Modeling the viscoelastic function of asphalt concrete using a spectrum method

A continuous spectrum method is proposed and applied for modeling the time-domain viscoelastic function of asphalt concrete materials. This technique, employing a Wiechert model for the relaxation function and a Kelvin model for the compliance function, is found to substantially enhance accuracy and consistency compared to existing methods. Furthermore, this paper shows how to determine a time-domain Prony series representation, which can be used efficiently for numerical analysis, such as finite element analysis, from the complex modulus in the frequency domain, based on the continuous spectrum method.     

A multiscale model for predicting the viscoelastic properties of asphalt concrete

It is well known that the accurate prediction of long term performance of asphalt concrete pavement requires modeling to account for viscoelasticity within the mastic. However, accounting for viscoelasticity can be costly when the material properties are measured at the scale of asphalt concrete. This is due to the fact that the material testing protocols must be performed recursively for each mixture considered for use in the final design.In this paper, a four level multiscale computational micromechanics methodology is utilized to determine the accuracy of micromechanics versus directly measured viscoelastic properties of asphalt concrete pavement. This is accomplished by first measuring the viscoelastic dynamic modulus of asphalt binder, as well as the elastic properties of the constituents, and this comprised the first scale analysis. In the second scale analysis, the finite element method is utilized to predict the effect of mineral fillers on the dynamic modulus. In the third scale analysis, the finite element method is again utilized to predict the effect of fine aggregates on the dynamic modulus. In the fourth and final scale analysis, the finite element method is utilized to predict the effect of large aggregates on the dynamic modulus of asphalt concrete. This final predicted result is then compared to the experimentally measured dynamic modulus of two different asphalt concretes for various volume fractions of the constituents. Results reveal that the errors in predictions are on the order of 60 %, while the ranking of the mixtures was consistent with experimental results. It should be noted that differences between the “final predicted results” and the experimental results can provide fruitful ground for understanding the effect of interactions not considered in the multiscale approach, most importantly, chemical interactions.     

Universal linear viscoelastic approximation property of fractional viscoelastic models with application to asphalt concrete

The paper presents a comprehensive linear viscoelastic characterization of asphalt concrete using fractional viscoelastic models. For this purpose, it is shown that fractional viscoelastic models are universal approximators of relaxation and retardation spectra. This essentially means that any spectrum can be mathematically represented by fractional viscoelastic models. Characterization of asphalt concrete is performed by constructing the dynamic modulus master curve and determining the parameters of the generalized fractional Maxwell model (GFMM). This procedure is similar to the widely used one of determining the master curve of asphalt concrete using a statistical function such as the sigmoidal model. However, from the GFMM, the relaxation modulus, creep compliance, continuous relaxation spectrum, and Prony series parameters can be determined analytically. A further advantage of the GFMM is that unlike the sigmoidal model, which only gives a representation of either the dynamic modulus or the storage modulus, the GFMM gives a representation of both the storage modulus and loss modulus (and therefore also the dynamic modulus and phase angle). The procedure was successfully applied to ten different mixes used in the State of Virginia.     

Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori-Tanaka homogenization scheme and a discrete microscopic retardation spectrum

In this paper the macroscopic creep and relaxation functions of a heterogeneous viscoelastic porous medium are derived by using Mori-Tanaka homogenization scheme. Analytical and semi-analytical solutions can then be determined with a parametric number of heterogeneous phases embedded in a viscoelastic matrix whose behavior is described with a parametric number of analogical units. Under some simplifying assumptions, a solution strategy is presented in order to make explicit how the microscopic retardation and relaxation times of the viscoelastic matrix control the distribution of the retardation and relaxation times of the homogenized medium.     

Dynamic mechanical characterization of asphalt concrete mixes with modified asphalt binders

The primary objective of this work is to characterize and compare the dynamic mechanical behavior of asphalt concrete mixes with styrene butadiene styrene (SBS) polymer and crumb rubber modified asphalt binders with the behavior of mixes with unmodified viscosity grade asphalt binders. Asphalt binders are characterized for their physical and rheological properties. Simple performance tests like dynamic modulus, dynamic and static creep tests are carried out at varying temperatures and time. Dynamic modulus master curves constructed using numerical optimization technique is used to explain the time and temperature dependency of modified and unmodified asphalt binder mixes. Creep parameters estimated through regression analysis explained the permanent deformation characteristics of asphalt concrete mixes. From the dynamic mechanical characterization studies, it is found that asphalt concrete mixes with SBS polymer modified asphalt binder showed significantly higher values of dynamic modulus and reduced rate of deformation at higher temperatures when compared to asphalt concrete mixes with crumb rubber and unmodified asphalt binders. From the concept of energy dissipation, it is found that SBS polymer modification substantially reduces the energy loss at higher temperatures. Multi-factorial analysis of variance carried out using generalized liner model showed that temperature, frequency and asphalt binder type significant influences the mechanical response of asphalt concrete mixes. The mechanical response of SBS polymer modified asphalt binders are significantly correlated with the rutting resistance of asphalt concrete mixes.     

A direct method of viscoelastic analysis for ageing concrete

The paper outlines a direct method of linear creep analysis of ageing concrete structures based on the classical elastic-viscoelastic analogy. The method develops from a constitutive relationship which accounts for reversible and irreversible strain components. The reversible strains include the effects of ageing, and irreversibility due to both aging and applied stress is considered. The relationship between the components allows the time variables to be replaced by corresponding functions of the irreversible compliance: the integral equation representing the creep solution of many structure problems is then invariable in form and the elastic-viscoelastic analogy can be invoked. The effects of an immediate elastic modulus that varies with time may be analysed. The method compares favourably with other analyses requiring step-by-step computations in time, and with other available direct methods which lead to considerable errors for variable stress regimes or give only steady-state solutions.

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Résumé On donne les grandes lignes d'une méthode directe d'analyse des constructions en béton où le vieillissement de celui-ci s'accomplit avec un fluage. On examine le problème de poser des relations de base uni- et pluri-axiales réalistes, en tenant compte des effects du vieillissement et de l'évolution des contraintes, pour utiliser ensuite cette relation dans une méthode d'anlyse directe. La relation de base qu'on développe ici comprend les composantes du fluage tant réversible qu'irréversible, telles qu'on les obtient par les essais de fluage et de recouvrance. On prend en compte les observations faites auparavant que les deux composantes progressent à des vitesses en relation, quels que soient l'age, le moment de l'observation ou les contraintes antécédentes. Il en résulte que par une transformation non linéaire appropriée de l'échelle de temps, et le remplacement des variables de temps par des fonctions correspondantes de l'adaptation irréversible, il est possible d'obtenir un modèle réaliste de la déformation pluri-axiale du béton sous une forme essentiellement invariable par rapport au vieillissement. Ceci permet alors d'écrite sous une forme invariable l'équation intégrale qui traduit la solution de fluage de nombreux problèmes structuraux et de faire appel à l'analogie classique élastico-visco-élastique; on peut ensuite obtenir les solutions de fluage par une transformation inverse de Laplace d'une solution élastique correspondante. Etant donné qu'on a constaté expérimentalement que les adaptations de déformation totale aussi bien volumétriques que par distorsion sont proportionnelles à l'adaptation uni-axiale, la méthode peut servir à résoudre les problèmes en contrainte multiaxiale.

     

Characterization of linear viscoelastic properties of asphalt concrete subjected to confining pressure

This paper presents a triaxial storage modulus master curve model and a continuous relaxation spectrum model for characterizing the linear viscoelastic (LVE) properties of asphalt concrete subjected to confining pressure. The triaxial master curve model relates the reduced storage modulus to the reduced frequency by performing both horizontal and vertical shifting on storage modulus test results. The horizontal shifting is used to characterize the time- and temperature-dependent behavior of asphalt concrete, while the vertical shifting is to characterize the pressure-dependent behavior. The vertical shift factor has the form of a sigmoidal function and varies with the reduced frequency and confining pressure. A relaxation spectrum model as well as a long-time equilibrium modulus model has been derived from the triaxial master curve model via integral transforms of basic LVE equations and complex algebra. The model predictions agree well with the laboratory test results of three asphalt mixtures indicating the proposed models can accurately characterize the LVE behavior of asphalt concrete under confinement.     

Development of a stress-mode sensitive viscoelastic constitutive relationship for asphalt concrete: experimental and numerical modeling

Asphalt binder is responsible for the thermo-viscoelastic mechanical behavior of asphalt concrete. Upon application of pure compressive stress to an asphalt concrete specimen, the stress is transferred by mechanisms such as aggregate interlock and the adhesion/cohesion properties of asphalt mastic. In the pure tensile stress mode, aggregate interlock plays a limited role in stress transfer, and the mastic phase plays the dominant role through its adhesive/cohesive and viscoelastic properties. Under actual combined loading patterns, any coordinate direction may experience different stress modes; therefore, the mechanical behavior is not the same in the different directions and the asphalt specimen behaves as an anisotropic material. The present study developed an anisotropic nonlinear viscoelastic constitutive relationship that is sensitive to the tension/compression stress mode by extending Schapery’s nonlinear viscoelastic model. The proposed constitutive relationship was implemented in Abaqus using a user material (UMAT) subroutine in an implicit scheme. Uniaxial compression and indirect tension (IDT) testing were used to characterize the viscoelastic properties of the bituminous materials and to calibrate and validate the proposed constitutive relationship. Compressive and tensile creep compliances were calculated using uniaxial compression, as well as IDT test results, for different creep-recovery loading patterns at intermediate temperature. The results showed that both tensile creep compliance and its rate were greater than those of compression. The calculated deflections based on these IDT test simulations were compared with experimental measurements and were deemed acceptable. This suggests that the proposed viscoelastic constitutive relationship correctly demonstrates the viscoelastic response and is more accurate for analysis of asphalt concrete in the laboratory or in situ.     

A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material

A bilinear cohesive zone model (CZM) is employed in conjunction with a viscoelastic bulk (background) material to investigate fracture behavior of asphalt concrete. An attractive feature of the bilinear CZM is a potential reduction of artificial compliance inherent in the intrinsic CZM. In this study, finite material strength and cohesive fracture energy, which are cohesive parameters, are obtained from laboratory experiments. Finite element implementation of the CZM is accomplished by means of a user-subroutine which is employed in a commercial finite element code (e.g., UEL in ABAQUS). The cohesive parameters are calibrated by simulation of mode I disk-shaped compact tension results. The ability to simulate mixed-mode fracture is demonstrated. The single-edge notched beam test is simulated where cohesive elements are inserted over an area to allow cracks to propagate in any general direction. The predicted mixed-mode crack trajectory is found to be in close agreement with experimental results. Furthermore, various aspects of CZMs and fracture behavior in asphalt concrete are discussed including: compliance, convergence, and energy balance.     

A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete

An existing viscoelastic constitutive model which accounts for the effects of rate-dependent damage growth is described and applied successfully to characterize the uniaxial stress, constant strain rate behavior of asphalt concrete. The special case of an elastic continuum damage model with multiaxial loading, which is based upon thermodynamics of irreversible processes with internal state variables, is first reviewed and then it is shown how this model has been extended to a corresponding viscoelastic damage model through the use of an elastic-viscoelastic correspondence principle. The general mathematical model is next specialized to uniaxial loading. A rate-type evolution law, similar in form to a crack growth law for a viscoelastic medium, is adopted for describing the damage growth within the body. Results from laboratory tests of uniaxial specimens under axial tension at different strain rates are then shown to be consistent with the theory. The discussion of data analysis describes the specific procedure used here to obtain the material parameters in the constitutive model for uniaxial loading and how the method may be generalized for multiaxial loading.     

Validation of time temperature superposition principle for high modulus asphalt concrete in the linear viscoelastic and fatigue domains

Validation of time–temperature superposition principle (TTSP) in the fatigue domain for a high modulus asphalt concrete (HMAC) is presented in this paper. All tests were performed in tension-compression under strain control mode. First, TTSP was validated in the linear viscoelastic domain. Then, fatigue tests were performed under three loading conditions, 9.2°C and 5 Hz, 11.0°C and 10 Hz and 12.9°C and 20 Hz, which are equivalent regarding TTSP. Two fatigue protocols were adopted: continuous fatigue test (FT) and fatigue test with rest period (FTRP). For FT, three samples were tested at 180μm/m for each loading condition whereas for FTRP, one sample was tested at 100 μm/m. The data were analysed by comparing the dynamic modulus evolution as a function of time or the fatigue life duration. The results showed that HMAC with fatigue damage remains thermorheologically simple (i.e., respects the TTSP) in the studied temperatures range.     

Approximate expressions for the ageing coefficient and the relaxation function in the viscoelastic analysis of concrete structures

Creep analysis of concrete structures meets well-known computational difficulties when one needs to determine the relaxation function corresponding to a specified creep function through the inversion of Volterra's linear integral equation. For this reason, the recent proposals for the 1990 edition of the CEB Model Code introduce an approximate formulation of the relaxation function obtained from the expression of the ageing coefficient χ(t, t ₀) of the algebraic age-adjusted-effective modulus (AAEM) method assumed as a function of age at loading,t ₀, only. In the first part of this paper, an approximate algebraic expression, $\bar \chi $ for the ageing coefficient with reference to the MC 90 is presented. This formulation differs from previous proposals in that, in addition to the influence oft ₀, it also takes into account the influence of other parameters, such as relative humidity, characteristic strength of concrete and effective thickness. In the second part, this approximate algebraic expression of the ageing coefficient is used to obtain an approximate formulation for the relaxation function which is in very good agreement with the exact values of the relaxation function as obtained from the creep function through the solution of the basic integral equation. Finally, an example of structural calculation is provided (by applying the AAEM method and comparing the values obtained with those yielded by the general numerical method): the example clearly shows the advantages offered by the new approximate formulation of the ageing coefficient, $\bar \chi $ , compared to the expression proposed in the CEB MC 90.     

沥青路面动态粘弹性响应分析

沥青混合料是一种典型的粘弹性材料,在高速行车荷载作用下路面结构承受动态荷载,所以对沥青路面结构进行动态粘弹性分析,才能更客观的反映其行为特性。本文对典型的沥青路面结构建立了三维动态粘弹性有限元模型,分析了路表弯沉动态粘弹性响应及影响因素。结果表明弯沉峰值的大小及出现时间受计算点离轮载距离的影响。阻尼比对弯沉峰值的影响不大,但对弯沉振动衰减速率影响较大,阻尼越大,衰减的越快。温度对弯沉峰值及振动衰减速率均有较大的影响,温度越高,振动衰减的越快。比较了不同条件下动态分析和准静态分析得到的弯沉峰值,表明动态分析得到的结果平均要大12％左右,且弯沉峰值增加百分比受横向距离的影响。     

A time domain boundary element method for modeling the quasi-static viscoelastic behavior of asphalt pavements

A time domain boundary element method (BEM) is presented to model the quasi-static linear viscoelastic behavior of asphalt pavements. In the viscoelastic analysis, the fundamental solution is derived in terms of elemental displacement discontinuities (DDs) and a boundary integral equation is formulated in the time domain. The unknown DDs are assumed to vary quadratically in the spatial domain and to vary linearly in the time domain. The equation is then solved incrementally through the whole time history using an explicit time-marching approach. All the spatial and temporal integrations can be performed analytically, which guarantees the accuracy of the method and the stability of the numerical procedure. Several viscoelastic models such as Boltzmann, Burgers, and power-law models are considered to characterize the time-dependent behavior of linear viscoelastic materials. The numerical method is applied to study the load-induced stress redistribution and its effects on the cracking performance of asphalt pavements. Some benchmark problems are solved to verify the accuracy and efficiency of the approach. Numerical experiments are also carried out to demonstrate application of the method in pavement engineering.     

Investigation of layered elastic theory prediction accuracy for asphalt concrete pavement design using micromechanical viscoelastic finite element modeling

In this study, predictions of full-scale micromechanical (MM) finite element (FE) models, developed from X-ray computed tomography images of asphalt concrete samples that were sawn from the accelerated pavement test sections, were used to evaluate the accuracy of layered elastic theory (LET) models that are used in pavement design today. First, MM FE and LET models were both calibrated using the measured strain gauge responses. Predictions of calibrated models were compared to evaluate the reasonableness of LET model outputs at high temperatures. Second, asphalt concrete stiffnesses measured in the laboratory were directly used for LET model development without performing any strain gauge calibration to evaluate the actual predictive capability of LET models in pavement design by using the calibrated MM FE model outputs as the ground truth. Recommendations were also made for future use of the MM FE models to improve the predictive capability of LET models.     

Engineering characterization of recycled asphalt concrete and aged bitumen mixed recycling agent

A recycling agent is commonly used to restore the aged bitumen to a condition that resembles that of the virgin bitumen. Three reclaimed asphalt pavement (RAP) stockpiles were sampled, and the aged binders recovered from RAP binders were mixed with recycling agents at ten levels to produce bitumen blends. The blends using virgin bitumen as the softening agent exhibited a significantly different rheological behavior from ones using the rejuvenating agent. The addition of a recycling agent could shift up or down the master curve of the blend vertically, depending on the engineering properties of the recycling agent. A normalized viscosity ratio (NVR) model was used to characterize the rheological properties of aged bitumen mixed with softening and rejuvenating agents. An interaction parameter was introduced into the model to consider the physico-chemical reaction between aged bitumen and recycling agent. This mixing rule was compared to the method specified in the blending chart by the Asphalt Institute (AI). The blending chart was shown to be applicable to determine the amount of the softening agent required to meet the target viscosity. The NVR model appeared to be a better tool for the rejuvenating agent to predict the viscosity of a recovered bitumen blend than the AI chart.     

R-curves characterisation analysis for asphalt concrete

Fracture tests, especially at lower testing temperatures, have become quite popular in quantifying low-temperature cracking. However, current fracture testing analysis methods often use a single number, such as fracture energy or fracture toughness, to quantify cracking resistance. These tests do not capture both the initiation and propagation of the crack. The Resistance Curve, or R-curve, is widely applied in many fields, such as metal, polymer and composites. The R-curve considers cracking resistance as a function of crack extension, which includes initiation and propagation. In this research, three asphalt concrete mixtures, including hot mix, hot mix with reclaimed asphalt pavement (RAP) and warm mix with RAP were tested at two temperatures, three levels of ageing and two levels of moisture condition by the Semi-Circular Bend fracture test. R-curves were constructed using the data from the fracture test, and digital images were utilised to capture the crack extension. In addition to capturing the traditional fracture energy, two new parameters were explored using the R-curves: the cohesive energy and the propagation parameter energy rate. It was found that cohesive energy was always in a narrow range (approximately 500–1000 J/m²) compared to the fracture energy range (approximately 500–1700 J/m²) over all combinations of ageing and moisture conditions, which indicates that the crack initiation may not be as sensitive to temperature, ageing and moisture as fracture energy. The results of energy rate indicated that moisture and short-term ageing impact the crack propagation by reducing the resistance of crack growth. These results proved that R-curves are a potentially useful tool to quantify the cracking resistance of asphalt concrete in both crack initiation and propagation.     

Improved method of considering air void and asphalt content changes on long-term performance of asphalt concrete pavements

Mixture properties (aggregate gradation and volumetric quantities), rate of loading and environmental conditions are the most important factors that affect the |E*| values. The main objective of this study was to develop a rational approach to investigate and model the effect of air voids and asphalt content on the |E*| master curves and consequently predict pavement performance. In this study, |E*| tests were conducted on three asphalt concrete mixtures with the same aggregate gradation, but different binder grades. For each of these mixtures, the air void and asphalt contents were varied at three levels. It is found that the developed method provides a more accurate estimate of the effects of volumetric changes in hot mix asphalt. The application of the proposed approach would be most beneficial for quality control/quality assurance purposes, performance-related specifications and for estimating contractors' incentives and penalties, where |E*| is utilised to predict the pavement performance.     

Memory functions for mechanical relaxation in viscoelastic materials

The roles of memory functions for compliance retardation and modulus relaxation in viscoelastic materials are examined. It is shown that essential features of the mechanical responses are the components which occur instantaneously on the application of either a stress or a strain. Taking these features into consideration it is shown that at non-zero time the cooperative memory function of compliance retardation is the time differential of the modulus relaxation function and the cooperative memory function of modulus relaxation is the time differential of the compliance relaxation function for step up functions of stress and strain, respectively. Zero time singularities in the memory functions have been eliminated in the derivation of the reduced dynamical equations, whose memory functions are limited to non-singular contributions which are always present.