 4. Representation of Linear Viscoelastic Behavior by Spectral Response Functions.- 4.0 Introduction.- 4.1 The Continuous Spectral Response Functions.- 4.1.1 Continuous Time Spectra.- 4.1.2 Continuous Frequency Spectra.- 4.1.3 Determination of the Continuous Spectra.- 4.2 Methods for Deriving the Continuous Spectra from the Step Responses.- 4.2.1 The Transform Inversion Method: The Approximations of Schwarzl and Staverman.- 4.2.2 The Differential Operator Method.- 4.2.3 The Power Law Method: The Approximations of Williams and Ferry.- 4.2.4 The Finite Difference Operator Method: The Approximations of Yasuda and Ninomiya, and of Tschoegl.- 4.3 Methods for Deriving the Continuous Spectra from the Harmonic Responses.- 4.3.1 Differential Operator Method: The Approximations of Schwarzl and Staverman, and of Tschoegl.- 4.3.1.1 Approximations Derived from the Storage Functions.- 4.3.1.2 Approximations Derived from the Loss Functions.- 4.3.2 The Transform Inversion Method.- 4.3.3 The Power Law Method: The Approximations of Williams and Ferry.- 4.3.4 The Finite Difference Operator Method: The Approximations of Ninomiya and Ferry, and of Tschoegl.- 4.3.4.1 Approximations Derived from the Storage Functions.- 4.3.4.2 Approximations Derived from the Loss Functions.- 4.4 Comparison of the Approximation to the Continuous Spectra.- 4.5 The Discrete Spectral Response Functions.- 4.6 The Viscoelastic Constants.- 4.7 Problems.- References.- 5. Representation of Linear Viscoelastic Behavior by Ladder Models.- 5.0 Introduction.- 5.1 General Ladder Models.- 5.2 Regular Ladder Models with a Finite Number of Elements: The Gross-Marvin Models.- 5.2.1 The Respondances of the Gross-Marvin Models.- 5.2.2 The Equivalent Series-Parallel Models.- 5.2.2.1 Limit Values.- 5.2.2.2 Poles and Residues.- 5.2.2.3 Respondances.- 5.3 Regular Ladder Models with a Finite Number of Elements: The Regular Converse Ladder Models.- 5.3.1 The Respondances of the Regular Converse Ladder Models.- 5.3.2 The Equivalent Series-Parallel Models.- 5.4 Comparison of the Obverse and Converse Regular Ladder Models. Model Fitting.- 5.5 Regular Ladder Models with an Infinite Number of Elements.- 5.5.1 The Extended Gross-Marvin Models.- 5.5.2 The Extended Marvin-Oser Model.- 5.5.3 The Extended Regular Converse Ladder Model.- 5.5.4 Other Extended Regular Ladder Models.- 5.6 The Continuous Ladder or Material Transmission Line.- 5.6.1 The Continuous Gross-Marvin Ladder or Inertialess Material Transmission Line.- 5.6.2 The Material Transmission Line with Inertia.- 5.6.2.1 The Lossless Line.- 5.6.2.2 The Lossy Line.- 5.7 Problems.- References.- 6. Representation of Linear Viscoelastic Behavior by Mathematical Models.- 6.0 Introduction.- 6.1 Modelling by the Use of Matching Functions.- 6.1.1 Matching Functions of the Z-and S-Type.- 6.1.2 Matching Functions of the A-Type.- 6.1.3 Modelling of the Experimental Response Functions.- 6.1.3.1 The Step Responses.- 6.1.3.2 The Slope Responses.- 6.1.3.3 The Harmonic Responses.- 6.1.4 Modelling of the Respondances.- 6.1.4.1 Harmonic Response Models from Respondance Models.- 6.1.4.2 Step Response Models from Respondance Model.- 6.2 Models Based on Fractional Differentiation (Power Laws).- 6.3 Modelling of the Spectral Response Functions.- 6.3.1 Direct Modelling.- 6.3.1.1 The Box Distribution.- 6.3.1.2 The Wedge and Associated Power Law Distribution..- 6.3.2 Spectra Derived from Models for the Experimental Responses.- 6.3.2.1 Spectra Derived from Models for the Step Responses.- 6.3.2.2 Spectra Derived from Models for the Harmonic Responses.- 6.4 Problems.- References.- 7. Response to Non-Standard Excitations.- 7.0 Introduction.- 7.1 Response to the Removal or the Reversal of a Stimulus.- 7.1.1 Creep Recovery.- 7.1.2 Response to the Removal of a Constant Rate of Strain.- 7.1.3 Response to the Reversal of Direction of a Constant Rate of Strain.- 7.2 Response to Repeated Non-Cyclic Excitations.- 7.2.1 Staircase Excitations.- 7.2.2 Pyramid Excitations.- 7.3 Response to Cyclic Excitations.- 7.3.1 Non-Steady-State Response.- 7.3.2 Steady-State Response.- 7.3.2.1 In Terms of the Spectral Response Functions.- 7.3.2.2 In Terms of the Harmonic Response Functions.- 7.4 Approximations to the Spectra from Responses to Non-Standard Excitations.- 7.5 Problems.- References.- 8. Interconversion of the Linear Viscoelastic Functions.- 8.0 Introduction.- 8.1 Interconversion Between Relaxation and Creep Response Functions..- 8.1.1 Interconversion Between the Respondances.- 8.1.2 Interconversion Between the Harmonic Responses.- 8.1.3 Interconversion Between the Step Responses.- 8.1.3.1 Theoretical Interrelations.- 8.1.3.2 Numerical Evaluation of the Convolution Integrals.- 8.1.3.3 Empirical Interconversion Equations.- 8.1.4 Interconversion Between the Spectral Functions.- 8.1.4.1 Theoretical Interrelations.- 8.1.4.2 Approximate Interconversion.- 8.1.4.3 Approximate Calculation of the Spectra from the Step Responses.- 8.2 Interconversion Between Time- and Frequency-Dependent Response Functions.- 8.2.1 Theoretical Interrelations.- 8.2.2 Interconversions Requiring Numerical Integration.- 8.2.3 Interconversion by Kernel Matching.- 8.2.4 Empirical Interconversion Equations.- 8.3 Interconversion Within the Frequency Domain.- 8.3.1 Relations Between the Real and Imaginary Parts of the Harmonic Response Functions.- 8.3.1.1 The Kronig-Kramers Relations.- 8.3.1.2 Approximations to the Kronig-Kramers Relations.- 8.3.1.3 Using Mathematical Models.- 8.3.2 Relations Between the Absolute Modulus or Compliance and the Loss Angle.- 8.4 Problems.- References.- 9. Energy Storage and Dissipation in a Linear Viscoelastic Material.- 9.0 Introduction.- 9.1 Strain Excitation (Stress Relaxation Behavior).- 9.1.1 Step Response.- 9.1.2 Slope Response.- 9.1.3 Harmonic Response.- 9.2 Stress Excitation (Creep Behavior).- 9.2.1 Step Response.- 9.2.2 Slope Response.- 9.2.3 Harmonic Response.- 9.3 Hysteresis.- 9.3.1 Response to Harmonic Excitations.- 9.3.2 Response to Triangular Excitations.- 9.3.2.1 Single Cycle Response.- 9.3.2.2 Steady-State Response.- 9.4 Problems.- 9.5 References.- 10. The Modelling of Multimodal Distributions of Respondance Times.- 10.0 Introduction.- 10.1 Bimodal Distributions.- 10.1.1 Series-Parallel Models.- 10.1.2 Models for the Spectral Response Functions.- 10.1.3 Ladder Models.- 10.1.4 Mathematical Models.- 10.2 Prolongated Unimodal Distributions.- 10.2.1 Series-Parallel Models.- 10.2.2 Models for the Spectral Response Functions.- 10.2.3 Ladder Models.- 10.2.4 Mathematical Models.- 10.3 Problems.- References.- 11. Linear Viscoelastic Behavior in Different Modes of Deformation.- 11.0 Introduction.- 11.1 The General (Anisotropic) Viscoelastic Stress-Strain Relations.- 11.1.1 The Correspondence Principle.- 11.1.2 The General Stress-Strain Relations.- 11.2 The Isotropic Viscoelastic Stress-Strain Relations.- 11.2.1 The Viscoelastic Generalized Hooke’s Laws.- 11.2.2 The Viscoelastic Generalized Newton’s Laws.- 11.3 Linear Viscoelastic Behavior of Isotropic Materials in Different Modes of Deformation.- 11.3.1 Behavior in Shear.- 11.3.2 Behavior in Isotropic Compression.- 11.3.3 Behavior in Constrained Uniaxial Compression.- 11.3.4 Behavior in Uniaxial Tension and Compression.- 11.3.5 Lateral Contraction (Poisson’s Ratio).- 11.3.6 Wave Propagation.- 11.3.6.1 Wave Propagation in an Extended Medium.- 11.3.6.2 Wave Propagation in a Bounded Medium.- 11.4 Interconversion of the Isotropic Response Functions in Different Modes of Deformation.- 11.4.1 Behavior in Shear from Measurements in Uniaxial Tension Including Lateral Contraction.- 11.4.1.1 Exact Interconversion.- 11.4.1.2 Numerical Interconversion.- 11.4.1.3 Approximate Interconversion.- 11.4.2 Behavior in Bulk from Measurements in Uniaxial Tension and in Shear.- 11.4.3 Behavior in Bulk from Measurements in Uniaxial Constrained Compression and Shear.- 11.4.4 Behavior in Bulk from Measurements in Uniaxial Tension or Compression Including Lateral Contraction or Dilatation.- 11.4.4.1 Exact Interconversion.- 11.4.4.2 Numerical Interconversion.- 11.5 Problems.- References.- Appendix: Transformation Calculus.- A1 Introduction.- A 2 Special Functions.- A2.1 The Delta Function.- A2.2 The Step Function.- A2.3 The Gate Function.- A2.4 The Slope Function.- A 2.5 The Ramp Function.- A3 The Laplace Transform.- A3.1 Properties.- A 3.1.1 Linearity.- A3.1.2 Uniqueness.- A3.1.3 Translation.- A3.1.4 Real Differentiation.- A3.1.5 Real Integration.- A3.1.6 Real Convolution.- A3.1.7 Limit Values.- A3.1.8 Cut-off. The Finite Laplace Transform.- A3.1.9 Repetition.- A3.2 Inversion.- A3.2.1 The Residue Theorem.- A3.2.2 Inversion on the Real Axis.- A3.3 Laplace Transform Pairs.- A 3.4 The s-Multiplied Laplace Transform (Carson Transform).- A4 The Fourier Transform.- A 4.1 Properties.- A4.2 Inversion.- A 4.3 Generalized Fourier Transforms.- A 4.4 Generalized Fourier Transform Pairs.- A 5 The Stieltjes Transform.- A5.1 Properties.- A 5.2 Inversion.- A 5.2.1 The Jump.- A 5.2.2 Inversion on the Real Axis.- A5.3 The Fourier-Laplace Transform.- A 5.4 Stieltjes Transform Pairs.- A 6 The Hilbert Transform.- A 6.1 Inversion.- References.- Solutions to Problems.- 1.- 2.- 3.- 4.- 5.- 6.- 7.- 8.- 9.- 10.- 11.- Epilogue.- Notes on Quotations.- List of Symbols.- Author Index.