Dynamic Modulus Master Curves Research Articles

The Relationship Between Loading Frequency and Truck Speed on Asphalt Pavements

The mechanistic-empirical road pavement design methods necessitate knowledge of the dynamic modulus master curve of asphalt mixtures. This requires the designer to accurately define both the load frequency and the temperature of the pavement structure. The parameter of load frequency is of great importance in laboratory fatigue tests. However, it presents a challenge since road structures are loaded in the time domain, not the frequency domain. This complexity has prompted researchers to develop time-frequency conversion methods to estimate the corresponding frequency from the duration of traffic-induced stress or deformation pulses. Recent findings indicate that the time-frequency equivalence factor is considerably lower than previously assumed. In this study, we introduce a new four-parameter empirical RAMBO material model, derived from the Ramberg–Osgood equation, which offers parameters to determine the time-frequency equivalence factor for asphalt mixtures. The relationship between the moving wheel load velocity and the load impulse time is analysed, primarily based on experimental data and computational methods from existing literature. A method for determining the wavelength of the load impulse is proposed. The theoretical correlation indicates that a wheel load travelling at 100 km/h results in a dynamic deformation frequency of 10 Hz. However, for a strain impulse, the same speed corresponds to a much lower frequency of 3–4 Hz. This study offers a refined understanding of the time-frequency equivalence factor, which is of crucial importance for more accurate road pavement design and fatigue testing.

An alternative approach for characterization of the linear viscoelastic behavior of asphalt binders

This study aimed to develop an alternative approach to characterise the linear viscoelastic behaviour of asphalt binders. A rheogram model was firstly proposed based on the Carreau–Yasuda model; its accuracy was substantiated by high R 2 values. The dynamic shear modulus master curve model was then obtained using the relationship between the magnitude of complex viscosity and dynamic shear modulus and the two definitions of the time-temperature shift factor. Afterwards, the phase angle master curve model was derived according to the approximate Kramers–Kronig relation. The proposed approach was demonstrated to be capable of accurately characterising the linear viscoelastic behaviour of selected asphalt binders. In addition, the predicted zero shear viscosity (ZSV) could be potentially employed to evaluate the asphalt binder rutting resistance. Compared with the widely-used Christensen-Anderson-Marasteanu (CAM) model, the developed master curves models were better at constructing the phase angle master curves of selected asphalt binders.

Establishment and evaluation of dynamic viscoelasticity constitutive model for asphalt mixtures based on PCA model: Utilizing coal-based synthetic natural gas slag and phosphogypsum whisker as substitute fillers

To promote the secondary utilization of solid waste in asphalt pavement and reduce resource waste and environmental damage, the fine solid waste powder from the secondary treatment of solid waste was used as a substitute filler to prepare asphalt mixtures. To more precisely characterize the effects of filler type and substitution ratio on the dynamic viscoelastic mechanical characteristics of asphalt mixtures, the dynamic modulus test was conducted on asphalt mixtures, while the dynamic modulus master curve of asphalt mixtures was established based on the time-temperature equivalent principle and the Sigmoidal function. The improved generalized Sigmoidal (IGS) model, five-parameter fractional Zener (FFZ) model, six-parameter fractional Zener (SFZ) model and ten-parameter fractional Zener (TFZ) model were used to construct the shrinkage frequency curves of different asphalt mixtures under different constitutive relationships, whereas the applicability of the constitutive models was further evaluated by the principal component analysis (PCA) model. The research results showed that the dynamic modulus and phase angle of the mixture exhibited a similar change rule with the variation of temperature and frequency, and the use of solid waste-based fillers could effectively reduce the effect of frequency variation on the viscoelasticity of the mixture. The dynamic modulus master curve constructed by Sigmoidal function could accurately describe the functional relationship between the dynamic modulus with loading frequency and temperature, and the variation trend of dynamic modulus in a wider frequency range was flattened with the increase in frequency, and this change rule was not affected by the substitution ratio of the filler, while the use of solid waste-based filler could reduce the dependence of the mixture on the temperature. Compared to several fractional constitutive models, the IGS model exhibited a higher fitting accuracy for the dynamic viscoelasticity of several mixtures, and with the use of solid waste-based fillers, the deviation of fitting curves gradually decreased. A comparative study showed that the fractional constitutive model was difficult to accurately describe the dynamic viscoelasticity of the mixture, and the IGS model was preferred to be recommended as a calculation model to investigate the constitutive relationship on the dynamic viscoelasticity of the mixture.

Study on the Application of Kramers-Kronig Relation for Polyurethane Mixture.

Polyurethane (PU) mixture, which is a new pavement mixture, exhibits different dynamic properties compared to a hot-mixed asphalt mixture (HMA). This paper analyzed whether the Kramers-Kronig (K-K) relation and thermorheologically simple properties applied to the PU mixture. Based on the results, the PU mixture exhibited thermorheologically simple properties within the test conditions. The time-temperature superposition principle (TTSP) was applicable for the PU mixture to construct a dynamic modulus master curve using the standard logistic sigmoidal (SLS) model, the generalized logistic sigmoidal (GLS) model, and the Havriliak-Negami (HN) model. The Hilbert integral transformed SLS and GLS models for the phase angle can accurately fit the measured phase angle data with newly fitted shift factors and predict the phase angle within the viscoelastic range. The core-core and black space diagrams both displayed single continuous smooth curves, which can be utilized to characterize the viscoelastic property of the PU mixture. The K-K relation is applicable for the PU mixture to obtain the phase angle master curve model, storage modulus, and loss modulus from the complex modulus test results with the test temperatures and loading frequencies. The phase angle of the PU mixture at extremely high or low test temperatures cannot be derived from the dynamic modulus data.

Study on the characterization method of dynamic modulus for asphalt mixture under multi-factor coupling conditions

In the actual service process of asphalt pavement, the dynamic response of the pavement structure layer is induced by vehicle loads. The dynamic modulus is a crucial parameter for analyzing the mechanical response of asphalt pavement. Due to significant influences from temperature and loading frequency on test measurements, along with its time-temperature equivalence, it becomes imperative to investigate the behavior of dynamic modulus under different temperature frequencies. Furthermore, laboratory loading modes also exert substantial effects on dynamic modulus. Employing different loading modes during measurement often leads to significantly varied stress states in specimens, yielding distinct values for dynamic modulus. Therefore, this study conducts dynamic modulus tests using direct tension, uniaxial compression, and indirect tension under five test temperatures and six loading frequencies while examining its variations under different stress states and temperature frequencies. The corresponding master curves and prediction models of dynamic modulus are derived, providing a valuable reference for predicting the dynamic modulus under wide-area test conditions. Through comparative analysis of dynamic modulus values at different stress levels, the stress level is incorporated into the master curve to establish corresponding master curves of dynamic modulus. By applying relevant theories and research on time-temperature equivalence, we obtained dynamic modulus master curves for three loading modes by fitting and analyzing dynamic modulus test data at various temperatures and loading frequencies. Combining the Sigmoid model and Williams-Landel-Ferry(WLF) equation, we derived prediction models for dynamic modulus under three loading modes. Comparative analysis of dynamic modulus values under these three loading modes allowed us to establish transformation equations for the prediction models in different loading scenarios. Finally, representative dynamic modulus values for permanent deformation in typical pavement structure layers were predicted based on laboratory test results and the derived models. The research findings presented in this paper have significant implications in eliminating the influence of loading modes on dynamic modulus predictions, establishing a unified prediction model, and providing essential parameters for predicting permanent deformation and fatigue life equivalent temperature.

Application of dynamic complex stiffness modulus master curves of HMA with recycled materials in the mechanistic-empirical design of road pavement structures

The main objective of the paper is to present the possibilities and advantages of advanced methods for determining the viscoelastic properties of asphalt mixtures produced using recycled materials in individual mechanistic-empirical pavement design. The use of various types of materials in asphalt mixture production can significantly alter their functional parameters, such as the dynamic stiffness modulus. In addition, this parameter changes with the change in the temperature of the road pavement and the speed of vehicles. In order to determine the durability of the pavement structure in an informed manner and minimize the risk of error, it is necessary to investigate and describe as precisely as possible this variability in the behavior of asphalt layers as a result of changes in the properties of input materials, e.g. by the addition of recycled materials and under the influence of changes in temperature and load velocity of the pavement. The most effective approach involves the application of master curves of the dynamic complex stiffness modulus of asphalt mixture, constructed based on research results conducted at different temperatures and loading frequencies. This allows for the determination of stress and deformation states of the pavement construction under various climatic conditions, utilizing seasonal or average monthly temperatures and considering the specific characteristics of pavement operation, such as different vehicle speeds for main routes, connectors, or street intersections. The paper compares the changes in dynamic modulus master curves for different types of asphalt mixtures with and without the addition of waste or recycled materials such as reclaimed asphalt pavement (RAP), rubber from used tires, or synthetic fibres and changes impact on the estimated durability of the pavement structure due to different approaches to determining climatic conditions and movement speed.

Rheological and Aging Characteristics of Polymer-Modified Asphalt with the Addition of Sulfur

Crosslinking agents, notably sulfur, are used in asphalt binder modification to facilitate chemical bonding between polymer chains and the asphalt binder. Despite some prior research indicating the benefits of sulfur crosslinking in enhancing polymer-modified asphalt’s (PMA) stability, there is a lack of comprehensive understanding regarding its impact on rheological properties and its anti-aging potential. This study addresses these gaps by thoroughly investigating the effects of varying the sulfur content (ranging from 0.03% to 0.5% by total weight of binder) on PMA’s rheological properties. The research assesses the effectiveness of sulfur in enhancing PMA’s resistance to aging using various methods, including the Glover-Rowe parameter, FTIR analysis, and the examination of the dynamic modulus and phase angle master curves. The results indicated that the addition of sulfur, particularly up to 0.3%, bumps the high-temperature performance grade by one level, and significantly improves elasticity, allowing the PMA to support heavier traffic without experiencing rutting, all while maintaining resistance to low-temperature cracking. Furthermore, PMA with sulfur demonstrated an increase in resistance to aging, reducing the aging potential by approximately 15% with the best sulfur formulation. This enhanced durability can reduce the frequency of maintenance activities, leading to cost savings, reduced roadwork emissions, and prolonged pavement life.

An Approach for Adjusting the Laboratory-Determined Dynamic Modulus Master Curve of Asphalt Layers Based on Falling Weight Deflectometer Measurements

Structural assessment is critical for designing asphalt pavement overlays, estimating the remaining life of pavements, and selecting an appropriate rehabilitation strategy for existing pavements. The falling weight deflectometer (FWD) serves as the primary nondestructive test used for evaluating the in situ properties and structural capacity of asphalt pavements. The current procedure involves analyzing the FWD response and estimating layer moduli by assuming an elastic response. However, the response of asphalt layers is viscoelastic (i.e., temperature- and frequency-dependent). This study proposes an approach that combines FWD data with laboratory measurements of the dynamic moduli of field cores to determine the in situ viscoelastic properties of asphalt layers. This approach is implemented by analyzing FWD data from four pavement sections in Qatar. Furthermore, the paper includes a comparative analysis of the response of pavement sections in which the asphalt layers are modeled using dynamic modulus master curves obtained from laboratory tests and those obtained using the approach presented in this study. It was found that using the laboratory-based master curves overestimated pavement performance (i.e., underestimated pavement distresses). It is recommended to use the dynamic moduli from the method presented in this paper for a more accurate estimation of pavement response and performance.

Experimental Study on the Dynamic Modulus of an Asphalt Roadbed Grouting Mixture under the Influence of Complex and Multiple Factors

After long-term service, the ground will experience settlement and the stability of the roadbed will be lost. In order to effectively reinforce the roadbed, an asphalt roadbed grouting mixture has been applied to the filling of the roadbed. The rotary compaction method was used to prepare different gradation types of lime composite-modified oil sludge pyrolysis residue asphalt, mixtures Sup13, Sup19, and Sup25. This article takes the dynamic modulus of an asphalt roadbed grouting mixture as the mechanical index, and the uniaxial compression dynamic modulus test is carried out on three kinds of rotary compaction asphalt mixtures, Sup13, Sup19, and Sup25. The dynamic modulus master curves of different gradation composite-modified oil sludge pyrolysis residue asphalt mixtures are fitted to study the dynamic modulus of asphalt mixtures under different nominal maximum particle sizes, loading frequencies, and temperatures. The results show that (1) The dynamic modulus of different gradation composite-modified oil sludge pyrolysis residue asphalt mixtures increases with the decrease in temperature and the increase in frequency; (2) when other conditions are the same, the compound-modified asphalt mixture’s dynamic modulus decreases significantly under low-frequency and high-temperature conditions; (3) in the range of 4.4–37.8 °C and medium loading frequency, the dynamic modulus of the compound-modified asphalt mixture is more affected by temperature and loading frequency; (4) in the low-temperature and high-frequency range, the compound-modified asphalt mixture with a larger nominal maximum particle size has a higher dynamic modulus, and the asphalt mixture with better stability of skeleton structure has a higher dynamic modulus. The research results of this article will provide scientific guidance for the study of the mechanical properties of asphalt roadbed grouting mixtures.

Study on the Construction of Dynamic Modulus Master Curve of Polyurethane Mixture with Dense Gradation

The PU mixture considered here is a new kind of pavement material with excellent road performance, which lacks study into its dynamic mechanical and viscoelastic properties. In this study, the dynamic modulus of the polyurethane (PU) mixture was fitted by using five master curve models, five shift factor equations, and four error minimization methods. According to test results, the log–log plot form was able to more effectively display the differences between master curves. The solver method, the sum of square error minimization (≤0.02), proved to be more appropriate and accurate with higher fitting parameter results. The line of equality statistic and Pearson linear correlation analysis results demonstrated that WLF and Kaelble equations were appropriate for five master curve models with trend line R2 values higher than 0.98. The GLS and SCM model with the WLF equation had the most accurate master curve fitting results. The dynamic modulus master curve shape of the PU mixture did not follow the traditional smooth “S” shape and did not show the ultimate dynamic modulus at extreme frequency. The viscoelasticity of the PU mixture is quite different from that of the asphalt mixture. This study recommended the most accurate error minimization method, the master curve model, and shift factor equations for characterizing the dynamic properties of the PU mixture.

A Comprehensive Analysis of Fatigue and Healing Capacity of Sasobit Polymer-Modified Asphalt from Two Perspectives: Binder and FAM

Wax-based Sasobit synthetic hard wax can be designed for producing asphalt with less energy consumption and lower carbon emissions in the pavement industry. This study investigated the effectiveness of Sasobit synthetic hard wax on microscopic properties, fatigue-fracture behavior, and healing potential of polymer-modified asphalt (PMA) from two perspectives: asphalt binder and fine aggregate matrix (FAM). First, microscopic morphology, dispersion mechanism, and thermal stability of styrene-butadiene-styrene (SBS)– and crumb rubber (CR)–modified asphalt binders under the influence of Sasobit synthetic hard wax were characterized. Second, linear viscoelastic properties and evolution law of fatigue life of Sasobit synthetic hard wax/polymer-modified asphalt binders and FAM with and without healing interval were assessed under the framework of the simplified viscoelastic continuum damage (S-VECD) model. Third, the correlation between properties of the modified asphalt binder and FAM was statistically analyzed by radar chart. Thermodynamic analysis indicated that Sasobit synthetic hard wax enhanced the thermal stability of PMA but slightly reduced the low-temperature performance. Morphological observations showed that Sasobit synthetic hard wax had limited effectiveness on the dispersion of the polymer particles in PMA. Dynamic modulus master curves and fatigue and healing tests demonstrated that Sasobit synthetic hard wax contributes to the stiffness, fatigue resistance, and self-healing capability of the asphalt binder and mixture. Finally, statistical analysis revealed that the asphalt binder performance lacked a close correlation with the FAM owing to the complex composition system.

A time-temperature-ageing shift model for bitumen and asphalt mixtures based on free volume theory

ABSTRACT This paper aims to develop a general form of the time-temperature-ageing shift model for bituminous materials based on free volume theory and validate its application in the Finite Element (FE) modelling for predicting aged asphalt mixtures’ mechanical responses. The frequency sweep, dynamic modulus, and Fourier Transform Infrared Spectroscopy (FTIR) tests were used to validate the proposed time-temperature-ageing shift model. A generalised Maxwell model incorporating the proposed shift model of asphalt mixtures was used to predict the mechanical responses of aged asphalt mixtures using FE modelling, verified via laboratory cyclic load tests. Results indicate that the time-temperature-ageing shift function can model the complex shear modulus and dynamic modulus master curves of bituminous materials, addressing the modulus dependency of temperature and ageing. Temperature dominates the modulus change of bituminous materials in the ageing process. The coefficients of the time-temperature-ageing shift model have clear physical meanings and can potentially quantify the effects of asphalt mixtures’ air voids on their temperature and ageing susceptibilities. The time-temperature-ageing shift model can be incorporated into the constitutive relations of the FE model to effectively couple and efficiently predict the effects of temperature and ageing on the mechanical responses of aged asphalt mixtures.

The Stiffness Behavior of Asphalt Mixtures with Different Compactness under Variable Confinement.

The dynamic modulus is a key property determining the short- and long-term performance of asphalt pavement, and its strong dependence on confining pressure and material density (mixture compactness) has been clearly indicated in the literature. It is always challenging to reproduce three-dimensional in situ stress conditions in the laboratory. To alleviate this difficulty, in this study, a convenient experimental setup was developed, in which the lateral confinement was made present and variable as a concomitant reaction of the surrounding materials to the vertical loading. Three dense-graded mixtures were prepared to a set of four different densities and then subjected to the confined dynamic modulus test. The results indicated a significant dependence of the confined modulus on the three factors of temperature, frequency, and compactness and that the mixture with coarser gradation demonstrated a less sensitivity to these parameters. A mathematical model was developed for the dynamic modulus master curve unifying these factors by means of horizontal shifting due to the time-temperature superposition principle (validated against the variable confinement at different compactness) and the vertical shift factor as a function of reduced frequency and compactness. The adequacy of the model was demonstrated using the experimental data, and its potential application in field pavement compaction was discussed.

Structural optimization of steel bridge deck pavement based on mixture performance and mechanical simulation

The upper layer of modified SMA mixture (SMAM) and the lower layer of epoxy asphalt mixture (EAM) is a steel bridge deck pavement (SBDP) structure with application potential, but there is a lack of systematic research on it. In this study, the pavement material and structure of SMAM + EAM were optimized using mixture performance tests and finite element analysis, which provided guidance to facilitate its promotion and application. Firstly, the pavement performance and dynamic properties of SMAMs and EAMs prepared with different binders were compared to preferably select the appropriate SBDP material. Secondly, the dynamic modulus master curves of the mixtures were established to provide material parameters for pavement structure design. Finally, the FEA method was applied to comparatively analyze the dynamic mechanical response of the pavement structure for different structures, thickness combinations, and temperatures to optimize the SBDP structure purposefully. The results demonstrated that EAM had better high-temperature rutting resistance than SMAM, while SMAM had superior moisture damage and skid resistance. The modifier contents in the asphalt binder used for SMAM and EAM were recommended to be 12 wt% and 40 wt%, respectively. From the FEA results, the SMAM + EAM structure met the performance requirements of SBDP. Compared with double-layered EAM, the deformation resistance of the SMAM + EAM structure was reduced, but it was beneficial to avoid the occurrence of crack defects and interlayer shear damage. Furthermore, the recommended pavement thickness combination was 4.5 + 3.5 cm.

Use of the Modified Ramberg-Osgood Material Model to Predict Dynamic Modulus Master Curves of Asphalt Mixtures.

Dynamic modulus master curves are usually constructed by using sigmoid functions, but the coefficients of these functions are not independent of each other. For this reason, it is not possible to clearly identify their physical mean. Another way of describing the dynamic modulus master curves is to choose the Ramberg-Osgood (RAMBO) material model, which is also well-suited for modelling the cyclic behaviour of soils. The Ramberg-Osgood model coefficients are completely independent of each other, so the evaluation of the fitted curve is simple and straightforward. This paper deals with the application of the Ramberg-Osgood material model compared to the usual techniques for constructing a master curve, determining the accuracy in describing the material behaviour of asphalt mixtures, and seeking any surplus information that cannot be derived by traditional techniques. Because the dynamic modulus and phase angle master curves are strictly related, in the present study, the asymmetric bell-shaped frequency curve of Toranzos was used to describe the phase angle for four types of asphalt mixtures (RmB, PmB, RA, and NB). The results show that the RAMBO model is a good alternative to the sigmoid function in describing the master curve of the dynamic modulus. We successfully used the Toranzos asymmetric bell-shaped frequency curve to describe the phase angle master curve. We also found a promising relationship between the independent RAMBO model parameters and the physical properties of the investigated binders, but this requires further research.

Comparative study of indirect tensile test and uniaxial compression test on asphalt mixtures: Dynamic modulus and stress-strain state

Dynamic modulus is an essential parameter for the performance characterisation of asphalt materials for performance prediction and pavement design. The Indirect Tensile (IDT) test and the Uniaxial Compression (UC) test are both well-known experiments performed in the laboratory to characterise the dynamic modulus of asphalt mixtures over a range of temperatures and loading frequencies. A considerable amount of research has investigated the difference between two test modes, while few studies analysed the fundamental difference in stress–strain distributions for the two test setups. This work aims at comparing the effect of the two test methods on dynamic modulus of asphalt mixtures, as well as stress–strain state. For these purposes, two types of mixtures commonly employed to build road surface layers, namely Asphalt Concrete (AC) and Stone Mastic Asphalt (SMA), were created and tested. The specimens were prepared with a gyratory compactor. The master curves of dynamic modulus and the stress–strain states obtained from the two testing procedures were compared. The dynamic modulus and phase angle results are almost identical for medium frequency and temperature, whereas the results exhibit significant discrepancies for lower and higher frequency values. AC 11 and SMA 11 mixtures show differences in comparison between the two tests. Moreover, the strains measured by the IDT test are variable and the strains obtained from the UC test stabilise around 40 με. Similarly, both tests have poor strain control at 40 °C. The values of normalised stresses measured by the IDT test are approximately 3.26 and 2.34 times greater than the ones measured by the UC test for AC 11 and SMA 11 mixtures, respectively. In general, the results of the mechanical characterisation of the asphalt mixtures conducted using both tests are similar. The IDT test has the advantage of sample size for sample preparation methods in both laboratory and field, and the UC test has a better deformation control at low and medium temperatures.

Strain softening of natural rubber composites filled with carbon black and aramid fiber

Engineered rubber vulcanizates may contain a low content of short fibers and a high content of nanoparticles while the effects of the different fillers on the softening behavior are not yet explored. Herein, influences of carbon black (CB) and short aramid fiber (AF) on the Payne and Mullins effects of natural rubber composites are investigated for the first time by creating master curves of dynamic modulus or dissipation energy with respect to the straining responses of the matrix. It is revealed that the composite vulcanizates demonstrate the Payne effect characterized by decay of storage modulus, weak overshoot of loss modulus, and very weak high-order harmonics; this effect is mainly dominated by the rubber matrix experiencing microscopic strain amplitude enlarged by the filler. The composite vulcanizates exhibit the Mullins effect that becomes increasingly marked with increasing filler loading and is partially recovered by thermal annealing at relatively high temperatures. The energy dissipation during cyclic tensions is rooted in the viscoelastic deformation of the matrix and the filler-rubber interfacial debonding. The former is marked at room temperature where the rubber phase undergoes a crystallization-melting process during loading-unloading. The latter being marked in the presence of a small content of AF causes yieldinglike deformation for the virgin composites at low tensile strains, and its contribution to the softening is not recoverable during thermal annealing. The results show that the viscoelastic matrix is of importance in controlling the softening of the composite vulcanizates, which will be of guiding significance to conduct research studies on high-performance rubber composites products.

Evaluating fatigue resistance based on viscoelastic properties of asphalt mixtures

ABSTRACT In this study, the fatigue resistance of different asphalt mixture types subjected to the same aging condition was evaluated using linear viscoelastic properties of asphalt mixtures. The relaxation spectra were determined from the dynamic modulus test based on the linear viscoelastic theory, and the relationship between the parameters of relaxation spectra and the parameters of the storage modulus master-curves were developed. The Texas overlay test was conducted to evaluate the fatigue resistance with the number of load cycle to failure used as the measure of fatigue resistance. From the corresponding results, it is shown that the inflection point frequency of the dynamic modulus master-curves could be used to assess the fatigue resistance compared to other parameters. The Glover-Rowe parameter also exhibited a fair relationship with the fatigue resistance of the asphalt mixtures. The effect of the shape parameter (γ) of the dynamic modulus master-curves was quantified but seemed not very sensitive to fatigue resistance for the asphalt mixtures evaluated. In the study, a new parameter (VECIndex) was developed to evaluate the fatigue resistance, which considered the inflection point frequency and shape of the dynamic modulus master-curves, both of which influence the linear viscoelastic properties and fatigue resistance of asphalt mixtures.

Determining with High Accuracy the Relaxation Modulus and Creep Compliance of Asphaltic Materials in the Form of Sums of Exponential Functions from Mathematical Master Curves of Dynamic Modulus

According to the physical causality principle, the two components of complex modulus are not independent yet interrelated to each other via Kramers-Kronig relationships. Using two mathematical functions to predict simultaneously the dynamic modulus (DM) and phase angle (PA) generally leads to the violation of the causality principle. Therefore, the calculation of one component based on the other one is a vital task for using mathematical models to predict the linear viscoelastic (LVE) behavior of asphaltic materials. In addition, the relaxation modulus (RM) or the creep compliance (CC) in the form of sums of exponential functions (SEFs) is necessary for the efficient resolution of the stress–strain relationship in the time domain. This paper presented a novel method for calculating with high accuracy the RM and CC in the form of SEFs based on the input DM master curve. First, the midpoint integration scheme was used to calculate the PA based on the exact relationship between the PA and DM. After that, the storage modulus and compliance were computed and decomposed with high accuracy into SEFs using a special technique. Then, the closed-form formula of the RM and CC were derived thanks to the special form function of the storage modulus and compliance. Finally, the special technique was used once again to decompose with high accuracy the RM and CC into SEFs for possible uses in finite-element method (FEM) analysis in the time domain. The results showed that the relative errors of all LVE properties computed were lower than 0.1 percent at all calculating points.

Reliability of Calculation of Dynamic Modulus for Asphalt Mixtures Using Different Master Curve Models and Shift Factor Equations.

To develop a mechanistic-empirical pavement design system for Norwegian conditions, this paper evaluates the influence of the adoption of different models and shifting techniques on the determination of dynamic modulus master curves of asphalt mixtures. Two asphalt mixture types commonly used in Norway, namely Asphalt Concrete (AC) and Stone Mastic Asphalt (SMA) containing neat bitumen and polymer-modified bitumen, were prepared by the roller compactor, and their dynamic moduli were determined by the cyclic indirect tensile test. The dynamic modulus master curves were constructed using the standard logistic sigmoidal model, a generalized logistic sigmoidal model and the Christensen–Anderson–Marasteanu model. The shifting techniques consisted of log-linear, quadratic polynomial function, Arrhenius, William–Landel–Ferry and Kaelble methods. The absolute error, normalised square error and goodness-of-fit statistics encompassing standard error ratio and coefficient of determination were used to appraise the models and shifting methods. The results showed that the standard logistic sigmoidal model and the Williams–Landel–Ferry equation had the most suitable fits for the specimens tested. The asphalt mixtures containing neat bitumen had a better fit than the ones containing polymer-modified bitumen. The Kaelble equation and log-linear equation led to similar results. These findings provide a relevant recommendation for the mechanistic-empirical pavement design system.