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Revista ingeniería de construcción

versión On-line ISSN 0718-5073

Rev. ing. constr. vol.26 no.2 Santiago ago. 2011

http://dx.doi.org/10.4067/S0718-50732011000200001 

Revista Ingeniería de Construcción Vol. 26 N°2, Agosto de 2011 www.ing.puc.cl/ric PAG. 128-149.

 

Modeling and analysis of susceptibility to permanent deformation in asphalt mixtures

Modelación y análisis de susceptibilidad a la deformación permanente de mezclas asfálticas

 

Julián Vidal V.*1, Alexander Ossa*

* Universidad EAFIT, Medellín. COLOMBIA.

Dirección para Correspondencia


ABSTRACT

Permanent deformation of asphaltic mixtures in the Metropolitan Area of Valle de Aburra - Antioquia, which are built under IVIAS' specifications (Instituto Nacional de Vías) and Valle de Aburrá, were studied by employing a constitutive model proposed previously. This model had proved to be effective in predicting deformations of asphaltic mixtures in the United Kingdom under different kinds of loads; under uniaxial and triaxial conditions and temperatures at intervals between 0 °C and 40 °C. For the specific case of Valle de Aburra, temperatures ranged between 20 °C and 50 °C and the model was implemented in order to predict susceptibility to permanent deformation. By means of an experimental study on mixtures, it was found that their behavior under steady state condition followed the model modified by Cross2, thus mixtures showed a visco-linear and non-linear behavior at low and high stress levels, respectively. It was observed that under loading and unloading conditions, there is a temperature dependence on the material behavior, which was properly predicted by Arrhenius3 under the studied temperatures interval. When modeling mixtures behavior, it was found that that their strain susceptibility varied widely depending on the constituent materials, especially on the aggregate, in spite of the fact that the mixtures are similar.

Keywords: Asphalt, asphaltic mixtures, permanent strain, recovery, cyclic tests.


 

1. Introduction

In Colombia the use of high temperature asphaltic mixtures, as construction material for tread layers has been increasingly spread, due to its structural and functional characteristics.

In spite of the advantages offered by asphaltic mixtures, there are different factors that prevent them from thoroughly accomplishing their functions and finally leading to premature failures. The main types of failures on asphaltic layers are fatigue cracking and the accumulation of steady strains or rutting.

The factors that determine the origin of rutting on the tread layer are the magnitude and frequency of vehicular traffic loads and climatic conditions. When executing the SUPERPAVE4 method, it was observed that rheological characteristics of asphaltic binder also affected the mixture's plastic behavior, which is a determinant resistance factor under service loads. When asphalt is subjected to high temperatures or slow loads, it behaves as a viscose liquid; at low temperatures or shock loads it behaves as an elastic solid; and at intermediate temperatures it behaves as a viscose-elastic material5.

Rutting is a strain or longitudinal depression following vehicles' path, which generates higher risks on asphaltic pavement service conditions. In rainfall season, water is accumulated by depressions provoking accidents due to a hydroplaning phenomenon.

Due to diverse problems originated by rutting effect, several laboratory tests have been developed following national and international specifications (e.g. NAT, SUPERPAVE), and mathematical models - available ever since - have been employed in order to predict and to asses permanent strain or rutting on asphaltic tread layers (e.g. CORAL, L6).

Laboratory tests - useful to characterize asphaltic mixtures behavior - simulate some loading conditions undergone by pavements. Such is the case of monotonic uniaxial and triaxial testing, which control material temperature, stress and strain speed rate. In this way it is possible to analyze the behavior and characteristics of asphaltic mixtures under axial symmetry loading conditions to be then compared to established construction standards.

Regarding mathematic models, the firsts were mainly based on empirical relations, because of the material complexity and due to the lack of knowledge on the heterogeneous compounds behavior. The most common models have employed mechanical theories on continuum media and micro-mechanic models are frequently used nowadays (e.g. F. Martínez, S. Angelone7).

Different methods of strain modeling on asphaltic mixtures are generally based on linear viscose-elasticity theories; however such mixtures have a nonlinear viscose-elastic behavior applicable under uniaxial and triaxial conditions (Huang, 1967). Besides, some of them are only applicable on strains under stable conditions, thus demanding a high number of parameters or calibration tests; those restrictions make then poorly practical and useless.

In 2004 Ossa and co-workers developed a simple constitutive model to predict asphaltic mixtures behavior undergoing steady strains, which is based on the strain rate decomposition in elastic, viscose and recoverable components, according to classic theories of plasticity and viscose-plasticity.

The purpose of this study is to check the applicability and/or to expand, theoretically or experimentally, the constitutive model developed by Ossa and co-workers, on asphaltic mixtures manufactured by three production plants in Valle de Aburrá (Medellín, Antioquia), which are elaborated under the specifications by INVIAS and Valle de Aburrá (Regulating entities at national and local level, respectively). The tested mixtures were re-elaborated by using working formulas delivered by the plants.

In order to achieve an adequate determination regarding mixtures behavior and characteristics in the constitutive model, uniaxial monotonic testings were developed for controlled strain, thermo-fluency and recovery. From results of such rests and the adjustment of parameters included in the constitutive model, it was possible to determine and characterize asphaltic mixtures under steady strains.

2. Description of constitutive model

In spite of the availability of a great number of methods to predict the behavior of asphaltic mixtures under different loading conditions, the logical-phenomenon model developed by Ossa and co-workers was employed by this study, due to its easy implementation and capability to incorporate material characteristics such as non-linearization, thermal dependency and confining pressure effects. This model is based on the strain rate decomposition in elastic, viscose and recoverable components and on classic plasticity and viscose-plasticity theories. This model was brought about because of experimental observations developed on typical asphaltic mixtures in United Kingdom. A detailed description of this model and parameters obtained are available on Ossa et al. (2010).

The model is based on the decomposition of tensor strain rate (as well on plasticity and classic viscose-plasticity) on plastic and elastic components

(1)

Elastic tensor strain rate is represented by:

(2)

where v, corresponds to Poisson relation E, corresponds toYoung modulus δij , is Kronecker delta. For δij we have:

(3)

Plastic tensor strain rate is decomposed into a deviatoric component and into a mean or hydrostatic component

(4)

Plastic tensor of plastic deviatoric strain rate is given by:

(5)

where:

, is stress deviatory tensor

, is Von Mises equivalent stress

, is component of equivalent viscose strain rate, active when .

, is equivalent recovery strain rate.

The equivalent recovery strain rate is expresses as function of parameterized recovery strains and stiffness factor , such factor is defined as the relation of strain rate under stable condition, measured by means of triaxial test or creep testing, divided by strain rate under stable condition in uniaxial test.

Parameterized strains is expressed as:

(6)

provided that s is the expansion gradient, recovery constant and plastic or steady strain.

On the other hand, is given by the following implicit equation:

(7)

and m are material constants, is the plastic or non-recoverable portion of viscose equivalent strain rate and is the reference strain rate for loading stress, which is expressed as function of Von Mises equivalent strain and stiffness factor

The tensor of mean or hydrostatic strain rate is given by:

(8)

Dependency on reference strain rate due to a given temperature can be described by WLF equation.

(9)

where , is the reference temperature, and C1S and C2S are universal constants equivalent to 8.86 and 101.6 K, respectively.

Or Arrhenius equation, corresponding to:

(10)

where k, is Arrhenius constant, equivalent to the relation between thermal activated energy and gases universal constant, is the reference strain speed rate at 0°C and , is the reference strain rate when a material is loaded.

Under uniaxial loading conditions, the relation between strain speed rate and stress in the constitutive model is summarized as follows:

(11)

The 6 parameters in this model are easily obtained from monotonic testing (controlled strain and constant stress or creep) and from recovery under uniaxial conditions, as experimentally reported in this study.

2.1 Experimental procedure

In order to experimentally achieve parameter values in the constitutive model proposed by Ossa and co-workers, specimens were elaborated for further use in controlled strain tests and for Creep tests under uniaxial compressive strength and recovery.

Specimens were elaborated from asphaltic and aggregates samples delivered by three different companies in the Valle de Aburrá, which were identified as Company A, Company B and Company C. The following nomenclature was employed for samples M1, M3 and M5 in accordance to specifications provided by INVIAS (Instituto Nacional de Vías), version 2002; samples M2, M4 and M6 according to specifications stated by Valle de Aburrá, version 1994. Mixtures from M1 to M4 were elaborated from alluvial deposit materials and M5 and 6 from sedimentary rock materials.

The preparation of asphaltic mixtures used for the elaboration of samples was executed under standard specifications stated by INVIAS, article 450-02 and regulations for pavement construction of Valle de Aburrá.

2.1.1. Materials employed

2.1.1.1 Mineral Aggregates

Three types of stone aggregates were used, two of them from alluvial deposits and the other from rock exploitation deposit determining some properties specified by INVIAS. Test results are shown in Table 1. According to specifications, it is quite clear that aggregates do not meet such standards, in spite of the fact that these aggregates are employed in our field for asphaltic mixtures production.

Table 1. Physical property of employed aggregates

2.1.1.2 Asphaltic cement

In order to prepare asphaltic mixtures conventional asphalt was delivered by Ecopetrol. Table 2 shows the obtained results from physical properties determination. It is clear that it does not meet some properties, such as the case of penetration index, however it is well know that our country does not have a refinery and finding asphalt not complying with specifications is highly usual.

Table 2. Physical properties of employed asphaltic cement

2.1.1.3 Asphaltic Mixture

Design of asphaltic mixture was developed by applying Marshall Methodology, which was provided by the companies. Table 3 shows properties obtained from specimens. Although some mixtures do not meet the specifications, tests were conducted since the purpose was to asses mixtures in the same way they are employed in our field.

Table 3. Physical and mechanical properties of asphaltic mixtures

The elaboration of specimens, which approximate dimensions are 20cm height and 10cm diameter, was developed by means of static compaction by applying a steady stress of 20 MPa and employing a double piston rod, so as to obtain higher homogeneity in the specimen. Maximum unit weight variation alongside specimen's height did not exceed 2%.

2.1.2 Uniaxial tests

Creep testing was conducted under different loading and temperature conditions. Loads of 1 kN, 2 kN, 4 kN, 15 kN and 19 kN were employed; and temperatures at 20 °C, 30 °C, 40 °C and 50 °C. Three samples per each loading condition were tested, so as to assess the test repeatability.

This analysis recorded time and axial strain as function of time, maintaining steady load. Figure 1 shows the typical behavior of an asphaltic mixture during Creep testing, i.e. strain variation as function of time.

Figure 1. Typical behavior of an asphaltic mixture during Creep testing. Sample from Valle de Aburrá, T=50 °C and F=2 kN

The results obtained from this test express the evolution as function of time, for a specific load and temperature settled for this analysis. Such test has been widely employed to evaluate different characteristics in asphaltic mixtures. Nowadays, it has been constantly used to predict rutting effect on asphaltic layers.

Controlled strain tests were conducted at a temperature of 20 ° C and at 0.02 mm.s-1 steady strain speed rate. The selection of such conditions is arbitrary; different temperatures and strain speed rates could have been used without affecting the results.

In the recovery test a given load is quickly applied and it is maintained at a pre-established constant level which is equivalent to one of the loads employed in steady loading test (creep). The specimen is strained up to a total specified nominal value εT.

From such strain the stress is removed and compressive strain is registered until strain reaches approximately cero value (εY≈O ) . Strain at this point is a plastic strain, as shown in Figure 2a. This test was repeated for different values εT, temperatures and creep stress, as observed in Figure 2b.

Figure 2a. Evolution of unitary strain as function of time during a recovery test

Figure 2b. Evolution of radial and axial strain in function of time during recovery test for different mixtures

2.2 Results

So as to analyze the results, the INVIAS type mixture elaborated by Company B was chosen. The results obtained by the other mixtures and companies are not shown as their behavior is similar.

2.2.1 Behavior of asphaltic mixtures under monotonic testings

Strain speed rate for Creep testing is obtained by the axial unitary strain slope as function of time, in the steady condition zone, where strain speed is maintained constant (Askeland and Phule (2004); Ashby et al. (2007)). Figure 3 shows the typical behavior under Creep testing, in steady condition for mixture M1 under four loading conditions, the strain speed under steady condition is also represented by the slope of the curve.

Figure 3 shows that at higher temperature there is a greater asphaltic mixture strain. This situation proves the validity of temperature dependency (equation 9) included in the constitutive model, which describes the relation existing between both variables. Temperature is directly proportional to strain speed, i.e. when temperature T increases there will also be a strain speed εÝ increase, as a function of material strain.

Figure 3. Typical behavior of an asphaltic mixture under Creep Testing. Mixtures INVIAS

Figure 4 summarizes the behavior of asphaltic mixture M1 under stable monotonic condition over stress interval, strain speed and temperatures, by means of a graph showing strain speed under stable stress v/s stable stress condition.

Similar to the observations made on ordinary asphalt by Cheung and Cebon (1997), mixture M1 at a temperature of 20°C describes a non-linear viscose behavior at high stress level (trend line slope different to 1, m = 0.8) and linear behavior at low stress levels (m = 1.0) as indicated in Figure 4.

It is also observed that temperature dependency on mixture M1 behavior, under steady condition, is well characterized by the relation WLF (Equation 9). Experimental results complied with trend lines generated by iteration of parameters in the constitutive model, thus ensuring that values obtained from parameters correctly predict the mixture M1 behavior.

Figure 4. Mixture M1 behaviors under controlled strain and Creep testings

2.2.2 Behavior of asphaltic mixtures under recovery tests

Figure 5 shows unitary axial strain curves as function of time, which describe behavior of mixture M1 for two Creep and recovery testing conditions T = 20°C, δ= 0.5MPa and T= 40°C, δ = 0.5MPa. Dotted lines also indicate the predictions in the constitutive model (Equation 10).

Figure 5. Recovery tests for mixture M1

In Figure 5 the obtained curves from experimental results, as well as predictions in the constitutive model (Equation 10); initially show two typical behaviors (stages) for the mixtures under Creep testings. During the first stage, known as primary Creep, the mixture shows a strain speed decrease, because the material endures strain strength. Such phenomenon is graphically evidenced upon the decrease of the curve slop as time goes by. Then we have the secondary Creep, stage when strain speed is constant, which is made quite clear due to the linear shape the material shows in this section. Finally we have recovery stage where load is removed and asphaltic mixture tends to recover its initial condition; that is the reason why strain as function of time decreases until being stabilized.

For conditions established in this study, a total strain interval for mixtures M1 of 0,0044≤εT ≤ 0,006 was obtained. Besides, total strain is directly proportional to recovered strain, according to the following equation:

(11)

The other mixtures fulfill in a similar way the relation described between total and recovered strain. Figure 6 shows such behavior between both strains for asphaltic mixtures elaborated in accordance with INVIAS specifications. Those mixtures are: M1 (INVIAS-Company B) , M3 (INVIAS-Company A) and M5 (INVIAS-Company C).

Figure 6. Relation between total and recovered strain for mixtures M1, M3 and M5

The values obtained from mixtures recovery constants Ψ comply with the range 0 ≤ Ψ ≤ 1, and are in accordance with the Equation (11). The results obtained from the graph recovered strain v/s total strain, shown in figure 6, evidence that recovery constant is only dependant on the material maximum strain and its behavior is completely independent on temperature.

The parameters values in the constitutive model for all mixtures are registered in Table 4.

Table 4. Parameter Summary in the constitutive model for 6 mixtures under steady conditions

2.2.3 Analysis of asphaltic mixtures elaborated in accordance with INVIAS specifications

Mixtures of INVIAS type are identified as M1, M3 and M5. Although they are elaborated under the same standard, each one of them has different mechanical characteristics; situation that may be analyzed from values found in parameters εo, m, k, σo, Ψ in the constitutive model (Equation 10), registered in Table 4. The m value expresses linearization or non-linearization of mixture behavior and, besides, it describes the existing relation between strain speed and stress applied on the zone of high stress levels and high strain speed rates. Thus m = (n - 1)/n is a parameter relating (n) trend line resulting from both variants graphics, then when the material shows a high strain speed rate upon the increase of small stress levels. Asphaltic mixture M3 is the one having the highest strain speed rate when small stress changes take place, followed by mixtures M5 and M1. Consequently it is highly susceptible to permanent strain.

The reference stress σo is considerably different for the three mixtures.σo is referred to the limit point where material changes from a linear viscose behavior into a non-linear viscose behavior, which is similar to yield point in non-viscose-elastic materials. By drawing a tangent line to the non-linear zone in the trend curve and, other tangent line to the linear zone, there will be a point where both tangents shall intersect; this is the limit point between linear viscose and non-linear viscose range of the material, σo, as shown in Figure 4 for M1 mixture. For smaller σo values, the linear viscose material stage is shorter and for higher values the opposite effect takes place. The reference stress σo corresponds to the highest stress extent where the material keeps a linear viscose behavior. In the same way, M5 has a shorter linear stage, rapidly going into a non-linear stage where material is quickly strained upon a small increase of stress level.

k value represents mixtures' thermal susceptibility, i.e., material sensitivity to temperature conditions provided by the test. Such k constant is directly proportional to strain speed rate, i.e., when increasing its value there will be an increase of strain speed rate ε0 due to thermal effects (material is softer and therefore it is easily strainable) According to this analysis and by checking registered values of Arrhenius constant in Table 4, the most susceptible material to temperature, and consequently, having higher strain due to thermal conditions is mixture M1, followed by M5 and M3.

The reference strain speed rate is related to the material stiffness and indicates the speed at which material becomes strained. Therefore, in accordance with values registered in Table 4, mixture M3 is an INVIAS type mixture straining at higher speed. M1 is the mixture straining at lower speed and, consequently, material is stiffer.

Recovery constant Ψ represents material ability to recover itself after a loading stress. Greater Ψ values indicate a higher material recovery and smaller values represent the opposite effect, i.e., the higher Ψ, higher delayed elastic recovery will take place and vice versa. According to Table 4, mixture M5 has a higher recovery rate facing a controlled loading stress (ΨM5 = 0.43), followed by mixture M3 with a value of ΨM1 = 0.39 and finally mixture M1 with a value of ΨM1 = 0.34.

Recovery constant Ψ also reveals an existing relation between recovered strain and total strain, described by Equation (11), where total strain is directly proportional to recovered strain; therefore, a material with a higher recovery constant y value and with increases of total strain, obtains higher recovery strain than a material with lower recovery constant value. Such effect is shown in Figure 7, indicating the relation between both strains for mixtures type INVIAS. Practically, recovery constant Ψ determines the response of the asphaltic tread layer upon permanent loading stress that generates the rutting effect. Then, this is a key parameter to identify materials with higher susceptibility to such kind of failures.

The influence of recovery constant Ψ and the behavior of mixture M5 (ΨM5 = 0.43), are demonstrated by means of a fastest recovery than in mixtures M1 and M3. Figure 7 indicates the value of recovered strain for mixtures M1 and M5 on a graph of unit strain as function of time.

Figure 7. Unitary strain as function of time for mixtures M1 and M5, at T = 20°C and = 0.5 MPa

According to the behavior of asphaltic mixtures M1, M3 and M5 - revealed from a physical representation of parameters in the constitutive model - it is concluded that the most resistant asphaltic mixture under traffic loads is the one elaborated from materials delivered by Company C (M5). Table 2 indicates that, in spite of the short linear viscose stage of mixture M5 (small σ0 value), it has the higher recovery constant, which provides a greater mixture recovery under loading stress and it also has the lowest strain speed rate. It has intermediate temperature susceptibility and a relation between stress and strain speed also intermediate.

2.2.4 Analysis of asphaltic mixtures elaborated according to Valle de Aburrá standards

Asphaltic mixtures elaborated under Valle de Aburrá standards correspond to mixtures M2, M4 and M6. Although they are elaborated under the same standards, each one of them has different mechanical characteristics; situation that might be predicted from values obtained by parameters ε0, m, k, σ0, Ψ (Equation 10), registered in table 4.

Main behaviors observed from such parameters are the following.

 The mixture with higher m slope is M2, which is the mixture having lower strain speed when small stress changes take place.

 Mixtures M4 and M6 have a similar σ0 reference stress value between them, however it is remote and minor compared to mixture M2. Mixture M2 has the highest σ0 value of the three asphaltic mixtures, being the longest the mixture with linear viscose stage; consequently, it is the mixture that takes longer to achieve the non-linear viscose stage.

 In accordance with registered values for Arrhenius constant in Table 4, the material most susceptible to temperature and, consequently, having greater strain due to thermal conditions is mixture M4, followed by M2 and M6.

 Reference strain speed rate ε0 is related to material stiffness and expresses a given material strain speed rate. Therefore, according to registered values in Table 4, mixture M2 strains at higher speed, followed by mixtures M6 and M4.

In accordance with Table 4, mixture M2 has the highest recovery constant value\|/ (equation 11). It means that asphaltic mixture M2 has greater ability to recover itself after loading stress than mixtures M4 and M6. Recovery constant V|/ also reveals an existing relation between recovered strain and total strain. Figure 8 shows the relation between recovered strain and total strain for asphaltic mixtures elaborated under Valle del Aburrá standards.

Figure 8. Relation between total and recovered strain for mixtures M2, M4 and M6

According to behavior of asphaltic mixtures M2, M 4 and M6, revealed from parameters in the constitutive model, it is concluded that asphaltic mixture having greater resistance to traffic loads is the one elaborated from materials delivered by Company B (M2). Table 2 indicates that although Mixture M2 is strained at higher speed than the other two mixtures (M4 and M6) and has a m value (relation between stress and strain speed rate) higher than M4 and M6; it has the highest recovery constant, which means it is a highly recoverable mixture under loading stress, its Arrhenius constant has the lowest value among three mixtures and it is the less susceptible to thermal conditions. It has a longer linear viscose stage and, consequently, it is the mixture taking longer to achieve a non-linear stage.

3. Conclusions

In this study steady monotonic and recovering testings were conducted on specimens elaborated from materials delivered by three asphaltic mixtures production plants under two Colombian standards (INVIAS 2002 and Valle de Aburrá). In both cases mechanical behavior is different, in spite of the fact that asphaltic mixtures comply with the specifications.

Because of a proper accuracy existing between experimental curves and those generated in the constitutive model employed by this test, it was proven that the used model describes at an adequate extent the behavior of asphaltic mixtures under study.

In spite of the number of recovery and Creep testings conducted by this study, it was proven that for the implementation and calibration of constitutive model only few tests are required and optimum results are obtained any way.

Monotonic and recovery testings were conducted on all asphaltic mixtures under steady condition, all of them showed a behavior dependant on temperature, which is accurately described by Arrhenius relation.

Similarly, from obtained results, it was determined that asphaltic mixtures have changes of mechanical behavior under loading stress. They describe a non-linear viscose behavior at high loading stress and a linear viscose behavior at lower loading stress.

Parameters making up the constitutive model reveal remarkable characteristics of asphaltic mixtures. The recovery constant Ψ represents the material ability to recover itself after loading stress, and then from its value it is possible to determine the response of the asphaltic tread layer under steady loading stresses that cause rutting effect. This is a key parameter to identify the material with higher susceptibility to this kind of failure. The m parameter is obtained as the curve slope, when representing strain speed rate v/s stress, and then its value provides the relation existing between them. A higher m value reveals linearization or non-linearization of material behavior. The reference stress σ0 determines the limit point where material changes from a linear viscose behavior into a non-linear viscose behavior, thus the higher stress value is the extent where the asphaltic mixture maintains a linear viscose behavior.

The k value represents thermal susceptibility of mixtures by indicating material sensitivity under given temperatures in this test. The reference strain speed rate εÝ0, reveals material stiffness and expresses its strain speed rate.

In accordance with obtained results, the asphaltic mixture elaborated from materials delivered by Company C, under standards by the INVIAS (Instituto de Vias), has a higher endurance against rutting effect than mixtures elaborated from materials provided by Companies A and B. Probably this situation occurs because both standards establish the use of a number of asphaltic binders, which might differ from one plant to another; and due to variable stone aggregates gradation.

On the other hand, the asphaltic mixture elaborated under the specifications by Valle de Aburrá and employing materials from Company B, has better mechanical properties than the ones elaborated from materials provided by Companies A and C.

This study employed a model proposed and studied under specific material and temperature conditions typical in the United Kingdom, under typical conditions in Colombia, specifically Medellín. Although this model is based on phenomenological considerations, it has variants of physical interpretation that lead to differentiate susceptibility to steady strain on diverse asphaltic mixtures. It is worth mentioning that by means of this study it was possible to determine that although asphaltic mixtures of identical standards, complying with defined specifications, they have quite different responses under loading stress. This conclusion might be a beginning for the adjustment of specifications or testing procedures in order to ensure a higher standardization on asphaltic mixtures for specific loading stress.

4. Notas

2 Cross MM. Rheology of non-newtonian fluids: a new flow equation for pseudoplastic system. J Colloid Sci 1965; 20: 417-37

3 Wolfgang Stiller. Arrhenius Equation and Non-Equilibrium Kinetics: 100 Years (Paperback), 1989

4Harman, T., D'Angelo, J., Bukowski, J., Superpave Asphalt Mixture Design - Workshop Workbook, Federal Highway Administration, Washington, 2000.2 PAEZ, D., PEREIRA, H. (2001)

5Estudio del ahuellamiento de mezclas asfálticas. XIII Simposio Colombiano sobre Ingeniería de Pavimentos». Universidad de Los Andes, Bogotá

6 CORAL, L (2003). »Refinamiento de un modelo de elementos finitos (FEM) para la estimación de ahuellamiento en pavimentos». Trabajo de Grado. Ingeniería Civil. Universidad de Los Andes, Bogotá

7F. Martínez, S. Angelone. Un modelo micromecánico simplificado para mezclas asfálticas. XXXIII Reunión del Asfalto", noviembre de 2004, Mendoza. Argentina

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SHRP-A-369 (1994), Binder characterisation and evaluation. Vol 3: Physical characterisation (A-369). Strategic Highway Research Program, Washington, DC.

Tabor D. (1951), Hardness of metals. Oxford: Clarendon Press.

Van der Poel C. (1945), A general system describing the visco-elastic properties of bitumen and its relation to routine test dat. J Appl Chem; 4:221-36.

Van der Poel C. (1954), Representation of rheological propierties of bitumen over a wide range of temperatures and loading times. In: Proceedings of the 2nd international congress on rheology, p. 331-- .

Vidal, J. (2006), "Comportamiento dinámico de mezclas asfálticas". Revista Universidad EAFIT. N° 143, Vol 42, pp. -8-88. Medellín - Colombia.

Ward IM. (19-1), Mechanical properties of solid polymers. New York: Wiley/Interscience.

Wang, L. (2011), Mechanics of Asphalt, Microstructure and micromechanics. New York: Mc Graw Hill. Whiteoak D. (1990), The shell birumen handbook. IK: Shell Bitumen.

Wolfgang Stiller (1989), Arrhenius Equation and Non-Equilibrium Kinetics: 100 Years (Paperback).


E-mail: jvidal@eafit.edu.co

Fecha de recepción: 14/ 12/ 2010. Fecha de aceptación: 30/ 03/ 2011.

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Ossa, E. A. et al. (2006), "Dilation behaviour of asphalt mixtures" Int. Journal of Road Materials and Pavement Design. VOL 7/SI - 2006 - pp.93-109.        [ Links ]

Ossa, E. A. et al.(2010), "Triaxial deformation behaviour of asphalt mixes" Journal of Materials in Civil Engineering, ASCE. Vol 22(2). pp 124-135.        [ Links ]

SHRP-A-369 (1994), Binder characterisation and evaluation. Vol 3: Physical characterisation (A-369). Strategic Highway Research Program, Washington, DC.         [ Links ]

Tabor D. (1951), Hardness of metals. Oxford: Clarendon Press.        [ Links ]

Van der Poel C. (1945), A general system describing the visco-elastic properties of bitumen and its relation to routine test dat. J Appl Chem; 4:221-36.        [ Links ]

Van der Poel C. (1954), Representation of rheological propierties of bitumen over a wide range of temperatures and loading times. In: Proceedings of the 2nd international congress on rheology, p. 331-7.        [ Links ]

Vidal, J. (2006), "Comportamiento dinámico de mezclas asfálticas". Revista Universidad EAFIT. N° 143, Vol 42, pp. 78-88. Medellín - Colombia.        [ Links ]

Ward IM. (1971), Mechanical properties of solid polymers. New York: Wiley/Interscience.        [ Links ]

Wang, L. (2011), Mechanics of Asphalt, Microstructure and micromechanics. New York: Mc Graw Hill.         [ Links ]

Whiteoak D. (1990), The shell birumen handbook. IK: Shell Bitumen.        [ Links ]

Wolfgang Stiller (1989), Arrhenius Equation and Non-Equilibrium Kinetics: 100 Years (Paperback).        [ Links ]

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