Rheological Interconversions

Transcription

1 Technical White Paper Rheological Interconversions Fundamental Properties of Asphalts and Modified Asphalts III Product: FP 23 March 215 Prepared for Federal Highway Administration Contract No. DTFH61-7-D-5 By Ronald R. Glaser, Thomas F. Turner, Changping Sui, Stephen Salmans and Jean-Pascal Planche Western Research Institute 3474 North 3rd Street Laramie, WY

2 TABLE OF CONTENTS INTRODUCTION...1 Shifting...2 Prony Series Shear Relaxation from Complex Modulus...4 Shear Relaxation to Extensional Relaxation...1 Creep Compliance Deconvolution and Stiffness...12 Prony Series Distribution Improvement...12 EXPERIMENTAL...13 RESULTS AND DISCUSSION...13 Nifty-Shifty Results...14 Prony Herd Calculation Results...21 CONCLUSIONS...37 RECOMMENDATIONS...37 ACKNOWLEDGMENTS...38 DISCLAIMER...38 REFERENCES...38 i

3 LIST OF FIGURES Figure 1. Graph. Least squares and fractional exponent residual fits compared...3 Figure 2. Diagram. Alignment shifting method minimizing connecting line length...4 Figure 3. Graph. AZ1-3 complex modulus master curve reference temperature -12 C...14 Figure 4. Graph. AZ1-3 Arrhenius shift function...14 Figure 5. Graph. AZ1-3 WLF shift function...15 Figure 6. Graph. AZ1-3 polynomial Arrhenius shift function...15 Figure 7. Graph. AZ1-3 black space...16 Figure 8. Graph. MN1-5 complex modulus master curve reference temperature -18 C...16 Figure 9. Graph. MN1-5 Arrhenius shift function...17 Figure 1. Graph. MN1-5 WLF shift function...17 Figure 11. Graph. MN1-5 polynomial Arrhenius shift function...18 Figure 12. Graph. MN1-5 black space...18 Figure 13. Graph. YNP complex modulus master curve reference temperature -24 C...19 Figure 14. Graph. YNP Arrhenius shift function...19 Figure 15. Graph. YNP WLF shift function...2 Figure 16. Graph. YNP polynomial Arrhenius shift function...2 Figure 17. Graph. YNP black space...21 Figure 18. Graph. AZ1-3. Prony spectrum...22 Figure 19. Graph. AZ1-3. Measured and calculated phase angle...22 Figure 2. Graph AZ1-3. Complex modulus measured compared to complex modulus calculated from relaxation function convolution...23 Figure 21. Graph. AZ1-3. Shear relaxation from Prony series...23 ii

4 LIST OF FIGURES (continued) Figure 22. Graph. AZ1-3. Extensional relaxation from shear relaxation assuming.5 Poisson s ratio and time dependant Poisson s ratio...24 Figure 23. Graph. AZ1-3. Creep compliance from deconvolution calculation and reciprocal extensional relaxation approximation...24 Figure 24. Graph. AZ1-3. Stiffness from calculated creep compliance...25 Figure 25. Graph. MN1-5. Prony spectrum...25 Figure 26. Graph MN1-5. Measured and calculated phase angle...26 Figure 27. Graph MN1-5. Complex modulus measured compared to complex modulus calculated from relaxation function convolution...26 Figure 28. Graph. MN1-5. Shear relaxation from Prony series...27 Figure 29. Graph. MN1-5. Extensional relaxation from shear relaxation assuming.5 Poisson s ratio and time dependant Poisson s ratio...27 Figure 3. Graph. MN1-5. Creep compliance from deconvolution calculation and reciprocal extensional relaxation approximation...28 Figure 31. Graph. MN1-5. Stiffness from calculated creep compliance...28 Figure 32. Graph. YNP. Prony spectrum...29 Figure 33. Graph YNP. Measured and calculated phase angle...29 Figure 34. Graph. YNP. Complex modulus measured compared to complex modulus calculated from relaxation function convolution...3 Figure 35. Graph. YNP. Shear relaxation from Prony series...3 Figure 36. Graph. YNP. Extensional relaxation from shear relaxation assuming.5 Poisson s ratio and time dependant Poisson s ratio...31 Figure 37. Graph. YNP. Creep compliance from deconvolution calculation and reciprocal extensional relaxation approximation...31 Figure 38. Graph. YNP. Stiffness from calculated creep compliance...32 iii

5 LIST OF FIGURES (continued) Figure 39. Graph. DSR and BBR stiffness compared using a constant Poisson s ratio assumption in the interconversions...33 Figure 4. Graph. The stiffness curves generated using a range of temperatures in the DSR...34 Figure 41. Graph. The stiffness curves generated using a range of constant Poisson ratio value assumptions...35 Figure 42. Graph. The stiffness curves generated using a range of constant Poisson ratio value assumptions and time dependent Poisson s ratio...36 Figure 43. Graph. Poisson s ratio decay times as a function of temperature...37 iv

6 LIST OF TABLES Table 1. Poisson s ratio exponential decay parameters for three binders...36 v

7 RHEOLOGICAL INTERCONVERSIONS INTRODUCTION The topic of interconversion of the various rheological moduli functions has been of interest to rheologists for a considerable time. The utility of being able to measure a modulus function and then generate a more useful one provides a great deal of latitude in selecting measurement methods. In the area of asphalt binder rheology, measurements are taken at high and modest temperatures using the dynamic shear rheometer (DSR). Until recently, low temperature data was unobtainable with this method, and creep measurements of a beam using the bending beam rheometer (BBR) were employed to obtain low temperature measurements needed for performance grading (PG). This combination has been in routine use since it was developed under the Strategic Highway Research Program (SHRP), but is limited for some applications due to the large amount of material required to fabricate the binder beams. WRI developed the 4-mm DSR plate method with machine compliance corrections that allows the use of DSR on very small sample down to -4 C (Sui et al. 21). There are other advantages to using DSR in lieu of BBR. Input data for mechanical performance simulations generally require the time domain relaxation function, often expressed as a Prony series in material files, instead of dynamic modulus. Conversion of dynamic modulus data to relaxation is a one step process, while conversion from bending beam creep data involves several steps, and has been shown to be inconsistent with DSR testing and conversion on the same materials (Marasteanu and Anderson 1999). The mathematical manipulations required to convert beam creep modulus data to relaxation are quite involved, and approximate methods are often applied instead of the mathematically rigorous, but tedious methods. These are covered quite well in the Marasteanu (1999) thesis. We have developed a series of algorithms to make the conversions while adhering to fundamental solutions based on linear viscoelastic theory as much as possible. This effort has attempted, in part, to understand the discrepancies between relaxation as determined from BBR measurements and relaxation as determined from DSR measurements. The objectives were to produce more general and fundamental interconversions that would apply to highly modified asphalt binders, and to determine if the use of approximate methods may have played a role in discrepancies noticed in the literature. The computing power available in the average personal computer is such that more rigorous numerical methods are indeed practical. There is, however, always a concern over computation error propagation with some of these numerical methods. To compare DSR and BBR measurements, DSR data are converted to stiffness, and then compared to stiffness from BBR measurements. The computational sequence is: 1) Shift DSR isotherm data to a time-temperature superposed (TTS) master curve. 2) Obtain a Prony series representation directly from the complex modulus using an algorithm developed in this study that removes any error caused by erroneous phase angle measurements. 1

8 3) Convert shear relaxation to extensional relaxation (assuming flexural and extensional moduli are equivalent). These computations require the use of Poisson s ratio, and we employed a constant Poisson s ratio and two approaches to time dependant Poisson s ratio methods in the calculations. 4) Numerical time domain deconvolution was employed to convert the extensional modulus to creep compliance. 5) Creep compliance was inverted to get stiffness. 6) Compare the interconverted DSR stiffness to the stiffness obtained from BBR creep measurements. All of these computations are performed on two separate Microsoft Excel spreadsheets. The first of these is Nifty Shifty, which performs a wild shift according to a modulus sorted connecting line minimization routine The Prony Herd spreadsheet accepts the shifted master curve data and performs all of the interconversion steps described in this study. The Prony series computation was later modified by changing the form of the Maxwell element coefficient distribution. Shifting The SHIFT routine first sorts the data according to complex modulus. The data are then classified according to the isotherm temperature and then shifted using an alignment algorithm. This spreadsheet requires that all isotherms overlap to some extent. If you imagine lines connecting the data points, the unshifted data has many zigzag patterns in it. The alignment is accomplished by minimizing the relative dimensionless length of the connecting lines using a sum of fractional exponent residuals. This is a much better fitting criterion than the sum of least squares. The square of the residuals weights the aberrant points more heavily than the good points. The traditional equation for a dimensionless sum of squares residue is: n 1 i= 2 (( datai fiti) / datai) (1) (Note this form evaluates local relative error and works over any range of data, including the many decades encountered in rheological data.) While a better approach is the fractional exponent sum of residuals: n 1 i=.25 ( ABS(( datai fiti) / datai)) (2) The two methods are compared side by side in the following figure: 2

9 3 Dependent variable data relative fractional exponent (.25) standard least squares Independent variable Figure 1. Graph. Least squares and fractional exponent residual fits compared. The shift factors are adjusted for each point according to its isotherm temperature until the sum of the residuals (overlap sums) is minimized using Excel Solver. Good results are obtained with an exponent of.25. In usual math notation, the code line above is: Residual at a data point = ( (1 ω / ω ) ) 2 + (1 G* / G* ) 2 + (1 ω / ω ) 2 + (1 G* / G* ) n (3) where: ω = shifted frequency G* = complex modulus n = weighting exponent < 1 The shifted frequency is then computed. The following diagram (figure 2) geometrically illustrates the line length minimization principle: 3

10 Figure 2. Diagram. Alignment shifting method minimizing connecting line length. Minimization is accomplished using Solver programmatically by minimizing the residuals sums by adjusting the frequency. The shift is made to the warmest isotherm one isotherm at a time until all isotherms have been shifted. Once all the shift factors are computed, the shift functions are fit (the linear Arrhenius using standard least squares, and the WLF and quadratic Arrhenius using solver). Additional lines of code in the SHIFT subroutine manage the plots on the spreadsheet. Once the shift is accomplished and the various shift functions determined, the user can generate a new master curve at any desired temperature by entering the values on the spreadsheet. The curve is shifted according to the chosen shift function and temperature. The new temperature only changes the frequency column, and new reference temperature columns are provided on the spreadsheet. The glass transition temperature in this fit is not fixed, but is fit along with the other parameters. The master curves are displayed on a plot that also contains the original isotherms so the quality of the shift functions can be easily evaluated visually. A good shift function should place the shifted back data through the isotherm generated at the same temperature. Prony Series Shear Relaxation from Complex Modulus We start with the defining equation, the hereditary integral for visco-elastic materials: for a Maxwell Element the relaxation modulus is t g() t = ge β (4) r r where t = time (s) and β = decay constant (s) 4

11 The response of this element in a DSR is given by the hereditary integral: t ( t τ ) β dε (sin( ωτ ) σω () t = ge r dτ (5) dτ where: σ ω = stress (Pa) ω = angular frequency (rad/s) τ = variable of integration (s) ε = strain (m/m) The integral can now be evaluated: () t β 1 e ( ωsin( ωt) cos( ωt)) σ + ω β β () t = grω ε 2 1 ω + 2 β (6) Notice the exponential transient term. From a practical stand point, rheometric data should not be collected until the response stabilizes. Some preliminary plots indicate this happens quite quickly, usually in less than 1 cycle for a single Maxwell element. A rapid approach to steady state is not necessarily true for a series with long decay times, which can be misinterpreted as the equilibrium modulus for a visco-elastic solid. For our current inquiry, we are interested in steady state response, so we take the limit as t and luckily find that the exponential vanishes. 1 ( ωsin( ωt) + cos( ωt)) σ ω β () t = grω ε 2 1 ω + 2 β (7) Define A and B as 1 β B = 2 1 ω + 2 β (8) 5

12 ω A = 2 1 ω + 2 β (9) After a little algebra (surprisingly simple) one obtains, 1 1 ϕ = tan ωβ (1) So, the steady state response function is ( t) = grω A + B sin ωt+ tan ε σ ω ωβ (11) The full transient form is ( t) g A B sin ε t tan σ ω = rω + ω + ωβ () t β e β 1 ω 2 β (12) Considering the series form with several Maxwell elements, the integrals simply add. Now that we have the response for a single Maxwell element, most complex materials can be treated by describing the relaxation function as series of Maxwell Elements: n i g () t = g e β (13) r i= 1 ri, t The complex vector form of the response is most convenient to use, as we shall see when we calculate the phase angle of the series: 6

13 1 ( ωsin( ωt) + cos( ωt)) σ ω β () t = grω ε 2 1 ω + 2 β (14) g ω r 2 grω β = (sin( ωt)) + (cos( ωt)) ω + ω β β (15) where: [ A(sin( ωt)) ] [ B(cos( ωt)) ] = + (16) g r ω 2 g A r ω β = and B = ω + 2 β ω + 2 β So the series can be written as: n n σ ω = i ω + ε i= 1 i= 1 ( t) A(sin( t)) Bi(cos( ωt)) (17) Let A Σ n = A and i= 1 i B Σ n = Bi i= 1 then ϕ B 1 Σ = tan AΣ (18) In the complex plane A Σ represents the real part, and B Σ represents the imaginary part, and one can work this out with phasors. The next concern in the computation is getting the modulus, which occurs when the response is at its maximum. The maximum strain is at π/2. The stress peaks before the strain, so the proper angle at the peak is π/2 phase angle. Divide this by the frequency, ω, to obtain the time at peak stress: max σ ω G * ( ω) = = AΣ(sin( π / 2 ϕ)) + BΣ(cos( π / 2 ϕ)) ε (19) 7

14 So, the algorithm is fairly straight forward. For each frequency ω we calculate complex modulus G* and phase angle, ϕ, like this: Given the prony series constants, first sum the A and B terms. A Σ n = A, B i= 1 where: 2 giω Ai = 2 1 ω + 2 βi giω βi Bi = 2 1 ω + 2 βi Now get the phase angle: i Σ n = B (2, 21) i= 1 i ϕ B 1 Σ = tan AΣ (22) Then get the modulus: or max * σ ω ( ) G ω = = AΣ(sin( π / 2 ϕ)) + BΣ(cos( π / 2 ϕ)) ε σ 2 2 * = ω = Σ + Σ sin / 2 ε max G A B ( π ) (23) (24) σ ω G* = = A + B ε max 2 2 Σ Σ (25) σ ω 2 2 G* = = G' + G" ε max (26) Use the working equation above to fit the Prony coefficients using generalized gradient reduction (Excel Solver ). The Prony series to G(t) conversion is a straight-forward sum of exponentials. 8

15 The Prony Herd sheet constrains the coefficients to fit up to 3 log normal Gaussian distributions (Jongepier and Kuilman 1969). where: g j (27) t 3 6 βk, j Ecalc = gk, je prony series 3 distributions k= 1 j= 1 2 log1( βk) log1( centerk) log1( σ k ) 1 Gglassy ( relative H k ) e 2 log1( σ k ) = normalized distributed coefficients 3 6 gk, j k= 1 j= 1 The program solves for: relative H,log1( center ),log1( σ ), G k k k glassy by minimizing: n * * G data ( tk ) G calc ( tk ) errorfunc = G ( t ) * k = 1 data k 3 Using 3 log-normal distributions to describe the Prony series exponential decay coefficients works well for creating a relaxation function that re-convolves to the master-curve, and this approach was used for the comparison with BBR stiffness. Further studies with alternative distributions revealed that a generalized Gaussian distribution (the exponent now a variable instead of 2) would provide high quality relaxation computations for unmodified binders with only one distribution (3 adjustable parameters). The distribution is further constrained by setting the peak of the distribution at the decay time for the most rapidly decaying Maxwell element. The reduction in adjustable parameters results in a unique solution using non-linear fitting methods. g j 1 Gglassy ( relative H k ) e 2 log1( σ k ) = 3 6 gk, j k= 1 j= 1 β log1( βk) log1( centerk) log1( σ k ) (28) It is important to note that using non-linear regression methods, the phase angle is not required to obtain the shear relaxation modulus, and, can in fact be derived from the complex modulus alone. This fact is quite helpful when working with data containing poorly measured phase angles, a frequent occurrence with DSR measurements under low torque conditions. Once the Prony series distributions are fitted, and the Prony series spectrum known, the shear relaxation function is known and an x,y pair list can be generated to describe the material relaxation in time. 9

16 Shear Relaxation to Extensional Relaxation An approximate formula is presented by Lakes (1992) for the time dependency of Poisson s ratio, assuming the bulk modulus is constant: 1 Et () υ () t = (29) 2 6B where: υ(t) = time dependent Poisson s ratio E(t) = extensional modulus (relaxation) B = bulk modulus (assumed time invariant) So, the equation above combined with Et () = Gt ()2(1 + υ) (3) can be rearranged with a bit of algebra to produce: 3 Gt () Et () = 3 Gt () 1+ 6B (31) The above formulation does not work when comparing BBR data. In the BBR flexural setting, it appears that a Poisson s ratio model based upon the flow of a mobile phase does indeed reconcile the DSR and BBR data sets. An approximate formulation follows assuming Darcy flow: dv dt = KA dp η dh (32) where: V = volume t = time K = permeability A = area normal to flow η = viscosity dp / dh = pore pressure (stress) gradient and dlwh KLw dp dt = η dh (33) 1

17 where: w = width h = height of some small slice L = length Assuming L, h, K, η, and dp/dh don t change much or dw K dp = dt (34) w hη dh hη dh λ = (35) K dp η dh λ = (36) ' K dp where ' K = relative permeability, so w w dw = w t 1 dt λ (37) t wt () = we λ (38) ν = w/ L= Poisson's ratio (39) so, approximately or wt () we = L L t e λ t λ ν() t = ν (41) (4) which is about right if v() t = as t. More generally ( ν ) λ ν() t = v e + v (42) t 11

18 The source code can be easily modified to use other Poisson ratio models, including the incorrect constant assumption frequently employed by others. Creep Compliance Deconvolution and Stiffness From Ferry (198) the relationship between creep and extensional relaxation is given by: t Et ( τ ) D( τ ) d τ = t (43) A number of methods exist for deconvolving this equation to D(creep) based upon the idea that for small steps D can be considered a constant (or ramp) and taken outside the integral in small Euler or trapezoidal rule steps. E(t) is the extensional relaxation. A variety of combinations of trapezoidal or Euler rule to either kernel or both have been employed, but we have found that the double Euler is least likely to oscillate (Park and Kim 1999). The discrete form reduces to the following recursive formula. Note that the double trapezoidal rule form for t 1 is employed as the equal spaced form is very sensitive to this first step. Dt ( ) = 1/ Et ( ) (44) 3 Et ( 1)/ Et ( ) Dt ( 1) = Et ( ) + Et ( ) 1 (45) n 1 tn Dt ( i 1) Et ( n ti)( ti ti 1) 1 Dt ( n) = ( Et ( ) + Et ( t ))( t t ) n n 1 n n 1 (46) for equal spacing n 1 tn Dt ( i 1) Et ( n ti) t 1 Dt ( n) = ( Et ( ) + Et ( )) t 1 (47) The recursion starts at index 2, so the last equation is looped n=2 to n maximum to get all the point values for D. Stiffness is the inverse of creep compliance. Prony Series Distribution Improvement A second series of studies was performed in an effort to improve the relaxation computation. We replaced the Gaussian distribution of the Prony spectrum with a stretched exponential, improving fit quality while reducing the adjustable parameter count. The ill-posed nature of this 12

19 computation appears to be eliminated. Additional investigations of single isotherm relaxation computations were done to test the robustness of the method and to investigate the deviation of asphalt binders from ideal thermally simple behavior. EXPERIMENTAL This report consists of three separate studies. The first investigates the interconversion of low temperature DSR complex modulus data to stiffness, and then compares that to stiffness results from the BBR. The second study focuses on the relaxation computation alone using and improved Prony spectrum distribution shape. The third study investigates the limitations of timetemperature superposition and the use of relaxation modulus conversions of individual complex modulus isotherms. The data from the examination of 3 asphalts using the bending beam rheometer and the 4 mm plate dynamic shear rheometer were inter-converted using the methods described in the introduction and compared in this study. A detailed description of these measurements can be found in Sui et al. (21). These binders are MN1-5 from the Minnesota comparative pavement performance site, AZ1-1 from the Arizona comparative pavement performance site, and YNP from the Yellowstone comparative pavement performance site. These data were generated in support of the 4 mm plate DSR method, described in a topical report (Farrar et al. 215). RESULTS AND DISCUSSION To examine inter-conversion results for unmodified asphalts, three binders were examined where data already existed from 4 mm plate DSR testing and BBR runs on the same materials. These binders are all unmodified and were used in field performance test sites as well. The Minnesota test site binder, MN1-5, the Arizona test site binder, AZ1-3, and the Yellowstone test site binder, YNP were examined. The BBR testing on the MN1-5 binder was done at -18 C, the AZ1-3 binder was run at -12 C, and the YNP binder at -24 C. The first step in the interconversion process is to convert the DSR isotherms to a master curve shifted to the BBR test temperature. Three shift functions were then fit to the alignment based master curve. The standard Arrhenius plot was used to obtain a simple Arrhenius shift function, the WLF equation was fit without specifying the glass transition temperature. While the use of a known glass transition temperature is often employed, for asphalts this produces poor fits at low temperatures, and a curve fitted reference temperature (usually lower than the true glass transition temperature) provides excellent shift function fits. The free volume theory for the WLF does not appear to work very well for producing a precise shift function. An alternative to WLF is also used. This simple alternative is a quadratic fit on an Arrhenius plot, which essentially describes temperature dependant activation energy for viscous flow. All three methods work reasonably well. Figures 3-17 show the shifting results for these three binders. 13

20 Nifty-Shifty Results 1.E+1 shift G* Complex Modulus, Pa 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 1.E-9 1.E-4 1.E+1 1.E+6 1.E+11 Reduced time, s shift Tmax Data Shift New Arrhenius Shift New Poly Shift New WLF Figure 3. Graph. AZ1-3 complex modulus master curve reference temperature -12 C. Arrhenius Log at y = E+4x E-1 R² = E Series1 Linear (Series1) 1/(T )-1/(Tref ) Figure 4. Graph. AZ1-3 Arrhenius shift function. 14

21 WLF Log at data WLF T-Tref Figure 5. Graph. AZ1-3 WLF shift function. 3 POLY Log at data fit Poly. (data) y = 1.241E+6x E+4x E-2 R² = 9.996E-1-1.5E-3-1.E-3-5.E-4.E+ 5.E-4 1.E-3 1.5E-3 2.E-3 1/(T )-1/(Tref ) Figure 6. Graph. AZ1-3 polynomial Arrhenius shift function. 15

22 Black Space 6 5 Phase angle, degrees E+5 1.E+6 1.E+7 1.E+8 1.E+9 1.E+1 Complex Modulus, Pa Figure 7. Graph. AZ1-3 black space. 1.E+1 shift G* Complex Modulus, Pa 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E-9 1.E-5 1.E-1 1.E+3 1.E+7 1.E+11 Reduced time,s shift Tmax Data Shift New Arrhenius Shift New Poly Shift New WLF Figure 8. Graph. MN1-5 complex modulus master curve reference temperature -18 C. 16

23 Log at Arrhenius y = 1.31E+4x E-1 R² = E Series1 Linear (Series1) 1/(T )-1/(Tref ) Figure 9. Graph. MN1-5 Arrhenius shift function. Log at WLF data WLF T-Tref Figure 1. Graph. MN1-5 WLF shift function. 17

24 POLY Log at data fit Poly. (data) -5-1 y = E+6x E+4x E-4 R² = E-1-1.5E-3-1.E-3-5.E-4 1.E-17 5.E-4 1.E-3 1.5E-3 2.E-3 1/(T )-1/(Tref ) Figure 11. Graph. MN1-5 polynomial Arrhenius shift function. Phase Angle, degrees Black Space E+4 1.E+5 1.E+6 1.E+7 1.E+8 1.E+9 1.E+1 Complex Modulus, Pa Figure 12. Graph. MN1-5 black space. 18

25 1.E+9 shift G* Complex Modulus, Pa 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E+3 1.E+2 1.E-12 1.E-6 1.E+ 1.E+6 Reduced time, s 1.E+12 shift Tmax Data Shift New Arrhenius Shift New Poly Shift New WLF Figure 13. Graph. YNP complex modulus master curve reference temperature -24 C. 3 Arrhenius Log at Series y = E+4x E-1 R² = E-1 Linear (Series1) /(T )-1/(Tref ) Figure 14. Graph. YNP Arrhenius shift function. 19

26 WLF Log at data WLF T-Tref Figure 15. Graph. YNP WLF shift function POLY Log at data fit Poly. (data) -5-1 y = E+6x E+3x E-2 R² = E-1-1.5E-3-1.E-3-5.E-4 1.E-17 5.E-4 1.E-3 1.5E-3 2.E-3 1/(T )-1/(Tref ) Figure 16. Graph. YNP polynomial Arrhenius shift function. 2

27 8 Black Space 7 6 Phase Angle, degrees E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 1.E+8 1.E+9 Complex Modulus, Pa Figure 17. Graph. YNP black space. Prony Herd Calculation Results The Prony Herd spreadsheet does all of the interconversions of the DSR to finally arrive at stiffness. The spreadsheet contains graphical displays of these computations and Graphs of these for three binders are shown in figures for the Darcy flow model case of time dependent Poisson s ratio: 21

28 Prony Spectrum 9.E+7 8.E+7 7.E+7 6.E+7 g i, Pa 5.E+7 4.E+7 3.E+7 2.E+7 1.E+7.E+ 1.E-19 1.E-15 1.E-11 1.E-7 1.E-3 1.E+1 1.E+5 1.E+9 decay constant, s Figure 18. Graph. AZ1-3. Prony spectrum. 6 Phase angle 5 Phase angle degrees calc data 1.E-9 1.E-7 1.E-5 1.E-3 1.E-1 1.E+1 1.E+3 angular freq radians Figure 19. Graph. AZ1-3. Measured and calculated phase angle. 22

29 Complex Modulus G(w) 1.E+1 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E+3 1.E+2 1.E+1 1.E+ 1.E-9 1.E-7 1.E-5 1.E-3 1.E-1 1.E+1 1.E+3 Calc Data angular freq radians Figure 2. Graph. AZ1-3. Complex modulus measured compared to complex modulus calculated from relaxation function convolution. Shear Relaxation Modulus G(t) 1.E+9 G(t) Calc Calc 1.E+8.E+ 5.E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Figure 21. Graph. AZ1-3. Shear relaxation from Prony series. 23

30 Relaxation Modulus E(t) 1.E+1 G(t) 1.E+9 E(t) v=.5 E(t) using v(t) 1.E+8-1.E+1 4.E+1 9.E+1 1.4E+2 1.9E+2 2.4E+2 2.9E+2 time, s Figure 22. Graph. AZ1-3. Extensional relaxation from shear relaxation assuming.5 Poisson s ratio and time dependant Poisson s ratio. 6.E-9 Creep Compliance, D(t) 5.E-9 4.E-9 D(t) 3.E-9 2.E-9 Calc 1/E(t) 1.E-9.E+ 1.2E+ 5.1E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Figure 23. Graph. AZ1-3. Creep compliance from deconvolution calculation and reciprocal extensional relaxation approximation. 24

31 Stiffness, S(t) 1.E+1 G(t) 1.E+9 Calc 1.E+8.E+ 5.E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Figure 24. Graph. AZ1-3. Stiffness from calculated creep compliance. Prony Spectrum 1.E+8 1.E+8 8.E+7 g i, Pa 6.E+7 4.E+7 2.E+7.E+ 1.E-2 1.E-16 1.E-12 1.E-8 1.E-4 1.E+ 1.E+4 1.E+8 decay constant s Figure 25. Graph. MN1-5. Prony spectrum. 25

32 Phase angle Phase angle degrees calc data -1 1.E-9 1.E-7 1.E-5 1.E-3 1.E-1 1.E+1 1.E+3 1.E+5 angular freq radians Figure 26. Graph. MN1-5. Measured and calculated phase angle. Complex Modulus G(w) 1.E+1 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E+3 1.E+2 1.E+1 1.E+ 1.E-9 1.E-6 1.E-3 1.E+ 1.E+3 1.E+6 Calc Data angular freq radians Figure 27. Graph. MN1-5. Complex modulus measured compared to complex modulus calculated from relaxation function convolution. 26

33 Relaxation Modulus, E(t) 1.E+1 G(t) 1.E+9 E(t) v=.5 E(t) using v(t) 1.E+8-1.E+1 9.E+1 1.9E+2 2.9E+2 time, s Figure 28. Graph. MN1-5. Shear relaxation from Prony series. 1.E+1 Relaxation Modulus, E(t) G(t) 1.E+9 E(t) v=.5 E(t) using v(t) 1.E+8-1.E+1 4.E+1 9.E+1 1.4E+2 1.9E+2 2.4E+2 2.9E+2 time, s Figure 29. Graph. MN1-5. Extensional relaxation from shear relaxation assuming.5 Poisson s ratio and time dependant Poisson s ratio. 27

34 Creep Compliance, D(t) 1.2E-8 1.E-8 8.E-9 D(t) 6.E-9 4.E-9 Calc 1/E(t) 2.E-9.E+ 1.2E+ 5.1E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Figure 3. Graph. MN1-5. Creep compliance from deconvolution calculation and reciprocal extensional relaxation approximation. 1.E+1 Stiffness, S(t) G(t) 1.E+9 Calc 1.E+8.E+ 5.E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Figure 31. Graph. MN1-5. Stiffness from calculated creep compliance. 28

35 g i, Pa Prony Spectrum 1.E+8 9.E+7 8.E+7 7.E+7 6.E+7 5.E+7 4.E+7 3.E+7 2.E+7 1.E+7.E+ 1.E-16 1.E-12 1.E-8 1.E-4 1.E+ 1.E+4 1.E+8 1.E+12 decay constant, s Figure 32. Graph. YNP. Prony spectrum. Phase angle 8 7 Phase angle degrees calc data 1.E-12 1.E-1 1.E-8 1.E-6 1.E-4 1.E-2 1.E+ angular freq radians Figure 33. Graph YNP. Measured and calculated phase angle. 29

36 G(w) Complex Modulus 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E+3 1.E+2 1.E+1 Calc Data 1.E+ 1.E-12 1.E-1 1.E-8 1.E-6 1.E-4 1.E-2 1.E+ angular freq radians Figure 34. Graph. YNP. Complex modulus measured compared to complex modulus calculated from relaxation function convolution. Shear Relaxation Modulus, G(t) G(t) 1.E+1 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E+3 1.E+2 1.E+1 1.E+.E+ 5.E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Calc Calc Figure 35. Graph. YNP. Shear relaxation from Prony series. 3

37 Relaxation Modulus, E(t) 1.E+1 G(t) 1.E+9 E(t) v=.5 E(t) using v(t) 1.E+8-1.E+1 4.E+1 9.E+1 1.4E+2 1.9E+2 2.4E+2 2.9E+2 time, s Figure 36. Graph. YNP. Extensional relaxation from shear relaxation assuming.5 Poisson s ratio and time dependant Poisson s ratio. 8.E-9 7.E-9 6.E-9 5.E-9 Creep Compliance, D(t) D(t) 4.E-9 3.E-9 2.E-9 1.E-9.E+ 1.2E+ 5.1E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Calc 1/E(t) Figure 37. Graph. YNP. Creep compliance from deconvolution calculation and reciprocal extensional relaxation approximation. 31

38 Stiffness, S(t) 1.E+1 G(t) 1.E+9 Series1 1.E+8.E+ 5.E+1 1.E+2 1.5E+2 2.E+2 2.5E+2 3.E+2 time, s Figure 38. Graph. YNP. Stiffness from calculated creep compliance. The interconverted DSR stiffnesses were compared to the BBR flexural stiffnesses for the three binders studied. In general, the slope of the stiffness curve was higher for the BBR testing than from the DSR. An example of the DSR and BBR stiffness curves over a time frame relevant to the determination of m-value is shown in figure

39 S(t), Pa 1.E+9 9.E+8 8.E+8 7.E+8 6.E+8 5.E+8 4.E+8 3.E+8 2.E+8 BBR Stiffness -18C MN-5 Mn-5 DSR stiffness - 18C Poisson's=.5 1.E Time, s Figure 39. Graph. DSR and BBR stiffness compared using a constant Poisson s ratio assumption in the interconversion. The question of whether small errors in temperature measurements in either instrument affected results was investigated by repeating the stiffness calculations at different shift temperatures. Figure 4 shows that some of the error in absolute terms could be attributed to a few degrees measurement error in the temperature, but a difference in slope persists. A drift to warmer temperatures would correct this, but seems much more unlikely than a drift to colder temperatures. Notice, however, that a.5 C change over 2 minutes time would cause this much error. 33

40 1.E+9 9.E+8 8.E+8 S(t), Pa 7.E+8 6.E+8 5.E+8 4.E+8 BBR -18C DSR -18C DSR -17C DSR -16C DSR -15C 3.E time, s Figure 4. Graph. The stiffness curves generated using a range of temperatures in the DSR. A range of stiffnesses were computed based upon a range of assumed constant Poisson s ratio in the interconversion calculation (figure 41). 34

41 S(t) Pa 1.E+9 9.E+8 8.E+8 7.E+8 6.E+8 5.E+8 4.E+8 3.E+8 2.E+8 1.E+8 BBR -18C Poisson's=.5 Poisson's=.4 Poisson's=.3 Poisson's=.2 Poisson's=.1 Poisson's=. Poisson's=-.1.E Time, s Figure 41. Graph. The stiffness curves generated using a range of constant Poisson ratio value assumptions. Poisson s ratio for a visco-elastic material is time dependent, but generally is considered to increase as the material relaxes and flows (Tschoegl et al. 22). Certainly this is true if the material is nearly incompressible. However, the data from the BBR suggest that the Poisson s ratio is actually decreasing. This is only possible if the material consists of two phases, a mobile phase and a stationary phase. Compression of the top of the beam would drive flow toward the bottom, and tension at the bottom would pull flow toward it. An approximate Darcy s law was derived earlier in the introduction to describe this kind of flow resulting in a decaying exponential description of the Poisson s ratio in time. The decaying Poisson s ratio interconversion from DSR data is shown in figure

42 S(t) Pa 1.E+9 9.E+8 8.E+8 7.E+8 6.E+8 5.E+8 4.E+8 3.E+8 2.E+8 1.E+8 BBR -18C Poisson's=.5 Poisson's=.4 Poisson's=.3 Poisson's=.2 Poisson's=.1.E Time, s Poisson's=. Figure 42. Graph. The stiffness curves generated using a range of constant Poisson ratio value assumptions and time dependent Poisson s ratio. The time dependent Poisson ratio parameters for the three binders studied, ν, time zero value, ν, infinite time value, and λ, the decay constant, are tabulated below in table 1 for the three binders studied. Table 1. Poisson s ratio exponential decay parameters for three binders. Binder MN1-5 AZ1-3 YNP ν ν λ (s) The BBR data were collected at three temperatures depending on lower PG grade temperature. These correlate nicely (figure 43) with the decay times, suggesting that the decay function, no matter its physical meaning, might be used to reconcile BBR and DSR data, since at the same temperatures, the Poisson decay function would essentially be the same. It may be possible to match DSR derived stiffness to BBR stiffness consistently to obtain m values. 36

43 BBR Temperature, Celsius y = 1.2x - 3 R² = Figure 43. Graph. Poisson s ratio decay times as a function of temperature. CONCLUSIONS The rigorous approach to interconversion of DSR complex modulus to BBR stiffness does not improve agreement between the two methods over approximate approaches unless a time dependant Poisson s ratio is employed in the conversion from shear relaxation to flexural relaxation. The physical explanation for the Poisson s ratio time dependent form is not clearly understood, but might be due to flow of a fluid component in the asphalt. Other explanations, such as softening in the BBR solvent bath may be the cause for the differences. A stretched exponential to describe the Prony series coefficients in the computation of the relaxation from DSR complex modulis removes the uniqueness issues with this computational method. Non-linear regression can be used to get relaxation directly from complex modulis master curves, or individual isotherms, without phase angle data. The computational method can be used to compute phase angles that compare well with those reported from the instrument. RECOMMENDATIONS The current embodiment of this calculation exists as a Visual Basic for Applications (VBA) program in Microsoft Excel. A more user friendly stand alone application for routine use would be better suited for use by the asphalt industry and research community. 37

44 ACKNOWLEDGMENTS The authors gratefully acknowledge the Federal Highway Administration, U.S. Department of Transportation, for financial support of this project under contract no. DTFH61-7D-5. DISCLAIMER This document is disseminated under the sponsorship of the Department of Transportation in the interest of information exchange. The United States Government assumes no liability for its contents or use thereof. The contents of this report reflect the views of Western Research Institute which is responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views of the policy of the Department of Transportation. REFERENCES Christensen, D. W., and D. A. Anderson, 1992, Interpretation of dynamic mechanical test data for paving grade asphalt cements. J. Assoc. Asphalt Paving. Technol., 61: Farrar, M., C. Sui, S. Salmans, and Q. Qin, 215, Technical white paper FP 8 Determining the Low Temperature Rheological Properties of Asphalt Binder Using a Dynamic Shear Rheometer (DSR). Prepared for Federal Highway Administration, Contract No. DTFH61-7-D-5, Fundamental Properties of Asphalts and Modified Asphalts, III, March 215. Ferry, J. D., 198, Viscoelastic Properties of Polymers, 3rd ed., John Wiley and Sons, Inc., New York. Jongepier, R., and B. Kuilman, 1969, Characteristics of the rheology of bitumens. Proc., Association of Asphalt Paving Technologists, 38: Lakes, R. S., 1992, The time-dependent Poisson's ratio of viscoelastic materials can increase or decrease. Cellular Polymers, 11: Marasteanu, M. O., and D. A. Anderson, 1999, Improved Model for Bitumen Rheological Characterization. Eurobitume Workshop on Performance Related Properties for Bituminous Binders, Luxembourg, Park, S. W., and Y. R. Kim, 1999, Interconversion between Relaxation Modulus and Creep Compliance for Viscoelastic Solids. J. Mat. Civ. Eng., 11 (1): Sui, C., M. J. Farrar, W. H. Tuminello, and T. F. Turner, 21, New Technique for Measuring Low-Temperature Properties of Asphalt Binders with Small Amounts of Material. Transportation Research Record, 2179:

45 Tschoegl, N. W., W. G. Knauss, and I. Emri, 22, Poisson s Ratio in Linear Viscoelasticity A Critical Review. Mechanics of Time-Dependent Materials, 6: