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1 T his d issertation has been m icrofilm ed exactly as received PAGEN, C harles Anthony, AN ANALYSIS OF THE THERMORHEOLOGICAL RESPONSE OF BITUMINOUS CONCRETE. The Ohio State U n iversity, P h.d., 1963 Engineering, c iv il University Microfilms, Inc., Ann Arbor, Michigan

2 AN ANALYSIS OF THE THERMORHEOLOGICAL RESPONSE OF BITUMINOUS CONCRETE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Charles Anthony Pagen, B. C. E., M. Sc. The Ohio State University 1963 Approved by Adviser Department of Civil Engineering

3 ACKNOWLEDGMENTS I would like to express my appreciation to Professor Robert F. Baker for his most excellent services as adviser and teacher, as well as for his valuable suggestions concerning the content of this research and his full cooperation in providing the required equipment and materials. Special thanks are also due to Professor Richard W. Bletzacker and the other members of the Civil Engineering Department faculty for their valuable discussions and advice during my stay here. This study, or indeed any experimental program, could not have been completed without the services of skilled research assistants. I wish to thank Bee Ku, Research Assistant, and David Wu, Technical Assistant, in particular, and the many other members on the staff of the Transportation Engineering Center who contributed to the completion of this research. This dissertation was completed at The Ohio State University during tenure of a fellowship awarded by the National Bituminous Concrete Association, whose continued interest and cooperation are gratefully acknowledged. I would like to thank the Transportation Engineering Center and Mr. Gene Hayner and his staff for their assistance in reproducing the final report, and especially Mrs. Sandra Geiger for typing the manuscript. Finally, this page would be incomplete without a word of recognition to my wife, Elizabeth Ann, without whose patience and encouragement this study could not have been completed.

4 CONTENTS Page ACKNOWLEDGMENTS... TABLES... ILLUSTRATIONS... NOTATIONS... ii vii viii xii Chapter I INTRODUCTION Rheological Behavior of Materials... 1 Temperature and Time Dependence of Material Response... 6 Pavement Performance and Design... 8 Objective Procedure Experimental Scope D MATHEMATICAL METHODS OF SPECIFYING THE TIME-DEPENDENT RESPONSE OF MATERIALS General Elastic Response of Materials Phenomenological Treatment of Linear Viscoelastic Behavior Operator Equation Electric Analog Fundamental Mechanical Models Kelvin-Voigt Solid Maxwell Liquid Burgers Model Generalized Voigt and Maxwell Models Retardation Spectra Distribution of Retardation Times... 41

5 CONTENTS (contd) Chapter HI Page Distribution of Relaxation Times Retardation and Relaxation Spectra in Logarithmic Time Scale Dynamic Response of Materials The Boltzman Superposition Principle Complex Compliance Function Three-Dimensional Viscoelastic Behavior TRANSFORMATIONS TO FREQUENCY DOMAIN General Fourier and Laplace Transformations Interrelations Among the Viscoelastic Functions Relations Between Creep Compliance Function, Dynamic Complex Compliance, and Retardation Spectrum Relations Between Relaxation Function, Complex Modulus Function, and Relaxation Spectrum Determination of Viscoelastic Spectra from Experimental Viscoelastic Functions Application of Transforms IV A THEORY FOR THE TEMPERATURE DEPENDENCE OF THE VISCOELASTIC FUNCTIONS OF BITUMINOUS CONCRETE General Analysis of Four-Param eter Model; Temperature Dependence of Elements Flexible-Chain Theories Isolated Flexible-Chain Theory of Rouse The Thermodynamics of Rubberlike Elasticity The Concept of Thermolinearity Procedure and Criteria for Applicability of Reduced Variables Summary V MATERIALS TESTED AND EXPERIMENTAL TECHNIQUES General Materials Tested iv

6 CONTENTS (contd) Chapter VI Page Preparation of Samples Transient Tests Periodic or Dynamic Tests Dynamic Resonant Frequency Tests PRESENTATION AND ANALYSIS OF EXPERIMENTAL DATA General Homogeneity and Isotropy of Materials Tested Transient Linear Viscoelastic Response of Bituminous Concrete Dynamic Experimental Results Thermorheologically Linear Response of Bituminous Concrete Temperature Reduction Factor Master Creep Function at TQ Master Creep Compliance Function Generalized Voigt Model Prediction of Dynamic Complex Modulus Complex Transverse Modulus Prediction of Complex Transverse Modulus Viscoelastic Response of 300 Series Bituminous Concrete Mixture Vn DISCUSSION AND APPLICATIONS OF EXPERIMENTAL RESULTS Correlation of Dynamic and Transient Experimental Data Interpretation of Results Application to Three-Dimensional Response VHI SUMMARY AND CONCLUSIONS APPENDIX I BITUMINOUS CONCRETE MIXTURE DETAILS 201 APPENDIX n STRESS LEVELS AND TEMPERATURES IN UNEAR VISCOELASTIC RANGE v

7 CONTENTS (contd) Page APPENDIX m TRANSIENT AND DYNAMIC EXPERIMENTAL RESPONSE OF 300 SERIES BITUMINOUS CONCRETE MIX BIBLIOGRAPHY AUTOBIOGRAPHY vi

8 TABLES Table Page 1 Dynamic Experimental Results - Absolute Value of Complex Elastic Modulus and Phase Angle Dynamic Experimental Results - Absolute Value of Complex Transverse Modulus and Phase Angle Dynamic Experimental Results - Real and Imaginary Components of Complex Compliance Dynamic Experimental Results - Real and Imaginary Components of Complex Transverse Compliance Complex Modulus and Phase Angle from Torsional and Flexural Resonant Frequencies Specific Gravity and Penetration Grade of Bitumen Compaction Details, Bitumen Content and Bulk Specific Gravity of Mixtures Bituminous Concrete Mixture Proportions Mixture Details Density and Void Analysis Stress Levels in Linear Viscoelastic Range at Experimental Temperatures vii

9 ILLUSTRATIONS Figure Page 1 Fundamental Mechanical Models Axial Stress and Strain vs. Time Relations for Burgers Model Complex Mechanical Models Dynamic Viscoelastic Response and Vectorial Resolutions Hypothetical Reduced Creep Functions of a Thermolinear Viscoelastic Material vs. Time Plotted at Several Temperatures Reduced Transient Creep Functions of a Thermolinear Viscoelastic Material vs. Time for Nine Temperatures Temperature Reduction Function, f(t) Temperature Dependence of Shift Factor, a>p Transient Test Equipment Sanborn Recording of Transient Test Dynamic Test Equipment Sanborn Recording of Dynamic Test Sonic Equipment Showing E-Scope and Lissajou Test Pattern on the Oscilloscope Lihear Viscoelastic Response of 500 Series Bituminous Concrete Mixture in Transient Creep Test at Four Stress Levels viii

10 ILLUSTRATIONS (contd) Figure Page 15 Elastic Creep Modulus of 500 Series Bituminous Concrete Mix Reduced to 298 K, vs. Time at Eleven Temperatures Temperature Dependence of Shift Factor, a Developed from Elastic Creep Data Composite Master Creep Function Obtained by Thermolinearity Concept, Representing Viscoelastic Behavior over an Extended Time Scale at 298 K, 500 Mix Composite Master Creep Compliance Functions at Three Temperatures, 500 Mix Axial Strain vs. Time from Master Creep Function at 298 K, 500 Mix Normalized Retarded Elastic Response, ip (t), vs. Time at 298 K, 500 Mix Distribution Function of Retardation Time, L (hi t ) Obtained from Master Creep Modulus at 298 K, 500 Mix Real and Imaginary Components of Complex Compliance as Functions of Angular Frequency at 298 K, 500 Mix Absolute Value of Complex Elastic Modulus and Phase Angle as Function of Angular Frequency at 298 K, 500 Mix Transverse Creep Function of 500 Series Bituminous Concrete Mix Reduced to 298 K, vs. Time at Eleven Temperatures Temperature Dependence of Shift Factor, a-p, Evaluated from Elastic and Transverse Creep Data, 500 Mix 169 ix

11 ILLUSTRATIONS (contd) Figure Page 26 Composite Master Transverse Creep Function at 298 K. 500 M ix Composite M aster Transverse Creep Compliance Function at 298 K, 500 Mix Circumferential Strain vs. Time from Master Transverse Creep Data at 298 K, 500 Mix Normalized Retarded Elastic Behavior, 41(t), vs. Time, 500 Mix Distribution Function of Retardation Time, L (fn t ), Obtained from Master Transverse Creep Function at 298 K, 500 Mix Storage and Loss Components of Complex Transverse Compliance as Functions of Frequency at 298 K, 500 Mix Absolute Value and Phase Angle of Complex Transverse Modulus as Functions of Angular Frequency at 298 K, 500 Mix Temperature Reduction Factor vs. Original Asphalt Viscosity for 300 and 500 Series Bituminous Mixes Original Asphalt Viscosity vs. Temperature Elastic Creep Modulus of 300 Series Bituminous Concrete Mix Reduced to 298 K vs. Time at Eleven Temperatures Temperature Dependence of Shift Factor Developed from Elastic Creep Data, 300 Mix Master Creep Function at 298 K Normalized Retarded Elastic Response vs. Time x

12 ILLUSTRATIONS (contd) Figure Page 39 Distribution Function of Retardation Time Obtained from Master Creep Modulus at 298 K Absolute Value of E* and Phase Angle at 298 K, 300 Mix Transverse Creep Modulus of 300 Series Bituminous Concrete Mix Reduced to 298 K vs. Time at Eleven Temperatures Temperature Dependence of Shift Factor Evaluated from Elastic and Transverse Creep Data, 300 Mix Master Transverse Creep Function at 298 K Normalized Retarded Elastic Response vs. Time Distribution Function of Retardation Time from Composite Transverse Creep Function at 298 K Absolute Value of T* and Phase Angle at 298 K, 300 Mix xi

13 NOTATIONS Root-mean-square end-to-end distance per square root of number of monomer units Ratio of retardation times at two different temperatures, or temperature reduction factor Base width (sample) Adjustable parameter Area Sample coefficient or form factor Constant Degree Centigrade Constant Sample coefficient or form factor Partial derivative with respect to time Base of natural logarithms Deviatoric strain tensor Young's modulus of elasticity Elastic element Activation energy Transient creep modulus Stress relaxation modulus Complex elastic modulus Real part of E* Imaginary part of E*

14 Er2^> /E */ f (T) f o F Imaginary part of E* Absolute value of E* Temperature reduction function Friction coefficient of a submolecule Fourier transform Degree Fahrenheit Free energy Inverse Fourier transform Fc (t) F* F 1 (u>) F2 M G G* G j (4) g 2 h h H (t ) H (in t ) i j J (t) c ' 7 Transverse Creep Compliance Complex transverse compliance Real part of F* Imaginary part of F* Modulus of Rigidity Complex shear modulus Real part of G* Imaginary part of G* Height (sample) Root-mean-square end-to-end distance between ends of polymer chain Distribution function of relaxation time Logarithmic relaxation spectrum Dummy variable Number of chain backbone atoms in monomer unit Dummy variable The imaginary unit Transient creep compliance xiii

15 J* J x (w) J 2 **>> Ji J o K K* <p) Complex compliance Real part of J* Imaginary part of J* Compliance contribution of model element Glasslike or elastic compliance Retarded elastic compliance Bulk modulus Degree Kelvin Dummy variable Boltzman's constant Subscript for Kelvin Complex bulk modulus Real part of K* Imaginary part of K* L Laplace transform Inverse Laplace transform L (t ) L ( n t ) M M* (w) M2 <u>) n Distribution function of retardation time Logarithmic retardation spectrum Subscript for Maxwell Complex modulus Real part of M* Imaginary part of M* Summation index Number of polymer molecules per cc xiv

16 Number of submolecules in molecule Flexural resonant frequency Summation index Torsional resonant frequency Linear differential operator Linear differential operator Number of monomer units in submolecule Linear differential operator Heat energy transferred to the body Linear differential operator Radius Root-mean-square end-to-end length of macromolecule Universal gas constant Entropy Generalized Laplace frequency Subscript for steady-state Deviatoric stress tensor Thickness (sample) Time Absolute temperature Period Transient Transverse Creep Compliance Arbitrary reference temperature for reduced variables Complex transverse modulus Real part of T* Imaginary part of T*

17 Displacements in x, y, z directions, respectively Potential energy Running tim e variable Energy transferred to the system Weight Work of the internal forces Internal kinetic energy Coordinates Reduced time Reduced frequency Degree of polymerization Electrical impedance Positive constant Shear strain Small increment Dirac delta Kronecker delta Monomeric friction coefficient Strain Mean normal strain Strain tensor Amplitude of harmonic strain Constant strain Steady-state strain Axial strain Circumferential strain

18 Instantaneous strain Retarded elastic strain Secondary strain Total strain Coefficient of shear viscosity Viscous element Coordinate Coefficient of volume viscosity Poisson's ratio Complex Poisson's ratio Yield stress Density Reference density for reduced variables Stress Mean normal stress Root-mean-square end-to-end distance of a submolecule Stress tensor Amplitude of harmonic stress Constant stress Axial stress Circumferential stress Summation Relaxation or retardation time Dummy variable Relaxation or retardation time of element of mechanical model xvii

19 Kelvin retardation time Maxwell relaxation time Phase Angle Phase angle of E* Phase angle of G* Phase angle of T* Circular frequency in radians per second Resonant frequency Derivative with respect to time Infinity Frequency spectrum

20 CHAPTER I INTRODUCTION Rheological Behavior of Materials One of the fundamental purposes of the engineering sciences is to define those properties which, by custom and tradition, have become known as the m e chanical response of the m aterials. The mechanics of deformable bodies is concerned in general with the interrelations between the dynamical forces acting on a material body and the resulting time and temperature dependent kinematical deformations. In the mechanical response of ideally elastic m aterials, the deformation response is fully recoverable on release of the forces, and the only material constant needed to characterize the material is its elasticity. In the deformation of ideally viscous and plastic materials, the release of the forces is not accompanied by the recovery of expanded energy, and the strain caused by the forces is nonrecoverable. The behavior of real materials are composed of all of the above deformational responses in varying proportion, depending on the nature of the material and on the conditions under which the forces are applied. In studying the mechanical response of bituminous concrete mixtures, the factors of time and temperature must be taken into account, since the rheological behavior of this material is known to be both time and temperature dependent. In the context of this study the materials under investigation are

21 dense bituminous concrete mixtures, and the thermodynamics of the linear 2 viscoelastic behavior of this material will be emphasized. There are three fundamental levels which an engineering analysis can assume: (l)1 the atomic and molecular level, the structural level, and the phenomenological level. The first level of analysis is the microscopic in which the interactions of the elementary particles are observed and the mass behavior inferred from the molecular or atomic structure. At this level the material is considered to be a discontinuous and nonhomogeneous body made up of discrete particles. The phenomena that are responsible for the time and temperature dependence of the mechanical properties of a material originate at the atomic and molecular level of aggregation of elements. In the second or structural level of investigation, the m aterial is considered to be continuous but nonhomogeneous and made up of a continuous array of elements, which are formed of units of different properties and finite dimensions. In the highest level of investigation, the phenomenological or macroanalytical level, the material is considered continuous and homogeneous, and the material properties of the material are the same in all directions within the body. The analysis of the thermally sensitive viscoelastic response of bituminous concrete mixtures used in this study will be on the phenomenological level where the material is considered to consist of macroscopic dimensions and to be both homogeneous and isotropic. The introduction of homogeneity and isotropy to real materials can only be justified on a statistical basis in which the properties of the m aterial and average t " 1 " 1 Underlined numbers in parentheses refer to the Bibliography.

22 3 shapes of the components making up the body are considered. Bituminous concrete mixes are essentially composite materials made up of three separate and distinct phases: a mineral aggregate phase made up of particles of various sizes and shapes, enclosing voids filled with the bitumen, and air phases. In the case of such statistical isotropy and homogeneity, the relations between the material properties describing the average behavior on the phenomenological level and the properties describing the individual behavior of the components can only be statistical relations and are based on the condition that the dimensions of the largest particles are small when compared to the dimensions of the body under study. By the assumption of statistical homogeneity and isotropy of the continuous material, it is possible to develop a general phenomenological theory of deforma- tional response of all materials. The general phenomenological theory provides the transition between the classical theory of hydrodynamics and classical theory of elasticity and includes them as limiting cases in the general theory of the flow and deformation of m aterials. Although the deformation of real engineering materials is a complex phenomenon, similar processes of deformation can be produced in different materials by varying the experimental conditions. It can be shown that the mechanical response of materials on the phenomenological level is described by relatively few basic characteristics which, coupled in various combinations, are present in all materials (2). An effective quantitative theory of the behavior of materials can be formulated once the response of each of the constituent elements can be expressed in simple mathematical term s that are accessible to analytical

23 treatment. These relations are generally expressed as functions of the variables specifying dynamical conditions, the stress, and the kinematical conditions, the strain, and their time derivatives. The establishment of mathematical relations describing the response of real materials can be accomplished by a visualization of relatively few simple mechanical models. Classical mechanics of deformable m aterials, which provides the basis of engineering analysis, have been developed from two of these idealizations, namely, the Hookean or linear elastic solid and the Newtonian or ideal viscous liquid. A simple mechanical model representing the Hookean body may be considered to be a helical spring, which is characterized by proportionality between stress <r, and strain e. The Newtonian body represents a molecular process where the entire deformation energy is dissipated as heat. A dashpot, the mechanical model representing this behavior, consists of a piston which moves in a container filled with a viscous liquid. A stress applied to the piston causes it to move through the liquid at a definite rate of strain, e, proportional to the applied force. Elasticity and viscosity are fundamental properties of rheology, but to complete the picture we must add a third concept, the St. Venant body, which is the classical model of plastic flow. In the case of the St. Venant element the elementary mechanism consists of a weight resting on a horizontal surface that resists lateral movement at a given stress level due to friction. In order to move the weight, the frictional resistance between the weight and the supporting surface, which corresponds to the yield stress of the material, must be overcome. A force too small to do this will have no effect, but a sufficiently

24 larger force moves the weight across the surface. As in the case of the dashpot when the force is removed, the weight will come to rest in its new position as all of the deformational energy is dissipated. Accordingly, plastic behavior depicted by the St. Venant element is essentially the property of a real material, represented by a mechanical model consisting of a St. Venant element and a spring in series. This real material may be considered to be represented by an elastic element when the stresses are below the material's yield stress,, and to deform excessively without any additional increase in the stress once the yield stress of the material is exceeded. Upon the release of the stress, the elastic part of the deformation will be regained. Elasticity, plasticity, and viscosity are fundamental rheological properties depicted by abstract concepts, which can be combined to represent mathematically the response of real materials. Although the idealized materials are limited in their properties, it will be shown later that the abstract concepts of idealized materials can be combined to represent the complex behavior of real materials by arranging the mechanical idealizations in various configurations to any degree of approximation desired. This study constitutes an attempt to make use of the model theory of viscoelastic response and to investigate the temperature dependence of the phenomenological theory in describing the behavior of engineering materials. The rheological behavior of a material can be specified by an equation between stress and strain and their derivatives with respect to time, called the rheological equation of state of the material. The parameters appearing in the rheological equation which define the material are the rheological moduli or constants, while the stress and strains are the rheological variables. The

25 6 coefficients of proportionality are modulus of elasticity, E, and shear viscosity, r, for the Hookean and Newtonian bodies, respectively. Temperature and Time Dependence of Material Response A closer investigation of the behavior of real materials reveals that the deformation under a given constant stress can be divided conveniently into three distinct parts according to their nature with respect to time and temperature dependence. An estimate of the instantaneous elastic response, independent of the time, can be found by measurement of the strain immediately after loading and is represented by the Hookean elastic component of the material deformation. Newtonian flow may also be observed in materials by measuring the increase in strain at long loading times which, characterized by the shear viscosity, t], is proportional to the time of loading. The retarded elastic behavior is also a time-dependent deformation and is due to the interaction between the elastic and viscous elements which are used to represent real materials. Hence, the instantaneous response of a material is not the only recoverable elastic behavior of a material, and the viscous flow is not the only time-dependent response. The retarded elastic deformation is elastic in the sense that the deformation and the deformational energy are both fully recoverable provided enough time is allowed for the recovery to take place. The retarded elastic response is viscous to the extent that the deformation and recovery of the deformational energy stored is time-dependent. Although no real material in nature can be classified as ideally elastic, viscous, or plastic, it may be possible to represent the response of the real materials under given conditions by one or a combination

26 of the fundamental rheological elements to a useful degree of approximation. 7 In the previous discussion there has been no explicit mention of the effect of temperature on any of the viscoelastic functions, although it has been suggested that the rheological response of thermally sensitive materials such as bitumens and bituminous concrete mixtures are dependent upon both time and temperature. There are many reasons why the influence of temperature on the mechanical response of a viscoelastic material is of great interest. One is of a purely practical nature bituminous mixtures are used in a wide range of environmental and loading conditions, and a complete specification of the mechanical properties of these materials cannot be restricted to room temperature nor a narrow range of temperatures. Another important reason for studying the influence of temperature is due to the thermodynamic information which can be obtained about the molecular process occurring upon deformation. And finally, it will be shown that by use of the temperature dependence of viscoelastic deformation, a powerful tool would be available to extend experimental results to define the response of the material at any intermediate temperature in the tested range as well as to extend the test results to portions of the experimental time scale normally inaccessible by conventional methods. The three characteristic regions of viscoelastic behavior for thermally sensitive materials may be observed by varying the experimental testing temperature and by performing a constant stress test in the linear range. At freezing temperatures a glassy region is observed in which the magnitude of the creep modulus, Ec (t), defined as the constant stress, o-, divided by the timedependent strain, e (t), is quite high and the loading time effects are not

27 pronounced. A transition region of viscoelastic response in which the modulus varies rapidly with time and experimental temperature is observed at slightly higher temperatures. At still higher temperatures, the response of the material is in the flow region and the modulus changes very rapidly with time from a small finite value to values which approach zero. Pavement Performance and Design Despite the fact that highway pavements have been constructed from earliest civilizations, all methods of pavement design have the common disadvantage of lacking a completely rational basis. Even after thousands of years of road building and extensive theoretical studies, there is still considerable difference of opinion concerning the best approach to pavement design. The procedure used at this time consists of applying one of several design procedures (3), (4), (5), modified by engineering experience, judgement and resources available. One of the important requirements for development of a completely rationi al method of pavement design is the ability to predict the stresses and strains induced into the components of a pavement system under any type of applied loads, at any time or environmental condition. Another requirement is the determination of failure criteria for such a structure. Of equal importance is the knowledge of rheological properties of the materials involved and, in the case of bituminous concrete mixtures, the strain susceptibility under thermal variations. All of the available methods of pavement design and calculation of the stresses and strains in such a structure are based on different simplifying

28 assumptions which are applicable only within a given range of conditions. An outstanding contribution to the rational method of pavement design was made by Westergaard in 1925 (6), who considered a rigid pavement slab to act as a homogeneous, isotropic, elastic solid and assumed the reactions of the subgrade to be only vertical and proportional to slab deflections analogous to the heavy liquid theory of Winkler (7). By the introduction of these assumptions, the complex analysis was reduced to a problem for the mathematical theory of elasticity and did not include the effect of time or temperature on the response of the material to load. Hogg (8) modified Westergaard's solution to consider the top layer to act as an elastic slab resting on a subgrade, but treated the subgrade as an elastic half space. The Westergaard method is being used widely today with modifications by Pickett (4) to allow for temperature warping of the pavement slab and slab corner conditions. The concept of the subgrade support acting as a series of independent springs was first used by Winkler for the solution of the problem of beams on elastic foundations. The equations expressing the stress components caused by a point load surface force at points within an elastic, homogeneous, isotropic, infinite mass were attributed to Boussinesq (9). Burmister (5), (10) extended Boussinesq's one-layer theory to develop a two-layer theory applying the equations of elasticity to both layers. Refinements of Burmister's method have been made by several authors to include more than two layers and to cover a wider range of pavement thickness and elastic constants (11), (12), (13), (14).

29 The fundamental difference between the layered system and the W estergaard (heavy liquid) solutions for the stress and deformations of a pavement structure arises from the assumed nature of the supporting layer or subbase. The Burmister solution assumes a continuous elastic solid foundation in which the vertical strains are not necessarily proportional to the vertical stresses at that point. Further, the stresses at all points in the system are influenced by the stresses at every other point. The Westergaard solution assumes, in effect, that the supporting medium is made up of a series of independent springs in which the deformation at every point is proportional to the load at that point and independent of loads elsewhere. The reaction of a heavy liquid on a floating body is of a similar nature. The most important anomaly of the preceding discussions is that the basic theories are dealing with ideally elastic materials, and it is generally agreed that there is no perfectly elastic material (15). In order to apply the concepts of the theory of elasticity to the problem of pavement design, we must be cognizant of the deviations in the engineering materials from those assumed by the theory. The materials comprising the structural section of flexible pavements are no exceptions and are not truly elastic; rather they exhibit stressstrain characteristics which are both time and temperature dependent. While the elastic theory may provide valuable information regarding the stresses and strains in bituminous pavements under fast-moving wheel loads at freezing temperatures, the effects of slowly applied or static loads over a range of temperatures cannot be accurately considered by the elastic theory. In addition, the

30 11 progressive accumulation of small irrecoverable deformations under repeated wheel loads and subsequent rutting of bituminous concrete in the vehicle wheel path or the flow of the pavement surface layer parallel to the direction of traffic and fatigue effects cannot be accounted for by the elastic theory. Thus, a completely rational method of pavement design must incorporate the effects of temperature and loading times on the stress-strain properties of the material. In light of the preceding discussion, it would seem that the concepts of rheology would provide a better analytic framework for the complex problem of flexible pavement design. The principles of viscoelastic response have been applied to the study of layered bituminous concrete pavements under restricted ideal conditions by several authors (16), (17), (18), in an attempt to overcome the limitations imposed by the assumptions of the elastic theory. The deformations of such viscoelastic systems have also been analyzed using moving loads (19). At this time limited analytical methods appear to be available; however, the viscoelastic properties of the materials involved have often been evaluated at specific conditions and were often assumed in order to be used in the calculations. If the viscoelastic properties of the materials comprising the pavement structure can be established over a wide range of loading conditions and environmental conditions, the response of the resulting structure can be correlated with theoretical solutions for given conditions to place the design and investigation of this engineering structure on a more rational basis.

31 The present study will be concerned with defining the rheological response of bituminous concrete mixtures over a wide range of loading times or frequencies and with evaluating the temperature dependence of these properties. The linear viscoelastic response of high polymers have been successfully investigated by using the concepts of rheology (2), (20), (21), (22), and a great deal of fundamental information has been developed in this field. Analyses of the viscoelastic behavior of bituminous concrete mixtures have been performed using rheological concepts (17), (23). and, this basic research has served as a valuable reference to asphalt technologists in establishing the viscoelastic nature of bituminous concrete. The temperature dependence of the viscoelastic response of high polymers has been investigated using thermodynamic concepts (24), (25). However, the concepts of the kinetic theory and time-temperature superposition principle have not been applied to the analysis of the linear viscoelastic behavior of bituminous concrete mixtures in the time and frequency domain. Objective The broad objective of this dissertation was to analyze the kinetic theory and thermorheological linearity concept concerning the effect of temperature and loading frequency or time on the general viscoelastic equations which define the mechanical behavior of thermally sensitive materials. The purpose in applying these concepts was to determine if time and temperature may have an equivalent effect on the mechanical response of bituminous mixes, and if it would be possible to predict the behavior of the material by a relatively few experiments

32 13 for any temperature in the experimental range and at loading times both longer and shorter than are normally obtained experimentally. It is hoped that the application of the linear viscoelastic and thermodynamic concepts to the behavior of this material will provide a sound mathematical framework for the analysis of the stresses and deformations in flexible pavement systems and aid in the development of more rational pavement design procedures. The specific objective of the research was concerned with defining the mechanical response of bituminous concrete mixtures over a wide range of environmental and loading conditions. Procedure The analytical procedure consisted of the following phases: 1. The application of the linear viscoelastic theory to represent the re s ponse of the materials used in this study was investigated by direct experimentation on the macroanalytical level by using both transient and dynamic tests. 2. Methods were studied for defining the stress-strain-time-temperature behavior of bituminous mixes in terms of two independent viscoelastic functions and a single temperature function which for a homogeneous, isotropic, linear viscoelastic body is sufficient to describe fully its mechanical response. 3. Thermodynamic concepts were investigated to evaluate the temperature dependence of the viscoelastic functions. 4. Procedures were studied in order to define the mechanical behavior of the material at any experimental temperature in the time or frequency domain by using rheological principles, the kinetic theory of rubberlike elasticity, and the thermorheological linearity concept.

33 14 5. The concepts summarized in this study were applied in order to determine if the time and temperature dependent viscoelastic functions of bitumen-aggregate mixtures are directly related to the cooperative motions of the individual flexible-chain macromolecules of the bitumen component. If the above concepts can be verified experimentally, the complicated dependence of the viscoelastic functions of the material on the two independent variables of time and temperature can be separated to yield a function of time alone, and one of temperature alone, which are respectively, the viscoelastic function at a standard reference state, Tq, and the temperature reduction function, f (T). When a stress acts on a linear viscoelastic material, both a change in shape and volume of the material occurs. A hydrostatic pressure produces only a volume change with a constant shape while a shear stress produces only a change in shape at constant volume. It is known for bituminous concrete mixtures that both changes depend on time and temperature, thus a detailed knowledge of both the shear and bulk rheological properties over the temperature range to which the material be subjected is needed to characterize completely the linear viscoelastic properties of the material. The treatment of the general three-dimensional stress-strain behavior of viscoelastic materials was based on the following hypotheses: the material under consideration was assumed to be both homogeneous and isotropic, and the mechanical response of the material can be represented by that of a linear viscoelastic substance at the low levels of stress used. The validity and degree of accuracy possible by applica-

34 15 tion of the above assumptions was examined by direct experimentation on the materials used in this study, dense bituminous concrete mixtures, although sim ilar concepts may be applied to other materials. Temperature effects are not presently included in the study of bituminous mixtures under the linear viscoelastic theory and in structural theories of pavement design applied to this material. However, temperature variations should be included in the above theories as the mechanical properties of a given bituminous concrete are generally quite different when examined at the various temperature ranges in which it functions. The temperature dependence of the viscoelastic functions of bituminous mixtures was analyzed by using the concepts of the kinetic theory and the thermorheologically linearity principle. This was done in an attempt to define the stress-strain-time-temperature response of the material at any temperature under consideration within the tested temperature range. The concepts of the viscoelastic theory were applied to the analysis of the transient and dynamic behavior of bituminous mixtures, and the individual components of the deviatoric and isotropic stress tensor were related to the co rresponding component of the deviatoric and isotropic strain tensor by means of algebraic coefficients which are complex functions of angular frequency. The response of the material was studied in the frequency domain where the stresses and strains are related by easily evaluated algebraic coefficients. In the time domain the stresses and strains are related by differential operators which are difficult to evaluate, and it may be difficult to apply differential operators of a high order.

35 The Laplace transformations were used as a method of solution of the integrodifferential equations to be discussed in Chapter II where the solution of the operator equations or the equivalent mechanical models which they represent were presented in the time domain. The Fourier transforms or their extension, the Laplace transforms, were used to convert the problem from the time domain to the frequency domain where the frequency spectrum of the solution can be simply obtained by purely algebraic means. In order to find the solution as a function of time, the inverse transform is then required since the dynamic quantities are the Fourier or Laplace transforms of the static quantities, and conversely. Analytical and graphical procedures were developed to obtain fundamental properties of the material at any temperature within the tested temperature range. These procedures used a generalized Voigt model to describe the mechanical response of the material. A composite master creep curve was developed from the individual creep functions at various temperatures and was used to extend the experimental results to time intervals longer or shorter than normally available from the limited experimentation planned. The mechanical response of the material was investigated by direct experimentation in order to determine if time and temperature have equivalent effects in defining the linear viscoelastic mechanical response of bituminous mixtures. And finally, two independent complex moduli, the complex elastic and transverse moduli and their respective phase angles, were predicted in the frequency domain by the distribution functions of retardation time obtained from transient creep tests.

36 Dynamic tests were used to determine the complex moduli directly in the frequency domain, which were compared to the predicted experimental complex moduli from transient tests. A correlation of the dynamic and static tests results in the frequency domain was used in an attempt to provide a verification of the theory of viscoelasticity to describe the mechanical response of the material. Such a correlation would also verify that time and temperature have equivalent effects upon the rheological properties and the application of the tim e-tem perature superposition principle to this m aterial. Correlation of dynamic and transient results will also provide verification of the theory, assumptions, and experimentation used in this study. If the principles investigated in this study are applicable to the rheological behavior of bituminous mixtures, it would allow the variable of temperature to be reduced in importance in future studies and design procedures. Previous studies have dealt extensively with the rheological behavior of high polymers, and it is quite appropriate now to undertake an analysis of the time and temperature dependence of the mechanical behavior of mixtures of bituminous materials and aggregates in order to rationalize the response of such mixtures under stress and to incorporate the variable of temperature into pavement design procedures. There are several methods for describing the stress-strain-tim e behavior of a material whose response to stress obeys the Boltzman superposition principle and which can be represented by a model constructed from elements which obey Hooke's elastic and Newton's viscous laws. One of the fundamental ways

37 is the method of the operator equation which involves a single differential equation of the nth order relating either the isotropic stress-strain tensors or the deviatoric stress-strain tensors. The same number of independent parameters is required to characterize a material if the operator equation or model representation is used. The major effort of this study was to define a model describing the response of the material under stress at any time or temperature or to define the operators which is equivalent to defining a model. Finally, the response of the material expressed as complex functions of frequency was extended to the three-dimensional state of stress in an attempt to define completely the material. The highest degree of usefulness of the complex moduli lies in the fact that all relations which are valid for the Hookean elastic solid in the time domain may be extended to the frequency domain and be used to describe the three-dimensional response of a viscoelastic solid to a state of stress when the moduli are expressed in terms of complex quantities. Experimental Scope The experimental phase of this investigation consisted of applying the time-temperature superposition concept to determine the rheological response of bituminous concrete mixtures on the pheonomenological level using transient and dynamic tests. The laboratory experimentation evaluated the time-dependent response of the material by constant stress tests and determined the relatively fast response of the material by means of dynamic tests at several frequencies in the linear range. The two sets of data covering a wide range in time and temperature scale were synthesized, and the resulting description of the visco-

38 19 elastic properties covered the entire range of the time scale from the total duration of the creep tests to the fastest frequency of the dynamic tests. This method of experimentation was applied as a knowledge of the complex moduli at all frequencies and temperatures completely described the mechanical properties of the material. It should be noted that this study was concerned with only the linear viscoelastic response of materials, and "failure" criteria were not included in the scope of the research. The viscoelastic response of the material to fast rates of loading was tested by means of dynamic tests in which the material was subjected to a sinusoidally varying axial stress, and the resulting axial and circumferential periodic strains were studied. The response of the material to constant static loads was studied by creep tests in which a constant axial load was quickly applied and the resulting changes in the axial and circumferential strains recorded by a Sanborn recorder to one hundredth of a second. The bituminous concrete specimens were tested under isothermal conditions over a temperature range from 0 to 120 F. to develop the individual transient and dynamic rheological curves at each load and test temperature. The data from the individual creep tests were used to develop a master creep function curve which was used to define the response of the material at all temperatures in the tested range as a function of time or frequency. A detailed investigation of the mix variables upon the rheological properties of bituminous concrete mixtures was outside the scope of the study. The experimental phase was conducted using several types of mixes in which the type and

39 20 gradation of aggregates and the amount, source, and kinematic viscosity of bitumen are varied. By use of this method the qualitative effects of the mix variables upon the rheological response of the bituminous mixes were available, and the behavior characteristics of the mixes used in this study were also quantatively established.

40 CHAPTER n MATHEMATICAL METHODS OF SPECIFYING THE TIME-DEPENDENT VISCOELASTIC RESPONSE OF MATERIALS General The classical theory of elasticity deals with the mechanical response of perfectly elastic materials, where in accordance with Hooke's law, stress is always directly proportional to strain and independent of the rate of strain. The hydrodynamic theory deals with the properties of viscous liquids, where in accordance with Newton's law, the stress is proportional to the rate of strain and independent of the strain itself. However, these materials are idealizations and real materials existing in nature generally show deviations from the above behaviors by stress and time anomalies. A viscoelastic material is one which exhibits both elastic and viscous characteristics and stress is related to strain by a function of time in the linear viscoelastic range. Before the dynamics of viscoelastic response can be readily applied in dealing with the response of real materials, it will be helpful to review the concepts of stress and strain and the separation of volume and shear effects by using the classical theory of elasticity. Elastic Response of Materials The mechanical response of an ideal, isotropic, elastic material can be described completely by the use of any two independent elastic material constants, 21

41 22 for example, the shear modulus and bulk modulus. The classical theory of elasticity also suggests that any strain in an elastic material can be conveniently separated into two components which cause a change in shape at constant volume and a change in volume with a constant shape. Hence, if we use the shear modulus and bulk modulus or the Young's modulus and Poisson's ratio as the physical constants to represent the elastic properties, a simple mathematical separation of pressure-volume and shear effects can be obtained. Mathematically, the stress and strains are tensors and can be resolved into the sum of a mean normal tensor and a deviatoric tensor which Love (26) has shown to represent the pressure-volume and pure shear effects, respectively. The most useful definition of stress and strain and the most concise statements of stress-strain relationships are formulated in terms of tensors. The stress tensor, <r.., has nine elements, of which six are independent for o-.* = o-.., where 1J U Jl tr denotes the stress and i, j are free variables which may take on any of the three values x, y, and z. The stress tensor can be represented by o^, which symbolizes the entire set of nine stress components in the following matrix: ^ij ~ *xx *y <r. yx yy *zx zy <T where o-^, (ryy, and <rz7 are the normal stress components, and <rxy, trxz, e tc., are the shearing stress components. The mean normal stress c, may be represented by: II-l <r = ^xx * Pyy + a zz H-2

42 The mean normal stress has the physical significance of a hydrostatic pressure. The deviatoric stress tensor, Sjj, is obtained by subtracting the mean normal stress from each of the three normal stress components. si j " tr XX V ^zx - cr yy-0' x z yz 'zz"* II-3 or p -j " aij " &ij * where is the Kronecker delta, and gjj = 1 for i = j, and g^ = 0 for i ^ j. The mean normal stress, <r, tends to change the volume of the material while the deviatoric stress tensor, s ^, alters the shape of the material. The state of strain at a point in a body is completely specified once the three extensions in mutually perpendicular directions and the six components of the changes in the right angles in the three perpendicular directions are defined. The nine components form a matrix called the strain tensor, ey, represented by: Eij " ^xx Exy exz eyx Eyy eyz Ezx c*y Ezz H-4 where e is the strain and the other notations are the same as for stress. When the displacements of a point in the three mutually perpendicular directions, x, y and z are given by u, v, and w, respectively, the infinitesimal strain tensor can be defined as

43 The normal strains are denoted by e**, Cyy, czz while the others are shearing strains. It should be noted that the shearing components of the engineering strain tensor is equal to one-half of the corresponding classical shearing strains. The strain tensor, e^, can also be resolved into a mean normal strain, e, and a deviatoric strain tensor, e^. The mean normal strain is defined as: e = cxx + Eyy + gzz n-6 3 The deviatoric strain is obtained by subtracting the mean normal strain, c, from each normal strain: 24 lexx " e :yx : zx xy Eyy e ezy cxz 6 yz c zz c n -7 or eij = cij ~ &ij * e The mean normal strain has the physical significance of measuring the volumetric dilation of the strained material and e is equal to 1/3 of the fractional volume dilation. The deviatoric strain tensor, e^, measures the shape distortion of strained material. Using these equations to establish the relation between stress and strain, the bulk modulus, K, governs the volumetric behavior, and the mean normal stress is directly proportional to the mean normal strain, which is a pressurevolume relation. <r = 3 K e II-8

44 25 The shape deformation is determined by the shear modulus, G, which relates the deviatoric stress tensor to the deviatoric strain tensor (term by term). Sjj = 2 G e^ II -9 The corresponding stress-strain equations for a purely viscous liquid may also be obtained by making use of the elasticity-viscosity analogy and by replacing the strain tensors by their derivatives with respect to time, and the bulk modulus by the coefficient of volume viscosity X, and the shear modulus by the coefficient of viscosity, qt (27). Thus, the stress-strain relationships for a purely viscous liquid may be shown to be <r = 3 X n-10 and sij = 2t ij In the analysis of linear viscoelastic m aterials, it will still be convenient to resolve the response of the material into volumetric and shape changes. The stress and strain components will be functions of time, and the stress tensor as a function of time will determine the strain tensor as a function of time. The analysis will be somewhat simplified since the mean normal strain as a function of time is completely determined by the mean normal stress as a function of time, and the time-dependent component of the deviatoric strain tensor is determined by the corresponding time-dependent component of the deviatoric stress tensor.

45 26 Phenomenological Treatment of Linear Viscoelastic Behavior A linear viscoelastic material can be defined as a material such that if the stress is doubled, the resulting strain will also be doubled at a given loading time interval and in a given type of experiment. The mechanical response of such a material may be described by a combination of linear elastic springs and viscous dashpots. A portion of the deformational energy is stored in the material while a portion of the energy is dissipated since these materials exhibit properties which are found in both elastic and viscous materials. It can be shown for a linear viscoelastic material that the magnitude of the applied stress influences the time-dependent deformation by a constant factor, and the creep modulus, Ec (t), at a constant temperature, is a function of time only and independent of the stress level in the linear viscoelastic range. If the stress is doubled in a constant stress experiment at any given time, exactly twice the deformation will result so that the creep moduli at the two different stress levels, a- and 2o-, or at any stress, no-, in the linear viscoelastic range, are identical. Similar concepts may also be applied to define the linear viscoelastic behavior of any material in constant strain, dynamic, and other experimental methods. Mechanical models are often used to represent real materials and formed to duplicate the observed time dependence of their response. At sufficiently low levels of stress compared to their ultimate strength, many materials are known to behave in a linear viscoelastic manner, within a practical degree of approximation. To describe the response of a particular material using the model repre

46 sentation, a mechanical system is used consisting of Hookean springs, Newtonian dashpots, and St. Venant elements connected in a series or parallel in various configurations. The mechanical model exhibits the same relations between force, elongation and time as those between stress, strain and time for the linear viscoelastic material which it represents. The springs and dashpots depict the essential features of the molecular param eters of the material; a Hookean spring is an element which stores energy on deformation, while a Newtonian dashpot and a St. Venant unit dissipate energy on deformation. Generally at the present time only a qualitative and tentative mechanistic interpretation about the internal parameters of real m aterials can be discussed, except than that on deformation, some of the elements store energy in a reversible manner and others dissipate energy irreversibly. However, molecular interpretations in the case of complicated materials should not be avoided, but the approximation between phenomenological theory and molecula r theory of viscoelastic behavior should be recognized. t Operator Equation In addition to the model theory there are several other methods for specifying the response of a linear viscoelastic material (28). One such fundamental method for representing the stress-strain equations of a material is the operator equation where the corresponding stresses and strain are related by linear operators. In order to describe completely the response of an isotropic material, two equations must be used, one for the deviatoric stress-

47 28 strain tensors and one for the isotropic stress-strain tensors. The equations may be written as and P (<r) = Q (e ) P (stj) = 2Q (ey) where P, ^ P and Q are linear differential operators and the other terms have the same meanings as before. The linear differential operators may be of the form and p P = Z r Pr r=o Q = S q qr _ 8r H-15 r=o, 8tr 1 Equation represents the shear behavior in terms of the deviatoric stress and strain, while equation n-12 gives the relation between mean normal stress and volume change. These two relations, with the two independent pairs of viscoelastic operators P, Q and P, Q determine the relations between the twelve independent stress and strain components. Since the shearing component of the deviatoric strain tensor is equal to one-half of the corresponding classical shearing strain, the factor, two, in the above equation and other analogous equations results. Electric Analog A third basic method of specifying the time-dependent properties of linear viscoelastic materials is the electric analog. The time dependent behavior of the Kilvin and Maxwell elements is exactly analogous to the time-dependent

48 29 electrical response of combinations of resistances and capacities. There are several possible ways to set up the analog, the most satisfactory being to equate capacities to springs and resistances to dashpots. By the use of this analog, the energy storage and dissipative units correspond physically, but parallel and series mechanical connections correspond to series and parallel electrical connections, respectively. Extensive literature has been devoted to this topic (24), (29) and it will not be discussed here in detail. Fundamental Mechanical Models Although the laws of rheology are essentially mathematical, they can be visualized by analyzing the behavior of some simple mechanical models. The most basic elements of viscoelastic mechanical models are: the Hookean elastic solid, the Newtonian viscous liquid, and the St. Venant plastic solid, all of which are used to represent mathematically the behavior of real materials. The properties of these abstract concepts can be combined into various arrangements in series or parallel combinations to produce a mathematical expression defining the stress-strain-tim e behavior of the material at a given temperature. Hookean elastic deformation can be represented by a spring alone, while Newtonian viscous flow can be represented by a single dashpot; but for the description of the more complicated rheological deformations of real materials, the spring, dashpots, and St. Venant elements must be combined in various ways. A basic concept of rheology reasons that all real materials possess all rheological properties, represented by Hookean, Newtonian, and St. Venant

49 30 bodies, but in varying degrees (30), (31). However, the yield stress of many materials is infinitesimal, and the St. Venant element is often omitted from a model describing the mechanical response of the material. There are several simple models which are combinations of the elastic and viscous elements exhibiting viscoelastic behavior: the most familiar are the Maxwell and Kelvin models. Many authors (2), (24), (32). (33) have discussed extensively these schematic representations of viscoelastic behavior. Only the Maxwell, Kelvin, and Burgers models will be discussed here in detail, since they form the basic units of the more complicated models used in this study. Kelvin-Voigt Solid A rheological model, proposed by Kelvin in 1875, sometimes called the Voigt element, shown in Figure la, consists of a linear elastic spring with elastic modulus E^, connected in parallel with a dashpot containing a linear viscous liquid of viscosity r^. Under any external applied stress, the strain in the spring and dashpot will be equal. The total stress carried by the model is the sum of the stresses in the spring and dashpot. Making use of Newton's law for viscous flow and Hooke's law of elastic behavior, the following equation for the Kelvin model may be obtained <r = E] e + r j^ c H-16 where E^ and are the coefficients of the elastic and viscous elements, respectively. If the Kelvin model is subjected to a constant stress <r0, the

50 J ±1 V. (a) Kelvin Model (b) Maxwell Model E m l i l 1? 'm (c) B urgers Model Fig. 1. Fundamental Mechanical Models.

51 32 solution of equation n-16 is e = ^ 2 - ( l - e _t^t k ) E k where t ^ = % /E^ and e is the base of natural logarithms. The quantity r^ E k has the dimensions of time and is defined as the retardation time, t^, of the Kelvin element. The significance of the retardation time is brought out by considering a constant stress experiment in which the load is suddenly removed from the material and the strain diminishes. The recovery of strain follows a simple exponential relation for the Kelvin model, and the retardation time, t^, is the time required for the strain to decrease to l/e of its original value. Under a constant stress the Kelvin element exhibits a creep behavior, the deformation being of the form J. (t) = X ( i - e"t/tk) % where J c (t), the creep compliance is equal to c (t)/cr0. One of the simplest rheological experiments is the creep test, where at time t = 0 the material is subjected to constant stress, o-0, and its deformation is measured as a function of time. A pure Hookean material would respond with a constant deformation of magnitude J <r0. The time dependence of real materials manifests itself in a monotonically increasing function J c (t), and its time dependence determines the deviation from pure elastic behavior. Maxwell Liquid The Kelvin model does not account for the stress relaxation phenomena observed during a constant strain test. To represent this behavior, we must

52 consider another basic model, such as the mechanical model described by Maxwell in 1868 shown in Figure lb, which consists of a spring and a dashpot connected in series. In the Maxwell model the total strain observed is the sum of the strain in the spring and the strain in the dashpot, while the stress in the spring and dashpot are equal. The equation for the rheologic behavior of the model may be shown to be where Em and are the coefficients of the elastic and viscous elements, respectively. If the Maxwell model is subjected to a constant strain, eq, the particular solution of this equation is o- = (r0 e -t^t m where r m = r^ /E m and cr0 is the initial stress. The retardation time, Tm, is equal to the time required for the stress to relax to l/e of its initial value when the strain is held at a constant value. As the model undergoes a sudden constant deformation, all of the deformation energy is stored in the spring and only this element is stretched in the first moment. However, with increasing time, the dashpot flows and the energy is transmitted to the dashpot and dissipated as the stress relaxes. The stress relaxation modulus, Er (t), of the Maxwell model decays exponentially with time Er (t) = Em e~t/t m n-21 The Kelvin and Maxwell models represent the two basic classes of m a terials: solids and liquids. The Kelvin model represents the viscoelastic behavior of solids because it will deform to a limiting value under a given load

53 34 and all deformatlonal energy is recoverable. The Maxwell model illustrates the response of liquids to applied loads as the model deforms permanently due to the smallest loads and the deformational energy is not recoverable. Burgers Model The fundamental Kelvin and Maxwell models are insufficient to describe completely the viscoelastic behavior of real materials since the Kelvin unit represents retarded deformation and the Maxwell unit represents stress relaxation. To develop a model which behaves more like a real material, we need a combination of several basic units since most viscoelastic materials will exhibit both relaxation and retardation in various degrees. The Burgers model, shown in Figure lc, is a combination of the Kelvin and Maxwell units connected in series. The deformation of the model under a constant load is the sum of the elastic deformation of the single spring, the retarded viscoelastic deformation of the Kelvin element, and of the viscous flow of the single outside dashpot. The creep compliance yielded by the four parameter model becomes J c w - e"*/tk> n ' 22 where Em m and n 'm are the elastic and viscous elements of the Maxwell body and those of the Kelvin body are and r^. TTie general behavior of the Burgers model is determined by the differential equation between stress and strain, which can be derived from the instantaneous

54 35 elastic strain of the outside spring, e j,, the retarded strain of the Kelvin element, e and the strain due to the single dashpot, : ct = ee + c4/ + cri The total strain, c j, is made up of the strain of the spring, Kelvin element, and dashpot which conform respectively to the following equations: <r = Em ee ktk * Tk n "25 " = V % n ' 26 From the above equations we can derive the following general differential equation of the Burgers model pdk- +Hk ) + -^k_l a ~ L + e. 1e n-27 K d? ' Em V * Im J r * 2 ^ d J Figure 2 shows the response of this mechanical model to a constant stress applied at time, t j and removed at time t2. The solution of equation n-2 for a constant stress <r0, applied to the unstrained model is c (t) = 1. + fj L ( 1 - e~t/ Tk ) + n -28 m Ek qm Generalized Voigt and Maxwell Models In the previous section we have shown that the mechanical behavior of real materials may be approximated by models composed of a finite number of linear springs and dashpots. However, if the mechanical behavior of engineering materials is observed closely, it is immediately realized that the above picture is too simple to describe adequately their behavior and more complicated models

55 36 M M CO CO a> CO o X < Time,t N w c o X < Time,t Fig. 2. Axial Stress and Strain vs. Time Relations for Burgers Model.

56 37 must be used. A creep curve from a constant stress test shows not one but a number of viscoelastic transitions. In order to account for such a series of transitions, a generalized model must be used to describe the stress-strain time response of material bodies. Several authors (2)t (24) have proposed that it is possible to represent the mechanical behavior of viscoelastic bodies by using an infinite number of elements in the model. The generalization of the Voigt model is used when the creep behavior of a material is studied, or the generalized Maxwell model is used when dealing with stress relaxation. The generalized Voigt mechanical model in Figure 3a, shows instantaneous elasticity EQ, Newtonian flow with viscosity, r o, and n different retardation times t t 2,..., Tn due to n Kelvin elements connected in series. The Kelvin element with retardation time t has a spring with elasticity Ej and a dashpot with viscosity The experimental creep curve has instantaneous elasticity and viscous flow and n different transitions due to the Kelvin elements with elastic compliance equal to where n w J.(t)=E Jj ( 1 - e Ti) i=l The model is defined by 2n + 2 quantities, the n + 1 dashpot viscosities and the n + 1 elastic spring constants.

57 (a) Generalized Voigt Model E, > E, > Ebh \±iv, tjd7?, tp v, (b) Generalized Maxwell Model Fig, 3. Complex Mechanical Models.

58 39 In the special case above, two of the Kelvin elements have degenerated into two of the fundamental elements, a Hookean solid and a Newtonian liquid. This can be visualized by noting that as approaches zero in equation 11-16, the stress-strain-tim e relation becomes identical to the equation of a viscous dashpot, and as approaches zero, equation n-16 becomes identical to the equation for an elastic solid. Similar degenerate elements may also be found in the Maxwell model. The generalized Maxwell model shown in Figure 3b consists of n Maxwell elements connected in parallel and represents a continuous spectrum of relaxation times. In a parallel arrangement the stresses are additive and the viscoelastic functions are obtained by summing over all of the n elements to determine the stress relaxation response n Er (t) = 2 E. e t/ i H-31 i=l Each Maxwell element consists of a spring Ej and a dashpot r^, exhibiting a relaxation time t. = t^/e j. Retardation Spectra The previous sections have considered mechanical models composed of a finite number of springs and dashpots to approximate the rheological behavior of a real material at a specified temperature. A powerful extension of this method is the generalized Voigt model consisting of an infinite number of units with characteristic constants varying continuously with the retardation time of

59 each individual Voigt element. Such a network is not characterized by a finite number of constants, but instead by a continuous function of one independent variable. In the case of a generalized continuous Voigt model, this variable is called the distribution function of retardation times, and in the case of the generalized continuous Maxwell model, it is the distribution function of relaxation times. A major reason for the use of distribution functions to describe the viscoelastic response of materials may be observed by comparing the experimental creep curves of several materials with the theoretical creep curve calculated for one Kelvin element represented by equation n-18. The experimental creep compliance transition curve cannot be described accurately by the theoretical curve of equation n-18. It can be shown that the experimental creep curves of a bituminous concrete mixtures increase more gradually and extend over a longer time region that the theoretical Kelvin creep transition curve. The experimental curve may be approximated to a higher degree by the theoretical curve simply by increasing the number of elements in the model from which the theoretical curve is calculated. Later sections will present detailed graphical and mathematical methods for obtaining analytical expressions to represent the mechanical response of materials, which can be carried to any degree of accuracy desired for a particular use.

60 41 Distribution of Retardation Times A continuous model can be constructed starting from the generalized Voigt model as shown in Figure 3a. This model consists of a single spring of compliance J 0, a single dashpot r 0, and an infinite number of Kelvin units connected in series. The Kelvin elements can be arranged according to increasing retardation times, and the contribution of each individual compliance Jj can be considered as a function of the retardation time, Tj. In the continuous model the Kelvin units with retardation times between t and r + dr will contribute to the compliance. The function L ( t ) is called the distribution function of retardation times or the retardation spectrum where o < t < 0 0. The creep behavior of such a model may be obtained from the creep compliance of an infinite Voigt model containing an infinite number of Kelvin elements and two degenerate elements in series and is in the form o It has also been pointed out in the previous section that the creep compliance of a real material can also be approximated by a model which contains a finite number of Kelvin elements. Equation n-30 leads directly to the introduction of the retardation function, L (t ), to represent the delayed elastic component of the creep compliance. Since the delayed elastic component of the creep compliance increases monotonically from zero at time equal to zero to a limiting value at finite time, this component of viscoelastic deformation can be represented by an integral of the form presented in equation n -32.

61 42 The physical meaning of the retardation times is described in equation 11-32: the retarded elastic deformation (t) can be considered to be caused by an infinite number of retardation processes. The retardation spectrum gives the contribution per unit interval of the time scale of the retardation process with L (t ) close to t to the equilibrium compliance of retarded elastic deformation or primary flow. Distribution of Relaxation Times The relaxation modulus of a generalized Maxwell model can be found from a sim ilar equation: n. Er (t) = 2 E. e t/t i + E i=l This leads to a second auxiliary function to represent the stress decay modulus component of the relaxation modulus. From equation n -3 3 we may write Er (t) =[ H (t ) e_t/t dt + Eq J o If the relaxation spectrum, H (r), is defined for all times, the mechanical behavior of the model is completely determined. The constant value EQ, which is obtained by the model after an infinite time, corresponds to the elasticity of a single spring. This spring may be connected in parallel with the generalized model of Figure 3b, so that the model used to represent the mechanical behavior of a material will have a finite stress after an infinite time.

62 43 Retardation and Relaxation Spectra in Logarithmic Time Scale The retardation and relaxation spectra as described in previous sections are the appropriate quantities when studying the viscoelastic behavior on the linear time scale. However, it may be shown that the time or frequency dependence becomes more pronounced if we plot the rheologic curves in logarithmic time or frequency diagrams. The range of both the magnitude of J c (t) and the time scale are also so large that the only way to represent the material in a single graph is to make both coordinates logarithmic. In logarithmic scales, two other distributions must be used: the logarithmic retardation spectrum, L (in t ), and the logarithmic relaxation spectrum, H (in t ). The meaning of the logarithmic distributions L (in t ) and H (in r) is the contribution of the retardation or relaxation processes with characteristic times between in r and in t + din t to the creep or stress relaxation. Using the logarithmic distribution functions, the stress relaxation and creep response can be evaluated by means of the following equations: Once the distribution functions have been evaluated, the dynamic response of the material, such as the loss and storage compliance or the real and imaginary moduli in the frequency domain, can be obtained.

63 44 Dynamic Response of Materials In previous sections transient experiments were considered in which the applied stress or strain is essentially a step function and equal to zero up to a given instant and then changes discontinuously to a finite value. We will now consider the response of materials in another type of test in which the stress varies sinusoidally with time, and the material undergoes axial and lateral sinusoidal deformations at the same frequency as the stress, but lagging the stress by a phase angle, 4* (29), (34). The concept of the complex modulus applied here to define the viscoelastic response of materials is based on the fact that when a sinusoidal excitation is applied to a material, the response of the material will also be sinusoidal, but out-of-phase with the stress by a certain amount. A linear viscoelastic material subjected to a sinusoidal stress will reach a steady-state condition after a limited number of cycles, due to mechanical conditioning, and the amplitudes of the stress and strain are related by the absolute value of the complex modulus, /E*/- The absolute value of this complex number is obtained by dividing the amplitude of the sinusoidal stress, <r0, by the amplitude of sinusoidally varying strain, c0, and the phase lag, 4, shown in Figure 4a, is the angle by which the strain lags the stress. If a given sinusoidal stress is imposed on a material, the measurement of the amplitude of the strain and the angle by which it lags the stress will define the response of the material at a single frequency, w, By evaluation of the phase lag and absolute value of the complex modulus at all frequencies, the

64 : * m w O c ow CO Time i E* E,U) Edk*) 2t (o) S te a d y-s ta te Response of a Viscoelastic Moterial too Sinusoidal Strain or Stress (c) Vectorial Resolution of Modulus Components in Sinusoidal Deformation i Cos^ E* (b) Rotating Vector Representation (d) Vectorical Resolution of Compliance Components in Sinusoidal Deformation Fig. 4. Dynamic Viscoelastic Response and Vectorial Resolutions.

65 46 viscoelastic response of the material is completely defined. The complex modulus of a material is a function of frequency, and like all complex numbers, may be resolved into a real and imaginary part or an absolute value and a phase angle. Thus, the mechanical behavior of a viscoelastic material at any given frequency can be specified by two independent quantities which completely describe the m aterial, the real and imaginary components or the magnitude and phase of the complex modulus. Using the above conditions, we can decompose the stress vectorially into two components, one in phase with the strain and one leading the strain by -n/2 radians which are rotating counter clockwise with a frequency equal to the frequency of the applied sinusoidal stress, as presented in Figure 4b. An alternate procedure is to resolve the strain into two components, one in phase and one lagging the stress by 90, which is possible by the rotating vector representation. When the stress is divided into two components, one in-phase with the strain and one leading by ^ fl radians, the value of the real part of the complex modulus is obtained by dividing the amplitude of the in-phase component of stress by the amplitude of strain, and the imaginary component is obtained by dividing the amplitude of the out-of-phase component of stress by the amplitude of the strain. The real part of the complex modulus, (u), is a measure of the elastic part of the deformation or energy stored and is therefore called the storage modulus. The imaginary part is associated with the internal friction loss or dissipation of energy and is called the loss modulus, (w). It is interesting

66 47 to note that the smaller the phase lag between stress and strain, the more elastic the material; while the larger the phase lag, the more viscous the material. As an illustration of the previous concept, let an alternating stress tr = <rqsin u>t be applied to a linear viscoelastic material where <rq is the amplitude of stress and w is the angular frequency of vibration. The steady-state strain will vary sinusoidally e = e 0 sin (wt - 4) with the same frequency as the stress, with an amplitude eq and a phase lag, 4- Either the absolute value of the dynamic modulus, / E*/, and phase lag, or the real and imaginary parts of the complex modulus may be used to describe the dynamic response of the material, since these two methods of representation are equivalent and are related to each other. The relations between the dynamic response parameters for linear viscoelastic behavior are defined as follows /E */ = IS. Eo Ej, (w) = /E */ cos 4 E2 H = /E */ sin 4 n -39 n -40 n -4 i If any two of the above characteristic quantities are known, then all the others can be calculated. The complex modulus also may be expressed in terms of the real and imaginary parts E* = E^ (jj) + j E2 H 11-42

67 where j is the imaginary unit, and in terms of the absolute value of the complex modulus and angle of lag by E* = /E * / e ^ H-43 where e is the base of natural logarithms. The meaning of these formulus is illustrated in Figure 4c by the vectorial resolution of the dynamic values in the form of a complex vector diagram, where 4 is the phase angle, /E */ the hypotenuse, E^ i^>) and E2 {*>) the sides, and j and r the imaginary and real axes, respectively. The dynamic data may also be expressed in terms of a complex compliance which is the reciprocal of the complex modulus J* = 1p) - j J 2 n where the storage component, (00), is obtained from the strain in-phase with the stress, and the imaginary component, J 2 (w), is obtained from the strain tj/2 radians out-of-phase with the stress. Although J* is the reciprocal of E*, their individual components are reciprocally related, but are connected by the following equations derived by vectorial resolution shown in Figure 4d J * = Ej (u) + j E2 (w) Multiplying the conjugate, we obtain

68 49 Similar analytical relations can be written for the components of the complex bulk modulus, K*, complex shear modulusy G*, complex transverse modulus, T*, or any other complex modulus, M*, and their corresponding compliances. The Boltzman Superposition Principle The Boltzman superposition principle (22) is an extension of the principle of linearity and is identical to the latter if defined as the concept that the total strain due to several stresses is exactly equal to the sum of the strains caused by the individual stresses. Thus, by the use of the superposition principle, the effects of different stresses are superimposed by simple addition. If a constant stress (j-q is applied at t = 0, the strain as a function of time will be given by e (t) = «rc J (t) If an additional stress o-^ is applied at t = t 1 the total strain according to Boltzman1s superposition principle will be e (t) = <r0 J (t) + Oj J (t - tj) H-49 which is the superposition of the two strains at their respective elapsed loading times. There are innumerable possible variations of loading history, whose effects can be generalized by the basic equation Ui=t e (t) = 2 o-i J (t - ui) Ui= -oo The summation of equation may be expressed as an integral by letting the stress as a function of time u be given by <r (u). Thus, the increment of stress

69 from time, u, to time, u + du is ( d.?.(v) )du and equation may be written du e (t) = / SL lm J (t - u)du n-i n-5i J-oo du Equation relates the strain at time t to the previous stress history through the creep compliance. If the loading time starts at zero time, the integration limits will then vary from zero to t. This is the case in a constant stress experiment where the stress is zero for times smaller than zero and a constant value for times greater than zero, implying a discontinuity at time equal to zero. Integration by parts of equation leads to the alternate formula o- (t - K) d J (K) dk dk where K is equal to the elapsed time (t - u) and J Qis the value of the creep compliance at t = 0. The above equations express the strain in terms of the entire past stress history. Analogous relations for the stress in terms of the past strain history can be expressed by and dk n -54 where ro is the stress relaxation modulus at t = 0. When the stress or strain histories are discontinuous, the form of superposition principle in equations and may be used.

70 51 If the creep compliance is known for all values of time, the deformation of the material can be predicted for any arbitrary stress described as a function of time, <r (t). The deformation can be obtained by a form of the superposition principle given by equation n-51. If the stress relaxation modulus, Er <t), is known, the stress produced by an arbitrary deformation e (t) can be predicted by <r (t) = f *Jsa t J -» E_ (t - u) du n-65 du The concept of the superposition principle is seen in the above equations where the total effect e (t) and <r (t) is evaluated as the sum of the effects of a great number of causes. Each of the causes are an increase of stress or strain during the time interval between u and u + du. The form of the superposition principle in equation n-51 and n-55 contains the time derivatives of the stress history, a (t), and the strain history,! (t), in the integration and may be used in calculations for all cases where the prescribed stress or strain histories are continuous functions of time without discontinuous jumps. Complex Compliance Function To supplement the transient experiments, the dynamic behavior of materials was also considered in which the angular frequency, u>, is the independent variable in periodic or dynamic loading patterns at a constant temperature. In the previous transient constant stress and constant strain experiments under isothermal conditions, the loading time, t, has been the independent variable. The

71 52 dynamic response of a material whose rheological behavior is represented by the equation of a generalized Voigt model may be obtained by the form of the superposition principle suitable for phenomena where the stress is specified, as in equation n-51. If a sinusoidal stress, <r = a0 e^*, is substituted into equation n-51 and a partial integration performed, the complex compliance function can be obtained in the form J* = J 0 + _ 1 + f e ' j w T d (T> dt n -56 JWTlo J o dr Decomposing J* into its real and imaginary components and J, (.) = J + r d ^ - (T) oos» t dr H-57 1 Jo dt 00 d J i i (t ) g j n WT (jt + _ 1 _ dt WT q where the instantaneous component of the compliance is represented by J Q, -1 no is the component due to viscous flow, and (w) - j J 2 H is that part of the compliance associated with the retarded elastic creep phenomenon. Mathematically the above integrals are Fourier transforms and enable one to calculate the complex modulus function or the real and imaginary components of the complex modulus from the creep compliance function.. The values of the complex moduli in the frequency domain are independent of the applied stress or strain amplitudes at stress or strain levels in the linear range and depend only on the frequency, u. The rheological response of the

72 53 material to a sinusoidal stress at a given angular frequency is completely defined as both the phase lag, and magnitudes of stress and strain can be evaluated by experimentation. Three-Dimensional Viscoelastic Behavior The complete theory developed for the complex elastic modulus remains valid for all moduli and compliances. The elastic storage moduli E^, K^, and G}, represent those part of the stress in-phase with the strain; the loss moduli E2, K2, and G2 represent those parts of stress which are 90 out-of-phase with the strain. The storage Poisson's ratio is that part of the lateral extension which is in-phase with the sinusoidal longitudinal contraction; the loss Poisson's ratio is that part of lateral extension which is 90 out-of-phase with the axial contraction. Thus, we may write for a dynamic shear test the complex shear modulus in the form G* = Gj (d) + j G2 t>) where the modulus has been separated into an in-phase or real component and out-of-phase or imaginary component. We may also use similar relations to describe the dynamic properties of linear viscoelastic materials for the complex bulk modulus and complex Poisson's ratio K* = Kj (w) + j K2 (u) v* = vx * ) + j v2 (p) n -61 It is also possible to extend the above definitions to define other complex moduli and a specific complex modulus used in this study which relates axial

73 54 stress to the resulting lateral or transverse strain, and defines it as the complex transverse modulus, T*, where T* = T1 M + j T2 to H-62 For an isotropic, homogeneous linear viscoelastic material, all relations which are valid for an elastic material are also valid for the viscoelastic material when the complex'moduli are used in the frequency domain as the fundamental properties (2), (23). Thus, the response of a linear viscoelastic material can be defined by two independent complex moduli sim ilar to the classical elastic bodies, which are included as special cases in the previous concepts. The mechanical response of an isotropic, homogeneous, linear viscoelastic material can be completely analyzed by the use of any two independent complex moduli, i. e., the complex elastic and transverse moduli. The classical theory of elasticity (26) shows that any strain in an elastic material can be conveniently separated into two components to cause a change in shape at constant volume anid a change in volume with a constant shape. Hence, it may be argued that if we use the complex shear and bulk moduli to represent the strength properties of the material, a simple mathematical separation of volume and shear effects can be obtained in the frequency domain as well as in the classical elastic theory.

74 CHAPTER ID TRANSFORMATIONS TO FREQUENCY DOMAIN General In the previous chapter, the concept of the complex modulus was discussed in detail and found to be a complex function of frequency by which the stressstrain equations of linear viscoelastic m aterials are related by algebraic coefficients in the frequency domain. It is important to note that there are no simple analogous algebraic stress-strain equations for a viscoelastic material which can be obtained in the time domain. These equations which mathematically relate sinusoidal stress to sinusoidal strain by complex functions of frequency are identical in form to the stress-strain equations of the classical elastic bodies and include them as special cases. The complex moduli may be determined independently by two separate methods: a transient test in the time domain where the experimental results are transformed by Fourier o r Laplace transforms to the frequency domain, depending on which is applicable for the type of time function under consideration, to yield on analytical expression for the complex modulus and phase lag as a continuous function of frequency; or dynamic tests from which discrete values of the magnitude and phase lag of the complex modulus are obtained. 55

75 56 Since the constant stress test is easily performed and the resulting master creep compliance curve quickly transformed to the frequency domain, this type of test was primarily used in the analysis of the temperature effects on the viscoelastic properties of bituminous concrete mixtures. However, dynamic tests were also performed to provide an independent check of the concepts, experimentation, and analysis presented in this study. In fact as discussed earlier, once the stress relaxation or the creep compliance function of a material is defined, all its viscoelastic functions can be predicted for any stress-strain history, including sinusoidal stress or strain. Therefore, it is possible to express the dynamic viscoelastic functions from the interrelations between the viscoelastic functions in terms of the static functions. The dynamic functions are the Fourier or Laplace transforms of the static functions, and conversely the static quantities in the time domain can be obtained by the inverse Fourier or Laplace transforms of the dynamic response of the material in the frequency domain. Thus, if one of the viscoelastic functions are known for all times or frequencies, the others can be calculated in principle, although these calculations are not always simple in practice and in some cases may not be feasible. Fourier and Laplace Transformations The fundamental concepts and applications of the Fourier and Laplace transforms may be found in references on operational mathematics (35), (36), and electrical linear circuit analysis (34), (37), and will be reviewed briefly in this section.

76 57 The basing of the steady-state solution of viscoelastic integrodifferential equations on sinusoidal functions in the earlier sections imposed no loss of generality as all other waveforms physically encountered can be expressed as a Fourier series or Fourier integrals. The Fourier series can be used to deal with periodic waveforms, and the Fourier integrals are used to handle nonperiodic waveforms. In both cases the waveform is considered to be built up of sinusoids of the appropriate frequencies, phases, and amplitudes. The analysis of a waveform into a frequency spectrum using the superposition principle by which the total response of a linear viscoelastic material is the sum of the material's response to each sinusoid acting separately, furnishes a useful method for evaluating the behavior of linear materials. This method is generalized in the following sections to obtain a mathematical tool for handling transient as well as steady-state response of materials by the following Fourier tran s forms: 00 f (t) e"*** dt m -i The Fourier transform can be used to transform a periodic function of time into a sum of discrete frequency responses, and the Fourier integral can be used to transform a nonperiodic function of time into a continuous function of frequency, provided the integrals converge. The Laplace transforms are closely related to the Fourier transforms with the frequency concept extended to include the generalized frequency

77 58 s = a + ju. By introduction of the concept of generalized or complex frequencies, it is possible to utilize the theory of functions of complex variables. In place of pure sinusoids with constant amplitude, the expressions are of the form to include sinusoids with exponentially increasing envelopes for the case of a greater than zero, and sinusoids exponentially decreasing envelopes for the case of a less than zero. Pure sinusoids can also be included as a special case when a is equal to zero. By use of the generalized frequency, many time functions that cannot be dealt with because of lack of convergence of the Fourier integral can be treated by use of the Laplace transform. In contrast to the classical method of solution of the integrodifferential operator equations discussed in Chapter n in which the solution of the viscoelastic response was obtained in the time domain, the Laplace transform converts the problem from the time domain to the frequency domain. In the frequency domain the frequency spectrum of the solution can be obtained by simple algebraic means. To determine the solution as a function of time, the inverse Laplace transform is required. The integral of the Fourier transform of equation ni-1 may fail to converge for many of the time excitations, f (t), used unless f (t) approaches zero as t approaches infinity which cannot be assumed to be the case. It can be shown that the Fourier transform does not converge for a sinusoid of constant amplitude or a step function excitation which are common functions studied. Thus, the Fourier transformation method cannot be directly applied to some of the most frequently encountered excitation functions. In transient tests it is

78 59 often desirable to be able to introduce the initial conditions that hold at some instant of time. The initial conditions summarize the state of the energy storing elements at t = 0, and contain the relevant information regarding the past history of the system. In conjunction with the excitation applied after t = 0, they define the future behavior of the material. Examining the above considerations, it is advantageous to modify the Fourier transform in the following manner. If the functions are considered identically zero until t = 0, the limits of integration in the direct Fourier transform become 0 and <». In order to insure convergence of the integral at the upper limit, the integrand is multiplied by e, where a is a sufficiently large real constant. By application of these conditions the direct transform becomes m -3 which can be regarded as the direct Fourier transform of the function f (t) e -at so that the latter can be obtained from F (a + ju>) by the inverse Fourier transformation The expression for the time function then becomes III-5

79 60 The generalized frequency, s, can be defined as s = a + juand since a is a constant, we can write a = c and obtain ds = j du III-6 Thus, the equations of the Fourier transforms become F (s) = J o f (t) e~st dt = L Ff (t)j in -7 and / c+j«i r n F (s) e ds = L F (s) m -8 C - j oo which are the relations desired. Equations HI-1 and HI-2 have become the Laplace transforms. Returning to the consideration of convergence, the fac- tor e has been used to multiply the given time function in order to make its Fourier transform converge at the upper limit and is essentially the technique of the Laplace transform. The expression will now converge to zero at the upper limit for any real value of a;, no matter how small ot is allowed to become, provided f (t) does not increase as fast an an exponential function as t approaches infinity. Interrelations Among the Viscoelastic Functions The representation of the behavior of a viscoelastic material was presented in Chapter n either by an infinite number Kelvin elements in series with a single Maxwell unit, or by means of an infinite number of Maxwell elements connected in parallel. Such a group of Maxwell elements in parallel

80 61 represents a discrete spectrum of relaxation times where each t. is associated with a spring mechanism E^. in such a parallel representation the stresses are additive, and the stress relaxation viscoelastic function Er (t) is obtained by summing over the parallel Maxwell elements. If there are n elements, the expression for the generalized Maxwell model is obtained in the form of equation n A finite group of Kelvin elements in series connection represents a discrete spectrum of retardation times,where eacht j is associated with a compliance of magnitude Jj. In such a series arrangement, the strains are additive and the viscoelastic function is obtained by summing over all the series elements in equation 11-29; where there are n units, the equation of the generalized Voigt model is obtained. Any experimentally observed creep curve for the compliance which increases monotonically can theoretically be fitted with any desired degree of accuracy by a series of terms by taking n sufficiently large in equation n -29. As the number of units in the Maxwell model increase without limit, the result is a continuous spectrum in which each infinitesimal contribution to rigidity is associated with a relaxation time whose logarithm lies in the range between In t and In t + d In t. For the continuous spectrum equation n-31 becomes 00 m -9 In an entirely analogous manner the Voigt model can be made infinite in extent

81 and represented by a continuous spectrum of retardation times and defined by the continuous analog of equation t / T L (In t ) (1 - e ) d (In t ) m -io To obtain the dynamic response of the material at a specified temperature in the frequency domain, Laplace transforms are taken of the viscoelastic re s ponse of the material defined by the m aster creep compliance function and represented by the generalized Voigt model in transient experiments. To obtain the response of the material in the frequency domain, the transform of the creep compliance function is used to evaluate the complex storage and loss compliances. The reason for doing this is that the components of complex moduli are easily obtained from the creep compliance function by the transformations described later in this chapter. Relations Between Creep Compliance Function. Dynamic Complex Compliance, and Retardation Spectrum The transient response of a simple Kelvin element in a constant stress test may be represented by equation n-18. To obtain the dynamic response of the material represented by this element as a function of frequency in vibration experiments under sinusoidal stress, equations and n-58 are applied to obtain the storage compliance by 00 o III-11

82 63 and the loss compliance by J 2 fr,) = J o (t) sin wt dt + l/oorio III 12 At time, t = 0, the creep compliance function is zero. Applying equation III 11 which is essentially a Laplace transform to the Kelvin transient response in the time domain, we obtain the storage compliance dynamic response in the form J x (w) = 0 + " (Ek T k> ' x e - t/t k c o s Wt dt = i < ) r n -1 3 The dynamic loss compliance is also evaluated in a similar manner using equation III-12 to obtain ra-i4 The above relations can now be applied to predict the dynamic response of any m aterial represented by the Kelvin element at any angular frequency. The dynamic behavior of a generalized Voigt model consisting of an infinite number of elements connected in series can also be obtained using the previous direct process involving fundamental concepts, Laplace transformations, and the interrelations between the viscoelastic functions, and shown to be and n 1 III T 0 i= l * i n ^ 2 < ) = 1 / wt1o + s ~ i= l Ei U)T III 16

83 64 The expressions for the storage and loss complex compliances in the frequency domain are obtained by summing over all the series units representing the generalized model. The actual mechanical model according to earlier concepts is made up of n + 2 units in series. Two elements are made up of a single dashpot and a single spring, while n are made up of Kelvin units. In the analysis of experimental data, it is possible to resolve an arbitrary set of parameters to represent a finite model which would suffice to predict macroscopic response; however, the model would not be unique. It has been discussed in earlier chapters that as the number of elements describing the response of the material in dynamic measurements is made infinite in extent, it is possible to extend the generalization of the Voigt model and to characterize the infinite model by a continuous function of one independent variable. This continuous variable in the case of the generalized Voigt model in both the time and frequency domain is the distribution function of retardation times L (in t). Thus, the same continuous spectrum of retardation times can be used to describe both the transient and dynamic response of materials in which each contribution to rigidity is associated with a retardation time whose logarithms fall in the range between in t and in t + d in t. The storage and loss components of the complex compliance can be obtained by integrations which are equivalent to summing over all the elements of an infinite Voigt model. The real and imaginary parts of the complex compliances are obtained by x 5 d (in t) 1 + T U) III 17

84 65 and Each of these equations requires a know ledge of the spectrum over a wide range of the frequency time scale, depending on how fast the integrand converges. In principle, L (to r ) may be obtained from experimental data of either of the three viscoelastic equations defining J j (jj), J 2 (*>)> and Jc (t). If, for example, J c (t) is defined at a given temperature over a wide range of time, then L (in t ) can be evaluated at this temperature, and J j (j) as well as J 2 (*>) can also be obtained at this specified temperature. The storage and loss components of the complex stress relaxation modulus can be obtained in a similar manner by integrations which are equivalent to summing over all the elements of the generalized Maxwell model. The dynamic moduli are given by Relations Between Relaxation Function, Complex Modulus Function, and Relaxation Spectrum and III-20

85 66 Determination of Viscoelastic Spectra from Experimental Viscoelastic Functions In the previous section a large number of functions of time or frequency have been discussed, each of which is sufficient to describe theoretically the linear viscoelastic behavior of the material. Many of these viscoelastic functions can be measured immediately such as JQ (t), (u), and J 2 (*>), while others such as the retardation and relaxation spectra cannot be measured directly, and their physical meaning is related to the respective mechanical model used to represent material's viscoelastic response rather than to the material itself. However, many of the equations fail to meet practical needs for converting from one viscoelastic function to another by use of the retardation or relaxation spectra. Integrations from <» to - oo are usually required, and the function may not be available over a wide enough range of time or frequency to obtain convergence of the integral. Although it is relatively easy to obtain the viscoelastic functions once the spectra are known, the reverse process by inversion of these equations is tedious and may not be possible. For the above reasons, approximate methods have been developed for performing such calculations (29). The approximate procedures involve taking derivatives of either the initial function or of related functions, using graphical or numerical methods to obtain the retardation spectrum from transient or dynamic measurements. The approximate methods generally have an analytical foundation based on the properties of the integrands of the corresponding exact viscoelastic equations.

86 67 The important problem arises of how the L (fn r) and H (inr) functions are calculated from the transient experimental functions, which are presented in graphical and tabular form and applied to predict the dynamic response of the material. Since, the exact methods are of limited use, the application of approximate methods are derived in the following sections. An analytical approximation method will now be considered which was originated in a method proposed by Alfrey (24) and has been recently developed by Schwarzl and others (2), (38). This method depends upon the existence of approximations to delta functions or to step functions in the integrals containing L (In t ) or H (in t ), as in equations m-10, HI-17, and HI-18. By use of the preceeding concepts the retardation spectrum can be evaluated in term s of equation HI-10. At long tim es, J c (t) increases without limit due to the contribution from the flow term t/n0, but if the latter is subtracted, the remainder ft) - t/'ho* approaches a limiting value. To obtain the retarded elastic component of the creep compliance function, (t), both the instantaneous compliance, J 0, and the secondary flow, t/r^, are subtracted from J c (t). Using the nomenclature from Chapter II, the retarded elastic component of the creep compliance function may be written 00 - t / T L (In t ) (1 - e ) d (fn t ) III-21 Using the substitution and t = en ; n = fn t z, t = e ; z - t m

87 68 equation III-21 becomes r 00 (n) = I (-en_z) L (z) < 1 - exp ) dz m -22 ^ -00 For values of n less than z, the coefficient of L (z) in the integrand can be shown (39) to be approximately zero and it then rises steeply to a value of approximately unity for values of n greater than z. If the coefficient [l - exp (-en z)j is approximated by a step function going from zero to one at n = z, the integral in equation m -22 can be approximated by (n) = f L (z) dz HI-23 J -00 and the integral is not greatly changed. Differentiating equation ni-23 with respect to n, we obtain Jjj <n>= L <n> HI-24 dn T which is the relation sought. Thus, the retarded elastic component of the creep compliance is plotted against fn t, and the slope of this experimental plot can be used as an approximation to the value of L fn t ) for any particular value of In t. A similar proof of the previous concepts is obtained by differentiating equation HI-22 with respect to n to obtain v / 00 en"z L (z) [exp (-en_z)] dz -oo HI-25 In equation ni-25 the coefficient of L (z) in the integrand is zero except for a small range of values where n is equal to z. As the value of L (z) does not

88 69 change sharply over this range, it may be assumed that L (z) is constant and approximately equal to L (n). Equation ni-25 then becomes (n) = L (n) f en z [exp (-en z)j dz QI where the value of the integral can be shown to be unity, and equation IQ-24 is again obtained. The principle applied here is to approximate the coefficient in equation in by a unit step function at t = t, and the value of the integral is not grossly altered; as a result, the retardation spectrum at t = t is obtainable as an approximation to the slope of the retarded elastic component of the creep compliance. t / T In a similar manner to the above derivation, the function e occurring in equation m -9 can be considered as an approximation to a step function, and the approximation to H (in t ) is evaluated by differentiating Er (t) with respect to in t to obtain - j r iq = H (in t ) III-27 dint The calculation of the distribution functions from dynamic measurements can now be considered, which is similar to the above procedures for the creep compliance and stress relaxation functions. In the case of the storage component of the complex compliance, (o>), the substitution a) = e n ; n = - in co can be made and the function -----^ in equation HI-17 is a good approxima- 1 + T <j) tion to a step function when plotted against in u>. By the use of analogous methods to those applied to the creep compliance function, an approximate relation between

89 70 the retardation spectrum L(!n t ) and J j (w) can be obtained (61) = ---- f j i M l ni-28 dinu 1 J The relation between the retardation spectrum and (w) can be obtained by platting the function ^ 1 + W T 5- in equation ni-18 against In cjt. This type of plot produces an approximation to a delta function, and J 2 {*>) - l/u r may be considered to be proportional to L (In t ) to obtain the approximation L (In t ) = 2/tj- [ j2 (u) - l/wn^j III-29 The distribution of relaxation times, H (In t ), can be obtained from the components of the dynamic stress relaxation modulus of the generalized Maxwell model by the substitution of w= e n in equations and III-20. It 2 2 can be shown that the function ^-7T----- in equation is an approxima- 1+u t2 tion to a stem function when plotted against In co, and the function = 1 + T C in equation m-20 is an approximation to a delta function when plotted against InuT to obtain the following relations and H,lnT)=T ^7 [Eri H m-30 H (In t ) = 2 / 7 7 [Er2 (w)] ni-31 Schwarzl (2) has obtained relations by the use of higher differential coefficients of the experimentally observed viscoelastic transient and dynamic functions. In the Schwarzl relations the integrands contain functions which are better approximations to either delta or step functions, and therefore better approximations are obtained for the distribution function of retardation and relaxation tim es. Expressions containing higher derivatives of the experimental

90 71 viscoelastic functions are limited as the second or higher order derivatives are extremely sensitive to small experimental errors in the observed response curve. Equations HI-10, HI-17, and III-18 are essentially Laplace transforms, which can be inverted by the equation suggested by Widder (2). III-32 in which the spectrum is a function of t rather than in t. It is impossible to obtain the spectrum rigorously since the accuracy of (t) is limited, but by assigning values of 1, 2, etc., to K, approximations to the spectrum of higher accuracy can be obtained. Substituting K equal to one in equation HI-32, the approximation proposed by Alfrey is obtained III-33 d in t In order to obtain a higher second order approximation, K is set equal to two and the following relation is obtained (in t) HI-34 where t is equal to 2 t. A third approximation is evaluated by assigning K equal to three to obtain L (In t ) = - d J 4<(*n *) + 3 d2 JiL (in t) _ 1 d3 (in t) ni-35 i n t 2 d (J*n t)^ 2 d (^n t) 3 where t = 3 t. The index K denotes the order of the approximation which is the highest logarithmic derivative occuring in the formula. An inversion of equations HI-17 and HI-18 may be used to evaluate the retardation spectrum from dynamic measurements by sim ilar methods. A

91 first approximation which may be used to obtain L (in t ) from the storage compliance is of the form 72 L (,b t> = - t ^ M r a - 36 where t is equal to l/u>, and similarly from the loss compliance* an approximation of the zero order is given by L (in t ) = 2/r [j2 (u) - l / u n j HI-37 The approximation methods can be theoretically extended to utilize second or higher order derivatives* but the limited precision of the available experimental data* which makes the higher derivatives unreliable* and the limited range of time or frequency over which the data are available* generally restrict the application of the higher order approximations. However, the first order approximations can be used with confidence to evaluate the distribution functions in regions where the spectra are not changing rapidly. The distribution functions serve as intermediates in the calculation of the transient viscoelastic response from dynamic experiments and conversely. The use of an analytic function to represent the experimental data over the measured time or frequency range can be justified mathematically only if the data are precise enough to evaluate derivatives of the higher orders. The reproducability of the data available from laboratory experimentation usually does not justify more than one and at most two differentiations to evaluate L (in t ).

92 73 Application of Transforms The approximations considered in the previous equations are obtained by either successive differentiation of the response curves with respect to In u> or in t to obtain the retardation or relaxation spectra. The storage components of the complex compliance moduli used in this study were evaluated by a numerical integration of the retardation spectra evaluated from the transient experimental data, while the loss components of the complex compliance moduli were evaluated by the proportion between the respective retardation spectrum and loss component. In order to define the viscoelastic properties of the materials investigated in this study, only the retardation spectrum was employed. Thus, by application of the approximate relations summarized in this chapter to the transient viscoelastic functions covering a wide range of time, a single function, L (In t ), describing the mechanical properties, was evaluated and used to predict the dynamic response of the material over a wide range of frequency. The degree of error in an approximate calculation of the spectra can be evaluated in various ways. The original experimental function from which the spectrum was calculated can be reconstructed by numerical integrations and compared with the initial experimental data. Values of the spectrum obtained by approximations from different experimental functions, such as J c (t), J j (w), and M can be compared. This comparison is easily obtained from dynamic measurements as both J 1 (») and 1p) or Gj (u) and (u>) are determined

93 74 simultaneously. It should be noted that the previous formulas are easily used and may be rapidly applied to viscoelastic functions in order to form a convenient method for future rheological studies of bituminous concrete mixtures. f

94 CHAPTER IV A THEORY FOR THE TEMPERATURE DEPENDENCE OF THE VISCOELASTIC FUNCTIONS OF BITUMINOUS CONCRETE General The phenomenological theory of linear viscoelastic behavior which was summarized in Chapter II is of great value for interrelating different types of experimental measurements and for describing the mechanical response of materials in the time or frequency domain. However, it may be used to define the viscoelastic response of the material at only a specific temperature and provides no information for the molecular origin of viscoelastic response. It has been mentioned earlier that at low temperatures the viscoelastic elements, which represent various polymer molecule configurations, are largely immobilized, and the distinct changes in viscoelastic properties with time or frequency which characterize polymeric systems such as bituminous concrete mixtures are not as pronounced as the viscoelastic response at higher temperatures. It is hypothesized that the concept of the kinetic theory of rubberlike elasticity and the time-temperature superposition principle to be presented in this chapter can be applied to describe the mechanical response of thermoplastic bituminous concrete mixtures in the frequency and time domain. 75

95 A high polymer may be defined as a material consisting of large macromolecules built up by the repetition of smaller chemical units and includes natural and synthetic materials such as rubbers, plastics, asphalts, and fibers. The length of the polymer chain is specified in terms of the number of repeating units in the chain, which is called the degree of polymerization. The molecular weight of the polymer is the product of the molecular weight of the repeating monomer unit and the degree of polymerization. Many of the high polymers useful for plastics, rubbers, asphalts, or fibers have molecular weights of the order of 300 to 1,000,000. In some cases the repetition is linear, similar to a chain built up from identical individual links; however, in many cases the chains are branched and interconnected to form complex three-dimensional networks. Asphalts are essentially complex mixtures of hydrocarbons, although they contain nitrogen, sulfur, oxygen, and traces of metals either in metal-containing organic compounds or dispersed in oxides and salts. Bitumens are not true polymers because they may be made up of complex constituents as well as repeating monomer units. However, bitumens have no regular structure that can be called crystalline and so they are classified as amorphous materials. In organic chemistry, hydrocarbons are classified on the basis of chemical behavior as being unsaturated or saturated. Unsaturated hydrocarbons contain one or more double or triple bonds between the carbon atoms, while saturated hydrocarbons have no multiple linkages between carbon atoms. As a result of the multiple bonds

96 77 between the carbon atoms, unsaturated hydrocarbons have a groat reactivity with other elements. In the case of semi-solid asphalt cements, the molecular weight of the macromolecules may approach values of 300, while the molecular weights of the chain molecules in other asphaltic materials vary greatly and may be larger than It is known that many asphaltic materials may have the same consistency at a specific temperature and yet may vary widely in molecular weight. It is quite probable that the above asphaltic materials will react differently under load and also respond differently to changes in temperature. The chemical composition of die materials will, in all probability, vary in the different molecular weight distributions, depending on the source of the crude oil. Thus, each m a terial is quite complex in its molecular makeup as the construction and molecular weight of macromolecules in a given asphalt may have a wide range and vary chemically among sim ilar asphalts. This section demonstrates qualitatively that the prominence of the viscoelastic phenomena in bituminous mixes is dependent upon the versatility of movements of the flexible, threadlike, bitumen hydrocarbon macromolecules which are directly associated with temperature. The most interesting feature of material viscoelasticity lies in the transition zone from brittle to liquid-like consistency, where the change in magnitude of the viscoelastic functions is quite large, and theoretical studies have been primarily devoted to understanding the response in this region. It is in the transition zone that the dependence of the

97 viscoelastic functions on temperature is most spectacular, just as the material's dependence upon time or frequency is more pronounced in this same zone. Since bitumens are thermoplastic materials, temperature is known to effect the mechanical response of the bitumen component in bituminous concrete mixtures. Therefore, the rheological equation of state must be given for isothermal conditions, or the temperature should appear in the equation as an independent variable. The fundamental problem in the study of a m aterial's rheological re s ponse, regardless of which approach is used, is to determine the functional r e lation between the stress and strain and their time and temperature dependence. Any investigation is extremely complicated which analyzes the temperature dependence of viscoelastic materials by evaluating an analytical form of the creep compliance, J (t), as a function of temperature at a given time, or the loss and 78 storage components of the complex compliance modulus, J 2 M and (w), at a specified frequency. However, we can apply the concept of reduced variables which affords a valuable simplification by separation of the two principle independent variables of time and temperature on which the viscoelastic properties depend. This simplification is accomplished by expressing the mechanical properties of the material in terms of a single function of time and a single function of tem perature whose form can be evaluated experimentally. The rheological properties of materials may be conveniently handled when the properties are represented as a function of time or temperature in graphical or tabular form even if the viscoelastic functions cannot be expressed with facility in analytical form.

98 79 Analysis of Four-Parameter Model; Temperature Dependence of Elements In Chapter n the viscoelastic behavior of real materials was approximately represented by a four-parameter Burgers model, showing the three fundamental characteristic rheological deformations: instantaneous elastic, time-dependent retarded elastic, and Newtonian flow. When analyzing the response of a material represented by such a model in a creep experiment, it is apparent that the mechanical nature of the m aterial depends strongly upon the relative magnitude of the spring and dashpot constants EQ, Elf t^, and and the time or frequency conditions under which the model is loaded, as well as the temperature during the experiment. At extremely low temperatures, the values of and r 0 may become so large that the retarded elastic and viscous flow rheological mechanisms are effectively "frozen" and the response of the material is essentially that of a simple elastic m aterial with a modulus, E0. As the temperature increases, both the viscosities of and t )0 decrease, and the behavior of the v is cous and retarded elastic behavior of the material may become completely dominate when compared to the elastic response of the material. At still higher temperatures, even the retarded elastic response is overshadowed by the large viscous flow and the material appears to act as a simple viscous liquid. If the material represented by the above Burgers model was tested under isothermal conditions at a relatively high temperature but at high frequency, the response of the material would again be essentially elastic as the time-dependent elements would not have time to react to the applied stress. Thus, by varying either the time or temperature of experimental conditions, the same viscoelastic response I

99 can be obtained. This argument is not restricted to any particular model network and may be applied to other similar configurations. In studying the response of real materials, it is important to understand that not only the magnitudes of the model constants describing the behavior of the material are important, but also the temperature and time dependence of these elements. This behavior can be explained qualitatively on the basis that at low temperatures and high frequencies, practically no configurational changes of the polymer molecules, which make up the material, occur within the deformation period, and the complex storage compliance, J j ((*>), is approximately equal to instantaneous elastic deformation, J Q. Similarly, each transition of the creep modulus in logarithmic time can be attributed to a molecular process, where an increase in experimental temperature corresponds to an increase in the time scale and a decrease in the frequency scale of the predicted complex moduli. Flexible-Chain Theories Although the use of the principle of reduced variables developed empirically in advance of the coiling polymer theories which support it, this study shall introduce this concept as a logical consequence of the flexible-chain theories developed by Rouse (40) and Zimm (41). The viscoelastic response in polymers is due to extremely complicated molecular adjustments which take place on macroscopic mechanical deformation. In a polymer system each flexible threadlike macromolecule extends its influence over distances much greater than its

100 81 own atomic dimensions and is continually changing the shape of its contour as it moves with its thermal energy. The method of reduced variables will now be examined in detail in order to evaluate how the parameters which describe the temperature dependence of the viscoelastic functions are related to molecular makeup and structure of the material. The theory of Rouse, developed in 1853 for the linear viscoelastic response of dilute solutions of coiling polymers, shows that temperature susceptibility enters into the polymer system in several ways. Rouse's theory was the first successful detailed quantitative interpretation of viscoelastic properties of flexible polymers, and although derived for dilute solutions, it can be applied within limits to more concentrated systems. A polymer macromolecule is surrounded and pervaded by the diluting solvent and is constantly rearranging its configuration by random actions due to Brownian motions. The driving force for these random motions is thermal energy, and the motions are opposed by the viscous forces due to hydrodynamic resistance of the solvent and internal viscosity of the polymer. The Brownian motions of the assorted polymer configurations and the rates of configurational changes are slightly altered by the application of stress in the linear viscoelastic range. Any of the viscoelastic functions surveyed in Chapter II could be discussed, but it is convenient to choose the components of the complex dynamic compliance, (p) and Jg (w)* which will be evaluated in later chapters. For a given periodic

101 82 stress the relative amounts of energy stored and dissipated per cycle depend on what extent the random Brownian motions are correlated with the external stress. The configurational motions on an atomic scale must be accounted for by rotations around all the bonds in the polymer chain. It can be reasoned qualitatively that at very high frequencies or short loading times there will be a limited time for any molecule rotations within a loading cycle and the response of the material will be primarily elastic, and (jj) is small. The short time response to stress is made up primarily of stretching and bending of the chemical bonds. At lower frequencies, regions of the polymer chain have time to change their relative positions with the loading cycle, and there will be components of deformation in-phase and out-of-phase with the changing stress. The contributions to and Jg {*>) will both be significant in this range. At still lower frequencies, J 2 M increases greatly and (u) approaches a limiting value. A complete description of the polymer configurations requires a more detailed knowledge than is normally obtainable as well as mathematically intractable of the dimensions and shapes of the individual chains, interractions with the solvent molecules, and packing effects. A simplification can be made by utilizing a prediction from polymer chain statistics which states that two points on the macromolecules separated by 50 or more chain atoms will be related to each other in space in accordance with a Gaussian distribution of vectors (2 ). This form of the distribution holds regardless of the local packing effects, bond distances and angles, and interraction with solvents, all of which enter into a proportionality constant with dimensions of length. The root-mean-square

102 distance between two points separated by q monomer units is <r = with q i greater than 50, where i is the number of chain atoms per monomer unit and a depends on the geometric parameters and is usually the order of several times the length of a single chain bond. In the theories of Rouse and Zimm the flexible-chain molecule is subdivided into N submolecules, each with q monomer units. The root-mean-square length of the entire molecule is (r q ) 1 ^ 2 = <r'\fk = <T\fz where Z is the degree of polymerization. Isolated Flexible Chain Theory of Rouse If a diluted solution of polymer molecules is sheared, the flowing solvent distorts the material so that the assortment of vectors between two of the chain atoms q units apart is changed from a Gaussian distribution. The Brownian motion will result in a diffusion back to approach this distribution, but the viscoelastic response is determined by the interreaction between these two effects. In the development of Rouse1 s theory the molecule is divided into N submolecules, each consisting of q monomer units, and the hydrodynamic forces exerted by the solvent are assumed to be concentrated at the junctions between the submolecules. In the treatment Rouse assumed no hydrodynamic interaction between the motions of the submolecule junctions. The resistance encountered by a submolecule junction moving through its surrounding is described by a friction coefficient, f0, which is proportional to q. It is also assumed that the average value of f0 can be used and no intermolecular viscosity is taken into account. The monomeric friction coefficient, &Q, is defined as = fo/q- The simul-

103 taneous motions of all the segment junctions can be described as the sum of a series of cooperative modes. Each mode is considered to represent a discrete contribution to the relaxation spectrum. The results may be expressed in terms of the distribution of relaxation times spectrum, H ( t ), from which any of the experimental viscoelastic functions can be derived by the relations in Chapter II: N H (t ) = n KT S i=l TifcCr-Tj) IV-1 84 where: n = the number of polymer molecules per cc 5 = Dirac delta K = Boltzman's constant T = absolute temperature t j = relaxation times i = 1, 2, 3,...,N An expression for the retardation time, t j, is also obtained by Rouse and is in the form <r2 fo, t. = F ^ T IV K T sin fi -jj/z (N+2)l Equation IV-2 may be replaced by a more convenient form (25) in order to be used in the following analysis T. = % \ IV-3 If equation IV -1 is substituted into equations 11-34, , and , developed by using an infinite Maxwell model to represent the m aterial's viscoelastic response in the time and frequency domain, the following summations

104 85 may be obtained: G (t) = n K T S i=l e_t/ Ti IV-4 IV-5 N G2 H=nKTE i=l U)T j 1 + W2 IV-6 where G (t) is the transient shear modulus and (w) and G2 (u) are the complex shear storage and shear loss moduli, respectively. By applying calculations similar to those made by Gross (42), it can be shown that equations analogous to IV-1 and IV-3 can be developed for the retardation spectrum and retardation time, respectively. By defining the m aterial's linear viscoelastic response by a generalized Voigt model, calculations of J c (t), (u), and (w) can be made as summations sim ilar to equations IV-4, IV-5, and IV-6. The effect of hydrodynamic interaction between the moving submolecules was considered in the treatment by Zimm to develop the relaxation spectrum similar to Rouse's development, but internal viscosity of the polymer was not taken into account. In an undiluted solution each molecule carries out the same kind of motion that it does in a dilute solution, except that they are slower motions due to the increased frictional resistance. In one respect the solution is sim pler, since in the undiluted polymer each molecule chain is in the same type of environment, surrounded by other polymer segments. The macronomolecule motion of a segment through its surroundings involves a very complicated

105 86 readjustment of its own flexible chain as well as its neighbors'; the resistance in many cases can still be expressed by an average monomeric friction coefficient, 5 ot as in the theory of Rouse. In extending the theory for dilute solutions to undiluted polymers, there is no clear a priori whether the results of Rouse or Zimm can be applied. In the above theories the polymer molecule is divided into equal segments whose length is arbitrary, but it is long enough so that the distance between the ends of a segment follows a Gaussian distribution function of the theory of rubberlike elasticity. In the theory of Rouse the response of the system associated with each mode of cooperative motion corresponds to one infinite Maxwell model element in which energy is stored proportional to a modulus Gj and dissipated by relaxation with a time t * as represented by the discrete relaxation spectrum. The dependence of H (r) and related quantities on temperature as expressed in the previous equations can be easily seen and will be thoroughly discussed in later sections of this chapter when they are formulated in terms of reduced variables. The Thermodynamics of Rubberlike Elasticity A proper understanding of the nature of rubber elasticity was not possible until evidence was obtained concerning the molecular structure of rubber and other naturally occurring high polymers. Present conceptions are based on the view that the long range extensibility arises from the thermal fluctuations of the long flexible-chain macromolecules of which the material is composed. The

106 87 thermal fluctuations of the flexible-chains are assumed to result from unrestricted rotations about the primary valence bonds of the macromolecule. The elasticity of rubberlike materials based on this theory is kinetic in origin, similar to the volume elasticity of a gas and not due to specific attractive forces as in the elasticity of a normal solid. The fluctuations in length of the flexible-chain segment cause it to act like a Hooke's spring under tension, storing energy due to a decrease in entropy associated with restrictions on the assortment of configurations the molecule can assume. By using the concepts of the kinetic theory, it may be shown that the moduli of all the elastic mechanisms in a rubberlike material are directly proportional to the absolute temperature. The thermo-elastic effects have been subjected to extensive research (25), (43), and it is only recently that a reasonably clear understanding of the complete phenomenon has emerged, as the question has proved to be considerably more complicated, both experimentally and theoretically, than originally suspected. The first law of thermodynamics states that for a mechanically closed system, the energy is invariable. If the first law is extended to include the principle of equivalence of work and heat, the law of conservation of energy for a constant mass takes the form d_w = _dwk + d_u_ d t d t d t where energy of the system W is the sum of the kinetic energy and the IV_7 stored potential energy U. This section summarizes the derivation of thermodynamic relations which will enable the changes in the internal energy, U, and

107 the entropy, S, which accompany the deformation of an elastic body to be expressed in terms of the observed dependence of the stress on length and temperature. The change in internal energy in a process may be defined by <LU = d _ + dwi_ d t d t d t IV _g where d Q is the heat absorbed by the system and d Wj is the work done on the system. From the second law of thermodynamics, the irreversibility of a mechanical process is defined by the condition that the process or change of state be accompanied by an increase in the entropy of the mechanical system. The entropy which is a function of the momentary state of a system is given for the volume element at rest by where T is the absolute temperature. Substituting this definition into equation IV-8; IV-9 dlj = T ds_ + dwj d t d t d t IV-1 which is a fundamental equation of thermodynamic change of state showing the interrelation of temperature, deformation, entropy, and stress by use of the specific internal energy, U. In a real mechanical process the creation of heat by friction-like processes can never be prevented, and such a process cannot be perfectly reversible. The irreversibility of a change of state requires that d S/d t be greater than zero. For a system of discrete particles, its state is determined by the energy distribution over the particles. The entropy defining the state of such a system

108 is related to the energy distribution over the particles by the Boltzman relation S = K log P which defines the relation between classical and statistical thermodynamics where P is the thermodynamical probability of the energy distribution and K is Boltzman's constant. In the Boltzman relation the thermodynamical probability of a state is defined as the number of equally probable energy distributions over the particles associated with the thermodynamical state of a complete system specified by its entropy. Introducing the concept of free energy, F, defined by F = U - T S IV-11 for a reversible process d F = d U _T dj> _s d T d t d t d t d t If the n rocess takes Diace at a constant tem oerature S equation is obtained J m = 0, the following IV-12 d F d t d W d t d U _T d_s d t d t IV-13 which is the relation desired. The development of the theory of rubberlike elasticity consisted of essentially two stages. In the first stage the statistical properties of a single molecule were examined mathematically, and equations were derived for its entropy and hence its free energy as a function of its length. In the second stage the properties of various networks formed by the cross-linking of a group of such chains were considered.

109 90 However, the treatment of a single polymer molecule can serve as the prototype of all theoretical treatments of this subject as the many extensions of the kinetic theory all follow the same general pattern. The statistics of the random chain have been considered from many different points of view, and the essential problem is to express the configurational entropy of the polymer as a function of the strain. It is also desirable to calculate the relative number of configurations of the chain corresponding to a specific distance between the chain ends. In order to accomplish this purpose one end of the flexible-chain A must be fixed at the origin of the coordinate system and the probability of finding the other end B of the chain in the vicinity of the point (x, y, z) calculated. The approximate formula given by Kuhn (2) for this probability is P (x,y, z) dx dy dz = e ~ ^ (x2+y +z ) dx dy dz IV-14 7T which expresses the fact that the required probability is proportional to the size of the volume element (dx dy dz) within which the end B may lie. The function is the well-known Gaussian or normal e rro r function. It contains only one adjustable parameter, b, which is related to o, the length of the link and Z, the number of links by b2 = 3/2 Z a2 IV-15 To find the most probable length of the chain, it is necessary to consider the probability, P (h), of a given distance, h, between the chain ends, i.e., the probability that the chain end B shall lie within a spherical shell of thickness,

110 d h. This may be obtained by the product of the probability density and the size of the volume element, 4^ h2 dh so that The resulting h - distribution function is a maximum value at the point IV-16 a IV-17 which corresponds to the most probable value of h or the most probable distance between the ends. A more frequently required quantity is the root-mean-square value of h obtained by IV-18 which, like the most probable value, is proportional to the square root of the number of links in the polymer chain. The elasticity of a single flexible-chain is related to the number of configurations available to it or the probability function which can be obtained by Boltzmann s relation S = K log p. For the Gaussian function expressed by equation IV-14, the following relations are obtained S (x, y, z) = C - K b2 (x2 + y2 + z2) and S (h) = C - K b2 h2 IV-19 where C is a constant. If the ends of the polymer chain are held at fixed points separated by the distance h, the corresponding entropy is proportional to h2.

111 Assuming that all configurations have the same internal energy, the free energy is therefore 92 F = C + K T b2 h2 IV-20 The work required to move one end of the polymer chain from the distance h to h + dh with respect to the other end is equal to the change in free energy and is also equal to f *dh where f is the mean force acting on the ends of the chain. The force may be expressed by f= d_f = 3 K T h = 2 K T b2 h IV-21 d h Z a which represents a mean normal tensile force acting along the line joining the ends of the flexible-chain and proportional to the distance between the ends. Several important features of equation IV-21 should be noted. First, the tensile force, f, and therefore the elastic stress-strain modulus, E, is directly proportional to the absolute temperature. The use of a more complex molecule model may result in a different value for E, but cannot yield any different temperature dependence. The essential problem has been to calculate the configurational energy of a polymer as a function of the strain. The above treatment has used a single polymer macromolecule as representative of the polymer sample. The distance between the two ends of the flexible-chain molecule on an unstretched polymer sample was given by the most probable value of the distribution function. When the sample is stretched, the two ends of the polymer molecule are pulled apart and the statistical probability of the new stretched length was considered to be

112 93 given by the probability function. The Boltzman entropy equation was applied to the single molecule and the reduced probability of the stretched state was tran s lated into thermodynamic language as a lowered entropy. An expression which relates the tension f with the elongation dh for a single chain was obtained. Kuhn assumed that the entropy of the polymer was simple the sum of the entropies of the individual chains and obtained a relation between tensile stress, elongation, and temperature. The Concept of Thermolinearity In the transition zone between brittle and rubberlike consistency, the viscoelastic response of a polymer system is due to the cooperative motions of the individual macromolecule chains which are governed by a single average friction coefficient, fq. The theory of Rouse applied to a diluted polymer shows that the temperature dependence of viscoelastic response is accounted for in several ways. Each Maxwell or Kelvin element of the generalized models that are used to describe a material's response corresponds to an individual macromolecule which makes a contribution to the components of the viscoelastic functions at a given time or frequency. Each of the retardation times of the infinite Voigt model used to represent a material's response depends in an analogous manner on the macrpmolecule movement and the influence of temperature. The results are entirely consistent with the concept of reduced variables which was applied in this study.

113 Rouse's expression for the relaxation spectra, equation IV-1, contains a factor of the absolute temperature and also the density which decreases slightly with an increase in temperature due to thermal expansion. In the expression for the relaxation time, equation IV-3, there is an additional temperature-influenced factor, and the characteristic dimension, a, may change, especially if there are restrictions to free rotation in the macromolecule chain. The effect of temperature appears essentially in the monomeric friction coefficient, s 0, which usually decreases quite rapidly as the friction, which is the nature of viscosity, opposing segmental motion decreases. The effects of the monomeric friction coefficient, characteristic dimension, and temperature may be conveniently confined into one temperature reduction factor, a-p, which can describe the ratio of the relaxation or retardation time at any temperature, T, to that of the relaxation or retardation time at an arbitrary reference temperature, Tq, by: a T = <T)T Tq = (az Sq)t T0 IV-22 (T>To T (a2 Bo)To T The effect of a temperature decrease from T to TQon the retardation times is to shift the complete curve to the right by the temperature reduction factor, a-p, which in this case is a positive number, as the change in a*p is dominated by the increase of D. The general shape of the derived experimental curve in this procedure is unaltered. The application of these shifts to an experimental transient or dynamic moduli curve assumes that every contribution to the spectrum 2 is proportional to n K T and that every retardation time is proportional tos &a. T 94

114 95 The other features of the theory of Rouse may be subject to modifications due to the macromolecule molecular weight distribution and the hydrodynamic interaction between the motion of submolecule junctions. The method of r e duced variables allows the reduction of the thermolinear viscoelastic properties of materials into two general curves: the temperature dependence of a-p, and the dependence of the viscoelastic functions on reduced time or frequency. It is known that the various polymer families react qualitatively in the same manner due to a temperature change with essentially only the time scale transposed. Although the literature summarized has been developed for dilute polymers, it is not unreasonable to apply similar concepts to the viscoelastic response of bituminous concrete mixtures as the temperature dependence of the material is related to that of the bitumen. The mineral aggregate is infinitely more rigid than the bitumen component in an aggregate-bitumen mixture, and the strength of the mixture depends essentially on the rigidity of the bitumen and the volume ratio of bitumen to aggregate. At high aggregate concentrations the physical properties and grading of the mineral aggregates and the corresponding variations in voids will have a greater influence on the m e chanical properties of the mixture. At higher concentrations of aggregates the temperature dependence of the mixture is related to the thin films of bitumen coating the aggregate particles. The thin bitumen films which may only be a few microns thick are still made up of a very great number of molecules. It should be noted that only two independent elements are used in the generalized Voigt model, and all retardation times show the same temperature

115 dependence. The existence of only two types of identical elements, which are arranged in the form of a complex network to describe the material's viscoelastic response, lend a physical interpretation to the concept of thermorheo- logically simple substances. The spectra express the complexity of the arrangement of identical elements and not a diversity of elements of the material. Thus, viscoelastic functions at different temperatures can be transformed into each other by the thermolinearity concept. This argument is not restricted to the / particular model structure considered here and may be applied to other similar models. 96 Procedure and Criteria for Applicability of Reduced Variables The molecular theories developed from the macromolecule flexible polymer chains presented earlier in this chapter predict that a single composite curve can be obtained when the viscoelastic functions for a time or frequency experiment are shifted horizontally along the time scale axis if the chain motions reflected in the measured functions are controlled by a single average friction coefficient, fc. The empirical criterion for fulfilling the above conditions required that the shape of individual curves evaluated at different tem peratures over a wide range of time should coincide within experimental error after the horizontal shift is applied. The temperature reduction factor, a^, must be determined empirically for each temperature, but the requirement of superposition over a range of time or frequency does not permit an arbitrary selection in this choice because in both the transient and dynamic tests, the same

116 97 value of arp must be used to bring both the real and imaginary components of the complex modulus into superposition as well as to superpose all the viscoelastic functions. If the adjacent transient experimental curves are not parallel and different values of a-p must be used for different portions of the time scale, the transformed curves of (u) and M will fail to match in shape and different values of for the complex storage and loss compliance will result. Whenever the above criteria are not met, the applicability of reduced variables in this convenient form must be rejected, and it may be concluded that a change in experimental temperature has effects in addition to those previously described. A material which meets the above criteria is called thermorheologically linear (2), (25) and is defined as one in which a change in the experimental temperature changes only the position of the viscoelastic function in either the time or any operational mathematical domain while the general shape of the curve is not altered. The thermolinearity concept may also be applied to other materials which approximately meet the above criteria in order to investigate or to approximate their mechanical response. Even when the m aterial's re s ponse is only approximately thermolinear, it allows the material's viscoelastic behavior to be essentially represented by two simple functions instead of a complex three-dimensional representation in time and temperature space. This reduction scheme must be slightly altered to make it theoretically more satisfactory. Early applications ignored the factor.tq o which enters

117 98 into the coordinate scheme in Figure 5 because of the entropy-spring nature of the stored elastic energy in the flexible-chain theory as explained by both the kinetic theory of rubberlike elasticity and theory of Rouse. The kinetic theory shows that the equilibrium modulus is proportional to the absolute tem perature, and the quantity Tn rather than Ec (t) should be governed by the superposition principle. The density of the material is also dependent rr on tem perature due to thermal expansion or contraction. The factors and T - q_ lead to small corrections that are introduced due to theoretical reasons. P It should be noted that the phase of the complex moduli do not depend on the tem perature or density factors, ^opo, for reduction. T P If a hypothetical thermorheologically linear m aterial is tested in a constant stress experiment under identical conditons but at several different temperatures and the reduced creep modulus, TqPo E. (t) evaluated and plotted, curves T p sim ilar to Figure 5 would result. Examination of the creep curve at a constant temperature, T2, reveals that the three characteristic zones of viscoelastic response are present: elastic, retarded elastic, and secondary viscous flow. The transient creep modulus determined under identical conditions, but at a higher constant temperature, Tg, is plotted in Figure 5, and is parallel to the creep function curve evaluated at temperature T^ for all values of time, but the creep function has been shifted horizontally along the time scale to shorter times. The creep modulus evaluated at a lower temperature, Tlf is also parallel to the reference curve at all times in the transition zone, and has been shifted to longer times as shown. It should be noted that similar plots

118 u E.c v- O o> o o Ll I M i en _3 3 T3 0 2 Q. 1 O TJ 83T3 o» q: Fig. 5. Time ( Logarithmic) Hypothetical Reduced Creep Functions of a Thermolinear Viscoelastic Material vs. Time Plotted at Several Temperatures. CO

119 100 could be obtained for thermally sensitive materials by plotting the creep modulus as a function of temperature at a given time, where in such an experimental plot an increase in temperature corresponds to an increase in loading time. The creep modulus in equation IV-23 which is generally a function of time and temperature E_ (t, T) = C (t, T) IV-23 can be reduced in this case to a function of the combined quantity, X, or reduced time Ec (A11, T) = Ec (X) IV-24 where X = Ai t - f (T) IV-25 The function Ec (X) is the master curve of the material in the transient creep test, and by the use of this curve the shape of all experimental creep curves within the tested temperature range may be determined for this material in such a testing procedure. The temperature function, f (T), is a function of temperature only and is used to establish the position of the modulus curve on the time scale. It is hypothesized that if one of the viscoelastic functions of a material obeys the fundamental time-temperature relations stated above, then all other viscoelastic functions such as creep compliance, shear modulus, bulk modulus, and the complex moduli obey similar time-temperature relations. The results of the experimental investigation of this hypothesis are included in subsequent

120 101 chapters. The principle effect of an experimental temperature change is to shift only the time scale of the experimental viscoelastic functions. The transient moduli are functions of reduced time, X, while the dynamic moduli are functions of the reduced frequency, Y, the relations for complex elastic storage and loss moduli are E1 (inw, T) = Ex (Y) E2?n, T) = E2 (Y) IV-26 IV-27 where Y = J?nw+f(T) IV-28 Experimental curves similar to Figure 5 could be plotted in a logarithmic frequency scale for the magnitude and phase of any of the complex moduli or the storage and loss components of the complex moduli evaluated at different temperatures. The previous concepts can be applied to any thermolinear viscoelastic material and applied to other moduli besides those mentioned above. For example, one may define a complex modulus which relates axial stress and the resulting transverse strain, and name it the complex transverse modulus denoting it by T*, and etc. A general state of stress may be conveniently separated into an isotropic and a deviatoric component, and any infinitesimal state of strain may be separated into a pure distortion and a pure volume change. Thus, it can be shown as in the previous chapters that only two independent moduli and their respective temperature reduction functions are needed to define the behavior of an isotropic, homogeneous, thermolinear viscoelastic material.

121 102 Alternately* E* and G* alone, or E* and T* will describe the material fully for any time or frequency and temperature. The reduction equations for any thermolinear viscoelastic material's response for a given complex modulus, M*, may be expressed for the complex storage modulus as M1 (j«nq, T) = (Y) IV-29 and the complex loss modulus M2 (Jfn wf T) = M2 P) IV-30 where Y = fnu+f (T) IV-31 All experimental curves of thermorheologically linear materials may now be reduced to a standard reference temperature, TQ, of the master curve. The effect of a temperature increase from TQ to T on a logarithmic plot of the creep modulus, such as Figure 5, consists of a shift of the creep curve vertically downward by the factor, J?n T0p0 t and horizontally to the left by Ai a^. Tp The temperature reduction factor in this case is a negative number as the change is dominated by the decrease in friction coefficient, sq. Thus, the modulus curve has been shifted vertically and horizontally, but its general shape remains unaltered. A creep modulus function determined experimentally at temperature, Tj, can now be shifted to the exact position it would have occupied if it had been tested at lower temperature, T_, by plotting E (t) = ToPo E_ (t) against Tipi jfn t/a-p when temperature reduction factor and coefficient of thermal expansion are known. By a sim ilar procedure a complete series of experimental tests

122 over a wide range temperatures can easily be reduced to a single composite curve representing the evaluated modulus at temperature, TQ. Using this concept, experimental tests with a limited time range but over a wide range in temperature can be used to extend the experimental time scale of a linear material and also to predict the modulus at any temperature in the tested range. From the relations between the viscoelastic functions for any transient or dynamic modulus, M, discussed in Chapter II, all the viscoelastic functions can be shifted to the reference temperature, T, by use of the following or sim ilar general relations: 103 M. (t) = lo B l. Mn (t) vs. fn t / a r IV-32 c T p c (w) = Mj (p) vs. jfnwa-ji IV -33 M2 (w) = M2 (u>) vs. i nwa^ IV-34 using the same notations as before. The master curves developed are functions of time or frequency only and describe the viscoelastic properties of the material at a standard reference temperature, T. The results in temperature change consists in multiplying all retardation times by a common temperature factor, a^,, which is a function of temperature only and embodies the equivalence of time and temperature effects. To illustrate the previous concepts, Figure 6 has been prepared in which the hypothetical reduced transient creep moduli of a thermolinear viscoelastic material have been plotted over a relatively short time range from t^ to t2 but over a wide temperature range for Tj to Tg. To obtain the master curve over

123 Reduced Creep Modulus, *= Ec(t)(Logarithmic) Master Curve To^ Ecffl vs T P Fig. 6. Log of Time Reduced Transient Creep Functions of a Thermolinear Viscoelastic Material vs. Time for Nine Temperatures. 104

124 105 a wide time interval at a standard reference temperature, Tc, similar to Figure 5, which in this case equals Tg, the previously reduced adjacent curves are shifted horizontally by the respective factor of at until the curves coincide at temperature Tg to define the master curve at this reference tem perature. In an analogous manner the master curve could be obtained at any temperature, Tj, in the tested interval over a wide range of the time scale. It is interesting to note that by selecting the lowest temperature in the experimental range as Tq, the master curve is developed for exceedingly long loading times. However, if the short time response of the material is of interest, the master curve in this time range is developed by designating the highest temperature as the standard reference temperature. The temperature dependence of the temperature reduction factor, fh a-p, can be evaluated by plotting the relative shifts to obtain coincidence of adjacent curves versus the temperature differences which can be determined from Figure 5 at a given value of the reduced creep modulus. The temperature reduction function, f (T), has been constructed in Figure 7 by plotting the time shifts between experimental curves versus the temperature intervals of Figure 5 at a specific reduced creep modulus value, equal to 6. In Figure 7 the value of n a-p necessary to shift the creep modulus to any desired temperature for observation must first be evaluated and added or subtracted to the logarithm of time at the given modulus value in order to transform the viscoelastic function from the reference temperature at which it is defined to the desired temperature. The opposite procedure may be used to reduce the visco-

125 106 t, Time( Logarithmic) *0 ten T, To T, T em perature Fig. 7. Temperature Reduction Function, f(t), Evaluated from the Data of Figure 5 for -i -QPo Ec(t) equal to B.

126 107 elastic functions at any temperature to a standard reference temperature and to develop a master curve. The temperature dependence of the shift factor may also be obtained from Figures 5 or 7 by plotting the values In a-p evaluated to superpose the various experimental reduced creep moduli curves obtained at different temperatures to the standard reference temperature versus experimental temperature. The following procedure may be used to develop this more complete form of the temperature reduction function for a thermorheologically linear material. A standard reference temperature, TQ, is arbitrarily chosen somewhere in the experimental range, and logarithmic plots of TnPo E. (t) versus logarithmic time are prepared. The horizontal distance T p c A fn a-p, between adjacent curves is measured and recorded. The measured values of a Ai a-p are then added progressively from TQto obtain the reduction factor at each experimental temperature. Figure 8 is a plot of f (T) developed from the hypothetical data of Figure 6. It should be noted that the form of Sn a-p as a function of temperature is a continuous, decreasing function with no great fluctuations. The form of the temperature reduction function presented in Figure 8 may be used directly to shift the complete master curve to any temperature within the experimental temperature range. The standard reference temperature, T0, serves to designate a standard reference state, analogous to the reference states used in thermodynamics. Thus, if the modulus is defined at the reference temperature, the modulus at any other temperature can be readily evaluated by the reverse shifting. The previous concepts may be applied to any thermolinear viscoelastic material

127 t* to o to +4 «o Ul c o mm o9 o QC w 2 o t f(t) f i o _J -4-6 T, To T» Temperature Fig. 8. Temperature Dependence of Shift Factor, a, Evaluated from the Data of Figure 6. T

128 in both the time and frequency domain as the temperature dependence of all the viscoelastic functions can be described in term s of the temperature dependence of the factor a,. T 109 Summary It is hypothesized that the complicated dependence of the viscoelastic functions of bituminous concrete on the independent variables of time or frequency and temperature can be separated into a function of time, which is the viscoelastic function reduced to a purely arbitrary and convenient state, TQ and a function of temperature, which is the temperature reduction function. The thermolinearity concept was used in this study to extend the experimental results to longer and shorter portions of the loading time and frequency scale than normally can be obtained experimentally for the same amount or complexity of testing to determine the transient and dynamic moduli at any intermediate temperature in the experimental range. By using the previously discussed concepts, master curves of Ec (t), Jc (t), ^>), J 2 (w), and /E */ were developed from the transient and dynamic experimental data covering a relatively small portion of the time scale but over a wide range of temperatures normally encountered in pavements.

129 CHAPTER V MATERIALS TESTED AND EXPERIMENTAL TECHNIQUES General The experimental phase of this investigation consisted of applying the timetemperature superposition principle to determine the rheological response of bituminous concrete mixtures on the macroanalytical level by using transient creep tests and dynamic periodic loading tests. The material was considered to be homogeneous and isotropic, and the experimental results were analyzed to determine the accuracy of the approximation. The laboratory experimentation evaluated the time-dependent response of the material by transient constant stress experiments and determined the dynamic response of the material by means of periodic stress tests at several frequencies at stress levels in the linear ranje. The constant stress experiments evaluated the creep response of the material which was used to predict the complex elastic and transverse moduli in the frequency domain. The dynamic tests were used to obtain the same two independent complex moduli directly in the frequency domain. The results of the two types of tests were compared and correlated to provide verification of the accuracy possible with the linear viscoelastic assumption with this material. Application of the time-temperature superposition principle to this material was also evaluated because the concepts of linear viscoelasticity and thermolinearity 110

130 were required to transform the transient experimental results at any temperature to the frequency domain and directly compared to the dynamic test results. The goal of the dynamic and transient experimental tests was to obtain the complex modulus of elasticity, E*, and the complex transverse modulus, T*, as continuous functions of frequency at any experimental temperature used in this study, which are the two independent moduli selected to define completely the mechanical response of the material. Since the preceding viscoelastic functions as described in Chapter II can be calculated from the dynamic or static tests, one method of test can be used to obtain an independent check on the theoretical formulas and experimental procedures used in the other method of test. A single static test covering the total duration of the time scale or a series of dynamic tests covering the entire frequency range can be used to determine E* and T*. In principle, knowledge of either the static test creep compliance or the dynamic test complex elastic modulus and phase lag over the entire range of time or frequency scale permits calculation of the other corresponding rheological rheological property. The viscoelastic functions were traced out over a wide temperature range in order to increase the effective time and frequency scale by the thermolinearity principle. When properly applied, this principle yields experimental plots in term s of reduced variables which can be used with considerable confidence to predict viscoelastic behavior in regions of time or frequency scale not obtained experimentally. Ill

131 112 Materials Tested The experiments performed in this investigation were carried out on cylindrical and prismatic specimens of bituminous concrete mixtures. The prisms were used only in the resonant frequency dynamic tests and measured approximately six inches in length, one inch in thickness and up to two inches in width; this was found to be an optimum size for dynamic testing from previous tests on bituminous materials. In order to secure maximum uniformity in the specimens, the beam samples were cut from the frozen cylindrical specimens using a diamond-tipped, water-cooled saw. The cylindrical specimens which measured 2.8 inches in diameter and six inches in height were tested in the transient constant stress test and in the mechanical dynamic test. To obtain the maximum homogeniety and isotropy in the cylindrical samples, a gyratory compactor was used to mold the cylindrical specimens. The experimental samples were prepared in a uniform procedure from i several different mixtures varying from each other in compaction energy expended, type and gradation of aggregates, and source, amount, and kinematic i viscosity of bitumen. Approximately 210 specimens were prepared for this study consisting of five different types of mixtures. Preparation of Samples The proportioning of the aggregates and asphalt was performed on the basis of weight. The crushed limestone aggregates were first separated into

132 113 different sieve sizes, oven dried to a constant temperature and then combined into single batches of the exact proportions. The individual batches were heated along with the mixing bowl and molds in an oven maintained at 325 until used. During the mixing procedure, the exact amount of bitumen which had been heated to 275 F was combined with the heated aggregates, and then thoroughly mixed in a mechanical mixer for a minimum of two minutes or until all of the aggregate surfaces were coated with asphalt. The mix was then placed in the heated gyratory compactor mold by filling the mold two-thirds full and spading the mix vigorously 25 times allowing the spatula to pass through the top layer only. A fixed axial pressure was applied to the specimen being compacted and maintained at a constant value while the gyration angle was set at two degrees and the desired number of gyrations applied. The compaction mold was r e moved from the compactor and allowed to cool until the specimen had achieved sufficient strength to be extruded from the mold. The samples were stored at room temperature until the age effects (4^ had become negligible and then tested. To determine the homogeneity and isotropy of the materials, several cylinders were sawed into small cubes, and the bulk density of each cube was determined by both geometrical and water displacement methods. The ultimate compressive strength of the cubes cut from the larger cylindrical specimens was also obtained from a constant rate of strain test on different planes of the samples, under standard testing conditions. The uniformity of all the cylindrical and beam samples was checked by determining the bulk densities by

133 both geometrical and water displacement methods. The water displacement procedure was an independent check on the geometrical methods and also was used to determine the surface voids of the specimens. Identification tests were also performed on the aggregates and bitumens used in preparing the different mixes in order to perform void and density analyses of bituminous mixtures. The identification tests performed on the 114 bitumens included kinematic viscosity, penetration, and specific gravity, while the apparent and bulk specific gravities were determined for the 'aggregates used. The calculations carried out in the density and void analyses included: per cent of voids, maximum theoretical density, per cent of aggregate voids, per cent of bitumen and aggregate by volume. A summary is presented in Appendix I of the above density and void analyses, and the asphalt content and aggregate gradations used, as well as the gyratory compaction details. Also included in Appendix I are the identification tests performed on the bitumens: penetration, specific gravity, and the kinematic viscosity measured by a sliding plate microviscometer. The fundamental concept on which the sliding plate microviscometer operates is based on the classical definition of a viscous body. A layer of the material of a known thickness is sheared between two plates by the application of a constant shearing stress, and the rate of shear is measured. i Transient Tests To define the viscoelastic behavior of materials, it is necessary to obtain measurements of the stress and strain over a wide range of the time and tempera-

134 ture scales. Transient experiments are limited by inertial effects and by the impossibility of having a truly instantaneous application of stress or strain at the beginning of the experiment. However, constant stress and constant strain experiments are preferred for long loading times and are limited only by the precision of the apparatus and stability of the material. In the transient creep tests conducted, a constant axial stress was suddenly' applied without impact and maintained constant throughout the test by a standard soil consolidation apparatus shown in Figure 9. Continuous recordings of the axial and circumferential strains were obtained by means of a Sanborn Twin-Viso recorder with two strain gage amplifiers. The measurement of the strains was accomplished by four SR-4 electric strain gages attached to each specimen at mid-height, two horizontally and two vertically at diametrically opposite points on the periphery of the sample. The horizontal gages were connected in series and used to obtain the axial strain. Thus, simultaneous records of axial and circumferential strain versus time were made available. Four identical SR-4 gages attaches to a dummy specimen at the testing temperature completed the strain gage circuit. To conduct the tests under isothermal conditions, all the specimens were enclosed in a rubber triaxial membrane and kept in a constant temperature water bath for several hours prior to testing. The samples were then placed into a smaller cylindrical water bath with continuous circulation of water at the specified temperature with a maximum variation in temperature of t 0.1 C and tested. At low testing temperatures below

135 Fig. 9. Transient Test Equipment

136 117 0 C, methanol alchohol was added to the circulating water in the proper proportion to prevent freezing. A typical transient test recording obtained of the axial and circum ferential strains by the Sanborn recorder is reproduced in Figure 10. It can be seen that the load was rapidly applied without impact in less than one-tenth of a second. Sanborn recordings were obtained by running the recorder at its highest speed of ten cm. per second and by quickly applying and removing the load without impact. In the analysis of the viscoelastic results of the constant stress test, the axial and circumferential strains could be subdivided into three distinct zones to depict clearly the characteristic feasures of viscoelastic response. At very short times the deformation consisted primarily of the instantaneous elastic deformation, eg. At slightly longer times the retarded elastic deformation, e develops and becomes the dominating term in the deformation. After still longer times the linear material shows a small further increase in deformation due to Newtonian flow, e^. The total strain, e i n the specimen at any time is given by T = e E + el J + c T1 V -l In the preliminary testing, it was observed that the steady state of deformation or strain caused by Newtonian flow was reached in less than 15 minutes of testing for all mixtures and temperatures investigated in this study. Thus, in all of the constant stress experiments performed, the load was allowed to act for this length of time at the eleven temperatures and stress levels studied,

137 One Second TIME Sanborn Recording of Transient Test.

138 119 after which it was quickly removed and the instantaneous elastic rebound re corded. All tests were performed on previously untested specimens to elim i nate any changes in rheological properties of the material, which may have occurred due to an earlier stress loading history (45). Identical tests were performed on three specimens from the same mix series, and the results were averaged to obtain the axial and circumferential strains at each test tem perature. It was found that the cross-section of the specimens was not greatly altered at the low levels of stress used in this study, therefore, the stress initially applied was assumed to remain constant. The limits of the linear theory cover a wide range for different materials and may change for the same material at different temperatures. Bituminous concrete is such a material in which the limits of linear viscoelasticity vary due to temperature, and it was necessary to develop experimental curves similar to Figure 2 for each of the eleven temperatures used in this study to determine if the stress levels used were approximately within the linear range. The transient experimental axial and circumferential strain curves are p resented in the analysis. The experimental results of the dynamic tests were also analyzed to determine if the complex moduli in the frequency domain are independent of the stress level and may be represented by the linear viscoelastic theory at low stress levels. The transient experimental stress levels used were chosen to be sufficiently low (as compared to the ultimate strength of the material) so as to be within the range for which the linear viscoelastic theory might produce a good

139 120 approximation for the material at the temperature of test, and comparable to the stress levels used in the dynamic tests. A summary of the stress levels used at each temperature in the static and dynamic tests is presented in Appendix II. Periodic or Dynamic Tests In order to supplement the transient experiments and provide information about the dynamic response of bituminous concrete mixtures at very short time intervals, dynamic tests were performed in which the stress varied periodically with a sinusoidal alternation at a frequency, w. The dynamic tests were also performed to provide an independent correlation of the complex moduli obtained from transient tests in the time domain and to yield values of the complex elastic and transverse moduli and respective phase lags, directly in the frequency domain. Using the results of periodic tests, the accuracy of the linear viscoelastic theory and validity of the application of the thermolinearity principle to predict the response of the material were also investigated. If the sample size, consistency and frequency are such that the inertia can be neglected (following the criterion that the length should be small when i i. compared with the wave-length of the elastic waves propagated at the frequency of measurement), the most convenient way to determine the in-phase and out-of-phase viscoelastic responses of a material in the linear range is to measure both stress and resulting strains as functions of time in sinusoidal

140 121 deformations. Such measurements were performed by direct mechanical means by stressing the material in series with a load cell. In the dynamic tests performed a periodic stress was applied to the cylindrical specimens aqd the resulting axial and circumferential strains were r e corded. The strain of a linear viscoelastic material in a dynamic test will also alternate sinusoidally, but will be out-of-phase with the stress as discussed in earlier chapters. It was necessary to record five quantities describing the steady-state periodic deformation of the material in order to evaluate the two independent moduli from dynamic tests. These quantities obtained as accurately as possible from experimental tests were: the steady-state amplitude of axial stress, <rozz, the steady-state amplitude of axial strain eozz, the steady-state amplitude of circumferential strain, e qqq, and and ^T, the phase angles by which the axial and circumferential strains lagged the axial stress, respectively. The sinusoidal stress applied to the sample was measured by a load cell connected to the Sanborn recorder to obtain a continuous recording of the stress. In the testing procedure that followed, the load cell was connected in series immediately above the specimen as shown in the photograph in Figure 11 with the other essential parts of the testing equipment. The axial and circumferential strains of the 2. 8 inch in diameter by six inch in height specimens were also re corded using the Sanborn recorder to determine the strain amplitudes and phase lags. The amplitudes of the sinusoidal stress and strains and their respective phase angles were readily obtained from graphs made by the Sanborn recorder.

141 Fig. 11. Dynamic Test Equipment

142 123 Figure 12 contains typical Sanborn recordings showing a simultaneous recording of axial strain and axial stress at a slow recording speed, and the lateral strain and axial stress at a fast Sanborn recording speed at the same frequency. The sinusoidal stress was applied by means of a mechanical repetitive load apparatus developed at the Transportation Engineering Center. An electrical constant speed motor is used to turn a sinusoidal cam of double eccentricity through a set of interchangeable gears, by which it is possible to vary the frequency of applied load. The cam imparts a periodic vertical motion to the tip of a lever arm which compresses the spring and transmits a sinusoidal stress to the sample below. By varying the rigidity of the springs used and regulating the compression distance of the springs, it was possible to obtain a desired stress using the load cell to calibrate the system. It was not possible to apply direct tensile stresses to the specimens using the above mechanical system, and an additional static compressive stress was superimposed on the samples which was exactly equal to the maximum value of the sinusoidal stress. Thus, the stress on the specimen varied sinusoidally from zero to maximum value, <r0, as shown in Figure 4. The stress may be written in the form o- = cr0 + cr0 sin wt V-2 The compressive stress superimposed on the specimen did not influence the dynamic test behavior of the specimens, and the net behavior of the material was the sum of the dynamic and static stress separately. Actually this is an application of the superposition principle to the rheological response of m aterials.

143 124 One Second TIME iskin One Second TIME Fig. 12. Sanborn Recording of Dynamic Test.

144 All dynamic experiments were conducted under isothermal conditions at three representative temperatures covering the range of temperatures investigated in the transient test, namely: 5.4, 25.0, and 36.1 C with a maximum variation of t 0.1 C. In order to have a direct correlation of results, the temperatures and the stress levels were identical to those used in the static tests. The specimens were kept in a constant temperature water bath for several hours prior to testing and then tested in a similar manner to the static tests. The dynamic experimental results are completely summarized in Table 1, 2, 3 and 4 of the analysis of Chapter VI in terms of the absolute value of the complex elastic modulus, /E*/» and transverse modulus, /T*/» and their respective phase lags 4 g and 4 ^ as well as the real and imaginary components of the elastic and transverse complex compliances. The major variables investigated in this phase of the experimentation were frequency, temperature, stress level, and type of bituminous mixture. The three frequencies used were 103, 206, and 824 cycles per minute at temperatures of 5.4, 25.0 and 36.1 C. The values of the periodic stress used were and 32.5 psi at 5. 4 C; 4.06, 8.12, psi at 25.0 C; and 4.06 and 8.12 psi at 36.1 C. A total of 64 samples were tested in the dynamic series of experiments. Dynamic Resonant Frequency Test The methods of the previous section are subject to the usual restriction. for direct sinusoidal measurements the critical dimension of the sample must be small compared with the wavelength of the corresponding compressional wave.

145 126 When the wavelength is the same order of magnitude as the sample dimensions, standing waves can be set up within the sample at specified frequencies; i.e., it vibrates at its various characteristic modes. It is necessary that the phase lag be small, but for relatively hard viscoelastic materials such as bituminous concrete, this is usually the case at high frequencies. The available experimental frequencies are limited to discrete values corresponding to different modes of vibration, but several frequencies can be obtained by using specimens of different sizes and shapes. The two properties that were measured in the sonic experiments in order to characterize the dynamic behavior of the materials were the real component of the complex modulus and phase angle. The dynamic viscoelastic data obtained were the real components of the complex elastic moduli from vibrations of bituminous concrete prisms in flexure and the complex shear moduli from experiments in which the prisms are vibrated in torsion. Since viscous losses accompany the deformation, the complex modulus was to be used to express both the real and imaginary parts of the modulus as a complex number. There are several methods of expressing the loss tangent in terms of directly measured quantities, provided they are not too large (46). The amplitude of vibration will be observed at its maximum value when the impressed frequency equals the critical resonant frequency of the specimen, <*>0. The change in the impressed frequency, aw between two points on the resonance frequency curve where the amplitude is l/\j2 of the maximum value of u>0 were also

146 127 measured, and the loss tangent approximated by tan 4 - o V-3 It should be noted that at the high frequencies used in the sonic resonant tests, the phase angle approaches zero, while the storage part of the complex modulus is approximately equal to the absolute value of the complex modulus. Since the resonance of the specimen is due to its own inertia and there is / no added mass associated with the apparatus, the density of the specimen enters into all calculations of the dynamic storage modulus. The dimensions of the sample must be known to a high precision and the uniformity of the specimen carefully controlled. If the specimen is molded, it must be free from anis- tropy, and if the sample is machined as it was in this experimentation, alteration of the surface must be avoided. The determination of the (4 and phase angle from the fundamental resonant frequency vibrations were performed following ASTM Designation C (47) and other similar references, (48), (49) and will not be discussed in further detail. The evaluation of the E^ (w) from the resonant frequency, weight, and dimensions of the prism s was calculated by the following equation E1 (w) = C W N2 V-4 where Ej^ (u) = real part of complex elastic modulus W = weight of specimen *

147 Sonic Equipment Showing E-Scope and Lissajou Test Pattern the Oscilloscope.

148 N C = resonant frequency of sample = a factor which depends on size and shape of sample, mode of vibration and Poisson's ratio A similar formula was used to calculate the elastic part of the complex shear modulus, (j). (N1)2 V-5 Figure 13 is a photograph of the resonant frequency test in progress showing the sonic apparatus and Lissajou test pattern on the oscilloscope. The respective size and shape factors, resonant fundamental frequencies, and specimen weights are summarized in Table 5 of Chapter VI with the calculated values of the dynamic storage and loss parts of the elastic and shear complex moduli. The prisms were also checked for the existence of nodal points to ascertain if their vibration was at the fundamental frequency by probing the specimen. Since the beam vibration was at its fundamental frequency in flexure, two nodes were easily observed, and there was zero amplitude of vibration at the node, and the Lissajou circle showed inclination in opposite directions as the node was crossed to indicate the location of a node. The experiments were performed on the 300 and 500 series mixtures under isothermal conditions at 25 C in order to have an independent correlation of results at high frequencies with the complex moduli predicted by the static experiments. The frequencies investigated were the resonant fundamental frequencies of the specimens at low stress levels in the linear range.

149 CHAPTER VI PRESENTATION AND ANALYSIS OF EXPERIMENTAL DATA General The analysis of the temperature dependence of the viscoelastic functions of bituminous concrete mixtures was now considered, and the methods discussed in the previous sections were applied to the transient constant stress and the dynamic tests data. The experimental data on the macroanalytical level are presented in graphical and tabular form in an attempt to verify that the materials used in this stucfy may be represented by thermorheologically linear materials under the experimental conditions investigated, and to verify that every contribution of the bitumen flexible-chain macromolecules to the viscoelastic functions is proportional to a single temperature reduction factor, a,p. The applicability of the linear viscoelastic theory to represent the r e s ponse of bituminous concrete mixtures was investigated by both dynamic and transient experiments. The complex elastic and the complex transverse moduli and their re s pective phase angles were calculated from transient tests for the two types of bituminous concrete mixes investigated in this study, designated series 300 and 500 at three temperatures, namely 5.4, 25.0, and 36.1 C, at which a direct correlation was obtained between the dynamic periodic stress and transient test results. 130

150 Dynamic experiments were also performed on the 700, 800, and 900 series bituminous concrete mixtures in order to study the temperature and time dependent viscoelastic behavior of these materials directly in the frequency domain. The sample calculations for evaluating the complex elastic modulus and complex transverse modulus using the concepts discussed in earlier chapters, as well as an analysis of the experimental results, are now presented, while the significance of the experimental data are discussed in Chapter VII. Homogeneity and Isotropy of Materials Tested The degree of homogeneity of the experimental specimens prepared by the gyratory compaction procedure used in this study was analyzed by evaluation of the bulk densities of all the experimental beam and cylindrical specimens. In order to determine to what degree the samples were isotropic, the ultimate compressive strength of the sawed cubes was also obtained by p erforming a constant rate of deformation test under standard experimental conditions on different planes of the samples. An analyses at the phenomenological level of the bulk densities and ultimate compressive strengths of the experimental mixtures showed the specimens prepared for this study to be quite homogeneous and isotropic. Transient Linear Viscoelastic Response of Bituminous Concrete In order for the response of a material to be described perfectly by the theory of linear viscoelasticity, the creep modulus in a constant stress test at

151 a specific temperature must be independent of the stress level. In Figure 14 are plotted the results of several creep tests on the 500 series bituminous concrete mix which were performed on four identical samples under similar conditions. Only the axial stress in each test was varied by simple multiplies and the axial strain recorded and plotted. A simple criterion using the concept of linear viscoelastic response was used to determine experimentally whether the creep behavior of bituminous concrete mixtures could be represented accurately by linear viscoelastic behavior at low stress levels. As an example of this criterion in Figure 14 the transient creep experimental results are presented which were evaluated from four similar specimens of a 500 series bituminous concrete mix varying only the stress level. Inspection of Figure 14 reveals that the strain curves are proportional to the stress levels at all times, showing that the linear theory is a satisfactory approximation of the behavior of this material. All three components of the axial strains shown in Figure 14 are proportional to the constant axial stress, cr0, applied to the specimens. The elastic creep modulus, Ec (t), is defined as the value of stress divided by strain at any time for any of the four tests as shown E (t) = -To,. =... = VI-1 c fti (t) e1zz eqzz A similar relation may also be used to define the creep transverse modulus,

152 1,600 1,400 Axial Strainx ICf*in in./in. 1, IjOOO Time,tin Seconds Fig. 14. Linear Viscoelastic Response of 500 Series Bituminous Concrete Mixture in Transient Creep Test at Four Stress Levels. 133

153 Equations VI-1 and VI-2 can be generalized to include any stress level in the linear viscoelastic range by 134 cnzz IV-3 and cn9 IV-4 at any given time. For each material there is a certain limit to the linearity either in a constant stress, constant strain, or dynamic test under given conditions beyond which stress is not proportional to strain, and nonlinear or collapse mechanisms will result. Evaluation of the elastic creep function moduli from the transient data of Figure 14 using the above equations seems to yield the same value of the transient moduli at any given time. Thus, the criterion provides verification that the response of the material can be represented by a system of linear springs and dashpots at any time and the transient creep moduli are independent of the stress level in the linear viscoelastic range. Transient creep tests were performed on the 300 series mixture used in the study under identical conditions, except that the stress was varied by constant multiples in a manner similar to the method used in the above discussion. The axial and circumferential strains of the 300 and 500 mixtures which were evaluated from the above tests I illustrated that linear viscoelastic behavior is a good approximation to depict the response of bituminous concrete mixtures at low stress levels.

154 135 Inspection of the 300 and 500 mix experimental plots of the axial and circumferential strains revealed that the strains had increased by the same ratios used to vary the stress. Thus, for the stress levels used in the constant stress experiments, namely: 4.06, 8.12, 12.18, andl6.24psi, the resulting strains are very closely equal to e, 2e, 3c, and 4e at any given time. Dynamic Experimental Results The results of the direct dynamic experiments are completely summarized in the following tables in terms of absolute values of the complex moduli, /E * / and /T * /, and their respective phase lags, 4% anc*4j<t as well as the real and imaginary parts of elastic and transverse complex compliance. Three other bituminous concrete mixtures, namely the 700, 800, and 900 mix designations, were also investigated by the dynamic experiments. The principle variables investigated in this phase of the experiment were: frequency, temperature, stress level, and type of bituminous mixture. The frequencies used in this study were 103, 206, and 824 cycles per minute at temperatures of 5.4, 25.0, and 36.1 C. The amplitude of the periodic stress used in this phase of the experimentation were and psi at 5.4 C; 4.06, 8.12, and 6.24 psi at 25.0 C; and 4.06 and 8.12 psi at 36.1 C. Table 1 presents a summary of the complex elastic modulus results for all the dynamic tests performed. The absolute values of the complex modulus were determined by dividing the amplitude of the axial stress by the amplitude

155 TABLE 1 DYNAMIC EXPERIMENTAL RESULTS-ABSOLUTE VALUE OF COMPLEX MODULUS, /E * / AND PHASE ANGLE, E *l x 106 in p si Frequency, R adians/second T em perature, *C Amplitude of Axial S tre ss, p si Mix Designation Phase Angle, $ E in Degrees

156 TABLE 2 DYNAMIC EXPERIMENTAL RESULTS-ABSOLUTE VALUE OF COMPLEX TRANSVERSE MODULUS, /T * / AND PHASE ANGLE, J t *J x 10* in pet Frequency, R adians/second T em perature, *C Amplitude of Axial S tre ss, psi Mix Designation P h ase Angle, in D egrees W -q

157 TABLE 3 DYNAMIC EXPERIMENTAL RESULTS-REAL AND IMAGINARY COMPONENTS OF COMPLEX COMPLIANCE J, (w) x 10-6 In 1 psi Frequency, R adians/second T em perature, *C Amplitude of Axial S tre ss, psi Mix Designation J 2 <w) x 10-6 I n - i

158 TABLE 4 REAL AND IMAGINARY COMPONENTS OF COMPLEX TRANSVERSE COMPLIANCE Frequency, R adians/second T em perature, *C Amplitude of Axial S tre ss, psi Mix Designation F2 <w) x 10 ^ 5 1r 1 p si

159 140 axial strain obtained from the Sanborn recorder graphs. Several important facts are brought out by inspection of the data in Table 1, the most important being that the magnitude and phase of the complex modulus are independent of the stress level and depend only on the frequency and temperature for all of the five materials studied to provide verification that the materials used in this study may be represented by linear viscoelastic materials. At any given frequency the absolute value of the complex moduli are independent of the stress level in the linear viscoelastic range. If the amplitude of the applied stress is doubled, the resulting amplitudes of axial and circumferential strains are also doubled so that /E * / or /T */ are unchanged. Another important consideration brought out by the data in Table 1 is that the magnitude of the absolute value of the complex modulus increases with frequency or a decrease in temperature. Included in Table 1 is a summary of the phase of the complex modulus which is defined as the angle by which sinusoidal axial strain lags the sinusoidal axial stress. The values of the phase angle as well as the amplitudes of axial stress and strain were determined from the recordings similar to those presented in Figure 12. Again it can be noted that the phase of the complex modulus at any given test condition is independent of the stress level used; however, the value of the phase angle decreases with increasing frequency or a decrease in temperature. In the later analysis of the viscoelastic response in transient tests, it is shown that the absolute value of the complex elastic modulus decreases rapidly at lower frequencies and approaches zero, while at higher frequencies the modulus increases and approaches a maximum value asympotically. The value of

160 141 the phase angle also changes rapidly with frequency, and at relatively high frequencies used in the sonic resonance test, it approaches zero. In Table 2 the absolute value of the complex transverse modulus and its corresponding phase lag are summarized. The absolute value of the complex transverse modulus, /T * /, is the value of the amplitude of the axial stress divided by the amplitude of the lateral strain. The phase lag, is the angle by which the laterial strain lags the axial stress. Inspection of Table 2 also points out the linear response of bituminous concrete at the experimental conditions and the levels of stress used. The magnitude of the complex modulus and loss angle are independent of the stress and depend only on frequency and temperature. In Tables 3 and 4 the real and imaginary components of the complex elastic and transverse compliances are respectively recorded. It should be noted that for a given bituminous concrete mixture, the experimental data indicate that J 1 (*>)» ^2 Fi ^)»?uld f 2 are independent of the applied stress level and depend only frequency and temperature of test. The results of the sonic dynamic experiments are summarized in Table 5. The real part and phase of the complex elastic and complex shear moduli were calculated from the experimental results. The dimensions of the samples, size and shape factors, resonant fundamental frequencies, and specimen weights are also recorded in Table 5. The results of the above dynamic experiments and the transient tests presented earlier provide verification that the time-dependent

161 TABLE 5 ABSOLUTE VALUE OF COMPLEX MODULUS AND PHASE ANGLE FROM TORSIONAL AND FLEXURAL RESONANT FREQUENCIES Bitum inous C oncrete M ixtures T em perature 25*C Specimen No. Length in. B ase in. T hickness in. Weight lbs. Resonant Frequency C B E l i " ) * F lexural Torsional 10 p si N cpe N 'cps * E ta D egrees Eofw) x 10 p si E* x 106 psi GWw) x 10 p si <JG in D egrees G2 p ) x 10 psi G* x 106 psi

162 response of the materials used in this study may be represented by the concepts of linear viscoelastic behavior. 143 Thermorheologically Linear Response of Bituminous Concrete The experimental data from the constant stress tests of the 500 series mixtures are presented in Figure 15. The creep moduli have been evaluated at eleven different experimental temperatures varying from 0 to 120 F as indicated and plotted on logarithmic plots versus time. A reference temperature, TQ equal to 298 K was arbitrarily chosen from within the experimental range. The creep functions were reduced by the factor -Ho-, which enters T into the coordinate schemes because of the entropy-spring nature of the stored elastic energy as discussed in the section on the kinetic theory of rubberlike elasticity. The factor has been omitted from the calculations as it approaches unity and is within the experimental e rro r of the tests. The temperature factor -Ha produces a slight vertical shift in the creep functions evaluated T in the constant stress experiments. Each creep function curve at a given temperature was obtained by testing three 500 mixture samples under exactly identical conditions and averaging the experimental axial and circumferential strain data. Investigation of the eleven creep functions in Figure 15 shows that a change in the experimental temperature shifts only the position of the reduced creep modulus curves with respect to the logarithmic time scale, and the general shape of the curves are not altered when the experiments are performed at different temperatures. The horizontal distance between each pair of adjacent

163 10 Reduced Creep Modulus, -j- Ec(t)x IO*in PSI (Logarithmic) K * a Time,t in Seconds (Logarithmic) Fig. 15. Elastic Creep Modulus of 500 Series Bituminous Concrete Mix Reduced to 298 K, vs. Time at Eleven Temperatures.

164 experimental curves was measured with a pair of dividers and found to be the 145 same within experimental erro r. Thus, portions of the adjacent creep viscoelastic functions are essentially parallel lines and have the same values of the creep modulus at different portions of the logarithmic time scale. A change in temperature has changed only the position of the modulus curve on the logarithmic time scale as all the retardation times have the same temperature dependence to provide experimental verification that the bitumen-aggregate mixtures used in this study may be represented by thermorheologically simple materials. Exact matching of the shapes of adjacent viscoelastic functions has already been cited as one criterion for the applicability of the method of reduced variables. Such thermal sensitivity of a material allows the transient creep modulus, which is generally a function of time and temperature, to be reduced, in this case to be only a function of reduced time as discussed earlier, and is I of the form Ec (in t, T) = Ec (X) VI-5 where X = in t - f (T) VI-6 Figure 15 illustrates the principle that temperature and time have a similar effect on the response of bituminous concrete mixtures, and a change in temperature of test is equivalent to a change in the time scale of experimental plots. As the kinetic theory of rubberlike elasticity shows that the equilibrium modulus is proportional to the absolute temperature, the reduction scheme is

165 146 slightly altered to make it somewhat more satisfactory theoretically: thus, the quantity T0p0 Ec (fn t, T) rather than E (in t, T) itself should obey the Tp time-temperature relation. The density, p, of the material is also slightly dependent on temperature, and the factors and Po lead to small corrections introduced for theoretical reasons. All experimental data of the bituminous concrete mixtures investigated have been reduced to the standard temperature, TQ, of the master curve. The dynamic test and transient constant stress test moduli are reduced as follows: 2x»- Ec (in t x, T) = EcTo (in t) VI-7 T ~ (in co dip, T) J jiji (Ai VI 8 T0 o J 2 (in co dp, T) = J2 T ^ w) VI 9 To o The master curves Ecpo (in t), J i t q w)» an<i J 2TQ ^ are functions of the reduced time or reduced frequency only and describe the viscoelastic properties of the material at a single reference temperature, TQ. The effect of temperature change consists in multiplying all retardation times by a common factor, ax, which is a function of temperature controlled by the motion of the individual bitumen macromolecules and embodies the equivalent effect of time and temperature on the rheological properties of the material. The molecular - J theories based on thp flexible-chain macromolecules applied to bitumens predict

166 147 that a single composite viscoelastic function should be obtained as the macromolecule motions reflected in the measure function are controlled by a single average friction coefficients. Temperature Reduction Factor, at The thermolinearity principle was applied to the viscoelastic data of Figure 15 to determine the temperature reduction function, f (T), which was then applied to shift viscoelastic data at the experimental temperature to the standard reference temperature for study by a multiplicative transform of the time scale. The temperature reduction function, f (T), is a smooth, continuous, decreasing function of temperature and was constructed from the creep data of Figure 15 by plotting the relative time shifts, a *n at, versus the corresponding temperature difference as shown in Figure 16. The procedure used in this study to develop the time-temperature relation was as follows: a re ference temperature, Tq, within the range of experiments was first chosen, which in this case was 298 K, although any temperature from 0 F to 120 F could have been easily used. Then the logarithmic plots of Ta. (In t) T versus time were prepared from creep functions evaluated at eleven temperatures by transient experiments. The horizontal distance between each pair of adjacent curves was measured and recorded as a *n a-p. The measured values of a Ai ap were then added progressively from the standard reference temperature to obtain f (T) which was then applied to develop the viscoelastic functions at any temperature in the tested range. It is shown later that the appropriate

167 fo O +5 ro csi + 4 Mix D esignation S erie s R eference Temperature,Tos 298 K TJ f(t ) T em perature, T in K Fig. 16. Temperature Dependence of Shift Factor, a^, Developed from Elastic Creep Data.

168 values of In a^ obtained from the transient experimental results of Ec (t) can also be used to superpose the transient test results of T_ (t) and the dynamic 149 viscoelastic functions, / E*/» /T*/» ^ e» anc* in the frequency domain for the same 500 series bituminous mix, indicating that f (T) is a basic characteristic of the material. The most important requirement of the time-temperature relations is that all relaxation or retardation times are influenced in the same amount by a temperature change. The general shape of the retardation spectrum is not altered by temperature but is only shifted along the time scale. Thus, the temperature reduction factor, a T, can be defined as H where t 0 is the To retardation time at a selected reference temperature TQ, and t j is any retardation time at any absolute experimental temperature Tj. All logarithmic plots and logarithms presented in the viscoelastic equations used in this study are to the base e unless otherwise stated. Master Creep Function at T^ The viscoelastic data plotted in Figure 15 was also used to extend the experimentally accessible time region of the creep curve obtained at 298 K to develop a single master creep function at this reference temperature. Since time and temperature have been shown to have a similar influence on the creep modulus in Figure 15, it was possible to shift the remaining ten reduced creep modulus curves to the standard reference temperature by transforming them horizontally along the logarithmic time axis until the experimental points are

169 superimposed upon the reference curve at 298 K to give a continuously smooth plot. The amount of horizontal shift necessary to bring coincidence, In a T, to the creep curves is defined as the temperature reduction factor and is determined by referring to Figure 16. The factor a-p was evaluated empirically at each temperature, but the requirement of superposition over a range of time or fre quency does not permit an arbitrary selection in this choice. In static and dynamic tests the same value of the reduction factor must bring both the real and imaginary components of the complex elastic modulus or (j) and J 2 (*>) into superposition. 150 In Figure 17 the idealized m aster creep function for the 500 series bituminous concrete mixture was determined at 298 K by application of the tim e- temperature relation. The master creep function for any temperature within the tested range shown on the experimental plots in Figure 15 can easily be obtained by a sim ilar procedure. In Figure 17 the time scale of the experimental results was extended beyond the practical experimental range by superposi- 6 tion of eleven individual creep functions, and now extends from 10 approximately 10 seconds or twelve days. seconds to The composite curve represents the behavior of the bituminous mix at a reference temperature TQ; the master curve is actually composed of segments of the eleven creep functions which were evaluated at different temperatures. As transient creep experiments usually do not extend over more than six decades of logarithmic time nor dynamic experimental measurements over more than

170 Reduced Master Creep Modulus, ^ Ec(t)xicf PSI(Logarittimic) Ec(t) Log of Reduced T im e, ^ 3 ^ 3 *' ' n Seconde Fig. 17. Composite Master Creep Function Obtained by Thermolinearity Concept, Representing Viscoelastic Behavior over an Extended Time Scale at 298 K, 500 Mix.

171 152 three decades of logarithmic frequency, specific numerical tests of the accuracy of the temperature reduction procedure are usually restricted to distances of this order, although the time scale of the reduced viscoelastic functions may cover a much greater time range. Such limitations, however, do not detract from the usefulness of the reduced master viscoelastic functions and accompanying temperature reduction function, f (T), as an expression of data from which the viscoelastic properties can be evaluated over wide ranges of temperature and the time scale within reasonable limits. Master Creep Compliance Function Similar concepts were also applied to develop a master creep compliance curve, T 7 Jc (In t), from the experimental data of the constant stress tests of To Figure 15 by plotting the creep compliances at eleven temperatures. The identical master creep compliance curve was also obtained by fundamentals from the master creep function at 298 K from Figure 17. The composite master creep compliance function at 298 K was evaluated for the 500 series bituminous concrete mix and is presented in Figure 18. factor, By use of the appropriate value of in a>p from Figure 16, and the temperature T, the master creep compliance function at 298 K has been transformed to another single reference temperature at K by the thermolinearity concept, and the extention of the experimental time scale to an exceedingly long interval can be noted in Figure 18. A similar procedure may be used to develop the m aster viscoelastic functions at any temperature, T-, in the experimental

172 120 _ 110 -te ioo b I 4 80 TO u c o 60 Ck. 6 5 Q. *» o> o 321.8#K Jcfi) 278.4#K s Log of Reduced Time, ^ 303* Secon(ls Fig. 18. Composite Master Creep Compliance Functions at Three Temperatures, 500 Mix. 153

173 154 range. The master compliance curve at 298 K has also transformed to a reference state at K, and the experimental time scale has been extended to define the viscoelastic response of the material at extremely fast rates of loading as shown in Figure 18. Generalized Voigt Model Before transforming the transient viscoelastic functions to the frequency domain by use of Laplace transforms and obtaining a correlation of results with the dynamic test data directly in the frequency domain, it was first necessary to reduce all the transient data to a common reference state. This was necessary in order to extend the time scale of the transient experimental data to apply the graphical transformations used in the study. The reduction is essentially a two-step process: a small vertical correction due to the kinetic theory of rubberlike elasticity was first applied to the viscoelastic functions, followed by a horizontal shift in the time scale to allow for the effect of temperature on the retardation spectrum. Referring to the concepts summarized in earlier chapters, the viscoelastic response of bituminous concrete mixtures at any specified time and temperature in the tested range can be represented by a mechanical model consisting of an infinite number of linear springs and dashpots. The response of such a generalized Voigt model used to describe the material can be represented by a continuous model in which the Kelvin units with retardation times between fn t and fn t + d fn t will contribute to the creep compliance. From the logarithmic

174 155 distribution function the creep behavior may be calculated by equation n-32 Jc W = Jo + ^ o + J* M * n r) [ l - e " t/t] d (inr) IV-10 The storage and loss components of the complex compliance may also be calculated by the retardation spectrum evaluated from transient tests using equations II-5 7 and n-58 to obtain the storage component and the loss component, respectively. The storage and loss components of the complex compliance obtained from the transient tests data were evaluated and compared to the values of (a) and evaluated from dynamic test in order to obtain a correlation of results in the frequency domain. The master creep compliance function at a constant temperature may be written in the form of equation Jc «- Jo + J* «+ *lo V I" 11 where the normalized retarded elastic component function, i j (t), increases monotonically from zero at zero time, to unity at infinite time (O) = o, «4j (a) = 1 The retarded elastic portion, (t), of the creep compliance can be determined from the experimental data at 298 K in Figure 18 by subtracting the instantaneous elastic, J Q, and Newtonian flow, t/n0, components of compliance from J c (t). The resulting retarded elastic function can then be normalized and used to calculate the retardation spectrum. The master creep compliance plotted in Figure 18 at 298 K was multiplied by the axial stress, which by referring to Appendix II is psi in this case,

175 to obtain the axial strain curve at this temperature plotted in Figure 19 versus time. The steady-state axial strain approaches a limiting value at long loading times; however, it should be noted that the axial strain continues to increase without limit at a small but perceptible amount, due to the contribution from viscous flow, t/io- The steady-state rate of strain, egs, was evaluated from the data in Figure 19 and used to calculate the coefficient of Newtonian viscosity, ti0< The contribution of viscous flow t/r\0 was then subtracted from master creep compliance function. The values of J (t) at short loading times plotted in Figure 18 also V approached a limiting value, but a positive slope is still noted to exist in Jc (t) even at the shortest times evaluated. An approximation to J c (t) may be obtained at zero time by using the smallest value of J (t) plotted in Figure 18 as the instantaneous component of the compliance, J Q. 156 Inspection of Jc (t) in Figure 18 reveals that J 0 asymptotically approaches a limiting value at extremely short loading times. The elastic response of a material may be defined by the concept that the material is stressed and the resulting strains measured in such a short time interval that the time-dependent response can be neglected. The difficulty, therefore, in the precise evaluation of the elastic response of a material is purely an experimental one and not fundamental. The instantaneous compliance of all the materials investigated in this study approached zero at short loading times and may be subtracted from J (t) in order to obtain the retarded elastic response. However, it should be c noted that the values of J Qare small when compared to the retarded elastic

176 2, PSI 1,600 Axial Strain, «x 10 in in./in 1, T im e, t in Secondsx 10* Fig. 19. Axial Strain vs. Time from Master Creep Function at 298 K, 500 Mix. 157

177 component of J (t) and were actually neglected when evaluating the retarded c elastic response in future calculations. The normalized elastic component plotted in Figure 20 was evaluated by subtracting only the Newtonian flow component from the creep compliance. 158 The distribution function of retardation times, L (In t ), is of importance; first, because it represents the behavior of the material at a specified temperature in a more general way than the functions such as J (t), J ( o ), and J-> ( j ), C l which describe the behavior of the material in a specific test; and secondly, it can be used to calculate the time-dependent viscoelastic response once the frequency behavior of a linear viscoelastic material has been defined. The important problem now arisesr of how to calculate L (f n t ) from the constant l stress experimental data presented in tabular or graphical form. The direct inversion of equations defining Ec (t), J c (t), &), and (a) cannot be used directly in dealing with the experimental results to obtain L (in t ). Therefore, the graphical methods discussed in Chapter III were used to determine the retardation spectrum from static creep experiments. The graphical approximation method of the first order derived in Chapter III was used to evaluate the spectrum by the slope of the normalized retarded elastic component of J c (t) plotted in Figure 20 by a form of equation n i-33 Li<<nT)=7T^7 [V] VI'12 The results of the graphical procedure for determining the distribution function of retardation times is presented in Figure 21. It should be pointed

178 l.o NOTE: I* x 10"* 5 ^ 7 Normalized Retarded Elastic Response, f (t) Time, t in Seconds ( Logarithmic) Fig. 20. Normalized Retarded Elastic Response, i >(t)» vs. Time at 298 K, Obtained from Master Creep Function of 500 Mix.

179 0.6 Distributution Function of Retardation TimeJ-l^r) NOTE: 1=104.2 x 10 -pgj L (//»r) Retardation Time, r in Seconds (Logarithmic) Fig. 21. Distribution Function of Retardation Time, L ( ^ r ) Obtained from Master Creep Modulus at 298 K, 500 Mix.

180 161 out here that there is a practical limit to the use of higher derivatives of observed response curves, as the second and higher derivatives are very sensitive to small experimental errors in the experimentally recorded viscoelastic behavior of the material. The retardation spectrum was also calculated from the creep compliance by the second order approximation method of Schwarzl using a form of equation OT-34 where t = 2t >**T>-dfct M'Tfcf] V I ' 1 3 The retardation spectrum obtained by the second order approximation was found to have the same general shape as Lj (in t ) except that the peak area evaluated in 1*2 (in t ) was greater. However, the values of (in t ) and (in t ) were approximately equal in the region of the spectrum used to evaluat e the predicted values of the complex moduli to be compared with the complex moduli evaluated from dynamic tests. As a result the first order approximations were used in the actual calculations. Later sections will show that a good correlation of r e sults can be obtained by the use of (in t ) to predict the complex moduli. Generalized models were used in this study to represent the response of real materials since this method allows the mechanical response of the bituminous concrete to be adequately represented, and it is possible to deal directly with the experimental viscoelastic functions rather than using a model consisting of a finite number of elements to only approximate the complicated response of the material. The highest degree of accuracy in representing the viscoelastic

181 162 functions was obtained in this case by use of the infinite Voigt Model. The distribution of retardation time was calculated from the static experimental response, and by use of the interrelations among the viscoelastic functions, a method to evaluate the dynamic response of the material is available. Although the experimental viscoelastic functions can be approximated by models composed of a finite number of elements to a limited degree of accuracy, the model obtained will not be unique. The accuracy desired to represent the response of the material can generally be improved by increasing the finite number of elements of which the theoretical model is composed. However, the limitations of a model to a finite number of elements appear to be artificial. Prediction of Dynamic Complex Modulus The amplitude of the axial stress divided by the amplitude of the axial strain when both of these vary sinusoidally with time may be represented as the complex elastic modulus, E*, or its reciprocal, the complex compliance, J*, as discussed in Chapter 11 where J* = J (*>) - j {*>) = A- VI-14 1 ^ E* The real and imaginary parts of J* are then given respectively by equation III-17 and equation III-18. The dynamic response of a material exhibiting linear viscoelastic behavior can thus be related to the response of a transient constant stress test by L (in t ). The calculated L (in t ) was then used to evaluate the real and imaginary components of the complex compliance in the frequency domain by application of

182 163 equations m-36 and XII-37, respectively. The above equations relating the dynamic response of the material to the retardation spectrum are of the form and L(in 1/w) = - ^ f J. (w) 1 V I-15 d In w L 1 J L (In 1/w) - - J 2 fc>) - l/wr 0 J VI-16 where t = 1/w. The dynamic response of the material at 298 K in terms of (lo) and M evaluated by the previous equations are presented in Figure 22 in graphical form covering a large range of the frequency scale. At any specified frequency and temperature, the dynamic response of a material is defined by either (jj) and fo) or alternately the absolute value of the complex compliance, /J*/» and the loss tangent, </. The two methods XL of representation which are complex functions of frequency are equivalent and are related to each other by and the loss tangent /J V = [ J x M 2 + J 2 M 2 ] 1/2-7 ^ - VI-17 tan </-, = J2-M - = E *>) Ej M VI-18 Figure 23 is a plot of the absolute value of the complex modulus, /E * /, and phase angle,(/g, versus angular frequency at 298 K obtained by the previous algebraic relations. The storage and loss components of the complex compliance plotted in Figure 22 have been evaluated at a single reference state, 298 K, for the 500

183 0.9 Normalized Real and Imaginary Componenteof Complex Compliance 08 a? NOTE: I* x I O p L _ Frequency, *> in Rodians / Second (Logarithmic) Fig. 22. Real and Imaginary Components of Complex Compliance as Functions of Angular Frequency at 298 K, 500 Mix. 164

184 5 a Absolute Value Phase Angle, $ e, in Degrees Frequency, w in Radians/ Second ( Logarithmic) Fig. 23. Absolute Value of Complex Elastic Modulus and Phase Angle as Function of Angular Frequency at 298 K, 500 Mix. C5 U1

185 166 series mix and are unique functions of frequency at this temperature. To develop similar functions at any temperature in the experimental temperature range from 0 to 120 F., the appropriate value of I nat from Figure 16 and temperature factor may be selected and used to transform the dynamic viscoelastic functions to the temperature desired. An identical result could be obtained by first developing the master transient viscoelastic function in the time domain at the temperature desired, T., and transforming the transient function to frequency domain at this temperature. Thus, the same viscoelastic function can be obtained by two methods using the same value of the temperature reduction factor. Similar procedures may be applied to evaluated /E * / and as well as (o) and E2 (k>) at any temperature from 0 to 120 F using the results of this study. Other thermally sensitive materials may also be investigated using the concepts summarized herein to gain insight into their time and temperature dependent mechanical response. Complex Transverse Modulus An identical procedure was used to evaluate the complex transverse moduli from transient constant stress tests which were obtained by dividing the axial stress by circumferential strain for the same 500 series bituminous concrete specimens from which the complex elastic moduli.were determined. The experimental transverse creep functions determined at the eleven individual temperatures are plotted in Figure 24. The equivalence of time and

186 100 u wo9o -I (ft (L e O X U 2*- «* 3 a T> i a u I >e o * ' !' ;' T im e* t in S eco n d s ( L o g o n th m ic) Fig. 24. Transverse Creep Function of 500 Series Bituminous Concrete Mix Reduced to 298 K, vs. Time at Eleven Temperatures.

187 168 temperature effects on the viscoelastic properties of bituminous concrete can again be observed from the experimental graphs. The curves determined experimentally at temperature, T., were reduced to the position they would have occupied at reference temperature, 298 K, by application of the vertical temperature shift due to the kinetic theory of rubberlike elasticity. The density factor was again omitted from the reduction scheme. The experimental creep measurements at eleven different temperatures were used to calculate the values of a In a j which are presented in Figure 25. In Figure 25 the experimental results of a n evaluated from both E c (t) and T c (t) have been plotted and a straight line used to approximate f (T) for this 500 series bituminous concrete mix. The equation of the straight line on a semi-log plot is of the form y = b l0 mx where y is the value of temperature reduction factor, x denotes the temperature in K, m is the slope of the line, and b is equal to the y intercept. The good correlation of the temperature reduction function evaluated from the transient elastic and transverse creep moduli in Figures 16 and 25, respectively, verify that if one characteristic viscoelastic function is controlled by the time-temperature relations, then all other viscoelastic functions of the same material are controlled by the same tim e-tem perature relations. The master transverse creep function was determined at 298 K by plotting (t) vs. In t/dm in Figure 26. The choice of 298 K as the standard reference temperature is again purely arbitrary and is based on convenience.

188 Mix D esignation S erie s R eferen ce T em perature, Tos296 K o Ec (t) T c(t) w O + 3 o o + 2 u_ co u S «a: «w 3 O w a E I- H- o 9 O -I T em p era tu re,t in K Fig. 25. Temperature Dependence of Shift Factor, a^, Evaluated from Elastic and Transverse Creep Data, 500 Mix.

189 100 Reduced Moater Tronawee Craap Modulus,^ Tcfl) xio'in PSKLog.) QOI 0001 * I 0 I Log of Raducad Time, *n 2303 in Second! Fig. 26. Composite Master Transverse Creep Function at 298 K, 500 Mix.

190 171 Once the master curve is determined at the temperature, TQ, it can be readily obtained at any other temperature by application of the appropriate value of in a,p and temperature factor. Thus, starting from a complicated dependence on both temperature and time or frequency, as illustrated in Figures 15 and 24, these two independent variables can be separated to yield a viscoelastic function of time or frequency alone at a standard reference temperature and a temperature function. The temperature reduction function, f (T), and the viscoelastic function reduced to Tq can be used to completely define the material at any time or temperature. Figures 17 and 26 represent, respectively, Ec (t) and Tc (t) as they would have been measured at temperature, Tq, over an enormous range of time scale. In a rheological study, the temperature and time dependent response of a material may also be dealt with by including the independent variable of temperature as well as time in the rheological equation of state used to depict the quantitative mechanical behavior of the material. In Figure 27 the reciprocal of the m aster transverse creep function, Fc (t), is plotted. Figure,28 is a plot of the circumferential strain, obtained from F^ (t) at 298 K. The steady-state rate of strain has been evaluated from this plot and used to compute the coefficient of Newtonian flow. The secondary flow or Newtonian flow component of the transverse compliance was subtracted from the curve to obtain the delayed elastic response of the material, while the instantaneous elastic component again approached zero and was insignifi-

191 IE Log of Reduced Time. ^ ^ in Second*. Fig. 27. Composite Master Transverse Creep Compliance Function at 298 K, 500 Mix. -a to

192 1,600 1, PSI Circumferential Strain, «H x 10 in in / i n. 1, Time.t in Seconds x 10* 699 Fig. 28. Circumferential Strain vs. Time from Master Transverse Creep Data at 298 K, 500 Mix. 173

193 174 cant when compared to the larger retarded elastic response and was omitted from the calculations. The normalized delayed elastic component of the transverse compliance, 4* (t), is plotted in Figure 29 and is essentially a sigmoidal curve. The distribution function of retardation times was evaluated by the methods used in the previous section for an infinite Voigt model representing the response of the bituminous concrete in the transverse direction. A first order approximation was again used to evaluate L (in t ) and the results are plotted in Figure 30. Prediction of Complex Transverse Modulus The dynamic storage and loss moduli of the complex transverse compliance, F j (jj) and F2 (u), were determined by equations similar to HI-17 and HI-18 from the transient experimental creep data and are presented in Figure 31. The magnitude and phase of the complex transverse modulus were calculated from the dynamic compliances by similar methods used to calculate /J * / and where and the phase angle /T * / - [ t j + T2 *o)2] 1/2 = y i VI-19 tan = T2 H = f 2 H VI-20 T (o) Fx *>) Figure 32 is a plot of the absolute value of the complex transverse modulus and phase angle versus angular frequency at 298 K.

194 l.o QT NOTE: 1= x 10 PSI Normalized Retarded Elastic Response, i> (t) 0.6 QS *<t) 0 Z Tim e, t in Seconds(Logarithmic) Fig. 29. Normalized Retarded Elastic Behavior, i >(t), vs. Time, 500 Mix.

195 Distribution Function of Retardation Time,L(/rtr) NOTE: 1*84.03x10' PSI Retardation Time, r in Seconds (Logarithmic) Fig. 30. Distribution Function of Retardation Time, L(Jtnr), Obtained from Master Transverse Creep Function at 298 K, 500 Mix. 176

196 Normalized Real and Imaginary Componants of Complex Transverse Compliance Fi («) NOTE 1= x 10 pjjj Frequency,<u in R ad ian s/ Second (Logarithmic) Fig. 31. Storage and Loss Components of Complex Transverse Compliance as Functions of Frequency at 298 K, 500 Mix. 177