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Publication Number: FHWA-HRT-08-073
Date: September 2009

Development of A Multiaxial Viscoelastoplastic Continuum Damage Model for Asphalt Mixtures

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Chapter 5. MVEPCD Characterization and Verification in Compression

The permanent deformation in HMA is affected by several mixture factors, such as the resistance of the binder to flow, aggregate angularity and gradation, the amount of asphalt, the air void content, etc. A significant amount of research has been conducted to develop laboratory test methods, analysis techniques, and models to study the permanent deformation growth of HMA. The nature of the permanent deformation models available in the literature ranges from empirical to mechanistic-empirical to completely mechanistic. Further attempts have been made in recent years to develop a mechanistic permanent deformation model that involves fundamental material characterization.(20,52,53)

The objective of this research was to develop a constitutive model of HMA in compression that could predict HMA behavior under loading conditions and temperatures encountered in the field. The following sections present theories used in the modeling, the experimental program, the characterization process, and predictions for various loading histories. For this effort only, the ALF Control mixture has been used, and it should be understood that all figures in this chapter are related to this mixture.

5.1. Engineering Behavior of Asphalt Concrete in Compression

5.1.1. Constant Rate Compression

Figure 85 and figure 86 show the rate-dependent stress-strain curves of HMA at the confining pressures of 0 and 500 kPa, respectively. As observed from figure 85 and figure 86, the gross trends in stress-strain curves with 500 kPa confinement were quite similar to those with 0 kPa confining pressure; the overall strength of the material decreased with increasing temperature and with decreasing strain rates. Directly comparing the curves at different temperatures (figure 87 through figure 90), it was found that at 5 and 25 °C, the response was not sensitive to confining pressure. The strengths began to deviate during the 40 °C tests and were significantly different at 55 °C, with the confined tests showing higher strengths. This trend was similar to that observed in the confined dynamic modulus tests, which was described for tension-compression loading in subsection 4.2.2 and for compression loading in section 5.2. This finding suggests that when the material was very stiff, the effects of confinement were negligible.

Figure 85. Graph. Stress-strain curves for unconfined constant strain-rate tests. This figure shows the relationships between measured axial stress and axial strains, which are obtained from constant crosshead rate tests at 0 confining pressure. The x axis shows axial strain from parenthesis 0 to 4 percent close parenthesis, and the y axis shows stress from parenthesis 0 to 10,000 close parenthesis kPa. The graph shows that the tests performed at 5 degrees Celsius are overall stronger than those at 25 degrees Celsius, which are stronger than the ones at 40 degrees Celsius, which are stronger than the tests at 54 degrees Celsius. The faster rates at the next higher temperature overlap with the slower rates at the next colder temperature. The maximum and minimum peak stresses are observed from 5-1 and 55-4, respectively.

Figure 85. Graph. Stress-strain curves for unconfined constant strain-rate tests.

Figure 86. Graph. Stress-strain curves for 500 kPa confinement constant strain-rate tests. This figure shows the relationships between measured axial stress and axial strains, which are obtained from constant crosshead rate tests at 500 kPa confining pressure. The x axis shows axial strain from parenthesis 0 to 4 percent close parenthesis, and the y axis shows stress from parenthesis 0 to 10,000 close parenthesis kPa. The graph shows that the tests performed at 5 degrees Celsius are overall stronger than those at 25 degrees Celsius, which are stronger than the ones at 40 degrees Celsius, which are stronger than the tests at 54 degrees Celsius. The faster rates at the next higher temperature overlap with the slower rates at the next colder temperature. At the higher temperatures, the strength is less sensitive to strain rate. The maximum and minimum peak stresses are observed from 5-1 and 55-4, respectively.

Figure 86. Graph. Stress-strain curves for 500 kPa confinement constant strain-rate tests.

Figure 87. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 5 °C. This figure shows results from constant rate compression tests at different rates and 5 degrees Celsius under both confined and unconfined conditions. The x axis shows strain from parenthesis 0 to 2.5 percent close parenthesis while the y axis shows stress in kPa. The stress is shown from parenthesis 0 to 10,000 close parenthesis kPa, For the rates shown, there is no major difference between the confined and unconfined tests.

Figure 87. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 5 °C.

Figure 88. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 25 °C. This figure shows results from constant rate compression tests at different rates and 25 degrees Celsius under both confined and unconfined conditions. The x axis shows strain from parenthesis 0 to 2.5 percent close parenthesis while the y axis shows stress in kPa. The stress is shown from parenthesis 0 to 10,000 close parenthesis kPa, For the rates shown, there is no major difference between the confined and unconfined tests.

Figure 88. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 25 °C.

Figure 89. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 40 °C. This figure shows results from constant rate compression tests at different rates and 40 degrees Celsius under both confined and unconfined conditions. The x axis shows strain from parenthesis 0 to 2.5 percent close parenthesis while the y axis shows stress in kPa. The stress is shown from parenthesis 0 to 6,000 close parenthesis kPa, For the rates shown, the confined tests begin to show higher strength and ductility.

Figure 89. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 40 °C.

Figure 90. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 55 °C. This figure shows results from constant rate compression tests at different rates and 55 degrees Celsius under both confined and unconfined conditions. The x axis shows strain from parenthesis 0 to 2.5 percent close parenthesis while the y axis shows stress in kPa. The stress is shown from parenthesis 0 to 4,000 close parenthesis kPa, For the rates shown, the confined tests show substantially higher strength and ductility.

Figure 90. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 55 °C.

5.1.2. Repetitive Creep and Recovery Test

5.1.2.1. Creep and Recovery Test with VL

Figure 91 through figure 93 present the viscoplastic strains for the given VL stress histories. The viscoplastic strains caused by the first loading in each group, except for the first loading group, were close to zero or were at least very small because the deviatoric stress that caused the viscoplastic strains was quite small compared to that of previous loadings.

Figure 91. Graph. Viscoplastic strain versus cumulative loading time (unconfined VL). This figure shows the measured viscoplastic strain from variable load level test at  0 confining pressure that is plotted with respect to cumulative loading time for two replicate tests. The specimen to specimen variability is shown to be very small. The cumulative loading time is plotted on the x axis from parenthesis 0 to 18 close parenthesis seconds, and viscoplastic strains are shown on the y axis from parenthesis 0 to 4 percent close parenthesis. As the cumulative loading time increases, the viscoplastic strain increases. At 16 s, the viscoplastic strain is equal to about 3 percent.

Figure 91. Graph. Viscoplastic strain versus cumulative loading time (unconfined VL).

Figure 92. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VL). This figure shows the measured viscoplastic strain from variable load level test at 140 kPa confining pressure that is plotted with respect to cumulative loading time for two replicate tests. The specimen to specimen variability is shown to be very small. The cumulative loading time is plotted on the x axis from parenthesis 0 to 20 close parenthesis seconds, and viscoplastic strains are shown on the y axis from parenthesis 0 to 2.5 percent close parenthesis. As the cumulative loading time increases, the viscoplastic strain increases. At 19 s, the viscoplastic strain is equal to about 2 percent.

Figure 92. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VL).

Figure 93. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VL). This figure shows the measured viscoplastic strain from variable load level test at 500 kPa confining pressure that is plotted with respect to cumulative loading time for two replicate tests. The specimen to specimen variability is shown to be small but larger than the variability observed at 0 and 140 kPa. The cumulative loading time is plotted on the x axis from parenthesis 0 to 18 close parenthesis seconds, and viscoplastic strains are shown on the y axis from parenthesis 0 to 3 percent close parenthesis. As the cumulative loading time increases, the viscoplastic strain increases. At 16 s, the viscoplastic strain is equal to about 2.5 percent.

Figure 93. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VL).

5.1.2.2. Creep and Recovery Tests with VT and RVT

Figure 94 and figure 95 present the viscoplastic strains measured at the end of each rest period in unconfined VT and 500 kPa confinement VT, respectively. In both unconfined and confined VT testing, the overall reproducibility was sufficiently adequate to identify the effects of load level and loading sequence. In RVT testing, less viscoplastic strain was observed than in VT testing despite the fact that the test conditions were the same for both tests, with the exception of the sequence of loading, as described previously. This difference in viscoplastic strain indicated that the sequence of loading plays was an important role in viscoplastic strain development. In addition, a change of slope between the two groups of viscoplastic strain, which could not be explained by the concepts inherent of the conventional viscoplastic model, was observed. It seems the characteristic behavior of HMA was affected by viscoelastic relaxation.

Although quantifying the variations in the viscoplastic strain rate under a given loading condition is necessary for rigorous modeling work, no test protocol was available that could capture only the viscoplastic strain rate because HMA showed time-dependent viscoelastic strain, too. However, trends for viscoplastic strain rates developed in repetitive creep and recovery testing could be estimated by analyzing the VT test results. Figure 96 presents incremental viscoplastic strain rates (i.e., incremental viscoplastic strain divided by pulse time). As shown in figure 96, at 1-percent viscoplastic strain, viscoplastic strain rates from 0.05-s loadings (2.0 x 10-3) were much greater than those from 6.4-s loadings (8.0 x 10-5). These results indicated that most of the viscoplastic strain developed at the beginning of the loading period and that the viscoplastic strain that developed during the remainder of the loading period (i.e., at 6.4 to 0.05 s) was relatively small. Because the calculated viscoplastic strain rate was the average of the viscoplastic strain rates during loading, the actual viscoplastic strain rates at the end of the loading were much smaller than 8.0 x 10-5. This is another important behavior of HMA, along with the softening concept presented in the following section. Mathematically, this behavior can be formulated in a viscoplastic constitutive model with either increasing viscosity or increasing yield stress due to aggregate interlocking.

Figure 94. Graph. Viscoplastic strain versus cumulative loading time (unconfined VT testing). This figure shows the measured viscoplastic strain obtained from variable loading time tests at 0 kPa that are plotted with respect to time for two replicates. The variability is somewhat larger than that observed in VL testing. The cumulative loading time is plotted on the x axis from parenthesis 0 to 50 close parenthesis seconds, and viscoplastic strains are shown on the y axis from parenthesis 0 to 4 percent close parenthesis. As the cumulative loading time increases, the viscoplastic strain increases. At 45 s, the viscoplastic strain is equal to about 4 percent.

Figure 94. Graph. Viscoplastic strain versus cumulative loading time (unconfined VT testing).

Figure 95. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement for VT and RVT testing). This figure shows the measured viscoplastic strain obtained from variable loading time tests and reversed variable loading time tests at 500 kPa confining pressure that are plotted with respect to cumulative loading time. Cumulative loading time is shown on the x axis from parenthesis 0 to 200 close parenthesis seconds, and viscoplastic strain is plotted on the y axis from parenthesis 0 to 3.5 percent close parenthesis. Three load levels (1,600, 1,800, and 2,000 kPa) are shown for the variable loading time tests, whereas only level 1,600 kPa is considered in reversed variable loading time test. At 200 s, the VT test at 1,600 kPa shows 1.5-percent strain; the test at 1,800 kPa shows approximately 2 percent; and the test at 2,000 kPa shows approximately 3 percent. The reversed variable time test at 1,600 kPa shows approximately 1 ⅓- percent strain at 200 s.

Figure 95. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement for VT and RVT testing).

Figure 96. Graph. Incremental viscoplastic strain rate versus viscoplastic strain (500 kPa confinement, 1600 kPa deviatoric). This figure shows the incremental viscoplastic strain rate which is incremental viscoplastic strain divided by pulse time is plotted with respect to viscoplastic strain for multiple pulse times. On the x axis, the viscoplastic strain is plotted from parenthesis 0 to 2.5 percent close parenthesis, and viscoplastic strain rate is shown on the y axis from parenthesis 1 times 10 superscript -6 to 0.1 close parenthesis strain per second. For this analysis, the variable loading time test at 500 kPa with 1,600 kPa confining pressure is utilized. For all pulse times, the viscoplastic strain rate is shown to decrease with increasing viscoplastic strain. It is also shown that at the same viscoplastic strain level that the incremental viscoplastic strain rate is lower for the longer loading times. This observation implies most of permanent strain develops in the beginning of loading.

Figure 96. Graph. Incremental viscoplastic strain rate versus viscoplastic strain (500 kPa confinement, 1,600 kPa deviatoric).

5.1.2.3. Creep and Recovery Tests with Constant Load Level and CLT

Figure 97 presents the viscoplastic strain history for each loading condition with a confining pressure of 500 kPa. A smaller viscoplastic strain was observed as the pulse time increased. Even considering the ramp time of 0.005 s, which was not taken into account in the cumulative loading time, the difference in the viscoplastic strains was quite significant.

As shown in figure 97, more viscoplastic strain was observed in the CLT tests that consisted of shorter loading times at a given cumulative loading time. The differences were significant. For example, at 150 s of cumulative loading time, the viscoplastic strain in 1,800 kPa CLT testing with a 0.4-s loading time was over 3 percent, whereas it was around 1.5 percent in 1,800 kPa CLT testing with a 6.4-s loading time and was 2.2 percent with a 1.6-s loading time. One reason for this behavior could be related to the dynamic effects associated with the ramp to the target load because one difference in the tests, as they are presented in figure 97, is the number of load applications. Alternatively, because the tests are exposed to different total rest times at given cumulative loading times, the differences could be related to material softening. Furthermore, it is not beyond reason to suppose that this softening behavior could be rate dependent. For further study of this issue, two additional VT tests with 0.1- and 0.05-s rest periods were performed at a confining pressure of 140 kPa. These test results were compared with the results from VT testing with 200 s of rest. In these tests, the conditions were identical (i.e., the number of loadings and rest periods were the same for each), except for the length of the rest period. Figure 98 presents the viscoplastic strains measured at the end of the rest periods. For VT tests with 0.05 and 0.1 s of rest, 200 s of rest was allowed at the end of the testing to measure the pure viscoplastic strain because it was not possible to measure pure viscoplastic strain immediately after 0.05 or 0.1 s of rest. The deviatoric stresses were 827 and 552 kPa, and the confining pressure was 140 kPa.

Figure 97. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement CLT). This figure shows the viscoplastic strains in constant load level and time tests with three pulse times (0.4, 1.6, and 6.4 s) are plotted with respect to cumulative loading time. The x axis shows cumulative loading time from parenthesis 0 to 200 close parenthesis seconds, and the y axis shows viscoplastic strains from parenthesis 0 to 2.5 close parenthesis. The pulse time for each test is kept constant for an entire testing and deviatoric stress of 1,800 kPa are kept constant for all constant load level tests, as well as confining pressure of 500 k Pa. It is shown that at the same cumulative loading time that less viscoplastic strain develops for the longer pulse time. At a cumulative loading time of 200 s, the viscoplastic strain from the 6.4 second loading is about 1.6 percent; for the 1.6 second test, the strain is approximately 2 percent; and for the 0.4-second pulse time, it is approximately 3 percent. For comparison, the variable loading time test at 1,800 kPa deviatoric stress and 500 kPa confining stress shows approximately 2 percent strain at 200 s.

Figure 97. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement CLT).

Figure 98. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT). This figure shows the viscoplastic strains in variable loading time tests with three rest periods (0.05, 1, and 200 s) are plotted with respect to cumulative loading time. Cumulative loading time is plotted on the x axis from parenthesis 0 to 350 close parenthesis seconds, and viscoplastic strains are shown on the y axis from parenthesis 0 to two percent close parenthesis. The confining pressure and deviatoric stress in these tests are 140 kPa and 827 kPa, respectively. Only the viscoplastic strain at the very end of the tests with 0.05- and 1-second rest period are shown. As the rest period increases, the amount of viscoplastic strain developed increases. At a cumulative loading time of 300 s, the viscoplastic strain in variable loading time test with rest period of 0.05 s is about 1 percent; with a rest period of 1 second, the viscoplastic strain is approximately 1.2 percent; and the 200 second rest period has 1.5 percent.

Figure 98. Graph. Viscoplastic strain versus cumulative loading time
(140 kPa confinement VT).

As seen in figure 98, even though the loading histories were identical except for the length of the rest period, a smaller viscoplastic strain developed as the rest period became shorter. Furthermore, the effect of the rest period on viscoplastic development was significant considering the amount of viscoplastic strain when using 552 kPa deviatoric stress and 200 s of rest. This experimental observation did not demonstrate the effects of the dynamic loading ramp, but it did demonstrate the significant effect of rate-dependent softening during unloading. Therefore, the modeling effort focused on developing a viscoplastic model that accounted for rate-dependent hardening and/or softening.

5.1.3. t-TS with Growing Damage in Compression

The principle of t-TS is one of the fundamental and most important concepts for HMA in tension modeling because it provides a strong mechanical background and significantly reduces the experimental efforts. In order to verify the principle for the compression stress state, stress characteristic curves were constructed from constant strain-rate test results by utilizing shift factors determined from the dynamic modulus tests. Further verification was also performed using repetitive creep and recovery tests.

For the verification of the principle for monotonic loading, a wide range of eight reference-strain values were chosen according to the results of both the uniaxial and triaxial compressive constant strain-rate tests (figure 85 and figure 86). According to the procedure shown schematically in figure 55 through figure 57 and discussed in detail elsewhere, the stress and time values were determined for all of these tests at fixed strain levels.(13,20) These plots of stress versus time are shown in figure 99 through figure 114. Then, shift factors obtained from small-strain LVE testing were applied to determine the reduced time that corresponded to each physical time. If the t-TS principle was valid with growing damage, the resulting plots of stress and reduced time would appear continuous at all strain levels. This behavior was indeed observed for the compression tests under both the confined and unconfined conditions (figure 115 and figure 116). These results verified that the t-TS concept held true for mixtures subjected to compressive loading as well as to tensile loading, even if there was severe damage and viscoplastic strain. However, to verify that the principle held for the physical mechanisms behind the behavior of repetitive creep and recovery tests, more rigorous verification was needed-this verification compared VT test results at 40 and 55 °C with the same reduced time histories.

For this verification, VT testing was first performed at 55 °C (the 200 s rest period results are used here). Next, the time history was used with the t-TS shift factors to compute the equivalent reduced time history at 40 °C. However, because the testing time was estimated to take several days (the equivalent time to 200 s at 55 °C is approximately 3,265 s at 40 °C), the following analysis was performed to finish the VT testing within a reasonable time. In this analysis, the measured strain history during the unloading portion of several load pulses was used to compute the strain rate, which was plotted against the rest period time in figure 117. To avoid issues related to the initial loading of a test, the 0.05 s data were taken from the second load block, whereas the other pulse times were taken from the first loading block of the VT test. As shown in figure 117, most of the strain rates became quite small after around 40 s, except for the rest periods following the 1.6 and 6.4 s load pulses. For this reason, 40 s was used to compute the reduced time for pulse times less than 1.6 s (653 s at 40 °C); 50 s was used for a pulse time of 1.6 s (816 s at 40 °C); and 60 s was used for a pulse time of 6.4 s (980 s at 40 °C). Note that strain rates reached an asymptotic value of zero more quickly as the strain level increased, and thus, it was conservative to consider the times used in the first loading block as the reference times. The results of these two tests are shown figure 118. As the figure shows, viscoplastic strains measured at the end of rest periods were well matched to each other. This agreement confirmed that the t-TS principle was applicable regardless of loading sequence and the amount of damage and viscoplastic strain in asphalt concrete.

Figure 99. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0001 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.0001 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 99. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0001 strain level under uniaxial conditions.

Figure 100. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0005 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.0005 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 100. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0005 strain level under uniaxial conditions.

Figure 101. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.001 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.001 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 101. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.001 strain level under uniaxial conditions.

Figure 102. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.003 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.003 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 102. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.003 strain level under uniaxial conditions.

Figure 103. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.005 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.005 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 103. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.005 strain level under uniaxial conditions.

Figure 104. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.01 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.01 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 104. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.01 strain level under uniaxial conditions.

Figure 105. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.015 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.015 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 105. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.015 strain level under uniaxial conditions.

Figure 106. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.02 strain level under uniaxial conditions. This figure shows the stress versus time curves at a strain level of 0.02 and uniaxial conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 106. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.02 strain level under uniaxial conditions.

Figure 107. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0001 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.0001 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 107. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0001 strain level under 500 kPa conditions.

Figure 108. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0005 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.0005 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 108. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.0005 strain level under 500 kPa conditions.

Figure 109. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.001 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.001 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 109. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.001 strain level under 500 kPa conditions.

Figure 110. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.003 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.003 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 110. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.003 strain level under 500 kPa conditions.

Figure 111. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.005 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.005 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The  y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 111. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.005 strain level under 500 kPa conditions.

Figure 112. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.01 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.01 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 112. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.01 strain level under 500 kPa conditions.

Figure 113. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.015 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.015 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data show lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 113. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.015 strain level under 500 kPa conditions.

Figure 114. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.02 strain level under 500 kPa conditions. This figure shows the stress versus time curves at a strain level of 0.02 and 500 kPa conditions. Data at 5, 25, 40, and 55 degrees Celsius are shown. The x axis ranges from parenthesis 0.01 to 1 times 10 superscript 5 close parenthesis seconds. The  y axis ranges from parenthesis 10 to 100,000 end stress in kPa. At constant time, the higher temperature data shows lower stress in all figures. For each temperature, as time increases, the stress decreases.

Figure 114. Graph. Stress-time curves for the Control mixture before the application of time-temperature shift factors at a 0.02 strain level under 500 kPa conditions.

Figure 115. Graph. Stress mastercurves for the Control mixture under uniaxial conditions. This figure shows the data in figure 87 through figure 94 plotted in stress-reduced time space after applying shift factor determined from dynamic modulus test at 0 confining pressure. The x axis shows reduced time from parenthesis 1 to 1 times  10 superscript 6 close parenthesis seconds, and the y axis shows stress from parenthesis 10 to 100,000 close parenthesis kPa. At a constant reduced time, the higher strain levels show higher stress; at a constant strain level, the stress decreases as reduced time increases. The plotted curves show continuity between temperatures, which supports the concept of t-TS with growing damage.

Figure 115. Graph. Stress mastercurves for the Control mixture under uniaxial conditions.

Figure 116. Graph. Stress mastercurves for the Control mixture under triaxial conditions (500 kPa). This figure shows the data in figure 87 through figure 94 plotted in stress-reduced time space after applying shift factor determined from dynamic modulus test at 500 kPa confining pressure. The x axis shows reduced time from parenthesis 1 to 1 times 10 superscript 6 close parenthesis seconds, and the y axis shows stress from parenthesis 10 to 100,000 close parenthesis kPa. At a constant reduced time, the higher strain levels show higher stress; at a constant strain level, the stress decreases as reduced time increases. The plotted curves show continuity between temperatures which supports the concept of time temperature superposition with growing damage.

Figure 116. Graph. Stress mastercurves for the Control mixture under triaxial conditions (500 kPa).

Figure 117. Graph. Variation of strain rate during unloading. This figure shows strain recoveries during unloading respect to time for several different total loading times. The x axis shows recovery time from parenthesis 0 to 60 close parenthesis seconds, and the y axis shows strain rate from parenthesis 0 to 2 times 10 superscript -5 close parenthesis strain per second. These measurements are obtained from variable loading time test. The recovery time is longer for the longer pulse times, but most of strains developed are almost completely recovered in 40 s from the moment of unloading.

Figure 117. Graph. Variation of strain rate during unloading.

Figure 118. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT at 40 °C and 55 °C). This figure shows the viscoplastic strain developed at 40 degree Celsius under a variable loading time (confining pressure of 140 kPa with deviatoric stress of 827 kPa), as compared to a strain developed at 55 degree Celsius under the same loading condition. The cumulative reduced loading time is shown for a reference temperature of 55 degrees Celsius on the x axis from parenthesis 0 to 350 close parenthesis seconds, and the viscoplastic strain is shown on the y axis from parenthesis 0 to 2 percent close parenthesis. Given typical specimen to specimen variability, the tests at the two temperatures agree very well.

Figure 118. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT at 40 and 55 °C).

5.2. MVECD Characterization in Compression

5.2.1. Linear Viscoelastic Characterization

As presented in chapter 2, the first stage in MVECD characterization was the determination of the LVE properties of the material. Following the test protocols presented in chapter 3, frequency-temperature sweep tests were conducted in both unconfined and confined states. The results of this characterization under zero-maximum deviatoric stress conditions are shown for the unconfined stress state in table 16, for a confining pressure of 140 kPa in table 17 and for a confining pressure of 500 kPa in table 18. The shift factor function coefficients that resulted as part of the mastercurve construction process for each of the tests in these tables are shown in table 19.

Table 16 through table 18 show that a trend similar to that observed for the zero-mean deviatoric conditions was also observed for the zero-maximum deviatoric tests. Specifically, under confinement, the material modulus increased as a function of the confinement level. Additionally, under conditions where a given confinement state diverged from the unconfined state, the mastercurve moved toward a lower temperature and higher frequency at the higher confined stress. Additionally, a higher confining stress generally resulted in a lower phase angle (i.e., higher elasticity). Although it may be argued that increased aggregate interlocking was the cause of these behaviors, it was assumed that such an interpretation was flawed due to the strain magnitudes typically induced during the testing. The net effect of the increased dynamic modulus and phase angle was an increase in the relaxation modulus, which is shown for the different confining pressures in figure 119. These curves were obtained by the method outlined in subsection 2.2.1. Figure 119 shows that the relaxation modulus began to diverge at around 0.01 to 1 s in reduced time. These times generally corresponded to a physical time of between
1 x 10-5 and 2 x 10-7 s at 54 °C and of between 1 x 10-4 and 1 x 10-7 at 40 °C. Therefore, it was important to consider the effect of confining pressure because the time at which the relaxation modulus started to diverge was within the range of reduced time that was used in the rutting analysis.

Table 16. Linear viscoelastic characterization and variation for the Control mixture in unconfined compression state at selected frequencies and temperatures.

Temp. (°C )

Frequency (Hz)

Average |E*| (MPa)

Average Phase Angle (°)

|E*| Coefficient of Variation (%)

Phase Angle Coefficient of Variation (%)

-10

25

33,603

4.8

12.6

4.0

10

32,251

6.1

12.1

10.0

5

30,865

6.5

11.0

4.6

1

28,633

7.9

11.7

2.8

0.5

27,398

8.0

11.4

3.4

0.1

23,909

9.7

7.9

44.1

5

25

23,707

9.7

7.0

45.6

10

21,323

11.7

5.0

34.3

5

20,085

12.5

9.1

4.2

1

16,399

15.1

10.9

4.5

0.5

14,262

17.2

11.4

3.9

0.1

11,205

20.7

11.9

5.5

20

25

12,444

19.2

11.4

4.3

10

10,624

21.4

14.7

4.7

5

8,809

23.2

14.5

5.4

1

6,229

26.2

7.0

8.4

0.5

4,941

27.8

1.0

3.0

0.1

2,818

32.7

5.4

2.4

40

25

2,872

32.5

3.2

0.2

10

1,834

36.2

1.3

1.0

5

1,386

38.0

2.5

1.7

1

904

35.7

7.0

3.0

0.5

640

35.9

7.0

2.6

0.1

363

32.6

20.4

12.1

54

25

973

35.6

4.1

3.3

10

604

35.8

5.8

2.7

5

475

34.5

1.0

2.7

1

321

31.3

20.8

15.4

0.5

226

29.0

4.7

18.5

0.1

185

24.3

11.0

38.7

Table 17. Linear viscoelastic characterization and variation for the Control mixture in 140 kPa confined compression state at selected frequencies and temperatures.

Temp. (°C )

Frequency (Hz)

Average |E*| (MPa)

Average Phase Angle (°)

|E*| Coefficient of Variation (%)

Phase Angle Coefficient of Variation (%)

-10

25

34,501

9.0

1.6

3.7

10

33,555

7.8

0.4

4.5

5

32,396

7.8

0.0

5.7

1

29,413

8.5

0.0

1.1

0.5

28,174

9.1

0.5

2.4

0.1

25,164

10.6

0.9

2.4

5

25

24,048

11.1

1.4

0.4

10

21,655

12.7

1.4

1.6

5

20,663

13.4

1.2

2.0

1

17,333

16.3

2.4

2.6

0.5

15,097

17.9

2.8

1.9

0.1

11,904

20.6

2.6

1.2

20

25

12,773

19.7

3.2

1.3

10

10,458

22.4

3.2

0.5

5

9,013

24.3

3.0

0.6

1

6,278

29.2

4.7

0.2

0.5

5,176

31.9

5.6

0.3

0.1

3,376

33.2

1.7

2.2

40

25

3,489

33.1

1.2

2.0

10

2,473

33.5

2.8

4.0

5

1,978

33.4

3.9

5.1

1

1,397

30.0

5.0

3.4

0.5

1,276

27.3

4.8

0.2

0.1

904

22.4

7.8

16.9

54

25

1,459

32.2

5.6

8.4

10

1,328

28.0

3.9

0.2

5

1,005

27.4

10.6

18.8

1

896

19.9

4.5

14.0

0.5

752

21.9

11.5

27.5

0.1

740

14.8

5.1

30.4

Table 18. Linear viscoelastic characterization and variation for the Control mixture in 500 kPa confined compression state at selected frequencies and temperatures.

Temp. (°C )

Frequency (Hz)

Average |E*| (MPa)

Average Phase Angle (°)

|E*| Coefficient of Variation (%)

Phase Angle Coefficient of Variation (%)

-10

25

32,837

6.3

10.0

0.9

10

31,947

7.7

11.1

0.9

5

30,803

8.6

11.6

0.9

1

27,616

10.3

11.0

0.5

0.5

26,114

11.1

11.5

0.9

0.1

22,672

13.5

11.2

0.2

5

25

23,317

13.1

8.6

2.9

10

21,641

13.6

6.7

0.3

5

19,651

11.9

7.0

0.4

1

16,013

15.1

7.5

2.1

0.5

13,834

17.2

7.3

1.6

0.1

10,903

20.7

8.4

2.2

20

25

12,927

18.1

5.1

0.1

10

11,312

20.1

5.3

0.2

5

9,540

22.9

5.1

0.3

1

7,067

27.3

4.0

0.2

0.5

5,943

23.0

7.0

0.9

0.1

3,996

24.0

7.5

0.5

40

25

4,621

29.2

5.8

6.2

10

3,631

23.7

0.8

0.3

5

3,166

23.3

2.6

0.2

1

2,477

18.7

3.9

1.8

0.5

2,204

16.7

3.1

1.3

0.1

1,737

12.5

6.7

9.8

54

25

2,664

19.7

3.9

7.2

10

2,295

17.8

4.0

2.7

5

2,059

16.3

0.5

3.1

1

1,674

11.7

2.5

17.6

0.5

1,528

9.8

1.2

2.5

0.1

1,398

7.0

3.0

7.9

Table 19. Effect of confining pressure on shift factor function coefficients for the Control mixture in compression state.

Parameters

0 kPa

140 kPa

500 kPa

α

0.00071

0.00090

0.00062

α2

-0.15833

-0.17176

-0.14508

α3

0.77385

0.83629

0.71001

Figure 119. Graph. Confining stress effect on the relaxation modulus. This figure shows the relaxation modulus determined from dynamic modulus tests at three different confining pressures compared in log-log space. The x axis shows the relaxation modulus from parenthesis 1 to 1 times 10 superscript 8 close parenthesis kPa, and the y axis shows reduced time from parenthesis 1 times 10 superscript -17 to 1 times 10 superscript 18 close parenthesis seconds. It is shown that at very short reduced times the relaxation modulus at 0 kPa, 140 kPa, and 500 kPa are for all intents and purposes the same. At 0.01 to 1 second in reduced time, the relaxation modulus determined at 500 kPa and 140 kPa start to deviate from that at 0 confining pressure.

Figure 119. Graph. Confining stress effect on the relaxation modulus.

5.2.2. Comparison of Zero-Mean and Zero-Maximum Deviatoric Stress Results

Figure 120 through figure 123 present the LVE characteristics determined from the zero-mean and zero-maximum deviatoric stress states. Only the 0 and 500 kPa results are shown in the figures because they were consistent for the two test methods. Overall, the comparisons were very favorable, and little difference was seen between the zero-mean and zero-maximum deviatoric stress states. However, a noticeable difference was seen in the phase angle results, but these results were subject to higher variability. Further, the phase angle calculation was less precise due to limitations of the measurement instrumentation. These results indicated that at small strains, asphalt concrete was not bimodal, and it was only after some threshold strain had been exceeded that such behavior occurred. Such findings are consistent with the work presented elsewhere when care is taken in performing the experiments at sufficiently small strains (50 to 70 με).(54,55) When such care is not taken, conflicting data are found in the literature.(28,56) For modeling then, it is important that the LVE characterization is limited to the 50 to 70με range; in this range, either the zero-mean or zero‑maximum method may be used.

As a final comparison of the two test conditions, the model developed in subsection 4.2.2 was applied to the zero-maximum deviatoric stress state tests. The results of this analysis are shown in figure 124, and good agreement was observed between the model and measured data.

Figure 120. Graph. Comparison of zero-mean and zero-maximum deviatoric stress dynamic modulus mastercurves in semi-logarithmic scale. This figure shows the dynamic modulus, in absolute value signs E superscript star, from parenthesis 0 to 35,000 close parenthesis MPa in arithmetic scale on the y axis and reduced frequency, Hertz, from parenthesis 1 times 10 superscript -8 to 1 times 10 superscript 5 close parenthesis logarithmically on the x axis. All show decreasing modulus with decreasing frequency. The arithmetic scale is shown to examine the high modulus values, and no clear effect of confining pressure is observed. It is also shown that the modulus determined from zero-mean stress and zero-maximum stress tests are the same given specimen to specimen variability.

Figure 120. Graph. Comparison of zero-mean and zero-maximum deviatoric stress dynamic modulus mastercurves in semi-logarithmic scale.

Figure 121. Graph. Comparison of zero-mean and zero-maximum deviatoric stress dynamic modulus mastercurves in logarithmic scale. This figure shows the data on the y axis from parenthesis 100 to 100,000 close parenthesis logarithmically and reduced frequency, Hertz, from parenthesis 1 times 10 superscript -8 to 1 times 10 superscript 5 close parenthesis logarithmically on the x axis. All show decreasing modulus with decreasing frequency. This graph shows a major difference between the confined and unconfined tests at very low reduced frequencies. However, the zero-mean and zero-maximum stress tests still agree with the higher confining pressure.

Figure 121. Graph. Comparison of zero-mean and zero-maximum deviatoric stress dynamic modulus mastercurves in logarithmic scale.

Figure 122. Graph. Effect of test method on shift factor functions. This figure shows the influence of confining pressure on the time-temperature shift factor as a function of temperature. The logarithm of the time-temperature shift factor is shown on the y axis from parenthesis -8 to 4 close parenthesis, and the temperature is plotted on the x axis from parenthesis -20 to 60 close parenthesis degrees Celsius. With increased confining pressure, the magnitudes of the shift factor function are found to decrease. It is also found that the shift factor determined from zero-mean and zero-maximum stress tests agree.

Figure 122. Graph. Effect of test method on shift factor functions.

Figure 123. Graph. Comparison of zero-mean and zero-maximum deviatoric stress phase angle mastercurves. This figure shows the influence of temperature and frequency, as expressed through the reduced frequency, on the material phase angle as a function of confining pressure and test method. The data are shown with the y axis from parenthesis 0 to 50 close parenthesis degrees, and the reduced frequency, Hertz, is shown from parenthesis 1 times 10 superscript -8 to 1 times 10 superscript 5 close parenthesis logarithmically on the x axis. At each confining pressure, the phase angle increases with decreased reduced frequency until a point at which the phase angle begins to decrease. The material is shown to behave more elastically when subjected to higher confining pressure. Given the expected variability in phase angle, the zero-mean and zero-maximum confining pressure tests agree at both confining pressures.

Figure 123. Graph. Comparison of zero-mean and zero-maximum deviatoric stress phase angle mastercurves.

Figure 124. Graph. Application of stress state-dependent model to zero-maximum deviatoric stress tests. This figure shows how the model developed in subsection 4.2.2 is applied to describe confining pressure dependency of dynamic modulus. The reduced frequency is shown logarithmically spaced from parenthesis 100 to 1 times 10 superscript 5 close parenthesis MPa on the y axis, and the reduced frequency, Hertz, is shown from parenthesis 1 times 10 superscript -8 to 1 times 10 superscript 5 close parenthesis logarithmically on the x axis. Measured dynamic moduli at three different confining pressures (0, 140, and 500 kPa) in zero maximum deviatoric stress condition are compared with predicted moduli.

Figure 124. Graph. Application of stress state-dependent model to zero-maximum deviatoric stress tests.

5.2.3. MVECD Damage Function Characterization

Characterization of the VECD model in compression was identical in analysis and general experimental requirements to that in tension. Constant rate compression tests were performed under both unconfined and confined conditions, the results of which were then used to calculate the damage functions C11, C12, and C22. The process was discussed previously in subsection 4.2.3 and is shown schematically in the flow diagram shown in figure 125.

5.2.3.1. Characterization of C11(S)

Uniaxial constant rate tests were used to characterize the C11 damage function. Test results were used with equation 89 and equation 90 to compute the material function and damage parameter, respectively. Note that stress-hardening (equation 85) was used for this analysis. As with the tension damage function, this relationship was refined following the NCHRP 9-19 methodology.(22) This relationship is shown for the Control mixture in figure 126. This figure is constructed at a reference temperature of 5 °C and is only for the material under compressive loading. For this mixture, a1 = 7.679 x 10-6, a2 =1.6971 x 10-6, a3 = 7.6793 x 10-6, a4 = 4.6192 x 10-7, a5 = 1.5128 x 10-7, and a6 = 4.1017 x 10-8.

Figure 125. Diagram. Flow diagram for MVECD model characterization. This figure shows a characterization of the MVECD model, which consists of three different steps. At each step there are three components: test, quantity of interest, and material property. The first test to perform is the temperature and frequency sweep test, which results in the dynamic modulus as the quantity of interest and the relaxation modulus and time temperature shift factor as the material properties. The next test protocol is the unconfined constant rate test at 5 degrees Celsius. The two quantities of interest here are the pseudo strain and pseudo dilation, which require the relaxation modulus as input. The resulting material property is the first and second material integrity terms as functions of damage. The final step is confined constant rate tests at 5 degrees Celsius, which provides again pseudo strain and pseudo dilation and produces, when combined with the first and second material integrity terms, the third material integrity term.

Figure 125. Diagram. MVECD model characterization.

Figure 126. Graph. C11 versus S for compression for Control mixture (5 °C reference). This figure shows the damage characteristic relationship for the first material integrity parameter, C subscript 11. The material integrity is plotted on the y axis from parenthesis 0 to 1 close parenthesis, and damage, S, is plotted on the x axis from parenthesis 0 to 2.5 times 10 superscript 6 close parenthesis. Results from multiple tests are shown to collapse into a single relationship and the best fit model of these tests is also shown. The relationship is observed to follow an exponential decay pattern, when at low damage levels, the material integrity is equal to 1, and at near failure values of damage, it is found to equal approximately 0.20.

Figure 126. Graph. C11 versus S for compression for Control mixture (5 °C reference).

5.2.3.2. Characterization of C12(S)

In addition to using uniaxial constant rate tests to characterize the C11 damage function, these test results were also used to characterize the C12 damage function. As with the tension tests, complications arose due to the time-dependent nature of Poisson's ratio in asphalt concrete. Based on the analysis presented in subsection 4.2.3.2, an initial value of approximately 0.179 was assumed for C12. This assumption was not unreasonable based on the results of the characterization tests. A slight difference between the initial values of the compression and tension data shown previously was observed due to the use of slightly different mixtures for the two sets of tests (Control-2006 for tension and Control for compression). Tension data for the Control mixture will be shown in subsequent sections of this report. The C12 material parameter was defined from equation 91, and the damage parameter was calculated and refined as before. The functional form taken for the compressive C12 damage function is shown along with the data in figure 127. It was observed that at early damage stages, Poisson's ratio changed very little, but, in general, as damage increased, Poisson's ratio reduced, as indicated by an increase in C12. For this mixture, it was found that b1 = 3.5598 x 10-8, 1.6982 x 10-8, and 3.2756 x10-10.

Figure 127. Graph. C12 versus S for compression for Control mixture (5 °C reference). This figure shows the characteristic relationship for the second material integrity term, C subscript 12, as a function of damage, S, for the three characterization tests and for the best fit model. The material integrity term is plotted on the y axis from parenthesis -0.4 to 0.6 close parenthesis, and the damage term is plotted on the x axis from parenthesis 0 to 1 times 10 superscript 7 close parenthesis. The graph shows that the initial values of the material integrity term is approximately 0.4 and that the data decrease quickly to 0.15 before decreasing linearly. At localization, it is found that the second material integrity term, C subscript 12, has a value of approximately -0.25. The functional relationship is also shown on the graph where the second material integrity term, C subscript 12, is equal to parenthesis 0.228558 minus coefficient K subscript 6 close parenthesis multiplied by exponent to the coefficient K subscript 2 multiplied by damage, S, raised to the coefficient K subscript 3; plus coefficient K subscript 4 multiplied by damage, S, plus coefficient K subscript 5 multiplied by damage, S, squared, plus coefficient K subscript 6.

Figure 127. Graph. C12 versus S for compression for Control mixture (5 °C reference).

5.2.3.3. Characterization of C22(S)

Confined constant rate tests, along with the other two damage functions, were used to characterize the final C22 function. Following the study conducted for the tension characterization methodology, a visual basic macro was created in Microsoft Excel® to determine the damage function by optimization. The characterized C22 function is shown in figure 128 along with the model function. The parameters of the model were H1 = -0.504959 and H4 = 1.16247 x 10-13.

Figure 128. Graph.  C22 versus S for compression for Control mixture (5 °C reference). This figure shows the characteristic relationship for the second material integrity term, C subscript 22, as a function of damage, S, for the three characterization test and for the best fit model. The material integrity term is plotted on the y axis from parenthesis -120 to 0 close parenthesis, and the damage term is plotted on the x axis from parenthesis 0 to 5 times 10 superscript 6 close parenthesis. The graph shows that data decrease following a second order polynomial pattern from an initial value of approximately 0 to a final value of approximately -100. Also shown on the graph is the functional relationship used for this damage function; the third material integrity term, C subscript 22, is equal to coefficient H subscript 1 plus coefficient H subscript 4 multiplied by damage, S, squared.

Figure 128. Graph. C22 versus S for compression for Control mixture (5 °C reference).

5.2.4. Comparison of MVECD Behavior in Compression and Tension

Asphalt concrete, like other composite materials, is known to exhibit bimodal behavior. In particular, the strength of the material it is in and its ductility are much higher in the compression direction than they are in the tensile direction. Comparisons of the MVECD damage functions revealed some interesting behaviors. First, when comparing the primary axial modulus damage function, C11, between tension and compression, the material was substantially more resistant to damage in compression than it was tension. This behavior is shown in the same scale in figure 129. This behavior was explained by the notion that the damage parameter, S, represented some kind of crack density or volume-averaged cracked area. In tension, these cracks were oriented perpendicular to the principal loading direction, whereas, in compression, the cracks ran parallel to the principal loading direction. In constant rate loading, cracks only grew when they were under a tensile stress. Because tension was only induced (at the microscale level) in compression tests and not directly applied, the material was more resistant to a given cracking volume.

This physical interpretation for damage was also supported by the behavior of the second damage function, C12. These damage functions are shown for both tension and compression in figure 130. This figure has both the C12 function and Poisson's ratio between the symmetric axis and perpendicular axis (ν3132) for the convenience of the discussion. From the comparison shown in figure 130, it was observed that the primary Poisson's ratio showed an overall decreasing trend in tension, whereas the opposite trend occurred in compression. As with the C11 damage function, the tensile behavior of C12 was observed to be more sensitive to damage. A possible explanation of this behavior depended on understanding that Poisson's ratio indicated the degree to which the radius changed relative to a unit change in length. Applying the physical interpretation of damage in tension and compression, the physical implications of the patterns in both damage curves were identical. The tension curve was increasing, thus indicating that with higher levels of damage, the direction parallel to the primary cracking direction (radial strain) was less sensitive to changes in strain perpendicular to the primary cracking direction (vertical strain). For compression, the primary cracking direction was parallel to the loading direction and perpendicular to the radial direction, so with the same interpretation (of damage in tension and compression), an increase in Poisson's ratio was observed. In the most general terms, the increased opening of a microcrack within a body was not necessarily accompanied by a relative increase in the parallel to the crack direction dimension of the body. In fact, the results indicated that at higher levels of cracking or damage the material became less likely to change dimension as much when loaded perpendicular to the damage orientation.

Figure 129. Graph. Comparison of tension and compression of C11 damage function. This figure shows the characteristic relationship for the first material integrity term, C subscript 11, as a function of damage, S, for the two characterization test in compression and in tension. The material integrity term is plotted on the y axis from parenthesis 1 to 0 close parenthesis, and the damage term is plotted on the x axis from parenthesis 0 to 2 times 10 superscript 6 close parenthesis. The graph shows that at any given level of damage the first material integrity term, C subscript 11, in compression is larger than that in tension.

Figure 129. Graph. Comparison of tension and compression of C11 damage function.

The physical interpretation of the C22 damage function was not as straightforward as the other two damage functions. As seen in equation 45, the C11 and C12 damage functions can be directly translated to elements in the stiffness matrix. However, the C22 damage function entered in the denominator of multiple elements of the material stiffness matrix. Nevertheless, this damage function can be considered as a sort of volumetric compliance, which has little meaning in a transverse isotropic problem. From figure 131, though, it is seen that the material was more volumetrically compliant in compression than it was in tension.

Figure 130. Graph. Comparison of tension and compression of C12 damage function. This figure shows the characteristic relationship for the second material integrity term, C subscript 12, as a function of damage, S, for tension and compression loading directions. The material integrity term is plotted on the y axis from parenthesis -0.4 to 0.4 close parenthesis, Poisson's ratio is shown on a second y axis from parenthesis 0.7 to 0.3 close parenthesis, and the damage term is plotted on the x axis from parenthesis 0 to 2 times 10 superscript 6 close parenthesis. In tension, the primary Poisson's ratio shows an overall decreasing trend; whereas, in compression, it shows the opposite trend. As with the C subscript 11 damage function, the tensile behavior of the second material integrity term, C subscript 12, is observed to be more sensitive to damage.

Figure 130. Graph. Comparison of tension and compression of C12 damage function.

Figure 131. Graph. Comparison of tension and compression of C22 damage function. This figure shows the characteristic relationship for the second material integrity term, C subscript 22, as a function of damage, S, for the three characterization test and for the best fit model. The material integrity term is plotted on the y axis from parenthesis -20 to 0 close parenthesis, and the damage term is plotted on the x axis from parenthesis 0 to 2 times 10 superscript 6 close parenthesis. In compression, the third material integrity term, C subscript 22, has a somewhat smaller absolute value at the same damage level. However, the tension behavior localizes at a much smaller damage level than the compression data, and the ultimate value of the third material integrity term, C subscript 22, in compression has a much larger absolute value.

Figure 131. Graph. Comparison of tension and compression of C22 damage function.

5.3. Viscoplastic Modeling of Asphalt Concrete in Compression

5.3.1. A Phenomenological Model Considering Pulse Time Effects

As a first step toward developing a mechanistic material model for the behavior of HMA in compression, a series of analyses was performed on VT and VL test data, and a phenomenological model was developed. The modeling approach adopted in the phenomenological model was based on the existing strain-hardening model presented for describing the tensile behavior in section 4.3 and shown in the general form as follows:

Equation 158. Simple flow rule as a function of stress and viscoplastic strain. The viscoplastic strain rate, epsilon overdot subscript vp, is equal to a function of stress, f, parenthesis sigma, close parenthesis, divided by a viscoplastic function, g, parenthesis epsilon subscript vp, close parenthesis.       (158)

Where:

Epsilon overdot subscript vp = Viscoplastic strain rate.

σ = Stress.

εvp = Viscoplastic strain.

As shown in equation 158, the viscoplastic strain rate was represented by the combination of two functions, f(σ) and g(εvp), which allowed the stress rate dependency and strain hardening to be taken into consideration in the model. Equation 158 can be generalized as equation 159, which accounts for the effect of the pulse time.

Equation 159. Flow rule considering pulse time effect. The viscoplastic strain rate, epsilon overdot subscript vp, is represented as a function of parenthesis epsilon subscript vp, sigma, t subscript p, close parenthesis.       (159)

Where:

tp = The loading time.

The exact form of the function, F, is presented along with experimental data in the following sections.

5.3.1.1. Tests Performed in This Study

Three types of repetitive creep and recovery tests were performed for the phenomenological model development, including the creep and recovery test with VL, the creep and recovery tests with VT, and the creep and recovery test with a constant load level and CLT. All the tests were conducted at 55 °C under the confining pressure of 500 kPa. Experimental details for these tests are given in the following sections because this particular study uses different test conditions than those outlined in chapter 3.

5.3.1.1.1. Creep and Recovery Test with VL:

The creep and recovery test with a VL test was performed with 200 kPa as the starting load level. An incremental factor of 1.0245 was used for the subsequent load levels to increase the load level until the complete failure of the specimen. The loading time and rest period were 0.1 and 10 s, respectively.

5.3.1.1.2. Creep and Recovery Tests with VT:

The creep and recovery test with a VT test was performed with a loading block consisting of 30 different loading times. The loading time varied from 0.005 to 2.0 s with an incremental factor of 1.1356. The rest period for each load cycle was 30 times that of each loading time. The VT tests were performed at three different load levels, 1,800, 2,000, and 2,200 kPa.

5.3.1.1.3. Creep and Recovery Test with CLT:

In this test, a constant load level and constant loading time were used for each test. Load levels and loading times were changed between tests. Three load levels of 1,800, 2,000, and 2,200 kPa were used with the loading time of 1.6 s. For the 2,000 kPa load level, the creep and recovery tests were conducted with three additional loading times of 0.1, 0.4, and 6.4 s.

The VL and VT tests were used to understand the effects of load level and loading time on the permanent deformation behavior of HMA and to calibrate the phenomenological model. The CLT tests were used to verify the calibrated model.

5.3.1.2. Model Characterization

By observing the viscoplastic strain rate versus the viscoplastic strain VT in figure 132, a constitutive relationship between these viscoplastic media was defined, as shown in equation 160.

Equation 160. Definition of phenomenological model considering pulse time and load level in logarithmic form. The logarithm of rate of viscoplastic strain, epsilon overdot subscript vp, is equal to coefficient a, as a function of t subscript p, multiplied by logarithm of epsilon subscript vp, plus D, as a function of t subscript p and stress, sigma.       (160)

Where:

a(tp) = Material function of loading time.

D(tp, σ) = Intercept of the curve on viscoplastic strain rate axis.

Figure 132. Graph. Incremental viscoplastic strain rate versus viscoplastic strain (500 kPa confinement, 2,000 kPa). This figure shows the incremental viscoplastic strain rate, which is the incremental viscoplastic strain divided by pulse time. It is plotted with respect to viscoplastic strain. On the x axis, the viscoplastic strain is plotted logarithmically from parenthesis 0.001 to 0.1 close parenthesis, and viscoplastic strain rate is shown logarithmically on the y axis from parenthesis 1 times 10 superscript -5 to 0.1 close parenthesis strain per second. For this analysis, the variable loading time test at 500 kPa confining pressure and 2,000 kPa deviatoric stress is utilized. For all pulse times, the viscoplastic strain rate is shown to decrease with increasing viscoplastic strain. It is also shown that at the same viscoplastic strain level, the incremental viscoplastic strain rate is lower for the longer loading times. This observation implies most of permanent strain develops in the beginning of loading.

Figure 132. Graph. Incremental viscoplastic strain rate versus viscoplastic strain
(500 kPa confinement, 2,000 kPa).

In equation 160, a was a function of the loading time, and D was a function of the load level and loading time. Equation 160 can be represented in equation 161, which was a generalized form of equation 158.

Equation 161. Definition of phenomenological model considering pulse time and load level in normal form. The viscoplastic strain rate, epsilon overdot subscript vp, is equal to square bracket parenthesis epsilon subscript vp, close parenthesis, raised to the power of coefficient a, parenthesis t subscript p, close parenthesis, multiplied by parenthesis t subscript p, close parenthesis, raised to the power of coefficient b, multiplied by 10 raised to the power of coefficient d, square bracket multiplied by 10 raised to the power of coefficient c parenthesis stress, sigma, close parenthesis, end square bracket.       (161)

Where:

k(tp) = Function of loading time.

Equation 160 required the determination of a(tp) and D(tp, σ) to calculate the viscoplastic strain rate for a given viscoplastic strain. Values of a(tp) and D(tp, σ) for given loading times could be found by fitting log functions against each viscoplastic strain rate versus viscoplastic strain curve corresponding to given loading time. At this time, the values of a(tp) can be represented by the second logarithm function, as shown in equation 162.

Equation 162. Definition of function a. The function, a, parenthesis t subscript p, close parenthesis, is equal to coefficient a subscript 1 multiplied by logarithmic t subscript p plus coefficient a subscript 2.       (162)

Where:

a1 and a2 = Material-dependent constants.

In order to determine the form of D(tp, σ), it was assumed that D(tp, σ) could be represented by the summation of the loading time term and the load level term, as shown in equation 163.

Equation 163. Definition of function D. The function D, parenthesis t subscript p, stress, sigma, close parenthesis, is equal to coefficient b multiplied by logarithmic t subscript p plus function c, parenthesis stress, sigma, close parenthesis, plus coefficient d.       (163)

Where:

The function c(σ) was given by c1log(σ) when the stress was less than 1,000 kPa and by c when the stress was greater than 1,000 kPa. The coefficient b was determined by fitting equation 163 against D(tp, σ) from the VT test. At this time, it was assumed that c(σ) and d constitute one constant that accounts for the effect of load level. The function c(σ) was then fitted by the logarithmic function and the linear function. Finally, coefficient d was determined by fitting equation 163 for the viscoplastic strain rate versus the viscoplastic strain curve of the VL and VT tests.

Figure 133 and figure 134 show the fitting results and the coefficients that were determined. Figure 135 and figure 136 presents predictions for the VT and VL tests, which were then used for the characterization process.

Figure 133. Graph. Determined fitting results and coefficients of function a(tp). This figure shows that the function to incorporate the loading time effect on viscoplastic strain development is determined by fitting the function to measurement. The loading time is shown logarithmically on the x axis from parenthesis 0.01 to 10 close parenthesis seconds, and the function, a, is shown on the y axis from parenthesis -3 to 0 close parenthesis. The analytical expression is shown on the graph, and the function of pulse time, a parenthesis t subscript p close parenthesis, is equal to -0.66015 multiplied to logarithm of pulse time, t subscript p, minus 2.18288.

Figure 133. Graph. Determined fitting results and coefficients of function a(tp).

Figure 134. Graph. Determined fitting results and coefficients of function D(tp,  ). This figure shows that the function to incorporate the load level effect on viscoplastic strain development is determined by fitting the function to measurement. The stress function, D, is shown on the y axis from parenthesis -12 to 0 close parenthesis, and the stress is shown on the x axis from parenthesis 0 to 5 close parenthesis, MPa. Also shown on the graph are the piecewise analytical expressions for the stress function. For stress levels below 1 MPa the function D is equal to 4.66371 multiplied by the logarithm of stress, sigma, minus 7.18183. For stress levels greater than 1 MPa, the function D is equal to 1.01300 multiplied by stress, sigma, minus 8.13657.

Figure 134. Graph. Determined fitting results and coefficients of function D(tp σ).

Figure 135. Graph. VT predictions. This figure shows the prediction for variable loading time tests at 500 kPa confinement and 2,000 kPa deviatoric stress along with results from three replicate tests. The cumulative loading time is shown on the x axis from parenthesis 0 to 160 close parenthesis seconds, and viscoplastic strain is shown on the y axis from parenthesis 0 to 4 percent close parenthesis. Given the variability of the test results, the phenomenological model predictions are reasonable.

Figure 135. Graph. VT predictions.

Figure 136. Graph. VL predictions. This figure shows the prediction for variable loading level tests at 500 kPa confinement and 2,000 kPa deviatoric stress along with results from three replicate tests. The cumulative loading time is shown on the x axis from parenthesis 0 to 15 close parenthesis seconds, and viscoplastic strain is shown on the y axis from parenthesis 0 to 4 percent close parenthesis. Given the variability of the test results, the phenomenological model predictions are reasonable.

Figure 136. Graph. VL predictions.

5.3.1.3. Verification of the Model

As shown in figure 137 through figure 142, predictions were made for the CLT tests. Although the model was able to account for differences in the viscoplastic development for various loading times, overall predictions were not as accurate. Several causes for the discrepancy between viscoplastic strain predictions and measurements could be suggested. However, the inability of the model to consider strain history was a highly probable cause for this discrepancy, given that the fitting results for the VT and VL tests were acceptable.

Figure 137. Graph. CLT predictions (2.0 MPa deviatoric stress—0.1-s pulse time). This figure shows the measured and predicted values from a constant load time test at 2.0 MPa load level and a pulse time of 0.1 second. The x axis shows cumulative loading time from parenthesis 0 to 40 close parenthesis second, and the viscoplastic strain is plotted in percentage from parenthesis 0 to 4 close parenthesis. The prediction is within the observed variability.

Figure 137. Graph. CLT predictions (2.0 MPa deviatoric stress-0.1-s pulse time).

Figure 138. Graph. CLT predictions (2.0 MPa deviatoric stress—0.4-s pulse time). This figure shows the measured and predicted values from a constant load time test at 2.0 MPa load level and a pulse time of 0.4 second. The x axis shows cumulative loading time from parenthesis 0 to 40 close parenthesis second, and the viscoplastic strain is plotted in percentage from parenthesis 0 to 4 close parenthesis. The prediction is at the very bottom edge of the observed test variability.

Figure 138. Graph. CLT predictions (2.0 MPa deviatoric stress-0.4-s pulse time).

Figure 139. Graph. CLT predictions (2.0 MPa deviatoric stress—1.6-s pulse time). This figure shows the measured and predicted values from a constant load time test at 2.0 MPa load level and a pulse time of 1.6 s. The x axis shows cumulative loading time from parenthesis 0 to 120 close parenthesis second, and the viscoplastic strain is plotted in percentage from parenthesis 0 to 4 close parenthesis. The prediction is slightly below the observed test variability.

Figure 139. Graph. CLT predictions (2.0 MPa deviatoric stress-1.6-s pulse time).

Figure 140. Graph. CLT predictions (2.0 MPa deviatoric stress—6.4-s pulse time). This figure shows the measured and predicted values from a constant load time test at 2.0 MPa load level and a pulse time of 6.4 s. The x axis shows cumulative loading time from parenthesis 0 to 200 close parenthesis second, and the viscoplastic strain is plotted in percentage from parenthesis 0 to 4 close parenthesis. The prediction is slightly below the observed variability.

Figure 140. Graph. CLT predictions (2.0 MPa deviatoric stress-6.4-s pulse time).

Figure 141.Graph. CLT predictions (1.8 MPa deviatoric stress—1.6-s pulse time). This figure shows the measured and predicted values from a constant load time test at 1.8 MPa load level and a pulse time of 1.6 s. The x axis shows cumulative loading time from parenthesis 0 to 200 close parenthesis second, and the viscoplastic strain is plotted in percentage from parenthesis 0 to 4 close parenthesis. The prediction is within the observed variability.

Figure 141. Graph. CLT predictions (1.8 MPa deviatoric stress-1.6-s pulse time).

Figure 142. Graph. CLT predictions (2.2 MPa deviatoric stress—1.6-s pulse time). This figure shows the measured and predicted values from a constant load time test at 2.2 MPa load level and a pulse time of 1.6 s. The x axis shows cumulative loading time from parenthesis 0 to 40 close parenthesis second, and the viscoplastic strain is plotted in percentage from parenthesis 0 to 4 close parenthesis. The prediction is underestimating the measured strains by approximately half.

Figure 142. Graph. CLT predictions (2.2 MPa deviatoric stress-1.6-s pulse time).

5.3.2. HISS-Perzyna Model

The HISS-Perzyna model, suggested by the Delft University of Technology and the University of Maryland, was investigated with respect to the data set obtained from experimental tests.(26,28,36) A prediction using the Delft University of Technology model was not be made because of numerical problems. However, the characterization process using the t-TS principle and coefficients of the model are described in subsection 5.3.2.1.

5.3.2.1. Delft University of Technology Model

The model suggested by researchers at the Delft University of Technology required the development of several relationships between the material parameters and the strain rates obtained from constant strain-rate tests. In this research, the t-TS principle was utilized to simplify the modeling effort and to reduce the number of relationships required. With the assumption that the yield stress in deviatoric stress space was presented as circular, (β=0), equation 72 was be reduced to equation 164.

Equation 164. Definition of simplified HISS model. The square root of the second deviatoric stress invariant, J subscript 2, is equal to the square root of expression represented by softening parameter, gamma, multiplied by parenthesis the first stress invariant, I subscript 1, minus the tensile strength of material when deviatoric stress is 0, R, close parenthesis raised to the power of 2 minus the hardening parameter, alpha, multiplied by parenthesis first stress invariant, I subscript 1, minus the tensile strength of material when deviatoric stress is 0, R, close parenthesis, raised to power of function n.       (164)

Where:

Φ = Softening parameter.

α = Hardening parameter.

R = Tensile strength of material when deviatoric stress is 0.

n = Parameter that determines shape of yield stress.

Figure 143 shows peak stresses for a series of compressive and tensile constant strain-rate tests; the strain rates are listed in table 4. These peak stresses were used as fundamental quantities to develop relationships between the material parameters and the reduced strain rates. R and γ0 could be determined as functions of the reduced strain rate by plotting the compressive and tensile peak stresses obtained from the constant strain-rate tests under a certain strain rate and then taking the slope and x-intercept of the line. In the model, R and γ0 represented the tensile strength for hydrostatic stress and the softening of the material in the post-peak region, respectively. The parameter n governs the overall shape and size of the yield function and was related to the dilation of the material. The beginning of dilation was defined as the stress at the minimum plastic volumetric strain because the plastic deviatoric strain and elastic strain (or viscoelastic strain) was assumed to not be associated with the volumetric change of a material. In addition, because HMA specimens dilated after a little compression as the compressive stress increased, the dilation stress could be determined. Once the dilation stresses were determined for several strain rates, the value of n could be determined using equation 165.

Equation 165. Definition of n function. The power function, n, is equal to 2 divided expression represented with 1 minus second deviatoric stress invariant, J subscript 2 subscript dilation, divided by softening function, gamma, parenthesis first stress invariant, I subscript 1 subscript dilation, minus the tensile strength of material when deviatoric stress is 0, R close parenthesis, raised to a power of 2.       (165)

Where:

I1,diation = I1 at beginning of dilation.

J2,dilation = J2 at beginning of dilation.

Once n is determined, α0 can be readily determined using equation 166.

Equation 166. Definition of initial alpha function. The initial hardening function, alpha subscript 0, is equal to initial softening, gamma subscript 0 multiplied by parenthesis the tensile strength of material when deviatoric stress is 0, R, divided by atmospheric pressure, P subscript a, close parenthesis, raised to the power of 2 minus n.       (166)

The sigmoidal function was used to represent relationships between the reduced strain rate and the material parameters. The form of the function and the coefficients determined for each parameter are listed in equation 167 and table 20. Figure 144 through figure 147 show a comparison of measured values versus predicted values at various reduced strain rates.

Equation 167. The form of sigmoidal function for softening or function R. The logarithmic parenthesis of the initial softening function, gamma subscript 0, or the tensile strength of material when deviatoric stress is 0, R, close parenthesis, is equal to coefficient a plus coefficient b, divided by 1 minus the exponential of parenthesis coefficient d minus logarithmic parenthesis reduced strain rate, epsilon overdot subscript reduced, close parenthesis.       (167)

Figure 143. Graph. Compressive and tensile peak stress in SQRT(J2)—I1 space. This figure shows the compressive and tensile peak stresses measured from unconfined constant crosshead rate tests plotted in square root of second deviatoric stress invariant, J subscript 2, first stress invariant, I subscript 1, space. The first stress invariant, I subscript 1, is plotted on the x axis from parenthesis -10,000 to 6,000 close parenthesis kPa, and the second deviatoric stress invariant, J subscript 2, is shown on the y axis from parenthesis 0 to 6,000 close parenthesis kPa. The results from tension tests are shown to follow a pattern that increases linearly in zone 1 of the plot space. The results from compression tests are shown to increase linearly through zone 2 of the plot space.

Figure 143. Graph. Compressive and tensile peak stress in SQRT(J2) - I1 space.

Table 20. Delft material model coefficients functions.

 

a

b

d

e

R

316.86

17,074.55

-9.28

1.75

γ

0.20

-0.15

-13.91

2.13

n

2.00

2075.86

0.48

1.06

α

-0.63

-93.39

-5.01

1.35

Figure 148 shows the strain rate-dependent yield surface that was constructed using the characterized parameters when the viscoplastic strain was equal to zero (i.e., the initial yield surface). It was observed that the initial yield surface increased as the temperature decreased; the reduced strain rate increased. This behavior coincided with observations from constant strain-rate tests in which more viscoplastic strains were developed under a small, reduced strain rate (or higher temperature).

As shown in equation 164, the second term in the square root always had to be smaller than the first term in order to construct a valid yield surface. However, because of the numerical errors involved in the characterization process of α and n, the prediction program was often required to calculate the square root of a negative number during analysis. This situation was encountered without t-TS, as mentioned by others.(36)

Figure 144. Graph. Determined   parameter function. This figure shows the change in Delft model initial softening parameter, gamma subscript 0, with reduced strain rate. The reduced strain rate is shown on the x axis from parenthesis 1 times 10 superscript -10  to 1 times 10 superscript -2 close parenthesis strain per second. The initial softening parameter, gamma subscript 0, is shown on the y axis from parenthesis 0 to 0.25 close parenthesis. The initial softening term shows a sigmoidal shape with reduced strain rate where, at the lowest rates, its value is approximately 0.2, and at the highest rates it is approximately 0.05.

Figure 144. Graph. Determined γ0 parameter function.

Figure 145. Graph. Determined R parameter function. This figure shows the change in the Delft model parameter relating to the tensile strength of the material when the deviatoric stress is 0, R, with reduced strain rate. The reduced strain rate is shown on the x axis from parenthesis 1 times 10 superscript -10 to 1 times 10 superscript -2 close parenthesis strain per second. The tensile strength of the material when the deviatoric stress is 0, R, is shown on the y axis from parenthesis 0 to 18,000 close parenthesis. This parameter also displays a sigmoidal shape with the low reduced rate value of approximately 300 and a high reduced rate value of approximately 17,500.

Figure 145. Graph. Determined R parameter function.

Figure 146. Graph. Determined n parameter function. This figure shows the change in the Delft model power term, n, with reduced strain rate. The reduced strain rate is shown on the x axis from parenthesis 1 times 10 superscript -10 to 1 times 10 superscript -2 close parenthesis strain per second. The power term, n, is shown on the y axis from parenthesis 0 to 20 close parenthesis. Over the reduced rates shown, this function shows an exponential growth with a value of 2 at the slowest reduced rates.

Figure 146. Graph. Determined n parameter function.

Figure 147. Graph. Determined   parameter function. This figure shows the change in the Delft model initial hardening parameter, alpha subscript 0, with reduced strain rate. The reduced strain rate is shown on the x axis from parenthesis 1 times 10 superscript -10 to 1 times 10 superscript -2 close parenthesis strain per second. The initial hardening parameter, alpha subscript 0, is shown on the y axis from parenthesis 0 to 0.25 close parenthesis. This function varies in a sigmoidal manner from an initial value of approximately 0.20 to a final value of 0.

Figure 147. Graph. Determined α0 parameter function.

Figure 148. Graph. Rate-dependent initial yield surface. This figure shows the initial yield surfaces in square root of the second deviatoric stress invariant, J subscript 2, versus first stress invariant, I subscript 1, space as a function of temperature and rate. The x axis shows the square root of the second deviatoric stress invariant, J subscript 2, from parenthesis 0 to 2,500 close parenthesis kPa while the y axis shows the first stress invariant, I subscript 1, from parenthesis -20,000 to 20,000 close parenthesis. It is found that as the temperature decreases the size of the initial yield surface increases. The magnitudes increase in both the vertical, square root of second deviatoric stress invariant, J subscript 2, and first stress invariant, I subscript 1, directions.

Figure 148. Graph. Rate-dependent initial yield surface.

5.3.2.2. University of Maryland Model

As shown in equation 73, a simplified HISS-Perzyna model was suggested by researchers at the University of Maryland.(28) Equation 168 represents a general form of the hardening function used for the suggested model.

Equation 168. The definition of hardening function. The hardening function, alpha, is equal to alpha subscript 0 multiplied by exponential from parenthesis minus kappa multiplied by epsilon subscript vp, close parenthesis.       (168)

Where:

α0 and Κ = Material constants.

However, the observation made in subsection 5.1.2.3 indicates that a single hardening function, equation 168, was not sufficient to represent the characteristic behavior of the material, such as softening during unloading. Therefore, one more variable, the viscoplastic strain increment during loading, was introduced into equation 168. Equation 169 represents the modified functionα to incorporate the variation of the viscoplastic strain rate during the pulse time in the existing model. α1 and α2 in equation 169 describe general variations of α in terms of viscoplastic strain and a local variation of α in terms of incremental viscoplastic strain in a pulse, respectively.

Equation 169. The definition of hardening function to consider pulse time. The time considerate hardening function, alpha, is equal to kappa subscript 1 multiplied by alpha subscript 1 multiplied by alpha subscript 2.       (169)

Where:

Equation 170. The definition of coefficients in pulse time considerate hardening function. Alpha subscript 1 is equal to exponent of negative material coefficient kappa subscript 2 multiplied by the viscoplastic strain level, epsilon subscript vp.       (170)

Equation 171. The definition of coefficients in pulse time considerate hardening function. Alpha subscript 2 is equal to 1 minus the exponent of negative material coefficient kappa subscript 2 multiplied by the change in viscoplastic strain level, uppercase delta epsilon subscript vp.       (171)

Figure 149 presents the variation of α determined by using a modified alpha-viscoplastic relationship for five 6.4-s pulses with 1,800 kPa of load level. As shown, α was no longer a simple decreasing function of the viscoplastic strain, but had multiple decreasing functions of which independent variables were incremental viscoplastic strain during each load pulse and overall viscoplastic strain. The incremental viscoplastic strain was reset to zero each time the material was unloaded.

Figure 150 through figure 152 present measured and predicted viscoplastic strains by using a modified hardening function. The model was able to describe viscoplastic strain development for various loading conditions, such as VT, RVT, and CLT; this capability was not possible in the existing HISS-Perzyna model. However, even though incremental viscoplastic strain in a pulse described multiple hardening rates at certain viscoplastic strains, it was more reasonable to assume that the multiple hardening rates were caused by the viscoelastic property of the material, given the rate dependency of the softening. Therefore, a viscoplastic model with a rate-dependent hardening-softening function was developed and is presented in subsection 5.3.3.

Figure 149. Graph. Variation of   for 1,800 kPa CLT loading (500 kPa confinement). This figure shows the backcalculated hardening function alpha required to describe hardening-softening behavior of the material under constant loading time test at confining pressure of 500 kPa with deviatoric stress of 1,800 kPa as a function of viscoplastic strain for five different loading pulses. The percent viscoplastic strain is shown on the x axis from parenthesis 0 to 1.6 close parenthesis, and the hardening function, alpha, is shown on the y axis from parenthesis 0 to 4.5 times 10 superscript -2 close parenthesis. Overall, the value of alpha decreases as viscoplastic strain increases, but to describe the material behavior, it must increase between the final loading of the first pulse and the first loading of the next pulse.

Figure 149. Graph. Variation of α for 1,800 kPa CLT loading (500 kPa confinement).

Figure 150. Graph. Viscoplastic strain predictions for VT tests (500 kPa confinement). This figure shows the measured and predicted response for variable time tests at 500 kPa confinement. The model predictions are made using a conventional viscoplastic model characterized using only 1.8 MPa deviatoric and 500 kPa confining variable time test. The x axis shows time from parenthesis 0 to 5,000 close parenthesis seconds, and the y axis shows viscoplastic strain in percentages from parenthesis 0 to 1.6 close parenthesis. The predictions show that the model does not capture the overall experimentally observed trends regarding pulse time and does not capture the magnitude of the stress effect on the material response.

Figure 150. Graph. Viscoplastic strain predictions for VT tests (500 kPa confinement).

Figure 151. Graph. Viscoplastic strain predictions for CLT tests (500 kPa confinement). This figure shows the measured and predicted response for constant load and time tests at 500 kPa confinement. The model predictions are made using a conventional viscoplastic model characterized using only 1.8 MPa deviatoric and 500 kPa confining variable time test. The x axis shows time from parenthesis 0 to 5,000 close parenthesis seconds, and the y axis shows viscoplastic strain in percentages from parenthesis 0 to 1.6 close parenthesis. The predictions show that the model does not capture the overall experimentally observed trends regarding pulse time and does not capture the magnitude of the stress effect on the material response.

Figure 151. Graph. Viscoplastic strain predictions for CLT tests (500 kPa confinement).

Figure 152. Graph. Viscoplastic strain predictions for RVT tests (500 kPa confinement). This figure shows the measured and predicted response for reversed variable time tests at 500 kPa confinement. The model predictions are made using a conventional viscoplastic model characterized using only 1.8 MPa deviatoric and 500 kPa confining variable time test. The x axis shows time from parenthesis 0 to 5,000 close parenthesis seconds, and the y axis shows viscoplastic strain in percentages from parenthesis 0 to 1.6 close parenthesis. The predictions show that the model does not capture the overall experimentally observed trends regarding pulse time and does not capture the magnitude of the stress effect on the material response.

Figure 152. Graph. Viscoplastic strain predictions for RVT tests (500 kPa confinement).

5.3.3. Development of a Viscoplastic Model Using Rate-Dependent Yield Stress

The model developed in this research was capable of capturing both additional hardening that was due to aggregate interlocking and rate-dependent softening due to viscoelastic relaxation. Viscosity in Perzyna's evolution law is separated into a constant term and a viscoplastic strain-dependent term that together represent the change of viscosity in viscoplastic flow. The yield stress function that takes into account rate-dependent hardening and softening is also described in subsection 5.3.3.1.

5.3.3.1. Flow Rule and Yield Function for Developed Viscoplastic Model

As an expansion of equation 55, a general flow rule for materials exhibiting kinematic and isotropic hardening is represented in equation 172. m amplified or reduced the stress rate dependency of the model, and D determined the viscosity in the viscoplastic flow. When D was a constant, it was assumed that the effect of the change in viscosity on the response of the material was taken into account by the yield stress function. However, when variations of yield stress were affected by the viscoelastic property of the material, it seemed reasonable to consider the viscosity in the viscoplastic flow as a function that was not subjected to yield stress.

Equation 172. The form of general flow rule. The viscoplastic strain rate, epsilon overdot subscript ij superscript vp, is equal to Macauley bracket stress function, f, parenthesis stress, sigma, minus hardening function, alpha, close parenthesis, minus coefficient, r, divided by viscosity parameter, D, end Macauley bracket, multiplied by the  partial derive of stress function, f, with respect to stress, sigma subscript ij, in index notation.       (172)

Where:

α = Kinematic hardening function.

γ = Isotropic hardening function.

D = Viscosity parameter.

m = Rate-dependency parameter.

Therefore, a flow rule that takes into consideration the variations of viscosity in the viscoplastic flow is suggested in equation 173 by incorporating Perzyna's flow rule and Von Mises' yield criterion. In equation 173, D was the viscosity and represented the scalar hardening and softening as described above. The anisotropic behavior of the material was also integrated by using Dij. Meanwhile, Gij represented the orientation-dependent isotropic hardening function that reflected the viscoplastic and viscoelastic property of the material. Because the material was subjected only to compressive stress, the kinematic hardening rule was not introduced in this model. The viscosity (D) was related to aggregate interlocking and was represented as a function of the viscoplastic strain, as shown in equation 174. Because the function could represent both increasing and decreasing viscosity according to the viscoplastic strain, it had the potential to represent the behavior of HMA mixtures in the tertiary region as well as in the primary and secondary regions.

Equation 173. The general flow rule with von Mises’ yield concept. The viscoplastic strain tensor, epsilon overdot subscript ij superscript vp, is equal to 3 divided by 2 multiplied by 1 divided by coefficient eta subscript 0, multiplied by Macauley bracket the second invariant function, J, parenthesis stress, sigma subscript ij, minus the yield stress tensor, G subscript ij, close parenthesis, divided by viscosity function, D, end Macauley bracket, multiplied by parenthesis deviatoric stress tensor, s subscript ij, minus deviatoric back stress tensor, g subscript ij, close parenthesis, divided by the second invariant function, J, parenthesis stress, sigma subscript ij, minus the yield stress tensor, G subscript ij, close parenthesis. This also equals coefficient B, multiplied by Macauley bracket the second invariant function, J, parenthesis stress, sigma subscript ij, minus the yield stress tensor, G subscript ij, close parenthesis, divided by viscosity function, D, end Macauley bracket, multiplied by parenthesis deviatoric stress tensor, s subscript ij, minus deviatoric back stress tensor, g subscript ij, close parenthesis, divided by the second invariant function, J, parenthesis stress, sigma subscript ij, minus the yield stress tensor, G subscript ij, close parenthesis.       (173)

Where:

D = Viscosity related to aggregate interlocking, equation 174.

η0, m, α, β, γ = Material constants.

D0 = Initial viscosity.

sij = Deviatoric stress tensor.

gij = Deviatoric back stress tensor.

σij = Stress tensor.

Gij = Yield stress tensor.

J(σij -Gij) = The second invariant of (σij - Gij).

Equation 174. Definition of function D. The viscosity function, D, is equal to initial viscosity, D subscript 0 plus material constant alpha multiplied by sine square bracket material constant beta multiplied by 1 minus exponential of negative material constant gamma multiplied by epsilon subscript vp, close parenthesis, end square bracket.       (174)

However, as discussed in subsection 5.1.2.3 , rate-dependent softening, which implied the possibility of a multiple state of the material at certain viscoplastic strains, was observed when HMA was subjected to repetitive loading. In order to introduce the characteristic behavior of HMA into the viscoplastic constitutive model, equation 175, which was one of the simplest forms, was suggested as the hardening-softening function.

Equation 175. General definition of rate-dependent yield function. The yield stress rate, G overdot, is equal to function, g, parenthesis viscoplastic strain rate, epsilon overdot subscript vp, viscoplastic strain level, epsilon subscript vp, and the yield function value, G, close parenthesis.       (175)

In equation 175, the yield stress increased as the viscoplastic strain and viscoplastic strain rate increased during loading, whereas it decreased during unloading (when the viscoplastic strain rate was zero). Figure 153 presents a schematic concept of the variation of yield stress subjected to a creep and recovery loading condition. The remaining yield stress, which was yield stress at the asymptote, was governed by the viscoplastic strain at the end of the loading condition, ε0vp, and material constants E1 and E2. The decreasing yield stress during unloading allowed a multiple viscoplastic strain rate at a certain viscoplastic strain.

Figure 153. Illustration. Variation of yield stress (Standard Linear Solid model). This figure shows the concept of rate-dependent hardening-softening function. The yield stress, G, is shown on the y axis, and time, t, is represented on the x axis. The yield stress is plotted for a standard creep and recovery test using a standard linear model to represent the yield function and the mechanical analog in figure 5 to represent the total viscoplastic model. The standard linear model consists of an isolated spring with stiffness, E subscript 1, connected in series with a spring of stiffness, E subscript 2, and a dashpot of viscosity, eta subscript 1, which is connected to one another in parallel. During loading, the yield stress is shown to increase very quickly at first and then continue increasing a diminished rate. Upon unloading, the yield stress is shown to decrease very rapidly at first and then continue to decrease at a reduced rate until reaching an asymptotic value. This asymptotic value is shown to equal the quotient of the sum of the isolated spring stiffness, E subscript 1, and the parallel spring stiffness, E subscript 2, and the product of the isolated spring stiffness, E subscript 1, and the parallel spring stiffness, E subscript 2; multiplied by the initial viscoplastic strain for the respective cycle.

Figure 153. Illustration. Variation of yield stress (Standard Linear Solid model).

Because of their simplicity, equation 176 was suggested as a hardening-softening function to confirm the characteristics of the model for an arbitrary stress history, and equation 177 was suggested to predict the actual behavior of the HMA mixture. As shown in equation 177, the hardening-softening function was represented as the convolution integral, including the relaxation modulus. Material constants A and B were introduced into the relaxation modulus to develop a relationship between the relaxation modulus, which was the LVE property, and the viscoplastic yield stress. Additionally, by utilizing the material constants, the number of parameters needed to calibrate the model could be reduced. In calculating the yield stress, the state variable approach was used to reduce computational time, as shown in equation 178.

Equation 176. Definition of rate-dependent yield function for hardening and softening. The yield stress rate, G overdot, is equal to material constant E subscript 1 multiplied by material constant E subscript 2 divided by material constant nu subscript 2 multiplied by viscoplastic strain, epsilon subscript vp, plus material constant E subscript 1 multiplied by viscoplastic strain rate, epsilon overdot superscript vp, minus parenthesis material coefficient E subscript 1 plus material constant E subscript 2 over material constant nu subscript 2, close parenthesis, multiplied by yield function, G.       (176)

Where:

E1, E2, η1, η2 = Material constants.

Equation 177. Definition of rate-dependent yield function. The yield stress, G, is equal to the convolution integral of relaxation modulus as a function of reduced time, tau, minus the reduced time. The relaxation modulus is represented with coefficient A multiplied by Prony coefficient E subscript 0 plus coefficient B, multiplied by the summation of Prony coefficients, E subscript i, multiplied by the exponential of negative time, t, minus integration constant, tau, divided by relaxation time, rho subscript i, multiplied by the derivative of viscoplastic strain, epsilon subscript vp, with respect to the integration constant, tau, multiplied by the derivative of the integration constant dtau.       (177)

Where:

A, B = Material constants.

E0, Ei = Prony coefficients determined for the relaxation modulus.

ρi = Relaxation time.

Equation 177 was solved using the state variable approach to predict strains and calculate pseudo strains. This approach is shown mathematically in equation 178.

Equation 178. Yield function with state variable approach. The yield stress, G, at time step i , G superscript n plus 1, is equal to elastic state variable function, xi subscript 0 superscript n plus 1, plus the summation of the individual time varying state variable functions, xi subscript i superscript n plus 1 from 1 to m.       (178)

Where:

Equation 179. Elastic function in state variable approach. The elastic response of the state variable at the following time step, xi subscript 0 superscript n plus 1, equals material coefficient A, multiplied by the difference in the viscoplastic strain level at the next time step raised to the power of time, epsilon subscript vp superscript t superscript n plus 1, and the viscoplastic strain level at the current time step, epsilon subscript vp superscript t superscript n, multiplied by long time modulus, E subscript 0.      (179)

Equation 180. Time varying function in state variable approach. The viscoelastic response of the state variable at the following time step for the given Prony term, xi superscript n plus 1 subscript i, equals material coefficient B, multiplied by the product of exponent negative change in time, uppercase delta t, divided by Prony relaxation time, rho subscript i, and the time varying function for the previous time step for the given Prony term, xi superscript n subscript i, plus the product of exponent negative change in time, uppercase delta t, divided by 2 multiplied by the Prony relaxation time, rho subscript i, the difference in the viscoplastic strain level at the next time step raised to the power of time, epsilon subscript vp superscript t superscript n plus 1, and the viscoplastic strain level at the current time step, epsilon subscript vp superscript t superscript n, and the current Prony term, E subscript i.       (180)

5.3.3.2. Characteristics of the Developed Model for Arbitrary Stress History

In order to confirm the characteristics of the developed viscoplastic model, the following predictions were made for the arbitrary stress histories by using equation 173 and equation 176. In this study, D was considered a constant to simplify the calibration and prediction processes. Two sets of stress histories were generated, as shown in figure 154 and figure 157. Table 21 shows the material constants used in this analysis.

Table 21. Material coefficients used for the developed model analysis.

m

D

η

η2

E1

E2

2

3,000

10

50,000

500

200

5.3.3.2.1. Effect of Rest Period:

Figure 154 shows two different stress histories used to check the sensitivity of the viscoplastic model to the effects of rest periods. For both stress histories, stress levels and the cumulative loading time were fixed to 2,000 units less stress and 160 s, respectively. However, for the first stress history, 8.0 s of rest between the loading pulses were allowed, whereas only 1 s of rest was allowed for the second loading history. Figure 155 presents the variation of yield stress for each stress history, and as expected, the model showed different yield stress developments depending upon the rest period.

Figure 156 presents the viscoplastic strain developed by each stress history. It shows more viscoplastic strain for a longer rest period. This result corresponded to the experimental observations made from figure 98.

Figure 154. Graph. Stress histories for rest period analysis. This figure shows the two stress histories used to check the effect of rest period. The stress is shown on the y axis from parenthesis 0 to 2,500 close parenthesis, and the time is shown on the x axis from parenthesis 0 to 350 close parenthesis seconds. For both stress histories, stress level and the cumulative loading time are fixed to 2,000 (unit less stress) and 160 s, respectively. However, for the first stress history, 8 s of rest between the loading pulses are allowed, whereas only 1 second of rest was allowed for the second loading history.

Figure 154. Graph. Stress histories for rest period analysis.

Figure 155. Graph. Yield stress versus cumulative loading time (rest period analysis). This figure shows the modeled effects of rest period. The y axis shows yield stress from parenthesis 0 to 1,000 close parenthesis, and the x axis shows the cumulative loading time from parenthesis 0 to 180 close parenthesis seconds. The two different rest periods shown in the previous figure are shown. For the 1-second rest period data, the yield stress grows mostly consistently from an initial value of 0 to approximately 900. For the 8-second rest period data, the yield stress steadily increases during the loading, but upon unloading, the yield stress recovers and has a smaller value at the beginning of the subsequent loading than at the end of the previous loading.

Figure 155. Graph. Yield stress versus cumulative loading time (rest period analysis).

Figure 156. Graph. Viscoplastic strain versus cumulative loading time (rest period analysis). This figure shows the modeled effects of rest period on permanent strain growth. On the x axis, cumulative loading time is shown from parenthesis 0 to 200 close parenthesis seconds. On the y axis, viscoplastic strain is shown from parenthesis 0 to 5 close parenthesis. The two different rest periods shown in the previous figures are shown.  The viscoplastic strain development in the 1-second rest period test is approximately 3.2 at the end of loading, whereas the development in the 8-second rest period test is approximately 3.7. The cause of this difference is due to the yield stress recovery in the 8-s rest period test.

Figure 156. Graph. Viscoplastic strain versus cumulative loading time (rest period analysis).

5.3.3.2.2. Effect of Loading Time:

Figure 157 presents another set of stress histories used to check the effects of loading time. For these stress histories, the load level, rest periods, and cumulative loading time were fixed to 2,000 units, 4 and 66 s, respectively. However, the first loading history consisted of 6 pulses at 11 s long, and the second loading history consists of 22 pulses at 3 s long.

The analysis results for the given stress histories are shown in figure 158. The loading history with shorter individual loading times showed more viscoplastic strain, which was identical to the CLT test results. As shown, the viscoplastic model that incorporated the softening rule appeared to account for the pulse time effect. As shown in figure 156 and figure 158, the viscoplastic model with the rate-dependent hardening-softening capability could account for the effects of loading time.

Figure 157. Graph. Stress history for loading time analysis. This figure shows two stress histories used to check the effect of loading time. The stress is shown on the y axis from parenthesis 0 to 2,500 close parenthesis, and the time is shown on the x axis from parenthesis 0 to 250 close parenthesis seconds. For both stress histories, stress level, the cumulative loading time, and rest period duration are fixed to 2,000 (unit less), 66 s, and 4 s, respectively. However, the first stress history consists of six pulses 11 s long while the second loading history consists of 22 pulses 3 s long.

Figure 157. Graph. Stress history for loading time analysis.

Figure 158. Graph. Viscoplastic strain versus cumulative loading time (loading time analysis). This figure shows the modeled effects of loading time on permanent strain growth. On the x axis, cumulative loading time is shown from parenthesis 0 to 70 close parenthesis seconds. On the y axis, viscoplastic strain is shown from parenthesis 0 to 3 close parenthesis. The two different loading histories shown in the previous figure are shown. As a result of the decreasing yield stress, the stress history with six pulses 11 s long shows less viscoplastic strain development than stress history with 22 pulses 3 s long.

Figure 158. Graph. Viscoplastic strain versus cumulative loading time (loading time analysis).

5.4. Characterization and Verification of the Viscoplastic Model

5.4.1. Calibration

Prior to the calibration process, data points acquired during the unloading period were filtered to reduce the computational time. Strains measured at the end of the rest periods were defined as the objective function. The nonlinear optimization function (lsqnonlin) in MatlabTM was utilized to minimize errors between the measured and predicted viscoplastic strains. Based on the model calibrated for the VT and VL tests, the viscoplastic strains of the other loading conditions, such as CLT and VLT, could be predicted. Table 22 shows the coefficients determined from the calibration process for 140 and 500 kPa confining pressures.

Table 22. Compression viscoplastic material model coefficients.

Confining Pressure (kPa)

m

D0

α

Beta

γ

A

B

140

3.83

1,951.76

2,758.93

3.07

258.34

0

6.3E-06

500

9.99

3,460.62

3,569.28

1.87

156.96

0

5.2E-03

Figure 159 and figure 160 present the calibration results for VT and VL testing at 140 kPa confining pressure, and figure 161 and figure 162 present the calibration results for VT and VL testing at 500 kPa confining pressure. In general, the predicted and measured viscoplastic strains match very well, although there was a slight discrepancy in the VT and VL 500 kPa confining pressure results.

Figure 159. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT). This figure shows the viscoplastic strain predicted for variable loading time test with confining pressure of 140 kPa at deviatoric stress of 827 kPa compared with measurement. The cumulative loading time is plotted on the x axis from parenthesis 0 to 350 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 1 and 8/10 close parenthesis. The viscoplastic strain growth at 300 s is approximately 1.5 percent. The predicted strain matches the measured strains very well both at the end of loading and throughout the loading history.

Figure 159. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT).

Figure 160. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VL). This figure shows the viscoplastic strain predicted for variable load level test with confining pressure of 140 kPa compared with measurement. The cumulative loading time is plotted on the x axis from parenthesis 0 to 500 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 1.5 close parenthesis. The viscoplastic strain growth at 400 s is approximately 1.5 percent. The predicted strain matches the measured strains very well both at the end of loading and throughout the loading history.

Figure 160. Graph. Viscoplastic strain versus cumulative loading time
(140 kPa confinement VL).

Figure 161. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VT). This figure shows the viscoplastic strain predicted for variable loading time test with confining pressure of 500 kPa at deviatoric stress of 1,800 kPa compared with measurement. The cumulative loading time is plotted on the x axis from parenthesis 0 to 140 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 400 s is approximately 2 percent. Given the observed variability in the test results, the predicted strain matches the measured strains well both at the end of loading and throughout the loading history.

Figure 161. Graph. Viscoplastic strain versus cumulative loading time
(500 kPa confinement VT).

Figure 162. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VL). This figure shows the viscoplastic strain predicted for variable load level test with confining pressure of 500 kPa compared with measurement. The cumulative loading time is plotted on the x axis from parenthesis 0 to 18 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 16 s is approximately 2 to 2.5 percent. Given the observed variability in the test results, the predicted strain matches the measured strains well both at the end of loading and throughout the loading history.

Figure 162. Graph. Viscoplastic strain versus cumulative loading time
(500 kPa confinement VL).

5.4.2. Verification

Viscoplastic strain predictions for the 140-kPa confining pressure tests are presented in figure 163 through figure 166. Figure 163 shows the ability of the developed model to consider the effects of rest periods on the viscoplastic strain development even though the viscoplastic strains were slightly underpredicted. Figure 164 and figure 165 show the predictions for the VLT tests and a low load level VL test, respectively; these predictions were quite good. Figure 166 presents predictions for complex loading histories, which were a combination of VT test results and flow number test results. Up to 0.5-percent strain, the prediction of the viscoplastic strain matched well with the measured viscoplastic strain; however, the viscoplastic strain was underpredicted for the last strain level. This discrepancy could indicate a need to refine the softening function. Figure 167 to figure 172 present the viscoplastic strain predictions made for 500 kPa confining pressure tests. The overall prediction was good considering the complexity of the loading history.

Figure 163. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT). This figure shows the viscoplastic strains in variable loading time tests with 3 different rest periods (0.05, 1, and 200 s) that are plotted with respect to cumulative loading time. Cumulative loading time is plotted on the x axis from parenthesis 0 to 350 close parenthesis seconds, and viscoplastic strains are shown by percentage on the y axis from parenthesis 0 to 1.8 close parenthesis. The confining pressure and deviatoric stress in these tests are 140 kPa and 827 kPa, respectively. Only the viscoplastic strain at the very end of the tests with 0.05- and 1-second rest period are shown. As the rest period increases, the amount of viscoplastic strain developed increases. At a cumulative loading time of 300 s, the viscoplastic strain in variable loading time test with rest period of 0.05 second is about 1 percent; with a rest period of 1 second, the viscoplastic strain is approximately 1.2 percent; and the 200-second rest period has 1.5 percent. Also shown on this graph are the model predictions. The overall trend regarding rest period time is observed in these predictions, but the predictions underestimate the viscoplastic strain at the end of the tests with 1-second and 0.05-second pulse times.

Figure 163. Graph. Viscoplastic strain versus cumulative loading time
(140 kPa confinement VT).

Figure 164. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VLT). This figure shows the viscoplastic strain predicted for variable load level and loading time test with confining pressure of 140 kPa compared with measurements. The cumulative loading time is plotted on the x axis from parenthesis 0 to 90 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 80 s is approximately 2 to 2.5 percent. During the test, it is observed that the rate of viscoplastic strain growth increases with load level and with decreasing pulse duration. Given the observed variability in the test results, the predicted strain matches the measured strains well both at the end of loading and throughout the loading history.

Figure 164. Graph. Viscoplastic strain versus cumulative loading time
(140 kPa confinement VLT).

Figure 165. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT). This figure shows the viscoplastic strain predicted for variable loading time test with confining pressure of 140 kPa at deviatoric stresses of 552 kPa and 827 kPa compared with measurement. The cumulative loading time is plotted on the x axis from parenthesis 0 to 45,000 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 1.8 close parenthesis. For the 552 kPa deviatoric stress test, the viscoplastic strain growth at 43,000 s is approximately 0.75 percent while the 827 kPa deviatoric stress test has approximately 1.4 percent viscoplastic strain at 43,000 s. It is also observed that during the test the rate of viscoplastic strain growth increases with decreasing pulse duration. The predicted strain matches the measured strains well both at the end of loading and throughout the loading history.

Figure 165. Graph. Viscoplastic strain versus cumulative loading time
(140 kPa confinement VT).

Figure 166. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT + flow). This figure shows the viscoplastic strain predicted for the variable time with flow number testing in between blocks with a confining pressure of 140 kPa at a deviatoric stress of 827 kPa compared with the measurement. The cumulative loading time is plotted on the x axis from parenthesis 0 to 35,000 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 1 and 8/10 close parenthesis. The measured strains are shown to be approximately 1.7 percent at 30,000 s, whereas the modeled response suggests approximately  1.2-percent strain at the same time. Up to approximately 0.4 percent in viscoplastic strain, the prediction matches well with measurement, but the error caused during the flow number portion of the test increases as time increase. The difference between measurement and prediction is about 0.3 percent at a viscoplastic strain level of 0.6 percent, and it is about 0.45 percent at a viscoplastic strain level of 1.2 percent.

Figure 166. Graph. Viscoplastic strain versus cumulative loading time
(140 kPa confinement VT + flow).

Figure 167. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 1,600 deviatoric VT). This figure shows the viscoplastic strain predicted for variable loading time test with confining pressure of 500 kPa at deviatoric stress of 1,600 kPa compared with measurement. The cumulative loading time is plotted on the x axis from parenthesis 0 to 250 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 80 s is approximately 2 percent. The predicted strain matches the measured strains well both at the end of loading and throughout the loading history.

Figure 167. Graph. Viscoplastic strain versus cumulative loading time
(500 kPa confinement 1,600 deviatoric VT).

Figure 168. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 2,000 deviatoric VT). This figure shows the viscoplastic strain predicted for variable loading time test with confining pressure of 500 kPa at deviatoric stress of 2,000 kPa is compared with measurements. The cumulative loading time is plotted on the x axis from parenthesis 0 to 80 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 70 s is approximately 2 percent. Given the specimen to specimen variability, the overall predicted strain matches the measured strains well both at the end of loading and throughout the loading history. However, the local slope, incremental viscoplastic strain divided by cumulative loading time of the prediction, does not quite match with measurement.

Figure 168. Graph. Viscoplastic strain versus cumulative loading time
(500 kPa confinement 2,000 deviatoric VT).

Figure 169. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 0.4 second CLT). This figure shows the viscoplastic strain predicted for constant loading time test with confining pressure of 500 kPa at deviatoric stress of 1,800 kPa compared with measurement. The pulse time of this test is 0.4 s. The cumulative loading time is plotted on the x axis from parenthesis 0 to 60 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 48 s is approximately 2 percent. The predicted viscoplastic strain is less than the measured strains; at 48 s, the predicted strain is approximately 1.6 percent, and the difference between measurement and prediction increases as cumulative loading time.

Figure 169. Graph. Viscoplastic strain versus cumulative loading time
(500 kPa confinement 0.4 s CLT).

Figure 170. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 1.6 second CLT). This figure shows the viscoplastic strain predicted for constant loading time test with confining pressure of 500 kPa at deviatoric stress of 1,800 kPa compared with measurement. The pulse time of this test is 1.6 s. The cumulative loading time is plotted on the x axis from parenthesis 0 to 70 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 60 s is approximately 2 percent. The predicted viscoplastic strain growth at 60 s is also approximately 2 percent.

Figure 170. Graph. Viscoplastic strain versus cumulative loading time
(500 kPa confinement 1.6 s CLT).

Figure 171. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement 6.4 second CLT). This figure shows the viscoplastic strain predicted for constant loading time test with confining pressure of 500 kPa at deviatoric stress of 1,800 kPa compared with measurement. The pulse time of this test is 6.4 s. The cumulative loading time is plotted on the x axis from parenthesis 0 to 450 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis. The viscoplastic strain growth at 400 s is approximately 2 percent. The predicted viscoplastic strain is more than measured and the difference between measurement, and prediction increases as cumulative loading time. At a cumulative loading time of 350 s, the difference between the measured and predicted viscoplastic strains is about 0.7 percent.

Figure 171. Graph. Viscoplastic strain versus cumulative loading time
(500 kPa confinement 6.4 s CLT).

Figure 172. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VLT). This figure shows the viscoplastic strain predicted for variable load level and loading time test with confining pressure of 500 kPa is compared with measurements. The cumulative loading time is plotted on the x axis from parenthesis 0 to 120 close parenthesis seconds, and the viscoplastic strain is plotted in percentages from parenthesis 0 to 2.5 close parenthesis.  The predicted viscoplastic strain is more than the measured viscoplastic strain, and the difference between them increases as cumulative loading time increases. At a cumulative loading time of 80 s, the difference between the two curves is approximately 0.8 percent.

Figure 172. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VLT).

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