Yusoff Et Al. 2011.pdf

Construction and Building Materials 25 (2011) 2171–2189

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Review

Modelling the linear viscoelastic rheological properties of bituminous binders Nur Izzi Md. Yusoff a,⇑, Montgomery T. Shaw b, Gordon D. Airey a a b

Nottingham Transportation Engineering Centre, University of Nottingham, UK Polymer Program, Institute of Materials Science, University of Connecticut, USA

a r t i c l e

i n f o

Article history: Received 21 September 2010 Received in revised form 4 October 2010 Accepted 13 November 2010 Available online 5 January 2011 Keywords: Modelling Linear viscoelastic Mathematic algebraic equations Mechanical element approaches Thermorheologically simple

a b s t r a c t An extensive literature review on the modelling of the linear viscoelastic (LVE) rheological properties of bitumen over the last six decades is presented in this paper. The use of reliable models can, in general, be considered as a valuable alternative tool for estimating the LVE rheological properties of bitumen. These properties are normally presented in terms of complex modulus and phase angle master curves at a particular reference temperature. The review in this paper consists of three nonlinear multivariable models, 13 empirical algebraic equations and four mechanical element approaches. The details as well as the advantages and disadvantages of the models are discussed. In general, all the models are able to predict the LVE rheological properties of unmodified bitumen as well as follow the time–temperature superposition principle (TTSP). However, the observations suggest a lack of agreement between predicted and experimental LVE rheological properties for materials that contain a phase transition, such as found for highly crystalline bitumen, structured bitumen with high asphaltenes content and highly modified bitumen. Ó 2010 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3. 4.

5.

6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bitumen rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time–temperature superposition principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear multivariable models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Van der Poel’s nomograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Modified Van der Poel’s nomograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. McLeod’s nomograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical algebraic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Jongepier and Kuilman’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Dobson’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Dickinson and Witt’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Christensen and Anderson (CA) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Fractional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Christensen, Anderson and Marasteanu (CAM) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Modified Christensen, Anderson and Marasteanu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8. Al-Qadi and co-workers’ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Polynomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10. Sigmoidal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11. The LCPC Master Curve Construction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12. New complex modulus and phase angle predictive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13. Generalised Logistic Sigmoidal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical element models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Huet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Huet–Sayegh Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author. Tel.: +44 115 8468442; fax: +44 115 9513909. E-mail address: [email protected] (Nur Izzi Md. Yusoff). 0950-0618/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2010.11.086

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7.

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6.3. Di Benedetto and Neifar (DBN) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. The 2S2P1D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction

Water level Bitumen is traditionally regarded as a colloidal system consisting of high molecular weight asphaltene micelles dispersed in a lower molecular weight oily medium (maltenes) [1–11]. It is this colloidal structure that defines the rheological properties of bitumen ranging from sol (Newtonian dominated behaviour) to gel (non-Newtonian dominated behaviour). Although there are many methods available to determine the rheological properties, the cyclic (oscillatory) and creep tests tend to be the best two techniques for representing the uniqueness of bitumen behaviour [12]. However, recognising that testing is generally laborious, time consuming and expensive, requiring skilled operators, predictive models or equations can be a valuable alternative tool for quantifying the linear viscoelastic (LVE) rheological properties of bituminous binders [13,14]. Using this approach, the rheological parameters (complex modulus, phase angle, etc.) at any particular temperature and frequency (time of loading) of a bitumen can be estimated by means of the constitutive equations to an accuracy that is acceptable for most purposes. In the 1950s and 1960s, nonlinear multivariable methods (also known as nomographs) were used to represent the LVE rheological properties of bitumen [15,16]. However, these nomographs became obsolete with time due to the invention of computational techniques and tended to be replaced by empirical algebraic equations and mechanical element approaches. In the empirical algebraic (also known as mathematical or phenomenological or constitutive) approach, any suitable mathematical formulation is simply adjusted to the experimental main curve, with the quality adjustment being the sole criterion of choice of formulation. Meanwhile, in the mechanical element approach (or analogical model), use is made of the fact that the LVE properties of material can be represented by a combination of simple spring and dashpot mechanical models, resulting in a particular mathematical formulation [17]. Most of these models rely on the construction of stiffness/complex modulus and phase angle master curves and the determination of temperature shift factor. In other words, they imply that the time–temperature superposition principle (TTSP) holds for the bituminous binders [18–23]. This paper provides a detailed description of the vast number of LVE rheological models that have been developed for bitumens over the years. These models range from nonlinear multivariable models (known as nomographs) to empirical algebraic equations and mechanical element approaches. The advantages and disadvantages of the various models are discussed and their applications to different materials are described. In addition to the various models, the concepts of bitumen rheology and TTSP are initially discussed to aid the readers’ understanding of these important elements to data representation and modelling. It should be noted that the authors have used the word ‘‘bitumen’’ in the European sense throughout this paper and not ‘‘asphalt’’ or ‘‘tar’’ when referring to the binder. The word ‘‘asphalt’’ brings a similar meaning to ‘‘bitumen’’ in North America but in Europe, ‘‘asphalt’’ refers to the complex mixture composed of various selected aggregates bound together with different percentages of air voids. This composite is often referred to as ‘‘asphalt concrete’’ in North America. Meanwhile, ‘‘tar’’ is a liquid obtained when organic materials such as coal or wood are carbonised or destructively distilled in the absence of air [10].

Spindle

Water chamber cover

DSR base plate

2186 2186 2187 2187

Bitumen

DSR body

Fig. 1. Dynamic shear rheometer set-up [26].

2. Bitumen rheology Rheology involves the study and evaluation of the flow and permanent deformation of time– and temperature - dependent materials, such as bitumen, that are stressed (usually shear stress or extensional stress) through the application of force [9,24,25]. The word rheology is believed to be originally from the Greek words ‘‘qex’’, which can be translated as ‘‘the river, flowing, streaming’’, and ‘‘kocoo’’ meaning ‘‘word, science’’ and therefore literally means ‘‘the study of the flow’’ or ‘‘flow science’’ [26,27]. Therefore, the rheology of bitumen can be broadly defined as the fundamental measurements associated with the flow and deformation characteristics of the material, with considerable research having been undertaken over the last five decades in studying the rheology of bitumen and asphalt [28]. Understanding the flow and deformation (rheological properties) of bitumen in an asphalt is important in terms of pavement performance. Asphalt that deforms and flows too readily may be susceptible to rutting and bleeding, while those that are too stiff may be exposed to fatigue and cracking. Nowadays, the LVE rheological properties of bitumen are usually determined using an oscillatory type testing apparatus known as a dynamic shear rheometer (DSR). The DSR is a very powerful tool used to determine the elastic, viscoelastic and viscous properties of bitumen over a wide range of temperatures and frequencies often using the testing configuration shown in Fig. 1. The LVE rheological properties of bitumen are normally presented in the form complex modulus magnitude (|G|)1 and phase angle (d) master curves. |G| by definition is the ratio of maximum (shear) stress to maximum strain when subjected to shear loading. Meanwhile, d is the phase difference between stress and strain in harmonic oscillation. If d equals 90°, bitumen can be considered to be purely viscous in nature, whereas d of 0° corresponds to a purely elastic behaviour. Between these two extremes, the material behaviour can be considered to be viscoelastic in nature with a combination of viscous and elastic responses. |G| and d are defined as the following:

1 Many authors use G to mean |G| and refer to the magnitude as the ‘‘complex modulus’’.

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jG j ¼ G02 þ G002

ð1aÞ

or in complex notation as:

G ¼ G0 þ iG

00

2173

standing bitumen rheology is of major concern since the mechanical properties of binders are closely linked to the service behaviour of actual pavement mixtures [23].

ð1bÞ 3. Time–temperature superposition principle

with

d ¼ tan1 ðG00 =G0 Þ

ð2Þ

pffiffiffiffiffiffiffi where G0 and G00 are storage and loss moduli and i ¼ 1, with   0 00 G = |G | cos d and G = |G | sin d. Fig. 2 exhibits typical behaviour of the |G| and d master curves for unmodified bitumen. Under-

Fig. 2. The |G| and d master curves [22].

Work done by various researchers, has found that there is an interrelationship between temperature and frequency (or temperature and time of loading) which through time–temperature shift factors, can bring measurements done at different temperatures to fit one overall continuous curve at a reduced frequency (or time scale) [20,26,28,29]. This curve is called a master curve and represents bitumen or asphalt behaviour at a given temperature over a large range of frequencies. Indeed it is necessary to establish a method that extends the frequency scale of measurements taken over a limited range of frequencies, since conducting tests over this extended range is impractical and time consuming. The time–temperature superposition principle (TTSP) can be used to relate the equivalency between temperature and frequency (time) and thereby produce a master curve [13]. TTSP (also known as time temperature scaling, frequency temperature superposition principle, method of reduced variables, thermorheological simplicity, time temperature reducibility, time temperature equivalency, and interchangeability of time and temperature) represents a powerful and convenient tool for evaluating rheological data. Materials whose rheological properties can be shifted by TTSP to produce a smooth, continuous master curves are termed thermorheologically simple materials. Fig. 3 shows a standard procedure used to construct TTSP master curves. To construct a master curve by applying the TTSP concept, data needs to be collected over ranges of temperatures and frequencies.

Fig. 3. The construction of (a) |G| and (b) d master curves [20].

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Table 1 Various shift factor methods. Method

References

Manual shift William, Landel and Ferry (WLF) Arrhenius Log-linear Viscosity Temperature Susceptibility (VTS) Fox Laboratoire Central des Ponts et Chaussées (LCPC)

[30] [18–22,31–33] [22,34–40] [41,42] [43] [44] [45]

of loading into a nonlinear multivariable model [15,49]. This model, based upon 20 years of laboratory work [50], used the empirical tests, penetration and ASTM Ring-and-Ball (R&B) softening point (TR&B), as input parameters [10,15,49–53]. According to Van der Poel, a simple concept of Young’s modulus, E can be applied to viscoelastic materials and can be shown as the following [15,54]:

E¼

r tensile stress ¼ total strain e

ð3Þ

The ‘‘stiffness modulus’’ of bitumen, normally abbreviates as S is defined as the ratio between stress and strain [16,50,54]: A reference temperature (T0) needs to be selected and the rheological data at all the other temperatures shifted with respect to frequency until the curves merge into single smooth function. In many cases, T0 can be taken from one of the test temperatures, or as an arbitrarily chosen temperature within the range of the data. The shifting can be done using any of the rheological parameters; and if, TTSP is valid, the other viscoelastic parameters will all form continuous functions after shifting. The amount of shifting required at each temperature to form the master curve is termed the shift factor (aT). A plot of aT versus temperature, with respect to the reference temperature, is generally prepared in conjunction with the master curve [26]. There are various ways of describing the temperature dependence of the shifts, as shown in Table 1. According to Chailleux et al. [45], master curve construction only makes sense if there are, firstly, no major structural rearrangements with temperature and time, such as phase transformations and, secondly, tests are conducted within the LVE region. The TTSP can be applied to all materials that undergo a transition, such as the glass transition, as well as to heterogeneous materials, providing the disperse phase undergoes no structural change in the transition zone [3]. However, the principle does not hold across phase transition, as found in highly crystalline bitumens, structured bitumen with high asphaltenes contents and highly modified bitumens (Fig. 4) [8,23,26,45–48]. The disruption of the TTSP of these materials results in inconsistencies in the rheological properties and these materials are termed thermorheologically complex. 4. Nonlinear multivariable models 4.1. Van der Poel’s nomograph As early as the 1950s, Van der Poel introduced the concept of bitumen stiffness modulus as a function of temperature and time

ðSÞt;T ¼

r

e

ð4Þ

t;T

where S is denoted as stiffness modulus and depends on loading procedure, frequency (time of loading) and temperature. It is worth mentioning that the term S was firstly coined by Van der Poel and is now widely used among bitumen and asphalt technologists [9,54]. S is equivalent to E(t), the tensile modulus. In this work, a total of 47 bitumen samples from different resources were tested with Penetration Index (PI) values ranging from 2.3 to +6.3 at different temperatures and frequencies. In addition, Van der Poel indicated that S depends on four variables: (a) time of loading or frequency, (b) temperature, (c) hardness of bitumen and (d) rheological type of bitumen. The rheological stiffness property of bitumen could therefore be estimated by entering the following information: (a) temperature, (b) softening point, (c) loading time and (d) PI into the nonlinear multivariable model. The hardness of bitumen can be completely characterised by the Ringand-Ball temperature and the PI determines the rheological characteristics. Meanwhile, for purely viscous behaviour, differences in hardness can be eliminated by a choice of temperature where the viscosities of all bitumen are equal. The bitumen stiffness modulus, S, is a function of time of loading (s), the Tdiff = (TR&B  T) and PI. The PI can be used to characterise the rheological type and can be determined from the following equation:

  20  PI log 800  log penetration ¼ 50  10 þ PI T R&B  T

where the TR&B is Ring-and-Ball softening point temperature (°C) and T is penetration temperature (normally taken at 25 °C) and the penetration value of 800 dmm (tenths of millimetre) corresponds to the penetration at softening point temperature for bitumens. Therefore, obtaining test results from one penetration test and the Ring-and-Ball softening point test, S can be found from

1E+08

10C 45C

Complex modulus (Pa)

1E+07

15C 55C

25C 65C

35C 75C

1E+06 1E+05 1E+04 1E+03 1E+02 1E+01 1E+00 -8

-6

-4

ð5Þ

-2

0

2

4

Log reduced frequency (Hz) Fig. 4. A breakdown in TTSP due to the presence of semi-crystalline structure in polymer-modified bitumen |G| master curve [26].

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the nonlinear multivariable model for any given temperature. Van der Poel’s Nonlinear Multivariable Model is depicted in Fig. 5. It can be recapitulated that S at any temperature condition and time of loading, within a factor of two, can be predicted solely on penetration and softening point data [55]. Moreover, it is believed that Van der Poel employed the TTSP in the construction of his nonlinear multivariable model even though it was not clearly written in his paper [22]. According to Van der Poel, the accuracy of this nomograph which covers a temperature range of 300 °C, is amply sufficient for engineering purposes. It was found that at lower temperatures, all bitumens behaved elastically in the classical sense, with S being equal to 3 GPa. This value is equal to the glassy modulus, in extension or flexure, of bitumen [21]. As the temperature increased, S was observed to depend solely on the polar molecules of bitumens. The precision of mathematical functions used by Van der Poel in developing the nonlinear multivariable model was never described in any publication [22]. However, the following approximation formula matches the predictions from the nomograph [56,57]:

S ¼ 1:157  107  t 0:368  ePI  ðT R&B  TÞ5

ð6Þ

where S is bitumen stiffness (in MPa), t is loading time (s) and T is temperature (°C). The other symbols are as previously defined. However, this equation is restricted to a range of input parameters; t between 0.01 and 0.1 s; PI between 1.0 and +1.0; and temperature difference (TR&B  T) between 10 and 70 °C [56,58]. Van der Poel’s Nonlinear Multivariable Model has been widely adopted in pavement design by various researchers and in fact, efforts have been made to modify this model [54,59,60]. These modifications, however, are largely minor and cosmetic and will be discussed more fully in the following section [22]. As it stands, the nonlinear multivariable model is a convenient and easily accessible method for practical used. This model can be used to estimate S over a wide range of temperature and loading times to an acceptable accuracy [61]. Nevertheless, Van der Poel and other researchers found several shortcomings when using this nonlinear multivariable model. For

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example, the model is vividly unable to describe the behaviour of unmodified bitumen that contains more than 2% waxy elements [15,51]. This model is principally developed on unmodified bitumens and not suitable to be used for polymer-modified bitumen [10,22]. It is well known that modified bitumens are more complex in terms of their rheological behaviour and thus this nonlinear multivariable model could be misleading. The modified bitumen data lies almost exclusively below the equivalency line and clearly indicates the inability of the nonlinear multivariable model to predict the stiffness of modified bitumen from the penetration and softening point of these materials [9]. In addition, the discrepancies between stiffness values measured and predicted with the nonlinear multivariable model for modified bitumen tend to be more significant at the lower temperatures and longer loading times [22,51]. Moreover, this method is not readily amendable to numerical calculation since it involves the usage of a nonlinear multivariable model [18]. Although the nonlinear multivariable method seems to be effectively used, it is inconvenient for analysis involving computers due to the lack of mathematical expressions [62]. 4.2. Modified Van der Poel’s nomograph As mentioned above, Van der Poel’s Nonlinear Multivariable Model has been widely used by various researchers for predicting the stiffness of bitumen. In conjunction with this model, Heukelom and Klomp [16] developed a relationship between stiffness of bitumen, S and the modulus of asphalt mixture, Sm, irrespective of the combination of loading time (frequency) and temperature underlying the value of S. A semi-empirical formula of this relation can be shown as follows:

   n 2:5 Cv Sm ¼ S 1 þ n 1  Cv

Fig. 5. Van der Poel’s Nonlinear Multivariable Model [15].

ð7Þ

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where Sm is asphalt stiffness (N/m2 or Pa) and S is bitumen stiffness (N/m2 or Pa). Cv (volume concentration of aggregates) and n are calculated using the following equations:

Cv ¼

VA VA þ VB

ð8Þ

and

S m/S

4  104 n ¼ 0:83  log S

! ð9Þ

where VA and VB are percentage volume of aggregate and bitumen, respectively. Heukolem and Klomp [16] studied Van der Poel’s method in detail and then modified the relationship between S, Sm, and Cv using Eqs. (7)–(9), as shown in Fig. 6. However, the stiffness equation suffers from various shortcomings, one of which is that it is only applicable for air void contents of about 3% and Cv values from 0.6 to 0.9. It is recommended to use C 0v if the air voids content is larger than 3% [55,56,58]:

C 0v ¼

Cv 0:97 þ 0:01  ð100  ðV A þ V B ÞÞ

ð10Þ

This correction, however, is applicable only to asphalt having bitumen volume concentration factor, Cb satisfying the following equation [50]:

Cb P

S (kg/m2) Fig. 6. Sm/S as a function of S and Cv [16].

2 1  C 0v 3

ð11Þ

B where C b ¼ V AVþV ¼ 1  Cv . B In 1966, Heukelom reshaped Van der Poel’s Nonlinear Multivariable Model with a slight correction at very low PI values. This model can be found in Heukolem’s paper [54]. To validate the modification, Heukolem checked the model predictions against the data for hundreds of bitumens representing a variety of grades from different sources. He found that as in the original version, the accuracy is comparable with the distance between lines, or better [54]. This new revised nonlinear multivariable model was used to predict stiffness of bitumen for roughly 7 years until Heukelom found uncertainty when reviewing the original data on a few hundred bitumens. An extensive study of the penetration at the softening point revealed that considerable departures from 800-dmm

Fig. 7. Modified Van der Poel’s Nonlinear Multivariable Model [59].

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(blown bitumen is produced via an oxidation process that involves passing air through the short residue, either on a batch or a continuous basis, with the short residue at a temperature between 240 and 320 °C) [10]. Heukolem found that the results corresponded with the experimental values within 10–15%, which is the average of repeatability of the measurements. The use of TR&B and PI (pen/ R&B), in contrast, gave errors which were 4–10 times larger. The revised nonlinear multivariable model, according to Heukelom, can be used for dynamic loading conditions if the frequency, f is replaced by an effective loading time of t = 1/2pf. However, for a complete description of the dynamic behaviour of bitumen, the phase angle between stress and strain is needed; needless to say, its value cannot be obtained from the nonlinear multivariable model [59].

4.3. McLeod’s nomograph

Fig. 8. Correlation between viscosity at 135 °C and penetration at 25 °C [60].

penetration may occur with bitumen having high softening points and high PI values [59]. In the few cases where there was a real departure, Heukolem replaced softening point (TR&B) with temperature at 800-dmm penetration. Consequently, PI (pen/pen) is preferred to PI (pen/R&B). Fig. 7 shows the new revised nonlinear multivariable model [59]. Heukelom used this new model to predict the stiffness of bitumen, particularly for blown bitumen

McLeod found that it was impossible to obtain a fixed relationship between Pfeiffer and Doormaal’s PI for bitumens and low temperature transverse cracking performance of pavement due to penetration of many bitumens at their softening point varying widely from 800-dmm penetration [60]. He established a different method to measure a quantitative difference of the variation in temperature susceptibility for bitumen. McLeod [63] defines temperature susceptibility as the rate at which the consistency of bitumen changes with a change in temperature. This method uses the penetration of bitumen at 25 °C and its viscosity in centistokes at 135 °C (or in poise at 60 °C) [60]. Therefore, the term ‘‘pen-vis

Fig. 9. Suggested modification of Heukolem’s version of Pfeiffer’s and Van Doormaal’s nomograph [60].

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number’’ (PVN) is used by McLeod instead of PI as a quantitative measure of temperature susceptibility. The PVN has been designated because the temperature susceptibility of bitumen is based on penetration and viscosity values. The PVNs are numerically similar to, and must be numerically identical with PI values for most types of bitumen because of the way in which it is derived. The PVN can be obtained by using the following equation [60,63,64]:

PVN ¼ 1:5 

ðL  XÞ ðL  MÞ

ð12Þ

where X is log viscosity (centistokes) measured at 135 °C, L is log viscosity in (centistokes) at 135 °C for a PVN of 0.0 and M is log viscosity (centistokes) at 135 °C for a PVN of 1.5. Fig. 8 shows a correlation between viscosity at 135 °C and penetration at 25 °C for various PVN values. The following equations (based on a least squares line) could be used to calculate more precise values of L and M [25,63,65]:

L ¼ 4:258  0:7967 logðPenetration at 25  CÞ and

ð13Þ

M ¼ 3:46289  0:61094 logðPenetration at 25  CÞ

ð14Þ

The lower PVN values indicate that bitumens are more susceptible to temperature [65]. The PVN values for bitumen are normally between +0.5 and 2.0 with a good range between +1 and 1 [25]. For example, bitumen has a penetration of 100 dmm at 25 °C and a viscosity of 400 centistokes at 135 °C. From McLeod’s chart (Fig. 8), L and M are taken as 450 and 180 centistokes, respectively. Using Eq. (12), the PVN value is calculated equal to 0.19. Fig. 9 shows the suggested modification of Heukolem’s version of Pfeiffer and Doormaal’s chart used to obtain the base temperature. Supposed that the penetration of bitumen and the PVN are 90 and 1.0, respectively (calculated from Eq. (12)), a straight-line intercept gives a value of 20 °C which is 25 °C (the temperature at which the penetration test was made) below the base temperature of bitumen. Therefore, the base temperature for this sample is 45 °C. After having established the base temperature, the stiffness of bitumen for any specific temperature and rate of loading could be obtained using Fig. 10 [60]. Graphically, this model is more or less similar to Van der Poel’s Nonlinear Multivariable Model, but

Fig. 10. Suggested modification to determine stiffness modulus of bitumen [60].

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has a slight modification where McLeod used the PVN and temperature correlation to obtain stiffness of bitumen. For instance, it is assumed that the loading time is 20,000 s at 28.9 °C. The service temperature is 45 + 28.9 = 73.9 °C, with the base temperature of bitumen being 45 °C. By drawing a straight-line intercept at 20,000 s, 73.9 °C and the PVN of 1.0, S is found equal to 49 MPa. Finally, McLeod used another graph that correlates the stiffness of bitumen with the stiffness of asphalt via Cv curves [63,64]. McLeod’s suggestion of using the relationship between viscosity and base temperature rather than the Ring-and-Ball softening point are significant deviations from Van der Poel’s model. The disadvantage of this model is its inability to predict the LVE rheological properties of polymer-modified bitumen since it was not developed for that type of material. Although PVN can readily be determined from the specific tests, some researchers believe that the temperature susceptibility obtained over higher temperature ranges (25–135 °C) cannot be extrapolated to be applicable at temperatures below 25 °C. As discussed by Roberts et al. [65], one noticeable difference between PI and PVN is that the PI changes on ageing (during mixing and subsequently in service) whereas PVN remains substantially the same. In general, Van der Poel, Heukolem and McLeod’s models suffer from similar shortcomings and their use should be avoided if other, more rational and accurate methods of characterisation are available. Anderson et al. [22] found the discrepancies between measured and predicted values were more noticeable at lower temperatures and longer times of loading. When such nonlinear multivariable models are used, the proper metric and conversion values are essential [50]. 5. Empirical algebraic equations 5.1. Jongepier and Kuilman’s Model Various researchers have used explicit empirical algebraic equations to characterise master curves of complex modulus for bitumen. Among them, Jongepier and Kuilman developed an empirical algebraic model, suggesting that the relaxation spectrum of bitumen is approximately log normal in shape [22]. Based on this assumption, they derived various rheological functions [19,66,67]. The relaxation spectrum was derived from experiments using the Weissenberg rheometer operating from 20 to 160 °C at frequencies from 5  104 to 50 Hz (from 3  103 to 32 rad/s) [23]. A total of 14 samples from different sources such as ‘‘pitch type bitumens’’ (strongly temperature susceptible), ‘‘road bitumens’’ and ‘‘blown type bitumens’’ (rubbery grades) were used in this study. This approach, however, requires the use of integral equations and/or transforms, which can only be solved using numerical methods. The relaxation spectra of bitumen, although close to a log normal distribution at long loading times, deviated significantly from a log normal distribution at short loading times [22]. The Jongepier and Kuilman’s Model is based on the distributions of relaxation time and expressed mathematically using a relatively complex set of equations. First, the frequency has been replaced by a (dimensionless) relative frequency:

xr ¼

xg0 Gg

ð15Þ

where xr is reduced frequency (rad/s), g0 is zero shear viscosity (Pa s) and Gg is the glassy modulus (Pa). The logarithmic relaxation time distribution of the log normal type is given by:

 2 Gg ln s=sm HðsÞ ¼ pffiffiffiffi exp  b b p

ð16Þ

where H(s) is the relaxation spectrum distribution, b is the width parameter, s is relaxation time (s) and sm is a time constant (which

2179

determines the position of the spectrum along the relaxation time s axis at a given temperature). Gg is defined as [51]:

Gg ¼

Z

1

HðsÞd ln s

ð17Þ

1

and the parameter sm is given as:

sm

b2 ¼ exp Gg 4

g0

! ð18Þ

Jongepier and Kuilman found the parameter b depends strongly on the type of bitumen. Moreover, b for a particular bitumen can only be found by curve fitting. The storage and loss moduli are expressed by Eqs. (19) and (20) after the following substitution: u = ln xt and x ¼ b22 ln xr ,

 2 Z 1  2   b x  12 Gg u G0 ðxÞ ¼ pffiffiffiffi exp   exp  b 2 b p 0

cosh x þ 12 u  du cosh u

 2 Z 1  2   b x  12 Gg u  exp  G00 ðxÞ ¼ pffiffiffiffi exp  b 2 b p 0

cosh x  12 u  du cosh u

ð19Þ

ð20Þ

The tan d (loss tangent), is simply the ratio of G00 to G0 [19,22,66]. Jongepier and Kuilman numerically integrated Eqs. (19) and (20) for a range of b values to produce |G| and d master curves. The factor b was found to characterise the shape of the relaxation spectrum, and was thus determined to be a rational parameter for characterising bitumen. Additionally, b was observed to be strongly correlated with the composition of bitumen. It is reported that values predicted from this model fit the observed data to within the experimental error, but that the accuracy of the model was not as good for bitumen with large b values as for bitumen with small b values [22]. Jongepier and Kuilman relied upon the WLF equation to characterise the temperature dependence of bitumens. Generally the Jongepier and Kuilman’s model is reasonably accurate in its treatment of the viscoelastic properties of bitumens. Brodynan et al. [68] suggested that the relaxation times are not log normally distributed since they found relaxation spectrum highly skewed on a logarithmic scale distribution. It is also reported that the Jongepier and Kuilman’s model makes use of integral equations which makes practical calculations with this model impossible [21,22,69]. No details concerning the precise determination of the model parameters were presented by the researchers [22]. In addition, the determination of relaxation spectrum from experimental data is an ill-posed (ill-conditioned, no unique solution) problem [70,71]. 5.2. Dobson’s Model Dobson developed an empirical algebraic equations for describing a master curve, based on the empirical relationship between |G| and d for bitumen [18,67,72,73]. However, Dobson does not expressed modulus in term of frequency, but the reverse. Dobson presented the results in term of a universal master curve, with the intention to characterise bitumen by graphical comparison with this master curve [22]. He described how the stiffness of bitumen under any conditions of temperature and rate of loading may be calculated from three fundamental parameters; (i) a viscosity, (ii) temperature dependence and (iii) rate dependence [18,72]. The temperature dependence and viscosity parameters are obtained by the use of a new viscosity–temperature chart, and the

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rate dependent parameter is obtained from measurement of apparent viscosity at two levels of shear stress [72]. The viscosity measurements were made at 60 °C by a vacuum capillary viscometer and at 25 °C by a coni-cylindrical viscometer. The fundamental assumption of this model, which was based on the empirical observations of dynamic data on a range of bitumens, is that the log–log slope of the |G| with respect to loading frequency is a function of the loss tangent and the width of the relaxation spectrum [18,22]:

dy tan d ¼ dx ð1 þ tan dÞð1  0:01  tan dÞ

ð21Þ

where y = log(|G|/Gg), |G| is the complex modulus magnitude, Gg is glassy modulus, x = log(g0xaT/Gg), g0 is the steady state or Newtonian viscosity and aT is the shift factor. Dobson also observed a linear relationship between tan d and |G| which can be expressed in the following form:

logð1 þ tan dÞ ¼ by

ð22Þ

where b is a parameter describing the width of the relaxation spectrum. It may also be regarded as a shear susceptibility index and is related to the PI [18]. Eqs. (21) and (22) can be combined to give a new equation for relating reduced frequency and complex modulus:

" # 1 20:5  Gb b r logð1  Gr Þ þ log xr ¼ log Gr  b 230:3

ð23Þ

or by rearranging:

  20:5  Gb b r log xb r ¼ log Gr  1 þ 230:3

ð24Þ

where xr = g0xaT/Gg and Gr = |G|/Gg. xb is a unique function of r Gb r . All the equations above are applicable for the value of tan d 6 9.5. Dobson developed an instrument to measure the complex shear modulus of 45-mg bitumen samples over a frequency range from 2 to 200 Hz and over a continuously variable temperature range [18]. This equipment was based on a plate measuring geometry. Dobson used the value of Gg as 1  109 Pa (1  1010 dynes/cm2) at tan d P 9.5, dy/dx = 1. It was found that the model exhibit good agreement with experimental data. Like Jongepier and Kuilman, Dobson described the effect of temperature on viscoelastic properties using the WLF equation. He found that a single set of coefficients could be used to fit the shift factor data for a range of bitumens, but a different coefficient set was needed for two extreme temperatures. The simple form of the WLF equation with two sets of constants: For T  Ts < 0, C1 = 12.5 and C2 = 142.5 K has been used. Meanwhile for T  Ts > 0, use was made with C1 = 8.86 and C2 = 101.6 K. Ts is equivalent to the reference temperature, with Ts = Tg  50. Tg was probably determined from dilatometric measurements [22]. In general, Dobson’s method for characterising the temperature dependency appears to be reasonably accurate. He also presents a practical means for applying this part of his model to rheological data on bitumen. Nevertheless, this model also has several shortcomings. Maccarrone [74] studied the Dobson Model for describing temperature dependence of aT and found the WLF equation with Dobson’s coefficients over predicted the aT at temperature below 20 °C when applied to aged bitumens. It is difficult to assess the accuracy of Dobson’s Model since he only made a few comparisons of measured and predicted |G| and d values. In addition, the failure to express modulus as an explicit function of reduced frequency is a serious shortcoming, as is the lack of a well-defined procedure for determining the constants in his equation for the modulus [22]. In addition, the development of this model was not applied to modified bitumen.

5.3. Dickinson and Witt’s Model Dickinson and Witt [20] performed dynamic mechanical testing on 14 different bitumens and developed analytical expressions for the |G| and d in terms of their frequency dependencies. They proposed the following equation [20,22]:

log Gr ¼

h io 1 1n log xr  ðlog xr Þ2 þ ð2bÞ2 2

ð25Þ

where Gr is relative complex modulus at frequency, x (i.e.,   Gr ¼ jG j=Gg ), xr is the relative angular frequency (xr = xg0aT/Gg) and b is a shear susceptibility parameter, which is defined as the distance on a logarithmic scale between Gg and the modulus at xr = 1. Meanwhile the phase angle can be expressed as the following:

d ¼ d0 þ

p  2d0 4

 h i12 1  log xr ðlog xr Þ2 þ ð2bÞ2

ð26Þ

where d is the phase angle and d0 is the limiting phase angle at infinite frequency [22]. By eliminating log xr in Eqs. (25) and (26), a relationship between complex modulus and phase angle was established by Dickinson and Witt, treated as a hyperbola model:

 1

2ðd  d0 Þ 2 log Gr ¼ b p  2d

ð27Þ

Because of the limited temperature range examined, they used the same coefficients as Dobson to describe temperature dependence of the aT for their bitumen (see Dobson’s Model). The standard errors (SE) of fit of |G| obtained ranged from 0.008 to 0.025 on the logarithmic scale, corresponding to a maximum error of about 10%. The accuracy of the d was not reported by Dickinson and Witt although it can be seen that the accuracy was comparable to the experimental error in determining the phase angle [21,22]. In addition, Dickinson and Witt observed that the spectra were unsymmetrical with respect to the maximum value and disputed Jongepier and Kuilman’s assumption of a log Gaussian distribution of relaxation times. The reported values of log Gg (Gg in Pa), ranged from 9.5 to 10.6, which are similar to the values of 9.7 or 10, reported by other researchers like Dobson [18] and Jongepier and Kuilman [19]. Maccarrone [74] evaluated various models for predicting the dynamic properties of bitumen based on 39 aged and 2 original bitumen samples. It was found that the Dickinson and Witt’s Model fits the rheological data quite well with a standard error of estimate (Se) of log |G| ranging from 0.001 to 0.012. Generally, this model is more practical and simpler than either Dobson’s or Jongepier and Kuilman’s Model but Gg and viscosity in the model are determined statistically and in may cases, overestimated [9,22]. 5.4. Christensen and Anderson (CA) Model During the Strategic Highway Research Program (SHRP) A-002A study, Christensen and Anderson performed dynamic mechanical analysis (DMA) on eight SHRP core bitumens for the purpose of developing an empirical algebraic equation that described the viscoelastic behaviour of bitumen. The model will be referred to as the Christensen and Anderson (CA) Model. They noted that four primary parameters (the glassy modulus, Gg, the steady state viscosity, gss, the crossover frequency, xc, and the rheological index, R) are needed to fully characterise the properties of any bitumen [21]. Fig. 11 shows some of the parameters mentioned above. Detailed explanations for each parameter can be found elsewhere and are omitted in this section for brevity [8,21,22,46,75]. The CA Model is presented as a series of equations for the primary dynamic vis-

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coelastic functions. For |G|, the following mathematical function can be used:

" jG j ¼ Gg 1 þ

x ðlogR 2Þ

#

R log 2

c

ð28Þ

x

where xc is the crossover frequency and R is a rheological index. The other parameters are as defined previously. Bitumen with larger values of R exhibit wider relaxation spectra. Meanwhile d (in degrees) can be taken as:

d¼" 1þ

90 #  ðlogR 2Þ

ð29Þ

x xc

The other parameters are as defined above [22,46,62,76]. Christensen and Anderson combined Eqs. (28) and (29) to define R as:



R¼



ðlog 2Þ  log jGGg j

d log 1  90

Fig. 12. Modelling using the CA Model [76].

 ð30Þ

where the parameters are as previously defined. They found this equation is quite useful when the value of R is desired, but it is impossible to obtain data at the region when d = 90°. In using this equation to calculate the R value, Gg can be taken as 1 GPa in shear and 3 GPa in extension (or flexure). It was reported that Eq. (30) is reasonably accurate within the region where d is between 10° and 70° and the best results are obtained near the crossover point, where d = 45° [22]. According to Christensen and Anderson, this model is not recommended to be used at temperatures and frequencies where d approximates 90°. The CA Model was validated using the SHRP core bitumens (unaged), thin film oven test (TFOT) aged and pressure ageing vessel (PAV) aged. It can generally be used over a wide range of frequencies and temperatures extended well into the glassy region [21]. However Silva et al. [76] found the model presented lack of fit particularly at high temperatures and/or long loading times (Fig. 12). To surmount this inconsistency, Christensen and Anderson suggested calculating a second set of parameter values for the secondary region in which the value of R is set equal to 0.81 when Newtonian flow is approached. They have manipulated the equations above to generate a series of equations from which the LVE parameters for the secondary viscoelastic region can be calculated. The details of those equations can be found elsewhere and are omitted for brevity [21,22]. The WLF equation is used above the defining temperature, Td and in the Newtonian region. Td is a characteristic parameter for each bitumen. It is reported that the values of C1 = 19 and C2 = 92 K can be used for all bitumens. However, it is recom-

Log |G*| (Pa)

mended to obtain the C1 and C2 by optimisation process from experimental data. No universal values can, a priori, be applied. Meanwhile, for the temperatures below Td and in the Newtonian region, an Arrhenius function is used to describe aT. The activation energy (Ea) for flow below Td was reported to be 261 kJ/mol. In general, the CA Model is relatively simple in shape and reasonably accurate as compared to the previous models. However, the sensitivity analysis showed the use of only parameters related to the shape of the relaxation spectrum (rheological index) is not enough to describe bitumen behaviour [76]. In addition, the model is not able to describe the LVE rheological properties of modified bitumen [9]. 5.5. Fractional Model Stastna et al. [77] proposed a simple model for |G| and d to describe the behaviour of bitumen. The model, called the Fractional Model, is based on the generalisation of the Maxwell model [77– 84]. This model has a relatively low number of parameters and requires only half the parameters compared to the generalised Maxwell model. |G| represents the response function of viscoelastic materials and can be described by a general power of a rational fraction. Models of this type have been studied by Stastna et al. [77] and comparison with experimental data for polymeric solutions can be found in the study of Stastna et al. [78]. |G| can be shown as the following: 1 "Q #2ðnmÞ m ð1 þ ðlk xÞ2 Þ jG j ¼ g0 x Q1n 2 1 ð1 þ ðkk xÞ Þ

where lk and kk are the relaxation times (lk > 0, kk > 0) and m and n are the numbers of relaxation time (n > m). Stastna et al. found the Fractional Model is much more flexible than the generalised Maxwell model and is easily manageable. d can be expressed as:

d¼

Log Fig. 11. Definition of the CA Model [8].

(rad/s)

ð31Þ

p 2

þ

" # m n X X 1 a tanðlk xÞ  a tanðkk xÞ ðn  mÞ 1 1

ð32Þ

where a is the Fourier transform of the Dirac delta function. They performed DMA on 19 unmodified and 5 modified bitumen samples from different sources. For almost all cases and with the reference temperature set at 0 °C, the Fractional Model generates excellent fitting of |G| and d master curves, both for the unmodified and modified bitumens. Stastna et al. [80] in their study employed both the WLF and Arrhenius equations for shifting purposes but stress was given to the earlier equation for a better fit. Nevertheless, no details concerning the precise determination of the parameters were pre-

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sented in their papers. Marasteanu and Anderson [85] reported this model lacks statistical robustness because of the number of unknown parameters (10)–(15) approaches the number of observations. Indeed it is difficult to interpret a model with so many parameters on a phenomenological basis. However, they found the degree of flexibility offered by the model is very useful when simulating plateaus or other irregularities in the master curves. This, however, can also lead to the fitting of anomalous portions of the master curve that are the result of testing error, rather than real rheological properties of binders [85].

where Ge = |G|(f ? 0) with Ge = 0 for bitumen, Gg = |G|(f ? 1), fc is a location parameter with dimensions of frequencies and f0 is reduced frequency, function of both temperature and strain and k and me are the shape parameters (dimensionless). fc is equal to crossover frequency in the CA and CAM Models. Fig. 14 shows a schematic diagram of the Modified CAM Model. It is seen that Gg and Ge are horizontal asymptotes when frequencies approach infinity and zero, respectively. The third asymptote is the one with the slope of me. The Gg and me asymptotes intercept at fc. Ge and me asymptotes intercept at:

5.6. Christensen, Anderson and Marasteanu (CAM) Model

fc0 ¼ fc

Marasteanu and Anderson [85] developed a new model by modifying the CA Model to improve the fitting particularly in the lower and higher zones of the frequency range of bitumens. The model known as the CAM Model after Christensen, Anderson and Marasteanu, attempts to improve the descriptions of both unmodified and modified bitumen. The researchers applied the Havriliak and Nagami Model to the initial CA Model and proposed the following equation for |G| [85–88].

 x v wv c jG j ¼ Gg 1 þ

x

ð33Þ



Ge Gg

m1

e

ð36Þ

For binders, fc0 ¼ 0. According to Zeng et al., the distance (one logarithmic decade being unity) between |G|(fc) and Gg for bitumen is given by:

R ¼ log

2me =k me =k

1 þ ð2

 1ÞGe =Gg

For binders, R = me/k log 2. The distance between jG jðfc0 Þ and Ge is given by:

8 9  k=me #me =k =   " < Gg Gg R ¼ 1þ 1  1þ : ; Ge Ge 0

where

v = log 2/R and R is the rheological index. d is defined as:

d¼h

90w xc v i

1þ

ð34Þ

x

The introduction of w parameter addresses the issue of how fast or how slow the |G| data converge into the two asymptotes (the 45° asymptote and the Gg asymptote) as the frequency goes to zero or infinity [85]. During the work, Marasteanu and Anderson tested their model using 38 unmodified and modified bitumens. The fitted |G| values of the measured values for the CAM and CA Models were within 10–35%, respectively. They found that typically the lack of fit occurred at the two asymptotes of the master curves and it was believed that the departure from the thermo-rheologically simplicity is bitumen dependent and is strongly related to bitumen composition, especially the presence of waxy elements and higher asphaltenes. Marasteanu and Anderson did not specify clearly the equation for describing the temperature dependence of bitumen. The degree of precision in modelling |G| and d was greater than the original CA Model. However, anomalies were still seen in the construction of a smooth master curve when test bitumens behaved as thermorheologically complex materials. Like the CA Model, Silva et al. [76] found the CAM Model presented lack of fit particularly at high temperatures as shown in Fig. 13.

Fig. 13. Modelling using the CAM Model [76].

Zeng et al. [62] developed empirical algebraic equations to characterise modified bitumens and asphalt with bitumen modification under dynamic shear loading over a wide range of frequencies, temperatures and strains. The model is composed of four formulations for the |G| and d master curves, temperature and strain dependencies. This model is capable of modelling the behaviour of bitumen as a viscoelastic fluid, and for asphalt as a viscoelastic solid in a universal form. The |G| equation is based on a generalisation of the CAM Model and generalised law model [12,62]:

Gg  Ge  k mke 1 þ ffc0

ð38Þ

For binders, R0 = log 2. For brevity, the physical meaning of the parameters are not shown here and can be found elsewhere [62]. d (in degrees) is shown as:

5.7. Modified Christensen, Anderson and Marasteanu Model

jG j ¼ Ge þ 

ð37Þ

ð35Þ Fig. 14. Definition of Generalised CAM Model [62].

Nur Izzi Md. Yusoff et al. / Construction and Building Materials 25 (2011) 2171–2189

( d ¼ 90I  ð90I  dm Þ 1 þ



logðfd =f 0 Þ Rd

2 )md =2

2183

5.9. Polynomial Model

ð39Þ

where dm is the phase angle constant at fd, the value at the inflexion (or inflection point) for bitumens, f0 is the reduced frequency, fd is a location parameter with dimensions of frequency at which dm occurs and Rd and md are the shape parameters. Zeng et al. did not explicitly provide the definition of I but showed I = 0 if f > fd and I = 1 if f 6 fd for bitumen. For asphalt, I is always equal to zero. Eq. (39) satisfies the requirement that d varies from 90° to 0° when the frequency is elevated from zero to infinity for bitumen [62]. Like others, Zeng et al. used the WLF equation to describe the shift factor of bitumens. In addition, they have suggested the Arrhenius function should be used if low temperatures are involved. Zeng et al. conducted the dynamic test by means of the DSR and Simple Shear Test (SST) for bitumen and asphalt respectively. Analysis of data involving 9 modified bitumens, 36 asphalts and 4 types of aggregates over various ranges of frequency, temperature and strain indicated that the model fits the measurements very well. They found a good agreement between measured and predicted |G| master curves. Conversely, Zeng et al. observed that the d master curve was not as good as its |G| counterpart. This phenomenon may be attributed to some extent to errors in measurement and analysis. The source of the discrepancies still remain unknown but they are believed to be due to the presence of aggregates in asphalt as well as to minor to moderate differences in temperature dependency existing between bitumen and its mixtures with aggregate [62].

The Polynomial Model can be used to describe the |G| master curves of bitumen, even though it was originally developed for asphalt. As suggested by Mohammad et al. [14], for practical purposes, a simpler polynomial function may be used to express the |G| master curve constructed from a dynamic modulus test. The form of the Polynomial Model can be shown as the following:

log jG j ¼ Aðlog f Þ3 þ Bðlog f Þ2 þ Cðlog f Þ þ D

ð42Þ

where f is reduced frequency and A, B and C are the shape parameters and D is a scaling parameter. This model generally can fix the test data from low to moderate temperatures satisfactorily. However, as the temperature increases or decreases, the curves tend to skew, depending on the degree of freedom in the equation used [89]. Moreover, a single Polynomial Model cannot be used for fitting the whole master curve. The LVE rheological properties of bitumen seem ‘‘incomplete’’ since the Polynomial Model does not take the behaviour of d into account. 5.10. Sigmoidal Model A new dynamic modulus function, the Sigmoidal Model, has been introduced in the Mechanistic-Empirical Pavement Design Guide (ME PDG) developed in National Cooperative Highway Research Program (NCHRP) Project A-37A. In the ME PDG, the Sigmoidal Model is used to describe the rate dependency of the modulus master curve [89–91]. Mathematically, the Sigmoidal Model can be shown as the following:

5.8. Al-Qadi and co-workers’ Model Al-Qadi et al. in their research, proposed a new model of |G| and d for describing the rheological behaviour of straight run and modified bitumen in the LVE region. They performed DMA using a DSR with parallel plate geometry at frequencies between 0.01 and 30 Hz and temperatures ranging from 5 to 75 °C. The proposed |G| based on the Havriliak and Negami function as follows [69]:

2 6 jG j ¼ Gg 41  h

3 1þ

1 7  v iw 5

ð40Þ

x x0

where x0 is the scale parameter that defines the location of the transition along the frequency axis, v and w are the dimensionless model parameters. In addition, the proposed d model (in degrees) can be shown as:

90 d¼h   v iw 1 þ xx0

ð41Þ

where the symbols are as previously defined. Fig. 15 shows the comparison between measured and predicted |G| and d master curves. The WLF equation has been used by Al-Qadi et al. to show the temperature dependency of bitumens. From the study, they found a good agreement between the measured and predicted values for |G|. Meanwhile the d model was found to be adequately described for unmodified bitumen with small percentage errors (<5%). Nevertheless, anomalies can still be seen where the proposed model could not simulate the small plateau region observed in the d master curve of modified bitumen in an accurate way. The difference, however, between the predicted and measured values was less than 10% which was within the acceptable range in such a test [69].

Fig. 15. Comparison between measured and predicted (a) |G| and (b) d using AlQadi and co-workers Model [69].

2184

log jG j ¼ t þ

Nur Izzi Md. Yusoff et al. / Construction and Building Materials 25 (2011) 2171–2189

a 1 þ ebþcflogðxÞg

ð43Þ

where log x is log reduced frequency, t is the lower asymptote, a is the difference between the values of the upper and lower asymptote, b and c define the shape between the asymptotes and the location of the inflection point (inflection point obtained from x = 10(b/c)) [92]. In the ME PDG, aT is expressed as a function of the bitumen viscosity to allow ageing over the life of pavement. The definition of each parameter is shown in Fig. 16. The Sigmoidal Model has been widely used by many researchers in the paving industry and details of this model are found elsewhere [89–92]. Like the Polynomial Model, the Sigmoidal Model was only applied to the |G| master curve construction, without taking into account the behaviour of d. Hence the entire viscoelastic behaviour of the material cannot be described. Normally for unmodified and polymer - modified bitumens, Gg can be taken as 1 GPa, for most engineering purposes. Bonaquist and Christensen [89] have proposed the modification of Sigmoidal Model as:

log jG j ¼ t þ

Max  t 1 þ ebþcflogðxÞg

ð44Þ

where Max is limiting value of complex modulus. The other symbols are as previously defined. 5.11. The LCPC Master Curve Construction Method Chailleux et al. from the Laboratoire Central des Ponts et Chaussées (LCPC), France adapted an empirical algebraic equation in order to construct master curves from complex modulus measurements [45]. The researchers applied the Kramers–Kronig relations, based on previous work done by Booij and Thoone [93], linking |G| and d of a complex function. The integral transform relationships between the real and imaginary parts of this function are generally known as the Kramers–Kronig relations. Booij and Thoone [93] carried out experiments on a polyvinylacetate sample involving oscillation measurements in a Mechanical Spectrometer at five different frequencies and temperatures between 22.85 and 119.85 °C. Superimposed curves of both G0 and G00 versus frequency were produced. This was done by means of time–temperature shift factors on both master curves at a reference temperature of 34.85 °C. Booij and Thoone also tested this relation for other materials as well, including a number of dielectric data. It appears that the relation invariably holds with standard deviation (SD) never exceeding 5% [93]. The Kramers–Kronig approximations give the following equations for |G| and d [45]:

Fig. 16. Definition of the Sigmoidal Model [40].

log jG ðxÞj  log jG ð1Þj ¼ 

2

Z

p

0

1

u  dðuÞ  x  dðxÞ du u2  x2

ð45Þ

and

d¼

2x

Z

p

1

0

log jG ðuÞj  log jG ðxÞj du u2  x2

ð46Þ

where u is defined as a dummy variable. Eq. (46) becomes exactly:

d¼

p d log jG j  d log x 2

ð47Þ

Chailleux et al. used the following shift factor relationship to characterise temperature dependence of bitumen:

log að T i ; T o Þ ¼

j¼ref X





log GðT j Þ  log GðT jþ1 Þ

j¼i

ðT ;T Þ davrj jþ1



p 2

ð48Þ

where davr is the average of two angles measured at xj and xj+1. To validate the possible use of this methodology, they applied the model to three unmodified bitumens, one SBS-modified bitumen and two asphalts using DMA. They proposed to plot d log |G|/ d log x versus d/90 to verify both the Booij and Thoone equation and the Kramers–Kronig relations (Fig. 17). Earlier, Marasteanu and Anderson applied the same Booij and Thoone approximation to analyse the dynamic shear data for a set of 71 unmodified and modified bitumens [94]. The validity of the Booij and Thoone relation was examined by calculating the slopes of the logarithmatic plots of |G| versus x, as the ratio of the difference of the logarithm of two consecutive |G| values divided by the difference of the logarithm of their corresponding frequencies. The d was, subsequently, calculated as the average of the two corresponding phase angles. No LVE rheological model was assumed and the slope was obtained by means of simple calculations. Furthermore, the Booij and Palmen approximation was also used to calculate the relaxation spectra. This approximation can be shown as below:

HðsÞ ffi

1

p

½jG j sin 2dx¼1=s

ð49Þ

where s is the relaxation time and H(s) is the strength of relaxation at s for discrete spectra. From this study, a smooth master curve was produced when applying this equation to the data at different temperatures from frequency sweep tests. This approximation plays a significant role in modelling DSR data and in generating rheological master curves for bitumens. It is worth mentioning that

Fig. 17. Example of plot d log |G|/d log x versus d/90 [45].

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this model fits data ranging from intermediate to high temperatures satisfactorily [94]. 5.12. New complex modulus and phase angle predictive model To surmount the limitations of the current models used in the ME PDG, Bari and Witczak developed a new predictive model for |G| and d [95]. A database containing 8940 data points from 41 different unmodified and modified bitumens was used in this study. The equation for |G| is shown as follows:

jG j ¼ 0:0051f s gfs ;T ðsin dÞ7:1542  0:4929f s þ 0:0211fs2

ð50Þ

where fs is the dynamic shear loading frequency to be used with |G| and d, gfs ;T is viscosity of bitumen (cP) as a function of both loading frequency (fs) and temperature (T), and d is the phase angle (°). The value of |G| is limited to a maximum value of 1 GPa. d is obtained from a nonlinear optimisation technique in the form of the following equation: 0

Table 2 Criteria of the goodness-of-fit statistics [96]. Criteria

R2

Se/Sy

Excellent Good Fair Poor Very poor

P0.90 0.70–0.89 0.40–0.69 0.20–0.39 60.19

60.35 0.36–0.55 0.56–0.75 0.76–0.89 P0.90

where Max is the limiting value of complex modulus. The other symbols are as previously defined. Although the Generalised Logistic Sigmoidal Model improves the prediction of the non-symmetrical shape of the master curve, the model is still unable to predict the behaviour of highly modified bitumens, as similarly observed for the Sigmoidal Model. 6. Mechanical element models

0

d ¼ 90 þ ðb1 þ b2 VTS Þ  logðfs  gfs ;T Þ þ ðb3 þ b4 VTS Þ  flogðfs ; gfs ;T Þg2

ð51Þ

where VTS0 ¼ 0:9699fs0:0575  VTS, fs is loading frequency in dynamic shear (Hz), b1, b2, b3, and b4 are the fitting parameters (7.3146, 2.6162, 0.1124 and 0.2029). The fitting parameters will change slightly as a function of the type of bitumen (crude source and grade). To evaluate the model’s performance, Bari and Witczak used the ratio of standard error of estimates over standard deviation (Se/Sy) and coefficient of determination (R2) to measure the goodness-offit statistics between measured and predicted data. The criteria of the goodness-of-fit statistics used are shown in Table 2 [96].2 In general, Bari and Witczak found a good correlation between measured and predicted data. This new d model has a very good correlation for unmodified bitumens compared with that of the modified bitumens. They concluded that modified bitumens used in this study had higher variability in stiffness characteristics as a result of their type and amount of modification. However, the overall variation from all 41 bitumens is practically negligible and the predicted plots are very close to the equality line. More bitumen modifications are needed for future development as the current model only includes a small sample of modified bitumens [95].

It is useful to consider the simple behaviour of analogue models constructed from linear springs and dashpots to get some feeling for LVE behaviour of bitumen. The spring (Hooke’s Model) is an ideal elastic element obeying the linear force extension relation while the dashpot (Newton’s Model) is an ideal viscous element that extends at a rate proportional to the applied stress. A number of different models with various arrays of spring and dashpot arrangements, such as the Jeffery, Zener and Burgers’ Models, are available to facilitate the mathematical expression of the viscoelastic behaviour of engineering materials. However, as reported by Monismith et al. [29] none of these models is itself sufficient to represent the behaviour of bitumens. 6.1. Huet Model The Huet Model was initially conceived by Christian Huet in order to model the behaviour of both bitumen and asphalt [98]. This model consists of a combination of a spring and two parabolic elements (k and h) in series as illustrated in Fig. 18. According to Olard and Di Benedetto [88], the parabolic element is an analogical model with a parabolic creep function with equations for creep compliance and complex modulus as follows:

 h t JðtÞ ¼ a

5.13. Generalised Logistic Sigmoidal Model

ð54Þ

s

Rowe et al. introduced a generalisation of the Sigmoidal Model, called the Generalised Logistic Sigmoidal Model (or Richards Model) to predict the stiffness of asphalt. This equation is also applicable to bitumen and the model can be shown as the following [92,97]:

log jG j ¼ t þ

a ½1 þ keðbþcflogðxÞgÞ 1=k

ð52Þ

where the symbols are as previously defined. The k parameter allows the curve to take a non-symmetrical shape for the master curve. When k reduces to one, Eq. (52) reduces to the standard sigmoidal function as represented in Eq. (43). As the glassy modulus can be taken as 1 GPa for most engineering purposes, it is possible to modify and transform the equation into the following form:

log jG j ¼ t þ

Max  t ½1 þ

keðbþcflogðxÞgÞ 1=k

ð53Þ

2 These criteria are not very good for describing mismatch. The R2 is only applicable for linear models with a large sample [96]. In addition, these two columns (R2 and Se/ Sy) are not really independent, i.e. R2 1  [(n  k)/(n  1)](Se/Sy)2 (where n = the sample size and k = the number of regression coefficient).

and

G ¼

ðixsÞh aCðh þ 1Þ

ð55Þ

where J(t) is the creep function, h is the exponent such as 0 < h < 1, a is a dimensionless constant, C is gamma function, t is the loading time, s is the characteristic time (which value pffiffiffiffiffiffiffi varies only with tem2 perature), i is the complex number ði ¼ 1Þ and x is the angular frequency. This model, in addition, has a continuous spectrum and can be presented by an infinite number of Kelvin-Voigt elements in series or Maxwell elements in parallel [88]. The analytical expression of G can be shown as follows [88,98–100]:

G ¼

G1 1 þ oðixsÞk þ ðixsÞh

ð56Þ

where G is the complex modulus, G1 is the limit of the complex modulus, h and k are exponents such as 0 < h < k < 1, o is a dimensionless constant. The other symbols are as previously defined. The Huet equation accounts for the non-symmetric shape of the frequency response.

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The WLF equation has been used by Huet to describe the temperature dependency of bitumen. He presented the results obtained under monotonic loading by means of the Cole–Cole diagram. It can be seen that the Huet model is well suited for the description of this kind of loading. However, it is believed that this model is unable to model modified bitumen correctly. Another drawback is that the original model does not contain a viscous element for simulating permanent deformation, in contrast with the Burger’s Model, a combination of the Maxwell and a Kelvin-Voigt unit in serial connection (four parameters model) [101,102]. 6.2. Huet–Sayegh Model Sayegh developed a model based on the generalisation of the Huet Model but modified by adding a spring of small rigidity compared with G1 in parallel [99]. This model consists of the combination of two springs (G0 and G1–G0) and two parabolic creep elements (k and h) and can be presented as shown in Fig. 19. If the G0 is equal to zero, then the Huet–Sayegh Model is identical to the Huet Model. As a matter of fact, the Huet–Sayegh Model looks like a Zener Model but instead of one linear dashpot, it has two parabolic dashpots [101–103]. The model can be described mathematically using the following formula:

G ¼ Go þ

G1  Go 1 þ oðixsÞk þ ðixsÞh

account the LVE behaviour in the small-strain domain, as well as plastic flow for large strain values. To find a compromise between the complexity of the development and a close description of the material behaviour, the number of considered bodies must be reasonable. The DBN Model, which is depicted in Fig. 20, can also be used for describing the rheological properties of bitumens in the LVE region. The G function of the DBN Model can be written as:

ð57Þ 

with G0 is the elastic modulus and o is a dimensionless constant. The other symbols are as previously defined. a, b, and c can be determined implicitly using s which is referred to as the characteristic time and it is calculated using the following equation [101,102]:

ln s ¼ a þ bT þ cT 2

Fig. 19. The Huet–Sayegh Model [18].

ð58Þ

where a, b and c are regression parameters representing the material characteristics. This model was originally developed for asphalt, but it can also be used for unmodified bitumens. Unlike the Huet Model, no analytical expression of the creep function of the Huet–Sayegh Model is available in the time domain. Olard and Di Benedetto attempted to fit the data both on bitumen and asphalt using the Huet–Sayegh Model and they found the model is unsuitable for bitumens at the very low frequencies where it is equivalent to a parabolic element instead of a linear dashpot [88]. The model also has a lack of an element representing the permanent deformation characteristics of bitumen like the serial dashpot in the Burger’s Model [101–103]. 6.3. Di Benedetto and Neifar (DBN) Model The DBN Model is a rheological model specially developed by Di Benedetto and Neifar for asphalt [12,100,104–106]. The DBN Model, abbreviated from Di Benedetto and Neifar, takes into

G ¼

n 1 X 1 þ Go i¼1 Gi þ ixgi ðTÞ

ð59Þ

where Go is the elastic modulus of the single spring, gi is a viscosity function of the temperature (T) and x = 2pf. The number n of the elementary body can be arbitrarily chosen [12]. The DBN Model bears a resemblance with the generalised Kelvin-Voigt Model, except the DBN Model does not incorporate an ‘‘end’’ dashpot element. As comparison, the generalised Kelvin-Voigt equation is shown as:

G ¼

n 1 X 1 1 þ þ Go i¼1 Gi þ ixgi ðTÞ ixg0

!1 ð60Þ

Di Benedetto and Neifar used the 2S2P1D Model (see Section 6.4) to calibrate the DBN Model and found good agreement between the two models and experimental data using an optimisation process. However, as reported by Blab et al. [100] the calibration of the DBN Model based on the 2S2P1D Model means that uncertainties associated with the 2S2P1D Model will also lead to uncertainties in the DBN Model. 6.4. The 2S2P1D Model Olard and Di Benedetto developed a mechanical element approach for modelling the LVE behaviour of bitumens and asphalts [75,88,107,108]. This model, based on a generalisation of the Huet–Sayegh Model, consists of a combination of two springs, two parabolic elements, and one dashpot as shown in Fig. 21. The model introduced by Olard and Di Benedetto has seven parameters and G is shown as:

G ¼ Go þ

Fig. 18. The Huet Model [88].

!1

G1  Go 1 þ oðixsÞ

k

þ ðixsÞh þ ðixsbÞ1

ð61Þ

where o and b are dimensionless constants. The other symbols are as previously defined. Like the Huet–Sayegh Model, this model has a continuous spectrum (i.e. can be presented by an infinite number of Kelvin-Voigt elements in series or Maxwell elements in parallel) [86]. Olard

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Fig. 20. The Di Benedetto and Neifar (DBN) Model [106].

Fig. 21. Representation of the 2S2P1D Model [108].

and Di Benedetto, however, found G0 for bitumen was very close to zero; therefore, the parameters can be reduced to six for bitumen as G0 can be neglected. They conducted complex modulus tests on nine bitumens and four asphalts with one mix design. Olard and Di Benedetto used the WLF equation to show the temperature dependency of bitumen and the C1 and C2 constants were obtained by an optimisation process from the experimental data. Delaporte et al. used a similar model and found an excellent relation between measured and predicted data of bitumen-filler mastics [109]. In general it can be inferred that the 2S2P1D Model can significantly reduce testing effort, at least for the purpose of describing bitumen behaviour in the LVE region. However, anomalies were still seen in the Olard and Di Benedetto study where the model did not perfectly fit the experimental data for d of between 50° and 70°. A third parabolic element was suggested by Olard and Di Benedetto, but this would have significantly increased the complexity of the model. Another disadvantage of the model is that it was unable to fit experimental data for modified bitumen, particularly at high temperatures. To account for modified bitumen, Olard and Di Benedetto introduced the partial time–temperature superposition principle (PTTSP) based on the observation that modified bitumen tended to conform only to the TTSP at temperatures below 10 °C. The PTTSP is therefore a shifting procedure that is only applied to the construction of a continuous |G| master curve.

7. Summary This paper aims to offer bitumen and paving engineers with an explanation on modelling methods used to predict the linear viscoelastic (LVE) rheological properties of bitumens. The LVE rheological properties of bitumen are generally and conveniently represented in terms of complex modulus magnitude and phase

angle master curves. The use of empirical models to describe these curves can be classified into three groups; nonlinear multivariable models, empirical algebraic equations and mechanical element approaches. However, nonlinear multivariable models have tended to become obsolete with time due to the invention of computational techniques and have been replaced by empirical algebraic equations and mechanical element approaches. The advantage of these approaches is that the elements might be relatable to structural features. This advantage might also apply to certain empirical algebraic equations. In general, all the models seem to be able to describe the rheological behaviour of bitumens satisfactorily if there are no major structural rearrangements with temperature and time, such as phase changes and, secondly, the tests are conducted within the LVE region. However, the behaviour of bitumen becomes more complex with the presence of waxy elements, high asphaltene contents and crystalline structures, as well as polymer modification, all of which can render a breakdown of the time– temperature superposition principle (TTSP). These materials are termed as thermorheologically complex. The introduction of a partial time–temperature superposition principle (PTTSP) appears to be as an alternative to the concept of thermorheologically complex behaviour of material.

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