Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results

In capturing visco-elastic behavior, experimental tests play a fundamental rule, since they allow to build up theoretical constitutive laws very useful for simulating their own behavior. The main challenge is representing the visco-elastic materials through simple models, in order to spread their use. However, the wide used models for capturing both relaxation and creep tests are combinations of simple models as Maxwell and/or Kelvin, that depend on several parameters for fitting both creep and relaxation tests. This paper, following Nutting and Gemant idea of fitting experimental data through a power law function, aims at stressing the validity of fractional model. In fact, as soon as relaxation test is well fitted by power law decay then the fractional constitutive law involving Caputo’s derivative directly appears. It will be shown that fractional model is proper for studying visco-elastic behavior, since it may capture both relaxation and creep tests, requiring the identification of two parameters only. This consideration is assessed by the good agreement between experimental tests on creep and relaxation and the fractional model proposed. Experimental tests, here reported are performed on two polymers having different chemical physical properties such that the fractional model may cover a wide range of visco-elastic behavior.     

The use of direct tension tests for the assessment of low temperature properties of bituminous binders

In France, tensile tests on bituminous binders are usually performed on vertical electromechanical loading frames, the test samples being placed in an insulated temperature control chamber and the deformations determined from the displacement of the cross-head. The paper presents the results of an inter-laboratory study of which the objective was to evaluate the suitability of this kind of equipment for the performance of direct tension tests at low temperatures. Most of the test parameters, used for this study complied with the procedure proposed by SHRP as from 1995 (AASHTO TP3). The various operating factors, and more particularly those affecting the determination of true sample elongation (mechanical plays, deformation of the gripping systems, ....), are discussed and a number of correcting procedures are proposed. Nevertheless, the results remained scattered. Considering the “flat” evolution of strain at failure with temperature in the domain of brittle behaviour, this makes it rather difficult to accurately define the temperature for failure at 1% elongation (T_ε=1%). This is also the case when this temperature is located in the zone of transition between brittle and ductile behaviour. The authors suggest that, together with the corresponding tensile strain, this “transition temperature” could be used as an alternative criterion which has the advantage of being linked to a physical behaviour of the material.     

Time dependent viscoelastic rheological response of pure,modified and synthetic bituminous binders

Bitumen is a viscoelastic material that exhibits both elastic and viscous components of response and displays both a temperature and time dependent relationship between applied stresses and resultant strains. In addition, as bitumen is responsible for the viscoelastic behaviour of all bituminous materials, it plays a dominant role in defining many of the aspects of asphalt road performance, such as strength and stiffness, permanent deformation and cracking. Although conventional bituminous materials perform satisfactorily in most highway pavement applications, there are situations that require the modification of the binder to enhance the properties of existing asphalt material. The best known form of modification is by means of polymer modification, traditionally used to improve the temperature and time susceptibility of bitumen. Tyre rubber modification is another form using recycled crumb tyre rubber to alter the properties of conventional bitumen. In addition, alternative binders (synthetic polymeric binders as well as renewable, environmental-friendly bio-binders) have entered the bitumen market over the last few years due to concerns over the continued availability of bitumen from current crudes and refinery processes. This paper provides a detailed rheological assessment, under both temperature and time regimes, of a range of conventional, modified and alternative binders in terms of the materials dynamic (oscillatory) viscoelastic response. The rheological results show the improved viscoelastic properties of polymer- and rubber-modified binders in terms of increased complex shear modulus and elastic response, particularly at high temperatures and low frequencies. The synthetic binders were found to demonstrate complex rheological behaviour relative to that seen for conventional bituminous binders.     

Evaluation of the anti-rutting potential of polymer-modified binders by means of creep-recovery shear tests

In the experimental study reported in this paper a creep-recovery shear test method was used to evaluate the anti-rutting potential of different polymer-modified bituminous binders. The effects of several factors related to modification were investigated, such as polymer type (SBS, styrene–butadiene–styrene vs. ethylene–vinyl–acetate), composition (styrene content), structure (linear vs. radial SBS) and dosage (3–6 % by weight of the base bitumen). The effect of short-term aging was also considered by comparing binder response before and after treatment with the rolling thin-film oven test. Following previous work on the development of the test protocol, experimental results were analyzed by referring to shear modulus curves G(t) and to permanent compliance (J _P), obtained by dividing residual strain at the end of the unloading phase by the stress applied during creep loading. Results indicated the effectiveness of the proposed method in discriminating between the behavior of the different polymer-modified binders and in capturing the effects caused by the factors considered in the investigation. Reliable rankings of the binders were established and were explained by referring to the specific behavior of employed modifiers.     

Effect of ageing on different modified bituminous binders: comparison between RTFOT and radiation ageing

Standard laboratory ageing methods of bitumen only take into account the effect of thermo-oxidation during the service life of a pavement but the effect of high energy cosmic radiation on site is not simulated in these procedures. The aim of the present work is to compare the laboratory simulated short term bitumen ageing (rolling thin film oven test) with ageing produced by short exposures of bitumen samples to Ultra Violet and gamma radiation. The influence of ageing agents on the thermal properties and rheological performance of the pristine and modified bitumen binders has been evaluated in this study. The thermal behavior of various aged bitumens is characterized by both isothermal as well as non-isothermal thermogravimetric analysis. The thermoanalytic investigations on bituminous samples are carried out to evaluate the thermal stabilities and activation energies of the binders and the life time prediction of the materials is made with the help of the kinetic information. It is found that modified bituminous binders are more resistant to heat and radiation. Different rheological tests are conducted by dynamic shear rheometer to examine the effect of ageing in terms of bitumen oxidation and polymer phase degradation which has a major consequence on high temperature rutting or low temperature cracking. Type of modifier is found to be of decisive importance. Creep and recovery tests show that the structure-time dependency of pristine aged bitumen is influenced much by stress and temperature than in the case of modified aged bitumens. The study has revealed that the elastomeric modifier protects the bituminous binder more than plastic modifier or nano filler. Finally, a fair correlation has been made between standard RTFO ageing and radiation aging.     

On the use of fractional calculus for the probabilistic characterization of random variables

In this paper, the classical problem of the probabilistic characterization of a random variable is reexamined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of α$\alpha$-stable random variables.     

Combination of a standard viscoelastic model and fractional derivate calculus to the characterization of polymers

Polymeric materials are known to be more or less dispersive and absorptive. In the field of mechanical vibrations, dispersion has for consequence that the dynamic modulus is frequency dependent, and absorption is exhibited by the fact that these materials have the ability to absorb energy under vibratory motion. The phenomenon of dispersion in conjunction with the notion of complex Modulus of Elasticity, permits to establish the relation between the real and the imaginary components of the Modulus of Elasticity, i.e. respectively the dynamic and loss moduli. The loss factor is simply determined through taking the quote of these two components of the Modulus of Elasticity. The theoretical background for the interrelations between the dynamic modulus and the loss modulus is found in the Kramers–Kronig relations. However, and due to the mathematical difficulties encountered in using the exact expressions of these relations, approximations are necessary for applications in practical situations. On the other hand, several simple models have been proposed to explain the viscoelastic behaviour of materials, but all fail in giving a full account of the phenomenon. Among these models, the standard viscoelastic model, or more known as the Zener model, is perhaps the most attractive one. To improve the performance of this model, the concept of fractional derivates has been incorporated into it, and which results in a four-parameter model. Applications have also shown the superiority of this model when theoretical predictions are compared to experimental data of different polymeric materials. The aim of this paper is to present the results of applying this model to rubber, both natural and filled, and to some other selected more general polymer. Electronic Publication     

Practical test methods for measuring the zero shear viscosity of bituminous binders

When the zero shear ciscosity (ZSV) is derived from a creep test, the creep time should be sufficiently long to obtain a steady state with a constant shear rate. The creep test works well for most conventional binders. Unfortunately, for polymer modified binders (PmBs), the long waiting times required to obtain a steady state are not always acceptable from a practical point of view and three are problems concerning the repeatability. This paper compares the steady state creep test to alternative test methods for the determination of the ZSV. One method is a short time creep/recovery test, with a short creep period and a longer recovery period, where the ZSV is derived from the recovery part. The other method is an oscillation test at low frequencies, where the complex viscosity is extrapolated to zero frequency to obtain the ZSV. Measurements are presented on a pure 70/100 binder and on a highly SBS-modified binder. While all methods give comparable and repeatable results for the case of the pure binder, complications occur for the highly modified binder. The oscillation method seems to be the most promising, method.

---

Résumé Lorsque la viscosité à taux de cisaillement nul (VTCN) est dérivée d'un essai de fluage, le temps de fluage doit être suffisamment long pour obtenir un état stable où la vitesse de cisaillement est constante. L'essai de fluage convient pour les liants conventionnels. Malheureusement, pour les bitumes modifiés (BmP), les longs temps d'attente nécessaires pour atteindre cet état stable ne sont pas toujours acceptables d'un point de vue pratique et il existe des problèmes concernant la répétabilité. Cet article compare l'essai de fluage avec attente de l'état stable à des méthodes d'essai alternatives pour déterminer la VTCN. Une des méthodes consiste en un essai fluage/recouvrance, avec une période de fluage courte et une période de recouvrance plus longue. La VTCN est alors dérivée de la partie recouvrance. L'autre méthode consiste en un, essai d'oscillation à basse fréquence. La viscosité complexe est extrapolée à la fréquence zéro pour obtenir la VTCN. Des résultats de mesure sont présentés pour un liant conventionnel (70/100) et pour un BmP à teneur élevée en SBS. Tandis que toutes la méthodes donnent des résultats comparables et répétables pour le liant conventionnel, beaucoup de complications apparaissent pour le bitume modifié à teneur élevée en SBS. La méthode d'oscillation semble la méthode la plus prometteuse.

---

---

Editorial Note Presented at the 6^th International RILEM Symposium on Performance Testing and Evaluation of Bituminous Materials (PTEBM'03), held on 14^th–16^th April 2003, in Zurich, Switzerland, this paper was selected as an outstanding communication and peer-reviewed by the Scientific Committee of the JournalMaterials and Structures.     

Non-linear viscoelastic model based ranking of modified binders for their rutting performance

Rutting is one of the most common distresses observed in bituminous pavements. A number of test procedures, associated parameters, and correlations have been developed for the binder and the mixture to evaluate their rutting performance. In this investigation, the rutting performance of an unmodified, and three modified binders (Elastomer, Plastomer and Crumb rubber) was evaluated separately for the binders and mixtures. Creep-recovery and stress relaxation experiments were carried out on binders in the temperature range of 46–70 \({^{\circ }}\)C. The mixtures were subjected to dry rut wheel testing at 50 and 60 \({^{\circ }}\)C. A nonlinear viscoelastic fluid model was used to capture the response of the binder. Using the material parameters identified separately for creep-recovery and stress relaxation experiments, the apparent viscosity was computed for each binder at different stress levels. It was seen that the ranking of mixtures evaluated from rut wheel testing was captured by the apparent viscosity predicted using the non-linear viscoelastic model.     

Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach

Fractional power-law nonlinear drift arises in many applications of engineering interest, as in structures with nonlinear fluid viscous-elastic dampers. The probabilistic characterization of such structures under external Gaussian white noise excitation is still an open problem.This paper addresses the solution of such a nonlinear system providing the equation governing the evolution of the characteristic function, which involves the Riesz fractional operator. An efficient numerical procedure to handle the problem is also proposed.     

On development of fractional calculus during the last fifty years

Fractional calculus generalizes integer order derivatives and integrals. During the last half century a considerable progress took place in this scientific area. This paper addresses the evolution and establishes an assertive measure of the research development.     

基于分数阶黏弹性模型的温拌改性沥青低温性能

基于分数阶黏弹性模型,通过低温弯曲梁流变试验（BBR）分析RH和Evotherm温拌剂对苯乙烯-丁二烯-苯乙烯嵌段共聚物（SBS）改性沥青低温性能的影响。结果表明：利用BBR试验数据拟合出的分数阶黏弹性模型参数可较好地预估温拌改性沥青的蠕变柔量和蠕变速率; RH温拌剂可以提高SBS改性沥青的阻尼比和耗散能比,但Evotherm温拌剂在低于-18℃时不能够提高;因7种沥青按照阻尼比和耗散能比排序相同,故也可以将阻尼比作为温拌改性沥青低温性能的一种评价指标; RH可以改善沥青的蠕变柔量,提高黏性流动柔量,且掺量越高改善效果越好,但不能提高黏弹性比,故不能用黏弹性比评价温拌SBS改性沥青的低温性能,而Evotherm温拌剂的规律性不强;两种温拌剂都在低于-24℃时对SBS改性沥青的低温性能改善不佳。     

The use of fractional exponential creep kernels for long-term behavior of laterally loaded piles in permafrost

A theory for long-term prediction of laterally loaded piles in a hereditary-elastic medium is presented. A mixed formulation approach with the aid of calculus of variations ensures continuity of pile displacements, bending moments and soil reactions. Finally, an asymptotic estimate of the pile behavior for very long time (t) is obtained using a fractional exponential operator derived from pile tests.     

The effect of physical ageing on the relaxation time spectrum of amorphous polymers: the fractional calculus approach

Empirical models corresponding to a constitutive equation with fractional derivatives are proposed for linear viscoelastic polymers. For these models, the relaxation modulus, the dynamic moduli, the relaxation time spectra, and other material functions can be calculated as a function of a few parameters that characterise the behaviour of a viscoelastic polymer. The fractional calculus approach allows us to calculate the relaxation time spectrum H() via the Stieltjes inversion in the linear viscoelastic zone. Polymethylmethacrylate (PMMA) is chosen as a model amorphous polymer in a temperature range from T_g + 90°C to T_g + 25°C. This polymer is characterised by a non-equilibrium state between at least the and relaxations. The structural recovery of PMMA has been investigated using dynamic mechanical thermal analysis (DMTA) by varying the preparational history. The effect of time and temperature on the model parameters and on the relaxation time spectra are also investigated in the neighbourhood of the glass transition.     

A modified Wenzel model for hydrophobic behavior of nanostructured surfaces

A modified Wenzel model was proposed to explore the influence of pore size distributions (PSDs) on water repellency of nanostructured surfaces. Rough surfaces with different porous structures, including surface areas and PSDs, were fabricated by stacking different solid ratios of TiO₂ nanoparticles. These fluorinated surfaces exhibited an excellent hydrophobic performance with the highest value of contact angle ∼ 165°. The PSDs of these surfaces, determined from Dubinin-Stoeckli equation, were found to vary with the solid ratios. The modified Wenzel model incorporated with the PSDs gave a fairly good prediction in describing the variation of contact angle with surface roughness, which is very close to the experimental data. These results demonstrated that the heterogeneity of surfaces caused by different PSDs would induce the hydrophobic behavior.