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This article was downloaded by: [Sohm juliette]On: 05 November 2012, At: 03:05Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
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Creep tests on bituminous mixtures andmodellingJuliette Sohm a , Thomas Gabet a , Pierre Hornych a , Jean-MichelPiau a & Hervé Di Benedetto ba IFSTTAR, LUNAM Université, Route de Bouaye CS4, 44341,Bouguenais Cedex, Franceb DGCB department, University of Lyon/ENTPE – DGCB, RueMaurice Audin, 69518, Vaulx en Velin Cedex, FranceVersion of record first published: 05 Nov 2012.
To cite this article: Juliette Sohm, Thomas Gabet, Pierre Hornych, Jean-Michel Piau & Hervé DiBenedetto (2012): Creep tests on bituminous mixtures and modelling, Road Materials and PavementDesign, DOI:10.1080/14680629.2012.735795
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Road Materials and Pavement DesigniFirst, 2012, 1–18
Creep tests on bituminous mixtures and modelling
Juliette Sohma*, Thomas Gabeta, Pierre Hornycha, Jean-Michel Piaua and Hervé Di Benedettob
aIFSTTAR, LUNAM Université, Route de Bouaye CS4, 44341 Bouguenais Cedex, France; bDGCBdepartment, University of Lyon/ENTPE – DGCB, Rue Maurice Audin, 69518 Vaulx en Velin Cedex, France
In order to study the permanent deformations of bituminous mixtures, a temperature-controlledtriaxial test has been set up. By means of triaxial creep tests at imposed stress, the influence ofdifferent parameters on the behaviour of bituminous materials (confining pressure, deviatoricstress and temperature) has been investigated. A viscoelastic–viscoplastic model has beendeveloped to simulate the creep tests and has shown good ability for simulating the test results.
Keywords: bituminous mixtures; triaxial test; rutting; creep; viscoelastic–viscoplasticmodelling; time–temperature superposition principle
1. IntroductionPermanent deformations are one of the main distress mechanisms affecting bituminous pavements.Their presence is revealed by longitudinal profile irregularities and especially by transverse profiledeformations in the wheel paths (Verstraeten, 1995). These permanent deformations or ruts affectriding comfort and safety of road users. For these reasons, LCPC has financed a PhD thesis(Sohm, 2011) and developed an extensive research programme on permanent deformations ofroad materials. One of the main objectives was the development of a homogeneous, thermo-controlled triaxial test, suitable for studying permanent deformation behaviour of bituminousmixtures (Sohm, Hornych, Gabet, & Di Benedetto, 2010).
This type of test has been chosen for several reasons. Stress paths generated in pavements bymoving wheel loads are difficult to reproduce in the laboratory for many aspects (Gabet et al., 2011;Perraton et al., 2011). These loads are cyclic, triaxial with stress rotations and show a complexevolution. Empirical tests like the French wheel tracking test (WWT) are used for the design ofbituminous mixes. They attempt to simulate moving wheel loads, but they are non-homogeneousin nature and very difficult to interpret in terms of rheological behaviour of the material. Forcomparative studies, such tests are interesting, but they cannot be used for development of con-stitutive models. Thus, triaxial homogeneous tests are more adapted to a rheological approach.Different authors underline the importance of considering triaxial behaviour with confining pres-sure (Clec’h, Sauzeat, & Di Benedetto, 2009; De Visscher, Maeck, & Vanelstraete, 2006; Ebels &Jenkins, 2006; Taherkhani & Collop, 2006; Taherkhani, Grenfell, Collop, Airey, & Scarpas, 2007),as it is the case for most geomaterials (Gabet, 2006).
At present, modelling and predicting the behaviour of asphalt concretes in the nonlinear andirrecoverable domain, under 3D stress states, still remain a challenge. Thanks to a homogeneoustest, several important aspects of the behaviour of bituminous materials can easily be studied like
*Corresponding author. Email: [email protected]
ISSN 1468-0629 print/ISSN 2164-7402 online© 2012 Taylor & Francishttp://dx.doi.org/10.1080/14680629.2012.735795http://www.tandfonline.com
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the sensitivity to axial stress, confining pressure, temperature, cyclic loading, while others willbe neglected, such as stress rotation (rotation of the main directions of stress tensor at a givenpoint, under moving loads). Such a test is also particularly adapted to the study of static loads onbituminous mixtures, for the prediction of the behaviour of industrial platforms or parking areas.
It was decided to develop a triaxial apparatus for bituminous mixes, able to perform a largerange of tests: monotonic creep tests, complex modulus tests and cyclic permanent deformationtests at small and large strain levels (from 50 μ strain up to 5%) and at different temperatures (from5◦C to 60◦C), different frequencies (up to 10 Hz) and different levels of confining pressures. As afirst step of this research on permanent deformation behaviour, it was decided to focus first on thecreep behaviour of bituminous mixtures before studying cyclic effects, in order to understand theeffect of different levels of loading, confining pressure and temperature. Ranges of both loadingand confining pressure have been set up by means of a numerical analysis performed using thesoftware Viscoroute®. This software allows to simulate a standard French tyre on a standardpavement (Sohm, 2011). This range of temperature is typically encountered in France.
As a first result, it is shown in this paper that the time–temperature superposition principle(TTSP) can be applied in the nonlinear domain even when considering triaxial loading. Thisprinciple, due to the viscoelastic behaviour of bituminous mixtures, is usually considered as validin the small strain linear domain. It assumes an equivalence between a change of temperatureand a change of time scale, and allows to transpose results obtained at one temperature to othertemperatures. As a second result, a viscoelastic–viscoplastic model for the prediction of creepbehaviour under 3D stress states (with confinement) is proposed. This model, developed forstatic load conditions, appears interesting, in practice, to simulate the response of bituminouspavements under static loads, such as those occurring at bus stops or vehicle parking areas or onstorage platforms.
2. Triaxial apparatus with temperature controlThe triaxial apparatus has been developed using an existing servo-hydraulic loading frame.The pressure cell, the temperature control system and the instrumentation have been developedspecifically for triaxial tests on bituminous mixtures (Figure 1).
The apparatus is designed for specimens with a diameter of 80 mm and a height of 160 mm. Themaximum axial load is 25 kN, the maximum confining pressure is 700 kPa and the temperature
Figure 1. Scheme of the triaxial cell.
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Road Materials and Pavement Design 3
range is from 5◦C to 60◦C. Monotonic and cyclic axial loading can be applied, keeping a constantconfining pressure. The maximum loading frequency is 10 Hz.
2.1. Triaxial cell and loading frameThe loading frame used to apply the axial load is a SCHENCK frame, with a servo-hydraulicactuator, with a load capacity of ±100 kN and a ±100 mm axial stroke. The loading frame isequipped with a monitoring control system, which allows one to perform a large range of loadingtests (monotonic or cyclic) with load or displacement control.
The triaxial cell for bituminous materials has been developed by GDS Instruments (Figure 1). Itis made of aluminium, has an external diameter of 300 mm and can accept 80 or 100 mm diameterspecimens. The confining fluid that is also used for temperature regulation can be water or air.Water allows a better homogeneity of the temperature. The cell is equipped with an internal loadtransducer of 25 kN capacity, and a 1000 kPa pressure transducer, placed outside the cell, on theconfining fluid supply circuit.
2.2. Temperature control systemThe temperature in the triaxial cell is regulated using a double heating and cooling system(Figure 1). The first system consists of a coil, situated inside the triaxial cell, in which the heat-ing and cooling fluid circulates (water with anti-freeze). This coil is connected with an external,removable heating and cooling system. To improve the control of the temperature in the cell,a thermal housing is added around the triaxial cell. A thermocouple placed inside the cell atmid-height of the specimen is used to control the temperature.
To ensure stabilisation of temperature before testing, specimens are stored in the triaxial cellwith thermal housing at the testing temperature for at least 4 h prior to testing.
2.3. Axial and radial displacement transducersThe axial and radial strains of the specimen are measured using a specific system of linear variabledifferential transformer (LVDT) sensors. The axial strains are measured on the central part of thespecimen using two LVDTs, placed vertically, on opposite sides of the specimen (Figure 2). The
Figure 2. View of specimen with the axial and radial LVDT sensors.
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Table 1. Characteristics of the tested bituminous mixture.
Size of aggregates (mm) Percentage (by mass)
0/2 27%2/4 10%4/6 25%6/10 35.5%Limestone filler 2.5%Penetration grade bitumen 50/70Bitumen percentage 5.50%
average of the signals of the two sensors is used to calculate the axial strain. The homogeneity ofthe strain field is also checked by comparing the values given by each sensor.
The radial strains are measured using an articulated ring, equipped with an LVDT, placed atmid-height of the specimen. The radial LVDT measures the opening of the ring, and thus thevariations of the specimen diameter. The axial LVDTs and the articulated ring are attached tothe specimen using metallic clamps, glued onto the specimen. The LVDTs have a measurementrange of 5 mm and a resolution of 1 μm.
3. Experimental procedures3.1. Material and sample preparationThe asphalt concrete tested in this study (using the triaxial apparatus) was a French dense-gradedbituminous mix, named BBSG or “Béton Bitumineux Semi-Grenu”. This material is typicallyused for wearing courses in France. The characteristics of this mix are given in Table 1. It hasbeen designed with a void percentage of 4.5%. The void percentage was measured by a nuclear(gamma ray) method (EN 12697-7, 2003). The mean of the void percentage, on 60 specimens,was 4.54% of voids and the standard deviation was 0.62% of voids (13.6%).
Once mixed, the material is compacted using a slab compactor with rubber-tired wheels (EN12697-33 + A1, 2007). The dimensions of the plates are 600 mm by 400 mm by 120 mm thick.After a minimum of 2 weeks of storage, six cylindrical specimens (80 mm in diameter and 160 mmhigh) were cored horizontally, from the central part of each slab. It would have been more relevantto core them vertically, in the compaction direction, but the thickness of the plates did not allowone to do that. The specimens were stored in a room at a controlled temperature of 18◦C beforetesting.
3.2. Preparation of triaxial test specimenTo perform tests in compression and tension, a metallic platen is glued at each end of the specimen.These platens are attached to the cell base and to the piston. To ensure homogeneity of the stressstates in the specimen, a good alignment of the specimen and end platens is required. A specificdevice, for gluing the specimens, has therefore been developed.
Then, the LVDT clamps are glued on the specimen. A latex membrane is placed around thespecimen, above the clamps and finally, the LVDT sensors are fixed on the clamps (Figure 2).
3.3. Characterisation of the bituminous materialTwo classical tests were made to characterise the bituminous mixture: the French WWT (large sizedevice, T = 60◦C, pressure of the tyre is 6 bar and the wheel frequency is 1 Hz) (EN 12697-22,
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Figure 3. Experimental values of complex modulus and master curve at the reference temperatureof 10◦C.
2007) and the complex modulus test (EN 12697-26, 2004). The WTT test gives a rut depth of7.2% at 30,000 cycles. Rutting resistance of the chosen bituminous mixture is considered asmedium. The complex modulus test is a two-point bending test on trapezoidal samples. Tem-peratures of the test are −10◦C, 0◦C, 10◦C, 15◦C, 20◦C, 30◦C and 40◦C. Frequencies of thetest are 1, 3, 10, 25, 30 and 40 Hz. The results are presented in Figure 3, as isotherms and mas-ter curve at the reference temperature of 10◦C. The reference complex modulus of the mix, at15◦C and 10 Hz, is 10,400 MPa. The TTSP is used to characterise the thermal sensitivity ofthe material. The TTSP says that changing the temperature is equivalent to changing the timescale. This comes to consider the secant (or complex modulus) as a function of time t/aT (orfrequency aT ω) where aT is a shift factor defined by aT = tT /t0. tT is the time that is requiredto give a specified response at temperature T and t0 is the time required to give an identicalresponse at the reference temperature. The classical WLF law (William, Landel, & and Ferry,1955) is used to determine the shift factor aT , where Tref is the reference temperature, and C1and C2 are two constants to fit (Equation (1)). C1 and C2 have been determined using the soft-ware viscoanalyse®, developed for the interpretation of complex modulus tests (Chailleux, Such,Ramond, & de La Roche, 2006). For the materials considered here and Tref = 10◦C, C1 = 32.2 andC2 = 217.6◦C
log(aT ) = − C1(T − Tref)
C2 + T − Tref. (1)
4. Testing programme and resultsThe aim of the experimental investigation is to study the influence of temperature, deviatoricstress and confining pressure on the creep response of our material. During a creep test, a constantdeviatoric stress q and a constant confining pressure (or radial stress) σrad are applied to thespecimen at the start of the test, and the evolutions of the axial and radial strains, εax and εrad, aremeasured (Figure 4). Five seconds are needed to apply loads.
For the analysis of the tests, the chosen sign convention is the soil mechanics convention(positive stress/strain in compression/contraction). The stress state is described using the meanstress p and the deviatoric stress q. For axisymmetric triaxial test conditions, these stresses are
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Figure 4. Principle of an axisymmetric triaxial creep test. Definition of the variables q and σrad. The soilmechanics sign convention is used.
defined by:
p = 2σax + σrad
3, (2)
q = |σax − σrad|, (3)
with σax being the axial or vertical stress and σrad being the radial or horizontal stress.Axial strains εax and radial strains εrad are measured during the tests. Volumetric strains εvol
and deviatoric strains εdev are deduced from these measures, using Equations (4) and (5).
εvol = εax + 2εrad, (4)
εdev = 23 (εax − εrad). (5)
Preliminary tests (not described in this paper) have been performed to study the repeatabilityof the triaxial creep tests. Five creep tests at q = 400 kPa, σrad = 0 kPa and T = 20◦C have beenperformed. In the framework of our tests, the absence of confining pressure represents the worstcase in terms of repeatability. In these conditions, a maximum variation of 15% has been obtainedbetween all the results for the axial and radial strains. This result gives an order of magnitude ofthe accuracy of the results presented below.
The tests performed and presented in this study are indicated in Figure 5. The tests wereperformed at:
• four levels of temperature: 10◦C, 20◦C, 30◦C and 40◦C;• five levels of confining pressure, from 0 to 200 kPa;• five levels of deviatoric stress, from 200 to 2000 kPa.
4.1. Loading and unloading behaviourIn order to separate the recoverable part from the irrecoverable part of the strains, unloadingphases were carried out at the end of eight of the creep tests. The unloading was carried out intwo stages. First, the deviatoric stress q was set to 0, and the strains were measured until therecoverable strains have reached a steady state. Then, the confining pressure σrad was also set to0 kPa, and the strains were recorded again.
An example of strain response during loading and unloading, for a creep test at q = 500 kPa,σrad = 100 kPa and T = 20◦C, is shown in Figure 6. It can be seen that the resilient strain afterunloading is very small in comparison with the irrecoverable strain. The results were similar in all
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Road Materials and Pavement Design 7
Figure 5. Summary of all the test conditions (deviatoric stress, confinement and temperature) applied inthis study.
(a)(b)
Figure 6. Loading and unloading curves for a creep test performed at q = 500 kPa, σr = 100 kPa andT = 20◦C. (a) Global curve and (b) zoom on the unloading part.
the tests, the recoverable strain never exceeds 6% of the irrecoverable strain. The results obtainedduring unloading will be used to separate the viscoplastic and viscoelastic strain components.
4.2. Influence of the deviatoric stressFive creep tests, at 20◦C, without confining pressure, were performed to study the influence ofthe deviatoric stress level. The five chosen levels of deviatoric stress are 200, 400, 600, 1000 and2000 kPa. Results are presented in Figure 7. Figure 7(a) shows the influence of the deviatoricstress on the axial strain as a function of time. In these tests, without confining pressure, no steadystate of the strains was reached at the end of the tests. Figure 7(b) emphasises the volumetricbehaviour. In all the tests, a dilatant behaviour is observed (i.e. volume increase). It can be notedthat the higher the deviatoric stress, the lesser the material is the dilatant, for a given level of axialstrain. This behaviour may be mainly explained by an influence of strain rate. Indeed, a higherdeviatoric stress leads to a higher axial strain rate, and due to viscous effects, the Poisson ratiodecreases, and this does not allow the material to creep at the same rate in the radial direction.
4.3. Influence of the confining pressureFive tests have been performed at 20◦C, at a deviatoric stress of q = 400 kPa and at five differentconfining pressures: 0, 25, 50, 100 and 200 kPa. Figure 8(a) shows that the higher the confiningpressure is, the lower the axial strain is. For confining pressures of 100 or 200 kPa, axial strainsseem to tend towards a constant value. Figure 8(b) shows that the material can be contractant forhigh confining pressures (volume decrease), but it remains mainly dilatant for confining pressures
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Figure 7. (a) Axial strains as a function of time and (b) volumetric strain as a function of axial strain, fortests at 20◦C, σrad = 0 kPa and for different deviatoric stresses.
(a) (b)
Figure 8. (a) Axial strains as a function of time and (b) volumetric strain as a function of axial strain, fortests at 20◦C, q = 400 kPa and different confinements.
lower than 100 kPa. This dependence to the confining pressure is typical of the behaviour ofgranular materials.
This variation of the behaviour with the confining pressure clearly confirms the benefits oftriaxial tests for understanding and modelling the permanent deformation behaviour. Triaxialtests with confining pressure clearly show different trends than uniaxial tests, like a steady stateof the axial strain at high confining pressure and a change of volumetric behaviour.
4.4. Influence of temperatureThree tests have been performed at a deviatoric stress of q = 400 kPa, without confining pressure,at three different temperatures, 10◦C, 20◦C and 30◦C. The influence of the temperature on thestrain levels reached is clearly shown in Figure 9(a). In Figure 9(b), the volumetric response doesnot present any significant trend or any ranking with temperature, despite the observed differences.
The same study has been performed at different confining pressures, respectively, σrad = 100and 200 kPa. Results are presented in Figures 10 and in 11. Figures 10(a) and 11(a) show thatthe axial strain tends towards a constant value, except at 10◦C. The higher the temperature is,the faster the axial strain reaches this plateau. The volumetric response (Figures 10(b) and 11(b))seems to depend more on the confining pressure than on the temperature. The influence of thetemperature on the volumetric response of the material is difficult to assess.
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Road Materials and Pavement Design 9
(a) (b)
Figure 9. (a) Axial strain as a function of time and (b) volumetric strain as a function of axial strain, fortests at q = 400 kPa, σrad = 0 kPa and different temperatures.
(a) (b)
Figure 10. (a) Axial strain as a function of time and (b) volumetric strain as a function of axial strain, fortests at q = 400 kPa, σrad = 100 kPa and different temperatures.
(a) (b)
Figure 11. (a) Axial strain as a function of time and (b) volumetric strain as a function of axial strain, fortests at q = 400 kPa, σrad = 200 kPa and several temperatures.
4.5. Validation of the TTSPAs mentioned before, the TTSP is commonly considered as valid in the linear domain for mostbituminous mixtures, due to bitumen. It is a powerful concept, because it makes it possible to limitthe experimental conditions (temperature or frequency range) and then to predict the behaviourfor other frequency or temperature conditions, using the master curve of the material. It is alsorecalled that the value of aT of the mix is close to that of its bituminous binder (Di Benedetto,
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Delaporte, & Sauzéat, 2006). An important objective of this study was to assess if the TTSP couldalso be applied in the nonlinear domain, where permanent strains are important, and under triaxialstress states, with confinement. More precisely, the purpose was to was see if the shift factor aT ,which is easily determined in the linear domain, from classical complex modulus tests, is stillapplicable when permanent strains are present.
The bituminous mix used in this study has been characterised by complex modulus tests, and theWLF law parameters (Equation (1)) have been determined from these tests. The equivalent time,defined as the time t divided by the shift factor aT of the WLF law, has then been used to normalisethe results of creep tests performed at different temperatures. Then, the normalised “strain versusequivalent time” curves obtained from these tests at different temperatures have been compared.This approach was used to compare results of tests performed for three different temperatures,10◦C, 20◦C and 30◦C (and in one loading condition also 40◦C), and for different conditions ofdeviatoric stress and confining pressure. It has to be noted that for this range of temperatures,the function aT varies by a factor of more than 104. The results are presented in Figure 12–15.Figure 12 shows results of tests performed at q = 400 kPa and σrad = 0 kPa. Figure 13 correspondsto tests performed at a higher deviatoric stress, q = 600 kPa and σrad = 0 kPa. Figures 14 and 15present results of tests performed with a deviatoric stress q = 400 kPa, and two levels of confiningpressure σrad = 100 kPa and 200 kPa.
For all these cases, for axial strains, the application of the TTSP leads to a difference betweenthe normalised axial strain versus equivalent time curves which does not exceed 15%. This is
(a) (b)
Figure 12. Axial (a) and radial (b) strains as a function of equivalent time t/aT , for tests at q = 400 kPa,σrad = 0 kPa and different temperatures.
(b)(a)
Figure 13. Axial (a) and radial (b) strains as a function of equivalent time t/aT , for tests at q = 600 kPa,σrad = 0 kPa and different temperatures.
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Road Materials and Pavement Design 11
(a) (b)
Figure 14. Axial (a) and radial (b) strains as a function of equivalent time t/aT , for tests at q = 400 kPa,σrad = 100 kPa and different temperatures.
(a) (b)
Figure 15. Axial (a) and radial (b) strains as a function of equivalent time t/aT , for tests at q = 400 kPa,σrad = 200 kPa and different temperatures.
of the same order of magnitude as the repeatability of the tests (see Section 4, introduction).In Figure 14, for instance, the ratio of time scales between the tests performed at 10◦C and40◦C is around 10,000. Despite this ratio, on an equivalent time scale, axial strains at bothtemperatures are very close to each other. For radial strains, a good agreement is also obtainedon Figures 13(b) and 14(b). On Figure 12(b), one test gives higher radial strains than the othertwo. On Figure 15(b), corresponding to tests with the confining pressure of 200 kPa, measurementproblems were encountered for the radial strains, due to friction in the radial strain measurementring. This led to irregular radial strain variations, and may explain the difference between theradial strain curves.
According to all these results, it can be concluded that for the test conditions of this study(in particular, temperatures between 10◦C and 40◦C), the TTCP could be used in the nonlineardomain, and for loading conditions with a confining pressure, with the purpose to assess behaviourat other temperatures, by means of the shift factor aT determined from the linear response of thematerial. In that sense, we consider that the TTSP is applicable in the nonlinear domain. Moreover,the fact that the shift factor aT determined from complex modulus tests also applies here indicatesthat the time-dependent response of the mix is strongly related with the viscosity of the binder,even in the nonlinear domain.
These results are supported by other research related to the extension of the TTSP. Di Benedetto,Nguyen, Pouget, and Sauzéat (2008) and Nguyen, Pouget, Di Benedetto, and Sauzéat (2009)have already shown that the TTSP could be extended to the nonlinear domain, without confining
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pressure. Di Benedetto, Sauzéat, and Sohm (2009) also showed, using ultrasonic wave propagationtests, that this principle is also valid in the very high-frequency domain.
5. Modelling of the creep testsDifferent approaches for the prediction of permanent deformations can be found in the literature:most empirical models are one-dimensional like the Esso model (Aussedat, 1977) or the modeldeveloped by the BRRC (Francken, 1977).
Neifar and Di Benedetto (2001) developed the DBN model (Di Benedetto and Neifar): a one-dimensional thermo-visco-elasto-plastic model. Comparisons with experimental results show thatthe DBN model predicts well the viscoelastic behaviour of bituminous mixtures (Di Benedetto,Olard, Sauzéat, & Delaporte, 2004), restrained thermal shrinkage tests with monotonic or cyclictemperature variations (Olard & Di Benedetto, 2005), and also cyclic sinusoidal compressiontests (Neifar, 1997). A three-dimensional version of the DBN model is proposed (Di Benedetto,Neifar, Sauzéat, & Olard, 2007).
Nguyen, Nehdar, and Tamagny (2006) have developed a three-dimensional cyclic elasto-viscoplastic model for asphalt concrete materials. The model is based on viscoplasticity andfocuses on the coupling of a Drucker–Prager-type criterion with a customised quadratic criterion.The consistency of the model has been demonstrated by numerical simulation of various loadingconditions and the authors validated the model by using experimental results of cyclic sinusoidalcompression tests. However, the bituminous mixture used for the tests had a high void percentageof 10%, and further studies indicated difficulties to predict the behaviour of bituminous mixtureswith lower void contents.
So, the results of the experimental programme have been used to develop a new model to predictof the monotonic behaviour of bituminous materials, based on the viscoplastic formulation ofPerzyna (1966). This model is based on the following assumptions:
– Classically, the total strain tensor is decomposed into a viscoelastic part and a viscoplasticpart:
–
εtot = εve + εvp (6)
– the viscoelastic part is described using the Huet–Sayegh model (Sayegh, 1965), which hasbeen used successfully for many years in France to model the viscoelastic behaviour ofbituminous mixes;
– the material follows the TTSP;– recoverable strains are very small in comparison with irrecoverable strains;– the volumetric behaviour depends on the confining pressure. It is dilatant at low confining
pressure and contractant at high confining pressure;– axial and radial strains stabilise if the confining pressure is high enough.
5.1. Viscoelastic componentHuet–Sayegh’s model (Sayegh, 1965) was chosen to represent the viscoelastic component. Thismodel, represented in Figure 16, is an analogical model constituted by a combination of a spring(stiffness Einf − E0) and two parabolic elements (J1(t) = ath and J2(t) = btk ). A spring of smallstiffness E0 compared with Einf is added in parallel. E0 is the static modulus when ωt → 0. Theeight parameters of the Huet–Sayegh model, obtained by means of the software viscoanalyse®,
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Figure 16. Representation of the Huet–Sayegh model.
Table 2. Parameters of the Huet–Sayegh model at the reference temperature of 20◦C.
δ k h E0 (MPa) Einf (MPa) τ0 C1 C2 (◦C)
2.26 0.24 0.71 27 32509 0.01 30.7 225.5
for the bituminous mix of this study, are presented in Table 2. However, simulations performedwith the Huet–Sayegh parameters showed that the viscoelastic strains calculated with the modelwere larger than the experimental viscoelastic strains obtained during the unloading phase ofthe creep test. To improve the modelling, it was necessary to increase the static modulus E0,and good predictions were obtained for a static modulus E0 = 400 MPa. It is interesting to notethat such a value of static modulus is of the same order of magnitude as the modulus obtainedin compression on unbound granular road materials (typically 100–500 MPa) (Balay, GomezCorreia, Hornych, Jouve, & Paute, 1998; El Abd, 2006). The static behaviour of the bituminousmaterial in compression at a loading rate tending to zero or at very high temperature is thereforevery similar to that of its granular skeleton. A possible improvement of the model could consist inreplacing the elastic part of the Huet–Sayegh model by a nonlinear unilateral elastic model suchas the Boyce model (Boyce, 1980; Hornych, Kazai, & Piau, 1998) used for unbound granularmaterials (for stress states in compression).
5.2. Viscoplastic componentThe viscoplastic strain rate is described by means of a Perzyna-type model (Perzyna, 1966) usingthe following viscoplastic flow:
ε̇vp = 1η(T )
ψ(f )∂f∂σ
. (7)
In terms of volumetric and deviatoric strain rates, this becomes
ε̇vpvol = 1
η(T )ψ(f )
∂f∂p
(8)
and
ε̇vpdev = 1
η(T )ψ(f )
∂f∂q
, (9)
where f is the yield surface and η(T ) is the viscosity parameter function of the temperature T .
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ψ(f) is defined by
ψ(f ) =⟨
ff0
⟩N
, (10)
where 〈f /f0〉 is the positive part of f /f0 and N a positive constant.The yield surface has a parabolic shape (Figure 17), and is defined by the following equation:
f (p, q) = p2 − p · pC + q · pC
2 · xm, (11)
where pc and xm are two hardening parameters, supposed to depend on εvpvol and ε
vpdev taken as
hardening variables. As a first attempt, pc and xm are supposed to be linear functions of εvpvol
and εvpdev:
pC = pC0(1 + cεvpvol + eεvp
dev) (12)
andxm = xm0(1 + dε
vpvol + bε
vpdev). (13)
Finally, in order to respect the TTSP, η(T ) is assumed to be equal to η/aT , with aT being theshift factor. Indeed in doing so, for a given strain obtained at a final time t2f and a temperatureT2, the same strain can be obtained at a temperature T0 at a final time t2f /aT , as shown below:
εvp(T2) =∫ t2f
0ε̇vp(T2, t) dt =
∫ t2f
0
ε̇vp(T0, t)aT2
dt =∫ t2f /aT2
0ε̇vp(T0, t′) dt′. (14)
pc0 , c, e, xm0 , d and b are constants, some of them not only such as pc0 , xm0 at least but also c andd are expected to be positive. In order to fit these parameters, eight creep tests were used. Foreach creep test, values of axial and radial strains at three different times (48 experimental valuesin total) were used to obtain the 8 parameters. A least-squares minimisation approach was usedto determine the values of the eight parameters, presented in Table 3.
It can be observed that c, e, d and b are all positives and significant. The significant evolutionof xm during the simulation is rather unusual for granular materials. It is probably related with thefact that the shear behaviour of bituminous materials is not purely frictional, but also depends oncontacts between aggregates through bitumen films, whose properties vary with the changes ofthickness linked to the strains ε
vpvol and ε
vpdev.
Figure 17. Parabolic yield surface of the Perzyna-type model.
Table 3. Parameters of the viscoplastic model.
pc0 (MPa) c e xm0 b d N η
0.38 0.36 0.59 0.42 1.32 0.73 2.92 0.54
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Road Materials and Pavement Design 15
5.3. Validation of the viscoelastic–viscoplastic modelThe results presented below illustrate the capacity of the model to simulate the experimental tests.The effects of deviatoric stress, confining pressure and temperature are considered.
5.3.1. Influence of deviatoric stressThree creep tests, at 20◦C, without confining pressure, have been simulated to study the influenceof different levels of deviatoric stress (200, 400 and 600 kPa). Results are presented in Figure 18.Figure 18(a) shows the influence of the deviatoric stress on the axial strain rate and the axial strainlevel. The model predicts correctly the increase in axial strains for deviatoric stresses q = 200and 400 kPa. For q = 600 kPa, the model predictions are lower than the experimental results.
For volumetric strains, a dilatant behaviour (i.e. a volume increase) is observed in all the tests,and this is well predicted by the model: the lower the deviatoric stress, the more dilatant thematerial, for a given level of axial strain.
5.3.2. Influence of confining pressureTests with four different confining pressures: 0, 50, 100 and 200 kPa, at 20◦C and at a deviatoricstress of q = 400 kPa have been simulated in Figure 19(a). When the confining pressure increases,the axial strains decrease, and for high confining pressures (100 or 200 kPa), they seem to tendtowards a constant value. The model predicts well these results, except for the test without con-fining pressure, where the difference with the experimental results increases. Figure 19(b) shows
(a) (b)
Figure 18. Experimental and predicted axial strains (a) and volumetric strains (b), for tests at σr = 0 kPa,T = 20◦C and different deviatoric stresses.
(a) (b)
Figure 19. Experimental and predicted axial strains (a) and volumetric strains (b), for tests at q = 400 kPa,T = 20◦C and different confining pressures.
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(b)(a)
Figure 20. Experimental and predicted axial strains (a) and volumetric strains (b), for tests at q = 400 kPa,σr = 100 kPa and different temperatures.
that the volumetric behaviour is contractant for high confining pressures (volume decrease), anddilatant for confining pressures lower than 100 kPa. This volumetric behaviour is well predicted.
5.3.3. Influence of temperatureFour tests at different temperatures, with q = 400 kPa and σrad = 100 kPa, have been modelled.Results are presented in Figure 20. As expected, when the temperature increases, the axial strainsincrease rapidly. For the applied stress level, a stabilisation of the axial strains is also observedfor the temperatures of 30◦C and 40◦C. The volumetric versus axial strain curves are similar forall temperatures. The model is able to reproduce these different trends.
6. ConclusionTo study the permanent deformation behaviour of bituminous mixes under monotonic and cyclicloading, a thermo-controlled triaxial test has been developed. Homogeneous tests on cylindricalspecimen have been performed in order to assess and understand the behaviour of a classicalbituminous mixture under creep loading. The triaxial creep tests have been carried out at differenttemperatures (from 10◦C to 40◦C), at different levels of deviatoric stresses (from 200 to 2000 kPa)and at different confining pressures (from 0 to 200 kPa).
Several findings concerning the three-dimensional response of asphalt concretes have beenobtained. As a typical result, it has been shown that a high deviatoric stress increases the level ofaxial strains obtained, and also the axial strain rate. The level of confinement also modifies theresponse of asphalt concretes. Indeed, when the confinement increases, the strain rate and the levelsof strain reached during the test decrease. When confinement is high enough, strains tend to sta-bilise, and reach a constant value. As it is the case for granular materials, the confinement also hasa strong influence on the volumetric behaviour of bituminous mixtures. For a given level of axialstrain, volumetric strain is dilatant at low confining pressures. As the confining pressure increases,the volumetric strains decrease, and become eventually contractant for high confining pressures.
This testing programme has also highlighted the great influence of temperature on the creepresponse of asphalt concretes. As it is the case in the viscoelastic domain, an increase in temperatureleads to higher strain rates. When a plateau is reached during the test, the temperature does notmodify this plateau; it only changes the strain rate to reach this plateau. The temperature does notseem to significantly affect the volumetric behaviour of our bituminous mixture. Our results showthat the TTSP, usually considered as valid in the linear domain for most asphalt concretes, couldbe extended to the nonlinear domain, even under confinement. Indeed, creep tests performedwith the same loading conditions of pressure and deviator, but at different temperatures, give
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Road Materials and Pavement Design 17
similar responses in “equivalent time scale”. The associated shift factor aT is the same as the onedetermined in the linear domain by means of classical complex modulus tests. This importantresult confirms previous results presented in the literature.
A viscoelastic–viscoplastic model has been developed in order to simulate the creep response ofour material, and especially to simulate the influence of the deviatoric stress and of the confiningpressure on the volumetric and the temporal response of the material. Another key point is that themodel follows the TTSP. The viscoelastic part is based on the Huet–Sayegh model. It can easily befitted by means of a complex modulus test and using the software viscoanalyse®. The viscoplasticpart is based on a Perzyna formulation. The viscoplastic parameters have been fitted by meansof several creep tests under different loading conditions. This model presents a good capacity tosimulate the triaxial response of the material in the studied domain. It shows both good temporaland volumetric responses, and in particular reproduces well the dilatant/contractant aspects,despite some limitations for the higher deviators without confinement.
From an experimental point of view, the next objective is to study the influence of cyclingloadings on the response of the material. The parameter frequency must be studied, consideringall the other parameters (levels of deviator, confining pressure and temperature). It will also beimportant to assess the TTSP for such tests. The model must also be optimised in terms of yieldsurface and hardening laws, to simulate as well as possible all the creep tests, with a minimumof tests to fit the parameters. This model must also be assessed in terms of capacity to simulatecyclic loadings. Applying it to structural calculations is presently considered, in order to modelpavement structures submitted to static loading and possibly to derive a “fast” testing procedure(type of test to be performed, and relevant stress and temperature levels) for selecting bituminousmaterials for such an application.
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