Viscoelastic properties and constitutive modelling of bitumen

Fuel xxx (2013) xxx–xxx

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Viscoelastic properties and constitutive modelling of bitumen Ehsan Behzadfar, Savvas G. Hatzikiriakos ⇑ Department of Chemical and Biological Engineering, The University of British Columbia, 2360 East Mall, Vancouver, British Columbia, V6T 1Z3, Canada

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

" The linear viscoelastic behaviour of

bitumen is studied in detail – relaxation modulus. " The nonlinear rheological behaviour of bitumen was studied using the minimum number of rheological testing. " A constitutive model was identified that captures the rheology of bitumen in a number of deformation histories.

a r t i c l e

i n f o

Article history: Received 15 September 2012 Received in revised form 9 December 2012 Accepted 10 December 2012 Available online 25 December 2012 Keywords: Bitumen Viscosity K-BKZ constitutive equation Viscoelastic properties

a b s t r a c t In this paper, the viscoelastic behaviour of bitumen is studied and an appropriate and suitable constitutive equation is identified to describe its rheological behaviour. First, the generalised Maxwell model has been utilized to represent the relaxation modulus of bitumen and found to be in excellent agreement with experimental data over a wide range of temperatures (30 °C to 90 °C). Furthermore, the time–temperature superposition principle was found to be applicable over this temperature range. The K-BKZ constitutive equation has been shown to represent accurately the rheological properties of bitumen. Analysis of experimental results revealed that either the Papanastasiou or the Marucci form of the damping function can be used in the K-BKZ constitutive equation. Moreover, the damping function was found to be independent of temperature (0–50 °C). Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Bitumen is a viscoelastic multiphase material with colloidal microstructures consisted of elastic solid aggregates and a viscoelastic matrix. Based on the solubility parameters, bitumen components can be classified into four classes of saturates, aromatics, resins and asphaltenes. Whilst the dispersed solid phase in bitumen is mostly composed of high molecular weight asphaltenes, the matrix is a mixture of the maltenes (saturates, aromatics and resins) [1]. Depending on the thermo-mechanical conditions, these components are able to form distinctive structures inside bitumen,

⇑ Corresponding author. Tel.: +1 604 822 3107.

which are the origin of complications in rheological properties of bitumen, varying from entirely viscous behaviour to purely elastic behaviour [2,3]. Bitumen has found a plethora of applications due to its unique properties related to hydrophobicity, adhesion, and thermo-processability. The most obvious application is in road paving industry where it is used as a binder of asphalt mixtures. The broad window of applications makes it necessary to have sufficiently accurate rheological models to describe the flow properties of bitumen at different processing conditions, which is the main target of the present work. Although the linear viscoelasticity of bitumens and heavy oils has been of interest to many researchers [4–9], there have been few efforts in studying the nonlinear viscoelasticity of these materials, mainly reporting their shear thinning behaviour

E-mail address: [email protected] (S.G. Hatzikiriakos). 0016-2361/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.fuel.2012.12.035

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(viscous models) for bitumen [10–14]. However, based on the viscoelastic behaviour of these materials, to the best of our knowledge no studies exist on their viscoelastic constitutive modelling. The first attempt to model the linear viscoelasticity of bitumen was performed by Van der Poel [4] followed by that of Heukolem and Klomp [5]. They utilized a nonlinear multivariable models (nomographs) to predict the stiffness modulus of bitumen having temperature, softening point, loading time and penetration index as inputs to their models. Improvements to this model were performed by McLeod in a series of papers [15]. Jongepier and Kuilman [16] developed an empirical algebraic equation to predict the rheological properties assuming that the relaxation spectrum of bitumen is log normal. Several other authors proposed empirical models to evaluate the linear viscoelastic properties of bitumen [6,7,17–23]. These models are basically algebraic equations, and their parameters have in general no physical meaning. This causes difficulties in gaining an understanding of the rheological response of these materials. On the other hand, mechanical model offer a basic understanding of rheological behaviour and they are more attractive to use. Some of these mechanical models which have been used to model the rheological behaviour of bitumen are depicted in Fig. 1. These models include combinations of an ideal elastic spring (elastic modulus), a viscous dashpot (viscosity) and a parabolic element (creep compliance response) [24]. A generalised Burger model (Fig. 1a) is constructed by placing a Maxwell model in series with a number of Kelvin–Voigt models [25]. Huet [26] proposed a model in which a spring is in series with two parabolic elements (Fig. 1b). Sayegh [27] modified the Huet model (the Huet–Sayegh model) by putting a spring in parallel to the Huet model (Fig. 1c). Olard and Di Benedetto [28] constructed the DBN model by eliminating the Maxwell model’s dashpot from the generalised Burger model (Fig. 1d). In other works, researchers have added a dashpot to the Huet–Sayegh model in series with the parabolic elements and in parallel with the separate spring to build the 2S2P1D model (Fig. 1e) [29,30]. These models may successfully model one type of experimental response (stress relaxation or creep), however, it is impossible to represent the complete rheological behaviours of the material by means of a simple mechanical model. In the present study, the viscoelastic behaviour of bitumen is examined over a wide range of temperatures in an attempt to

understand its complete rheological behaviour from elastic, to viscoelastic and to viscoelastoplastic in some cases. The generalised Maxwell model is used to model the linear viscoelastic properties of bitumen, which serves as a basis for our study of the nonlinear viscoelasticity. An appropriate constitutive equation is proposed to account for the rheological responses of bitumen at both linear and nonlinear viscoelastic regions over a wide range of temperatures. The proposed constitutive equation is examined and tested for a number of different deformation histories in order to determine its ability in predicting comprehensively the viscoelastic flow properties of bitumen.

2. Materials and methods 2.1. Material characterisation The bitumen used in this study was obtained from Athabasca oil sands, Alberta, Canada with specific gravity of 0.969 at 22 °C. The asphaltene and maltenes (saturates, aromatics, resins) content of the bitumen were determined based on their solubility in npentane. The bitumen was mixed with n-pentane on the weight ratio of 1:40 and stirred overnight. The mixture was filtered twice by using paper filters of different pore sizes, namely 1–5 lm and 0.2 lm, respectively. Filtrations were accompanied with continuous vacuuming and extra solvent was used to make the filter paper colourless. The retentates were dried in an oven at 60 °C for 30 min and left in a vacuum oven at room temperature for 48 h. The permeates were placed in the rotary evaporator at 60 °C to remove out the n-pentane. The evaporator was kept running until no more npentane was collected. Both retentates and permeates were weighted to obtain the weight percentage of the asphaltenes and maltenes. Elemental analysis of the sample was performed using the 2400 Perkin–Elmer CHNS/O Analyzer by combustion at 1000 °C. The oxygen content was calculated from subtraction based on the weight contents of the other elements. Metal analysis data were provided from the supplier based on ASTM-D5600. The bitumen was stored at ambient temperature before testing and neither phase dissociation nor evaporation occurred. The compositional and elemental analysis of the bitumen sample are summarised in Table 1.

Fig. 1. Different viscoelastic mechanical models used for bitumen. (a) The generalised Burger model. (b) The Huet model. (c) The Huet–Sayegh model. (d) The DBN (Di Benedetto and Neifar) model. (e) The 2S2P1D model.

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describes the linear viscoelastic behaviour of the material which can be expressed as follows:

Table 1 Compositional and elemental analysis of the bitumen. Fraction

Mass fraction

C5 asphaltenes Maltenes

18.32 81.68

Element

Mass fraction

C H N S Oa Atomic H/C

83.42 10.1 0.52 4.77 0.79 1.45

Metal

ppm

Fe V Mn Ni

1.99 126 0.15 34.4

mðt  t0 Þ ¼

2.2. Rheological measurements

GðtÞ ¼ Rheological measurements were performed using the Anton Paar MCR501 (Anton Paar, Austria), a stress/strain controlled rheometer equipped with the parallel plate and cone (1°) and plate geometries with diameter of 25 mm. Amplitude sweep tests were performed to determine the linear viscoelasticity region of the sample at different temperatures, preceding the frequency sweep experiments. The temperature range of the experimental testing was varied from 30 °C to 90 °C (±0.1 °C). Above 90 °C, bitumen is practically a low viscous Newtonian fluid. The elastic and storage moduli along with complex viscosity data were determined from dynamic rheological measurements (frequency sweep experiments of small amplitude oscillatory shear) at different temperatures applying frequencies from 0.005 to 500 rad/s. Stress relaxation experiments were performed to determine the damping function and were run with different strain values at the temperature range from 0 °C to 50 °C. Start-up of steady shear experiments were performed at 10 °C with different shear rates whilst enough time was allowed for the material to reach steady state. This was followed by cessation of steady shear stress experiments to determine the relaxation behaviour of the bitumen. Measurements were repeated to check for evaporation, thermal degradation or phase separation effects during the experiments. The experimental data were used to obtain the parameters of the K-BKZ model and, furthermore, to test this model in predicting the rheological response of bitumen at different deformation histories.

hðIC1 ; IIC1 Þ ¼

t

1

0 mðt  t0 ÞhðIC1 ; IIC1 ÞC 1 ij ðt  t Þdt

0

ð1Þ

where m(t), hðIC1 ; IIC1 Þ and C 1 ij ðtÞ are the memory function, the damping function and Finger strain tensor, respectively. The IC1 and IIC1 are the 1st and 2nd invariants of the Finger strain tensor which are essentially strain-dependent [34]. The memory function

ð3Þ

a ða  3Þ þ bIC1 þ ð1  bÞIIC1

ð4Þ

where a and b are adjustable parameter to be determined from the regression process on the experimental data and are dependent on the chemical structure, molecular weight and molecular weight distribution of the material. For simple shear flows, Eq. (4) reduces to:

hðcÞ ¼ The K-BKZ model is one of the most popular constitutive equations used in the rheology of polymer melts and it was proposed by Kaye [31] and Bernstein et al. [32]. A decade later, Wagner [33] proposed a simplified version of the K-BKZ model, discarding the second normal stress difference and this made the model more successful in accurately representing the rheological response of polymer melts. This equation can be written as follows:

    N N1 X X t t  G0 þ Gi exp Gi exp ki ki i¼1 i¼1

where Gi and ki (=gi/Gi) are the relaxation moduli or strengths and relaxation times of the ith element of the total number N of the modes. G0 is the relaxation modulus or strength of the dashpot with the practically infinite relaxation time depicted in Fig. 2c. As seen from Eq. (1), the separability assumption of the memory function is utilized for the modelling (memory function is the product of a time-dependent and a strain-dependent function) which has been found to hold for most viscoelastic materials like molten polymers [36,37]. The damping function is a measure of non-linear behaviour of the viscoelastic material as it accounts for the departure of the rheological response from linear viscoelasticity. Rolon-Garrido and Wagner [38] reviewed the various damping functions which are of interest in rheology. The Wagner [39] exponential equation and the Papanastasiou et al. [40] equations are the most popular equations used as damping functions for the case of viscoelastic materials such as molten polymers. Papanastasiou et al. [40] proposed the following universal equation as damping function:

3. Theory

Z

ð2Þ

where G(t) is the time-dependent linear relaxation modulus of the material. As described above, due to their unique structure (including solid and liquid phases as can be seen in Fig. 2) bitumen are viscoelastic materials. The linear relaxation modulus can be represented by the generalised Maxwell model [35], already used for most viscoelastic materials including molten polymers (Fig. 2b). As it will be shown in the experimental section, the use of the generalised Maxwell model in modelling of bitumen rheology, results into an infinity viscosity for one of the dashpots, and the model practically reduces to the Zener model (Fig. 2c). In other words, the generalised Maxwell model or the Zener model can interchangeably be used to model the linear viscoelastic behaviour of bitumen. Simply one of the relaxation times of the spectrum is much greater than the rest. Then, the relaxation modulus can be expressed in the following form:

a Calculated from the difference of mass fractions of other elements from 1.

rij ¼

dGðt  t 0 Þ dt

a 1 ¼ a þ c2 1 þ a1 c2

ð5Þ

We also found that the Marucci damping function represents the stress relaxation data of bitumen well [41]. For simple shear this can be written as (no adjustable parameters):

hðcÞ ¼

6 4 þ c2 þ ð4 þ c2 Þ1=2

ð6Þ

In the current study, we will examine both models to verify their suitability in describing the rheological response of bitumen, particularly using shear stress relaxation and startup of steady shear experiments [41]. Simple shear experiments will also be used to test the proposed constitutive equation. For such a test (constant shear rate), Eq. (1) reduces to the following form:

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Fig. 2. (a) The schematic of bitumen structure including solid and liquid phases. The shell around the asphaltenic cores shows the interphase layer. The red arrows depict the short range connections which form the weak network, (b) the generalised Maxwell model and (c) the Zener model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Z

þ1

s  mðsÞðhðcÞÞc¼c_ s ds

0

ð7Þ

The stress growth and decay coefficients in the simple shear flows can be computed through the shear relaxation modulus values whose comparison with the experimental data would be a quantitative tool to test the model’s capabilities in predicting different material functions. First the stress growth coefficient Equation reduces to [42]:

gþ ðt; c_ Þ 

rþ ðt; c_ Þ ¼ c_

  Z tX N s d Gi exp  ðchðcÞÞc¼c_ s ds ki dc 0 i¼1

ð8Þ

Similarly, the shear stress decay coefficient for the stress relaxation after cessation of steady shear can be calculated by [42]:

g ðt; c_ Þ  ¼

r ðt; c_ Þ c_

( Z N X Gi i¼1

k2i

0

1

  )   s t _ sds ki exp  hðcsÞ exp  ki ki

ð9Þ

where rþ ðt; c_ Þ and r ðt; c_ Þ are stress at the steady shear and stress relaxation experiments, respectively. 4. Results and discussion It is generally accepted that bitumen is a colloidal suspension composed of maltenes and asphaltenes where maltenes encompass saturates, aromatics and resins. The major difference of bitumen with commonly known colloids and suspensions is the strong temperature-dependence of both solid and liquid phases [43]. As asphaltenes are large molecules and solid at most temperatures, they account for the particulate part of the colloid whilst maltenes are assumed to form the liquid matrix. Asphaltenes start to melt at 67 °C whilst melting temperature of maltenes spans from 50 °C to 30 °C [35,44]. Accordingly, maltenes are likely transferred from the liquid to solid phase as the temperature decreases, whilst asphaltenes remain in solid phase for the most practical temperatures. This complication makes the study of the bitumen flow properties more complex since different parameters must be considered simultaneously to analyse the rheology of bitumen.

of a material in dynamic experiments can be determined by an amplitude sweep test in desired frequencies, preferably high frequencies to ensure the linearity at lower frequencies as well. The linear viscoelastic behaviour of bitumen was studied by means of small amplitude oscillatory shear experiments at small strain values over a wide range of temperatures from 30 °C to 90 °C. These results are presented in this section. The storage, G0 , and loss, G00 , moduli of bitumen are depicted in Figs. 3 and 4, respectively, at selected temperatures. As shown, the loss modulus, G0 (viscous behaviour) is more dominant at higher temperatures, whilst the measured storage modulus, G0 , at lower temperatures exhibits greater values. In fact at 30 °C, G0 is greater than G00 indicating the elastic behaviour of the material. It can also be observed that by increasing the temperature, G0 drops more rapidly than G00 , demonstrating the dominance of the viscous behaviour at higher temperatures. Whist the bitumen acts as a truly elastic material at 30 °C, it behaves like an entire viscous liquid at 90 °C and above. Increasing the temperature, more maltenes migrate from the solid phase to the interlayer region and advance to the liquid phase (Fig. 2a). This migration is also accompanied with the mobility increment of the bitumen molecules improving its tendency to mechanical energy dissipation rather than energy storage. A plateau can be noticed in the storage modulus values, which is more evident at higher temperatures where the stress values are quite small. The plateau reveals of the presence of a weak network in the bitumen. Whilst this delicate network is easy to deform and break at very small strain, it is also recoverable at very

108 G' G' G' G' G' G'

107 106 105

G'(ω) (Pa)

r12 ¼ c_

104

-30°C 0°C 10°C 30°C 60°C 90°C

103 102 101 100 10-1

4.1. Linear viscoelasticity Bitumen, like other viscoelastic materials, shows linearity between applied stress and induced strain at sufficiently low deformations (linear viscoelastic region). The linear viscoelastic region

10-2 10-2

10-1

100

101

102

103

104

105

Angular Frequency, ω (rad/s) Fig. 3. Storage modulus, G0 , of the bitumen versus frequency, x, at selected temperatures.

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105 G'' -30°C G'' 0°C G'' 10°C G'' 30°C G'' 60°C G'' 90°C

106

G''(ω) (Pa)

105 104

104

Horizontal shift factor, aT

107

103 102 101 100 10-1

103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6

10-2 10-2

10-1

100

101

102

103

104

10-7 -40

105

Experimental data Arrhenius model (Ea=134.5kJ/mol) WLF model (Tr= 10°C,C1=17.57,C2=184.82) -20

Angular Frequency, ω (rad/s)

short times after deformation. The influence of this formed structure will also be seen in the relaxation spectrum. Although some researchers believe that bitumen shows no yield stress [12], our finding is in agreement with authors who confirmed the existence of the three-dimensional network in bitumen [13,45]. Using the principle of time–temperature superposition the master curve of dynamic moduli and complex viscosity were constructed at the reference temperature of 10 °C. These are depicted in Fig. 5. It appears that the time–temperature superposition (TTS) principle is applicable for the bitumen over the wide range of temperatures from 30 °C to 90 °C which covers the temperature range for most practical applications. Although there is no consensus in the literature that the time–temperature superposition (TTS) principle holds true for bitumens [12,46,47], our finding confirmed the applicability TTS. It is noted that this superposition was performed by using only horizontal shifting and application of vertical shift did not improve significantly the quality of the superposition. The values of the shift factors, aT calculated to produce the master curves depicted in Fig. 5, are plotted in Fig. 6. The shift factor values follow both the Arrhenius, aT ¼ exp½Ea =Rð1=T  1=T r Þ where Ea is the activation energy and Tr is the reference temperature and the WLF [47], logðaT Þ ¼ C 1 ðT  T r Þ=ðC 2 þ ðT  T r ÞÞ equations. Using the Arrhenius equation, we calculated activation energy for flow of 134.5 kJ/mol considering 10 °C as the reference

108

107

107

106

10

2

104 103 102

101

101

|η*(ω)|/aT (Pa.s)

G'(ω) & G"(ω) (Pa)

103

100 100

10-1

10-1

10-2

60

80

100

temperature. Our finding of the activation energy was consistent with reported values in the literature [13]. As seen from Fig. 5, the complex viscosity exhibits a significant shear-thinning behaviour whilst the upturn of the viscosity curve at very low frequencies confirms the presence of a small network in the bitumen. This upturn in the complex viscosity is accompanied by the plateau in the storage modulus at diminishingly small frequencies, also manifesting the existence of some structured networks with very long relaxation times. It should be mentioned that the viscosity values at small shear rates are similar to those reported by Bazyleva et al. [48]. Some differences are attributed to different chemistry and ageing of the samples [49]. In order to model the linear rheological behaviour of the bitumen by means of discrete relaxation time spectra, the parsimonious (PM) model [50,51] was used with the dynamic rheological data in which the relaxation time spectrum is shown in terms of Maxwell modes (the generalised Maxwell model). Equivalently, the Zener model can be used since one of the relaxation times in the generalised Maxwell model posses a very long relaxation time which can practically be considered as infinite leading to the Zener model. The dynamic moduli can be written as [50,51]:

G0 ¼

N X Gi ðxki Þ2 i¼1

10-3 10-2 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107 108

Reduced angular frequency, ω aT (rad/s) Fig. 5. Master curves of dynamic moduli (storage and loss) and complex viscosity of the bitumen against reduced frequency at the reference temperature of 10 °C. The lines show fits of the generalised Maxwell model.

1 þ ðxki Þ2

N X Gi ðxki Þ i¼1

105

105 G' G" |η*|

40

Fig. 6. The shift factor values, aT, at different temperatures obtained from the master curves at 10 °C. The full and dashed lines show the Arrhenius and WLF models, respectively.

G00 ¼

104

20

Temperature, T (°C)

Fig. 4. Loss modulus, G0 , of the bitumen versus frequency, x, at selected temperatures.

106

0

1 þ ðxki Þ2

ð10Þ

ð11Þ

where G0 , G00 , x, Gi and ki are the storage modulus, the loss modulus, the frequency, the ith relaxation strength and the ith relaxation time, respectively. Following the analysis proposed by Winter [52], the optimum number of modes (mode density) was found to be 15. The agreement between the calculated parsimonious (PM) model with 15 modes (continuous lines) and the experimental data is excellent as can be seen in Fig. 5. The computed relaxation moduli, Gi, and times, ki , of the model are depicted in Fig. 7 and the values are listed in Table 2. The number of modes is in agreement with the usual analyses in rheology where approximately one mode corresponds for every decade of frequency. A few past studies on the rheology of bitumen have reported on the analysis of the relaxation spectrum in terms of complex viscosity or shear modulus in order to describe the linear viscoelastic properties of bitumen [6–8,16]. However, the relaxation spectrum concept in terms of the generalised Maxwell (Zener) model is a well-developed concept in polymer rheology, which is also used

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Table 3 Parameters of the BSW spectrum for the bitumen.

Linear G(t) BSW spectrum

ng

0 ngGN 0 n eG N

λ0

-8 -7 -6 -5 -4 -3 -2 -1 0

1

2

3

λmax

4

5

6

7

8

9 10

λ i (s) Fig. 7. The relaxation spectrum of the bitumen at 10 °C obtained from the generalised Maxwell model along with the BSW spectrum. Decomposition of the BSW spectrum is also presented.

Table 2 Parameters of the generalised Maxwell model. Relaxation times (s)

Relaxation strength (Pa)

1.00  107 1.50  106 1.83  105 1.84  104 1.46  103 8.34  103 0.05 0.55 3.93 27.14 1144.49 11294.91 120403.99 1097863.45 220300543.94

14843332.28 12362273.82 10973131.27 4876916.70 2312245.10 776419.25 194903.82 10978.29 906.21 58.41 0.36 0.10 8.32  103 0.04 0.05

HðkÞ ¼

Value

G0N (Pa) ng ne k0 (s) kmax (s)

0.30 1.13 0.20 3272.33 2.20  108

k ne G0N ðkmax Þne þ ng G0N ðkk0 Þng

for kl < k < kmax

0

for k > kmax

calculated to be 1.13, which is slightly above the values commonly reported for polymers showing the broad relaxation times with diverse relaxation strength in bitumen microstructure. However, the value of ne, unlike other parameters, is in the range similar to that reported for polymeric substances [54,55] whilst k0 was also far above the values reported for polymers (107 s) [56]. Despite of dissimilarities in the parameters obtained for the bitumen with typical polymers, arising from the structural differences, the BSW spectrum is a useful tool to describe the linear viscoelastic behaviour of bitumen satisfactorily and possibly differences between various bitumen specimens can be reflected in terms of these spectrum. Fig. 8 compares the experimentally determined relaxation modulus with that predicted by the multimode Maxwell model using the relaxation spectrum depicted in Fig. 7. As it can be deduced, the agreement between the experimental data and calculated function based on the dynamic rheometry is excellent. The existence of a delicate structure slows down the relaxation considerably at longer relaxation times (about 100 s) with a tendency of obtaining an equilibrium modulus at even longer times. This is also shown by the maximum relaxation time of the last mode of the spectrum listed in Table 2. 4.2. Nonlinear viscoelasticity

here to analyse it. Baumgaertel et al. [53] used the following continuous representation of the relaxation spectrum with an abrupt cut-off to define the maximum relaxation time (hereafter called BSW spectrum):

(

Parameter

ne

ð12Þ

where ne and ng are the slopes of the spectrum in the long-time and short-time regions, respectively, kl , k0 and kmax are the shortest observation time, the Rouse time (the relaxation time for the onset of the glass transition) and the longest relaxation time, respectively, G0N is the plateau modulus. The first term represents the entanglement and flow region, whereas the second term describes the high frequency region. The decomposition of the BSW spectrum is provided in Fig. 7 for the case of bitumen which seems to fit its relaxation spectrum. The continuous BSW spectrum is plotted in Fig. 7 along with the discrete data obtained from the generalised Maxwell model. The agreement between the discrete and continuous data confirms the interchangeability in the representation of the relaxation spectra, as proposed by Baumgaertel et al. [51]. This allows one to benefit from the accurate manifestation of the linear viscoelastic behaviour by using the minimum possible number of modes. The values for the parameters obtained from the BSW spectrum are presented in Table 3. The value of G0N is much lower than that has been reported for typical polymers (>1 MPa) which is because of the smaller size of bitumen molecules compared to those of polymers. ng was

Once the magnitude of the strain or strain rate exceeds a critical value, the material structure is affected and the applied stress does not follow a linear relationship with the imposed strain (nonlinear viscoelasticity). In the context of the K-BKZ model, the damping function should be determined in order to capture the nonlinear viscoelasticity of bitumen. This can be done by means of stress relaxation experiments after the imposition of sudden strains. Fig. 9a shows the relaxation modulus of the bitumen at 10 °C and at different applied strains, c, ranging from 0.2 to 10. As shown, the behaviour remains in the linear region for all strain magnitudes

104

Relaxation mudulus, G(γ,t) (Pa)

Gi (Pa)

6

Linear G(t) Predicted G(t)

103

102

101

100

10-1

10-2 10-1

100

101

102

103

104

Time, t (s) Fig. 8. The linear shear relaxation modulus of the bitumen, G(t), at 10 °C (c = 1). The dashed line shows the prediction of the generalised Maxwell model.

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(b)

(a)

Relaxation mudulus, G(t,γ)/h(γ) (Pa)

Relaxation mudulus, G(t,γ) (Pa)

10000

1000

100 γ =0.10 γ =0.50 γ =1.00 γ =2.00 γ =4.00 γ =6.00 γ =8.00 γ =10.0

10

1

0.1

0.01 0.1

1

10

100

105

104

103 γ =0.10, h=1.00 γ =0.50, h=1.00 γ =1.00, h=1.00 γ =2.00, h=0.62 γ =4.00, h=0.29 γ =6.00, h=0.11 γ =8.00, h=0.08 γ =10.0, h=0.06

102

101

100

10-1 0.1

1

Time, t (s)

10

100

Tim e, t (s)

Fig. 9. (a) The shear relaxation modulus of the bitumen, G(t, c), after the imposition of sudden strain values, c, at 10 °C. (b) Superposition of the stress relaxation modulus data of (a) to determine the damping function.

0.1

0.01 0.01

The start-up of steady shear and cessation of steady shear flow experiments were conducted and the results were compared to the predictions of the K-BKZ model in order to test the capabilities of this model in capturing the rheological behaviour of bitumen. 4.3.1. Start-up of steady shear experiments In the start-up experiment, a desired shear rate is applied at t = 0 to the sample already at equilibrium for a certain period of time, whilst the shear stress rise is monitored. Fig. 11 plots the shear stress growth coefficient, gþ ðt; c_ Þ, at 10 °C along with the K-BKZ model prediction (Eq. (8)) using the Papanastasiou and Marucci models (Eqs. (5) and (6)). At every shear rate the material behaviour initially follows linear behaviour. At some point, it deviates from linear behaviour, exhibiting an overshoot (typical viscoelastic behaviour) before settling to its steady-state value. The linear viscoelastic region can be found at progressively shorter times depending on the level of the shear rate. The agreement is overall good, although the model underpredicts the strong experimentally observed overshoot.

4x104

. γ (s-1) 0.01 0.05 0.10 0.50 1.00

.

0°C 10°C 30°C 40°C 50°C Papanastasiou model (1983) α = 5.40 FBN model (2000)

0.1

4.3. Model assessment

3x104

+

Damping factor, h (γ)

1

predictions of the material rheological response in various deformation histories.

Stress growth coefficient, η (t,γ) (Pa.s)

less than about c = 2 whilst at c > 2 the relaxation modulus starts showing strain dependence and deviates from the linear viscoelastic modulus also shown by the continuous line. In order to determine the damping function, the various curves were shifted upward to superpose to the relaxation modulus in the linear viscoelastic region. The resulted superposition is depicted in Fig. 9b, and the shift factors are listed in the legend of Fig. 9b. It can be inferred from Fig. 9b that the superposition is perfect, confirming that the time-deformation separability assumption used to write Eq. (1). The damping function was determined at various temperatures, namely 0 °C, 10 °C, 30 °C, 40 °C and 50 °C to examine possible dependence on temperature. Similar to the data at 10 °C, in all cases excellent superposition was obtained, confirming again the applicability of the separability assumption. The calculated values of the damping function at different temperatures are plotted in Fig. 10. As it can be deduced, the data follow a unique pattern independent of temperature. The Papanastasiou and Marruci functions (Eqs. (5) and (6)) have also been plotted along with the data points showing an excellent agreement. As shown, both forms of the damping function (Eqs. (5) and (6)) can be utilized to properly describe the damping function of the bitumen. However, it must be pointed out that the Marruci model does not contain any adjustable parameter [41]. Once the damping function is known, the KBKZ model (Eq. (1)) is complete and ready to use in order to make

1

10

Time, t (s)

2x104

Papanastasiou model (1983) FBN model (2000) 104 0.01

0.10

1.00

10.00

100.00

Time, t (s) Fig. 10. The damping function of the bitumen, h(c), at temperatures from 0 °C to 50 °C. The dashed and full lines show the fit of the Papanastasiou model using a = 5.4 and the prediction of the Marucci FBN (Force Balance Network) model, respectively.

Fig. 11. The stress growth coefficient of the bitumen, gþ ðt; c_ Þ, at different levels of shear rate, c_ , at 10 °C. The continuous lines represent the predictions of the K-BKZ model using the Papanastasiou and Marucci damping functions.

Please cite this article in press as: Behzadfar E, Hatzikiriakos SG. Viscoelastic properties and constitutive modelling of bitumen. Fuel (2013), http:// dx.doi.org/10.1016/j.fuel.2012.12.035

8

E. Behzadfar, S.G. Hatzikiriakos / Fuel xxx (2013) xxx–xxx 106

|G*(ω)| (Pa) or σ (Pa)

Steady shear LVE 105

104

103

102

101 10-3

10-2

10-1

100

.

101

-1

ω (rad/s) or γ (s )

Stress decay coefficient, η−(t,γ) (Pa.s)

Fig. 12. Testing the applicability of Cox–Merz rule by comparing the flow curve determined from steady shear experiments with dynamic complex modulus values, jG j, versus angular frequency, x, (dashed line) at 10 °C.

.

.

several types of experiments and theoretically in terms of modelling its rheological response by means of a continuum constitutive equation, namely the K-BKZ. The generalised Maxwell model was found to be an excellent equation that fits the linear response of the bitumen in dynamic flow fields such as small amplitude oscillatory shear. The parameters obtained from the generalised Maxwell model (relaxation spectrum) were fed into the K-BKZ constitutive equation to predict the linear viscoelastic behaviour of the bitumen which was shown to be excellent. The time–temperature superposition was found to apply in the case of bitumen over a wide range of temperatures, namely from 30 °C to 90 °C, essentially covering the range for most practical applications. The damping function data, h(c), was also determined over a wide range of temperatures. It was found to be independent of temperature. The Papanastasiou and the Marruci models were utilized to represent the the effect of shear strain on the damping function which was found to be adequate in most cases. The implementation of these two damping function models into the K-BKZ constitutive equation predicted the flow behaviour of the bitumen both in the linear and nonlinear viscoelastic regions adequately in most cases.

Papanastasiou model (1983) FBN model (2000)

10000

References . γ

1000

0.01s-1 0.05s-1 0.10s-1

100 0.50s-1 1.00s-1

10 0

5

10

15

20

25

30

Time, t (s) Fig. 13. The shear stress decay coefficient, g ðt; c_ Þ, of the bitumen at different shear rate values, c_ , at 10 °C. The continuous lines represent the predictions of the K-BKZ model using the Papanastasiou and Marucci damping functions.

Having the steady shear values of the viscosity at different shear rates, the applicability of the Cox–Merz rule can be tested. Fig. 12 depicts the flow curve of bitumen determined from steady shear experiments (Fig. 11) in comparison with the linear viscoelastic data (LVE) plotted as a flow curve (jG j versus x) at 10 °C, respectively. As it can be seen the agreement is excellent implying (i) the applicability of the Cox–Merzat least at 10 °C and (ii) the absence of any possibly strong slip effects [57]. 4.3.2. Cessation of steady shear flow experiments In this experiment, the material which undergoes a steady shear flow is subjected to cessation of the steady shear flow at t = 0. As the motion is suddenly halted, the stress starts dropping with time until complete relaxation. This decrease is typically represented in terms of the shear stress decay coefficient, g ðt; c_ Þ, defined by dividing the instantaneous decaying shear stress by the shear rate preceding the relaxation part of the experiment. The results for several levels of shear rates are plotted in Fig. 13. The continuous lines represent the predictions of the K-BKZ model (Eq. (9)) using the Papanastasiou and Marucci damping functions (Eqs. (5) and (6)). Overall the representation is satisfactory. 5. Conclusions The rheological behaviour of bitumen as a viscoelastic material has been studied in this work, experimentally by performing

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Please cite this article in press as: Behzadfar E, Hatzikiriakos SG. Viscoelastic properties and constitutive modelling of bitumen. Fuel (2013), http:// dx.doi.org/10.1016/j.fuel.2012.12.035