Application of the theory of viscoelasticity to evaluate the resilient modulus test in asphalt mixes

Construction and Building Materials 149 (2017) 648–658

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Application of the theory of viscoelasticity to evaluate the resilient modulus test in asphalt mixes Luciano Pivoto Specht a,⇑, Lucas F. de A.L. Babadopulos b, Hervé Di Benedetto b, Cédric Sauzéat b, Jorge Barbosa Soares c a b c

Graduate Program in Civil Engineering, Universidade Federal de Santa Maria, Brazil École Nationale des Travaux Publics de l’État, Université de Lyon, France Graduate Program in Transportation Engineering, Universidade Federal do Ceará, Brazil

h i g h l i g h t s ⁄


A continuum spectrum linear viscoelastic model (2S2P1D) is calibrated with complex modulus (E ) data.
A linear viscoelastic analytical method is proposed to simulate indirect tensile resilient modulus (IT RM) tests.
Numerous IT RM test (different T and f) results are compared with simulation.
Necessary hypotheses for the analysis of IT RM tests are thoroughly discussed.
Bituminous pavement analyses show the importance of reliable stiffness measurement.

a r t i c l e

i n f o

Article history: Received 17 January 2017 Received in revised form 19 April 2017 Accepted 5 May 2017

Keywords: Viscoelasticity Indirect tensile resilient modulus Complex modulus Asphalt pavement analysis

a b s t r a c t This paper deals with the analysis of indirect tensile resilient modulus (IT RM) tests for asphalt mixes. In Brazil, for instance, asphalt mix stiffness is obtained from a classical elastic analysis of the referred test, which may not be suitable for a viscoelastic material. A method is proposed to compare IT RM results from classical experimental analysis and from tests simulated with viscoelastic models. Four different asphalt mixes with four different binders were analyzed using IT RM tests, with varying loading time and temperatures. Complex modulus (E⁄) tests were performed to characterize the linear viscoelastic behavior of the investigated materials on a wide range of temperature and frequency. 2S2P1D model and general Kelvin Voigt (GKV) model were then fitted to the E⁄ data. IT RM tests results were calculated using an analytical method and the viscoelastic GKV models obtained for the four mixes. Differences found for the stiffness results obtained from the distinct procedures are discussed, and pavement analyses showed that such differences substantially affect the structural response. This is to be expected as a consequence of the missing of fundamental considerations on the material behavior. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Mechanistic pavement design requires estimation of stress and strain fields on the pavement structure. Such fields acting in the pavement structure are used to estimate material progressive damage with load repetitions (time) and, consequently, pavement performance [59,60,35,44,38]. Pavement analysis requires each layer material’s stiffness and Poisson’s ratio.

⇑ Corresponding author. E-mail addresses: [email protected] (L.P. Specht), [email protected] (Lucas F. de A.L. Babadopulos), [email protected] (H. Di Benedetto), [email protected] (C. Sauzéat), [email protected] (J.B. Soares). http://dx.doi.org/10.1016/j.conbuildmat.2017.05.037 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.

In Brazil, pavement materials characterization is usually performed in laboratory using triaxial tests for soils and unbound materials, and indirect tensile modulus tests for asphalt and stabilized mixes. The indirect tensile resilient modulus test (IT RM) is derived from the indirect tensile strength test, developed in Brazil in the 1940s for concrete [11], and subsequently adapted to stiffness measurements in the 1970s and 1980s [50,48,49]. For the stiffness measurement, solicitation is cyclic (periodic repetitions), and one cycle is composed of a loaded part (usually considered as a triangular or a haversine function, [34] and an unloaded part (rest period). For asphalt mixes, the test is commonly performed at 25 °C and 0.1 s of loading (commonly considered as equivalent to a loading frequency of 10 Hz) [8] and 0.9 s of rest.

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Indirect tensile tests are relatively simple to perform and are considered by some authors to be effective at characterizing bituminous mixtures’ stiffness and fatigue behaviors [54]. It is also less costly and allows testing of field cored and lab samples [9,55,51]. However the use of a single temperature and a single loading frequency neglects the viscoelastic (time or frequency dependency) nature of the asphalt mix and its temperature susceptibility, inherited from the asphalt binder. Different procedures and sample sizes may conduct to different results (in Brazil, Marshall specimens are used, presenting 100 mm diameter and 63.5 mm height). The viscoelastic nature of asphalt mixes in pavement design methods is considered in countries such as France since the 1960s [52,31] and in the United States since the 1990s [1], where the complex modulus (E⁄) test is commonly used. In Brazil, the current official asphaltic pavement design method is based on North American empirical procedures (USACE and the AASHO Road Test) adapted in the 1960s [53]. The sublayer materials are characterized by the California Bearing Ratio (CBR), and the procedure was proposed in a completely different scenario with respect to roadway loads and available laboratory equipment. A national research and technical effort involving several universities, the national authority for highway administration (DNIT) and the national oil company (Petrobras) should result shortly in the release of a new Brazilian pavement design guide, with mechanistic considerations. Despite the fact that it will be a more advanced method, in its basic level, asphalt mix stiffness may be determined from the indirect tensile resilient modulus (IT RM) test at 25 °C. A systematic improvement of the method is planned, with future considerations for viscoelasticity in the pavement analysis. In that context, the present paper intends to apply the theory of viscoelasticity to model the IT RM test in asphalt mixes and to investigate the impact of the use of different experimental procedures in structural analysis. A thorough discussion on the test analysis is presented. The plan for this work contains the following steps: a) performing IT RM and E⁄ tests for 4 different asphalt mixes, prepared using Brazilian commercial unmodified and modified binders (description in Section 2 and results in Sections 2.3 and 3.3); b) modeling viscoelastic behavior of the mixes using E⁄ results, with 2S2P1D continuum spectrum model and Generalized Kelvin-Voigt model (GKV) (Section 3); c) obtaining RM values from viscoelastic IT RM test simulation, using analytical solution for GKV model and comparing with experimental data (Section 4); d) evaluating the difference between modeled and experimental RM results, for a typical layered structure of asphalt pavement (Section 5).

Elastomeric), and TLAFlex (AC 50/70pen + 25% TLA + 4.5% SBS). Those binders present the following Superpave specifications [2,4], respectively: PG 58-16, PG 70-22, PG 76-22, and PG 76-10. The asphalt mixes were designed according to Superpave using 19 mm dense mix grading curve [20]. The binder content was determined targeting 4.0% air voids for 100 gyrations on the Superpave Gyratory Compactor. The binder content was: AC 50/70pen, 5.4%; TLA, 5.9%; SBS 60/85E, 5.6%; and TLAFlex, 5.9%. Samples used for IT RM and E⁄ tests were prepared stopping the gyratory compaction to obtain 5.5% voids in the asphalt mix, considering that this is the mean air voids presented in the field in the beginning of service life.

2. Experimental investigation

e ðixtÞ ¼ e0 eiðxtþue Þ

2.1. Materials The materials employed for this study are commonly used in pavement applications in Brazil. The aggregate (materials and grading curve) in all asphalt mixes was the same, while each asphalt mix contained a different binder. The mineral aggregate was obtained from a volcanic rock (Dacite). Coarse aggregate (>4.76 mm) presents a soundness test [16] result of 0.3%, Los Angeles abrasion [18] of 10.26%, apparent specific gravity of 2.828, bulk specific gravity of 2.689 and water absorption of 1.83% [19]. Fine aggregate presents a sand equivalent [17] of 68.74%, and apparent specific gravity [15] of 2.597. Four binders were selected for the investigated asphalt mixes. There is one unmodified (AC 50/70pen) and three modified binders: TLA – Trinidad Lake Asphalt (AC 50/70pen + 25% TLA, by total weight of binder), SBS 60/85E (AC 50/70pen + 4.5% StyreneButadiene-Styrene; ring an ball > 60 °C and elastic recovery > 85%,

2.2. Stiffness tests Fig. 1 presents the device and a scheme of the loading used for the tests. It is to be observed that IT RM tests are conducted using loading in a perpendicular direction with respect to the sample compaction direction [5,21], while E⁄ tests are conducted in the same direction as compaction [3]. Sample’s geometry is cylindrical with 100 mm diameter in both cases, and approximately 63.5 mm height in the case of IT RM and 150 mm height in the case of E⁄. RM test was conducted on 10, 25 and 35 °C, and 10 Hz, while E⁄ was performed on 4, 20 and 40 °C, and 0.1, 0.5, 1, 5, 10 and 25 Hz. 3. Viscoelastic behavior modeling (E⁄ Tests) 3.1. Viscoelastic parameters definition Under small strain hypothesis, many viscoelastic materials such as asphalt mixes can be considered to behave linearly [22,14]. Although nonlinear effects may be present in the behavior of asphalt mixes even without damage [42,12,37,7], linear viscoelastic (LVE) hypothesis can still be used as a first approximation to describe viscoelastic materials response. Different methods can be used to identify linear viscoelastic parameters. Some lead to properties on the time domain (relaxation modulus and creep compliance) and others on the frequency domain (complex modulus – E⁄). To characterize LVE properties in the frequency domain, complex modulus tests were performed for different temperatures and frequencies. The complex modulus is obtained from sinusoidal stress and strain measurements in a steady state, which are expressed in complex form as in Eqs. (1) and (2).

r ðixtÞ ¼ r0 eiðxtþur Þ

ð1Þ ð2Þ 2

where i is the imaginary number (i ¼ 1), r0 is the amplitude for stress and e0 for strain, ur the phase angle for stress and ue for strain, and x the angular frequency. Complex modulus is then obtained from stress and strain as in Eqs. (3) and (4).

E ¼ jE jeiuE with jE j ¼ r0 =e0 and uE ¼ ur  ue

ð3Þ

E ¼ E1 þ i:E2 with E1 ¼ jE j cos uE and E2 ¼ jE j sin uE

ð4Þ

3.2. Viscoelastic modeling In order to model the results of complex modulus tests, the 2S2P1D (2 springs, 2 parabolic elements and 1 dashpot) model was used [43] (cf. Fig. 2f and g). This model presents a continuous viscoelastic spectrum. It can be applied to asphalt binders, mastics and mixes [43]. It gives analytically the complex modulus (E ), at a given temperature, of a material from a set of 7 constants (Eq. (5)).

L.P. Specht et al. / Construction and Building Materials 149 (2017) 648–658

Displacement

Load 0

b) Actuator

tp

Actuator

Loading strip

tc t (s)

Sample

LVDT (x2)

(10cmx6.35cm)

/

Axial Strain Amplitude

a) Po

Axial Stress Amplitude

650

T=2 /

t (s)

Top cap

Sample (10cmx20cm)

LVDT (x3)

Loading strip Bottom cap

Fig. 1. Test device for resilient (a) modulus (IT RM) tests (b) and complex modulus (E*) tests (b).

E

a)

e) For i=1 to n

1D Diff. Eq.:

E0

η

b) 1D Diff. Eq.:

c)

ηi

ηn

Ei

En

E0-E00

log aT ¼ C 1

h

h

E00

1D Fractional Diff. Eq.:

d)

In order to capture the change in behavior due to temperature, the Time-Temperature Superposition Principle (TTSP) has been used, considering a WLF (Williams-Landel-Ferry) equation [58]. The shift factor aT is expressed in Eq. (6) using two constants C 1 and C 2 .

f)

For i=1 to n

Ei

En

ηi

ηn

sðTÞ ¼ aT ðTÞ  sðT ref Þ

η

E0

g)

DðtÞ ¼ Dg þ Fig. 2. Mechanical analogs – a) spring, b) dashpot, c) parabolic element, d) GKV, e) GM, f) 2S2P1D – and g) 2S2P1D model interpretation in Cole-Cole plot.

E0  E00 1 þ dðixsÞ

k

þ ðixsÞ

h

ð7Þ

In order to obtain the analytical response of the model in the time domain, an analytical function for relaxation modulus or creep compliance is needed and it is not available from 2S2P1D model due to its mathematical complexity [43,47]. Therefore, a discretization of the model was performed. It is obtained using the Generalized Kelvin-Voigt (GKV) model. Any number of elements can be used: a high number in order to decrease the difference between the models or a lower number to simplify calculation. The least squares procedure to fit a generalized Kelvin-Voigt model to the 2S2P1D model is described in detail by Tiouajni et al. [56]. The GKV model (cf. Fig. 2d) gives analytically the creep compliance D(t) as in Eq. (8).

τi=ηi/Ei

E ðixÞ ¼ E00 þ

ð6Þ

The characteristic time (s) only depends on the temperature. It is expressed in function of aT as in Eq. (7).

k

ρi=ηi/Ei

ðT  T ref Þ C 2 þ ðT  T ref Þ

1

þ ðixbsÞ

ð5Þ

The two elastic constants, E00 and E0 , define minimum and maximum values for the norm of complex modulus. The maximum value of the norm of complex modulus can be obtained experimentally from complex modulus tests at low temperatures or from ultrasonic wave propagation tests [24,41,32]. The two parabolic elements constants drive the slopes of the Cole-Cole plot for low (h) and high (k) frequencies. The linear dashpot imposes a right angle for a frequency tending to zero. b is a dimensionless parameter that is proportional to the linear dashpot viscosity (g ¼ bðE0  E00 Þs) and it affects complex modulus mainly at low equivalent frequencies. The parameter d works as a shape factor. The parameter s is a characteristic time.

n X Dj ð1  et=sj Þ

ð8Þ

j¼1

In Eq. (8), Dg represents the glassy compliance (obtained when time of loading tends to zero), Dj and sj define respectively the compliance (inverse of the modulus constant of the spring) and the characteristic time (ratio between dashpot’s viscosity and spring’s modulus) of the Voigt element j, and j represents each of the n Voigt elements associated in series (cf. Fig. 2d). The set of elastic constants Dj associated with the set of corresponding characteristic times (sj ) form what is called a discrete viscoelastic spectrum [28]. Theoretically, an infinite quantity of viscoelastic elements would be needed to represent the LVE behavior of a material or, alternatively, a continuous viscoelastic spectrum. Fig. 2 presents elements (spring, dashpot, parabolic element in Fig. 2a, b and c, respectively) and models used for viscoelasticity: GKV model, Generalized Maxwell (GM) model and 2S2P1D model in Fig. 2d and f. It can be shown that GKV and GM models are equivalent and interconvertible [45,56]

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3.3. LVE characterization and modeling results

4. Viscoelastic analysis of IT RM test

Figs. 3 to 6 present the viscoelastic characterization results from E⁄ tests for the 4 tested asphalt mixes in a Cole-Cole plot (Fig. 3), Black diagram (Fig. 4), and master curves for |E⁄| (Fig. 5) and phase angle (u) (Fig. 6), respectively. The Cole-Cole diagram expresses the relation between the real part (E1 = |E⁄|cosu) and the imaginary part (E2 = |E⁄|sinu) of the material complex modulus, while the Black diagram expresses the relation between |E⁄| and u. Simulated curves for each of the tested materials, also plotted in the figures, were obtained using the 2S2P1D model (Eq. (5)). The fitted model constants are presented in Table 1 for each of the tested materials. The existence of a unique curve in Cole-Cole and Black diagrams validates the Time-Temperature Superposition Principle (TTSP) for all mixes evaluated in this paper, meaning that they can then be considered as ‘‘thermorheologically simple” [28]. Therefore, WLF equation (Eq. (6)) can be used to obtain results for different temperatures. It is possible to observe that the lower maximum values of E2 (Cole-Cole plot in Fig. 5) and of u (Black diagram in Fig. 6) indicate that the viscous portion of the behavior of the mix with polymermodified binder is relatively less important when compared to the unmodified ones (there is relatively less dissipated energy under loading). Unmodified mix’s u reaches a maximum close to 45°, which is more than 7° higher than the maximum u of mixes with SBS 60/85E. These results may be associated to a positive effect of TLA and SBS addition on asphalt mixes that is observed in field applications. Simulated curves for each material, also plotted in the figures, were obtained using the 2S2P1D model (Eq. (5)) and the GKV model (Eq. (8)). The fitted model constants are given in Table 1 for each material. In the Appendix, Table A.1. presents the fitted GKV model parameters with 35 viscoelastic elements used for calculations for each of the materials studied in this paper. It is seen on Black and Cole-Cole diagrams that some fluctuations are obtained in the measurement of complex modulus. Those fluctuations are related to the accuracy of the determination of phase angle after stress and strain signals analyses. The use of a fundamental material model with continuous viscoelastic spectrum (such as 2S2P1D, used in the paper) mitigates those fluctuations and produces accurate description of material behavior. After that (which can be seen as a fundamental pre-smoothing of the experimental curves, as sometimes presented in the literature), it is possible to obtain the parameters of a viscoelastic model simpler to use for viscoelastic analysis in the time domain, such as the GKV model used in the paper for IT RM test simulation and calculation of RM.

4.1. Linear elastic solution of IT RM test The IT RM tests were first analyzed in a classical manner using the linear elastic analytical solution by Hondros [33], including the effect of the loading strips. Eq. (9) is originated from the solution of the elastic problem associated to the indirect tensile (as the IT RM) test, for linear elastic, isotropic, homogeneous material loaded in the direction of a diameter using load strips, which keep the ratio of 0.125 of the strip’s width with respect to the sample’s diameter.

D2R ¼

P ðt þ 0:2699Þ Eh

In Eq. (9), D2R is the displacement between the most external points following a perpendicular diameter with respect to the loading line. The load is represented by P. E is the material Young’s modulus (for the elastic material, 1=E is its compliance), h is the cylinder’s height, and t is the Poisson’s ratio. Eq. (9) is the basis for the calculation of RM (which plays the role of Young’s modulus in the equation). 4.2. Viscoelastic solution from linear elastic solution of IT RM test The solution of the correspondent linear viscoelasticity problem (same geometry and boundary conditions) has been determined by applying the Elastic-Viscoelastic Correspondence Principle (EVCP). That principle states that the solution for the linear elasticity problem is the same as the one for the linear viscoelasticity problem in Laplace-Carson transformed space [13]. In order to apply the EVCP, the Poisson’s ratio of the linear viscoelastic material has been considered as a constant. For a viscoelastic medium, Eq. (9) turns into Eq. (10) after applying the EVCP [57,6].

D2R ðtÞ ¼

ðt þ 0:2699Þ h

Z

t

Dðt  uÞ  0

@PðuÞ du @u

AC 50/70 @4°C AC 50/70 @20°C AC 50/70 @40°C TLA @4°C TLA @20°C TLA @40°C 60/85E @4°C 60/85E @20°C 60/85E @40°C TLAFlex @4°C TLAFlex @20°C TLAFlex @40°C GKV AC 50/70pen GKV TLA GKV 60/85E GKV TLAFlex 2S2P1D AC 50/70pen 2S2P1D TLA 2S2P1D 60/85E 2S2P1D TLAFlex

3500 3000 2500 2000 1500 1000 500 0 0

5000

10000

ð10Þ

In order to solve Eq. (10) and simulate IT RM tests, the necessary inputs are the cylindrical sample’s height h, material properties t (constant) and DðtÞ (creep compliance), and the loading PðtÞ. Such loading function, PðtÞ, presents a loading period for a time interval of tp and a rest period completing the cycle repetition with an interval of t c (cf. Fig. 1a). The loading period can be modeled by a haversine with a peak value of P 0 , while the rest period presents load equal to zero, as represented by Eq. (11).

4000

E2(MPa)

ð9Þ

15000

20000

25000

30000

35000

E1(MPa) Fig. 3. Experimental values (from E* tests) and simulation with GKV and 2S2P1D (parameters in Table 1) on Cole-Cole plot.

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L.P. Specht et al. / Construction and Building Materials 149 (2017) 648–658

100000

|E*| (MPa)

10000

1000

100

10

AC 50/70 @4°C

TLA @4°C

60/85E @4°C

TLAFlex @4°C

AC 50/70 @20°C

TLA @20°C

60/85E @20°C

TLAFlex @20°C

AC 50/70 @40°C

TLA @40°C

TLAFlex @40°C

60/85E @40°C

GKV AC 50/70pen

GKV TLA

GKV 60/85E

GKV TLAFlex

2S2P1D AC 50/70pen

2S2P1D TLA

2S2P1D 60/85E

2S2P1D TLA Flex

1 0

10

20

ϕ (o)

30

40

50

Fig. 4. Experimental values (from E* tests) and simulation with GKV and 2S2P1D (parameters in Table 1) on Black diagram.

25000 AC 50/70pen TLA 60/85E TLA Flex GKV AC 50/70pen GKV TLA GKV 60/85E GKV TLAFlex 2S2P1D AC 50/70pen 2S2P1D TLA 2S2P1D 60/85E 2S2P1D TLAFlex

|E*| (MPa)

20000

15000

10000

5000

0

0,0001

0,001

0,01

0,1

1

10

100

1000

10000

Reduced frequency (Hz) Fig. 5. Master curves of |E*| (reference temperature of 20 °C): experimental data (from E* tests) and simulation with GKV and 2S2P1D (parameters in Table 1).

50

AC 50/70pen TLA 60/85E TLA Flex GKV AC 50/70pen GKV TLA GKV 60/85E GKV TLAFlex 2S2P1D AC 50/70pen 2S2P1D TLA 2S2P1D 60/85E 2S2P1D TLAFlex

45 40 35

ϕ (0)

30 25 20 15 10 5 0 0.0001

0.001

0.01

0.1

1

10

100

1000

10000

Reduced frequency (Hz) Fig. 6. Master curves of u (reference temperature of 20 °C): experimental data (from E* tests) and simulation with GKV and 2S2P1D (parameters in Table 1).

Table 1 2S2P1D model parameters (Equation (5)), C1 and C2 WLF equation parameters (Equation (6)) at a reference temperature of 20 °C. E*

Mix

AC 50/70 pen TLA SBS 60/85E TLAFlex

WLF

E00 (MPa)

E0 (MPa)

k

h

d

sE (s)

b

C1

C2 (°C)

75 90 130 110

24,000 29,000 19,300 23,500

0.315 0.235 0.290 0.250

0.660 0.570 0.559 0.570

2.70 2.60 2.10 2.10

0.06 0.15 0.08 0.09

100 200 90 500

20.72 30.40 20.86 20.28

151.47 210.44 151.54 151.30

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L.P. Specht et al. / Construction and Building Materials 149 (2017) 648–658

PðtÞ ¼

P0 2

h



 1  cos 2tpp  t

i

if 0 6 t < t p if t p 6 t < t c

0

where D2Ri ðtp Þ is the displacement corresponding to the viscoelastic element i at the end of the loading period (t ¼ t p ). From Eqs. (12), (13) and (14), it is possible to analytically simulate the result of IT RM tests with an isotropic, homogeneous, linear viscoelastic material in terms of the displacement D2R between the most external points within a perpendicular diameter with respect to the loading line. RM results can be obtained from that solution in an approximate way. Fig. 7 represents a simulated test result for one cycle (load from Eq. (11) and displacement from Eqs. (12), (13) and (14)).

ð11Þ

The GKV model (Eq. (8)) was used to obtain the creep compliance DðtÞ. Results of E⁄ tests were fitted using the 2S2P1D model. Parameters of GKV model with 35 elements were then obtained from the procedure explained in Section 3.2. From those assumptions, the solution of Eq. (10) for a material whose mechanical behavior is represented by a GKV model leads to Eqs. (12), (13) and (14) [57]; [6]. For 0 6 t < t p :

(  )   n X P0 Dg 2p D2R ðtÞ ¼ ðt þ 0:2699Þ t þ p Di F i ðt; t p ; si Þ 1  cos h 2 tp i¼1

4.3. Determination of RM from measurements and LVE analysis The measured RM values have been obtained from the tests and calculated using Eq. (15), directly deduced from the linear elastic solution.

ð12Þ where

F i ðt; t p ; si Þ ¼

   1 2p 1  cos t 2p tp      2 t 2ps 2p tp 2p t  sin t þ 2 2 i 2 e si þ cos tp 2psi tp 4p si þ tp

RM ¼

For t p 6 t < tc : n tp t X D2Ri ðtp Þ  e si

ð15Þ

where D2Rr represents the measured resilient displacement, defined following the standard or the procedure adopted herein [21]. For this paper, resilient displacement is the difference between the peak value of the displacement and the value of displacement at the end of a given cycle [21]. As done from experimental results, the calculated RM values have been calculated using Equation (15) and the linear viscoelastic simulation results, from which the value of D2Rr was deduced.

ð13Þ

D2R ðtÞ ¼

P0 ðt þ 0:2699Þ D2Rr  h

ð14Þ

i¼1

7.E-06

0.35

(m)

Displacement -

tc=1s

tp=0.1s

6.E-06

0.30 Displacement - Eq. 10

5.E-06

0.25 Displacement - Eq. 12

4.E-06

0.20 AC 50/70pen mix (2S2P1D model in Table 1) (GKV model parameters in Annex) temperature= 35°C IT RM loading frequency = 10Hz

3.E-06 2.E-06

0.15 0.10

1.E-06

Load - P (kN)

(

0.05

0.E+00

0.00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t (s) Fig. 7. Simulated IT RM test result with an isotropic linear viscoelastic material (Kelvin-Voigt model were obtained from 2S2P1D model fitted on E* results for AC50/70 mix. 2S2P1D parameters are given in Table 1, and GKV parameters are given in Table A.1 in the Appendix) for one cycle at T = 35 °C.

20000 T=10oC

RM Measured= 1.29RM Calculated R² = 0.93

18000

RM Measured (MPa)

16000 Line of equality

14000 12000 T=25oC

10000 8000 6000 T=35oC

4000 2000

AC 50-70pen @5Hz

AC 50-70pen @10Hz

AC 50-70pen @25Hz

TLA @5Hz

TLA @10Hz

TLA @25Hz

60/85E @5Hz

60/85E @10Hz

60/85E @25Hz

TLAFlex @5Hz

TLAFlex @10Hz

TLAFlex @25Hz

0 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

RM Calculated (MPa) Fig. 8. Comparison between calculated and measured RM values (points represent mean values, while error bars represent one standard deviation, obtained from three replicates).

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L.P. Specht et al. / Construction and Building Materials 149 (2017) 648–658

Fig. 8 presents a comparison between the results of the application of the theory of viscoelasticity to model the IT RM test in asphalt mixes (Eqs. (11)–(14)) and the measured values at the same conditions of temperature and frequency. Error bars (one standard deviation long) for the measured RM values (y-axis), obtained from three replicate tests, are presented in Fig. 8. It can be noticed that the measured values are approximately 29% higher, than the calculated ones, in average. In the literature, it is possible to find various reasons likely to contribute for the difference between calculated and experimental results. Those reasons are qualitatively discussed hereafter. The first possible reason for differences in results is the variability of the experimental measurements. However, it would be

expected that random error to produce results above and below the line of equality, and not a systematic difference as it is observed. Another reason is related to test non-homogeneity of stress and displacement fields in the specimen during IT RM tests (each point presents different states of stress and strain), due to test configuration (that would happen even if the material was perfectly homogeneous). Since that happens, a structural solution (strain fields estimated from test geometry, boundary conditions and a constitutive law for the material) for the test is needed prior to the analysis of its results. For that, supplementary hypotheses about constitutive behavior are needed [22,23], such as that of isotropic linear viscoelasticity, constant Poisson’s ratio, and no damage (as assumed in this paper). Using the structural solution, only the indirect access to the constitutive behavior of the material is possible (choice of better model parameters that fit experiments, even if the model is not suitable for the tested material). Still due to test geometry, horizontal displacement measured on a diameter of the specimen is used to estimate strain field within the sample (which is given from the structural solution for the test). The horizontal strain in the test (D2R ðtÞ=ð2RÞ) is a mean strain, averaged over the horizontal diameter. In fact, the strain at each point is different. If the material behavior is strain-dependent, for example, the simplification hypotheses used to analyze test results do not apply. Fig. 9 shows schematically how strain (from elastic solution) changes in the horizontal diameter as a function of position (RM value for that example was 5000 MPa and Poisson’s ratio was considered to be 0.35). Average horizontal strain is approximately half of the strain in the midpoint of the sample. From Fig. 9, it can be seen that changes in the test itself could be done to obtain a more consistent result, such as testing larger specimens and taking displacement measurements near their center.

100

2500

Fig. 9. Horizontal strain distribution and D2R =ð2RÞ for two possible LVDT positions (isotropic linear elastic calculation – [33] – with RM of 5000 MPa and Poisson’s ratio of 0.35).

80

2000

60

1500

40

1000

20

500

Load - P (N)

Load

2

(2 ) (μm/m)

Extension Strain

0

0 0 to 5s

100 to 105s

200 to 205

300 to 305s

400 to 405s

Time (s) 30

2400

2

Load

25

2000

20

1600

15

1200

10

800

5

400

0

Load - P (N)

(2 ) (μm/m)

Extension Strain

0 0

1

2

3

4

5

Time (s) Fig. 10. Typical measurements on IT RM test (Fig. 1a) at 25 °C with external measurements of displacements (0.1 s loading – P 0 ¼ 2kN – and 0.9 rest for each cycle; cylindrical sample with 10.16 cm diameter and 6.35 cm high).

655

However, the other problems presented in the paper for this test would still remain, as well as the lack of capturing temperature susceptibility. Samples preparation (compaction) may also be a factor for the observed differences. Compaction tends to produce mixes with aggregate particles presenting a preferential orientation, producing anisotropy. Di Benedetto et al. [26] tested samples extracted from different directions of compaction in plates and observed that, for the tested material, the modulus obtained in the direction perpendicular to compaction could be approximately 10% higher than the one for the direction of compaction for high equivalent frequencies (TTSP shifted frequencies at a given temperature). The same has been observed by Di Benedetto et al. [24]. So, as in this paper E⁄ tests are performed in the direction of compaction and IT RM tests are performed in the direction perpendicular to compaction, measured value for RM may be higher than the calculated one. However, when testing asphalt mixes obtained from gyratory compaction, Di Benedetto et al. [24] observed that specimens presented the modulus obtained in the direction perpendicular to compaction about 10% lower than the one for the direction of compaction for high equivalent frequencies, which is the inverse of the general observations in the present paper. According to Di Benedetto et al. [24], the change in anisotropic behavior is due to a different geometrical organization of the granular skeleton. Similar conclusion was presented by Ezaoui and Di Benedetto [27] from results on dry sand prepared using different compaction methods. Another important observation is that stiffness measurements need to be conducted under small strains domain (of the order of a maximum 150 lm/m peak-to-peak strain amplitude, as defined in Fig. 1) to avoid structural changes in the material. Fig. 10 presents a typical result (average horizontal strain DD/D and applied load) of an IT RM test. A zoom over the first 5 cycles is presented. It can be seen that strain accumulates during the test, and that should be limited to small values respecting linear viscoelasticity limits.

5. Pavement structural analyses In order to evaluate the differences between calculated and measured RM, pavement structural analyses are performed. Fig. 11 presents the cross-section geometry for a pavement structure analyzed in this paper. Two results are evaluated in detail: the extension strain at the bottom of the asphalt layer (commonly associated to fatigue life) and the contraction strain at the top of the subgrade (commonly associated to rutting). The tool used for

Extension strain - RM measured (μm/m)

L.P. Specht et al. / Construction and Building Materials 149 (2017) 648–658

400 AC 50-70pen @5Hz AC 50-70pen @10Hz AC 50-70pen @25Hz TLA @5Hz TLA @10Hz TLA @25Hz 60/85E @5Hz 60/85E @10Hz 60/85E @25Hz TLAFlex @5Hz TLAFlex @10Hz TLAFlex @25Hz

350 300 250 200 150

Line of equality y = 0.77x R² = 0.94

T=35oC

100 T=25oC

50 T=10oC

0

0

100

200

300

400

Extension strain - RM calculated (μm/m) Fig. 12. Comparison between horizontal extension strain in the bottom of the asphalt layer as obtained using calculated RM and measured RM values.

the structural analysis was AEMC (Multilayer Elastic Analysis Software – programa de Análise Elástica de Múltiplas Camadas) [29]. AEMC uses Burmister [10] solution for multilayer continuum systems under the following hypotheses: a) materials are homogeneous, isotropic and linear elastic; b) layers weight is negligible; c) layers are infinite in horizontal directions; d) layers present a finite thickness, except the subgrade, which is semi-infinite; e) upper layer is statically loaded in the vertical direction only in the loaded area; f) load area (tire contact) is circular; g) stress and displacements are nil at great distances from the load; h) interface between surface course and base layer may be considered either as fully bonded or as perfectly sliding. The analysis was performed for each result of RM (measured or calculated), in different conditions of temperature and loading frequency (in order to simulate different traffic speeds). The layer interface was considered as fully bonded. Four wheels axle load, represented in Fig. 11 (half axle), is modeled by 80 kN vertical force uniformly distributed on rigid circular area of radius equal to 0.108 m. The distance between the 2 wheels (from the center of their contact with the pavement) is 32.4 cm. Fig. 12 presents a comparison between the horizontal extension strain in the bottom of the asphalt layer calculated considering simulated RM results and measured RM data. Fig. 13 presents a comparison between the vertical contraction strain on the top of the subgrade calculated considering calculated RM results and measured RM data. It is possible to observe the impact of the RM value used in the analysis, and also the impact of using a calculated

Axle load = 80kN Wheel load = 20kN Tire inflation pressure = 0.56MPa

Asphalt mix

= 0.30

VARIABLE RM

16cm

Granular base

= 0.35

RM = 200MPa

15cm

21cm

Granular sub-base

= 0.35

RM = 250MPa

Semi-infinite subgrade

= 0.45

RM = 80MPa

Fig. 11. Analyzed pavement structure.

Fig. 13. Comparison between vertical contraction strain on the top of the subgrade as obtained using calculated RM and measured RM values.

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L.P. Specht et al. / Construction and Building Materials 149 (2017) 648–658

or a measured value. When it comes to analyzing the importance of the temperature in the pavement behavior, clear differences are presented in Figs. 12 and 13. While at 10 °C the observed extension strain in the bottom of the asphalt layer is approximately 75 lm/m, it is approximately 300 lm/m (4 times higher) at 35 °C, a common temperature for asphalt pavements. For the contraction strain on the top of the subgrade, the results varied from approximately 175 lm/m at 10 °C to 375 lm/m at 35 °C. When it comes to the effect of loading speed (frequency), for the tested range of IT RM test loading frequencies (5 to 25 Hz), the effect is less important than temperature, but still not negligible. That change in frequency may be associated to a change in vehicle speed from 11 to 55 km/h, using correlations from the literature [30,46]. Finally, when it comes to the differences between structural analysis results using calculated or measured values of RM, the extension strain in the bottom of the asphalt layer is 23% lower using measured values than using calculated values, while for the contraction strain on the top of the subgrade that difference is 13%. It is clear that considering only stiffness results at 25 °C and 10 Hz for pavement analysis is an oversimplification that cannot represent asphalt pavement behavior. Many authors recommend the viscoelastic analysis of pavement structures [39,23,44,25], acknowledging the importance of load-time history effects [36]. However, reliable viscoelastic material properties must be used, obtained from experimental tests capable of capturing material behavior with acceptable and as few as possible simplification hypotheses. In addition, the effects of temperature and frequency should also be considered for the fatigue behavior of asphalt mixes [40] when designing asphalt pavements.

lems in using the classical analysis of the IT RM test for bituminous mixes, which are viscoelastic. The second contribution is to highlight the consequences of the missing of fundamental considerations on the material behavior on pavement analysis. IT RM tests and cyclic uniaxial E⁄ tests were performed for four asphalt mixes. Their linear viscoelastic behaviors (represented by E⁄) were modeled using a continuous spectrum model available in the literature (2S2P1D). The Elastic-Viscoelastic Correspondence Principle was then applied to simulate the viscoelastic solution of IT RM test. For the different tested conditions (temperatures from 10 to 35 °C and loading period frequencies from 5 to 25 Hz), IT RM values obtained from experiments were about 30% higher in average than what was observed from simulation. A typical pavement structure was analyzed using stiffness results from both simulated and experimental RM to evaluate not only the effect of simulation versus experiments, but also to evaluate the pavement behavior sensitivity to temperature and loading frequency (associated to traffic speed). The horizontal extension strain in the bottom of the asphalt layer and the vertical contraction strain at the top of the subgrade are approximately 23% and 13% lower for the analysis using the measured RM values. As discussed in this paper, some reasons linked with the elastic analysis of IT RM tests may explain the differences between measured RM and calculated RM, with focus on test procedures, interpretation and limitations of the linear viscoelastic analysis of the IT RM tests. That justifies efforts towards developing methods to characterize the fundamental viscoelastic behavior of asphalt mixes.

Acknowledgments 6. Conclusions Recent efforts for developing the new Brazilian asphaltic pavement design method indicate that viscoelasticity will eventually be incorporated, which is also true for other countries following the mechanistic-empirical path for pavement design. This paper investigates the indirect tensile resilient modulus test (IT RM). The first contribution of the paper is to clearly present the theoretical prob-

The authors would like to thank the Brazilian agencies CAPES (BEX 2573/15-6 and BEX 13551/13-2) and CNPq (302677/20151) for the research grants and Rede Temática de Asfalto/Petrobras.

Appendix A

Table A.1 GKV model parameters (with 35 viscoelastic elements) at 35 °C for the four studied mixes (2S2P1D model and WLF equation parameters in Table 1) used for calculation. AC 50-70pen

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

TLA

SBS 60/85E

TLAFlex

si (s)

Di (MPa1)

si (s)

Di (MPa1)

si (s)

Di (MPa1)

si (s)

Di (MPa1)

2.90E-17 1.00E16 3.45E16 1.19E15 4.11E15 1.42E14 4.89E14 1.69E13 5.82E13 2.01E12 6.93E12 2.39E11 8.24E11 2.84E10 9.81E10 3.39E09 1.17E08 4.03E08 1.39E07 4.80E07 1.65E06 5.71E06 1.97E05

1.05E08 4.64E09 6.36E09 1.13E08 1.46E08 2.38E08 3.26E08 5.10E08 7.23E08 1.10E07 1.59E07 2.39E07 3.49E07 5.21E07 7.66E07 1.14E06 1.69E06 2.52E06 3.75E06 5.64E06 8.53E06 1.31E05 2.04E05

2.55E17 8.80E17 3.04E16 1.05E15 3.61E15 1.25E14 4.30E14 1.48E13 5.12E13 1.77E12 6.09E12 2.10E11 7.25E11 2.50E10 8.63E10 2.98E09 1.03E08 3.55E08 1.22E07 4.22E07 1.46E06 5.02E06 1.73E05

5.92E08 3.42E08 3.39E08 3.85E08 6.08E08 7.16E08 1.06E07 1.31E07 1.87E07 2.38E07 3.31E07 4.30E07 5.89E07 7.75E07 1.05E06 1.40E06 1.90E06 2.54E06 3.46E06 4.69E06 6.45E06 8.94E06 1.26E05

2.81E17 9.68E17 3.34E16 1.15E15 3.98E15 1.37E14 4.73E14 1.63E13 5.63E13 1.94E12 6.70E12 2.31E11 7.98E11 2.75E10 9.50E10 3.28E09 1.13E08 3.90E08 1.35E07 4.64E07 1.60E06 5.53E06 1.91E05

1.83E08 9.34E09 9.03E09 1.89E08 2.08E08 3.65E08 4.50E08 7.24E08 9.52E08 1.46E07 1.99E07 2.96E07 4.14E07 6.08E07 8.62E07 1.26E06 1.81E06 2.64E06 3.84E06 5.67E06 8.40E06 1.27E05 1.93E05

3.22E17 1.11E16 3.83E16 1.32E15 4.56E15 1.57E14 5.43E14 1.87E13 6.47E13 2.23E12 7.70E12 2.66E11 9.16E11 3.16E10 1.09E09 3.76E09 1.30E08 4.48E08 1.55E07 5.33E07 1.84E06 6.34E06 2.19E05

4.36E08 2.50E08 2.21E08 3.46E08 4.39E08 6.31E08 8.26E08 1.16E07 1.55E07 2.15E07 2.89E07 3.98E07 5.39E07 7.42E07 1.01E06 1.39E06 1.90E06 2.63E06 3.63E06 5.08E06 7.17E06 1.03E05 1.50E05

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L.P. Specht et al. / Construction and Building Materials 149 (2017) 648–658 Table A.1 (continued) AC 50-70pen

24 25 26 27 28 29 30 31 32 33 34 35 g

TLA

SBS 60/85E

TLAFlex

si (s)

Di (MPa1)

si (s)

Di (MPa1)

si (s)

Di (MPa1)

si (s)

Di (MPa1)

6.80E05 2.34E04 8.09E04 2.79E03 9.63E03 3.32E02 1.15E01 3.95E01 1.36E+00 4.71E+00 1.62E+01 5.60E+01

3.26E05 5.36E05 9.18E05 1.65E04 3.12E04 6.29E04 1.34E03 2.72E03 4.01E03 2.71E03 7.75E04 3.95E04 4.17E05

5.98E05 2.06E04 7.12E04 2.46E03 8.47E03 2.92E02 1.01E01 3.48E01 1.20E+00 4.14E+00 1.43E+01 4.93E+01

1.82E05 2.72E05 4.19E05 6.72E05 1.13E04 1.97E04 3.59E04 6.78E04 1.30E03 2.34E03 3.13E03 2.76E03 3.45E05

6.58E05 2.27E04 7.83E04 2.70E03 9.32E03 3.22E02 1.11E01 3.83E01 1.32E+00 4.55E+00 1.57E+01 5.42E+01

3.01E05 4.78E05 7.79E05 1.30E04 2.24E04 3.95E04 7.14E04 1.28E03 1.99E03 1.89E03 6.59E04 1.41E04 5.18E05

7.55E05 2.61E04 8.99E04 3.10E03 1.07E02 3.69E02 1.27E01 4.39E01 1.52E+00 5.23E+00 1.80E+01 6.23E+01

2.26E05 3.50E05 5.61E05 9.36E05 1.62E04 2.91E04 5.36E04 9.78E04 1.63E03 2.11E03 1.72E03 1.36E03 4.26E05

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