Continuous relaxation and retardation spectrum method for ...

Mech Time-Depend Mater DOI 10.1007/s11043-011-9162-9

Continuous relaxation and retardation spectrum method for viscoelastic characterization of asphalt concrete Sudip Bhattacharjee · Aravind Krishna Swamy · Jo S. Daniel

Received: 16 March 2011 / Accepted: 20 October 2011 © Springer Science+Business Media, B. V. 2011

Abstract This paper presents a simple and practical approach to obtain the continuous relaxation and retardation spectra of asphalt concrete directly from the complex (dynamic) modulus test data. The spectra thus obtained are continuous functions of relaxation and retardation time. The major advantage of this method is that the continuous form is directly obtained from the master curves which are readily available from the standard characterization tests of linearly viscoelastic behavior of asphalt concrete. The continuous spectrum method offers efficient alternative to the numerical computation of discrete spectra and can be easily used for modeling viscoelastic behavior. In this research, asphalt concrete specimens have been tested for linearly viscoelastic characterization. The linearly viscoelastic test data have been used to develop storage modulus and storage compliance master curves. The continuous spectra are obtained from the fitted sigmoid function of the master curves via the inverse integral transform. The continuous spectra are shown to be the limiting case of the discrete distributions. The continuous spectra and the time-domain viscoelastic functions (relaxation modulus and creep compliance) computed from the spectra matched very well with the approximate solutions. It is observed that the shape of the spectra is dependent on the master curve parameters. The continuous spectra thus obtained can easily be implemented in material mix design process. Prony-series coefficients can be easily obtained from the continuous spectra and used in numerical analysis such as finite element analysis. Keywords Complex modulus · Continuous spectrum · Relaxation · Retardation · Integral transform · Master curve S. Bhattacharjee () Civil Engineering, Alabama A&M University, P.O. Box 367, Normal, AL 35762, USA e-mail: [email protected] A.K. Swamy · J.S. Daniel Civil Engineering, University of New Hampshire, Durham, NH 03824, USA A.K. Swamy e-mail: [email protected] J.S. Daniel e-mail: [email protected]

Mech Time-Depend Mater

1 Introduction The relaxation and retardation spectra, defined as the distribution of relaxation and retardation time, respectively, are the fundamental characteristics of a linearly viscoelastic material from which other viscoelastic functions and responses can be obtained (Tschoegl 1989). These functions describe the fundamental response behavior of the viscoelastic material and thus it is important to properly determine and characterize them from given experimental observations. There are several techniques available in the literature for numerical computation of the spectra from the experimental data which involve the computation of the spectral magnitudes at discrete relaxation or retardation time instances. The linear regression and collocation method (Schapery 1961) uses a discrete set of observed data points to solve for the unknown modulus and compliance values. However, it is an ill-posed problem, the solution is not unique and the quality of the solution depends on the spacing of the discrete points (Honerkamp and Weese 1989). Several authors have used Tichonov regularization method for better convergence of the solution (Honerkamp and Weese 1989, 1993; Elster et al. 1991). Baumgaertel and Winter (1989) used a nonlinear regression method where the number of relaxation modes was considered a variable and kept small to avoid the ill-posedness of the problem. The recursive method by Emri and Tschoegl (1993a, 1993b, 1995) involves selection of subset data from experimental observations, obtaining the spectra and improving the solution in subsequent iterations. Park and Kim (2001) presented the method of presmoothing of experimental data via power-law-series representation followed by fitting of Prony series to the pre-smoothed data. Liu (2001) developed nonnegative lest square method of obtaining the discrete positive spectrum directly from the creep data. Other approaches of obtaining discrete spectrum include the regularization method with quadratic programming (Ramkumar et al. 1997), Marquardt–Levenberg procedure (Mustapha and Phillips 2000), method of Bayesian analysis (Hansen 2008) and approximation of relaxation spectrum by means of piecewise cubic Hermite spline polynomials (Stadler and Bailly 2009). Instead of the discrete representation, the continuous form of the spectrum can be used. However, the extraction of the continuous spectrum using the integral relations between the spectrum and the experimental viscoelastic function involves inversion of the integrals in the complex plane and is generally a challenging problem. Bažant and Xi (1995) obtained the continuous retardation spectrum using inverse Laplace transform and higher order derivatives of the creep compliance function. Mun and Zi (2010) used similar approach in which frequency domain data were converted to time domain using approximate method, developing a master curve in time domain and then using the master curve function to obtain higher order derivative. Baumgaertel et al. (1990) proposed a continuous relaxation spectrum of linear flexible monodisperse polymers by observing the relaxation behavior. Since the continuous form of the spectrum represents the fundamental nature of the relaxation process in many materials, it will be useful to determine such spectrum from the experimental data. This paper presents a simple and practical approach to obtain the continuous relaxation and retardation spectra of asphalt concrete directly from the complex (dynamic) modulus test data under uniaxial loading condition. Asphalt concrete is a commonly used pavement material which displays viscoelastic behavior under wide range of temperature and frequency conditions. In this method, the continuous spectra are directly derived from the frequency domain master curves obtained from the experimental data. No interconversion between time and frequency domain is required. This method does not require any numerical optimization or regularization routine and provides the relaxation and retardation spectra as continuous functions of relaxation and retardation time directly from the observed dynamic


test data. This is expected to be a simple and straightforward approach since master curves are easily obtained from the standard frequency sweep viscoelastic test. The effects of master curve parameters on the spectra are discussed and the comparison between the continuous spectra and the existing approximate solutions is presented.

2 Characterization of linearly viscoelastic (LVE) behavior of asphalt concrete The mixture used in this study was prepared with 19 mm Nominal Maximum Aggregate Size granite aggregates with 4.5% PG 76-22 asphalt binder (PG ≡ performance grade) by mass. Mix design was performed according to the Superpave guidelines (Brian et al. 2010). The target air void was 7% ± 0.5%. The LVE frequency sweep testing for this study was done with a closed-loop servo-hydraulic system manufactured by Instron (model 8800). The testing apparatus included a loading fame with 20 kip hydraulic actuator and 5 kip load cell. An environment chamber was used to control the temperature which used liquid nitrogen for cooling and an electrical heating element for high temperature. The National Instrument data acquisition system with Labview™ was used to acquire the test data. Matlab™ was used subsequently to post process the acquired data and fit master curves. Several specimens were fabricated using the Superpave gyratory compactor and were cut and cored to 75 mm diameter and 150 mm tall specimens. Steel plates were glued to the ends of the uniaxial specimens using plastic epoxy glue in a gluing jig designed to align the specimen vertically. Four LVDTs spaced 90◦ apart around the circumference of each specimen were attached to the surface using a 100 mm gage length. Figure 1 shows the LVE characterization test set up in the environment chamber. The tests were conducted with temperatures between −10°C and 30°C at 10°C increment and the following frequencies: 0.1, 0.2, 0.5, 1, 2, 5, 10 and 20 Hz at each temperature. To keep the material behavior within the viscoelastic domain the applied load was controlled to keep the induced strain within 70 microstrain for each test. A rest period was provided between tests at two different Fig. 1 Dynamic frequency sweep test setup

Mech Time-Depend Mater Fig. 2 Complex modulus (|E ∗ |) at different temperatures as function of reduced frequency (fR )

frequencies to allow the material enough time to relax. The sinusoidal tension-compression stress wave with zero mean stress was applied. The acquired stress and strain data were post-processed using Matlab™ to fit sinusoidal curves and the phase angle was calculated from the fitted curves. The absolute value of the complex (dynamic) modulus was calculated as the ratio of the peak stress to the peak strain. The un-shifted complex modulus values determined at various temperatures are shown in Fig. 2. The dynamic modulus master curve was obtained by calculating the reduced frequency fR = aT × f , where aT is the shift factor, f is the loading frequency (in Hz), and shifting the individual curve at each temperature according to the time-temperature superposition principle. The validity of the time-temperature superposition principle for asphalt concrete has been demonstrated by many authors (e.g. Chehab et al. 2002). Figure 3 shows the complex modulus master curve and the shift factor as function of temperature. The straight line behavior of the ln(aT ) versus 1/T plot indicates the presence of Arrhenius type thermally activated process which is described via the equation: f = f0 exp(−Ea /RT ), where f is the frequency corresponding to a peak (e.g. the loss peak), f0 is a characteristic frequency, R (= 8.314 J K−1 mol−1 ) is the gas constant, T is the temperature (in K) and Ea is the activation energy (in J mol−1 ) (Ferry 1980). The activation energy was calculated from the slope of the ln(aT ) versus 1/T plot which resulted 230 kJ/mole. The complex tensile modulus (E ∗ ) is expressed in terms of the storage modulus (E ) and the loss modulus (E ) as in (1): E ∗ = E + j E ,

E = |E ∗ | cos φ,

E = |E ∗ | sin φ

and

tan φ = E /E , (1)

where φ is the phase angle and |E ∗ | is the magnitude of the complex modulus. The complex compliance (D ∗ ) and its real (storage compliance, D ) and imaginary (loss compliance, D ) parts were obtained using (2): D∗ =

1 = D − j D E∗

D = D =

(2a)

E 2

E + E 2

(2b)

E 2

E , + E 2

(2c)

Mech Time-Depend Mater Fig. 3 Complex modulus (|E ∗ |) master curve at 20°C reference temperature as function of reduced frequency (fR ) obtained from all specimens; sigmoid function coefficients: k1 = 1.262, k2 = 3.198, k3 = 0.928, k4 = 0.532, R 2 = 0.987; Inset: shift factor (aT ) as function of temperature (T ); ln(aT ) = 27.708 × 103 × (1/T ) − 94.355, R 2 = 0.999, T in K

Fig. 4 Storage modulus (E ) and storage compliance (D ) master curves as functions of reduced frequency (fR ) obtained from all specimens; E fitting coefficients: k1 = 1.436, k2 = 2.997, k3 = 0.726, k4 = 0.609, R 2 = 0.985; D fitting coefficients: k1 = −0.358, k2 = 4.147, k3 = 1.349, k4 = 0.430, R 2 = 0.987

√ where j = −1. Similar to |E ∗ |, the E and D master curves of the mix were also obtained from the observed data using (1) and (2) and are shown in Fig. 4. The master curves in Fig. 4 were obtained using the observed data from all specimens while Fig. 5 shows the master curves obtained from one specimen. The sigmoid function described in (3) has been used to model the |E ∗ |, E and D master curves: log χ = k1 ±

k2 , 1 + exp(k3 − k4 log fR )

(3)

where χ ≡ |E ∗ |, E or D . The coefficients ki (i = 1, 2, 3, 4) were determined from the experimental data. It should be noted that to have a complete description of the linearly viscoelastic relaxation behavior of asphalt concrete, a complex functional model of E ∗ = E ∗ (j ω) is required. However, it is difficult to obtain such a model for asphalt concrete. The complex functional models, such as Cole–Cole, Davidson–Cole and Havriliak–Negami, have been widely used to characterize the relaxation behavior of dielectric materials (Havriliak and Negami 1966). The observed mechanical relaxation behavior of asphalt concrete is different from the dielectric relaxation behavior and there is no complex functional form that can model the type of asphalt concrete behavior observed in this study. Thus a complete complex representation of E ∗ (as a function of j ω) is not always possible for asphalt


Fig. 5 (a) Storage modulus and (b) storage compliance master curves as functions of reduced frequency (fR ) obtained from testing one specimen; R 2 = 0.9985 for storage modulus and R 2 = 0.9991 for storage compliance

concrete. The coefficients ki and the shift factors at various temperatures were determined simultaneously by minimizing the error between the predicted and observed values. The opposite nature of the frequency dependency of the master curves shown in Figs. 3 and 4 is described via the ± sign in front of k2 , which indicates that log(χ ) is an increasing function of fR for |E ∗ | and E master curves, and is a decreasing function for D master curves. The positive sign in front of k2 is used when χ ≡ E or |E ∗ | and the negative sign is used when χ ≡ D . The master curves in Fig. 4 are used to determine the continuous relaxation and retardation spectra of asphalt concrete.

3 Continuous relaxation and retardation spectra of asphalt concrete The continuous relaxation spectrum H (τ ) as a function of the relaxation time τ , and the continuous retardation spectrum L(ρ) as a function of the retardation time ρ, can be defined in both time and frequency domain. In the time domain, H (τ ) and L(ρ) of linearly viscoelastic solid materials are defined via the relaxation modulus E(t), and the creep compliance D(t), respectively, according to (4a) (Tschoegl 1989): ∞ H (τ ) 1 − exp(−t/τ ) d ln τ, E(t) = Eg − −∞ ∞

D(t) = Dg +

−∞

L(ρ) 1 − exp(−t/ρ) d ln ρ.

(4a)

The spectra can also be defined in the frequency domain via the storage modulus and storage compliance function as in (4b): ∞ ∞ 1 1 H (τ ) d ln τ, D (ω) = D + L(ρ) d ln ρ. E (ω) = Eg − g 2τ 2 1 + ω 1 + ω2 ρ 2 −∞ −∞ (4b) In these equations H (τ ) d ln(τ ) can be interpreted as the contribution of the relaxation time between ln(τ ) and ln(τ ) + d ln(τ ) and similarly for L(ρ). The viscoelastic constants Eg and Dg are the glassy modulus and compliance, respectively, when ω → ∞ and t → 0. These viscoelastic constants are related to the spectra according to (4c): ∞ ∞ H (τ ) d ln τ = Eg − Ee , L(ρ) d ln ρ = De − Dg , (4c) −∞

−∞


where Ee and De are the equilibrium modulus and compliance, respectively, when ω → 0 and t → ∞. Equation (4c) indicates that the area under the curve H (τ ) versus ln(τ ) and L(ρ) versus ln(ρ) is normalized by dividing the area by (Eg − Ee ) and (De − Dg ), respectively. Equation (4b) indicates that the observed storage modulus and compliance depend on the relaxation and retardation spectrum. These equations are Fredholm integrals of first kind. The extraction of the true spectra directly from (4a) and (4b) is challenging and requires the inversion of the equations in the complex plane. Provencher (1982) developed a general purpose computer algorithm for inversion of noisy data using inverse Laplace transform, which was later used by researchers to obtain dielectric relaxation spectrum from experimental data (Alvarez et al. 1991). However, a careful study of the viscoelastic material functions in the frequency domain indicates that it is possible to extract the continuous functions H (τ ) and L(ρ) from the above equations if proper method of integral transforms are used. 3.1 The continuous spectra from the integral transform of viscoelastic functions The theory of integral transforms can be used to establish relations between various linearly viscoelastic functions and H (τ ) and L(ρ) can be extracted from those relations (Tschoegl 1989). In principle, H (τ ) and L(ρ) can be obtained from the complex modulus as follows: H (τ ) = ±π −1 Im E ∗ τ −1 exp(±j π) ,

L(ρ) = ∓π −1 Im D ∗ ρ −1 exp(±j π) ,

(5a)

where Im indicates the imaginary part of the quantity written right to it. Equation (5a) indicates that if the functional forms of E ∗ (j ω) and D ∗ (j ω) are available, H (τ ) and L(ρ) can be obtained by replacing the variable j ω in the respective complex function by τ −1 exp(±j π) for H (τ ) and ρ −1 exp(±j π) for L(ρ), and then retaining the imaginary part only. However, it is difficult to determine a complex functional form of the modulus and compliance directly from the experimental data for the asphalt concrete used in the study, i.e. to write both E ∗ and D ∗ as functions of j ω. On the other hand, the storage modulus and storage compliance as functions of reduced frequency, fR , are available from the experimental data by fitting master curves. A similar derivation (Tschoegl 1989, shown in Appendix A) yields the following form of H (τ ) and L(ρ) from the storage modulus and storage compliance functions, respectively: jπ jπ 2 2 H (τ ) = ± Im E τ −1 exp ± , L(ρ) = ∓ Im D ρ −1 exp ± . π 2 π 2 (5b) Equation (5b) indicates that if a functional form of E (ω) and D (ω) is available, H (τ ) and L(ρ) are obtained by replacing the variable ω in the expressions by τ −1 exp(±j π/2) for H (τ ) and ρ −1 exp(±j π/2) for L(ρ), and then retaining the imaginary part only. This approach has been used to determine the continuous relaxation and retardation spectra from the experimental data. 3.2 The continuous spectra from the storage modulus and storage compliance master curves The use of storage modulus master curve is particularly attractive since the coefficients of the function relate to several macroscopic properties of the materials, such as air voids, asphalt


content, gradation, etc. (NCHRP 2004). The following derivations use the generalized form of the master curves used in (3), which is re-written in (6): log χ (f ) = k1 ±

k2 1 + exp[k3 − k4 log f ]

or

ln χ (ω) = A ±

B , (6) 1 + exp(C − D ln ω)

where χ ≡ E or D , log f = log aT + log fobs and log aT = k5 − k6 T . Here, ω (rad/sec) or f (Hz) is the reduced frequency, aT is the temperature shift factor, T is the temperature and fobs is the testing frequency at a given temperature. In (6), the same equation is written with respect to logarithm of base 10 and natural logarithm (base e). To apply (5b) it is beneficial to use the natural logarithm instead of base 10. The coefficients A, B, C and D are related to the coefficients k1 , k2 , k3 and k4 as follows: A = 2.3026k1 , B = 2.3026k2 , C = k3 + 0.7982k4 , and D = 0.4343k4 . The positive sign in front of B is used when χ ≡ E and the negative sign is used when χ ≡ D . From (6), the storage modulus or storage compliance is given by B χ (ω) = exp(A) exp ± . (7) 1 + exp(C − D ln ω) In (7), ω is replaced by ω = τ −1 exp(±j π/2) for χ ≡ E , and by ω = ρ −1 exp(±j π/2) for χ ≡ D . Using (5b) the relaxation and retardation spectra are thus expressed as follows. Relaxation spectrum:

2 B H (τ ) = ± exp(A) Im exp (8a) π 1 + exp{C − D ln(τ −1 e±j π/2 )} 2 =± exp(A) Im exp Z(τ ) . (8b) π Retardation spectrum:

−B 2 exp(A) Im exp L(ρ) = ∓ π 1 + exp{C − D ln(ρ −1 e±j π/2 )} 2 =∓ exp(A) Im exp Z(ρ) . π

(8c) (8d)

In (8) Z is a complex quantity and can be further rearranged and simplified using complex algebra and the principal branch of the logarithm; the final form of Z is given by

X=

Z(τ ) = X(τ ) ± j Y (τ ),

(9a)

Z(ρ) = −X(ρ) ∓ j Y (ρ),

(9b)

B[e−a + cos(Dπ/2)] , e−a + ea + 2 cos(Dπ/2)

Y=

B sin(Dπ/2) e−a + ea + 2 cos(Dπ/2)

(9c)

where a = C + D ln τ , for relaxation spectrum, and

(9d)

a = C + D ln ρ, for retardation spectrum.

(9e)


Substituting these expressions into (8b) and (8d) and retaining only the imaginary part, following simplified forms of the relaxation and retardation spectra are obtained: H (τ ) = (2/π)eα1 sin β1 ,

and

α1 = α1 (τ ) = A + X(τ ), β1 = β1 (τ ) = Y (τ ),

L(ρ) = (2/π)eα2 sin β2 ,

(10a)

α2 = α2 (ρ) = A − X(ρ),

(10b)

β2 = β2 (ρ) = Y (ρ),

(10c)

where X and Y are given by (9c). These equations provide the relaxation and retardation spectra as continuous functions of relaxation and retardation time, respectively, using the storage modulus and storage compliance master curve functions. Therefore, given a set of experimental data on dynamic modulus test, the continuous form of the relaxation and retardation spectra can be obtained via (10). This can be easily performed using simple programs such as Excel™ spreadsheet or Matlab™. This approach has been used in this study to obtain the continuous H (τ ) and L(ρ) from the observed data. First, the storage modulus and compliance values are calculated from the observed complex modulus and phase angle data and sigmoid functions in the form of (6) are fitted to them to obtain the storage modulus and compliance master curves. The sigmoid functions are then used to determine the parameters α and β using (9) and (10). Finally the spectra are obtained from (10). The relaxation and retardation times vary as 0 < τ < ∞ and 0 < ρ < ∞. The two spectra are shown in Figs. 6 and 7, respectively, using (10). 3.3 Comparison with the discrete spectra Discrete spectra were generated from the observed data and compared with the continuous spectra to verify that the continuous spectra are the limiting case of the discrete spectra. The discrete relaxation spectrum can be described via the generalized Maxwell model characterized by n number of Maxwell elements in parallel. Similarly the discrete retardation spectrum is described via the generalized Voigt model with n number of Voigt elements in series. Each of the relaxation and retardation time is associated with a particular modulus and compliance value, respectively. When H (τ ) and L(ρ) are discrete distributions of τ and ρ, the integrals in (4b) and (4c) are transformed into (11a) and (11b), respectively: E (ω) = Ee +

Ek ω 2 τ 2 k , 1 + ω2 τk2 k

Eg = Ee +

D (ω) = Dg +

k

De = Dg +

Ek ,

k

Dk , 1 + ω2 ρk2

Dk ,

(11a)

(11b)

k

where Ek and Dk are the kth modulus and compliance associated with the kth relaxation and retardation times τk and ρk , respectively, k = 1, 2, 3, . . . , n. Equation (11a) is the Pronyseries representation of (4b). For comparison purpose, the Prony-series coefficients Ek and Dk were obtained using the collocation method (Tschoegl 1989). The storage modulus and compliance in (11a) are written in the matrix form: {P } = [M]{N }, where the matrices are given by Pj = E (ωj ) − Ee ,

Mj k =

ωj2 τk2 1 + ωj2 τk2

,

Nk = E k ,

k, j = 1, 2, . . . , n,

(12a)

Mech Time-Depend Mater Fig. 6 Continuous relaxation spectrum H (τ ) from the storage modulus master curve; inset: the continuous spectrum as limiting case of the discrete spectrum with different numbers of Maxwell elements (n)

Fig. 7 Continuous retardation spectrum L(ρ) from the storage compliance master curve; inset: the continuous spectrum as limiting case of the discrete spectrum with different numbers of Voigt elements (n)

Pj = D (ωj ) − Dg ,

Mj k =

1 , 1 + ωj2 ρk2

Nk = D k ,

k, j = 1, 2, . . . , n.

(12b)

The relaxation and retardation times are chosen at decade intervals. The number of decades and number of elements can be chosen in such a way that the whole spectrum of the frequencies is obtained. The Matlab™ optimization toolbox was used to obtain the solution for Ek and Dk with the condition {N } ≥ 0 and (11b). Then solution of (12a) and (12b) provides the values of Ek s and Dk s. The values of Ek and Dk depend on the choice of the values of τk and ρk , respectively. To consider the effect of particular choice of τk and ρk , data were sampled at three different decade intervals: 2 decades, 1 decade and 0.5 decade. Thus, for the master curves shown in Fig. 4, three sets of values of Ek and Dk were obtained. Figure 6 inset shows the three discrete distributions of relaxation time where the Prony coefficients have been plotted against the relaxation times. The continuous relaxation spectrum obtained from (10) is also shown on the same plot as the continuous line. Similar plot is shown in Fig. 7 inset for the retardation time. It is observed from these figures that when the relaxation and retardation time intervals are larger (i.e. fewer number of elements), a very distinct discrete distribution deviating largely from the continuous one is obtained. The discrete spectrum approaches to the


continuous spectrum as the number of elements in Maxwell and Voigt models is increased. This confirms the validity of the continuous spectra obtained in (10). It is to be noted that extensive computational effort, time and software packages are required to extract Prony-series coefficients or the discrete distribution from the observed data as discussed in Introduction. On the other hand the continuous form of the spectrum (10) obtained from dynamic test data can be more advantageous in this aspect. There are some software packages available for this purpose (Baumgaertel and Winter 1989; Winter and Mours 2005). The following discussions focus on the continuous model of the spectra in (10) which can be easily obtained from a simple LVE characterization test of asphalt concrete.

4 Effect of master curve parameters on the spectral properties The parameters k1 and k2 of the sigmoid function (see (6)) are associated with the asymptotic values of the storage modulus and compliance. The parameter k3 determines the location of the inflection point of the curve and k4 corresponds to the steepness of the curve; thus both k3 and k4 affect the frequency range and the shape of the master curve. Therefore it is important to observe the effect of the parameters k3 and k4 on the spectra. To determine the effect, a set of master curves were generated with different values of k3 and k4 while keeping k1 and k2 constant. Figure 8 shows the generated master curves where the lines 1 → 2 → 3 correspond to increasing k3 and the lines 1 → 4 → 5 to increasing k4 , all other parameters remaining constant. The effect on the relaxation spectrum is shown in Fig. 9, and their effect on the retardation spectrum is shown in Fig. 10. For both storage modulus and storage compliance master curves the increase in k3 shifts the inflection point of the curves to the left, i.e. towards the lower frequency range. This results the shift of both H (τ ) and L(ρ) towards the higher τ and ρ as seen in Figs. 9 and 10. However, the width (spread) of the distribution does not change as the width of the dominating frequency range does not change due to change in k3 . The situation is different for k4 where the increase in k4 decreases the frequency range of the master curves and thus reduces the contributing relaxation and retardation time range in the spectra. Thus the increase in k4 also increases the peak value of the spectra since the area under the curve remains constant according to (4c). It is evident that the shape of the distribution function (the peak value and the spread) is affected by both k3 and k4 . Fig. 8 Storage modulus (E ) (increasing with frequency) and storage compliance (D ) (decreasing with frequency) master curves as functions of reduced frequency (fr ) for various values of k3 and k4 while k1 and k2 remain constant

Mech Time-Depend Mater Fig. 9 Effect of the parameters k3 and k4 on the relaxation spectrum with constant k1 and k2

Fig. 10 Effect of the parameters k3 and k4 on the retardation spectrum with constant k1 and k2

To quantify the change in the shape of the spectra, statistical quantities of the spectra can be calculated. Using the normalized form for the spectra, the αth moment of the logarithm of the relaxation or retardation time can be defined using (4c) as (Tschoegl 1989; Zorn 2002): ∞ (ln x)α F (x) d ln x, (13) (ln x)α = K −1 −∞

where x represents relaxation or retardation time, F (x) represents the corresponding spectrum and K represents (Eg − Ee ) or (De − Dg ). The average logarithmic relaxation or retardation time ((ln x)avg ) is thus obtained by setting α = 1: (ln x)avg = ln x , and the variance (σln2 x ) and the skewness (γ1 ln x ) of the spectra are obtained as σln2 x = (ln x)2 − ln x 2 , γ1 ln x =

(ln x)3 3ln x ln x 3 − − 3 . σln x σln3 x σln x

(14)

(15)

(16)

Mech Time-Depend Mater Fig. 11 Effect of k3 and k4 on the average relaxation time, (ln τ )avg , and the average retardation time, (ln ρ)avg

Fig. 12 Effect of k3 and k4 on the variance of the relaxation 2 , and the time distribution, σln τ variance of the retardation time 2 distribution, σln ρ

Numerical integrations have been performed to evaluate the moments in (14), (15) and (16). The results are shown in Figs. 11, 12 and 13 for different values of k3 and k4 . The change in k3 has no effect on the variance (spread) and skewness of the spectra. However, the increase in k3 increases the average logarithmic relaxation and retardation time linearly. Therefore the overall effect of k3 is to shift the distribution function to the right (as seen in Figs. 9 and 10). In case of relaxation spectrum, the increase in k4 increases the average logarithmic relaxation time, decreases its variance and makes the distribution more negatively skewed. In case of retardation spectrum, the increase in k4 decreases the average logarithmic relaxation time, decreases its variance and makes the distribution more positively skewed. Thus in general the parameter k4 affects both skewness and width of the distribution.

5 Time-domain viscoelastic functions from the spectra The continuous spectra in (10) can be used for computation of relaxation modulus or creep compliance via (4a). The integrals in (4a) are replaced by summations which are equivalent

Mech Time-Depend Mater Fig. 13 Effect of k3 and k4 on the skewness of the relaxation time distribution, γ1 ln τ , and the skewness of the retardation time distribution, γ1 ln ρ

Fig. 14 Relaxation modulus (curves decreasing with time) and creep compliance (curves increasing with time) computed from the continuous spectra (see (10)) with various values of parameters k3 and k4

to Prony-series representations of E(t) and D(t): E(t) = Eg −

n

Ek 1 − exp(−t/τk ) ,

k=1

D(t) = Dg +

n

Dk 1 − exp(−t/ρk ) , (17a)

k=1

Ek = H (τk )d ln τk ,

Dk = L(ρk )d ln ρk ,

(17b)

where ln(τk ) and ln(ρk ) vary from −∞ to +∞. The relaxation modulus and creep compliance computed from the continuous spectra are shown in Fig. 14 for the values of k3 and k4 used in the model master curves in Fig. 8. The curves in Fig. 14 display the direct relationship between the spectra and the time-domain functions as expected. The shifting of the spectra towards higher relaxation and retardation time domain due to the increase in k3 (Figs. 9 and 10) results in shifting of E(t) and D(t) parallel to the time axis towards higher time domain. The reduction of spectral width due to the increase in k4 results in shorter time regime of these functions (i.e. steeper curves).


6 Comparison with approximate solutions The validity of the continuous form of the spectra was determined by comparing them with the spectra obtained from approximate methods. Tschoegl (1989) used the differential operator method to develop approximate solutions for H (τ ) and L(ρ) for several orders of ˆ accuracy. The second-order approximations of H (τ ) and L(ρ) (denoted by Hˆ (τ ) and L(ρ), respectively) are given by dE 1 d 2 E Hˆ (τ ) = ± , d ln ω 2 d(ln ω)2 1/ω=τ √2 or τ/√2

(18a)

dD 1 d 2 D ˆ L(ρ) =− ∓ . (18b) d ln ω 2 d(ln ω)2 1/ω=ρ √2 or ρ/√2 √ The positive sign and 1/ω = τ 2 √ is used when the slope of H -vs-ln ω is positive. Similarly, the negative sign and 1/ω = ρ 2 is used when the slope of L-vs-ln ω is positive. This method is applied to the data from the master curves shown in Fig. 4. The approximate relaxation spectra are calculated from the storage modulus and storage compliance according to (18), and are compared with the continuous form from (10). Figure 15 shows the plots of the continuous and approximate solutions for H (τ ), and the inset shows the error between them. The approximate spectrum from (18) matches very well with the continuous spectrum (solid line). The maximum absolute error between the two types of spectrum shown in Fig. 15 is about 4%, which appears near the peak of the distribution. The comparison for the retardation spectrum is shown in Fig. 16, which shows the maximum absolute error of about 15% appearing near the initial lower magnitude portion of the distribution; the error near the peak is around 1%. The relaxation spectrum obtained from the storage modulus was compared against that obtained from the loss modulus. The following first-order approximation was used to compute the relaxation spectrum from the loss modulus master curve: dE (ω) ˆ , H (τ ) = (2/π) E (ω) ± d ln ω 1/ω=τ √3 or τ/√3 Fig. 15 Comparison between the continuous and the approximate relaxation spectrum; inset shows the associated error

(18c)

Mech Time-Depend Mater Fig. 16 Comparison between the continuous and the approximate retardation spectrum; inset shows the associated error

Fig. 17 Comparison between the relaxation spectrum obtained from the storage modulus and the loss modulus master curves; the arrows indicate the range of observed data

√ where the positive sign and 1/ω = τ 3 is used when the slope of H -vs-ln ω is positive. The comparison is shown in Fig. 17 which indicates that the two forms match well within the observed data range (indicated by the arrows in the figure). For a complete match, data from a wider range of frequencies, especially from higher frequencies, need to be collected. However, mechanical cyclic load testing above 20/25 Hz produces undesirable vibrations in the specimens which produce noise in the data. Therefore tests were conducted up to 20 Hz. Next, the relaxation modulus and creep compliance functions calculated from the continuous spectra are compared with the ones obtained from the approximate spectra. Equations (17a) and (17b) are used to calculate E(t) and D(t) from the continuous and the approximate spectra. Sufficiently large value of n can be selected to obtain the accurate values of the time-domain functions. Figure 18 shows the comparison for E(t) where the maximum error is less than 0.4% and Fig. 19 shows the comparison for D(t) with maximum error less than 2.5%. In general, Figs. 15–19 show very good comparison between the continuous form and the approximate form of the spectra and the material functions derived from them.

Mech Time-Depend Mater Fig. 18 Comparison of the relaxation modulus E(t) derived from the continuous spectrum (see (10)) and the approximate spectrum (see (18a)); inset shows the associated error

Fig. 19 Comparison of the creep compliance D(t) derived from the continuous spectrum (see (10)) and the approximate spectrum (see (18b)); inset shows the associated error

7 Summary and conclusions The continuous forms of the relaxation and retardation spectra of asphalt concrete from LVE test data have been developed. The storage modulus master curve has been used to obtain the relaxation spectrum via integral transform and complex algebra. The storage compliance master curve has been used in a similar way to obtain the retardation spectrum. The relaxation and retardation spectra thus obtained are the continuous functions of the relaxation and retardation time. The major advantage of this method is that the continuous form is directly obtained from the master curves which are readily available from the standard LVE tests. The continuous form of the spectra does not require sophisticated numerical computations and they match very well with the discrete spectra as the limiting case. The calculated relaxation modulus and creep compliance functions from the spectra also matched very well with the approximate solutions. The continuous functions can be easily used in the discretized form, e.g. Prony series. The master curve properties such as the location of the inflection point and the frequency range affect the shape of the distribution functions. Since these master curve parameters are directly related to the material parameters such as the air void percentage (Rowe et al. 2009), grain size distribution and asphalt binder percentage, the continuous form of the spectra provides a quick way to evaluate the material properties.


The simple form of the spectra allows easy implementation in the material mix design and evaluation process. Acknowledgement The financial support to this study was provided by the National Cooperative Highway Research Program (NCHRP). The guidance from Dr. M.A. Alim (Electrical Engineering) on understanding the relaxation mechanisms of dielectric materials is specially acknowledged.

Appendix A Introducing the change of variable τ = λ−1 in H (τ ), one obtains the following relations: H1 (λ) = H (λ−1 ) = H (τ )|τ =λ−1 , where H1 is same as H with the variable τ replaced by λ−1 . Then, the complex modulus is given by E ∗ (ω) = Eg −

∞

H (τ ) −∞

1 d ln τ = Eg − 1 + j ωτ

∞

H1 (λ) 0

1 dλ, λ + jω

(A1)

which can be re-written in the following way: Eg − E ∗ (ω) =

∞

H1 (λ) 0

1 dλ = FLT H1 (λ); j ω = H¯ 1∗ (ω) = H¯ 1 (ω) − j H¯ 1 (ω) λ + jω (A2)

where FLT represents the Fourier–Laplace Transform of the function H (λ−1 ), which is denoted by H¯ 1∗ (ω). Again changing the variables, τ 2 = λ−2 = ξ −1 , and ω2 = v in (A2) and equating the real part from both sides, we obtain Eg − E (ω) = H¯ 1 (ω) =

∞

0

1 λH1 (λ) dλ = λ2 + ω 2 2

∞ 0

√ H1 ( ξ ) dξ . ξ +v v=ω2

(A3)

√ Replacing H1 ( ξ ) by a function φ(ξ ), (A3) can be written in the following form: 2 Eg − E (ω) ω=√v =

∞ 0

φ(ξ ) ¯ dξ = STI φ(ξ ); v = φ(v), ξ +v

(A5)

√ where φ(ξ ) = H1 ( ξ ) = H1 (λ) = H (λ−1 ) = H (τ )|τ =1/λ = H (τ )|τ =1/√ξ and STI is the Stieltjes transform of φ(ξ ). Inversion of the above equation provides the following: ¯ φ(ξ ) = ∓π −1 Im φ¯ ξ exp(±j π) = ∓π −1 Im φ(v)| v=ξ exp(±j π ) −1 = ∓2π Im Eg − E (ω) ω=√v=√ξ exp(±j π/2) 2 jπ =± ξ exp ± Im E . π 2 Since

(A6a)

(A6b)

√ ξ = τ −1 and φ(ξ ) = H (τ )|τ =1/√ξ , (A6b) leads to jπ 2 Im E τ −1 exp ± . H (τ ) = ± π 2

Similar derivations with D result in L(ρ) in (5b).

(A7)


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