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11 interlaboratory d2s values are roughly twice single-operator three replicates. Because the statistics were calculated based values, the interlaboratory precision for compliance would upon replicate measurements conducted within each labo- be 40 to 60 percent, which is clearly too high. However, ratory, the results correspond approximately to single-operator improvements in test equipment and procedures can proba- conditions; between-laboratory variability would be larger. bly substantially improve the precision of this test. The statistics on strength are based upon the data reported Similar trends are observed in d2s precision for m-value and by Anderson and McGennis, as discussed previously (3). Poisson's ratio. The variability for Poisson's ratio seems to be The statistics for IDT creep data are based upon data sub- particularly high, given that the average values for were from mitted by six different laboratories, as described previously. 0.35 to 0.40. The poor precision for the determination of Pois- The precision estimates for critical temperature were cal- son's ratio and the fact that this property is in general not con- culated without the data from Labs L3 and L6, because of sidered critical in determining resistance to low-temperature the anomalously high variability in this statistic for these cracking, indicate that measurement of is not necessary for laboratories. low-temperature characterization of asphalt concrete. It is of The precision for the IDT strength test appears to be accept- course a necessary part of the IDT procedure. able. The precision for the IDT compliance procedure, on the The d2s precision values for critical temperature (Figure 4) other hand, needs to be improved as part of the implementa- range from slightly less than 2°C for Lab L2 to almost 12°C tion process. Ruggedness testing for this procedure is the next for Lab L3. However, the compliance data for Lab L3 was logical step in the development of this procedure. This testing determined at very low temperatures, which is probably the should identify items in the procedure and equipment that sub- reason for the extreme variability in critical temperatures. This stantially affect the test precision. emphasizes the importance of performing compliance mea- surements at appropriate temperatures. As discussed previ- ously, the compliance measurements for Lab L6 were done THEORY OF IDT TESTING AND ANALYSIS using an electro-mechanical test system and should also not There are a number of theoretical considerations in the be considered representative of the data quality possible with evaluation of both the IDT and uniaxial creep and strength a good servo-hydraulic system. Eliminating these two labo- tests. These include the following issues: ratories, the average d2s precision for critical temperature was 2.9°C, equivalent to half a binder grade. Again, this is probably somewhat high considering that this is a single- · Linearity, operator value, but it should be possible to improve this value · Homogeneity, by further standardization of the low-temperature test equip- · Anisotropy, ment and procedure. · Poisson's ratio, · Coefficient of thermal contraction, and · Estimation of relaxation modulus from creep compliance. Summary of Findings on Precision and Bias A summary of the various estimated statistics on the IDT These issues impact the test methods and analysis in a vari- creep and strength tests is presented in Table 7. These sta- ety of ways. They are discussed in detail in the following tistics are based upon a test consisting of the average of sections. TABLE 7 Summary of statistics on IDT creep and strength tests (n3 replicates) Property Statistics Lowest Temp. Middle Temp. Highest Temp. Average, kPa 2870 Std. Error, kPa 200 Strength N/A N/A C. V., % 7.0 d2s, % 19.7 Average, 1/GPa 0.0463 0.0986 0.2809 Std. Error, 1/GPa 0.0042 0.0115 0.0413 Compliance C. V., % 7.9 9.9 11.3 d2s, % 22.3 28.0 32.0 d log D Average 0.143 0.238 0.355 m(t ) = Std. Error 0.021 0.018 0.018 d log t d2s 0.061 0.051 0.052 Average 0.350 0.376 0.376 Poisson's Ratio Std. Error 0.049 0.042 0.042 d2s 0.139 0.118 0.118 Average, °C -26.3 Critical Temp. Std. Error, °C 1.0 d2s, °C 2.9
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12 Linearity this test during SHRP that should account for the triaxial loading conditions, end effects should not be a concern for The issue of linearity is of great practical importance. Intu- this test. On the other hand, the typical 50-mm specimen itively, asphalt concrete at low temperatures is expected to thickness for the IDT is very small, as is the LVDT gage behave in a linear manner through loading approaching the length of 37.5 mm. Without a detailed finite element analy- point of failure, because of the high stiffness of asphalt con- sis (which was beyond the scope of this study), it cannot be crete under these conditions and the very low strains. It is, concluded with certainty whether RVE requirements have however, important to verify that the loads used in the IDT been met for either the IDT or uniaxial creep and strength test are appropriate--as high as possible, to ensure large tests at low temperature. deflections and good repeatability, while still remaining in One indication of whether the specimen size for the IDT the linear viscoelastic region. AASHTO T322 calls for a and uniaxial tests is adequate is the precision of these meth- maximum strain of 500 × 10-6 mm/mm, or 0.05 percent. This ods compared to what is possible for similar tests where spec- value is consistent with work performed by Mehta and Chris- imen homogeneity is not an issue. For the IDT precision study tensen (11), who reported that deviations from linearity began described previously, the single-test coefficient of variation to occur at the same strain level of 0.05 percent. This aspect (C.V.) ranged from 14 to 20 percent. In the experiment phase of AASHTO T322 probably does not need revision. of this project, presented later in this chapter, the single test C.V. for the IDT creep test was found to be 16 percent, which is in excellent agreement with the results of the interlabora- Homogeneity tory precision study. Mehta and Christensen reported C.V. values of 9 to 17 percent for relaxation tests performed using Homogeneity is the degree to which the properties of a the IDT geometry (11). Pellinen reported in detail on the pre- material are the same at any given point. Most mechanical cision of uniaxial dynamic modulus measurements using analyses, including those used to estimate mechanical prop- cylinders 100 mm in diameter by 150 mm high; her analysis erties from both the IDT and uniaxial creep and strength tests, broke the variability of the data into within- and between- assume that a material is homogenous--that is, the properties specimen components (13). She reported overall C.V. values are the same at any given location within the object consid- for a standard specimen with two LVDTs to be 15 to 21 per- ered. However, asphalt concrete is clearly not truly homoge- cent for 12.5-mm mixtures and 17 to 24 percent for 19-mm nous, because it is composed of three distinct phases--asphalt mixtures (13). As a comparison, Christensen and Anderson binder, aggregate, and air. The question is, therefore, whether reported C.V. values for complex modulus of asphalt significant errors are involved in the assumption of homo- binders--a very homogenous material compared to asphalt geneity in the analysis of IDT and uniaxial tests and in the concrete--measured using a dynamic shear rheometer to analysis of thermal cracking in general. In general, the larger range from 10 to 17 percent (14). Because of the complexity a specimen is compared to any nonuniformity it contains, the of preparing asphalt concrete specimens for testing, a some- more accurate the assumption of homogeneity. For this rea- what higher level of variability should be expected compared son, homogeneity is probably a very good assumption when to data on asphalt binders. Therefore, it would appear that typ- analyzing an entire, intact paving system. However, for test ical precision levels for both the IDT and uniaxial modulus specimens of relatively small size, homogeneity might not be tests are consistent with a reasonable degree of homogeneity. even approximately obtained. Further improvements in test precision would provide addi- Weissman and associates presented a detailed analysis of tional confidence that specimen homogeneity is not an issue the effects of specimen dimension on the results of permanent with these procedures. deformation tests (12). They discussed the concept of the rep- resentative volume element (RVE), which in simple terms can be thought of as the minimum acceptable specimen dimension Anisotropy for a given test and material in order to ensure that the assump- tion of homogeneity is met. Meeting RVE requirements is an Isotropic behavior is generally assumed when analyzing important contribution to test precision. Weissman and his test data or performing stress and strain analyses. That is, it coauthors suggested an RVE of 125 mm for an asphalt concrete is assumed that the mechanical properties of the material in mixture containing 19-mm nominal maximum size aggregate. question are independent of direction and sense. Because of In order to ensure that end effects were insignificant, this the manner in which asphalt concrete specimens and pave- would mean that a uniaxial creep test would require a speci- ments are compacted, it is quite possible that asphalt con- men about 350 mm high by 125 mm in diameter (12). How- crete is anisotropic, that its properties vary depending upon ever, this analysis dealt with permanent deformation tests, the direction of loading. Support for anisotropic behavior is where the modulus of the aggregate is much greater than that seen in the relationships between uniaxial and shear moduli of the surrounding mastic, a situation that greatly increases values reported by several researchers (4, 15 ). For an isotropic, RVE size. For low-temperature tests, the RVE should be sig- linear elastic material, the uniaxial modulus E should be nificantly smaller. Furthermore, because of the strip loading 2(1 + ) G , where is Poisson's ratio, typically ranging used in the IDT test and the correction factors developed for from about 0.3 to 0.5 for asphalt concrete and G is the
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13 shear modulus. Although the behavior of asphalt concrete tion of low-temperature cracking, again suggesting that per- under the conditions in question is linear viscoelastic rather haps uniaxial creep tests could provide the needed data more than linear elastic, this relationship should apply quite well. simply and more directly than the IDT creep test. However, it However, E values are normally much higher compared should be kept in mind that in order to properly analyze IDT to G than predicted by this relationship, indicating that creep data, it is essential to determine strains in both the verti- there is substantial anisotropy in the behavior of asphalt con- cal and horizontal directions, so that calculation of Poisson's crete mixtures (4). It appears that because of preferential ratio is an inherent part of the IDT procedure. orientation of aggregate particles during the shearing that occurs with compaction, the shear stiffness of mixtures per- pendicular to the plane of compaction is relatively low com- Coefficient of Thermal Contraction pared to the uniaxial compressive stiffness in the direction of compaction. The thermal stress developed when a pavement cools is The results of the laboratory testing performed as part of directly proportional to the coefficient of thermal contrac- this project (described later in this chapter) indicate that the tion. An equation was developed during SHRP for estimat- low-temperature creep compliance of asphalt concrete mix- ing the coefficient of thermal contraction, which is based tures does in fact exhibit anisotropy. The creep compliance upon mixture composition and the coefficient of thermal measured in the diametral plane using the IDT test is less than contraction values for the binder and aggregate (1): that measured along the length of the specimen (uniaxially) in compression, which in turn is less than the compliance mea- AC VMA + Agg (100 - VMA) mix = (1) sured in uniaxial tension. The effect of this anisotropy on the 300 results of IDT and uniaxial stiffness tests is, however, not clear. On one hand, the IDT test has an advantage compared where to the uniaxial test because the IDT procedure primarily mea- mix = linear coefficient of thermal contraction for mix- sures creep compliance in the horizontal plane, which is most ture, m/m/C; important in thermal cracking. On the other hand, the analy- AC = volumetric coefficient of thermal contraction for sis of the IDT test is based upon an assumption of isotropic asphalt binder, m3/m3/C; behavior, and so the analysis is probably not completely accu- Agg = volumetric coefficient of thermal contraction for rate, because it appears that the properties in tension and aggregate, m3/m3/C; and compression are not identical. However, because the tensile VMA = voids in mineral aggregate. strains in the IDT test are quite low, the difference in the As detailed by Kwanda and Stoffels, the SHRP equation is not tension and compression compliance values is probably accurate (8). Values of mix measured by Kwanda and Stoffels small, resulting in a small effect on the IDT data analysis. and as calculated using the SHRP equation are plotted in Fig- In view of the apparent anisotropy in asphalt concrete mix- ure 5. Another problem with the SHRP approach is that the tures, the IDT test geometry is probably the most effective coefficient of thermal contraction of the aggregate must be of the available methods for determining low-temperature known. The values estimated during SHRP were made using creep compliance. Agg values estimated based upon typical values for the aggre- gates used in each mixture. In most cases, this information will Poisson's Ratio Another issue in the IDT test procedure is whether it is truly necessary to determine Poisson's ratio when characterizing the 3.0E-05 mechanical behavior of HMA at low temperature. Poisson's SHRP Mix Alpha, m/m/C ratio represents the ratio of lateral to axial deformation under uniaxial loading. It is theoretically necessary to know Pois- son's ratio when performing stress analyses in two or three dimensions. However, in performing simple, one-dimensional 2.0E-05 stress analyses, such as those used in the Superpave thermal cracking analysis, Poisson's ratio is not used. Furthermore, for most materials, Poisson's ratio falls between about 0.2 and 0.5. For asphalt concrete, Huang states that values typi- 1.0E-05 cally fall in a narrower range, from 0.3 to 0.4 (16 ). Huang 1.0E-05 2.0E-05 3.0E-05 goes on to state, "Because Poisson's ratio has a relatively Measured Mix Alpha, m/m/C small effect on pavement responses, it is customary to assume a reasonable value for use in design, rather than to Figure 5. Coefficient of thermal contraction values for determine it from actual tests" (16 ). It appears as though SHRP mixtures, as measured by Kwanda and Stoffels (8) determination of Poisson's ratio is not critical to the predic- and as predicted using the SHRP equation.
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14 not be readily available to engineers and technicians perform- 2.5E-04 Calc. Binder Alpha, m/m/C ing mixture design and analysis. Although laboratory proce- y = 5.34E-04x + 7.72E-05 dures for measuring the coefficient of thermal contraction 2.0E-04 of asphalt concrete mixtures do exist, these methods have not been widely used and are of unknown precision (9). An 1.5E-04 improved, simpler approach is needed for estimating mix for use in analyzing low-temperature IDT and/or uniaxial creep 1.0E-04 and strength test data. 5.0E-05 In examining the thermal contraction data on the SHRP mixtures for this project, it was found that the mixture coeffi- 0.0E+00 cient of thermal contraction as predicted using the SHRP equa- 0.00 0.05 0.10 0.15 0.20 tion is largely independent of Agg. This is because the Mix m at -20 C coefficient of thermal contraction for binders--typically around 1.15 × 10-4 m/m/°C (linear) at temperatures above the Figure 6. Calculated binder coefficient of thermal glass transition--is much, much larger than the typical value contraction as a function of mixture m-value at -20°C for for construction aggregates, about 7 × 10-6 m/m/°C. Further- SHRP mixtures (R2 = 42%). more, it was found that mixture coefficient of thermal con- traction was much more strongly related to binder volume rather than VMA, as assumed in the SHRP equation. There- lated using Equation 3 as a function of mixture m-values. There fore, a more appropriate equation for estimating the coeffi- is a definite relationship, though it is only weak to moderate in cient of thermal contraction for asphalt concrete mixtures strength (R2 = 42 %). However, it should be kept in mind that would be: the measurement of the coefficient of thermal contraction of the mix is difficult and somewhat variable. The standard deviation AC Vbe + 7 × 10 -6 (100 - VMA) for mix for the data reported by Kwanda and Stoffels (8) was mix = (2) 1.5 × 10-6 m/m/°C, corresponding to a standard error (n = 2) of 100 1.1 × 10-6 m/m/°C and a d2s precision of 3.0 × 10-6 m/m/°C. Where Vbe is the volume percentage of asphalt binder in a For the average mix value of 2.0 × 10-5 m/m/°C, this last mixture, and the coefficient of thermal contraction for the value corresponds to a precision of 15 percent as a percent- binder is a linear value, in m/m/°C. Equation 2 can be re- age of the mean response. Also note in Figure 6 that the cal- arranged to give AC in terms of the mixture composition and culated values of AC are relatively large, typical for tem- coefficient of thermal contraction: peratures above Tg. It can be concluded that it is generally not necessary to account for the decrease in AC that occurs mix 100 - 7 × 10 -6 (100 - VMA) at and below Tg. AC = (3) Vbe Using the relationship shown in Figure 6, AC values were estimated for the SHRP mixtures and then used along with Using Equation 3, the SHRP mixture composition data pro- Vbe and VMA values to predict mix values, using Equation vided by Lytton and his associates (1), and the mixture coeffi- 2. These calculations can be combined into one equation for cient of thermal contraction values measured by Kwanda and estimating the coefficient of thermal contraction of asphalt Stoffels (8), AC values were estimated for the SHRP mixtures. concrete mixtures: Because coefficient of thermal contraction values for binders are largely a function of the glass transition temperature (5.3 × 10 -4 m + 7.7 × 10 -5 )Vbe of the binder, there should be an approximate relationship + 7 × 10 -6 (100 - VMA) mix = ( 4) between binder stiffness at low temperature and this estimated 100 value of AC. However, the binder data on the SHRP mixtures were very limited and based on measurements on extracted Where m is the log-log slope of the mixture creep compli- binders that are probably not highly reliable. Also, low- ance with respect to time (t), from = 5 to 100 seconds at the temperature binder data might not always be available when lowest test temperature, normally -20°C. The resulting val- analyzing data on asphalt concrete mixtures at low tempera- ues are compared to those measured by Kwanda and Stoffels ture. Therefore, it was felt that mixture stiffness data would (8) in Figure 7. Although the predictions are not highly accu- provide a more practical means of estimating AC. Mixture rate, they are substantially better than those made using the creep compliance is a function of binder stiffness, aggregate SHRP equation (Figure 5). Furthermore, compared with the modulus, and mixture composition, and so is not a good choice d2s confidence limits for measured mix (included in Figure for relating to binder coefficient of thermal contraction. Mix- 7 as horizontal error bars), the accuracy of the predictions is ture m-value (d [log (D)]/d [log(t)]), where D is the creep com- probably as good as can be expected and appears to be com- pliance, is a much better choice for correlation to AC, because parable in accuracy to values determined experimentally. It this should be largely independent of aggregate properties and is suggested that Equation 4 be used to estimate mix values mixture composition. Figure 6 is a plot of AC values calcu- when analyzing low-temperature creep and strength data on
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15 3.0E-05 N +1 E (t r ) = Ei e - i t r (7) i =1 Predicted mix, m/m/C where 2.0E-05 E(tr) = relaxation modulus at reduced time tr Ei = modulus for Prony series element i i = relaxation time for Prony series element i Both Equation 5 and Equation 7 represent mechanical ana- 1.0E-05 logues for describing linear viscoelastic behavior. Equation 5 1.0E-05 2.0E-05 3.0E-05 represents a generalized Maxwell model, whereas Equation 7 Measured mix, m/m/C represents a generalized Kelvin model. The procedure described above was used to calculate relax- Figure 7. Mixture coefficient of thermal contraction values ation modulus using data for six different mixtures, as sub- as measured by Kwanda and Stoffels (8) (with d2s precision mitted by Lab L4 of the IDT creep precision study described limits for measured values) and as predicted by Equation 4. previously. In fitting the Prony series to the creep compli- ance data, five evenly spaced relaxation times were assumed, asphalt concrete mixtures, rather than either using the covering a time range slightly larger than that for the entire SHRP equation (Equation 1) to estimate values or deter- master curve. Then the compliance values for each element mining values experimentally. were determined using simultaneous equations, resulting in exact agreement between the measured and fitted Prony- Estimation of Relaxation Modulus series compliance values at each of the selected relaxation from Creep Compliance times. The Prony series parameters for the relaxation mod- ulus were calculated using the collocation method described Another potential problem in the analysis of low- by Christensen (17 ); a detailed description of this method is temperature creep data is the estimation of relaxation modu- beyond the scope of this report, but it also relies on simultane- lus from creep compliance. This is an essential step in the ous equations to determine the series parameters. An example calculation of thermal stress in a pavement using either IDT of the Prony series fit to the creep compliance is shown in or uniaxial creep data. In the approach developed by Roque Figure 8, which is for mixture 1 from Lab L4 of the precision and his associates during and after SHRP (1,5), a master curve study. Note that the compliance values predicted by the Prony of creep compliance is developed from creep data at three tem- series approach diverge dramatically from the power law fit peratures, normally -20, -10 and 0°C. Then, an exponential, or Prony, series is fit to these data: to the master curve at a reduced time of about 1 × 106 seconds. This represents the end of the experimentally determined mas- N ter curve and the longest relaxation time for the Prony series. D(tr ) = D0 + Di (1 - e - tr i ) + tr (5) Figure 9 shows the corresponding predicted relaxation modu- i =1 lus; the creep modulus, 1/D(t), which is a rough approximation where to the relaxation modulus; and an estimate of the relaxation modulus based upon Christensen's method for approximate D(tr) = creep compliance at reduced time tr, inversion of the Laplace transform (10,17 ). As with the creep D0 = glassy compliance at tr = 0, Di = compliance for Prony series element i, compliance, the relaxation modulus values predicted using the i = relaxation time for Prony series element i, and Prony series approach diverge dramatically from other esti- = viscosity as tr . mates at long reduced times. Some irregularities in the Prony series values are also evident as waviness in the master curve The relaxation modulus is related to the creep compliance at long reduced times, which is due to the discrete nature of through the Laplace transform: the Prony series. If a very large number of elements are used L[ D(t )] L[ E(t )] = s 2 (6) at closely spaced relaxation times, these irregularities become insignificant. It would appear from this and other comparisons where that Christensen's approximate method is somewhat more accurate and reliable than the Prony series method, although L[D(t)] = the Laplace transform of the creep compliance, the differences are small except at long reduced times. L[E(t)] = the Laplace transform of the relaxation modulus, The six mixtures submitted by Lab L4 in the precision study and were analyzed as described above. Furthermore, each analy- s = the transform parameter. sis was performed using the full set of creep data (-20, -10, An exponential series for the relaxation modulus can be cal- and 0°C data), using data at -20 and -10°C only, and using culated once the Prony series parameters for the creep com- data at -20°C only. This was done to evaluate the effect of pliance are known: mixture stiffness on the Prony series error, because the error
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16 1.E-01 1.E-02 Compliance, 1/psi 1.E-03 Prony Series Power Law 1.E-04 1.E-05 1.E-06 1.E-07 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 Reduced Time at -20°C, s Figure 8. Creep compliance for Mixture 1 from Lab L4 of the precision study: Power law and Prony series fits to the master creep curve. is likely to become more severe as the mixture becomes stiffer. series approach, three additional estimates of Tc were made The results shown above were typical, and the analysis con- using relaxation modulus values estimated using the Prony firmed that the extent of this error became larger as the asphalt series approach, with full data, data to -10°C, and with -20°C concrete stiffness increased. To quantify the magnitude of data only. The results of this analysis are summarized in Fig- this error, Christensen's version of the SHRP analysis method ure 10. Note that the critical cracking temperatures estimated was used to estimate a critical cracking temperature (10). using the Prony series approach are always lower than those This method is essentially identical to the SHRP method, but calculated using Christensen's (17 ) approximate method for Christensen's (17 ) approximate method for estimating relax- estimating the relaxation modulus. Furthermore, the error ation modulus from creep compliance is used rather than the increases as the data used in the analysis become stiffer. The Prony series approach, and the calculation stops at estimat- errors however are generally small--only a few degrees-- ing the critical cracking temperature, Tc, defined as that at though for the -20°C data, the errors often exceed 3°C. which the thermal stress in the pavement reaches the tensile This error can be corrected in several ways. The number of strength. In this case, a typical mixture coefficient of thermal relaxation times used in the Prony series could be increased expansion of 1.1 × 10-5 m/m/°C was assumed for all mixtures. (say doubled) and the time covered by the Prony series also Similarly, a typical tensile strength of 3.0 MPa was also increased. Another approach would be to use an approxi- assumed. However, to estimate the potential error in the Prony mate method for estimating the relaxation modulus, which, 1.E+07 E(t) from Prony Series E(t) Approximate 1/D(t) 1.E+06 E(t) or 1/D(t), psi 1.E+05 1.E+04 1.E+03 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 Reduced Time at -20°C, s Figure 9. Relaxation modulus for mixture 1 from Lab L4 of the precision study: Values estimated using Prony series approach, Christensen's approximate method, and the inverse of the creep compliance.