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DO 0T CIRCU TE DO NO CIRCULA E MATERIAL FUNCTIONS FROM LARGE-AMPLITUDE OSCILLATORY SHEARING OF POLYISOBUTYLENE IN CETANE BY A MODIFIED R-l6 WEISSENBERG RHEOGONIOMETER BY David James Henry Cross A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for a degree of MASTER OF SCIENCE Department of Chemical Engineering A3722i97 ABSTRACT MATERIAL FUNCTIONS FROM LARGE-AMPLITUDE OSCILLATORY SHEARING OF POLYISOBUTYLENE IN CETANE BY A MODIFIED R-l6 WEISSENBERG RHEOGONIOMETER BY David James Henry Cross Early investigators who used the unmodified Weissenberg Rheogoniometer (WRG) encountered some inadequacies in the machine design which permits the cone—plate gap to open. The enhancements are the replacement of the bending cantilever beam by a stationary piezotron load cell and the use of weights to prevent the dove-tail slide and torsion—head assembly from moving. The objective is to collect material functions from oscillatory shearing in the nonlinear region of polyisobutylene in cetane by using a modified R—l6 WRG. There is a considerable amount of scatter in the data presumably due to variations in room temperature; nevertheless, the following trends are apparent. For nonlinear behavior the dynamic storage moduli and loss moduli are related to the odd components of the Fourier series for shear stress. The frequency response over a range of strain amplitudes shows that the storage modulus increases with the increase in frequency. At each test frequency, the strain response curve shows that the storage modulus decreases linearly with the increase in shear strain. The loss modulus is a function of frequency only. ACKNOWLEDGEMENTS It is a pleasure to acknowledge with gratitude the assis- tance and cooperation of several people associated with this project directly or indirectly: Dr. K. Jayaraman for proposing this topic, his guidance as thesis advisor, and equally importantly, for his ready accessibility for suggestions and discussions during this re- search. His personal enthusiasm and interest have been most inspiring. . Dr. R.F. Blanks as academic advisor for most of my gradu- ate studies. Dr. B.W. Wilkinson as academic advisor for much of my undergraduate studies. Dr. C.M. Cooper for his many helpful comments and discus— sions during my stay in the department. Messrs. R. Rose and D. Childs for performing the neces- sary electronic and mechanical repairs. Mrs. D. Breuss and Ms. Jean Cobb for typing the final draft. The many graduate students, faculty and staff who made my stay at Michigan State a very pleasant experience. Michigan State University and the Macromolecule Research Group for providing a graduate assistantship and research funds, and finally, My parents for their encouragement and support. ii TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . . . . . . . . . LIST OF NOTATION . . . . . . . . . . . . . . . . . CHAPTER I - INTRODUCTION 1.1 Background on Materials . . . . . . . . . . 1.2 Rheometry . . . . . . . . . . . . . 1.3 Basic Equations & Assumptions . . . . . . . 1.4 Motivations . . . . . . . . . . . . . . . . 1.5 Objective . . . . . . . . . . . . . . . . . CHAPTER II - METHOD OF APPROACH 2.1 Material Functions for the Unmodified WRG . 2.2 Enhancements to the R— 16 WRG . . . . . . 2.3 Material Functions for the Modified WRG. . . CIiAP PT 3.1 Unmodified R-l6 Weissenberg Rheogoniometer . 3.2 Modified R- 16 Weissenberg RheOgoniometer . . 3. 3 Laboratory Techniques . . . . . . 3. 4 Experiments for the Unmodified WRG . . . . . 3.5 Experiment for the Modified WRG . . . . . . CHAPTER IV - ANALYSIS OF DATA PROCESSING 4.1 Dynamic Material Functions and Formulas . . 4 2 Visicorder Calculation Methods and Problems 4. 3 Computer Program and Fourier Analysis . . . CHAPTER V - RESULTS OF THE INVESTIGATION 5.1 Material Function from the Unmodified WRG . 5.2 Material Functions from the Modified WRG . CHAPTER VI - SUMMARY AND CONCLUSION 6.1 Highlights . . . . . . . . . . . . . . . . 6.2 Inferences . . . . . . . . . . . . . . . . . CHAPTER VII - RECOMMENDATIONS 7.1 Temperature Control . . . . . . . . . . . . 7.2 Graphics Terminal . . . . . . . . . . . . . LIST OF REFERENCES . . . . . . . . . . . . . . . . ER III - DESCRIPTION OF APPARATUS AND EXPERIMENTS 37 44 49 54 56 66 68 72 85 86 88 90 92 Appendices A. Listing of the Program B. Collection of Data iv .-.4' . . Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 1a lb 14b 15 l6 17 LIST OF FIGURES Simple Shear Strain .... ...... ................ Simple Shear Stress ..... .... ....... ... ........ Ideal Cone & Plate Gometry . ........ . ..... ..... Truncated Cone ...... ........... ....... ........ Cantiliver Beam and Clamp ........ ......... .... Cross-section Area of Beam .......... . ...... ... Sketch of C-Ring .............................. Weissenberg Rheogoniometer Internals .......... Normal Force Measuring Assembly ... ............ Harmonic Motion Mechanism .... ........ ......... Measurement Paths ..................... . ....... Sketch of Modified Normal Force Measuring System ........................................ Piezotron Calibration ............... . ......... Pair of Cosine Waves Forms ...... . ............. Visicorder Output Measurement ................. Oscilloscope Measurements ..................... Shear Stress Growth ........................... Normal Stress Growth . ......................... Frequency Response of Dynamic Viscosity and and Storage Modulus for 1490 Polyisobutylene in Cetane .............. . ............. . ........ Shear Stress and Normal Stress Growth of 1490 Polyisobutylene in Cetane from a Modified WRG.. Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 0.5 Strain Amplude from a Modified WRG ..... 29 29 31 38 40 42 46 63 65 71 73 Figure 18 Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 1.3 Strain Amplitude from a Modified WRG -~-- 74 Figure 19 Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 2.1 Strain Amplitude from a Modified WRG ---- 75 Figure 20 Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 3.3 Strain Amplitude from a Modified WRG .--- 76 Figure 21 Strain Amplitude Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at the Frequency of 0.6 and 3.77/ second from a Modified WRG ..................... 78 Figure 22 Strain Amplitude Response of the Loss Stor- age Modulus for 1490 Polyisobutylene in Cetane at the Frequency of 0.6 and 3.77/ second from a Modified WRG .................. ... 80 Figure 23 Calcomp Plot of the Large—Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/sand Strain Amplitude of 3.3 ........... . ............ 81 Figure 24 Calcomp Plot of the Large-Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 2.17 ....................... Bl Figure 25 Calcomp Plot of the Large-Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 1,31, ...................... B3 Figure 26 Calcomp Plot of the Large—Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s Strain Amplitude of 1.31 ....................... B5 Figure 27 Calcomp Plot of the Large—Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 1.31 ................ ....... B7 Figure 28 Calcomp Plot of the Small—Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5515 ..................... B9 vi Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 29 30 31 Calcomp Plot of the Small—Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.53613 ................. Calcomp Plot of the Small-Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5340 ..... ............. Calcomp Plot of the Small—Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5309 .................. Calcomp Plot of the Small—Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5097 ................... Calcomp Plot of the Small-Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5035 ................... Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Fre— quency of 0.38 c/s and Strain Amplitude of 0.5488 .................................... Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Fre- quency of 0.38 c/s and Strain Amplitude of 0.5390 .................................... Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Fre— quency of 0.38 c/s and Strain Amplitude of 0.5200 .................................... Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Fre— quency of 0.38 c/s and Strain Amplitude of 0,5090 .................................... Calcomp Plot of the Large—Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 3.4123 ................... .. B29 L 1 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 48a 48b Calcomp Plot of the Large-Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane atthe Frequency of 0.6 c/s and Strain Amplitude of 3_3088 ..................... Calcomp Plot of the Large-Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 3.3048 ..................... Calcomp Plot of the Large-Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 2,160 ...................... Calcomp Plot of the Large-Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 2.1363 --------------------- Calcomp Plot of the Large-Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.3168 ..................... Calcomp Plot of the Large-Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.2982 ..................... Calcomp Plot of the Large-Amplitude Oscil— latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.2973 ..................... Calcomp Plot of the Large—Amplitude Oscil- latory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.2678 ..................... Calcomp Plot of Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 0.5194 ...... Calcomp Plot of Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 0.5139 for Shear Stress Response ...................... Calcomp Plot of Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 0.5139 for Normal Stress Response ..................... viii B52 BS4 Figure 49 Calcomp Plot of Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 0.5128 BS6 ix Table Table Table Table Table Table Table Table Table Table Table 10 LIST OF TABLES Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 3.3. ............................ Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 2.17 ............................ Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 1.31 ..................... Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 1.31. .................... Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 1.31 ..................... Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 0.5515 ................... Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 0.53613 .................. Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 0.5340 ------------------- Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 0.5309 ------------------- Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 0.5097 ------------------- Fourier Components and Material Functions at the Frequency of 0.0952 c/s and the Strain Amplitude of 0.5097 ------------------- Table ‘Table Table Table Table Table Table Table Table Table Table Table 12 13 14 15 16 17 Fourier Components and Material Functions at the Frequency of 0.38 c/s and the Strain Amplitude of 0.5488 ................. Fourier Components and Material Functions at the Frequency of 0.38 c.s and the Strain Amplitude of 0.5390 ----------------- Fourier Components and Material Functions at the Frequency of 0.38 c/s and the Strain Amplitude of 0.5200 ----------------- Fourier Components and Material Functions at the Frequency of 0.38 c/s and the Strain Amplitude of 0.5090 ................. Fourier Components and Material Functions at the Frequency of 0.6 c/s and the Strain Amplitude of 3.4123 ................. Fourier Components and Material Functions at the Frequency of 0.6 c/s and the Strain Amplitude of 3.3088 Fourier Components and Material Functions at the Frequency of 0.6 c/s and Strain Amplitude of 3.3048 Fourier Components and Material at the Frequency of 0.6 c/s and Strain Amplitude of 2.160 Fourier Components and Material at the Frequency of 0.6 c/s and Strain Amplitude of 2.1363 Fourier Components and Material at the Frequency of 0.6 c/s and Strain Amplitude of 1.3168 --. -------------- Fourier Components and Material at the Frequency of 0.6 c/s and Strain Amplitude of 1.2982 Fourier Components and Material at the Frequency of 0.6 c/s and Strain Amplitude of 1.2973 xi the ooooooooooooooooo Functions the Functions the Functions the Functions the ooooooooooooooooo Functions the Table Table Table Table 24 25 26 27 Fourier Components and Material at the Frequency of 0.6 c/s and Strain Amplitude of 1.2678 ................ Fourier Components and Material at the Frequency of 0. 6 c/s and Strain Amplitude of 0. 519 4 ................ Fourier Components and Material at the Frequency of 0. 6 c/s and Strain Amplitude of 0. 5139 ................ Fourier Components and Material at the Frequency of 0.6 c/s and Strain Amplitude of 0.5128 ................ Functions the Functions the Functions the Functions the LIST OF NOTATION apparent viscosity limiting viscosity as rate of shear approaches zero complex dynamic viscosity dynamic viscosity imaginary part of complex dynamic viscosity dynamic rigidity loss modulus strain rate of strain, shear rate shear stress shear rate matrix rth, 9.EE element of shear rate matrix shear stress matrix rth, a th element of shear stress matrix angular velocity, angular frequency in oscillatory shear flow linear velocity density torque total normal force spherical coordinate - radius spherical coordinate - polar angle spherical coordinate — azimuthal angle phase lag xiii restoring constant of torsion bar moment of inertial isentropic pressure imaginary number Fourier component for displacement Fourier series Fourier series time Temperature resonant frequency pii = 3.14159 frequency in cycles/sec shear stress magnitude thermal conductivity gap angle of cone and plate first normal stress coefficient second normal stress coefficient radius of cone or plate amplitude of oscillation first normal stress difference second normal stress differences complex normal stress coefficient gram/cm .real part of C? gram/cm imaginary part of C? gram/cm normal stress displacement function gm/cm CHAPTER I INTRODUCTION 1.1 Background on Materials Two extreme laws for the behavior of materials are Newton's law of constant viscosity and Hooke's law of constant elasticity. For example, water at constant temper— ature and pressure obeys Newton's law, while rubber obeys Hooke's law quite accurately to large deformations. Yet, materials which obey one of these two laws are regarded as common everyday materials, and they pose no problem to the engineer. Modern engineering is increasingly involved in the processing of non—Newtonian fluids such as suspensions, polymer solutions and melts which behave much differently than water or rubber does. Latexes, polymer solutions and melts are examples of pseudoplastic behavior. A contrasting behavior is dilatant fluids which are particulate dispersion such as concentrated suspensions, slurries, and resins in plasticizer. Pseudoplastic behavior shows a decrease in viscosity with shear rate, while dilatant behavior shows an increase in viscosity with shear rate. The shear rate is the instantaneous rate of strain. A simple shear strain, shown in Figure la, is similar to a pack of playing cards. This deformation causes successive layers of the volume element to move in their planes relative to the reference plane in such a way that the displacement of a layer is proportional to its distance from the reference plane. The dimension perpendicular to the plane of shear, SHEAR STRAIN AND STRESS B _ ,4 ,7 A 11/ , , + I, L—-—c-———o——--———-u——v—a—LZ I II/ " ' I l - Figure la. Simple Shear Strain Figure lb. Simple Shear Stress such as length AB in Figure la remains constant. The relative displacement of the top and bottom layer divided by their separation, Yz/Y, is called simple shear strain. This term may be abbreviated to 'shear strain', 'strain', or 'shear' for the deformation in Figure la. The angle of shear,Y , is related toll /£ by tan”Y=6£ /£ . If the deformation is small,y , expressed in radians, equals the shear strain. The force providing a shear stress is shown acting on the top surface of the volume element of material in Figure lb. An equal opposite force must be applied to the bottom surface if the element is to remain at rest or steady motion. Also, equal opposite forces must be applied to the other two surfaces, as shown in Figure lb, if the element is not to experience angular acceleration. The forces that are parallel to the surfaces are known as shear stress, while forces that are normal to the surfaces are known as normal stresses. The term stress implies a force per unit area and has units of pressure. If the stress is removed for the deformation shown in Figure 1b, the strain may or may not return to zero. Flow occurs when the strain does not eventually return to zero. If flow occurs for an infinitesimal stress the material is a liquid, otherwise it is considered a solid. Many seemingly solids, such as clay, will flow above a certain "yield" stress. The flow in Figure l is a particular example of streamline flow. The fluid elements at that point follow the same path which need not be a straight line. Inelastic materials show no recovery of strain or energy. Some liquids such as many adhesives demonstrate partial recovery of strain and energy; these liquids are called elastic liquids. If the deformation and recovery of the material is instantaneous, then it is ideally elastic. Some responses can be quite slow as with many polymers which are referred to as viscoelastic. Walters1 strictly uses this term as viscoelastic solid, but most researchers use it for liquids. Viscoelastic liquids that are sheared in their linear region obey Hooke's law. Williams 2 gave the, concise definition of linearity, that "the ratio of stress to strain for any history is a function of time only." The strain, whether constant or not, imposed on this liquid at all times before time zero had been increased by a factor, the stress at earlier time would have increased by the same factor. Likewise, if the stresses had been applied to the earlier times the strains would have been proportional to the stresses. Linearity implies the principle of superposition, which can be interrupted in two ways. The first way is that when simultaneous small strains are imposed on the viscoelastic liquid, the resultant stress is proportional to the sum of the individual strains acting separately. The second way is successive strains imposed on the viscoelastic liquid, which may cause certain types of nonlinear behavior to also appear as discussed by Williams. Normally, non—linear behavior appears by imposing a large strain on a viscoelastic liquid. When a strain rate is imposed suddenly on a fluid, the initial stress may not be maintained for two reasons, besides inertial effect. The first reason is the linear versus non—linear region, because part of the mechanical energy supplied to the material may be stored as elastic energy. The stored energy appears as elastic strain recovery when the stress reduces to zero. The second reason for a change in stress is that the structure changes. Perhaps weak bonds between suspended particles are broken, or long chain molecules become aligned. No elastic strain recovery is observed when stress reduces to zero. The stress usually decreases, but may increase with time. An irreversible loss of viscosity indicates a permanent degradation of the fluid. If the viscosity returns to its original value after the material has relaxed long enough with no strains imposed, the behavior is either rheopexy or thixotropy. Rheopexy behavior is the increase in viscosity with time of shearing, while thixotropy behavior is the decrease in viscosity with time of shearing. Bauer and Collins3 have given the history of the use of thixotropy. Thixotropic and viscoelastic behaviors could be confused if changes in stress are caused by changes in temperature. Or, viscoelasticity could be confused with rheopexy if the recovery of the material was not observed when strain is removed. Other differences between elastic and thixotropic material are the initial stress of a viscoelastic material is controlled by the inertia of the fluid, whereas the initial stress of a time dependent material depends on initial viscosity. Most polymers exhibit both elastic and time dependent behavior. 1.2 Rheometry Rheometry is the science of measuring the deformation and flow of fluids, and a rheometer such as the Weissenberg Rheogoniometer, is a measuring instrument. Measurements of fluids exhibiting both elastic and time dependent behavior are best made on a rotational rheometer where shearing can be done as long as desired. Figure 2a show the ideal cone and plate geometry that is often used on a Weissenberg Rheogoniometer. A cone with a radius R has its axis perpen— dicular to a plate, the vertex of the cone being in the surface of the plate. The cone rotates or oscillates with a relative angular velocity of w. The angle 00 between the cone and the plate is usually less than 5° and may be as small as 0.3°. Large angles are not normally used because the analysis of the results for non—Newtonian fluids is complex. Some theoreticians, such as Cheng,“ have derived explicit formula for the shear rate at the cone in time- independent non—Newtonian fluids for large cone angles, but the assumption that the free fluid surface forms part of a sphere may not be justified. Small gap causes the shear rate to be uniform, the inertia of the sample to be less, the temperature rise to be minimized, and a small sample to be sufficient. For streamline flow the shear rate at any point CONE AND PLATE‘GEOMETRY Figure 2a. Ideal Cone and Plate Geometry Figure 2b. Truncated Cone is approximately given by: y = rm = w r sin 00 00 where for small cone angle sin 00 = 00. Shear rate is independent of position in the gap. This property gives the cone and plate an enormous advantage when studying time dependent behavior because all elements of a sample have the same shear history. For uniform shear rate the shear stress, W, is a constant throughout the small gap. The torque, W, on the plate is the summation of narrow rings between radius r and r +6 r which gives SW = T2nr25r The total torque is w = f2 2nrzrdr = 2nR3T/3 In most practices the tip of the cone is ground flat as shown in Figure 2b to a radius R 1 R 2nr21dr=2n (R3- R3)T/3. R1 1 , which gives W = f If Rl equals 0.1 R the torque is reduced by 0.1%. The total torque is reduced by less than 0.1% because the parallel section of the cone contributes to the torque on the plate. 1.3 Basic Equations and Assumptions The cone and plate rheometer is a popular apparatus, because experiments with this geometry measures forces generated by known velocity profiles that have only material functions as unknowns. Velocity profiles and torque—stress relationship can be derived from the equation of motion, which is Newton's second law of motion, and from the equation of continuity, which is the 'conservation of mass' principle. For a geometric volume element, V, fixed in space and bounded by a surface S, the rate of change of momentum with V and across S are controlled by the body forces throughout V and the surface forces over S. The relevant momentum balance can be expressed as: rate of rate of rate of sum of momentum = momentum - momentum + forces of (1.3.1) accumulated in out system Bird, Stewart, and Lightfoot5 consider the rates of flow of the component direction of momentum into and out of the volume element. Momentum flows into and out of the volume by convection and by molecular transfer. There are nine components of the convective momentum flux pvv which is the "dyadic product" of the mass velocity vectorpvv and the velocity v. Similarly, there are nine stress components to the stress tensor IL: The single vector-tensor equation for the momentum balance equation (1.31) is 8 DV : _ V'OW -V'T _Vp+pg (1.3.2) ’51;— _ rate of increase rate of mom. rate of mom. sum of of mom.per unit gain by gain by viscous other volume convection transfer per forces per unit vol. unit volume on sys. The rate of momentum gain by convection term can be combined with the rate of accumulation term by means of the substantial time derivative, D/Dt. The substantial time 10 derivative is the derivative following the streamline flow of velocity vector v . The equivalent equation to (1.3.2) is pDv = -V - T- Vp + pg Dt (1.3.3) This form of the equation of motion is a statement in the form of mass times acceleration equals the sum of forces; Newton's second law of motion. The arbitrary volume element moving with the fluid is accelerated because of the forces acting on it. The equation of continuity is developed by writing a mass balance over the geometric volume element. By a similar method, the conservation of mass is as follows: C "O = — p(V ' V). U H- (1.3.4) The term,Dp /Dt is the substantial derivative of density. The corresponding equations of (1.3.3) and (1.3.4) for spherical polar coordinates are given in Bird, Steward, and Lightfootf In the analysis of spherical flow the use of spherical coordinates allows the description of Velocity in terms of fewer velocity components and results in a simplification of the boundary conditions. The equations in are general for all problems of spherical flow and for Newtonian or non-Newtonian fluids. For the cone and plate geometry the assumptions are as follows: (1) Flow is strictly tangential, sov¢=f(r,e) andvr=ve = O, (2) No bulk flow occurs, (3) "Inertia" effects are negligible, (4) Gap angle between cone and plate is less than 5°, (5) Cone and plate are of radius R, (6) Free surface of the liquid is part of a sphere of radius R with its center at the cone vertex, (7) Surface—tension forces are negligible. After applying these assumptions to the generalized equa— tions, the three components of the equation of motion for steady state reduce to _ _ 3’15: _ _ 1- 8(1‘2Trr) - r comp. p _ 1 __ + (Tee Tod) (1.3.5) r T r ___F____— e_comp. — ovzocotO = — 1 §_(Teesino)+w¢coto (1.3.6) r WEE)— O r ¢-camx O = — l §_(rzrr¢)-l areo -Tr¢ -2cot0 (1.3.7) r2 3r r Be r r with the following boundary conditions: at O = n/2 , v¢ = 0 (1.3.8) at O = n/2 + 61 , v¢ = rw sin 01 (1.3.9) at r = O , v¢ = 0 (1.3.10) Equation (1.3.9) is the condition for steady rotational shearing. For oscillatory shearing, (1.3.9) would be at O = fi/2 + 91 v¢ = rw sin Olsin wt (1.3.11) 9 12 The equations (1.3.5) through (1.3.11) are still general for Newtonian and non—Newtonian fluids. In order to use these equations to derive velocity profiles, however, the various stresses must be substituted with expressions for velocity gradients and fluid properties. Nally7 derives the velocity profile in steady location shear as an infinite series involving Bessel and associated Legendre functions for Newtonian fluids. For non—Newtonian fluids, the derivation of the velocity profile would require a rheological model such as the Power—law model.8 The equation of energy is developed by an energy balance over geometric volume element. The relevant balance can be expressed as: rate of net rate of net rate net rate internal internal and of heat of work and kinetic = kinetic + addition - done by (1.3.12) energy energy in by by system accumulated convection conduction on sur- roundings Although this first law balance does not include all forms of energy, it generalizes the work and kinetic energy effects. The kinetic energy is the mfl/Z on a per unit-volume basis for the fluid in motion. The internal energy is the random translational, rotational, and interaction energy of the molecules which depends on the local temperature and density of the fluid. The single vector-tensor equation for the energy balance equation (1.3.12) 159 13 8 fl 9 (U + v2/2) = - (V-pV(U + v2/2)) — (V°q) rate of energy rate of energy rate of energy gained in by in b convection conduction +p (v-g) - (V-pv) — (V-(l'v)) rate of work rate of work rate of work (1.3.13) done on fluid done of fluid done on fluid by gravity by pressure by viscous forces Each of the terms in equation (1.3.13) is on a per unit volume basis. By mathematical manipulation of equation (1.3.13) and use of the equation of continuity and motion, the rate of energy gained by convection can be combined with the rate of accumulation term by means of the substantial time derivative, D/Dt. The substantial time derivative is the derivative that follows the streamline flow. The equivalent equation to (1.3.13) is p %% = - (V°q) - 0(V'v) - (l:Vv) (l 3.14) where the double dot product(livv) = (V-(l-v))—(v-(V-l)). The (:2VV) is the viscous dissipation term which is left when pD(v2/2) is substituted by the equation of mechanical energy. The equation of mechanical energy is written as D(v2/2) Dt = — (v-Vo) - (v-(V-l))+ 0(V°g) The terms, such asKV-Vp)and p(v-g) , cancel out of equation (1.3.13) after the substitution. 14 For the calculation of temperature rise, the equation (1.3.14) for thermal energy is more useful in terms of heat capacity and fluid temperature than in terms of internal energy. Again by mathematical manipulation and the use of the equation of continuity, the total derivative of internal energy U, and the fact that the substantial derivative is linear operator, the equation of energy becomes OCV = g—: = - (V-q) — T (g—fi)v(V-V) - (l:\7V) (1.3.15) The corresponding equation of (1.3.15) in spherical polar coordinates are given in Bird, Steward, and Lightfoot.10 In a rheogoniometer, the same sample of material can be sheared indefinitely, the temperature rise due to viscous shear heating is frequently a problem at low shear rates. Bird and Turianllmade very good approximations of the temperature rise distribution for both Newtonian and non-Newtonian fluids in a cone—plate instruments. The energy equation describing the temperature profile in the fluid region between the cone and plate is obtained from the general energy equation by the appropriate simplification. For small gap angles the equation of motion has an approximate solution in which it is assumed the components of the velocity have the form V¢= rf (e),vr=o,ve= o, The energy equation for constant thermal conductivity k reduces to l E)— 28_T 1 8 sinOBT 1 8V — cotev. :0 kl:r2 310 (r 8r)+rzsinO 3—0( m” -Te¢(; 35 T \b) (1.3.16) 15 The term reoin equation (1.3.16) is the heat generated by irreversible mechanical energy degradation which is the (-1:Vv) term in equation (1.3.14). The normal stress componentstrr,ree,r¢¢, which are generally not zero, do not contribute to the (-T:VV) , because the associated components of the dyadic Vv are identically zero for the assumed velocity profile. For the torque W that is applied to rotate the cone at an angular velocity w, thee¢ - component of the Viscous portion of the stress tensor 3 and the¢ - component of the velocityxlare approximately given by 19¢ E 3w/2n R3 (1.3.17) V¢ E wr(n/2—0)/eo (1.3.18) A further approximation is sinOSJ. and cotGEO , sinceO is nearly equal to w/Z. Equation (1.3.16) has the following boundary conditions: at 0 =Tr/2 T =To (1.3.19) at 0 =fi/2 + 00 T = To (1.3.20) at P = 0 T = To (1.3.21) at P = R BT/Br = 0 (1.3.22) The boundary conditions (1.3.19), (1.3.20) and (1.3.21) state that the metallic surfaces of the cone and plate stay at temperature To where as the last boundary condition (1.3.22) 16 indicates that no heat loss occurs at the liquid-air interface. Although the geometry is such that an exact solution of the equation of motion and heat conduction is fairly difficult, Bird and Turian satisfactorily estimated the temperature rise by use of calculus of variations. In the derivation they did not need to use any specific rheological model to obtain the following formula: (T - To)max E 3Ww00/16 n k R (1.3.19) This equation estimates the maximum temperature rise from experimental conditions even for non—Newtonian fluid with normal stresses. A rheogoniometer typically has a maximum speed of 1000 r.p.m. and maximum torque bar of 1200 g (force) cm. A cone and plate may have a radius R = 1 cm and gap angleOormmeo radian. Most organic fluids have a thermoconductivity on the order of 0.001 cal/cm/seC/C. For these values and using the conversion factors 980 erg/g-cm and 4.186 x 107 erg/cal, the equation (1.3.19) gives (T—To) max = 3C. 1.4 Motivations Data on materials showing linear viscoelastic behavior with experimental error under small shear is well documented since the advent of Dr. Weissenberg'§2first design of a rheogoniometer in 1948. Many authors such as Ferry13 and Lodgelkhave written on linear viscoelastic theory. Although 17 the linear region gives useful information, manufacturers apply large shear rates during processing, so mathematical models for large deformation is a necessity which makes nonlinear models more practical than linear models. Until a decade ago there was little attempt at theoretical analysis of the nonlinear behavior in a form suitable for practical application. More recently proposed nonlinear theories, such as Acierno's,22 involve parameters which must be evaluated from experiments at large deformations. An obvious way to increase the rate of shear for oscillatory or rotational shearing in a rheogoniometer is to decrease the cone—plate gap angler , or to increase the angular velocity w. In oscillatory shearing an increase in the amplitude of the oscillation increases the angle of shear or the shear strain Yo. Large-amplitude oscillatory shearing is a way to increase the shear rate without increasing the angular velocity which can throw the test fluid out of the cone—plate gap by centrifugal force. Another way of testing the dynamic behavior of a material under nonlinear conditions is to shear it steadily and to super-impose a low-amplitude oscillatory shear so that linear methods can be used to relate the shear strain to the stress variations. Various superpositions will produce different waveforms for measuring the primary and secondary stresses which depends on the type of polymers under test. WalterslS has summarized this type of test and its theory. 18 In 1969 MacDonaldf Marsh, and Ashare made large- amplitude oscillatory tests using a Weissenberg rheogoniometer, but it was necessary to watch for waveform distortion caused by instrument defects. The waveform distortions prompted many investigators to modify their rheogoniometer. In 1970, Lee"7 , et. al. published a paper on modifications on their R—16 Weissenberg rheogoniometer. They installed a versatile oscillatory mechanism that allowed both amplitude and frequency variation. Also, they replaced the solenoids for torque and normal force measuring with piezoelectric load cells. Their modifications were not completely satisfactory, so new designs were being developed 6 . intro- at the time of their publication. In 1972 Higman1 duced a new torque and normal thrust measuring system for both the R—16 and R-18 Weissenberg rheogoniometers. The modifications by Higman consisted of a torque and normal thrust piezoelectric transducer which replaced the air bearing displacement transducer for the torque measurement and the servo—cantilever displacement transducer for the normal thrust measurement in the standard rheogoniometer. Also in 1972, Meissner19 published a new machine design of a cross—beam support to increase axial rigidity in the rheogoniometer. In 1973, MacDonald20 used the rheogoniometer with the cone—plate geometry for superimposing a low- amplitude harmonic strain during steady shearing. MacDonald discussed in his paper some problems in the test, such as, those associated with the slackness in the gears. In 1977, l9 Crawley and Greassley21 incorporated both the piezoelectric load cells and the cross-beam support for an enhanced axial and torque measuring system. The industrial need for nonlinear models has led many investigators to collect data by a variety of high shear rate experiments. The Weissenberg rheogoniometer is a versatile machine which allows the testing of nonlinear models by such experiments. Unfortunately, the recent venture into the nonlinear region has shown the need for mechanical enhance- ments to the Weissenberg rheogoniometer. The original or unmodified R-l6 Weissenberg rheogoniometer had measuring devices called linear variable differential transducer or LVDT, that requires a movement for recording forces. The opening of the cone-plate gap and the twisting of the stationary plate violates the spherical geometry and the no "slip" boundary condition of the basic equations and assumptions. The movement of rigid machine members is a design flaw that became only noticeable because the recent testing of polymers in the nonlinear region produced much larger forces than earlier tests in the linear region. Data that were collected by unmodified rheogoniometer in the nonlinear region of viscoelastic fluids are questionable. Before models such as the set of differential equations proposed by Acierno at. al.22 can be used to predict the stresses in materials subjected to large deformation, the models need to be evaluated against data collected from modified rheogoniometers. 20 1.5 Objectives Walters23 distinguishes between two objectives which are related for rheological measurements. Objective 1 - ...involves a straightforward attempt to determine the behavior of non—Newtonian liquids in a number of simple rheometrical flow situations using suitably defined material functions. The simple desire here is to seek a correlation bet— ween molecular structure and material behavior or alternatively between material properties and observed behavior in practical situations. Objective 2 — ...is more sophisticated and decidely more difficult. It involves the prediction of behavior in non—simple flow situations from the results of simple rheometrical experiments. Fortunately, many industrial process involve simple flow geometrics and the material functions that are determined can be used for similar applications. The progress that has been made on Objective 2 is for certain types of materials such as viscoelastic liquids. The most reliable data has been collected from the linear region. This data has been used to develop constitutive equations for use in the stress equation of motion and continuity to predict behavior for practical flows. Contrarily, the data from the nonlinear region is questionable because the measuring machines such as rheogoniometer will bend some stationary parts because large forces are exerted by some polymers that are undergoing large shearing. The R—l6 Weissenberg Rheogoniometer needs to be modified so that the data collected can be used to develop constitutive equations for the nonlinear region for shearing. 21 with respect to the collection of data from the nonlinear region, the state of the art is closer to Objective 1. Consequently, the objective of this thesis is to collect material functions from oscillatory shearing in the nonlinear region of polyisobutylene in centane by using a modified R—16 Weissenberg Rheogoniometer. Polyisobutylene in cetane was chosen because it is available from the National Bureau of Standards (NBS) as a Viscoelastic fluid. Also, a number of rheologist have reported the data on it. A comparison with reported data would verify our modifications as a bonafide approach for measuring nonlinear behavior. Our R—l6 Weissenberg Rheogoniometer is a gracious gift from Dow Chemical Company of Midland, Michigan. Unfortunate- ly, the rheogoniometer had received a thorough usage at Dow Chemical and many electronic parts need replacing. Since we had limited funds, we decided to do as much of the repairs and modifications ourselves. Our research funds were spent on a piezoelectric load cell and a torsion bar assembly. The piezoelectric cell was used for the normal stress measuring, because the normal force servo mechanism was completely inadequate. The materials of interest in our work are polymer solution of high concentration, and these have time constants of a few seconds. Spriggs, at. al.2Hhave shown that the unmodified normal force servo mechanism can be used only on materials with a time constant of a minute or more. The new torsion bar assembly allows us to change the torsion 22 bar to a stiffer bar without changing the sample fluid and resetting the cone—plate gap. The torsion bar assembly was bought because there was not enough funds to buy both the piezoelectric load cell and a temperature controller. The money that was spent on the torsion bar assembly was well spent, because the experiments can always be conducted by only allowing the minimum movement of the plate to measure the shear stress. The temperature controller was not working because the heating element was burnt out. If the on—off temperature .i controller was working, it would not benefit the experiment ’ because previous data on polyisobutylene in the literature were collected at 25 to 30C which is ambient. Also, the controller had a very course temperature scale which could not indicate the temperature better than plus or minus 5C. By necessity, the experiments were conducted at room temperature during the Winter. Since the electronic tube equipment dissipated heat, sufficient time was given before shearing to allow the room temperature to become stable. During the summer months the experiments were conducted after sundown, because the room temperature would go down and the heat dissipated from the electronic equipment would compen— sate to help keep the temperature leveled. This method for room temperature control is not a substitute for a room thermostat which the old laboratory does not have. Consequently, this thesis is more of a feasibility study to determine the quality of the data collected from the nonlinear region. CHAPTER II METHOD OF APPROACH 2.1 Material Functions from an Unmodified WRG The Viscosity,n , defined by the canonical equation 0 w) = we (w/ y 2.1 is a material function. Material functions are physical properties which may depend on strain, strain rates, or shear stress, etc. Also, for steady shear flow of non-Newtonian fluids the remaining stress distribution is Tr¢ =Te¢ = O 2.2 Trr ~Tee = 01(?) = yle (M) 2.3 2.4 Tee -T¢¢ = v2(Y) = y2N2 (+) The flow in the cone-plate gap is described by three material functionsn ,v The variablerqis best called the shear— 1") 2 . dependent viscosity or "apparent viscosity"; V1, v2 are called the first and second normal stress difference, and N1, N2 are by definition called the first and second normal stress coefficients. For Newtonian liquids the apparent viscosityn is a constant and the normal stress differences v1, and v2 are zero at all shear rates. The elastic liquids will behave as Newtonian liquids if shear rate is small enough, because the normal stress difference tends to zero faster than the apparent viscosity goes to a constant or "zero shear" viscosity. For most viscoelastic fluids, the 23 24 apparent viscosity is a monotonic decreasing function which decreases from a zero-shear value to a lower value at higher shear rates. The lower value may not be observed, since it can be several orders of magnitude lower than the zero-shear viscosity. In a preliminary investigation, the steady state shearing offers little information on needed enhancements for dynamic measurements. Stress growth or stress relaxation curves give "diagnostic" information on the machine's capability to respond to a sudden change of shear rate such as a step input. The time constant is the time for the shear stress to reach 67% of its steady state value, and the shorter the time constant the better. The unmodified Weissenberg Rheogoniometer or WRG appears at first to be well suited for this test. The cone or lower platen can be rotated for as long as necessary to achieve steady conditions, and the clutch and brake mechanism should stop the rotation in 10 milliseconds (ms.) The rotation of the plate or upper platen should be negligible for most materials or very small as relaxation proceeds. However, Batchelor, Berry and Horsfallzshave found three potential flaws during studies on stress buildup and decay in polyisobutylene. Firstly, the impact of the clutch striking the driving plate may introduce a small spurious torque. Secondly, a fast switch must be used to switch off the clutch current and switch on the brake current otherwise the lower plate may "free wheel" for a time. And lastly, the motion of the upper 25 platen as the torque falls rapidly may introduce a strain rate which is comparable to the initial shear rate. The best way to check this last flaw is to repeat the measurement using a. stiffer torsion bar‘ to detect any differences. During start—ups the normal force causes a slight separation of the cone and plate, and radial flow of the sample does occur. Part of the normal force generated by the fluid goes to overcome the shear stresses of the radial flow so a fraction of the force is not measured. The normal force will build up slower than the real rate of increase should be. Meissner26 increased the response of the unmodified WRS by increasing the cone—plate gap angle. The 8 degree cone-plate angle preferred by Meissner reduces the time for peak stress to be reached, but the edge of the fluid breaks apart at much lower shear rates. Also, Galvin and Whorlow27have studied the change of the cone-plate angle and normal force buildup jJi polyethylene. Chang, Y00, and Hartnett28 studied a series of normal stress measurements with several cantilevers to obtain data which show that the normal force in transient experiment approach asymptotic values as the cantilever rigidity increases. These asymptotic values were taken as representing the material response. Kearsley and Zapaszghave concluded that even when mathematical correction to all known errors are taken into account, the transient normal stress measurement are not reliable on the unmodified WRG. Another dynamic test besides transient stress growth is 26 small-amplitude oscillatory shearing. The input shear rate by the cone with the motion given by equation (1.3.11) is approximately described by the following equation - ”YO sin(nt (2.5) - YO Sin(ut (2.6) where yo is the strain rate amplitude. The corresponding stress distribution for viscoelastic materials is V Tre =1 (—n' sin(nt +-g cos wt) (2.7) Tr¢ =Te¢ =trr — Tee = Tee - r¢¢ = O (2.8) where n ' is the 'dynamic viscosity' and G' is the 'dynamic rigidity'. For a Newtonian fluid, equation (2.7) implies that the stress is proportional to the shear rate so G' = 0 and n' is the constant Viscosity. For Hookean solid the converse is suggested by equation (2.7). Generally, most of the literature shows the mathematics with complex variables. The equation (2.7) is now given by Ire = yn* exp (iu)t) (2.9) where n='= = n' - 1 El = n' — i n" (2.10) which n* is called "complex dynamic viscosity" and n" is the "loss viscosity'. Similarly, the literature uses the definition G9" 3 G' ‘1' 1G" (2.11) as the 'complex modulus' and G" ' “'1' (2.12) 27 where G" is called the 'loss modulus'. The assumption of small-amplitude oscillatory-shear experiment is that the material deforms linearly. Equation (2.5) and (2.7) indicate that the harmonic strain results in a harmonic stress of amplitude proportional to the strain amplitude with a phase lag which is independent of amplitude. This assumption is tested by varying the strain amplitude on the amplitude ratio,To/Yo , while shearing at a common frequency. For all viscoelastic liquids in the linear region the complex viscosity 0* goes to the zero shear viscosity no as the frequency of oscillation goes to zero. The amplitude ratio goes to zero and the phase lag goes to n/2 as the frequency of the oscillation goes to zero. Some experiments are conducted to identify the system's natural frequency, wo, because machine's resonance at the natural frequency voids all measurements. The machine's resonance causes the amplitude ratio to go to one and the phase lag to go to zero, hence, the machine is insensitive to the material properties of the test fluid. Fortunately, the natural frequency can be changed by using a different torsion bar. The most common practice is to collect data above and below the natural frequency and to draw a smooth curve through the discontinuity. Although there is a "natural frequency" in the normal force direction, it seems to be at a high enough frequency to unaffect the stress measurement, regardless, whether a cantilever beam or a piezoelectric load cell is used. 2.2 Enhancements to the R—16 WRG Normal stress growths in simple rotational shear and first normal stress difference in oscillatory shear are impractical experiments with the unmodified WRG, because the normal force measurement uses the cantilever beam. In the nonlinear region the normal stress differences oscillates with the frequency 2d) at a displacement level which independent of time. A peak force during oscillatory shearing of polyisobutylene might be 10 N(Newtons). Such a produced force may move the upper platen (plate) upward about 1 um (micron) and may move the lower platen (cone) and cantilever beam downward about lOu m. Figure 3a shows a force, F, acting downward on the cantilever beam. The depression due to its own weight at any point on a uniform beam which is rigidly clamped horizontally at one end is yl = WX2(X2 — 4Lx + 6L2) / (24 BI) (2.14) where E is Young's modulus, W is the weight per unit length, and I is the second moment of area of the cross section. Figure 3b shows a rectangular cross section bar, which is bent in a plane perpendicular to the edge of length b. The second moment is given by _ 3 I — bd /12 (2.15) The load, F, at the end increases the depression by y2 = FX2 (3L - x) / (6 El) (2.16) 29 UNMODIFIED WRG Figure 3a. Cantilever Beam and Clamp [:lJ d Figure 3b. Cross-Section Area of Beam 30 F at x = L, becomes yl + y2 = 1.3 (W3 + WL/8) / (EI) (2.17) The cantilever beam is replaced as an enhancement by a piezoelectric load cell, which has a movement of about 0.1 um. Naturally, the upper platen does move upward by more than 111m. A force greater than or equal to 10N is required to restrict the upper platen to 111m. This enhancement is simply accomplished by standard weights that are usually used for a balancing scales. The most benefit from added weights is placing the weights as close as possible to the upper platen. Figure 4 shows a C—ring which can be placed on the {g upper platen. Depending on the gage of metal, the C-rings are cut to a diameter for a specific weight. A pair of C—rings affords more symmetry, but the C—rings do increase the moment of inertia. The increased moment of inertia affects the acceleration term in the equation of motion of the upper platen. The equation of motion is approximately given by, _ dzcb 211113 G” dd) 211R3 , Wo coswt I Idt2 + 300 13 E + ( 300 G + Cm (2.18) wherew 0 cos wt is the harmonic torque, I is the moment of inertia, ¢ is the angle the platen rotates through, and C is the spring constant for the torsion bar. The C-rings are temporary and inexpensive device, which minimizes the upward movement of the upper platen. Usually a Honeywell Visicorder is used to trace the harmonic stresses and strain on ultraviolet sensitive paper. 3" MODIFIED WRG Figure 4. Sketch of C-ring 32 The analysis of the galvaniometer recording is tedious and not very accurate. Although the galvaniometer gives qualitative information, a computer gives the means to analysis the signals from the transducers. The IBM 1800 is an analog/digital computer which uses a digital voltmeter, DVM, to convert the electrical signals between —10 volts to +10 volts into discrete numerical values for the computer program. The Fourier method, which is programmed for tabular data gives the equations for the shear stress and normal stress in terms of a series of Sines and cosines. These 1. equations are used to compute the material functions. The IBM 1800 computer is interfaced with a Calcomp plotter, which provides a trace of the signal for qualitative purposes. 2.3 Material Function from a Modified WRG The modified WRG is best evaluated by varying only one enhancement at a time. All stress growth and relaxation experiments use the IBM 1800 computer and the Calcomp plotter which offer more consistency during comparisons. With the piezoelectric load cell installed, the tests are conducted with and without C-Rings. If the stress growth curve is shaped more similar to the step increase function with the C—ring, then the modification is an enhancement. Naturally, the stress relaxation curve is shaped as a step decrease function. The modified WRG does not offer any advantage for small amplitude oscillatory shearing, but an experiment that was done by the unmodified WRG needs to be repeated by the 33 modified WRG for comparison. The computer interface does permit the calculation of more difficult material functions. Williams and Bird3°discussed the time-dependent behavior of normal stresses exhibited by fluids is small-amplitude oscillation. They solved the equation of motion for the cone-plate geometry to relate the amplitude and phase relationship of the oscillating stresses to experimental measurements. The results are expressed in terms of a "complex normal stress coefficient," ;*, and a "normal stress displacement function," rd , which are given by definition, C* _ 18¢ — Tee ‘ ' n a v “a? 53—) (2.19) d: _ Re{dcb—de} 92 ‘2 >593 ‘2 (2.20) 81’ r9 where (2.21) cf: : C' — i Q", U and;" are the real and imaginary parts (2.22) w ='n/2-+0, (2.23) Tjj = Re {dj + 133' exp(2i wt)} for j = r,e,<;b (2.24) T1? is the complex amplitude of the stress, and Q is the amplitude of the angular velocity of the cone, radian/sec. The equation of motion for oscillatory shearing are the same as the equations for steady rotation except the equation 34 (1.3.7) for the ¢— component is ~av¢ _ 1 a (rztr¢)_ Bred) m) 2cotOTe¢ p—_—— _— ...—.— Bt r Sr 80 r r (2 25) since v¢ = v¢ (r, 0, t). For small amplitude oscillatory 1 r shearing, the TI ¢terms are assumed to be small when compared to other terms in equation (2.25). Another assumption is that the normal components of stress which are perpendicular to the flow direction are equal. When these two assumptions are made, the equation of motion becomes: 3| 2 . p v ¢ = _ Bree + ted - T86 I r 8r r (2.26) _ 39: '_ _ l Bree cote , p r cot e - r §E——-+ ’37—'(T¢¢ ~tee). , (2.27) Bub _ 1 816(1) _ 2cot0 p — “‘ F —se ‘7 — “’15 ' (2.28) The boundary conditions are: At 0 =fi / 2 V0 = 0 r (2.29) At 0 :n /2 + 00 v¢ = r 0 Sin wt-l (2.30) At r = 0 ¢ = O . (2.31) The modified WRG is best suited for large-amplitude oscillatory shearing. MacDonald, Marsh, and Asharealstudied the rheological behavior for large-amplitude motion. Since shear stress is a continuous function of time, the Fourier 35 expansion is _ 2n+1 19¢ - 2 A2n+1 Cos (wt+ ¢) n—o (2.32) where ¢ is the phase shift occurring between the input strain and the output stress. The assumption of a linear velocity profile in the fluid has been shown valid for small cone angles of less than 4 degrees and angular velocity less than —1 119 sec . The large-amplitude complex viscosity is defined by n‘" (w ,Y°) = — 10— n'(w,Yo) - in" (wmo) yo (2.33) With large strain amplitudes or high frequency, shear stress measured on the plate shows higher odd harmonics. Equations (2.32) and (2.33) reduce to small-amplitude shear stress and complex viscosity in the limit of small strain amplitudes. If the higher harmonics can be determined, then experiments may be used to fit fluid models to experimental results. Walters and Jones32 have concluded that the amplitude of the third harmonic could be very small because the spring constant and the moment of inertia can be large. The exception is near one—third of the natural frequency, which may resonant similar to the natural frequency for the first harmonic. Their experiments on Newtonian fluid and a viscoelastic liquid clearly indicate a third harmonic content at the one—third of the natural frequency. A similar 36 resonance occurs near one—fifth of the natural frequency which is caused by the fifth harmonic. If the nonlinear effects are not noticeable, the oscillatory experiment could be performed at frequency close to wo/(2M + 1), M = 1, 2, 3... The normal stresses have nonlinear effects that are caused from large—amplitude shearing. Akers and Williams33 used the total force method to determine the first normal stress difference, which was complicated by machine- compliance problems. Christiansen and Leppard3” used flushed—mounted transducers to investigate the first and second normal stress differences. Tanner35 correlated the normal stress data for polyisobutylene solutions from 28 papers. The first normal stress data are correlated as a function of concentration, molecular weight, and shear stress. CHAPTER III DESCRIPTION OF APPARATUS AND EXPERIMENTS 3.1 Unmodified R-16 Weissenberg Rheogoniometer The Weissenberg Rheogoniometer to be described in Figure 5 is the model R—16 manufactured by Sangamo Controls Ltd?6 It is an intermediate successor to a series of machines, developed from the original ideas of Weissenberg, which were intended to measure not only shear stress in steady rotation but also oscillatory stresses and normal stresses. A 1 hp, 1800 rpm, synchronous motor drives a 60—speed gearbox covering about six decades of angular velocity in approximately logarithmic index so that the period of oscillation of the platen can be varied from 0.0165. to 1.325 x 104 s. It is necessary to stop the motor to change gears. The output shaft of the gearbox is connected to a drive box containing an electromagnetic brake-clutch unit which allows the stopping and starting of the platen within 10 ms while the gearbox is still running. Also the drivebox comprises a variable sinewave generator for oscillatory tests. The horizontal output shaft of the drive box has a worm gear engaging with a worm wheel on the main vertical drive shaft of the machine. The test fluid is held between the platen attached to the top of the drive shaft and a platen attached to the bottom of the air bearing rotor in the torsion head. Normally the upper platen is a flat plate and the lower platen is the cone, but they may be interchanged for thinner fluids. 37 38 WEISSENBERG RHEOGONIOMETER Torque Torque bar transducer Air bearing Constant Temperature ____:”--_1 oven Upper platen Clutch and L—---J Lower platen brake Oscillation drive position Lower platen holder Worm I)E% Top bearing l [T— __—__—_ --__ I l _—_— —__— | l l \ I: ll Drive box housing | — ': :: I I 744 ll I’I\\\~Linear ball races 4.4 .. H Gear box FII ||4_A__.Bottom bearing output ll shaft Figure 5. Weissenberg Rheogoniometer Internal (from Sangamo Controls Ltd.) 39 The torsion head consists of a torque bar, which is available in a wide range of stiffnesses, that is clamped at the top and attached at the bottom to the rotor of an air bearing which holds the upper platen. The twist in the torque bar is measured with a linear variable inductance transducer. The armature is connected to a radius arm 100 mm long clamped to the bottom of the torque bar. With the Farol electronic equipment the range of measurable torques ranges 8 Nm to 20 Nm. the entire torsion head from about 2 x 10— assembly including the air bearing and torque transducer can be moved vertically along a dove—tail slide, its movement near the lower platen is measured by a second transducer so the platens may be separated for cleaning and returned back to the same cone-plate gap. The tip of the cone is usually grounded away by a known amount. The location of the cone relative to the plate is critical for the best operation of WRG. For many years the only normal force measuring system for the WRG was the servo arrangement shown in Figure 637 The lower platen is driven through a beryllium copper diaphram which has a high torsional rigidity but allows free vertical movement. A rod connected to the lower platen holder passes down through a hollow drive shaft and ends in a ball. This ball is kept centered in a conical bearing mounted on the cantilever. When a downward force is exerted to the lower platen, the movement is detected by a null—detecting transducer below the cantilever and the original position of 4O UNMODIFIED WRG 1M4 ——Lower platen // holder / — ——— — '—_:)————\—— .\ . \ . Exten51on ,11~’ S-Diaphragm piece ___1 l L_ __| l _________Hollow drive shaft Platen Servo drive rod ~1 ~1 connection Micrometer Normal force beam Spring loaded Clamp plunger Null detecting transducer Force measuring transducer Figure 6. Normal Force Measuring Assembly (from Sangamo Controls Ltd.) 41 the platen is returned by lifting the end of the cantilever. The end is moved by a plunger that is loaded by a strong spring below the cantilever and is controlled by a micrometer above the cantilever. The movement of the micrometer is proportional to the force on the platen. The micrometer can be adjusted by hand. In practice a servomotor which is controlled by the null-sensing transducer makes the necessary adjustment. The movement of the end of the cantilever is measured by another transducer. The servo—system returns the middle of the cantilever spring back to its original position within 0.1 mm is a couple seconds, but the cone—plate gap may not return back to 0.1 m. The WRG uses a variable amplitude, variable frequency, harmonic rotation which can be superimposed on a steady rotation. Figure 73%hows a circular cam eccentrically mounted on a shaft driven through bevel gears and a gearbox to give oscillation frequencies up to 60 Hz. The oscillation thimble rotates a lever arm, which by a transfer slide, rotates the actuator. The lever pivot can be moved from the indicated position until it is line with the transfer slide, which reduces the oscillation to zero. The platen shaft is driven via a worm gear which oscillates by the actuator or rotates by a separate motor and gearbox. The details that are not shown include spring to keep the worm shaft in contact with the actuator and to keep the worm in firm contact with the worm wheel. The spring prevents backlash. Also, the axial cam on the end of the camshaft which contacts Output to end of worm shaft I“\ .Actuator \ 42 UNMODIFIED WRG Cam shaft Oscillation thimble {\III‘\Adjustable pivot Figure 7. Transfer slide Harmonic Motion Mechanism (from Sangamo Controls Ltd.) 43 a spring loaded follower is not shown in Figure 7. The cam provides an opposing load on the oscillatory drive motor and gearbox to the load which is produced by the main cam. Hence, backlash in the gears is avoided which would be caused by cyclic bad variations. The sinewave that are produced for worm shaft amplitudes between 0.025 and 1.0 mm are free of distortions. For temperature control the platens may be surrounded by an electric oven. At lower temperature the oven may be cooled or heated by circulating silicone fluid through the double walled chamber from a thermostat. The enclosure may be filled with an inert gas. Although a thermocouple may be attached to the upper platen, it is doubtful that the actual fluid temperature is being measured at the extreme temperatures of -50°C to 400°C, which the machine was designed. The temperature controller and pen recorder were inoperable, so experiments were conducted at room temperature during the entire research. The R—16 WRG is a second hand, machine that. was graciously donated from Dow Chemical. After the WRG was installed and running, some peculiar waveforms occurred during oscillatory shearing of an NBS calibration oil. A spike appeared on the shear stress waveform at large strain amplitude and at all frequencies. While using the Honeywell Visicorder, the set screw that is located on the front side of the WRG was turned counterclockwise to adjust the worm gear and the helical bevel gear until the spike on the 44 waveform was eliminated. The gears can introduce vibrations if they are not meshing properly. During the moving of the WRG from Dow Chemical, the torsion head became misaligned from the centerline of the hollow drive shaft. Using a precision indicator that can measure to 0.0001 of an inch, the air-bear rotor of the torsion head was centered for trueness at the top and bottom of the rotor to within 0.0002 inches. 3.2 Modified R-16 Weissenberg Rheogoniometer An overall simplification of the measuring system is given in Figure 8. Each block represents a possible electronic fault in the system. The new system has fewer blocks, hence, the number of possible sources of errors and the time required for searching is reduced. Low-pass filters are often used in oscillatory work; Van Rijn39 pointed out that they should be carefully matched. The piezoelectric crystal system that was used to replace the LVDT transducer is a Solartron Corporation model piezotron and a charge amplifier. The piezotron, which is a fast response transducer is shown in Figure 9. The electric charge is proportional to the applied force on the cell. The piezotron is very rigid with an operating range from 1000 gm of compression to 500 gm of tension. The operating range for temperature environment is fronl below —50°C to 250°C. However, the piezotron is very sensitive to temperature variations. When the ambient temperature changes by 5-10°C the electrostatic output from the cell becomes unstable, but 45 BLOCK DIAGRAM Transducer Torsion bar Recorder Servo Servo Meter .ervo Motor Transducer n plif, Unit Transducer a.) For the Unmodified System Torsion bar Transducer Normal/ Thrust Charged Transducer Amplifier b.) For the Modified System Figure 8. Measurement Paths E 7% Hm j///////m .. /. G R _ W D m 6 4 F L I D O M t.t 0n _ v.1 .10 Pp e r u g .1 . / 333/ /////////////// . e k s 9 f Modified Normal Force Mea //// / 47 it has been proposed to install temperature and humidity control for the room. In order to amplify the charge from the piezotron, a Solartron Calibration Charge Amplifier is used. It converts the electrostatic output from the piezotron cell to a current and voltage output. The WRG low-pass and the IBM 1800 Analog/Digital computer with the Calcomp plotter or the Honeywell Visicorder is used to display the output. The amplifier has five different scales for operating purposes and a short-long time constant positions. The selection permits measurements for static response by using a high input resistance, and for drift—free dynamic operation by using a low input resistance. When the GND button switch is pressed, the residual charges from the measuring system is removed. During the loading of the test polymer an unwanted charge can build up. The calibration which is shown in Figure 10 is linear and was made on the four scales. As discussed in Chapter II the C—ring which is shown in Figure 4 provide the counterweight to oppose the normal force exerted by the viscoelastic fluid. The C-rings that are used in our research are made from aluminum with a diameters from 3 to 5 inches. The slots are cut wide enough to accommodate the diameter of the air rotor. The C—rings are not used with the oven, because the weight of oven changes the cone-plate gap. The IBM 1800 Analog/Digital computer has about a 60 K memory, which doesn't allow for a very sophisticated program. Weight 48 MODIFIED WRG 500 _ 250 1 100 _ 50 _ 10 J I u l 10 50 100 250 500 Weight per 1 Volt, grams/volt Figure 10. Piezotron Calibration 49 The programs which are given in Appendix A reads the voltages from the analog input and converts these voltages into physical parameters. Only the digital voltmeter DVM is used from the analog portion of the computer system. This voltage reading is accomplished by a “canned" subroutine which is accessed by CALL HFAI. The trunk number, which is the terminal connection, and the number of lines is indicated in the call statement. The pause between voltage readings is accomplished by another canned subroutine which is accessed by CALL HYDLY in terms of milliseconds. The other support equipment is the card reader, the 1443 line printer, and the typewriter which are on—line terminals that are accessed by the file numbers in the Fortran read and write statements. The Calcomp plotter is wired through teh analog output and the program accesses the plotter by CALL HYPLT. The plotter inherently draws straight lines between successive points. The sinusoidal waves obtain a "curvature" by using many points. The plotted signals give a qualitative description. A better plot of transient data is recorded by the Honeywell 1508 Visicorder. The recorder has a maximum recording speed of 80 inches per second, and it gives time lines in intervals of 10, 1, 0.1, or 0.01 seconds. The M100—120A galvanometer, which has a response time of 0.01 seconds, is used during the recording of transient experiments. 3.3 Laboratory Procedures and Techniques The platens were cleaned after every experiment. Whenever the next run within an experiment was at a reduced 50 shear rate from the previous run, the platens were cleaned and reloaded with a new test sample. Most of the bulk used sample was removed by paper towel, and any film on the surfaces and edges of the platens are cleaned by Kimwipes with acetone. The lower platens were checked for concentricity and tilt before sample loading, while the upper platen was checked only between experiments. Whenever the platens were changed, the platen holder was checked for concentricity. The lower platen extension piece was centered at the bottom first then at the top to ensure squareness. Transducer Calibration. In order to ensure the accuracy of the data it was necessary to perform calibration checks periodically for the proper function of the rheometer. The following calibration procedure was generally adopted for all transducer, transducer meter and amplifier on a monthly basis or immediately after repairs. The oscillation input, torsion head, and gap set transducer and transducer meter unit were calibrated with the help of a micrometer jig for known linear movements of the transducer armature. 1. The mechanical set zero of the transducer meter is described in the rheogoniometer operating manual by Sangamo Controls. 2. Set the transducer meter/amplifier range switch to "cal" position and adjust to full scale deflection on the meter by means of "set cal." 3. Adjust the position of the transducer body and amplifier zero-offset to obtain a null reading on the 51 transducer meter. 4. Next adjust the transducer meter unit of the amplifier so that i 100, 1 35, and i 10 micron movement of the armature results in a corresponding identical reading on the amplifier meter with the meter range switch set on "100 " range. 5. Repeat the above procedure for each of the other meter range setting 100, 25, 10, 2.5 and 0.25. By this method the calibrations were verified to be linear and to be within a maximum error bound of 5 percent on the most sensitive range 0.25, and generally better on the other ranges. gap Setting. The piegotron cell is an expensive enhancement, so the gap setting must be done carefully. A white piece of paper provides a good contrasting background. The alligator leads of a voltmeter are connected to the adjusting screw of the top and bottom platen. The torsion head is lowered as close as possible to the bottom platen, then the voltmeter is watched while slowly turning the torsion—head handle in clockwise rotation. If a "hard" contact is made, then the torsion head should be raised high enough to eliminate any backlash in the lead screw of the torsion—head slide. When the voltmeter indicates an electrical circuit is made, the platens have made contact. The thumb setting screw for the gap setting transducer is turned to depress the transducer armature by the required cone—plate gap beyond the null or zero micron point on the 52 gap setting meter. The torsion head is then raised high enough to load the sample. Sample Loading. Although only 1— to 2— c.c. of the test sample is required, care must be exercised during loading. When the sample, such as polyisobutylene in cetane, is poured on the bottom platen, air bubbles tend to become trapped in the liquid. A Chemist's spatula can be used to burst the bubbles from the sample. After all the bubbles have been removed, the top platen or plate is lowered to just kiss the convexed puddle of sample. If the sample puddle is concaved, then an air bubble will be trapped and a new sample must be loaded. To detect the air bubble a pen flashlight can shine light through a translucient sample against a contrasting background of a white piece of paper. As the top platen is being lowered to the null point of the gap meter, the polymer sample is squeezed out to the edge of the platens, and it will start to drip from the cone—plate gap. The spatula can be used to trim the excess sample during the gap setting. The polymer sample needs to relax for about 30-minutes to relieve any stresses that have been induced. 3.4 Experiments for an Unmodified WRG The stress growth experiments used a 7.5 cm diameter plate with a 1.533 degree cone~p1ate angle. The tests with the unmodified WRG used a cantilever of 403 gm/micron, torsion bar of 0.943 dynes—cm/micron and a gearbox setting of 1.6 or 11.3 rpm. During the shear stress growth the Visicorder is 53 set at a chart speed of 8 inches/second with a timer at 0.1 second. Since the normal stress growth is much slower, the Visicorder is operated at 1/2 inch/second with the timer on 1.0 second. The small-amplitude oscillatory shearing comprises the frequency response and the strain amplitude response experi— ments. During the frequency response experiments the strain amplitude is held constant and the frequency is varied. The strain amplitude dial is set at a low enough indication such that operating at the highest frequency permits shearing in the linear viscoelastic region. Shearing of the polymer is begun at the lowest frequency and is increased in steps to the highest frequency. The machine must be stopped to change the gearbox index for the new frequency. At a strain amplitude of 1.0 the literature indicates a frequency of about 0.6 cycle/ second as an upper limit for linear viscoelastic region. Since the upper limit is an a posteriori fact, the frequency is usually increased beyond the upper limit to also collect data in the nonlinear region. Similarly for strain—amplitude response experiments, the strain amplitude dial is changed from zero to the upper limit for the linear viscoelastic region while the frequency is held constant. The strain amplitude dial can be moved slowly while the machine is running. Likewise the strain amplitude is in- creased above the upper limit for the linear region. The cone can be changed from 1.533 degree gap angle to 0.5533 degrees to increase the strain amplitude if necessary. 54 3.5 Experiments for a Modified WRG The stress growth experiments used a 7.5 cm diameter plate with a 0.5533 degree cone—plate angle. These experi- ments test the benefit of using C—rings. The modified WRG uses the piezotron load cell, torsion bar of 0.943 dyne-cm/ micron and a gearbox setting of 3.0 or 0.45 rpm. To convert the computer time into the real time corresponding to 0.45 rpm, a delay factor of 37 is used in the TESTY program. The TESTY program simply reads and prints the voltages for the torsion, oscillatory, and piezotron tranducers. For these tests the shear rate is 9.8/second. For oscillatory shearing experiments the following input data is required for the WEISS program, 1.) Number of cycles for averaging (eg 1—5) Number of the highest harmonic (eg 1—6) JMAX for the Rhomberg integration (eg 6—7) Cone angle in degrees (eg 0.5522) Platen diameter in centimeters (eg 7.5) Delay/point (eg 9, 37, 109) Torsion bar constant in dynes—cm/thousandth (eg 2.395E04) Strain amplitude dial indication (eg 16.4) Frequency in cycles/second (eg 0.0952 Range on torsion meter (eg 2.5) Range on oscillatory meter (eg 100) Transfer function for piezotron in volts/gm (eg 0.02). 55 The program asks for the above information in this order, and the format for the entries is printed by the program before the input is required. The number of cycles for averaging depends on how well the signal reading is synchronized to the number of points read per cycle. A typical range for cycle averaging is from 1 to 5. The number of points for the cycle is determined by JMAX which is used in the Rhomberg integration. JMAX is equal to 1 plus 2 raised to the JMAX + 1 power, so JMAX of 7 equals 257 points for the cycle. For a given frequency, the delay/ point is chosen from experience to read the 257 points in the period. For example, at a frequency of 0.0952 c/s a delay/ point of 109 will cause the digital voltmeter on the IBM 1800 analog/digital computer to read 257 voltages in about 10.5 seconds. CHAPTER IV ANALYSIS OF DATA PROCESSING 4.1 Dynamic Material Functions As mentioned in section 2.1 for small-amplitude oscillatory shearing, the input shear rate is approximately described by equation (2.6), y =-?o sin wt (4.1) where Yozwyo ' Figure 11 shows a pair of sinusoidal waves for stress and strain. Point 0 is the point of origin for the strain waveform, which is given by Y='Yo cos wt It is assumed the strain produces a shear stress T: To cos (wt + 6) where is the phase shift, so I :To cos 6 cos wt — To sin 6 sin wt — Yo (G' cos wt — G" sin mt) G”. =G'Y+w—Y The dynamic storage modulus G' = (To cos 6)/Yo gives the phase stress amplitude and loss modulus, G" = (To sin6)/ yo gives the quadrature stress amplitude. 56 (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) 2'lT o E .< ———_ ———— __..———— —___ Figure 11. Pair of Cosine Waveforms 58 The dynamic viscosity and loss viscosity arises if the shear rate is considered, as is usually done for essentially fluid systems. Equation (4.4) becomes in terms of the strain rate amplitude Yo: T=Yo (—n'sinwt + n" cos wt) (4.9) where 0': (To sin 6)/Y o and n"=(TO cos 6)/ Yo (4.10) The factorYon'gives a measure of the component of stress in phrase with the strain rate. In the above discussion for Figure 11, point 0 is an arbitrary origin for the measurement of time. Point 0' could be taken as the origin with respect to shear stress waveform, so that the strain y = Yo cos (wt -5) (4°11) produces a shear stress (4.12) T: To cos wt The phase lag 6 and the strain amplitude ratioro/yoare a function of the material, and can be regarded as material properties for linear viscoelasticity, but these quantities generally will vary with frequency. Only two frequency— dependent quantities are required to determine the stress for a harmonic strain, so a variety of different pairs of such quantities are commonly used depending on the particular situation. 59 Equation (4.11) may be rewritten as Y: yo cos 6 cos wt + YOsin6sinwt, (4-13) where yocos6 is the amplitude of the strain component which is in phase with the stress, andyosin6 is the amplitude of the strain component which is out of phase with the stress. For stored and dissipated energy the following quantities are defined, J' = (yo cos6)/ TO and J" = (yosin6)/To (4.14) so that y: TO (J' cos wt + J” sin wt) (4.15) § and tan a: J”/J' (4.16) The J' is sometime called the dynamic storage compliance and J" is called the loss compliance. For ideally elastic solid, the stress and strain are in phase, 5: 0, J" = 0 and J' is related to the elastic energy stored in the material with no energy lost. Similarly for a Newtonian fluid, the stress and strain rate are out of phase, 506 = 90 degrees, J' = O and J" is the rate of energy dissipated. Generally for viscoelastic fluid both J' and J" are nonzero for measuring the degree of stored and dissipated energy. By differentiating and using equation (4.12) equation (4.15) becomes 'Y : J'T - —_ T (4.17) and also differentiating equation (4.15) Y = J'T + mJ"T. 60 The Euler's equation, which is coswt + i sin wt: exp(i wt) where i = V-11 , is used for relationship based on the previous expressions for stress, strain, and shear rate because sin wt and cos wt are tedious to manipulate. Equations (4.11) and (4.12) are written as I = Re (To exp (1 wt)) (4.18) and y = Re (yo exp (i(wt — 6)) = Re (yo exp (-16 ) exp (iwt) (4.19) i The complex strain amplitude is yoexp(—i6) , Which is sufficient to establish both the physical amplitude and the phase lag of the strain. If the complex compliance J* of the material is known, where J* = yo exp (-i6 )/ TO (4.20) then the stress can be calculated for any strain amplitude. Also, 3* = 32 (cos 6- i sin6) (4.21) IO = J' — i J" (4.22) so that J* may be found from the previously defined compliances. J' is the real part of the complex compliance, but J" is not the imaginary part of the complex compliance. Although -iJ" is called the imaginary part, J" is real and positive. In equation (4.20) the complex stress amplitude is real because the stress was used as the phase reference. The complex modulus, G* is sometime defined as G* = 1/J* (4.23) 61 So 6* = T——_° eXP (i ‘5) = G' + iG" (4.24) Yo ’ GI : J'/(J'2 + JHZ ) (4.25) G” = J"/(J'2 + JHZ ) (4-26) J) = G'/(G'2 + Gn2 ) (4.27) J" = GH/(G12+ GHZ ) (4.28) and tan 6: J"/J' = G”/G' (4.29) The shear rate is obtained by differentiating the complex strain in equation (4.19) to give 9 = Re (iw Yo exp (i6) exp(iwt)) = —w yo sin (wt - 6) (4.30) The complex dynamic viscosity is sometimes defined by : n 1': : TAO , iwyo exp (~16 ) (4.31) and using equation (4.10), it follows ”*z n' ‘ i n" (4 32) The advantage of using complex variables is that time appears in the equation in the form exp (iwt), so such terms cancel out when stress-strain ratios are taken. At any point in the calculation in which the complex amplitude of stress and strain occurs, the physical amplitude and phase shift may be determined by multiplying the complex amplitude by exp (iwt) and taking the real part of the expression. 62 By analogy to the mechanical equivalent of the "Q-factor", this may be defined as Q = 2 n Maximum stored elastic energy Energy dissipated per cycle For an initially undeformed ideally elastic solid that is sheared by a strainyo, the work done is T dy = G'yzo /2. (4.33) Apparently, the stored energy is recovered if the strain is slowly reduced to zero. For viscoelastic fluid, the G'Y%/2 ~ is a measure of stored energy, but the work done in the deformation and the energy recovered does depend on how the strain varies with time. For harmonic motion the total work done on a unit volume per cycle is, 2n/w f0 1? dt = f :n/w_w Y20(G‘cos wt —G" sin wt)sin wt dt (4.34) = TT 'Yzo G"- From previous definitions, it follows Q = G'/G” = 1/tan6 (4.35) 4.2 Visicorder Calculation Methods and Problems Figure 12 shows a typical Visicorder record for strain and stress waveforms which should have their centerlines parallel but not necessarily coincident. The distances corresponding to a complete cycle and to the phase shift can be measured from the curves respective centerlines. 0 Alternatively, Walters” shows that the measurements can be UNMOD IFI ED WRG On / Figure 12. Visicorder Output Measurement 64 made along any line parallel to the centerlines such as ABCDE. The phase difference would be calculated as 6 =n (AB + CD)/AE (4.36) Obviously, the ABCDE line must intersect both curves, and it should be well away from the peaks for reasonable accuracy. The accuracy is somewhat affected when the peaks for the wave form are small and the frequency is low. The reason is the curves become more horizontal so the intersection becomes ambiguous. The lengths for AB, CD and AE can be measured a little longer or shorter than the actual length such that a 110% error will be introduced into the phase lag calculation. This method tends to be tedious and time consuming; it is more practical to have the signal interfaced with a computer. Smith et.al.u1 used an accurate method on an oscillo— scope to measure the plase shift. Figure 13a shows elliptical traces that are analyzed by measuring the distances correspond- ing to 2x0, 2FO and 2 x0 sin 6. The force and displacement vol- tages are analyzed on separate axes. If the phase shift is small, then Figure 13b shows the major and minor axis of the ellipse are measured rather than 2>gasin 6. Two expressions for the area of the ellipse are equated as §Fdx = fFOcos(wt+6')wasinmt dt = Foxosin6 = nab thhat Sin6 = abTr/FoxO Some oscilloscopes include a calibrated phase shift net- work in the circuits which is adjusted instead of an ellipse. UNMODIFIED WRG ZFP ///// :1 r Displacement a.) Large phase shift Force 2b / // =:Displacement I / / b.) Small phase shift Figure 13. Oscilloscope Measurements 66 If a distorted line appears then higher harmonics exist in the stress wafeform. 4.3 Computer Programs and Fourier Analysis The theory of Fourier analysis and formulaes for compu— ting the Fourier coefficients are given in a text by Church- hill.“2 For shear stress and strain the phase shift with re- spect to the origin is computed as phase shift = arctan(-A1/B1) and the amplitude is computed as amplitude = (A? + 89* where A1 and B1 are the first Fourier coefficients of the sine and cosine series. Since the normal stress has a displacement and has twice the frequency of the shear stress waveforms the phase shift with respect to the origin is calculated as phase shift = arctan(-A2/B2) and the amplitude is calculated as amplitude = (A: + B§)% where A2 and B2 are the second Fourier coefficients of the sine and cosine series for the normal stress waveform. The dis- placement or total normal force is equal to A0 which is the integral average of the tabulated normal stress signal. The phase lag for shear stress or normal stress is simply the phase shift for the input strain oscillation minus the phase shift of the output oscillation for the shear stress or normal stress. The necessary integration of the tabulated function is computed by the Rhomberg extrapolation method which is program— 67 med in a text by Carnahan and Wilkes.“3 IBM also provides a utility program that calculates the Fourier coefficients based on a recursion formula. Unfortunately, for the same accuracy on recovery of the Fourier coefficients the IBM program required 10 times the number of points as required by the Rhomberg inte— gration. An accuracy of 98% for the recovery was achieved by numerical simulation of an analytical sine and cosine series which was not generated with the use of the digital voltmeter and the WRG. The fewer number of points is particularly impore tant for oscillations with short periods, because the digital volt meter may not be able to read the analog voltage signal at a fast enough speed. Consequently, experiments must be limited to shearing frequency of less than 1 c/s. CHAPTER V RESULTS OF THE INVESTIGATION 5.1 Material Functions from an Unmodified WRG Figure 14a shows a typical shear stress growth from the unmodified WRG, which employed the cantilever spring but not the C-rings. The shear stress growths always showed over- shooting with vibrationssuperimposed. The vibrations are sus- pected to be caused by the separating movement of the the torsion—head assembly and dove—tail slide. The lack of a brace or added weight permits the tosion-head assembly and the dove-tail slide to bend in an arc, hence, increasing the cone— plate gap. The response time for the shear stress growth is 0.01 seconds. Figure 14b shows a typical normal stress growth. The normal stress growth has a response time of 0.6 seconds. The response time for normal stress relaxation is 5.2 seconds. Figure 15 is a frequency response plot of the dynamic viscosity and the storage modulus that summarizes one of the oscillatory shearing experiments by the unmodified WRG. The test conditions can be found in section 3.4. The original pur- pose was to find the natural or resonance frequency. Although the phase lag did not become zero, the resonance frequency is estimated to be 42 cycles/second. A plot of phase lag versus frequency, which is not shown, has inflections at 14 and 8 cy— cles/second. Figure 15 uses 8 out of 15 data points because the use of anymore points would be too close to the natural frequency. 68 Stress Stress 69 UNMODIFIED WRG Figure 14a. Time (Arbitrary) Shear Stress Growth Figure 14b. Time (Arbitrary) Normal Stress Growth 7O I IITIWH l lllllll llllll I DYNES/CM3 I‘ I 1 G ||||||| PO/SE I 7 f? I ||||l||| l0 lllllllll llllllll c.1350" Figure 15. Frequency Response of Dynamic Viscosity and Storage Modulus for 1490 Polysobutylene in Cetane 71 MODIFIED WRG shear stress normal Stress 0‘) U) o 3.) 4.) n time a.) Plot at 26°C shear stress . normal stress 9 / U) o u 4.) U] time b.) Plot at 26.5°C 3 inear stress 0 u 4..) In Shormal stress time c.) Plot at 27°C shear stress 0‘) U). I Q) .— 5 /#/~A~WJ”¢~MFJ‘«~m~v~ 1—¢~Fr~n~s U) normal stress time d.) Plot at 280C Figure 16. Shear Stress and Normal Stress Growths of 1490 Polyisobutylene in Cetane from a Modified WRG. 72 5.2 Material Functions from a Modified WRG Under the test conditions that are given in section 3.5, there was not a significant enhancement in the response time for the stress growth experiments. The differences in the test conditions are the C-rings with weights and the room tem- peratures. The test without the use of C-rings was conducted first from a temperature of 23° to 25° C, while the test with the use of C—rings with weights was conducted second from 26° to 28° C. Figure 16 shows the plots of the four runs from the second test. The temperatureincreases approximately a 0.500 per i plot from plots "a" to "d". Although the response time from plot "a" did not change from the first test, it did become longer as the room temperature increased during the second test. Much of the noticeable differences is in the normal stress growth since the piezotron as mentioned earlier is sen— sitive to temperature changes. For the shear stress growths, there are small differences which may be attributed to the temperature effects on polyisobutylene. Figures 17, l8, l9, and 20 are frequency response plots of the storage modulus that summarizes the oscillatory experi— ment done on a modified WRG. Figure 17 has a strain amplitude from 0.5 to 0.53. Since there is a large amount of scatter on the semilog plot, the curve through the points is not a statis- tical curve fit. The large deviation of some points is caused by room temperature effects. The experiments were conducted on different evenings, so the room temperature varied from 230 DYNES/CMZ I 7 G 5 /O (u 73 MODIFIED WRG ) l I _ A ... l l l 2 g 4 w ,SEC Figure 17. Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 0.5 Strain Amplitude from a Modified WRG. 74 MODIFIED WRG l [03 : F L. N: : C) 93 E Q Q) [HI I l ‘ L Figure 18. 2 3_, Cu .850 Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 1.3 Strain Amplitude from a Modified WRG. 75 MODIFIED WRG lllll II DYNES/CM2 | 1 I 2 N G B lllll I l l I l / 2 3 4 w ,SEC" Figure 19. Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 2.1 Strain Amplitude from a Modified WRG. 76 MODIFIED WRG /O DYNES/CM2 I I 7 N G 5 l / 2 3 4 w ,SEC" Figure 20. Frequency Response of the Dynamic Storage Modulus for 1490 Polyisobutylene in Cetane at 3.3 Strain Amplitude from a Modified WRG. 77 to 35° C. Any experiment that was conducted above 37°C is not included because of obvious electronic failures that had occurred. The shearing was conducted at only three frequen— cies because the input parameter for the delay per point was obtained by trial and error. Although more frequencies are needed for a better curve fit, the data does suggest the drawn curve. Figure 18 is a frequency response plot of the storage modulus at a strain amplitude from 1.29 to 1.32. Although there are points at only two frequencies, a line with a small curva- ture is drawn through the two sets of points because the pre- vious figure suggested a curve. Experiments at a frequency of 2.38/second had failed because the room temperature was too high. Similarly, Figures 19 and 20 are frequency response plots of the storage modulus at the strain amplitude of approx— imately 2.1 and 3.3 respectively. If the curves of Figure 17 through 20 were plotted on one figure, there would be a family of curves such that the posi- tion of each of the curves becomes lower with the increase in the strain amplitude. This is quantified better in the follow- ' ing figure. Figure 21 is a strain amplitude response plot of the stor— age modulus at the frequencies of 0.6/second and 3.77/second. Since there is information at only one strain amplitude for the frequency of 2.38/second, a line is not drawn for it. The data for the frequency of 0.6/second suggest a straight 78 MODIFIED WRG 4 ’0 IIIIIIHI I IIIllfi ‘ _ 3 A /O: A : N — - __ g L- CU=3.77SEC’ —_— E — A _ 2 "' .— >~ _ Q — / G. l D > I /0‘2 ' 1 I : w=0.6OSI-:c" : 3 /0’_l I I IIIIIIIo l I IIIIIII /0 /0 l0 STRAIN, x, Figure 21. Strain Amplitude Response of the Dynamic / Storage Modulus for 1490 Polyisobutylene in Cetane at the Frequency of 0.6 and 3.77/ second from a Modified WRG 79 line, so a straight parallel line is also drawn through the data for the frequency of 3.77/second. The storage modu— lus decreases with the increase of strain amplitude. An extrapolation of the straight lines indicates a "zero shear" storage modulus of 100 dynes/sq cm and 1000 dynes/sq cm for the frequencies of 0.6/second and 3.77/second respectively. Figure 22 is a plot of the loss modulus versus strain amplitude. The data for the frequency of 0.6/second suggest a straight horizontal line, so a similar line is drawn for the other frequency. This figure indicates that the loss modu- lus is a function of frequency only and not a function of strain amplitude. Appendix B contains the data that was used in the pre— vious figures. Each table which comprises the Fourier com— ponents, material functions, and computer input is preceded by its Calcomp figure which is a plot of the raw data. The Calcomp plots show electrical noise that the low—frequency pass filters did not remove and the electrical noise that is induced by long shielded wire from the WRG electronics to the computer. The computer is located 60 feet away from the WRG. All the Calcomp plots have a labeled strain curve which is a single—cycle waveform that goes through the origin. Shear Stress. Figure 23 is a Calcomp plot of a large— amplitude oscillatory shear experiment at 0.0952 cycle/second and a strain amplitude of 3.3. The shear stress waveform is similar to the strain waveform, but the stress waveform has a slight phase shift with respect to the origin. The Fourier 80 MODIFIED WRG / 4 0_ I IIIIIIII .I ITllltt /O : ~— : w=3.77SEC" : N __ AL 1“ I: s _ . A e - A — U\) —— _ Lu _ A __ E §Q _ a .l _ to” + a—a—L ! w=0.603£0" IO: —- —— A Z _ .3 I l I I l I [III I I l l I ll ’0/07 loo /0’ STRAIN, 4/0 Figure 22. Strain Amplitude Response of the Loss Storage Modulus for 1490 Polyisobutylene in Cetane at the Frequency of 0.6 and 3.77/second from a Modified WRG. shear stress odd harmonics shear strain 81 MODIFIED WRG m m w H m u o m -a £1 H0 ME Em OJ: C. s m > m U) 5 H 0.: '0 ‘U 0 Figure 23. only even harmonics Calcomp Plot of the Large-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 3.3. normal stress phase shift normal stress normal stress 82 MODIFIED WRG Fourier Components and Material Functions Table 1. harmonic l 2 stress An 516.04 4.3158 Bn 120.02 —7.0896 strain An -0.1188 0.0285 Bn 3.2871 -0.0369 normal An 118.75 —466.02 Bn —133.l7 -13.384 shear stress A0 = -l3.81l shear strain A0 = —0.0768 normal stress A0 = 360.05 Strain Amplitude = 3.2892 Shear Stress stress amplitude = 529.81 amplitude ratio = 161.08 phase shift = 1.3784 dynamic viscosity = 264.32 imaginary part = 51.492 dynamic rigidity = 30.798 loss modulus = 222.03 Normal Stress normal stress displacement amplitude displacement function coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/pOint 257 II II II II II O'\ 109 real part of imaginary part of 3 4 7.8768 -2.7144 —l6.338 0.4610 0.0081 -0.0014 0.0318 -0.0203 21.578 76.512 -21.425 9.4735 dynes/sq cm dynes/sq cm dynes/sq cm dynes/sq cm radians poise poise dynes/sq cm dynes/sq cm II || || || II II torsion head range oscillatory range peak voltage cone angle frequency 5 6 5.5888 3.7059 -3.1261 —l.09ll -0.0050 0.0095 0.0064 -0.0080 -3.9225 -20.l75 —2.2631 —5.1019 360.05 dynes/sq cm 466.21 dynes/sq cm 0.8072 radians gm/cm gm/cm gm/cm 25.0 100.0 0.25 0.5522° 0.0952 c/s 83 series is plotted as a superimposed waveform on the raw data signal. Usually only the odd harmonics are used to evaluate the Fourier series for the shear stress curve, because the data substantiates the theory. Table 1 contains both the odd and even harmonics of the Fourier components. The magni— tude of the first component is larger than the magnitude of the secondcomponent and the third is larger than the fourth,and so on. By starting with equation (4.3) for only the odd harmon— ics a similar derivation for equation (4.5) gives the follow- ing, I = §Y{G' cos(2n+l)wt — G" sin(2n+l)wt} o 2n+1 2n+1 where I G2n+1 = A2n+1/Y° I G2n+1 * -B2n+1/Y0 are related to the Fourier An and Bn series for the odd har— monics. For n=0 , we obtain the equation (4.5) for linear Viscoelastic theory. Normal Stress. Figure 23 also shows the normal stress waveform which is a plot of the data signal. Unlike the shear stress waveform, the normal stress is a two cycle waveform. A normal force is produced for each directional movement of the bottom platen. Unfortunately, the even harmonics do not curve fit the raw data as well as they should. There is a labeled superimposed curve for the even harmonics and a labeled super- imposed curve for both odd plus even harmonics. The superim— posed curve of odd plus even harmonics always goes through the normal stress signal by staying within the bounds of the elec— 84 trical noise. The closer the match between the superimposed curves, the better the experimental technique. The unlevel heights of the peaks are mostly due to the lower platen not being in the null position before loading the polyisobutylene sample. With the exception of 82 Table 1 indicates the relative domance of the even harmonics over the odd harmonics for the normal stress signal. Generally, the normal stress displacement increases with the increase of strain amplitude. For the linear region, the effects are almost null because the diaphram may not be flexiable enough. For the nonlinear region, the effects of increasing strain amplitude is very pronounced. Earlier normal stress experiments which are not given in Appendix B compared low-frequency filters of the Piezotron amplifier. The first filter, 545A, is a low pass filter of 150 kHz, and the second filter 545A16 is a low pass filter of 1 kHz. Both the first and second Piezotron filters were not adequate, because these filters did not diminsh the electronic noises. The filter system of the Farol electronics was used for all the experiments in Appendix B. CHAPTER VI SUMMARY AND CONCLUSION 6.1 Highlights Since large shear rates are applied during the proces- sing of polymers, industries are demanding mathematical models to describe the large deformation. Early investigators who used the WRG encountered some inadequacies in the machine de- sign which permit the cone—plate gap to open. The most impor— flyv’w r tant enhancement to the unmodified WRG is the replacement of the bending cantilever beam by a stationary piezotron load cell. The second enhancement is the use of C-rings or weights to prevent the dove-tail slide to rock on pivot. The objec- tive is to collect material functions from oscillatory shear— ing in the nonlinear region of polyisobutylene in cetane by using a modified R—l6 WRG. Besides characterizing the fluid, stress growth and stress relaxation experiments give a "diag- nostic" information of the machine's ability to reSpond to a sudden change of shear rate. The small— and large-amplitude oscillatory shearing experiments identify the transition from linear to nonlinear deformation. Similarly, experiments over a range of frequencies will identify the transition to the nonlinear region and identify the natural or resonance fre— quency on the WRG. From these experiments, strain amplitude and frequency response curves are constructed to characterize the fluid for formulating models by others. 85 86 6.2 Inferences The stress growth experiments did not show a signifi- cant improvement in the response time for the modified WRG. Instead of using C-rings which do not have enough weight for the enhancement, a one kilogram weight on t0p of the torsion head would suffice as a substitute for a brace. For shear stress growths, there are small differences in the curves which may be attributed to the temperature effects on poly— isobutylene. The main results of this thesis are the frequency and strain response curves. Unfortunately, there is some scatter in the data due mostly to variation in room temperature. At a given shear strain the dynamic storage modulus increases with the increase in frequency or shear rate. For a given frequenCy the dynamic modulus decreases with the increase in shear strain. The loss storage modulus increases with the increase in frequenCy, but it is not a function of shear strain. For nonlinear behavior the derivation in Chapters IV and V is that each odd component of the Fourier cosine series for shear stress is related to a distinct dynamic modulus and that each odd component of the Fourier cosineseries is related to a distinct loss modulus. Since the odd components only show domance in the linear region for oscillatory shearing, the dynamic moduli and loss moduli are necessary for nonlinear mathematical models. As shown in Chapter IV, the other mate— rial functions are related to the dynamic and loss modulus; it follows that the other nonlinear material functions are v~~9m5é from the dynamic moduli and loss 6 linear regiOn. CHAPTER VII RECOMMENDATIONS 7.1 Temperature Control There are some existing problems and needed repairs. First, the laboratory needs an individual room thermostat that is capable of heating or cooling the room. The tube elec- tronics of the Farol power supply increases the room tempera- ture which has an adverse effect on operations. When the room temperature reaches 35° C the electronics fail to Operate and the experiment must be terminated. Secondly, the oven that surrounds the cone and plate has a broken resistance wire or heating element, because the circuit in the oven does not pass any electricity. Thirdly, the thermocouples that contact the cone or plate are frailed and they need to be replaced. And, last, the oven pinches on the frictionless air rotor so the oven needs to be adjusted. For precise control at non—ambient temperatures there are many commercial controllers which use either an electric heater or gas thermostat. Electric heating. For high—temperature in excess of 100°C, heater windings can be used in the cone and plate. Raha, Wil— liams, and Lambuudesign an independent heated cone and plate for polymer melts. A 250 watt element in the plate is the main heater and a 22 watt element in the cone is the secondary heater. The cone and plate reachs temperatures of 300°C in 88 89 about 30 minutes from a cold start. It has an accuraCy of i l0 C. For on-off controllers, the heated parts should be massive to smooth the fluctuations. The temperature control- ler should be near the heaters and thethermocouplesmeasuring the test temperature should be close as possible to the test sample. Gas Thermostat. For temperatures below ambient, air which has been bubbled through liquid nitrogen can be passed through the oven chamber surrounding the sample. Another method which gives better control involves evaporating liquid nitrogen with a small electric heater and then warming the cold gas to the required temperature. Van der Wal, gt alfs have described an automatic control system based on this principle covering the range —1800 C to +3000 C with an accuracy of : 1°C and long term stability of 0.050C. For temperatures slightly above am- bient such as 30°C some compressed filtered air which has been split from the air supply before the air bearing can be warmed by a small heater. Regardless of the mode of heat transfer, a variable con- troller is preferred over an on-off controller. For accurate work, the resistance of a platinium resistance thermometer placed in the oven is compared with the standard resistance of a Wheatstone bridge circuit. The out of balance current is prOportional to the error signal between the temperature of the oven and the temperature to be maintained. The current through the oven element is controlled by a semiconductor device known as a thyristorLi6 This conducts current only when both a posi- 90 tive voltage is applied and a positive trigger voltage is ap- plied to a control grid. These trigger signals are applied at an interval after the start of each positive a.c. voltage cycle. By varying the delay before the trigger fires the thyristor,only a fraction of the current cycle is passed. Hence, the average current through the oven element is varied over a four to one range. This delay is controlled by an er- ror signal. The correction applied to the oven current is prOportional to the temperature error. The oscillation of the oven temperature due to the time lag between the change of current and the corresponding change in temperature is reduced to a minimum. Also, there is no loss of power in the control circuits as there would be with a simple rheostat. 7.2 Graphics Terminal The College of Engineering has some graphic terminals that are portable. Some terminals that have already been in— terfaced to the IBM 1800 computer are also available with ther- moprinters. It would be more convenient to use a graphic terminal instead of the IBM typewriter that is currently interfaced with the computer. For convenient use in the laborar tory, it is necessary to run another line from the computer room to the laboratory. With the graphics terminal next to the WRG corrections and adjustments can be made before the hardcopy of the waveforms or the calculation of the Fourier components. The 12 input parameters that are given in section 3.5 can be entered via the graphics terminal. The request for a plot or plots of shear stress, normal stress, or shear strain can be entered as well as starting a new run. LIST OF REFERENCES 1 2 92 LIST OF REFERENCES Walters, K. Rheometry, Chapman and Hall, 1975. Williams, J. G. Stress Analysis in Polymers, Longman, 1973. Bauer, W. H. and Collins, E. H., Rheology: Theory and Applications, Vol.4 (edited by F. R. Eirich), pp. 423—59, Academic Press, 1967. “Cheng, D.C.H. Brit. J. Appl. Phys. 1_7_, pp. 253-63, 1966. Bird, R. B., Stewart, W. E. and Lightfoot, E.N., Transport Phenomena, Chap. 3, John Wiley, New York, 1960. Ibid, p.86 Nally, M.C. Brit. J; of Appl. Phys., 16, p.1023, 1965. 8 De Wacle, A. J. Oil Colloid Chem. Assoc., 6, p. 33, 1925. 11 12 Bird, R. B., Stewart, W. E. and Lightfoot, E. N., op cit. Chap. 10. Bird, R. B., Steward, W.E. and Lightfoot, E.N., op. cit., p.318. Bird, R. B. and Turian, R. M., Chem. Engng. Sci., 11, pp. 331—334, 1962. Weissenberg, K. Int. Cong. Rheol., 1, p. 27, 1948. Ferry, J. D. Viscoelastic Properties of Polymers, 2nd ed. Wiley, 1970. Lodge, A.S., Elastic Liquid, Academic Press, 1964. Walters, op. cit., Chap. 6. MacDonald, I.F., Marsh, B. C. and Ashare, E. Chem Engng. Sci., 24, pp. 1615—25, 1969. Lee, K. H., Jones, L. G., Pandalai, K. and Brodkey, R. 8., Trans. Soc. Rheology, 14:4, pp. 555—72, 1970. Higman, R. W. Rheol. Acta, 12, pp. 533—9, 1973. II 93 19Meissner, J. J. Appl. Polym. Sci, lg, pp. 2877-99, 1972. 2°MacDonald, I. F. Trans Soc. Rheol., 11: pp. 537—55, 1973. 21Crawley, R. L., Graessley, W. W., Trans Soc. Rheol., 21:2, pp. 19-49, 1977. 22Acierno, D., La Mantia, F. P., DeCindio, B. and Nicodemo, L., Trans. Soc. Rheol., 21, pp. 261—71, 1977. 23Walters, K., op. cite, p. 3. 21'Spriggs, T. W. Huppler, J. D., and Bird, R. B. Trans. Soc. Rheol., 10:1, p. 191, 1966. 25Batchelor, J., Berry, J. P. and Horsfall, F. Rheol Acta, 8, pp. 221-5, 1969. 26Meissner, op. cite. 27Galvin, P. T. and Whorlow, R. W. J. Appl. Polym. Sci., 19, pp. 567—83, 1975. 28Chang, K. I., Yoo, S. S., and Hartnett, J. P., Trans. Soc., Rheol., 19:2, pp. 155—171, 1975. 29Kearsley, E. A. and Zapas, L. J., Trans Soc. Rheol., 29, pp. 623-37, 1976. 3°Williams, M. C., and Bird, R. B., I & E.C. Fund., ;, 1, pp. 42-49, 1964. 31MacDonald, I. F., Marsh, B. D., and Ashare, E., Chem. Enging. Sci., 24, pp. 1615-1625, 1969. 32Walters, K. and Jones, T.E.R. Proc. 5th Int. Cong. on Rheol., 4, p. 337, 1970. 3aAkers, L. C. and Williams, M. C., J. Chem. Phys., 51, p. 3834, 1969. 3L'Christiansen, E. B., and Leppard, W. R., Trans Soc. Rheol., 18, p. 65, 1974. 35Tanner, R. 1., Trans. Soc. Rheol., 17:2, pp. 365-373, 1973. 36The Weissenberg Rheogoniometer Manual, Sangamo Controls Ltd., Bognor Regis. 37Ibid. 3BIbid. 94 39 Van Rijn, C.F.H., Rheol. Acta., 6 pp. 295—6, 1967. no Walters, Op. cit., Appendix A. 141 Smith, J.R., Smith, T.L. and Tschoegl, N.W. Rheol. Acta., 9, pp. 239-52, 1970. 1&2 Churchill, R.V., Fourier Series and Boundary Value Prob— lems. 2nd ed. McGraw Hill, New York, 1969. #3 Carnahan, B., Luther, H.A., Willies, J.O. Applied Numeri- cal Methods, John Wiley, New York, 1969. an Raha, S., Williams, M.J. and Lamb, R.L. AICHE 20, pp. 474- L, 84, 1974. 5 Van der Wal, C.W., Nederveen, C.J. and Schwippert, G.A. Rheol. Acta. 8, pp. 130—3, 1969. 46 Levitt, B. P.(ed), Findlay's Practical Physical Chemistry, 9th ed., Halsted Press, John Wiley, New York, 1972. ‘ APPENDICES APPENDIX A APPENDIX A COMPUTER PROGRAM LISTINGS As mentioned in Chapter 3 this appendix comprises the listings for the WEISS and TESTY programs. For oscillatory shearing experiments the following input data is required for the WEISS program, 1.) 2.) ll.) 12.) Number of cycles for averaging (eg. 1—5) Number of the highest harmonic (eg. 1-6) JMAX for the Rhomberg integration (eg. 6—7) Cone angle in degrees (eg. 0.5522) Planten diameter in centimeters (eg. 7.5) Delay per point (eg. 9, 37, 109) Torsion bar constant in dynes — cm/thousandth (eg. 2.395 E 04) Strain amplitude dial indication (eg. 16.4) Frequency in cycles/second (eg. 0.0952) Range on torsion meter (eg. 2.5) Range on oscillatory meter (eg. 100) Transfer function for piezotran in volts/gm (eg. 0.02). The program asks for the above information in this order, and the format for the entries is printed by the program before the input is required. The usual practice is to bring the IBM 1800 computer up from a cold start. The cold start is to press the computer ON button, install magnetic tape cartridge into the A2 tape driver, wait for 10 minutes for the green GO light, feed the two start cards through the card reader. The READY light on the computer should turn on. Now, the WEISS Fortran pro— gram can be fed through the card reader. Items 1 through 6 usually remain constant during the experiment, while items 7 through 12 may change between runs of an experiment. Also, it is not essential for the WRG to be running to answer items one through six. Item 7 through 12 can only be answered after entering the guess for the peak voltage which is typically be- tween 0.4 and 0.6. The peak voltage activates the reading of the voltage signal by the digital voltmeter, so the WRG needs to be running at this time. // FOR WEISS *NONPROCESS PROGRAM *ONE WORD INTEGER *IOCS (CARD 1443 PRINTER TYPEWRITER) *LIST ALL DIMENSION A(6).B(6).TORSH(257).OSCIL(257).PIEzD(257) c READ AND PRINT MAXIMUM NUMBER OF POINTS AND HARMONIC NUMBER READ(2.2000) NAVEC.NHARM.UMAX,CONEA.PTDIA 2000 FORMAT(5x.I10.5x.I10.5x.110.5x.P5.O.5x,F5.O) NMAX=2**(JMAX+1)+1 2 WRITE(1.2100) 2100 FORMAT(1X.50H DELAY=***** USE FORMAT IS RIGHT JUSTIFIED COL. 1 ,/. 0' TORSION BAR CONSTANT...USE E11 4',/,' STRAIN AMPLITUDE DIAL ..US EE FORMAT F5.0’./.’ FREQUENCV IN C/S...USE FORMAT F5.0’././.’ TORSI ION RANGE. OSCIL RANGE. PIEZO TRANS FUNC...USE F5.0/F5.0/F5.0’) WRITE(1.2150) 2150 FORMAT(’ WHAT IS PHASE VOLTAGE... USE F5.0’) READ(6.2200) IDLAY:TORKB.SAMPD.FREQ.RANGT.RANGO.TRANF.PHASD 2200 FORMAT(IS./.E11.4,/.F5.0,/.F5.0./.F5.0,/,F5.0./.F5.0./.F5.0) IF(I0LAY) 100.500.500 500 WRITE(3.3000) NMAX.NHARM,JMAX.FREQ.NAVEX.IDLAY.TORKB.SAMPD 3000 FORMAT(63x.23H INTERVAL SPACING NO. =,I10./.63x.22H MAXIMUM HARMON SIC NO.=.I10./.63X,15H RHOMBERG dMAX=.IS./.63X,11H FREQUENCY=.E14.7 R,/.63X.21H CVCLE AVERAGING NO.=,110./.63x.14H DELAv/POINT =.110./. A63x,' TORSION BAR CONSTANT =’.E11.4./.63X,’ STRAIN AMPLITUDE DIAL s=',P5.2) WRITE(3.3100) RANGT,RANGO.TRANF.CONEA.PTDIA 3100 FORMAT(63x,' TORSION HEAD RANGE =’F6.2./.63X.’ OSCILLATORY RANGE = G'.F6.2./.63X,’ PIEzOTRON TRANSFER FUNCTION =',F6.2./,63x.' CONE AN FGLE=’ F10 5. /. 63x , PLATEN DIAMETER=’.F10.5) CALCULATE AND INITIALIZE CONSTANTS GM/ML NMAX1=NMAX-1 SCALE=1./NAVEC FACTP=981.8./TRANF/PIE/PTDIA/PTDIA/3276.7/.4O FACTO= RANGO/. 83/22. 334/CONEA/3276. 7 IF(RANGT-. 26) 3 3 3 FACTT= 3. 82*RANGT‘TORKB/. 93/(PTDIA**3)/3276. 7*. 40 GO TO 3333 333 FACTT=3.BZ‘RANGT*TORKB/.90/(PTDIA*‘3)/3276.7*.40 3333 CONTINUE C INITIALIZE ARRAYS O 4 d=1,NHARM A(d)=0.0 B(J)=0.0 4 CONTINUE DO 5 N=1.NMAX TORSH(N)=0.0 A4 OSCIL(N)=0.0 PIEZO(N)=0.0 5 CONTINUE WRITE(1.2250) - 2250 FORMAT(1X.’WHAT IS THE PEAK VOLTAGE VOLTO...USE FORMAT F5 0') READ(6.2260) PEAKV 2260 FORMAT(F5.0) WRITE(3.2270) PEAKV 2270 FORMAT(63X,’ PEAK VOLTAGE VOLTO=’,F10.5) WRITE(3.2300) 2300 FORMAT(63X.7HWAITING,/) 7 CALL HFAIR(7.1.VOLTO) IF(-PEAKV+VOLTO) 7.7.8 8 CONTINUE WRITE(3.1300) 9 CALL HFAIR(7.1,VOLTO) IF(VOLTO-PHASD) 10.9.9 10 CONTINUE DD 15 M=1.NAVEC DO 12 N=1,NMAX1 , CALL HFAI(6,3,KT0RS,KOSCI.KPIEz) TORSH(N)=TORSH(N)-KTORS OSCIL(N)=OSCIL(N)-KOSCI PIEZO(N)=PIEZO(N)+KPIEZ _ CALL DELAY(IDLAY) 12 CONTINUE 15 CONTINUE TORSH(NMAX)=TORSH(1) OSCIL(NMAx)=OSCIL(1) PIEzo(NMAx)=PIEZO(1) C SCALE AND CONVERSION FACTOR DO 20 N=1,NMAx TORSH(N)=TORSH(N)*SCALE*FACTT OSCIL(N)=OSCIL(N)*SCALE*FACTO PIEZO(N)=PXEZO(N)*SCALE*FACTP 20 CONTINUE . wRITE(3.1900) 1900 F0RMAT(57X,63HAVERAGED VALUES FOR SIGNALS TO BE USED FOR FOURIER A ANALYSIS... ) WRITE(3.1500) WRITE(3.1400) (I.TORSH(I).OSCIL(I).PIEZO(I).I=1.NMAx) WRITE(1.1700) READ(6.1800) ICONT IF(1-ICONT) 1.25.2 25 CONTINUE 1700 FORMAT(1x,27HCHECK DATA FOR CONSISTANCY. ./.34H TO CONTINUE. TYPE 11 IN COLUMN ONE /) 1800 F0RMAT(I1) C CALL ON FOURIER ANALYSIS 1000 FORMAT(1X./.48X.17.5112./.48X.6E12.5/48X.6E12.5.//) 1100 26 57 58 65 27 29 67 68 31 33 A5 FORMAT(1X.’ ERROR...NMAX NOT GREATER THAN OR EQUAL TO NHARM...ERRO OR OR ’ ERROR...NHARM LESS THAN ZERO...ERROR CALL FORIT(TORSH2 NMAx NHARM A0. A B IER JMAX FREQ. PIE) IF(IER- 1) 26 27 wRITE(3 1000) (I2 1= 1 6) (A(I) I= L 6). B(IL I=L 6) wRITE(1 17oo) READ(6.1800) ICONT IF(1-ICONT) 1.57.58 CONTINUE CALL CALCM1TDRSH.A0,A.B.PIE,FREO.XSCAL.YSCAL,NMAX.NHARM) CONTINUE PHASD=ATAN(-A(1)/B(1)) THOMX=SQRT(A(1)*A(1)+B(1)*B(1)) WRITE(1.1700) READ(6.1300) ICONT IF(1-ICONT) 1.65.2 CONTINUE GO TO 29 ‘ WRITE(3.1100) CALL FORIT(OSCIL.NMAX.NHARM.A0.A.B.IER,UMAX.FREQ.PIE) IF(IER-1) 30,31. wRITE(3.1ooo) (I.I=1,6).(A(I).I=1.e).(B(I).I=1.6) WRITE(1.1700) READ(6.1800) ICONT IF(1-IC0NT) 1.67.68 CONTINUE CALL CALCM(OSCIL.AO.A.B.PIE.FREQ,XSCAL.YSCAL.NMAx,NHARM) CONTINUE PHASG=ATAN(-A(1)/B(1)) GAMMO=SQRT(A(1)*A(1)+B(1)*B(1)) AMPRO=THOMX/GAMMO/2./FREO/PIE PHASD=PHASG-PHASD DYVIS=THOMX~SIN1PHASD)/GAMMO/2./PIE/FREQ STORM=THAMX*COS(PHASD)/GAMMO WRITE(1.1700) READ(6.1800) ICONT IF(1-ICONT) 1.75.2 CONTINUE GO To 33 wRITE(3.1100) CALL F0R1T1P1Ezo3 NMAX NHARM A0 A B. IER OMAx FREQ PIE) IF(IER~1) 34 35. WRITE(3.1000) (131=1 6) (A(I). I=1 6). (8(1). 1: 1. 6) WRITE(1.1700) READ(6.1800) ICONT IF(1-ICONT) 1.77.78 CONTINUE CALL CALCMIPIEZO.AO,A_B,PIE.FREQ.XSCAL.YSCAL.NMAx,NHARM) CONTINUE WRITE(3.SOOO) GAMMO.AMPRO.PHASD.DYVIS.STORM,THOMx A6 5000 FORMAT(50X.’ STRAIN AMPLITUDE =’.E12.5,/.50X.’ AMPLITUDE RATIO = D’.E12.5./.50X.’ SHEAR PHASE DIFFERINCE='.E12.5.RADIANS',/.50X, TDYNAMIC VISCOSITY=',E12.5.'POISE'./,50X,' STORAGE MODULUS=’.E12.5 H.’DYNES/CM2’./.50X.’ SHEAR STRESS AMPLITUDE=’.E12.5,’DYNES/CMZ’) CALCULATE THE MATERIAL FUNCTIONS. .COMPLEX NORMAL STRESS COEFFICIENT C AND NARMAL STRESS DIDPLACEMENT FUNCTION COMMIT . USE B ARRAY FOR EFFICIENCY ACCORDING WILLIAMS BIRD C 3(1): H $ 3(2)= K s B13)— 0P s 3(4)=0M s B(5)=s $ BIG)=T s A12)=A PHASD= ATAN(- A(2)/B(2 2))/2 A(2)= SQRT(A(2)*A(2)+B(2)*B(2)) PHASD= PHASG PHASD B(2)= RHO*2 *PIE*FREQ*PTDIA*PTDIA*DYVIS/4. B(2)= B(2)/(DYVIS*DYVIS+(STORM/2 /PIE/FREO)**2 ) 13(2)= RHO PTDIA»PTDIA*STORM/4. /(DYVIS*DYVIS+(STORM/2/PIE/FREO)**2) 3(3)= SQRT(SQRT(B(1)*B(1)/2 +B(2)*B(2))+B(2)/2 )*CONEA*PIE/180. B(4)=SQRT(SORT(B(1)*B(1)/2.+B(2)*B(2))-B(2)/2.)*CONEA*PIE/180. 8(5)=(((EXP(B(4))-EXP(-B(4)))/2.)*C0$(B(3)))**2. B(5)=B(5)-(((EXP(B(4))-EXP(-B(4)))/2.)*SIN(B(§)))**2. B(6)=(EXP(2.*B(4))-EXP(-2.*B(4)))*SIN(B(3))*COS(B(3))/2. ZD=COS(CONEA*PIE/180.) ZD=-A0/((2*PIE*FREQ*GAMMO)**2)/SQRT(B(1)*B(1)+B(2)*B(2))/ZD/ZD ZD=ZD*((((EXP(B(4))-EXP(—B(4)))/2.)YCOS(B(3)))**2.+(((EXP(B(4))-EX PP(-B(4)))/2.)*SIN(B(3)))**2.) ZP=A(2)/(2*PIE*FREO)**2/(B(1)*B(1)+B(2)*B(2))/(COS(CONEA*PIE/180.) *t«2_) ZPP=ZP*((B(2)*B(5)-B(1)*B(6))*SIN(2*PHASD)-(B(1)*B(5)+B(2)*B(6))*C SOS(2.*PHASD)) ZP=ZP*((B(2)*B(5)-B(1)*B(6))*COS(2*PHASD)-(B(1)*B(5)+B(2)*B(6))*SI PN(2.PHASD)) C -------- PRINT OUT THE ANSWERS ---------- WRITE(3.6000) PHASD.A(2),AO.A.2D.2P.2PP 6000 FORMAT(50X.’ NORMAL PHASE DIFFERENCE=’E12.5.’RADIANS’./.50X.’ AMPL NTIUDE OF NORMAL STRESS’.E12.5.’DYNES/CM2’./.50X.’ AVERAGE DISPLACE MMENT OF NORMAL STRESS’.E12.5.’DYNE/CM2’./.50X.’ NORMAL~STRESS DISP FLACIMENT FUNCTION’,E12.5,'GM/CM’./.SOX.’ REAL PART OF COMPLEX NORM TAL STRESS COEFFICIENT’.E12.5.’GM/CM’,/.50X.’ IMAGINARY PART OF NOR 1MAL STRESS COEFFICIENT’,E12.5.’GM/CM’) GO TO 1 35 WRITE(3.1100) 100 CONTINUE 1300 FORMAT(63X.16HSIGNALS READ... 1400 FORMAT(63X,15,5X.E12.5.5X,E12.5,5X,E12.5) 1500 FORMAT(63X.1HI,9X.8HTDRSH(1).9X.8HOSCIL(I),9X.3HPIEZD(1) ) CALL EXIT END 1:221“ A7 // FOR FORIT *NONPROCESS PROGRAM *ONE WORD INTEGERS *LIST ALL SUBROUTINE FORIT(FNT,NMAX,NHARM.AO,A.B.IER.UMAX,FRE0,PIE) DIMENSION A(6).B(6).FNT(257).FI(7.7) CHANGE ORIGINAL NARRAY TO INTEGRAND ............. DO 1 N= 1. FNT(N)= FNT(N)*FREO 1 CONTINUE 0 CHECK FOR PARAMETER ERRORS EPS=0.0005 IE IF(NHARM) 2.3.3 IE 2 RETU RN IF(NHARM- NMAX) 5.5 4 IER= 1 50) RN COMPUTE EAO - AVERAGE VALUE OVER SYMETRICAL INTERVAL 5 BEL DW=-1. 0/2. /FR UPPER=- BELOW DO 9 d=1.dMAX DO 9 I=1.dMAX FI(I,J)=0.0 9 CONTINUE CALL ROMBG(BELOW.UPPER.FNT.EPS.dMAX.d.FI.NMAX,ANSWR) AO=ANSWR . WRITE(2 6) 0. AD WRITE(3 7) (d d= 1 dMA WRITE(3. a) (I (FI(I. d). )0: 1 7) I: 1 7) DO 10 a: 1 JMAX DO 10 I=1,UMAX FIII.U)=0.0 1o CONTINUE COMPUTE A(N) AND B(N) FOR N=1.2.3....NHARMONIC DD 11 N=1.NMAX TIME= (N- (NMAX+1)/2 )/(NMAX—1)/FREO 11 FNT(N)= FNT(N)*COS(2. *PIEPFRE0*TIME)*2 CALL ROMBG(BELDW UPPER FNT EPS JMAX U. FI. NMAX ANSWR) A(1)= ANSWR WRITE(3.6) U.A(1) WRITE(3,7) (O.U=1,7) WRITE(3.8) (I.(FI(I.d).d=1,7).I=1.7) DD 16 M=2.NHARM DO 12 N=1.NMAX TIME=(N-(NMAX+1)/2.)/(NMAX—1)/FRE0 12 FNT(N)=FNT(N)*COS(2.*M*PIE*FREO*TIME)/COS(2.*(M-I)*PIE*FREOPTIME) 14 16 (1)46)” M A8 CALL ROMBG(BELOW.UPPER.FNT.EPS.dMAX.d,FI.NMAX.ANSWR) A(M)=ANSWR WRITE(3.6) d.A(M) WRITE(3.7) (d.d=1.7) WRITE(3.8) (I.(FI(I,d).d=I.7).I=1.7) DD 15 U=1.JMAX DO 14 I=1.dMAX FI(I.J)=0.0 CONTINUE CONTINUE MHALF=(NMAX+1)/2 SAVE= FNT(MHALF) DO 17 N= 1 NMAX TIME=(NT(NMAX+1)/2. )/(NMAX—1)/FREO FNT(N)= FNT(N)»SIN(2 *PIEYFREQiTIME)/COS(2.*NHARM*PIE+FREO*TIME) CALL ROMBG(BELOW.UPPER.FNT.EPS.dMAX.d,FI,NMAX.ANSWR) B(1)=ANSWR WRITE(3 6) J B(1) WRITE(3 7) (d.d=17) WRITE(3. 8) (I (FI(I d). 0= 1 7) I= 1 7) LHALF= (NMAX+1)/2- 1 NHALF=(NMAX+1)/2+1 DO 20 M=2,NHARM DO 18 N=1.LHALF TIME=(N-(NMAX+1)/2.)/(NMAX-1)/FREO FNT(N)=FNT(N)*SIN(2.*M*PIEPFREQ*TIME)/SIN(2.*(M-I)*PIE*FREO*TIME) DO 19 N=NHALF,NMAX TIME=(N-(NMAX+T)/2.)/(NMAX-1)/FREO FNT(N)= FNT(N)nSIN(2. *M*PIE*FREQ*TIME)/SIN(2 *(M-1)*PIE*FREO*TIME) CALL RDMBG(BELOW UPPER FNT EPS OMAX 0 F1 NMAX ANSWR ) B(M)= ANSWR WRITE(3.6) d.B(M) WRITE(3.7) (d.d=1.7) WRITE(3,8) (I.(FI(I,O).O=1,7).I=1,7) CONTINUE .RESTORE ORIGINAL ARRAY FNT ...... DO 21 N=1.LHALF TIME=(N-(NMAX+1)/2.)/(NMAX-1)/FREO FNT(N)=FNT(N)/SIN(2.wNHARMtPIErFRE0*TIME)/2./FRE0 FNT(MHALF)= SAVE/2. /FREo DO 22 N= NHALF NMAX TIME=(H—(NMAX+1)/2. )/(NMAX—1)/FRE0 FNT(N)= FNT(N)/SIN(2. iNHARM*PIE*FREO*TIME)/2. /FRE0 FORMAT(30X 3H 0:. I5 5X 10H INTEGRAL= E14 FORMAT(30X 3H0 K 7112) FORMAT(30X.13.1X,7F12.5) RETURN END Z\9 // FOR ROMBG *NONPROCESS PROGRAM *ONE WORD INTEGERS *LIST ALL mum SUBROUTINE ROMBG(A,B,F,EPS.dMAX.d.FI.NMAX.ANSWR) DIMENSION FI(7.7).F(257) PRESET CONSTANTS DO 7 d=1,dMAX IBY=(NMAX+1)/2*td IBEGN=IBY+1 M=NMAX-2 DO 2 I=IBEGN.M.IBY F2=F2+F(I) IBEGN=IBY/2+1 F4=0.0 M=NMAX-1 DO 3 I=IBEGN.M.IBY F4=F4+F(I) F1(d.1)=(B-A)*(F(1)+F(NMAX)+2.*F2+4.*F4)/5-/N IF(d-1) 6,6,4 CONTINUE KM=d-1 DO 5 K=1.KM dMK=d-K KPO=K+1 UMKpO= d- K+1 FI(dMK KPO)= (4. **K*FI(JMKPO K)- FI(dMK K))/(4. **K- 1. IF((ABS(FI(JMKPO KM)- FI(UMK KM))- EPS1) 8.8.6 CONTINUE CONTINUE ANSWR=FI(UMK,KPO) RETURN END ) A10 // FOR CALCM *NONPROCESS PROGRAM *ONE WORD INTEGERS I"LIST 60 5050 5060 5070 91 92 53 54 61 500 501 64 1900 2000 200 ALL SUBROUTINE CALCM(ARRAY.AO,A,B.PIE.FREQ.XSCAL,YSCAL.NMAX.NHARM) DIMENSION A(6).B(6),ARRAY(257) DRAWS ORIENT ABSCISSA ...... CALL HYPLT(O..O..O) CALL HYPLT(-3.75.0..1) CALL HYPLT(3.75.0..1) CALL HYPLT(O.,2.5.2) CALL HYPLT(O..-2.5.1) CALL HYPLT(O..O..1) CONTINUE WRITE(1.5050) FORMAT(’ wHAT Is XSCALE AND YSCALE... USE F5.0/F5.0’./) READ(6.5060) XSCAL,YSCAL FORMAT(FS.O./.F5.0) WRITE(3.5070) XSCAL.YSCAL FORMAT(60X,’XSCALE FOR PLOTTER+’.F10.5.’ YSCALE FOR PLOTTER=’,F10. 55) N=1 TIMME=(N-(NMAX+1)/2.)/(NMAX-1)/FREQ ARAY=ARRAY(N) CALL HYPLT(O..0.,2) CALL HYPLT(XSCAL.YSCAL.3) wRITE(1.91) FORMAT(1X.’DO YOU wANT PLOT OF EXPERIMENTAL DATA.’./.’ TYPE 1 IN c NNE’./) READ(6.92) ITRY FORMAT(I1) IF(ITRY-1) 61.53.53 CONTINUE CALL HYPLT(TIME.ARAY.2) DO 54 N=2.NMAx TIME=(N-(NMAX+1)/2.)/(NMAX-1)/FREO ARAY=ARRAY(N) CALL HYPLT(TIME.ARAY,1) CONTINUE CALL HYPLT(O..O..2) CONTINUE WRITE(1.500) FORMAT(1X.’DO YOU wANT EVEN AND/OR ODD HARMONICS...’./.’ USE FORMA TT I1. ODD IS 1.... EVEN ID 2..... BOTH Is 3'./) READ(6.501) IPART FORMAT(I1) N=1 ' TIME=(N-(NMAX+1)/2.)/(NMAX-1)/EREo CALL FUNCT(TIME.AO.A.B.PIE.FREQ.G.IPART.NHARM) ARAY=G CALL HYPLT(TIME.ARAY.2) DD 64 N=2,NMAX TIME=(N«(NAmx+1)/2.)/(NMAx—1)/FREo CALL FUNCT(TIME.A0,A.B.PIE.FREQ.IPART.NHARM) ARAY=G CALL HYPLT(TIME,ARAY.1) CONTINUE CALL HYPLT(0..0..2) WRITE(1.1900) ' FORMAT(1X.’DO YOU wANT To TRY THIS ONE AGAIN... 1 YES. 0 N0’./) READ(6.2ooo) ITRv EORMAT(I1) IF(ITRY-1) 200.60.60 CONTINUE ' CALL HYPLT(0..O..-1) RETURN END All // FOR FUNCT *NONPROCESS PROGRAM *ONE WORD INTEGERs *LIST ALL SUBROUTINE FUNCT(TIME A0 A B PIE FREQ G IPART NHARM) C NEED THE FOURIER COMPONENTS DIMENSION ANSUM= 0.0 BNSUM=0.0 CALCULATION FOR FOURIER SERIES IF(IPART-z) 1.2.3 1 MSTAR=1 IBv=2 GO TO 4 2 MSTAR=2 IBY=2 GO TO 4 3 MSTAR=1 13v=1 4 CONTINUE DO 5 N=MSTAR.NHARM.IBY ANSUM=ANSUM+A(N)*COS(2.*N*PIE*FREQ*TIME) BNSUM= EBNSUM+B(N)*SIN(2.EN*PIE*FREQ*TIME) 5 CONTIN G= A0+ANSUM+BNSUM RETURN END APPENDIX B shear stress normal stress Bl MODIFIED WRG : ...] . m if H u U) _ u m n 0') Figure 24. Calcomp Plot of the Large—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 3.3. Table 2. harmonic stress An Bn strain An Bn normal An Bn shear stres shear strai normal stre Strain Ampl Shear Stres stress ampl amplitude r phase shift dynamic vis imaginary p dynamic rig loss modulu Normal Stre normal stre normal stre phase shift normal stre normal stre MODIFIED WRG B2 Fourier Components and Material Functions 1 2 362.99 1.5835 90.843 —4.4929 —0.0534 0.0148 2.1708 -0.0225 115.83 —401.33 -18.462 —9.1217 5 A0 = 2.0177 n A0 = 0.0328 55 A0 = 367.03 itude = 2.1714 s itude = 374.18 atio = 172.32 = 1.3501 cosity = 281.10 art = 63.053 idity = 37.714 5 = 232.66 ss 55 displacement ss amplitude SS SS Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 25 displacement function coefficient: 3 4 5 6 6.0514 —1.1082‘ -1.4292 1.0232 -6.5130 —1.3807 0.6366 0.4978 0.0044 0.0015 -0.0013 0.0008 0.0205 -0.0124 0.0070 -0.0053 18.440 101.08 —8.3871 —27.029 —7.2524 13.278 10.972 —18.245 dynes/sq cm dynes/sq cm dynes/sq cm dynes/sq cm radians poise poise dynes/sq cm dynes/sq cm = 367.03 dynes/sq cm = 401.43 dynes/sq cm = 0.7986 radians = —0.0143 gm/cm real part of = 0.0696 gm/cm imaginary part of = 0.1454 gm/cm 7 torsion head range = 2.5 7 oscillatory range = 25 6 peak voltage = 0.6 1 cone angle = 0.5522° 9 frequency = 0.0952 c/s 10 shear stress normal stress Figure 25. B3 MODIFIED WRG Calcomp Plot of the Large-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 1.31. MODIFIED WRG B4 Table 3. Fourier Components and Material Functions harmonic 1 2 3 4 5 6 stress An 188.44 0.8677 -0.5596 0.1367 —0.2660 0.2248 Bn 94.975 —0.8953 -1.3039 0.0229 -1.6931 -0.2310 strain An -0.1745 0.0065 —0.0016 0.0006 0.0004 —0.0003 Bn 1.3006 0.0038 0.0034 0.0001 -0.0019 -0.0009 normal An —48.366 —105.08 1.1228 25.907 -1.1684 —2.7346 Bn -20.439 —31.539 3.5849 23.028 -3.9873 —1l.003 shear stress A0 = 17.839 dynes/sq cm shear strain A0 = -0.0001 normal stress A0 = -49.884 dynes/sq cm Strain Amplitude = 1.3122 Shear Stress stress amplitude = 211.02 dynes/sq cm amplitude ratio = 160.81 dynes/sq cm phase shift = 1.2373 radians dynamic viscosity = 254.04 poise imaginary part = 88.005 poise dynamic rigidity = 52.641 dynes/sq cm loss modulus = 151.95 dynes/sq cm Normal Stress normal stress displacement = —49.884 dynes/sq cm normal stress amplitude = 109.71 dynes/sq cm phase shift = 0.7730 radians normal stress displacement function = gm/cm normal stress coefficient: real part of = gm/cm imaginary part of = gm/cm Computer Program Input interval spacing number = 257 torsion head range = 1.0 rhomberg jmax = 7 oscillatory range = 25.0 maximum harmonic = 6 peak voltage = 0.42 cycle averaging number = 10 cone angle = 0.5522° delay/point = 109 frequency = 0.0952 c/s shear stress shear strai B5 MODIFIED WRG Figure 26. Calcomp Plot of the Large-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 1.31. Table 4, harmonic stress An Bn strain An Bn normal An Bn shear stres shear strai normal stre Strain Ampl Shear Stres stress ampl amplitude r phase shift dynamic viscosity imaginary part dynamic rig loss modulu B6 MODIFIED WRG Fourier Components and Material Functions 1 2 3 4 5 6 195.93 2.6662 0.8991 0.3121 -1.0033 -0.4566 87.777 -0.7516 —0.0935 0.1268 -l.4421 ,0.2120 -0.1520 0.0068 -0.0014 0.0011 -0.0004 -0.0005 1.3031 0.0031 0.0041 -0.0004 -0.0020 -0.0005 —52.388 -120.14 —0.0225 29.245 —2.228 -3.2796 -32.496 -25.830 3.7987 23.061 —4.5396 -9.4770 5 A0 n Ao ss Ao 19.129 dynes/sq cm 0.0003 —9.8511 dynes/sq cm itude = 1.3119 S itude = 214.69 dynes/sq cm atio = 163.65 dynes/sq cm = 1.2657 radians = 260.96 poise = 70.042 poise idity = 49.158 dynes/sq cm s = 158.20 dynes/sq cm SS Normal Stre normal stre normal stre phase shift normal stre normal stre —9.8511 dynes/sq cm 122.89 dynes/sq cm 0.7957 radians ss displacement ss amplitude II II II II II II Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point ss displacement function gm/cm ss coefficient: real part of gm/cm imaginary part of gm/cm = 257 torsion head range = 1.0 = 7 oscillatory range = 25.0 = 6 peak voltage = 0.42 = 10 cone angle = 0.5522° = 109 frequency = 0.0952 c/s normal stress Figure Shear stress B7 MODIFIED WRG shear strain Calcomp Plot of the Large—Amplitude Oscillatory Shearing of Polyisobutylene. in Cetane at the Frequencey of 0.0952 c/s and Strain Amplitude of 1.31. Table 5. harmonic stress An Bn strain An Bn normal shear stres B8 MODIFIED WRG Fourier Components and Material Functions 1 2 204.26 0.98174 69.819 —1.5688 -0.0777 0.0068 1.3080 0.0035 -52.419 -129.47 -48.800 2.1693 s Ao shear strain A0 = 0.0001 normal stre Strain Ampl Shear Stres stress ampl amplitude r phase shift dynamic vis imaginary p dynamic rig loss modulu 55 A0 itude = 1.3103 s itude = 215.86 atio = 164.74 = 1.3008 cosity = 265.43 art = 73.460 idity = 43.941 5 = 158.77 SS Normal Stre normal stre normal stre phase shift normal stre normal stre ss displacement ss amplitude 3 4 5 1.8865 0.0462 -1.0 —0.9018 -0.3374 -0.3 -0.0002 0.0007 -0.0 0.0034 0.0003 -0.0 1.5563 36.224 —4.8 -10.752 25.410 —9.1 17.225 dynes/sq cm 2131.7 dynes/sq cm dynes/sq dynes/sq radians poise poise dynes/sq dynes/sq ss displacement function 55 coefficient: real part of imaginary part of Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point II II II II II 0‘1 torsion head range oscillatory range cm CITl cm 2131.7 129.48 6 181 -0.3273 436 0.6133 007 -0.0006 019 0.0001 126 -6.6001 877 -2.6846 dynes/sq cm dynes/sq cm -0.7177 radians peak voltage cone angle frequency gm/cm gm/cm gm/cm 1.0 25.0 0.42 0.5522° 0.0952 c/s shear stress normal stress shear strain B9 MODIFIED WRG Figure 28. Calcomp Plot of the Small-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the FrequenCy of 0.0952 c/s and Strain Amplitude of 0.5515. Table 6. harmonic stress An Bn strain An Bn normal An Bnm - shear stress 1 71.816 29.917 0.0037 0.5515 8.8982 63.805 AO shear strain Ao normal stres Strain Ampli Shear Stress stress ampli amplitude ra phase shift dynamic Visc imaginary pa dynamic rigi loss modulus Normal Stres normal stres normal stres phase shift normal stres normal stres 5 A0 tude tude tio osity rt dity S MODIFIED WRG B10 Fourier Components and Material Functions 5 displacement s amplitude s coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 2 3 4 5 6 0.2139 0.8271 —0.0380 -0.5480 0.0433 —0.5894 0.6821 —0.1767 -0.l904 —0.2567 0.0028 -0.0004 0.0002 -0.0008 0.0003 -0.0052 0.0058 —0.0027 0.0006 -0.0020 —136.66 4.066 18.955 0.9066 —9.4814 27.305 -1.7252 —8.054 2.7868 -3.7458 = 0.9234 dynes/sq cm = 8.1466 = 22.979 dynes/sq cm = 0.5515 = 77.798 dynes/sq cm = 141.07 dynes/sq cm = 1.1692 radians = 217.09 poise = 92.18 poise = 55.131 dynes/sq cm = 192.84 dynes/sq cm = 22.979 dynes/sq cm = 139.33 dynes/sq cm = -0.6936 radians s displacement function = -0.0139 gm/cm real part of = -0.0457 gm/cm imaginary part of = —0.0156 gm/cm = 257 torsion head range = 1.0 = 7 oscillatory range = 10.0 = 6 peak voltage = 0.4 = l cone angle = 0.5522° = 109 frequency = 0.0952 c/s Bll MODIFIED WRG /Z\\\\\\\\\\\\\\ Figure 29. Calcomp Plot of the Small—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.53613. shear stress shear strain B12 MODIFIED WRG Fourier Components and Material Functions Table 7 0 harmonic 1 2 stress An 85.692 —0.0056 Bn 50.322 -1.0898 strain An -0.0744 0.0023 Bn 0.5309 0.0017 normal An 25.510 -116.51 Bn 22.448 53.571 shear stress A0 = 5 0617 shear strain A0 = —0 0036 normal stress A0 =- 10.793 Strain Amplitude = 0.53613 Shear Stress stress amplitude = 99.375 amplitude ratio = 185.36 phase shift = 1.1791 dynamic viscosity = 286.4 imaginary part = 118.30 dynamic rigidity = 111.67 loss modulus = 171.32 Normal Stress normal stress displacement normal stress amplitude phase shift normal stress normal stress coe fficient: -0.2720 —3.6896 -4.9099 3 0.0000 0.0010 0.4003 dynes/sq cm dynes/sq cm dynes/sq dynes/sq radians poise poise dyneS/sq dynes/sq displacement function real part of imaginary part of Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point torsion head range oscillatory range 4 -0.5486 2.5522 0.0003 0.0004 42.566 15.929 cm CI'fl cm CH1 peak voltage cone angle frequency 5 6 0.6556 0.3557 -0.2151 0.1208 -0.0001 —0.0006 -0.0003 0.0000 —4.4282 0.6154 -8.4051 -13.l86 10.793 dynes/sq cm 128.23 dynes/sq cm 0.4307 radians gm/cm gm/cm gm/cm 1.0 25.0 0.5 0.6522° 0.0952 c/s II || || 11 ll shear stress hear strain Figure 30. B13 MODIFIED WRG Calcomp Plot of the Small-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5340. Table 8. harmonic stress An Bn strain An Bn normal An Bn shear stres shear strain Ao normal stre Strain Ampl Shear Stres stress ampl amplitude ratio phase shift dynamic viscosity imaginary part dynamic rig loss modulu Normal Stre normal stre normal stre phase shift normal stre normal stre B14 MODIFIED WRG Fourier Components and Material Functions 1 2 91.132 0.3810 62.900 —0.7124 —0.1776 0.0024 0.5036 -0.0032 17.772 —157.96 2.9049 —4.1664 5 A0 = 2.0103 = —0.0030 55 A0 = 123.38 itude = 0.5340 s itude = 110.73 = 207.35 = 1.3057 = 334.54 = 90.822 idity = 54.326 5 = 200.11 ss 53 displacement 88 SS SS amplitude Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/pOint 3 4 0.7035 —0.0528 1.1992 —0.3267 -0.0009 0.0003 0.0041 -0.0020 8.3752 12.123 —4.4100 40.225 dynes/sq cm dynes/sq cm dynes/sq cm dynes/sq cm radians poise poise dynes/sq cm dynes/sq cm 257 7 ’_l 001 108 displacement function coefficient: real part of imaginary part of torsion head range oscillatory range peak voltage cone angle frequency 5 0.1602 -0.1240 0.0003 0.0015 -1.2140 —5.2268 II 11 II || II 1.0 25.0 0.50 0.5522° 0.0952 c/s 6 -0.2194 -0.2399 0.0000 -0.0015 7.6909 -4.3855 123.38 dynes/sq cm 158.01 dynes/sq cm 1.1112 radians gm/cm gm/cm gm/cm shear strain shear stress normal stress Figure 31. B15 MODIFIED WRG Calcomp Plot of the Small-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5309 MODIFIED WRG B16 ‘ Table 9, Fourier Components and Material Functions harmonic 1 2 3 4 5 6 stress An 84.413 0.4778 0.1091 —0.3929 0.5825 —0.5014 Bn 43.781 -0.1251 —5.7583 0.2431 0.0462 0.3822 strain An 0.0107 0.0027 0.0005 0.0002 -0.0006 -0.0019 Bn 0.5308 0.0048 -0.0017 0.0014 -0.0027 0.0019 normal An 25.065 —98.127 —0.0835 37.733 -l.7162 -5.4318 Bn 16.622 91.675 —0.3135 —10.018 4.8512 -3.4076 shear stress A0 = 0.1636 dynes/sq cm shear strain A0 = 0.0037 normal stress A0 = 16.278 dynes/sq cm Strain Amplitude = 0.5309 Shear Stress stress amplitude = 95.091 dynes/sq cm amplitude ratio = 179.11 dynes/sq cm phase shift = 1.0722 radians dynamic viscosity = 262.99 poise imaginary part = 143.19 poise dynamic rigidity = 85.649 dynes/sq cm loss modulus = 157.30 dynes/sq cm Normal Stress normal stress displacement 16.278 dynes/sq cm 134.29 dynes/sq cm -0.4298 radians normal stress phase shift amplitude normal stress displacement function gm/cm normal stress coefficient: real part of gm/cm imaginary part of gm/cm Computer Program Input interval spacing number = 257 torsion head range = 1.0 rhomberg jmax = 7 oscillatory range = 25.0 maximum harmonic = 6 peak voltage = 0.50 cycle averaging number = 10 cone angle = 1.533° delay/point = 108 frequency = 0.0952 c/s B17 MODIFIED WRG shear stress shear strain normal stress Figure 32. Calcomp Plot of the Small—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequencey of 0.0952 c/s and Strain Amplitude of 0.5097 B18 MODIFIED WRG Table 10. Fourier Components and Material Functions harmonic 1 2 3 4 5 6 stress An 66.746 0.1938 0.6951 -0.1101 -0.3502 —0.1118 Bn 30.660 —0.7381 0.7541 —0.2300 -0.2856 -0.2260 strain An -0.0112 0.0029 -0.0004 0.0005 -0.0006 -0.0002 Bn 0.5096 -0.0054 0.0059 —0.0026 0.0007 -0.0021 normal An —7.039 —157.64 5.0370 24.693 -2.679 -2.1033 Bn —143.52 63.296 -13.979 5.274 —8.724 2.864 shear stress Ao =-1.l761 dynes/sq cm shear strain A0 = 0.0069 normal stress A0 = 44.082 dynes/sq cm Strain Amplitude = 0.5097 Shear Stress stress amplitude = 73.451 dynes/sq cm amplitude ratio = 144.11 dynes/sq cm phase shift = 1.1622 radians dynamic viscosity = 221.07 poise imaginary part = 95.721 poise dynamic rigidity = 57.256 dynes/sq cm loss modulus = 132.24 dynes/sq cm Normal Stress 44.082 dynes/sq cm 169.88 dynes/sq cm -O.5725 radians -0.0312 gm/cm -0.0022 gm/cm —0.0441 gm/cm normal stress displacement normal stress amplitude phase shift normal stress displacement function normal stress coefficient: real part of imaginary part of II II II II II II Computer Program Input interval spacing number = 257 torsion head range = 1.0 rhomberg jmax = 7 oscillatory range = 10.0 maximum harmonic = 6 peak voltage = 0.4 cycle averaging number = l cone angle = 0.5522° delay/point = 109 frequency = 0.0952 c/s B19 MODIFIED WRG : ".1 a 3 _ .. S (U . a) 1 a U1 U) U) D n p 0‘) r-l w E -: u o s Figure 33. Calcomp Plot of the Small—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.0952 c/s and Strain Amplitude of 0.5035 Table 11. harmonic stress An Bn strain An Bn normal An Bn shear stres shear strain Ao normal stre Strain Ampl Shear Stres stress ampl amplitude ratio phase shift dynamic viscosity imaginary part dynamic rig loss modulu Normal Stre normal stre normal stre phase shift normal stre normal stre B20 MODIFIED WRG Fourier Components and Material Functions Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 1 2 3 4 5 6 -0.0053 0.0023 0.0004 0.0002 -0.0009 0.0004 0.5034 —0.0050 0.0047 -0.0024 0.0008 —0.0015 21.315 —133.53 -1.8270 23.904 -2.0541 -3.5800 -104.66 38.570 -5.0959 —4.8236 0.2457 0.6979 5 A0 = dynes/sq cm = 0.0072 85 A0 = -40.271 dynes/sq cm itude = 0.5035 s itude = dynes/sq cm = dynes/sq cm = radians = poise = poise idity = dynes/sq cm s = dynes/sq cm ss 55 displacement = 40.271 dynes/sq cm 58 amplitude = 138.99 dynes/sq cm = -0.6342 radians ss displacement function = 0.02725 gm/cm ss coefficient: real part of = -0.0014 gm/cm imaginary part of = -0.0455 gm/cm = 257 torsion head range = 1.0 = 7 oscillatory range = 10.0 = 6 peak voltage = 0.4 = 1 cone angle = 0.5522° = 109 frequency = 0.0952 c/s 'B21 MODIFIED WRG normal stress shear strain Figure 34. Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.38 c/s and Strain Amplitude of 0.5488 B22 MODIFIED WRG Table 12. Fourier Components and Material Functions harmonic 1 2 3 4 5 6 stress An 52.184 35.040 43.115 2.7908 14.493 2.4823 Bn 1136.6 75.346 52.487 17.300 -0.7683 5.1879 strain An —0.0274 0.00056 0.0014 0.0003 0.0008 0.0003 Bn 0.5481 0.0061 0.0020 0.0001 -0.0001 0.0001 normal An 42.898 -274.89 -10.521 31.629 0.5354 8.898 Bn 36.525 -63.680 5.7919 19.427 4.7852 1.8845 shear stress Ao shear strain Ao normal stress Ao 30.333 dynes/sq cm 0.0012 50.905 dynes/sq cm Strain Amplitude — 0.5488 Shear Stress stress amplitude = 1137.8 dynes/sq cm amplitude ratio = 2073.2 dynes/sq cm phase shift = 0.0958 radians dynamic viscosity = 83.040 poise imaginary part = 864.33 poise dynamic rigidity = 2063.7 dynes/sq cm loss modulus = 198.31 dynes/sq cm Normal Stress 50.905 dynes/sq cm 282.17 dynes/sq cm 1.3931 radians normal stress displacement normal stress amplitude phase shift normal stress displacement function gm/cm normal stress coefficient: real part of imaginary part of = gm/cm Computer Program Input interval spacing number = 129 torsion head range = 25.0 rhomberg jmax = 6 oscillatory range = 25.0 maximum harmonic = 6 peak voltage = 0.5 cycle averaging number = 10 cone angle = 1.533° delay/point = 36 frequency = 0.38 c/s 823 MODIFIED WRG shear strain / shear stress Figure 35. Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.38 c/s and Strain Amplitude of 0.5390 Table 13. harmonic stress An Bn strain An - Bn normal An Bn shear stress shear strain normal stres Strain Ampli Shear Stress stress ampli amplitude ra phase shift dynamic visc imaginary pa 1 307.13 538.46 0.0177 0.5387 A0 A0 5 A0 1| tude tude tio osity rt dynamic rigidity loss modulus Normal Stres normal stres normal stres phase shift normal stres normal stres S F 015 B24 MODIFIED WRG ourier Components and Material Functions 2 47.720 20.420 0.0052 0.0048 3 4 46.307 12.62 —17.415 -2.020 0.0013 0.000 -0.0011 0.000 .9025 dynes/sq cm .0004 dynes/sq cm .5390 619.89 1150.1 0.5512 252.28 410.35 979.75 602.32 s displacement s amplitude s displacement function 5 coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point II II II II II 0\ real part of imaginary part of dynes/sq cm dynes/sq cm radians poise poise dynes/sq cm dynes/sq cm II I! II II II II torsion head oscillatory r peak voltage cone angle frequency 5 6 7 -2.4911 2.5900 3 -13.691 -3.9300 1 0.0002 —0.0003 8 -0.0012 0.0010 dynes/sq cm dynes/sq cm radians gm/cm gm/cm gm/Cm range = 25.0 ange = 25.0 = 0.5 = 1.533° = 0.38 c/s B25 MODIFIED WRG shear stress shear strain normal stress Figure 36. Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.38 c/s and Strain Amplitude of 0.5200 Table 14. harmonic stress An Bn strain An Bn normal An Bn shear stress 1 301.59 397.81 0.0387 0.5186 42.000 -60.32 Ao shear strain A0 = normal stres Strain Ampli Shear Stress stress ampli amplitude ra phase shift 5 A0 tude = tude tio dynamic viscosity imaginary part dynamic rigidity loss modulus Normal Stres normal stres normal stres phase shift normal stres normal stres S s disp s ampl s coef B26 ' MODIFIED WRG Fourier Components and Material Functions Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 2 3 4 5 6 5.8643 0.3053 0.6018 0.2710 0.1732 5.7880 0.2840 2.7608 —2.8230 1.2150 0.0036 0.0015 0.0009 —0.0011 —0.0002 0.0090 -0.034 0.0030 -0.0027 0.0023 —l79.34 -11.019 34.310 5.638 -5.832 8 48.406 0.7473 —8.359 1.426 —2.087 -5.997 dynes/sq cm 0.0066 56.200 dynes/sq cm 0.5200 = 499.21 dynes/sq cm = 960.02 dynes/sq cm = 0.5742 radians = 218.39 poise = 337.60 poise = 806.05 dynes/sq cm = 521.45 dynes/sq cm lacement = 56.200 dynes/sq cm itude = 185.76 dynes/sq cm = —0.7281 radians s displacement function = gm/cm ficient: real part of = gm/cm imaginary part of = gm/cm = 257 torsion head range = 2.5 = 7 oscillatory range = 10.0 = 6 peak voltage = 0.4 = 1 cone angle = 0.5522° = 109 frequency = 0.38 c/s B27 MODIFIED WRG shear strain shear stress normal stress Figure 37. Calcomp Plot of the Oscillatory Shearing of Polyisobutylene Cetane at the Frequency of 0.38 c/s and Strain Amplitude of 0.5090 B28 MODIFIED WRG Table 15. Fourier Components and Material Functions harmonic 1 2 3 4 5 6 stress An 129.50 1.7977 0.4191 0.2459 0.1962 0.0555 Bn 153.56 2.1543 0.1778 1.0867 -1.2885 0.5033 strain An 0.04996 0.0025 0.0008 0.0007 -0.0006 0.0000 Bn 0.5065 0.0085 -0.0029 0.0035 —0.0035 0.0022 normal An 39.028 -196.71 -11.573 37.630 4.9411 -5.9970 Bn -62.249 59.193 3.4550 -16.585 2.4960 2.6140 shear stress Ao =—3.0753 dynes/sq cm shear strain A0 = 0.0078 normal stress A0 = 30.951 dynes/sq cm Strain Amplitude = 0.5090 Shear Stress stress amplitude = 200.87 dynes/sq cm amplitude ratio = 394.64 dynes/sq cm phase shift = 0.6023 radians dynamic viscosity = 93.641 poise imaginary part = 136.20 poise dynamic rigidity = 325.20 dynes/sq cm loss modulus = 223.58 dynes/sq cm Normal Stress 30.95 dynes/sq cm 205.42 dynes/sq cm -0.7376 radians normal stress displacement normal stress amplitude phase shift || II II II II II normal stress displacement function gm/cm normal stress coefficient: real part of gm/cm imaginary part of gm/cm Computer Program Input interval spacing number = 257 torsion head range = 2.5 rhomberg jmax = 7 oscillatory range = 10.0 maximum harmonic = 6 peak voltage = 0.4 cycle averaging number = l cone angle = 0.5522° delay/point = 37 frequency = 0.38 c/s shear stress B29 MODIFIED WRG / : -H m n .1.) U) u m m a U1 1 Figure 38. Calcomp Plot of the Large—Amplitude OsCillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 3.4123 Table 16. harmonic stress An Bn strain An Bn normal An Bn shear stres shear strai normal stre Strain Ampl Shear Stres stress ampl amplitude r phase shift dynamic vis imaginary p dynamic rig loss modulu 1 507.83 1076.86 -0.1094 3.4106 126.59 -180.15 5 A0 n A0 55 A0 itude = s itude atio cosity art idity s SS Normal Stre normal stre normal stre phase shift normal stre normal stre B30 MODIFIED WRG Fourier Components and Material Functions 3. 0. 14 3. 2 17.971 13.226 0.0476 0.0393 338.44 16.670 3 4 5 12.789 2.6355 3.7482 30.436 2.8214 1.6487 0.0122 -0.0041 0.0004 -0.0086 0.0107 —0.0154 4.0180 49.779 1.1075 -16.688 -5.6510 —9.1564 8435 dynes/sq cm 0221 05.3 dynes/sq cm 4123 1190.6 348.91 0.4727 42.140 82.402 310.65 171.80 55 displacement ss amplitude dynes/sq dynes/sq radians poise poise dynes/sq dynes/sq ss displacement function 55 coefficient: real part of imaginary part of Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 129 II II II II II 1001mm torsion head range oscillatory range cm cm cm Cm peak voltage cone angle frequency 6 -0.8826 2.8480 —0.0034 0.0113 -l3.858 -3.9132 1405.4 dynes/sq cm 338.85 dynes/sq cm 0.7287 radians gm/cm gm/cm gm/cm 035522° 0.6 c/s II II 1| II II O as shear strain B31 MODIFIED WRG shear stress Figure 39. Calcomp Plot of the Large-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 3.3088 Table 17. harmonic stress An 15 En 65 strain An -0. Bn 3. normal An 14 En ~22 shear stress Ao shear strain Ao normal stress Ao Strain Amplitu Shear Stress stress amplitu amplitude rati B32 MODIFIED WRG Fourier Components and Material Functions phase shift dynamic viscosity imaginary part dynamic rigidity loss modulus Normal Stress normal stress normal stress phase shift normal stress normal stress Computer Program Input interval spaci rhomberg jmax maximum harmon cycle averagin delay/point l 2 3 4 5 6 71.01 24.476 69.183 —3.8817 -8.7468 5.8667 2.42 —14.578 -58.537 -39.804 —17.486 —3.8251 0331 0.0492 0.0093 —0.0077 0.0015 0.0036 3086 0.0356 -0.0086 0.0057 —0.0193 0.0050 0.32 —82.240 7.7130 —8.6480 0.5716 4.8585 .332 -158.18 2.7542 19.682 1.6368 0.1037 = -29.051 dynes/sq cm = 0.01034 = 1149.31 dynes/sq cm de = 3.3088 de = 1701.09 dynes/sq cm 0 = 514.11 dynes/sq cm = 1.1872 radians = 126.46 poise = 51.038 poise = 192.41 dynes/sq cm = 476.75 dynes/sq cm displacement = 1149.30 dynes/sq cm amplitude = 178.35 dynes/sq cm = 0.2561 radians displacement function = gm/cm coefficient: real part of = gm/cm imaginary part of = gm/cm ng number = 129 torsion head range = 10.0 = 6 oscillatory range = 25.0 ic = 6 peak voltage = 0.6 g number = 5 cone angle = 0.5522° = 9 frequency = 0.6 c/s B33 MODIFIED WRG shear stress shear Strain\ normal stress—1 Figure 40. Calcomp Plot of the Large-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 3.3048 ' Table 18. harmonic stress An 1536.64 Bn 71.487 strain An —0.1747 Bn 3.3001 normal An 135.74 Bn 4.7617 shear stress Ao shear strain Ao normal stress Ao Strain Amplitude Shear Stress stress amplitude amplitude ratio phase shift dynamic viscosity imaginary part dynamic rigidity loss modulus Normal Stress normal stress normal stress phase shift normal stress normal stress II II II B34 MODIFIED WRG Fourier Components 2 3 24.313 74.615 -12.509 -50.855 0.0448 0.01316 -0.0068 0.0018 0.0419 —0.0099 -81.452 9.1036 -216.14 5.0297 -29.297 dynes/sq cm 0.0128 1198.16 dynes/sq cm 3.3048 = 512.82 dynes/sq = 1.1883 radians = 126.20 poise = 50.771 poise = 191.43 dynes/sq = 475.77 dynes/sq displacement amplitude displacement function coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point II II II II II oLnoxm 1694.79 dynes/sq real part of imaginary part of II II II II II II 4 -3.0491 -3.5270 —11.949 19.390 cm cm cm and Material Functions 5 6 -8.0004 4.9087 —20.359 -3.3800 -0.0007 0.0037 -0.0249 0.0086 0.7236 2.7453 2.6911 -0.4144 1198.15 dynes/sq cm 230.98 dynes/sq cm 0.2331 radians 129 torsion head range oscillatory range peak voltage cone angle frequency gm/cm gm/cm gm/cm 10.0 25.0 0.6 0.5522° 0.6 c/s II II II II II shear strain B35 MODIFIED WRG normal stress Figure 41. Calcomp Plot of the Large—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 2.160 Table 19. harmonic 1 stress An 378.8 Bn 815.2 strain An 0.042 Bn 2.167 normal An 78.97 Bn -213.1 shear stress Ao shear strain Ao normal stress Ao Strain Amplitude Shear Stress stress amplitude amplitude ratio phase shift dynamic viscosity imaginary part dynamic rigidity loss modulus Normal Stress 7 7 9 5 5 4 II II II B36 MODIFIED WRG Fourier Components and Material Functions 2 3 11.652 5.3933 10.321 13.675 0.0252 0.0051 0.0248 -0.0061 -205.08 —2.311 41.551 -13.143 - 5.0905 dynes/sq cm normal stress displacement normal stress amplitude phase shift 4 5 6 1.3597 0.4375 0.6955 3.0624 -0.6369 1.6052 0.0027 -0.0025 0.0025 0.0086 -0.0077 0.0047 17.973 9.5153 -4.5368 11.832 -0.8590 3.3143 0.0274 398.86 dynes/sq cm 2.160 = 898.99 dynes/sq cm = 416.20 dynes/sq cm = 0.4153 radians = 44.4 poise = 101.01 poise = 379.44 dynes/sq cm = 167.93 dynes/sq cm 398.86 209.25 —0.7051 normal stress displacement function normal stress coefficient: real part of imaginary part of Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point cone ang II II II II II \DU’lmCh 129 torsion head range oscillatory range peak voltage 1e frequency II II II 11 || dynes/sq cm dynes/sq cm radians gm/cm gm/cm gm/cm 10.0 25.0 0.4 0.5522° 0.6 c/s B37 MODIFIED WRG shear strain shear stress normal stress ./2 Figure 42. Calcomp Plot of the Large-Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 2.1363 Table 20. harmonic stress An Bn strain An Bn normal An Bn shear stres shear strain Ao normal stre Strain Ampl Shear Stres stress ampl amplitude ratio phase shift dynamic viscosity imaginary part dynamic rig loss modulus B38 MODIFIED WRG Fourier Components and Material Functions 1 2 1153.93 17.497 555.78 -7.825 -0.0101 0.0224 2.1363 0.0198 106.11 —69.034 —27.803 —120.26 5 A0 =—21.688 = 0.02514 55 A0 = 626.65 itude = 2.1363 5 1280.8 599.54 1.1267 143.61 68.327 257.60 541.39 itude idity Normal Stress normal stre normal stre phase shift normal stre normal stre ss displacement ss amplitude dynes/sq cm dynes/sq cm 3 40.838 —l9.004 0.0075 -0.0050 2.4597 —1.9765 4 2.9242 1.2127 0.0013 0.0079 -1.2699 12.633 dynes/sq cm dynes/sq cm radians poise poise dynes/sq cm dynes/sq cm 55 displacement function 55 coefficient: real part of imaginary part of Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 12 9 \OU’IONON torsion head range oscillatory range peak voltage cone angle frequency 5 —3.2 —4.7 -0.0 -0.0 1.0 0.5 626.651 138.67 0.2653 II II II II II 6 570 4.9478 481 0.5273 005 0.0034 065 0.0062 045 -0.2438 818 -2.0195 dynes/sq cm dynes/sq cm radians gm/cm gm/cm gm/cm B39 MODIFIED WRG shear stress normal stress shear strain Figure 43. Calcomp Plot of the Large—Amplitude Oscillatory Shearing of Polysobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.3168 Table 21. Fourier Components and Material Functions harmonic 1 2 3 4 5 6 stress An 715.40 11.649 17.281 0.6266 —l.7119 —0.3267 Bn 493.27 1.0388 3.3791 2.6989 —2.7457 1.9895 strain An -0.0677 0.0113 0.0006 —0.0016 -0.0002 -0.0012 Bn 1.3150 0.0163 —0.0020 0.0046 —0.0043 0.0037 normal An 77.058 -31.903 2.5572 —4.2238 —1.2758 3.7524 Bn —10.780 -67.866 —4.3538 12.301 -1.1712 —0.0144 shear stress A0 = 1.1833 dynes/sq cm shear strain Ao =—0.0062 normal stress A0 = 341.65 dynes/sq cm Strain Amplitude = 1.3168 Shear Stress stress amplitude = 868.98 dynes/sq cm amplitude ratio = 659.92 dynes/sq cm phase shift = 1.0186 radians dynamic viscosity = 149.03 poise imaginary part = 91.823 poise dynamic rigidity = 346.16 dynes/sq cm loss modulus = 660.77 dynes/sq cm Normal Stress normal stress displacement = 341.65 dynes/sq cm normal stress amplitude = 74.991 dynes/sq cm phase shift = 0.2712 radians normal stress displacement function = gm/cm normal stress coefficient: real part of = gm/cm imaginary part of = gm/cm Computer Program Input interval spacing number = 129 torsion head range = 2.5 rhomberg jmax = 6 oscillatory range = 25 maximum harmonic = 6 peak voltage = 0.4 cycle averaging number = 5 cone angle = 0.5522° delay/point = 9 frequency = 0.600 c/s B40 MODIFIED WRG B41 Figure 44. Calcomp Plot of the Large—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.2982 B42 MODIFIED WRG mmmuum Human camnum Hmonm \ mmouum adage: Table 22. harmonic stress An Bn strain An Bn normal An Bn — shear stress shear strain Ao normal stress Ao Strain Ampli Shear Stress stress ampli 1 804.60 300.40 0.2244 1.2787 74.449 24.458 A0 tude = tude amplitude ratio phase shift dynamic viscosity imaginary pa rt dynamic rigidity loss modulus Normal Stres normal stres normal stres phase shift normal stres normal stres S B43 MODIFIED WRG Fourier Components and Material Functions 2 8.7434 ~5.l876 0.0121 0.0080 —61.236 ~46.387 25.100 0.3038 263.20 1.2982 858.85 661.57 1.0398 151.32 570.47 335.02 88.865 II II II II II II II s displacement s amplitude 3 16.160 ~8.082 0.0039 -0.0012 0.0109 -5.6975 4 5 6 2.5677 -1.8406 2.0686 -0.8924 -l.94l3 -l.2144 0.0016 -0.0008 0.0015 0.0032 -0.0043 0.00181 4.5209 -0.7214 0.02536 13.330 0.4320 —0.2122 dynes/sq cm dynes/sq cm dynes dynes /sq cm /sq cm radians poise poise dynes dynes s displacement function 5 coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 12 9 \DU‘10‘10'1 real part of imaginary part of torsion /sq cm /sq cm 263.20 76.821 0.2876 || II I! II II I! head range oscillatory range peak vo 1tage cone angle frequency dynes/sq cm dynes/sq cm radians gm/cm gm/cm gm/Cm 10.0 25.0 0.4 0.5522° 0.6 c/s B44 MODIFIED WRG shear strain normal stress Figure 45. Calcomp Plot of the Large—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.2973 Table 23. harmonic stress An Bn strain An Bn normal An shear stres shear strai normal stre Strain Ampl Shear Stres stress ampl amplitude r phase shift dynamic vis 1 190.17 572.01 —0.0311 1.2969 62.847 -115.92 5 A0 n Ao ss Ao itude = s itude atio cosity imaginary part dynamic rig loss modulu idity 5 Normal Stre normal stre normal stre phase shift normal stre normal stre SS B45 MODIFIED WRG Fourier Components 5. 0. 34 1. II || II II II II II ss displacement ss amplitude 55 displacement function 55 coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point and Material Functions 2 3 ,4 5 6 5.013 -0.8333 0.3638 -0.6835 0.5678 8.056 4.5456 2.4911 -0.4537 1.3098 0.0117 0.0009 0.0003 -0.0020 -0.0002 0.0157 -0.0023 0.0047 -0.0041 0.0043 109.16 —7.8308 33.580 3.7231 -12.282 10.086 -3.0701 -8.4400 1.5108 0.0467 9346 dynes/sq cm 0019 6.01 dynes/sq cm 2973 602.797 dynes/sq cm 464.66 dynes/sq cm 0.3450 radians 41.680 poise 115.99 poise 437.30 dynes/sq cm 157.14 dynes/sq cm = 346.01 dynes/sq cm = 109.63 dynes/sq cm = -0.7153 radians = gm/cm real part of = gm/cm imaginary part of = gm/cm = 129 torsion head range = 10.0 = 6 oscillatory range = 25.0 = 6 peak voltage = 0.4 = 5 cone angle = 0.5522° = 9 frequency = 0.6 c/s shear strain B46 MODIFIED w-Rq normal stress Figure 46. Calcomp Plot of the Large—Amplitude Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 1.2678 Table 24. harmonic 1 stress An Bn strain An 0.023 Bn 1.267 normal An 63.34 Bn -125.7 shear stress Ao shear strain Ao normal stress Ao Strain Amplitude Shear Stress stress amplitude amplitude ratio phase shift dynamic viscosity imaginary part dynamic rigidity loss modulus Normal Stress 4 6 2 8 B47 MODIFIED WRG Fourier 2 0.0099 0.0193 -92.228 19.720 -0.0011 446.51 1.2678 normal stress displacement normal stress amplitude phase shift Components and Material Functions 3 4 5 0.0018 0.0005 —0.0004 —0.0074 0.0066 —0.0039 -8.9380 24.967 2.3603 0.2509 -12.281 -2.5140 dynes/sq cm dynes/sq cm dynes/sq dynes/sq radians poise poise dynes/sq dynes/sq normal stress displacement function normal stress coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point 1 real part of imaginary part of 29 \OU’IONON torsion head range oscillatory range cm Cm cm peak voltage cone angle frequency 6 —0.0005 0.0045 -6.9247 3.0130 446.51 dynes/sq cm 94.313 dynes/sq cm 0.6986 radians gm/cm gm/cm gm/cm || II II II ll 0 4:. Figure 47. Calcomp Plot of the oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Amplitude of 0.5194 Canaan yawn nuonum Human fr_._. mmwhgm Husnon\\\\\ MODIFIED WRG B50 Table 25. Fourier Components and Material Functions harmonic 1 2 3 4 5 6 stress An 49.375 0.9597 -0.4332 —0.0591 -0.0247 0.1878 Bn 251.45 1.3207 1.5757 0.5509 0.6856 0.2675 strain An —0.0269 0.0025 0.0009 —0.0002 —0.0004 0.0001 Bn 0.5189 0.0029 0.0018 0.0014 —0.0004 0.0001 normal An 49.031 -213.70 -6.166 45.695 3.2922 -7.764 Bn -63.090 2.532 —8.618 5.976 0.3984 -1.326 shear stress A0 = 0.2002 dynes/sq cm shear strain A0 = 0.0095 normal stress A0 = 154.20 dynes/sq cm Strain Amplitude = 0.5194 Shear Stress stress amplitude = 256.25 dynes/sq cm amplitude ratio = 493.17 dynes/sq cm phase shift = 0.2458 radians ' dynamic viscosity = 31.828 poise imaginary part = 126.89 poise dynamic rigidity = 478.37 dynes/sq cm loss modulus = 119.99 dynes/sq cm Normal Stress normal stress displacement = 154.20 dynes/sq cm normal stress amplitude = 213.72 dynes/sq cm phase shift = -0.7277 radians normal stress displacement function = gm/cm normal stress coefficient: real part of = gm/cm imaginary part of = gm/cm Computer Program Input interval spacing number = 257 torsion head range = 2.5 rhomberg jmax = 7 oscillatory range = 10.0 maximum harmonic = 6 peak voltage = 0.4 cycle averaging number = 1 cone angle = 0.5522° delay/point = 9 frequency = 0.6 c/s Figure 48a. Calcomp Plot of the Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 C/S and Strain Amplitude of 0.5139 for Shear Stress Response Figure 48b. Calcomp Plot of 0xcillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Ampli- tude of 0.5139 for Normal Stress Response mmeHm Hfiwflm Figure 49. Calcomp Plot of Oscillatory Shearing of Polyisobutylene in Cetane at the Frequency of 0.6 c/s and Strain Ampli- tude of 0.5128 ./ muonum HmEHo: Table 26. harmonic 1 stress An 456.8 Bn 723.6 strain An 0.094 Bn 0.505 normal An -8.502 Bn 72.20 shear stress Ao shear strain Ao normal stress Ao Strain Amplitude Shear Stress stress amplitude amplitude ratio phase shift dynamic viscosity imaginary part dynamic rigidity loss modulus Normal Stress 8 7 5 1 7 7 II II II B57 MODIFIED WRG Fourier Components and Material Functions 2 37.668 35.052 0.0067 0.0080 -169.90 317.00 3 4 5 6 46.969 5.7976 1.4019 1.1561 -8.677 7.0847 —5.1237 2.8878 0.0013 0.0010 —0.0001 0.0001 -0.0041 0.0034 -0.0031 0.0024 -8.3708 —6.9140 -2.0253 -0.4645 23.056 18.450 -2.6840 1.4916 -108.91 dynes/sq cm -0.0020 91.539 dynes/sq cm 0.5139 855.83 1665.5 0.3783 163.15 410.55 1547.8 615.14 II II II II II II II normal stress displacement normal stress amplitude phase shift normal stress displacement function normal stress coefficient: Computer Program Input interval spacing number rhomberg jmax maximum harmonic cycle averaging number delay/point II II II II II 0 real part of imaginary part of dynes/sq cm dynes/sq cm radians poise poise dynes/sq cm dynes/sq cm 359.66 0.0611 II II II || || || torsion head range oscillatory range peak voltage cone angle frequency 3 91.539 dynes/sq cm dynes/sq cm radians gm/cm gm/cm gm/cm 100.0 25.0 0.5 1.533° 0.6 c/s B58 MODIFIED WRG :Hmuum HMOLM \ mmwnum “worm Table 27. harmonic stress An Bn strain An Bn normal An Bn shear stress shear strain normal stres Strain Ampli Shear Stress stress ampli amplitude ra phase shift dynamic viscosity imaginary pa dynamic rigidity loss modulus Normal Stres normal stres normal stres phase shift normal stres normal stres normal stres Computer Program Input interval spacing number B59 MODIFIED WRG Fourier Components and Material Functions 1 368.83 3 636.51 2 0.0974 0 0.5035 0 24.698 -1 68.525 -3 A0 = —4. A0 = -O. 5 A0 = —13 tude = 0. tude tio rt II II II II || II II S 2 1.943 3.316 .0061 .0079 07.86 44.54 3 4 5 27.865 5.1873 0.2399 -38.805 1.4524 -5.3167 0.0012 0.0006 0.0004 -0.0040 0.0030 -0.0028 -8.1107 —2.0691 -2.5524 -18.385 20.494 -2.9447 7462 dyneS/sq cm 0019 1.75 dynes/sq cm 5128 735.65 1434.6 0.3341 124.79 359.50 1355.3 470.45 s displacement s amplitude 5 function 5 displacement function 5 coeffici rhomberg jmax maximum harmonic cycle averag delay/point ing number dynes/sq cm dynes/sq cm radians poise poise dynes/sq cm dynes/sq cm = —131.75 = 361.03 = 0.0394 ent: real part of = imaginary part of = 129 || II II II II 0‘1 torsion head range oscillatory range peak voltage cone angle frequency dynes dynes 6 0.6240 1.3730 0.0005 0.0020 -0.0677 2.0509 /sq cm /sq cm radians gm/cm gm/cm gm/cm gm/cm 100.0 25.0 0.5 1.53 30 0.6 c/s ANS “11111111111[111)11111111111‘1‘“ 3 1