It is known that asphalt concrete in its linear viscoelastic range is thermorheologically .... Information on the materials and mixture design are documented in a ...
Time-Temperature Superposition Principle For Asphalt Concrete With Growing Damage in Tension State Ghassan R. Chehab Y. Richard Kim Richard A. Schapery Matthew W. Witczak and Ramon Bonaquist Abstract It is known that asphalt concrete in its linear viscoelastic range is thermorheologically simple. This paper presents the experimental/analytical research that demonstrates the validity of the principle of time-temperature superposition even with growing damage and viscoplastic straining by conducting constant crosshead rate tests on specimens that were pulled apart in tension until failure. Direct benefits and applications from this finding include reduction in testing program conditions, development of strength and corresponding strain mastercurves as a function of reduced time, a methodology for the prediction of stress-strain curves for any given crosshead strain test, and simplification of thermorheological analysis of pavement structures. Key Words: Time-Temperature Superposition, Tensile Damage, Linear Viscoelastic Introduction It has been shown in earlier research that asphalt concrete in linear viscoelastic state is a thermorheologically simple material (1,2,3). That is, given that the asphalt concrete is in its undamaged state, time-temperature superposition can be applied. As an application of that principle, data from complex modulus testing conducted within linear viscoelastic limits at different frequencies and temperatures should yield a single continuous mastercurve for dynamic modulus and phase angle as a function of frequency at a given reference temperature by horizontally shifting individual curves along the logarithmic frequency axis. However, for comprehensive material modeling, laboratory testing often extends to the damaged state where micro- and macro-cracks in the asphalt concrete matrix develop and grow. It has not yet been shown that the time-temperature superposition
principle holds when the damage varies with time. If verified, one of the most important implications would be the reduction of the required laboratory testing program for comprehensive material characterization of asphalt mixtures. The purpose of this study, which is a subtask of project NCHRP 9-19 (4), is to determine whether asphalt concrete with time-dependent damage can still be considered a thermorheologically simple material so as to simplify the complex testing program required in this research project; and more generally, to simplify characteristic and structural analysis of asphalt pavements. For that purpose, a series of tests were conducted that consisted of a linear viscoelastic complex modulus test followed by a constant crosshead rate test until failure in uniaxial tension mode at different temperatures and strain rates. The shift factors for the undamaged state were first determined by constructing the dynamic modulus mastercurve for a reference temperature; then, those shift factors were applied to the monotonic test data to successfully construct a continuous stress versus log reduced time mastercurve for a given strain level; in this paper, theory shows that in constant strain rate tests (for local or crosshead based strains), mastercurves may be constructed at constant strains. The validity of the superposition principle for damaging material was shown using on-specimen LVDT strains as well as crosshead-based strains. As a direct application of time-temperature superposition for damaging material, tensile strength and the corresponding peak strain mastercurves were constructed as a function of reduced strain rate. As another application, a methodology using the shift factor was presented for the prediction of stress-strain curves for any desired constant crosshead strain rate. Theoretical Background Asphalt concrete is a viscoelastic material that exhibits time and temperature dependency, and, except at low temperatures, viscoplastic non-recoverable strain. It is also known that when in its linear viscoelastic range, asphalt concrete is thermorheologically simple (TRS); that is, the effects of time or frequency and temperature can be expressed through one joint parameter. As such, the same material property values can be
obtained either at low temperatures and long times or at high test temperatures but short times. The viscoelastic material property as a function of time (or frequency), such as relaxation modulus, and creep compliance, at various temperatures can be shifted along the horizontal time axis (log scale) to form a single characteristic mastercurve of that property as a function of reduced time at a desired reference temperature. Thus, for the relaxation modulus at a certain time and temperature E (t, T) = E (ξ) t where ξ = aT t = time before shifting for a given temperature, T, ξ = reduced time at reference temperature T0 , and aT = shift factor for temperature T.
(1) (2)
The well-known WLF equation developed by William, Landal, and Ferry (5) estimates the shift factor as: t c (T − T0 ) (3) log aT = log T = 1 t T0 c 2 +T − T0 where c1 and c2 are constants dependent on the reference temperature To expressed in Kelvin. The WLF equation can only be applied to temperatures above the glass transition temperature, which is around –30C for asphalt. In this research, the WLF equation was not used; instead, the shift factors were determined experimentally through graphical shifting of dynamic modulus curves and were later refined through error minimization using fitting techniques. More details are provided in the subsequent sections. With time-dependent damage; i.e., when linear viscoelasticity conditions are violated, it has not yet been proven that asphalt concrete is thermorheologically simple (TRS). The derivation showing the validity of TRS for asphalt concrete in the damaging state is based on Dr. Schapery’s work on solid rocket propellant (6). The derivation is presented in Appendix B.
Sample Preparation and Testing Equipment Specimens used in this study were fabricated from 12.5-mm Maryland State Highway Administration Superpave mixtures (PG 64-22). Information on the materials and mixture design are documented in a separate report (9). Specimens used were 75 mm in diameter and 150 mm in height, cut and cored from 150 by-175 mm specimens compacted using the Superpave gyratory compactor, ServoPac. A specimen geometry study conducted by Chehab et al. (10) on specimens compacted using ServoPac showed that the cut and cored 75- by 150-mm specimen exhibits relatively uniform air void variation along the height and diameter of the specimen. Specimens were fabricated at the Arizona State University laboratory and shipped in sealed plastic bags to North Carolina State University where they were stored sealed inside a closed cabinet at room temperature to minimize aging. A gluing gig was used to ensure that the glued endplates and the specimen are aligned along a single vertical axis. The testing machine used was UTM-25, a 25-kN servohydraulic closed loop machine fabricated by IPC. Displacements were measured using spring-loaded LVDTs; two with 75-mm gage length and two with 100-mm gage length attached to the middle section of the specimen at equal distances from the ends. Using two different gage lengths enables the determination of the onset of localization since the opening of the major cracks that start to form in the asphalt matrix between the gage lengths would be numerically divided by two different gage lengths thus leading to two different strain values (11). Testing Program The testing program adopted in this research consisted of a series of complex modulus tests followed by a constant crosshead strain rate test in tension until failure of the specimen at several testing conditions. To check the applicability of time-temperature superposition with growing damage for a wide range of testing conditions, the number of testing conditions was increased and the number of test replicates was minimized, instead of conducting more test replicates over a narrow range of test conditions.
Complex Modulus Test The complex modulus test was conducted first to obtain the linear viscoelastic properties of the specimen being tested and to determine the time-temperature shift factors for the undamaged state by constructing the dynamic modulus mastercurve as a function of reduced time. Sinusoidal loading in tension and compression (with zero mean) sufficient to produce total strain amplitude of about 70 micro-strains was applied at six different frequencies. The 70 micro-strain limit was proposed in the NCHRP 9-19 Task C project conducted at ASU and was shown not to cause damage even at high temperatures and low frequencies (12,13). Targeting less than 70 micro-strains at high temperatures and low frequencies introduces a high noise to signal ratio and difficulty in controlling the stress waveform; while targeting higher strain levels may cause damage at high temperatures and low frequencies. Using tension and compression with mean stress of zero minimizes the accumulated strain at the end of cycling, which in turn minimizes the possibility of damage and needed rest period before the constant strain rate test is conducted. The testing commences with 10 Hz preconditioning cycles, followed by 20, 10, 3, 1, 0.3, and 0.1 Hz cycles. Enough cycles are applied to reach steady state conditions at each frequency with a 300-second rest period allowed between frequencies to relieve any accumulated strain. Tests were conducted at four temperatures –10, 5, 25 and 40C. The 70 micro-strain limit and 300-second rest were deemed not to cause any damage even at high temperatures as evident by the absence of significant accumulation of permanent strain at the end of the test. However, more study needs to be done to set a protocol for determining appropriate strain levels for particular temperatures and frequencies. Testing conditions for the complex modulus test are summarized in Table A1 in Appendix A. Constant Crosshead Rate Tests After allowing enough time for any accumulated strain from the complex modulus testing to be recovered, each specimen was pulled at a constant crosshead rate until failure. Testing temperatures were the same as those of the complex modulus test, while crosshead strain rates varied between 0.000019 to 0.07 per second.
Determination of Crosshead Strain Rates If the time-temperature superposition principle is applicable to asphalt concrete with growing damage, then the construction of a stress-log reduced time mastercurve for a given strain level should be feasible (Refer to Appendix B). To attempt that, common strain levels resulting from the various testing conditions need to exist so that the corresponding mastercurves can be constructed. However, due to the viscoelastic nature (rate and temperature dependency), if the same loading rates are used for all the testing temperatures then it may not be possible to obtain strain levels common to all conditions. For example, for a slow loading rate at 40C, the resulting strains will be much larger in value than the maximum strain resulting for the same strain rate at 5C; consequently, mastercurves could only be constructed for those small strain levels common to both temperatures and smaller than the failure strain at 5C. To overcome that problem and obtain strains of comparable magnitudes at different testing conditions, different ranges of strain rates had to be used for different temperatures. Assuming that time-temperature superposition holds with growing damage, those rates can be determined according to the following scheme. For a given stress-log reduced time crossplot corresponding to a particular strain level, two points corresponding to temperatures T1 and T2 overlap if they have the same stresses and same log reduced times (ξ’s). Thus for a given stress, log (ξ 1 ) = log (ξ 2 ). However, log ξ = log t a and t = ε k ′ , where ε is the strain and k ′ is the T
strain rate. Since the crossplot is for a constant strain level ε, log ( k ′ 1 x aT 1) = log (k ′ 2 x aT 2), or k1′ aT 2 = k ′2 aT 1 Thus, knowing the strain rates for 25C, Equation (4) can be used to determine strain rates at 5C and 40C that ensure overlap in the stress-log reduced time crossplot for a given strain level at the reference temperature of 25C. The lowest rate at 5C can be set to overlap with the second highest at 25C and the highest at 40C can be set to overlap with the second lowest rate at 25C. Similarly, the above equation can be used to determine strain rates at –10C that yield overlap with the 5C data in the crossplot. Since it is proposed
(4)
that time-temperature superposition is valid with growing damage in the analysis, shift factors from dynamic modulus may used to estimate the specimen strain rates. Table A2 in Appendix A presents the crosshead strain rates that resulted from this method and used in the testing program. Experimental Results and Analysis Complex Modulus Test Stress and strain data from the complex modulus test are fitted to the functional forms shown in Equations (5) and (6), respectively, using least squares method. Dynamic modulus and phase angle values are then calculated according to Equations (7) and (8) using fitted data from the last six cycles, where a steady state condition is achieved. σ = σ0 + σ1 cos( 2πft + φ1 )
(5)
ε = ε0 + ε1 t + ε2 cos( 2πft + φ2 )
(6)
E∗ =
σ1 ε2
φ = φ2 − φ1 where σ and ε = stress and strain respectively, t and f = time and frequency respectively, σ0 , σ1 , ε 0 , ε 1 , ε 2 , φ 1 , and φ 2 = regression constants, and |E*| and φ are dynamic modulus and phase angle respectively. The regression constants σ0 , ε 0 , ε 1 are needed in case there is an offset from the mean of zero, and in case there is a drift in strain as the test proceeds. Figures 1 and 2 are plots of dynamic modulus and phase angle, respectively, as functions of frequency for all the tests. Shift factors, aT , used to shift the dynamic modulus, |E*|, and phase angle, φ, versus frequency curves at –10, 5, and 40C along the frequency axis to form a continuous master curve at 25C, are defined as follows:
(7) (8)
100000
|E*| (MPa)
-10 C 5C 10000
25 C 40 C 1000 Mastercurve at 25 C after shift
100 0.001
0.1
10
1000
100000 10000000
Frequency (Hz)
Figure 1. |E*| as a function of frequency for different test temperatures
Phase angle (Deg)
100 40 C 25 C
10
Mastercurve at 25 C after shift
5C
-10 C
1 0.001
0.1
10
1000
100000
10000000
Frequency (Hz)
Figure 2. Phase angle as a function of frequency for different test temperatures Log (f R) = log (f x aT ) where f R = reduced frequency at the reference temperature (25C); f = frequency at a given temperature T before shifting; and
(9)
aT = shift factor for temperature T. Shift factors are determined by first assigning initial trial values and then using least squares technique to refine them through error minimization between actual |E*| values and those fitted using a sigmoidal function of the form shown in Equation (10): b
log E * = a + 1+ exp
1 (d + e log f R )
(10)
where f R is the reduced frequency, a, b, d, and e are regression coefficients, and |E*| is the dynamic modulus. If shift factors for temperatures other than those incorporated in the testing program are required for the same material, they can be interpolated from the log shift factor vs. temperature plot shown in Figure 3. After applying the shift factors, the resulting mastercurves for |E*| and φ shown in Figures 4 and 5 are obtained. As mentioned previously, the shift factors for the undamaged state will be used to determine the crosshead rates to be applied at the different temperatures and to check the validity of timetemperature superposition with growing damage. It is interesting to see from Figure 5 that phase angle increases with the decrease in reduced frequency which is explained by the fact that asphalt concrete exhibits more viscous behavior at lower frequencies. However, at reduced frequencies lower than 0.1 Hz, phase angle starts to decrease. This may be due to the fact that at high temperatures/slow frequencies, the asphalt concrete matrix weakens and thus individual aggregate properties start to exhibit a more significant effect on the overall asphalt concrete behavior. Since aggregates are elastic and thus exhibit no phase angle, the overall phase angle of the asphalt mix starts to drop as the reduced frequency reduces (1,3). However, non-crosslinked polymers without filler exhibit this behavior due to entanglement of the long chains; thus, there may be additional physical sources within the asphalt matrix contributing to this behavior (14).
6 5.3 2
y = 0.0008x - 0.164x + 3.5635 Log aT
3 2.72 0
-1.75
0
-3 -10
0
10
20
30
40
Temperature (C)
Figure 3. Shift factor vs. temperature from complex modulus tests
100000
|E*| (MPa)
-10 C
10000
25 C 5C
1000 40 C
100 0.001
0.1
10
1000
100000
10000000
Reduced Frequency (Hz)
Figure 4. Dynamic modulus mastercurve at 25C
100 40 C
(Deg)
25 C
5C
10
1 0.001
0.1
10
-10 C
1000
100000
10000000
Reduced Frequency (Hz)
Figure 5. Phase angle mastercurve at 25C Constant Crosshead Rate Tests Stress-Strain Curves A total of 20 tests were conducted at –10, 5, 25, and 40C. All three tests at –10C failed in a brittle mode while loading, while at 5C, only the two fastest rates failed in a brittle mode. Figures 6-9 are plots of stress-strain curves for the tests conducted at the four testing temperatures. Strains shown are those measured using 75-mm GL LVDTs mounted to the middle section of the specimen. As observed from Figure 6, the stress-strain curves at –10C are very similar. The peak stress and its corresponding peak strain for the three tests are very comparable in value although the strain rates are very different, the fastest rate being 700 times faster than the slowest. This suggests that the rate dependence (viscoelastic behavior) is minimal at such low temperature. Figure 7 is a plot of stress-strain curves at 5C. Tests at this temperature exhibit both failure modes, brittle and ductile. Tests conducted at a rate of 0.000056 exhibit a transitional failure mode; i.e., brittle fracture in the unloading stages (post-peak), where a single macro-crack develops abruptly after peak stress is reached and separates the specimen into two pieces. Figure 8 shows stress-strain plots of tests conducted at 25C. There is a close match between the curves of replicates at the same rate. It is worthy of noting that for tests with failure occurring outside the gage length of the LVDTs, the
strain measured using the LVDTs decreases because of strain recovery as the crack outside the LVDT grows. This can be observed for tests at strain rates of 0.0015, 0.0045, and 0.0135 per second. Comparing the stress-strain curves for all the temperatures, it is noted that strains corresponding to the peak stress are comparable for all strain rates when failure occurs in a ductile mode (Figures 8 and 9).
Stress (kPa)
4500
3000 Crosshead Strain Rates: 0.0135 0.0005
1500
0.000019
0 0
0.0001
0.0002
0.0003
0.0004
75 mm GL LVDT Strain
Figure 6. Stress-strain plot at –10C (1 specimen at each rate) Effect of Machine Compliance on Specimen Strains and Validity of Superposition Principle Because of machine compliance; i.e., deformation of certain machine components along the loading train under load, strains measured from the on-specimen and onend plates LVDTs are smaller than those measured using the crosshead LVDT. The difference increases at low temperatures and high strain rates due to the increased stiffness of the material being tested. Also attributed to the machine compliance is the nonconstant on-specimen strain rate, given that the crosshead strain rate remains constant throughout the test. For all tests, it was observed that the on-specimen LVDT strain rate followed a power law in time (up to a certain strain/time). Figure 10 illustrates this effect of machine compliance on specimen strain rates.
4500 Xh:0.008 Xh:0.0005
Stress (kPa)
Xh:0.000056-t2
3000 Xh:0.000056-t1
1500
Xh:0.00003-t2
Xh:0.00003-t1
Xh:0.000012
0 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
75 mm GL LVDT Strain
Figure 7. Stress-strain curves at 5C (Crosshead strain rate and replicate number indicated next to each curve)
3000 Crosshead strain rates: xh: 0.0135 0.0135-t2
Stress (kPa)
0.0135-t1-outside failure 0.0045-t2 xh: 0.0045
1500 xh: 0.0015
0.0045-t1-outside failure 0.0015-t1-outside failure 0.0005-t1 0.0005-t2
xh: 0.0005
0 0
0.01
0.02
0.03
0.04
75mm GL LVDT Strain
Figure 8. Stress-strain curves at 25C (2 replicates at each rate except for 0.0015)
Stress (kPa)
900
600 Crosshead Strain Rates: 0.07 0.0009 0.007
300
0 0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Strain from 75 mm GL LVDT
Figure 9. Stress-strain curves at 40C (1 replicate per strain rate) From the theoretical derivation, it is known that timetemperature superposition for damaged state can work, given that the specimen strain rate follows a pure power law, or more generally any strain that is defined by one time-scale parameter (Refer to Appendix B). However, for some tests a deviation from the power form occurs at the onset of strain localization if the top end plate displaces unevenly with respect to the horizontal plane. Experimentally, the onset of localization can be determined by comparing strains measured from LVDTs of different gage lengths. Strain localization is defined by the coalescence and development of microcracks to develop localized macrocracks. Due to the existence of the macro-crack, the method of obtaining average strain by dividing the displacement measured by the Figure 10. Difference between crosshead and on-specimen 75 mm GL LVDT strains for a monotonic test conducted at 25C and 0.0135 strains/sec 2500 Strain localization
Stress (kPa)
2000
75-mm 100-mm
1500
plates
Peak
1000
Post-peak
500
Pre-peak 0 0
0.005
0.01
Strain
0.015
0.02
Figure 11. Detection of strain localization for a strain rate of 0.00003 at 5C.
3000
LVDT 3766 (front side) LVDT 3767 (back side)
0.01
2500
stress
0.008
2000
0.006
1500 Can be used for superposition
0.004
Can not be used for superposition
0.002
1000
Stress (kPa)
Plate to Plate Strain
0.012
500
0 0
100
200
300
0 400
Time (sec)
Figure 12. Plate uneven displacement (just after 200 seconds) and effect on superposition for a test at a strain rate of 0.00003 at 5C Specimen LVDT Strain
0.1
can be used for superposition
0.01
0.001 cannot be used for superposition
0.0001
0.00001 1
10
100
1000
Time (sec)
LVDTs by the gage length is not valid anymore. Figure 11 shows the onset of strain localization for a test at 5C and strain rate of 0.00003. In this case, the onset of localization is the point where the stress-strain curves from the 75-mm GL, 100-mm GL, and plate to plate LVDTs start to deviate. After that deviation, data
from those tests can not be used for superposition applications if plate rotation occurs, and consequently the stain rate ceases to follow a pure power law. The corresponding uneven plate displacement (evident through the deviation of the two LVDT measurements, front and back) and effect on superposition is shown in Figure 12, while Figure 13 shows the resulting deviation of strain from the power functional form. It is worthy noting that the problem of uneven plate displacement was mainly present for tests run at 5C. This could be due to the high stiffness of the material at 5C compared to 25 and 40C. At –10C, specimens failed in a brittle mode without any localization prior to failure, and thus uneven plate displacement did not occur.
Figure 13. On-specimen LVDT strain deviation from pure power law (linear on log-log scales) and effect on superposition for the same test presented in Figures 11 and 12 0.020
3000
0.015 2000
Strain
Stress
0.010 1000 0.005 Specimen strain
0.000 0
0.5
1
0 1.5
Time (sec)
Time-Temperature with Growing Damage Asphalt concrete mixtures can be regarded as thermorheologically simple if, for a given strain level, a stress-log
Stress (kPa)
Crosshead strain
reduced time mastercurve can be constructed. Since the undamaged state is a special case of the damaged state, the shift factors determined earlier for constructing the dynamic modulus mastercurve should match those applied to construct the stress-log reduced time mastercurve. Moreover, the shift factors should only be a function of temperature and independent of strain level. The procedure begins by selecting several strain levels for which the mastercurves are to be constructed. The strain levels should be large enough to be representative of the damaged state of the mixture, as discussed in the previous section. The strain levels presented in this paper correspond to initial loading, pre-peak, peak and post-peak regions on the stress-strain curves. Even with very slow strain rates, the strain levels corresponding to the entire stress-strain curves at –10C were very small. Even at 5C, the fast strain rate tests yielded low strain levels. Thus, for high strain levels there were no data from those tests that could be included for the superposition of crossplots. For each selected strain level and testing temperature, the corresponding stress level and time from the tests conducted are obtained. The next step is to plot the stress-time crossplot for each strain level and temperature on one graph. Then, to construct the mastercurve at 25C for a given strain level, the stress-time crossplot for that strain level and for each temperature is shifted along the logarithmic time axis using the appropriate shift factor aT determined from the dynamic modulus testing. Figure 14 is a schematic illustrating the methodology to obtain the crossplots, which are shown in Figure 15: (a-l). As observed, the crossplots are smooth and continuous suggesting that superposition is valid with growing damage. For strain levels greater than 0.00019, there were no data from –10C tests due to early failure as discussed previously. For strains greater than 0.006, points from 5C tests start to deviate from the reduced crossplot due to plate rotation as discussed previously. For strains larger than 0.01, only data from tests conducted at 25C and 40C could be incorporated. For comparison of mastercurves, three strain levels corresponding to initial, pre-peak and post-peak regions on the stress-strain curves are plotted in Figure 16 on a single graph at a reference temperature of 25C.
Superposition of Stress-Time Crossplots Using Crosshead Strains In the previous section it was shown that by using the shift factors from the undamaged state stress-log reduced time mastercurves could be constructed for the desired LVDT strains. Thus, it can be stated that using LVDT strains, asphalt concrete is thermorheologically simple with growing damage. However, it still remains to be seen whether mastercurves can be constructed using crosshead strains. It was observed that strains measured from LVDTs connected to the specimen or the end plates of the specimen were smaller than the crosshead strains. This suggests that other parts of the loading train, such as load cell and various connections, are deforming under load (machine compliance). If that deformation is independent of loading rate and temperature and only depends on load, then it can be characterized as an elastic deformation and can be thus modeled as a spring. In such a scenario, the crossheadbased strains can be divided into two parts: viscoelastic and viscoplastic from the material on one hand and elastic from the material and other components of the loading train on the other hand. Shift factors characterizing the viscoelastic component will correspond only to the material and thus should be the same as those obtained using on-specimen LVDT strains. If the machine compliance can not be characterized as elastic deformation only but exhibits time and temperature dependency, then superposition using crosshead strains will not be valid since shift factors will correspond not only to the material but to the machine compliance as well. If deformation attributed to the machine compliance is linearly elastic, then that deformation divided by the load should be a constant for all testing conditions. This constant can be regarded as the stiffness of a spring that characterizes the machine. This was checked for several test conditions as shown in Figure 17. Plate to plate strain was subtracted from the crosshead strain and the resulting strain was divided by the stress. The result is approximately constant for all values of crosshead strain and rate until the peak in stress. After the peak, cracks in the specimen start to develop and plate to plate displacement can not be used for calculation of strain anymore. Therefore, it was concluded that, it
1. Select several strain levels ε k ’s (k: 1 to q) for analysis. 2. Find for every strain level ε k , the corresponding stress,σ k , and time, t k , at all T’s and for all rates, R’s.
3. For each strain level crossplot the stresses vs log times for each rate. 4. Repeat step 3 for all Temperatures. 5. Apply shift factors to construct a single σreduced log(t) mastercurve for each ε k .
ε
σ1,n
R1
Rn εq
ε
σ (t q,n ,σq,n)
ε
(t 1n ,σ1,n
R1
ε
σq,1 ε1
Ti
εq
Ti σ1,1 σq,n
Rn
t 1,n
t1,1 tq,n
σ
Ti
tq,
t
All Temperatures εq
(t q,1 ,σq,1)
ε
T1
(t 1,1,σ1,1)
T
Log t
T Log t
ε σ
q
Reference Temperature
ε 1
Reduced Log (t)
Figure 14. Schematic illustrating the methodology to obtain stress-reduced time crossplots
4200
a) 0.00015
Stress (kPa)
Stress (kPa)
4200 -10 C
2800
5C 25 C
1400
40 C d e f
5C
2800
25 C 40 C
1400 d ef
0
0 -6
-3
0
3
-6 -3 0 3 Log Reduced Time (sec)
Log Time (sec) 4200
4200 5C
c) 0.0006
Stress (kPa)
5C
Stress (kPa)
-10 C
b) 0.00015
25C
2800
40 C
1400 0
d) 0.0006
25 C
2800
40 C
1400 0
-6
-3 0 Log Time (sec)
3
-6 -3 0 3 Log Reduced Time (sec)
Figure 15. (a) and (b): Crossplots for 0.00015 LVDT strain before and after shift respectively; (c) and (d) for 0.0006 LVDT strain
4200
4200 f) 0.003
Stress (kPa)
Stress (kPa)
e) 0.003
2800 5C
1400
25
2800 5C
1400
25
40 C
40 C
0
0 -6
-3
0
3
-6
Log Time (sec)
-3
3
Log Reduced Time (sec)
4200
4200 g) 0.006
h) 0.006
Stress (kPa)
Stress (kPa)
0
2800 5C
1400
25
2800 5C
1400
25
40 C
40 C
0
0 -6
-3
0
Log Time (sec)
3
-6
-3
0
3
Log Reduced Time (sec)
Figure 15. Continued: (e) and (f): Crossplots for 0.003 LVDT strain before and after shift respectively; (g) and (h) for 0.006 LVDT strain
4200
4200
j) 0.01
Stress (kPa)
Stress (kPa)
i) 0.01
2800 25
1400
40 C
2800
40 C
0
0 -6
-3
0
-6
3
-3
0
3
Log Reduced Time (sec)
Log Time (sec) 4200
4200 k) 0.02
Stress (kPa)
Stress (kPa)
25 C
1400
2800 1400 25 C
l) 0.02
2800 1400 25
40 C
40 C
0
0 -6
-3
0
Log Time (sec)
3
-6
-3
0
3
Log Reduced Time (sec)
Figure 15. Continued: (i) and (j): Crossplots for 0.01 LVDT strain before and after shift respectively; (k) and (l) for 0.02 LVDT strain
22
4000
Strain level: Temperature 0.00015 : -10, 5, 25, and 40 C. 0.0004 : 5, 25, and 40 C. 0.02: 25 and 40 C.
Stress (kPa)
3000
2000
-10 C 5C 25 C 40 C
1000
0 -6
-4
-2
0
2
log Reduced Time (sec)
Figure 16. Stress-log reduced time mastercurves for selected strain levels
(Ram-Plate) Strain/Stress (1/kPa)
could be assumed in general that the machine compliance deformations are linearly elastic. This conclusion was later confirmed by conducting additional testing done on an aluminum specimen (11). Having checked that the machine compliance deformations are linearly elastic, shift factors determined from the crosshead strains should correspond to the specimen material and thus should match those obtained using the LVDT strains. Using those shift factors, the sadme procedure used before for constructing stress-log reduced time mastercurves for LVDT strains is repeated using crosshead-based strains. Mastercurves for selected strains are presented in Figure 18. 9.0E-07 8.0E-07 7.0E-07 6.0E-07 5.0E-07 :-10-0.000019
4.0E-07
5-0.00003-t2 5-0.000012
3.0E-07
25-0.0135-t3 40-0.07
2.0E-07
40-0.0078
1.0E-07 0
0.001
0.002
0.003
Ram Strain
0.004
0.005
23
Figure 17. Machine compliance evaluated at different temperatures and crosshead strain rates Applications Using Time-Temperature Superposition for Damaged State A direct benefit of the validity of time-temperature superposition with growing damage is the reduction in any testing program required for modeling purposes due to the consequent reduction in the testing conditions. However, the benefit is not limited to this but extends to other applications as well. Samples of possible applications are presented in this section. Superposition of Strength and Corresponding Strain One of the most important applications of the time-temperature superposition is the development of a mastercurve of strength as a function of reduced strain rate at a desired reference temperature 1200 Strain
Stress (kPa)
0.00015
800
0.0006 0.015
400
0 -7
-5
-3 -1 1 Log Reduced Time (sec)
3
Figure 18. Stress-log reduced time crossplots at 25C using crosshead strains (25C). Developing such a curve enables the determination of the strength of a material at any strain rate and temperature combination. The same holds true for the strain at the peak stress. The necessary theoretical derivation is documented in Appendix B. In addition, the strength mastercurve would be of great significance for thermal cracking applications, where strength could be compared to the stress buildup due to thermal contraction
24
to determine potential crack propagation. However, for thermal cracking applications, material properties, especially strength, need to be determined at very low temperatures. Since the lowest testing temperature investigated in this research thus far had been –10C, additional testing was conducted at –20 and –30C. Monotonic testing conditions and shift factors from complex modulus tests for these additional temperatures are presented in Table A3. For crosshead strains, which vary linearly with time, the strain rate is the slope of the specimen strain-time history. However, since LVDT strains do not vary linearly with time, the strain can be fit using the following power form up to the failure of the specimen; ε = k ′× t n
(11)
where the coefficients k ′ and n are regression constants. For subsequent analysis, the coefficient, k ′ , will be regarded as the specimen LVDT strain rate. Then, the reduced strain rates can be calculated as follows: For the crosshead strain in a linear form: ε = k′ × t ,
(12)
t ε = k ′ × aT × aT
,
(13)
ε = k ×ξ ,
(14)
where ε is strain, k ′ is the slope of strain vs. time at temperature T, aT is shift factor of temperature T, t is time, ξ is reduced time at reference temperature, and k is reduced strain rate at reference temperature. For the LVDT strain in a power form, as in the theory section, ε = k′ × t n , ε = k ′ × aT
n
(15) t × aT
n
,
(16)
25
ε = k × (ξ )n .
(17)
Therefore, for constant strain rate, the reduced strain rate is the slope multiplied by the shift factor; whereas, for strain in pure power form, the reduced strain rate is the coefficient multiplied by the shift factor raised to the power n. Figure 19 shows strength mastercurves as a function of reduced strain rates at 25C obtained using crosshead strain rates and LVDT specimen strain rates. The mastercurve is divided into three regions as described in Table 1. TABLE 1 FAILURE MODES Region
Temperatur e (C)
Loading Rate
Failure Mode
A
40, 25, 5
All at 40C, 25C Slow at 5C
Ductile
B
5
Intermediate
Ductile, brittle failure during unloading
C
5, -10, -20, -30
Fast at 5C, All at -10, -20, -30C
Brittle during loading
The strength mastercurve shown in Figure 19 indicates the increase in strength as the strain rate increases; i.e., the rate dependence of tensile strength. However, for a certain reduced strain rate range (1 to 1000 per seconds), the failure pattern changes from ductile to brittle and the rate dependence of the strength becomes insignificant. As the reduced strain rate increases further (greater than 10,000 per second), the strength starts to decrease. It is suggested that this is because at very low temperatures, the difference in thermal contraction coefficients of asphalt and aggregates (15) leads to local thermal stress-induced damage, consequently leading to the weakening of the asphaltaggregate matrix. As a result, less load is required to fail the specimen. However, this damage may significantly depend on thermal history, which would cause strength to depart from thermorheologically simple behavior. There are not enough data here to critically check this behavior at high reduced rates.
26
Figures 20 and 21 are plots of the mastercurve of strain at peak stress with respect to LVDT and crosshead-based strains respectively. Similarly, these mastercurves are divided into three regions according to the specimen failure mode. Thus, once several constant crosshead strain tests at different conditions are conducted, strength and corresponding strain mastercurves as a function of reduced strain rate can be constructed. Those mastercurves are instrumental in determining the strength and corresponding strain at any other given temperature and strain rate condition.
5000
Ductile Failure A
Brittle Failure During Loading C
Peak Stress (kPa)
4000
3000
2000 B
1000
Brittle Failure During Unloading Using crosshead strain rate Using LVDT strain rate
0 1.0E-05
1.0E-03
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+07
Reduced Strain Rate
Figure 19. Strength mastercurve as a function of reduced strain rate (crosshead and LVDT) at 25C Prediction of Stress-Strain Curves for Constant Crosshead Rate Tests Having constructed the stress-reduced time crossplots for various strain levels, it is possible to predict the stresses for any given constant crosshead rate test at other conditions (temperatures and strain rates). Those stresses can be predicted using either the crosshead strain rate or the specimen strain rate as long as it
27
0.014 Ductile Failure
LVDT Strain at Peak
0.012
Brittle Failure During Loading
A
C
0.01 Brittle Failure During B Unloading
0.008 0.006
-10 C 5C
0.004
25 C 40 C
0.002 0 0.00001
0.001
0.1
10
1000
100000
Reduced LVDT Strain Rate
Figure 20. Mastercurve of specimen strain at peak stress as a function of reduced LVDT strain rate at 25C 0.014 Ductile Failure
Crosshead Strain at Peak
0.012
Brittle Failure During Loading
A
C -30 C
0.01 B
0.008
-20 C
Brittle Failure During Unloading
-10 C 5C 25 C
0.006
40 C
0.004 0.002 0 0.00001
0.001
0.1
10
1000
100000 10000000
Reduced Crosshead Strain Rate
Figure 21. Mastercurve of crosshead strain at peak stress as a function of reduced crosshead strain at 25C
28
follows a pure power law. However, stresses can only be predicted for strain levels at which stress-log reduced time crossplots exist. The following procedure is used to predict the stress-strain curve. First, the procedure explained in previous sections is conducted to construct stress-log reduced time mastercurves after running several constant crosshead rate tests. Then for a particular test for which the prediction of stress is to be done, a given strain is selected and the corresponding time is calculated from the strain rate. Based on the temperature at which the prediction is needed, the time is divided by the appropriate shift factor to yield a reduced time at the reference temperature; e.g., 25C. For that reduced time and using the stress-log reduced time crossplot corresponding to the selected strain, the stress is determined. For accuracy, each crossplot from which the stress is to be determined is fitted to a polynomial function. This procedure is repeated for all strain levels for which the crossplots exist. After determining the stresses, stress-strain curves can be constructed. Figure 22 is a schematic illustrating the prediction methodology. In the case where the reduced strain rate yields brittle fracture, the prediction is carried out for strains less or equal to the maximum (ultimate) strain for that reduced strain rate. That maximum strain is obtained from the mastercurve of strain at peak stress as a function of reduced strain rate (Figure 20 or 21). The prediction procedure was applied to selected tests that were actually conducted in the testing program. In that way, predicted stress-strain curves can be compared to the actual. The on-specimen LVDT strain rate fitted to the pure power law was used to determine the time. Crossplots used were those constructed earlier corresponding to the specimen LVDT strains. Figure 23 shows the predicted and actual stress-strain curves for a test run at a crosshead strain rate of 0.0135 at 25C. There is an excellent match between the actual and predicted curves. Because the largest strain for which the crossplot was constructed was 0.02, stresses for strains beyond that value can not be predicted. Similarly, Figures 24 and 25 show actual and predicted curves for a test at 5C with a rate of 0.008 strains/sec and –10C with a rate of 0.0005. Since the reduced strain rates for both of these tests are predicted to yield failure in a brittle mode (Figure 19), the prediction of their stress-strain curves needs to be done for strains less or equal to those corresponding to their strength. Figure
29
26 shows both the actual and predicted stress-strain curves at 40C for a crosshead strain rate of 0.07. The match is good in the prepeak and post-peak regions; however, there is an over prediction of stress at peak, probably due to specimen to specimen variation. In general, it can be concluded that the stress prediction methodology seems to be promising. The errors in prediction are of the same order of magnitude as the difference in responses attributed to the specimen to specimen variability for the same testing condition. It should be emphasized that the predictions made here came from the use of mastercurves of constant strain rate data and the linear viscoelastic shift factor. A more detailed material model is needed to predict stresses for other types of strain histories. Development of such a model is the focus of current research. Conclusions and Recommendations It is concluded that asphalt concrete in tension is a thermorheologically simple material even with growing damage. Additional conclusions resulted from this study: • Stress-log reduced time mastercurves from constant crosshead-strain rate tests in tension until failure on cylindrical specimens were constructed using LVDT strains as well as crosshead strains. • Shift factors used for constructing stress-reduced time mastercurves (damaged state) were the same as those used in constructing the dynamic modulus mastercurve (undamaged linear-viscoelastic state). • Building on the validity of time-temperature superposition in the damaged state, by conducting several constant crosshead strain tests, strength and corresponding strain mastercurves were constructed as a function of strain rate. Those mastercurves can be used to predict the strength and corresponding strain at any other temperature and strain rate conditions. • Using the stress-reduced time mastercurves, which can be constructed by conducting several constant crosshead strain rates at different conditions (temperatures/strain rates), the stress-strain history of a crosshead-strain rate test
30
Stress-log red. Time at Reference
ε
ε2
σ
Strain Rate at Which Stresses are to be
εn
ε&
For ε 1 , ε 2 ,..ε n
εn ε1
Calculate t1 , t2 , .. tn
ε2 ε1
Log t R σ
t1
ε2
t2
tn
t
σ2 σ1 σn
εn
For t1R, t2R, .. tnR
ε1
Find σ1 , σ2 , .. σn t1R
t2R
tnR
Log t R
For the temperature of the test, use aT to determine the reduced times, t1R, t2R, .. tnR at reference temperature
Figure 22. Methodology for Predicting stresses for constant crosshead rate tests using stress-time crosspl ots
31
Stress (kPa)
3000
2000 Predicted Actual
1000
0 0
0.01
0.02 Strain
0.03
0.04
Figure 23. Predicted and actual stress-strain curves for a crosshead strain rate of 0.0135 at 25C
Stress (kPa)
4500
3000 Predicted Actual
1500
0 0
0.0002
0.0004
0.0006
0.0008
0.001
Strain
Figure 24. Actual and predicted stress-strain curves for a strain rate of 0.008 per sec at 5C
1200
32
Stress (kPa)
1000 800
Predicted Actual
600 400 5000 200
Stress (kPa)
0 0
0.01
0.02
Strain
0.03
0.04
2500 Predicted Actual
0 0
0.0001 0.0002 0.0003 0.0004 0.0005 0.0006
Strain
Figure 25. Actual and predicted stress-strain curves for a strain rate of 0.0005 per sec at –10C Figure 26. Actual and predicted stress-strain curves for a strain rate of 0.07 per sec at 40C •
conducted at other conditions and temperatures can be predicted.
•
Development of thermo-mechanical constitutive equations for strain histories other than constant rate is simplified considerably as a result of the thermorheologically simple behavior demonstrated in this paper. This work is currently underway; in which, Equations (B3) and (B5) provide its basis. Acknowledgments
The National Science Foundation (NSF) and NCHRP funded this work. It is part of the NCHRP 9-19 project sponsored by the
33
American Association of State Highway and Transportation Officials, in cooperation with the Federal Highway Administration, and was conducted in the National Cooperative Highway Research Program, which is administered by the Transportation Research Board of the National Research Council. The authors gratefully acknowledge this support. Disclaimer The opinions and conclusions expressed or implied in the report are those of the research agency. They are not necessarily those of the NSF, Transportation Research Board, the National Research Council, the Federal Highway Administration, the American Association of State Highway and Transportation Officials, or the individual states participating in the National Cooperative Highway Research Program. References 1. Y.R. Kim and Y.C. Lee, “Interrelationships Among Stifnesses of Asphalt Aggregate Mixtures”, Journal of the Association of Asphalt Paving Technologists, Vol. 64, 575609, (1995). 2. Y.R. Kim, B.O. Hibbs, Y.C. Lee, and E.H. Inge, “Asphalt Paving Material Properties Affected by Temperature”, Technical Report Submitted to North Carolina Department of Transportation under Contract No. 23241-93-6, March 1994. 3. J.L. Goodrich, “Asphaltic Binder Rheology, Asphalt Concrete Rheology and Asphalt Concrete Mix Properties”, Journal of the Association of Asphalt Paving Technologists, Vol. 60, (1991). 4. Superpave Models Team “Advanced Material Characterization Models Framework and Laboratory Test Plan-Final Report”, (September 1999), Department of Civil engineering, Arizona State University. 5. M.L. Williams, R.F. Landal, and J.D. Ferry, “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass Forming Liquid”,
34
The Journal Of The American Chemical Society, Vol. 77 (1955). 6. S.W. Park, and R.A. Schapery, “A Viscoelastic Constitutive Model for Particulate Composites with Growing Damage”, International Journal of Solids and Structures, Vol. 34, 931-947, (March 1997). 7. R. A. Schapery, “Fracture Mechanics of Solid Propellants”, Fracture Mechanics, Edited by N.Perrone, H. Liebowity, D. Mulville, W. Pilkey; University Press of Virginia, Charlottesville, 387-398, (1978). 8. R. A. Schapery, “Nonlinear Viscoelastic and Viscoplastic Constitutive Equations with Growing Damage”, Int. Journal of Fracture, Vol. 97, 33-66, (1999). 9. Superpave Models Team, “Inter-laboratory Testing Manual”, Superpave Support and Performance Models Management, (October 1998), Department of Civil Engineering, University of Maryland, College Park, MD. 10. G. Chehab, E. O’Quinn, and Y.R. Kim, "Specimen Geometry Study for Direct Tension Test Based on Mechanical Tests and Air Void Variation in Asphalt Concrete Specimens Compacted by Superpave Gyratory Compactor", Transportation Research Record 1723, 125132, (2000), National Research Council, Washington, D.C. 11. J. S. Daniel, G. Chehab, Y.R. Kim, “Machine and Instrumentation Issues Affecting the Measurement of Fundamental Asphalt Concrete Properties in the Laboratory”, Paper Submitted For Presentation And Publication at the 2002 Annual Meeting Of The Transportation Research Board. 12. K. Kaloush, “Simple Performance Test for Permanent Deformation of Asphalt Mixtures”, Ph.D. Dissertation, Arizona State University, Tempe, AZ 2001. 13. T.K. Pellinen, “Investigation of the Use of Dynamic Modulus As An Indicator of Hot-Mix Asphalt Performance”, Ph.D. Dissertation, Arizona State University, Tempe, AZ 2001. 14. J. Ferry, “Viscoelastic Properties of Polymers” Third Edition, New York: Wiley, 1980. 15. SHRP-A-357, “Development and Validation of Performance Prediction Models and Specifications for
35
Asphalt Binders and Paving Mixes”, National Research Council, 1993, Washington, D.C.
36
Appendix A TABLE A1 COMPLEX MODULUS TEST PARAMETERS Load (kN)
Frequency Cycles
(Hz)
Rest Period (sec)
Temperature (C) -10
5
25
40
100
+/-2.2
+/-1.5
+/-0.55
+/-0.15
300
20
200
+/-4.7
+/-3.25
+/-1.2
+/-0.35
300
10
100
+/-4.5
+/-3.0
+/-1.0
+/-0.28
300
3
100
+/-4.3
+/-2.75
+/-0.7
+/-0.15
300
1
60
+/-4.05
+/-2.35
+/-0.45
+/-0.13
300
0.3
30
+/-3.8
+/-2.05
+/-0.3
+/-0.11
300
0.1
15
+/-3.45
+/-1.55
+/-0.25
+/-0.1
300
Precond.
10
TABLE A2 CROSSHEAD STRAIN RATES (REPLICATES IN PARENTHESES) -10 0.000019 0.0005 0.0135
Temperature (C) 5 25 0.000012 0.0005 (2) 0.00003 (2) 0.0015 0.000056 (2) 0.0045 (2) 0.0005 0.0135 (2) 0.008
40 0.0009 0.0078 0.07
TABLE A3. TESTING CONDITIONS AT –20 AND –30C Temperature (C)
Shift factor
-30
108.9
-20
107.0
Crosshead rate (strains/sec) 0.007 0.01 0.005 0.01 0.2
Strength (kPa) 1995 2100 2670 2969 2770
37
Appendix B Solid propellant, which consists of a rubber matrix that is highly-filled with hard particles, has been found to be thermorheologically simple (TRS) not only when it is linearly viscoelastic, but also when it is strongly nonlinear due to microcracking (6). The shift factor is independent of the amount of damage. Experimental studies of macro-crack growth in solid propellant at several temperatures have demonstrated that the shift factor for this crack growth is identical to that for linear viscoelastic behavior (7). The physical basis for this behavior is that the time and temperature dependence of deformation and all crack growth in the rubber and at interfaces originates from the rubber, which is itself TRS. With this motivation, and the fact that TRS behavior of asphalt concrete exists in its linear range, experiments were conducted to determine if TRS extends to behavior with micro-cracking and viscoplasticity. By examining the basic structure of the underlying constitutive equations, one can identify a convenient test history and data reduction method for determining if asphalt concrete is TRS. Both deformation and failure behavior are addressed in this section. Structure of the Constitutive Equations: Using abbreviated notation, the total strain ε (including viscoplastic strain) and stress σ tensors are related as follows (8) for non-aging materials: ε = - ∂G/∂σ
(B1)
where G=G(σ, S, T) is the Gibbs free energy, T is temperature and S represents the set of all thermodynamic state variables that account for local effects on all scales (molecular motions, microdeformations, micro-cracking and macro-cracking (if any)). The set of evolution equations for S is: dS = f(σ, S, T ) dt where f comes from the intrinsic viscous behavior of asphalt concrete. There are as many equations in Equation (B2) as S
(B2)
38
variables. In principle, therefore, Equation (B2) may be solved to express S as a function of σ and T histories. This result may then be substituted into Equation (B1) to provide strain in terms of stress and temperature histories. A TRS material is one in which all effects of T in Equation (B2) appear as a common factor, which we denote as 1/ aT . In this case, Equation (B2) reduces to dS = F (σ, S ) dξ
(B3)
where dξ = dt / aT for constant or transient temperature; in the latter case, ξ = ∫ t0 dt / aT
(B4)
while Equation (2) applies in the former case. The effect of temperature in Equation (B2) is assumed to produce only thermal expansion strain, ε T say. Thus, we may write ε σ ≡ ε - ε T = - ∂Gσ / ∂ σ
(B5)
where ε σ is the “strain due to stress” and Gσ=Gσ(σ, S). Also, S comes from the solution of Equation (B3). It is important to observe that all time-dependent behavior for non-aging TRS materials comes from Equation (B3), and only reduced time, not physical time, appears. Thus, physical time enters mechanical behavior of non-aging asphalt concrete only through external inputs if Equations (B3) and (B5) are applicable. Application to uniaxial loading: A convenient series of tests that may be used to check for TRS behavior consists of a series of constant crosshead rates to failure at a series of constant temperatures using cylindrical bars; they should be sufficiently long that the stress state is essentially uniaxial. With such tests, the theory is needed to determine how to check for TRS behavior from analysis of the stress-strain data. In practice, the overall specimen strain, or local strain using for example LVDTs, may not increase at a constant rate even if a constant crosshead rate is specified, as discussed elsewhere in this
39
paper. A power law in time may describe much better the local or global axial strain (due to stress), εσ = k ′t n
(B6)
where n is assumed constant, but k′ is a variable because a series of different crosshead rates are imposed. Rewriting this strain input in terms of reduced time, ε σ = kξ n
(B7)
where k = k′anT
(B8)
If n=1, then k is the “reduced strain rate”; although k is not really reduced strain rate when n ≠ 1, we shall still use this name for ease of discussion. Next, specializing Equations (B3) and (B5) to uniaxial stressstrain behavior and inverting Equation (B5), The following is obtained σ = g (εσ , S )
(B9)
Also, dS /dξ = h( εσ , S )
(B10)
after using Equation (B9) in Equation (B3). Given Equation (B7) and solving (in principle) Equation (B10) for S, we obtain stress in the form σ = gˆ ( ξ, k,n)
(B11)
in which both ξ and k are “reduced” variables. In order to analyze data for TRS behavior it is helpful to eliminate k in favor of ε σ using Equation (B7). Thus, Equation (B11) may be written as σ = ˆf ( ε σ , ξ,n)
(B12)
40
Stress-strain data Equation (B12) provides the basis for checking stress-strain data for TRS behavior. It shows that if the material is TRS and if the strain history is that in Equation (B6), then plots of σ (or log σ) versus log t at any given constant ε σ (and for a set of temperatures) may be shifted by amounts of log aT to form a master curve. (These constant-strain curves are constructed by making cross-plots of the original stress-strain data taken at constant k′. In other words, for each ε σ the dependence on time and temperature is the same as for a linear viscoelastic material; in the latter case, it is helpful to shift curves of σ/ε σ because this quantity is independent of ε σ. It should be noted that we have not assumed the material non-linearity for all strain histories is a function of only current strain; it is the special history in Equation (B6) that produces the behavior in Equation (B12). Strength data In principle, Equation (B12) (or the much more general version, Equations (B3) and (B5), apply on a local scale even with strain localization if the strains do not change significantly on a scale comparable to a suitably defined average aggregate particle dimension; in the case of Equation (B12), the stress state must be essentially uniaxial. Alternatively, these equations may be used on a global scale, even with strain localization, because S can be used, in principle, to account for localization; thus, in Equation (B12) σ may be axial force divided by initial cross-sectional area while ε σ is crosshead-based stain for both pre-peak and full post-peak behavior. Let us first assume the failure behavior is “ductile”, in that specimens do not break until after a maximum stress is reached. For each rate k′ and temperature T, the maximum stress is given by the condition dσ/dt=0. In terms of Equation (B11) this implies that dgˆ / dξ = 0
(B13)
Thus, solving Equation (B13) the reduced time at the maximum in σ, say σm, is ξm = function ( k , n) and the corresponding strain, Equation (B7), is
(B14)
41
εm = kξmn = function(k , n)
(B15)
Similarly, from Equation (B11), the “ductile” strength is σm = function( k , n)
(B16)
Equations (B15) and (B16) show that, for the TRS material model employed, the strain at maximum stress and maximum stress may be expressed as master curves in terms of reduced strain rate, Equation (B8). If a specimen breaks before a maximum, dσ/dt=0, is reached, then the failure is usually called “brittle.” In this case, we may interpret failure to be the result of at least one crack that propagates the full specimen width. Taking one of the S variables as crack length, say Sl then when Sl reaches a critical size, Sc, brittle failure occurs. The latter corresponds to the specimen width or, more commonly, a size beyond which crack growth is dynamic. Denote the stress and strain at this time ξ c by σc and ε c , respectively. Solution of Equation (B10) for Sl, given Equation (B7), gives for ξ ≤ ξc , S1 = function(ξ, k , n)
(B17)
This equation came from a quasi-static analysis; unstable crack growth corresponds to predicting S1 → ∞ at some finite time ξ c; if the growth is not dynamic, then Sl ~ specimen width at ξ=ξ c. In either case, Equation (B17) implies ξ c =function (k, n) at the time of brittle failure. In turn, Equations (B7) and (B11) imply εc =function (k, n)
(B18)
σc =function (k, n)
(B19)
Thus, master curves in reduced strain rate may be developed, just as for ductile failure. However, the functional form of these curves will be different because they reflect different physical processes. It should be emphasized that the theoretical basis and method for analyzing experimental data to test for TRS with growing
42
damage, as described in the appendix, are believed new. Both deformation and failure behavior are addressed.
“Time-Temperature Superposition Principle for Asphalt Concrete Mixtures with Growing Damage in Tension State” Ghassan Chehab, Y. Richard Kim, R.A. Schapery, Matthew Witczak and Ray Bonaquist Discussions DR. GEOFFREY ROWE: I think this is a really interesting piece of work and I compliment you on doing it. The power of this technique is really quite large, that you were able to construct these ultimate property master curves. You have defined the stiffness master curves. You have defined the ultimate property master curves and now we can start to use approaches as suggested by Richard Schapery and start to look at fatigue computations and the like. Richard Kim has a very complete fundamental picture in terms of the way the mixture is behaving. I know in your paper you probably did not allude to that quite so much, but I think there is a great power in the techniques that you are developing here for taking these ultimate properties and the mastercurves and the viscoelastic properties to damage prediction in the field. Just for your information, we are looking at almost an identical sort of analysis approach in a little study that we have started with some binder producers. We are looking at trying to develop the same sort of procedures using the direct tension test to develop these ultimate property mastercurves. That is something we hope to be sharing with you in future years. It is a great piece of work. I would be interested in your comments on taking this forward into other types of damage prediction like fatigue. Once again I compliment you on the work.
DR. GHASSAN CHEHAB: Thank you very much. The immediate application that I am going to use this for basically is developing a model for the characterization of fatigue life. That will include incorporation of viscoplastic behavior, because as we know, asphalt content response is not only viscoelastic, it does have a viscoplastic component to it. I carried out the timetemperature superposition study because I knew beforehand that it was going to ease the modeling of viscoelastic and viscoplastic responses a great deal if I prove that it wo rks. DR. ROWE: Let me just ask you a follow-up on that. Do you think, in terms of the shape of the ultimate properties, you see the strain at break and the stress at break, whether we will be able to use some sort of functional form or shape description to describe the shape of that curve?
DR. CHEHAB: The mastercurve? DR. ROWE: Yes, in terms of the stress at break and the strain at break mastercurve that you are developing. I am thinking, looking at the work on polystyrene, the shape of your curve seems to be very similar to the ultimate properties you see with some of these other polymers. To interpolate in the model and to apply this rapidly within a model situation, what we are going to need eventually is some sort of functional form that will describe the ultimate properties at a break situation. Pretty much the same way we can use functional forms for describing mastercurves of stiffness, like Terhi Pellinen presented earlier.
DR. CHEHAB: I really have not fitted a model to it. I just wanted to show that we can build a continuous single mastercurve that is smooth and continuous and show that the data is shiftable. I think, based on what the shape looks like, we can fit a functional form to it. But I have not really tried it. DR. ROWE: Nice piece of work. I look forward to seeing what comes in the future. DR. CHEHAB: Thank you very much.
PROFESSOR EYAD MASAD: I like this work so much that I have to ask this question. First of all, congratulations, a very nice piece of work. The question is, when you shift at different strain rates, do you obtain one function to represent the shift factors as a function of temperature and can you fit it to one of these known equations for shift functions like the WLF equation? DR. CHEHAB: Yes. The WLF was the starting point for me to see what the shift factor values would be. But ultimately what I did to get the shift factors is basically use a sigmoidal function to fit the mastercurve. By minimizing the error between actual and fitted data the shift factors can be determined. PROF. MASAD: And are the shift factor versus temperature follows the same function at different strain rates? Dr. CHEHAB: At different strains? PROF. MASAD: Yes, strain rates. PROF. CHEHAB: The dynamic modulus gives you different strain. I do not know exactly what you are alluding to, different strain levels? Do you mean in my test if I do a dynamic modulus? PROF. MASAD: You have your shifting, the way you did your shifting as I understand it. DR. CHEHAB: Is this for the damage part or the linear viscoelastic part? PROF. MASAD:For the linear viscoelastic and the damage. Do you get the same fitting function for the shift factors versus temperature relationship DR. CHEHAB: Oh yes, it’s a single relationship. A single universal relationship exists for all strains. I used the strains up to failure and the same shift factor versus temperature relationship determined from linear viscoelasticity holds true. The same shift factors that are applied to strains just before breaking (the specimen breaks) are also used for strains just when the test starts. The strain level independence seems to be the power of it. This is the proof of it. This is the proof that time temperature supposition works. If it doesn’t work, then the shift factors are going to change with strain level. But the fact that time-temperature works regardless of the state that the mixture is in, that shows that time-temperature works for the asphalt mixture in tension regardless of strain level. PROF. MASAD: Based on the previous work of Schapery and Richard Kim, you followed the idea of establishing the master curve, and using the small strains to identify the linear range in the master curve This seems to be different approach than the previous work of Dr. Kim. Now you can predict the ultimate strength without having to calculate the pseudo stiffness and quantify the difference between the linear viscoelastic behavior versus the actual stress. DR. CHEHAB: I think the next presentation, by Jo Daniel, is going to be more focused on that approach. We did not use the pseudo stiffness here.
PROF. MASAD: Do you plan to compare the model’s results with the master curve? DR. CHEHAB: Yes. We want to combine both. Ultimately both have to match together. I told you we are working on the viscoplastic modeling and for viscoplastic modeling we use the viscoelastic model of Dr. Schapery and Dr. Kim. That model depends on pseudo stiffness, and it will be presented in the next presentation. We determine viscoelastic response and then we use it in a different model to determine viscoplastic response. We use time-temperature superposition and we integrate both together to get a complete model. MR. ASLAM AL-OMARI: Very good job, Ghassan. A nice presentation. I have a few comments or questions. First, it is about the phase angle (Μ). Did you look at the phase angle at different frequencies? DR. CHEHAB: Yes. MR. AL-OMARI: Do you have any idea how it behaves? Is it increasing or decreasing with frequency? DR. CHEHAB: It increases with decrease in reduced frequency and then when the effect of aggregate starts to be more dominant, the phase angle starts to decrease. So that is why one needs to use EN instead of E*, because the latter better conveys the real behavior. MR. AL-OMARI: Another thing. I saw that you used the stresses just before the specimen failed. Dr. CHEHAB: Stresses just before? MR. AL-OMARI: Yes. DR. CHEHAB: Yes. I can use any stress. It is just an example. MR. AL-OMARI: Those where point A, B, C, D? DR. CHEHAB: Yes. Let me go back to that slide. You are not talking about that you are talking about this. Actually if you look at this strain that stress is beyond peak. MR. AL-OMARI: I know that. I mean even after failure. DR. CHEHAB: No it is not after failure. Before complete failure but after the peak. That is the point where localization starts and coalesced microcracks develop and progress. MR. AL-OMARI: Do you think that the material there is behaving linear viscoelastic? DR. CHEHAB: No. Linear viscoelastic behavior stops before that. You see where the strain level 0.00015 is for the first test? It stops even below that. That is where linear viscoelastic ends. PROFESSOR Y. RICHARD KIM: I want to give a response to Prof. Masad’s question on my work on
viscoelasticity and continuum damaging mechanics. That model works well when there is no significant viscoplasticity. What is being done in the NCHRP9-19 project is to develop the model that handles the asphalt mixture’s behavior not only at low and intermediate temperatures but also at high temperatures. To develop such a model we need to include both the viscoelastic and viscoplastic models. So, the motivation behind this work is to reduce the number of tests required to develop the viscoelastoplastic model by applying the time-temperature superposition principle. And as Ghassan mentioned, Jo Daniel will present what happens if we limit ourselves within low to intermediate temperatures where viscoplasticity is not very significant. The same type of work has been done on compression under the direction of Dr. Chuck Schwartz at the University of Maryland. His work was presented at this year’s TRB meeting. It yielded basically the same conclusion as ours. The time-temperature superposition principle holds valid even all the way to the failure. This is a very important governing principle that we rely on in developing the advanced material characterization program under the NCHRP 9-19 study.
MR. LAITH TASHMAN: Congratulations, very nice work. Very quick question about this. If I want to use your work in a model where one of my inputs is the strain rate, does that mean I am going to have to do the testing again at different strain rates or can you predict the behavior of the material at any temperature and one strain rate? DR. CHEHAB: Excellent question. If you are using the same mixture, no. I have developed these cross-plots for a particular mixture. After doing that I can predict the stresses for any constant cross rate test at any temperature, that is any reduced rate. If you at Washington State want to follow that now, you will probably have to conduct two constant rate tests at three or four temperatures, then establish the stress versus log-reduced time master curves. Then, you can predict stresses for all other strain rates and at all other temperatures. So, at first you do have to run a couple of tests, like seven or eight to be just confident. Then, you can use the results to predict responses at any other condition. RICHARD DAVIS (Submitted Discussion): The reason that this was so interesting is that it is about the determination of the tensile strength of asphalt pavements. The primary purpose of a pavement historically has been the ability to resist the stresses applied by traffic without excessive distortion and to maintain a smooth surface over a long period of time. An advantage of asphalt pavements is the ability to reduce the required thickness of pavement by developing tensile strength under moving loads. Dry aggregate having the same gradation as the aggregate in a pavement would have air voids but very little tensile strength. It is evident that the tensile strength is primarily due to the properties of the asphalt. Establishing the tensile strength of asphalt itself is far from simple, but when aggregate is introduced to make a mixture, it becomes much more complicated. The uniformity and level of air voids is obviously very important to tensile strength since the tensile strength of air is very low. The writer’s approach to asphalt pavement design can be defined simply as “avoid failure”. Three important sources of asphalt pavement failure were defined many years ago as being thermal cracking, rutting, and inadequate structural strength. The first two were solved to the writer’s satisfaction years ago (1,2) but the third is the object of continued research. The core problem in this research is the determination of the tensile strength of the pavement under critical conditions. The questions that prompted this discussion may have been covered in the meeting but are being raised in case they were not. These questions are: • What are the estimates of average and upper and lower limits of air voids in the specimens tested in this paper?
•
What is the nature of the error minimization program employed in this study? The preprint mentions the importance of carefully controlling air voids but the writer found no actual values given. The preprint does mention that the specimens were fabricated from Maryland State Highway Superpave mixtures. One might infer from this that the carefully controlled air voids would range from about 3 to 5 percent. The preprint refers to this as the undamaged state. The writer considers all air voids as a measure of damage to the pavement so would consider all of the specimens damaged to the extent that they contained air voids. This is not a criticism of the paper but merely a question of definition of terms. The increase in air voids associated with cracking would be considered increased damage by either definition. One reason that this is so interesting to the writer is that it brings back memories of research conducted in the 1960s. When the effect of air voids and the spectrum of strain rates that would exist at any given crosshead strain rate in the asphalt in a pavement specimen was considered, it was decided to first run tensile tests on the asphalt binder alone in order to simplify the problem. This seemed practical since the binder was recognized as the source of tensile strength in the pavement. Tests were run at various temperatures to determine the tensile strength of asphalts with a variety of temperature susceptibilities. After some study, 4C was chosen as the critical temperature for asphalt strength in the pavement when the subgrade was strong in the summertime or in the wintertime when the subgrade was strong due to being frozen. But the tensile strength was really needed in the early spring when the frozen soils were thawing at about 4C. This is the critical condition where the subgrade is usually weakest and this is the condition where the tensile strength is most needed. When asphalt with the same penetrations at 25C but with a variety of temperature susceptibilities were tested at 4C, the tensile strengths varied widely. With the more temperature susceptible binders the machine would groan as the stress shot up and the specimen would snap immediately. With the less susceptible asphalts the stress would increase less rapidly and the specimen would stretch to the full length of the machine. This was very similar to the results shown in the paper being discussed. It was found that when a short, stiff spring was connected to the stiffest binders the build up in stress was not so steep and they too would stretch the full length of the machine just as with the softer asphalts. The obvious conclusion was that the stress build up was so rapid and the relaxation time was so long in the stiffer asphalts that the stress exceeded the ultimate strength of the specimens. The spring gave more time for the specimen to relax and with the relaxation of stress in the specimen it would stretch out to 150 mm. Factors of from one to two were assigned for the increase in tensile strength in pavement with different aggregate gradations and air void levels. The test results from tensile tests on actual pavement specimens give comparable values but this paper does not show as much variation as the writer had feared. The variation shown in ths paper s very encouraging even though it is for only a limited number of specimens and one pavement type. Further work would be most interesting. A research program is faced with great difficulty in arriving at the precision of its methods. If they are as imprecise as many methods are in the asphalt pavement field, their interpretation may lead to some misleading conclusions. There is usually too little data to make a realistic estimate of precision. This is why the second question is about the
minimization program for the reduction of error. This is something on which we can all agree. Let’s get rid of error. When the writer first came in contact with asphalt pavement construction the variability was staggering. One of the earliest problems was the question of whether an asphaltic material met the state specification. The engineer in charge told me that the test result from his laboratory was accurate. When asked the limits of accuracy, he said that his test result was accurate to at least eight significant figures after the decimal point. There was the widely held view that test methods were scientifically developed and, therefore, they were entirely accurate and any variability was due to operator or laboratory variability. As the writer became better informed in the asphalt pavement industry he came to realize that the variability of the test method was so much greater than the variability of the operator or the laboratory that the latter usually did not have a measurable effect. The test methods often contributed more to confusion than they did to enlightenment (2). Much of the variability was due to the wide variatio n in the specimens which were supposed to be identical. But if the test method could not reduce this variability, its use still contributed to the confusion. In my earlier discussion of my work in tensile strength determination, I feared that tensile testing of pavement specimens would be even more variable than stability tests. I am pleased that your work does not show this, but your use of an error minimization program raised questions in my mind as to how it determines error. The usual basis is outlier analysis where some values are declared to be from another statistical population because they are so far from the mean of he original population that the probability of there being so far out is very low. Of course, just because it is improbable for the test result to be from a given population does not mean that it is not. Only improbable. When we drop a specimen in the laboratory and we can see it is cracked halfway through, we have a good reason for not including it in our test results. While we all want to eliminate error, we must be careful not to eliminate data that shows the true extent of variation in our testing process. Many erroneous conclusions have been reached through the elimination of valid data. Especially in an industry which has proven to be as variable as the asphalt pavement industry. 1. R.L. Davis “Reduction of Low- Temperature Cracking in Asphalt Pavement Through the Use of Large Stone Mixtures” Proceedings of the Canadian Technical Asphalt Association, Volume 33, pp. 42-55, 1988 2. R.L. Davis Prepared Discussion of “A Comparison of HMA Field Performance and Laboratory Sensitivities” by Amy L. Epps and Adam J. Hand. Journal of the Association of Asphalt Paving Technologists, Volume 70, 2001.
