A new approach for determination of the coefficient of thermal expansion of asphalt concrete

Measurement 85 (2016) 222–231

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A new approach for determination of the coefficient of thermal expansion of asphalt concrete Tsung-Chin Hou a,⇑, San-Je Huang b,c, Chi Hsu c a

Department of Civil Engineering, National Cheng Kung University, Taiwan Materials Testing Laboratory, Directorate General of Highways, Taiwan c Department of Civil Engineering, National Kaohsiung University of Applied Sciences, Taiwan b

a r t i c l e

i n f o

Article history: Received 22 October 2014 Received in revised form 30 October 2015 Accepted 19 February 2016 Available online 27 February 2016 Keywords: Coefficient of thermal expansion Asphalt concrete Strain gauge Climate change Civil infrastructure Transportation system

a b s t r a c t A new approach for determining the coefficient of thermal expansion (CTE) of asphalt concrete (AC), based on thermoelasticity and using a distributed thermocouple and strain gauge apparatus, is proposed. The thermoelastic model for an instrumented hollow cylindrical AC specimen was established. Experiments were conducted with six scenarios of thermal flux for simulating different levels of heating on AC pavement. Three approaches for interpreting the CTE according to the readings of the thermocouple and strain gauge are provided, with different CTE measurements expected and observed. Terms associated with strain measuring accuracy derived from thermoelasticity are defined. The thermal calibration of instrument readings was shown to be necessary for obtaining accurate CTE readings for AC specimens. The CTE readings obtained in this study showed strong agreement with values reported in the literature. The proposed measuring approach is shown to be simple to install and theoretically founded. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Global warming indirectly influences the quality and service life of transportation infrastructure. For example, preliminary reports indicate that climate change has caused noticeable impacts on the Qinghai-Tibet Highway [1]. The growing thermal absorption of asphalt pavement has led to an increased deterioration of roads. Mills et al. argued that anthropogenic climate change challenges the assumption of a static climate on which current design criteria have primarily relied [2]. Asphalt-based materials absorb heat and moisture at a faster rate than other construction materials such as concrete do, and the viscoelastic nature that governs their mechanical properties is highly temperature sensitive. This may lead to significant changes in material moduli, strengths and some others, which ultimately alter the performance of asphalt under

⇑ Corresponding author at: No. 1 University Road, Tainan 701, Taiwan. http://dx.doi.org/10.1016/j.measurement.2016.02.035 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.

regular service loading [3,4]. Several studies have indicated that the deterioration rates of pavement are altered by climate change, and the associated factors that affect the material properties should be considered accordingly [2–5]. The coefficient of thermal expansion (CTE) is one of the primary parameters in designing the quality and performance of asphalt concrete (AC) pavement. Numerous studies have measured the CTE of metals and alloys [6]; however, relatively scant effort has been exerted in measuring the CTE of AC materials. The approaches for measuring the CTE of AC in laboratories include dilatometry [7–9], optical heterodyne interferometry [10–13], digital image correlation (DIC) [14–16], and strain gauge measurement [17–22]. For dilatometry, the test chamber in a push-rod dilatometer system generally consists of a tubed thermal drive such as a furnace, cryostat, or liquid bath for uniformly heating or cooling the sample material at a controlled rate along its longitudinal dimension. However, measuring the CTE by using a dilatometer is a delicate and

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demanding task, rendering it more suited for the materials of a scientific laboratory than for a typical experimental stress analysis facility [20]. Optical heterodyne interferometry has been shown to be capable of attaining the highprecision requirements for measuring the CTE of dimensionally stable materials at room temperature [10,12,13]. However, such a high-precision apparatus is considered excessively expensive for AC because its CTE is generally two to four digits. DIC is an emerging noncontact approach capable of sensing small strains that are then used for computing the CTE of solid materials [14–16]. The noncontact feature enables DIC to be performed under severe conditions (e.g., over 600 °C) [14,15]; however, it usually experiences convergence difficulties, particularly when a small strain measurement is required. Strain gauges have been widely used for mechanical tests of materials. They are applicable to various types of matter, geometry, and anisotropy and do not require expensive instruments, making them an economic alternative for measuring CTEs [20–22]. Ratanawilai et al. compared the CTEs of printed circuit boards measured using Moiré interferometry and an electric resistance strain gauge. They reported that consistent CTE measurements can be obtained with the difference being less than 0.8 ppm/°C [11]. De Strycker et al. investigated the CTEs of S235, SS304, and SS409 stainless steels measured using DIC and strain gauges, and similar results at temperatures lower than 120 °C were obtained [14]. All these preliminary studies have suggested that strain gauges are suitable for accurately measuring the CTE of AC over its operating temperature range in the field (room temperature to 120 °C). In addition to being economical and accurate, the strain gauge approach is preferred over other approaches because it requires less operation space for small strain measurements of objects. This study employed a strain gauge apparatus to measure the CTE of AC based on thermoelasticity theory. A simplified thermoelasticity model (hollow cylinder) was first derived. Similar to ASTM dilatometry [7], the use of hollow cylinders in this study enables us to uniformly heat the sample material at a controlled rate along its longitudinal dimension, and thus simplify the investigation of CTE of AC. The thermal probe and multi-strain-gauge experimental set-up was established and testing procedures were conducted, and the measured thermal strains of AC hollow cylinder specimens were then calibrated accordingly and used for interpreting the corresponding CTE and the coefficient of thermal contraction (CTC). Furthermore, the results obtained were compared with values from the literature to determine their validity and applicability. It should be mentioned that bonding strain gauges to nonhomogenous materials such as AC requires sophisticated preparations including the selection of proper gauge length, proper use of the adhesives, and prior processing of the bonding surface (typically, polishing).

2. Theoretical models The thermoelasticity models used in this study were primarily for objects with a hollow circular cylindrical

geometry with finite longitudinal (z) and radial (r) boundaries. Several assumptions were made as per rational considerations: 1. Cylindrical coordinates (r, h, z) and displacement fields (u, v, w) are used. 2. Materials are homogeneous, isotropic, and linear elastic, and the properties are temperature independent. 3. The surface forces (if any) along the longitudinal direction are constant; in other words, stresses and strains are independent of the longitudinal coordinate. 4. Torques are not present on the two ends of the cylinder; the shear stresses and shear strains in the longitudinal direction are zero (i.e., srz = shz = crz = chz = 0). 5. Temperatures are constant along the longitudinal direction; the thermal field is merely a function of the radius (i.e., T = T(r)). 6. Displacements are functions of the radial coordinate only; the generalized plane strain model can be simplified as a one-dimensional plane-strain problem (i.e. ez = e0 = constant). 7. The stresses are below the yield limits. 8. The initial thermal field is constant (T = TR, where TR is the reference temperature), and thermal stresses are analysed on the basis of prescribed boundary conditions T(ri) = Ti and T(ro) = To. 2.1. Thermal conduction of circular hollow cylinders For a circular hollow cylinder that enables only onedimensional (radial coordinate) steady-state heat transfer (Fig. 1), the equation of the thermal field in cylindrical coordinates can be expressed as [23]


 1 d dT r ¼ 0: r dr dr

ð1Þ

By integrating Eq. (1) and applying the boundary conditions T(ri) = Ti, T(ro) = To, the temperature distribution can be obtained as follows:

DT ¼ DT o  ðT o  T i Þ where DT = T  TR, temperature.

ln ðr=r o Þ ln ðr i =r o Þ DTo = To  TR,

ð2Þ and

TR = reference

2.2. Thermal strains of circular hollow cylinders For the one-dimensional steady-state plane-strain (ez = e0) thermoelasticity problem, the displacement fields can be simplified as u = u(r). Furthermore, by assuming that no forces are present on the inner and outer surfaces (i.e., free surfaces, rr(ri) = rr(ro) = 0), thermal stresses can then be expressed as [23]

"

aEðT o  T i Þ ln ðr=ro Þ ðro =rÞ2  1 rr ¼  2ð1  lÞ ln ðr i =ro Þ ðr o =ri Þ2  1

# ð3aÞ

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Fig. 1. Circular thick-wall hollow cylinder with surface temperatures and steady-state heat transfer.

"

#

aEðT o  T i Þ 1 ln ðr=ro Þ ðro =rÞ2 þ 1 rh ¼ þ þ 2ð1  lÞ ln ðri =ro Þ ln ðri =ro Þ ðro =ri Þ2  1

er ¼ ð3bÞ

"

rz ¼

ð3cÞ In Eq. (3), rr, rh, and rz are the radial, circumferential, and longitudinal stresses, respectively; l is the Poisson’s ratio; a is the CTE; and E is the modulus of elasticity. At the outer surface (r = ro) of the cylinder, the thermal stresses are obtained as follows:

ðrr Þr¼ro ¼ 0

aEðT o  T i Þ 1 1 ¼ þ ð1  lÞ 2 ln ðr i =r o Þ ðr o =r i Þ2  1 "

ðrz Þr¼ro ¼

eh ¼

ð1 þ lÞðr o =rÞ2  ð1  3lÞ ðro =r i Þ2  1

þ

2ð1  lÞðT o  T R Þ ðT o  T i Þ

# ð4bÞ

ð1 þ lÞðr o =rÞ2 þ ð1  3lÞ ðro =r i Þ2  1

þ

2ð1  lÞðT o  T R Þ ðT o  T i Þ

"

aEðT o  T i Þ 1 1 : þ ð1  lÞ 2 ln ðr i =r o Þ ðr o =r i Þ2  1

ð4cÞ

As depicted in Eqs. (4b) and (4c), it is clear that ðrh Þr¼ro ¼ ðrz Þr¼ro . Thermal strains can also be derived using Hooke’s law:

ð5aÞ

1 E

ð5bÞ

1 E

ð5cÞ

eh ¼ ½rh  lðrr þ rz Þ þ aDT ez ¼ ½rz  lðrr þ rh Þ þ aDT ¼ e0 :

ð6bÞ #

ð6cÞ Similarly, the thermal strains at the outer surface (r = ro) are then obtained:

!

al 1 2 ðT o  T i Þ ¼  ð1  lÞ ln ðr i =ro Þ ðr o =ri Þ2  1 þ aðT o  T R Þ

ðeh Þr¼ro ¼ a

ð7aÞ

! 1 1 ðT o  T i Þ þ aðT o  T R Þ þ 2lnðri =ro Þ ðro =r i Þ2  1 ð7bÞ

1 E

er ¼ ½rr  lðrh þ rz Þ þ aDT

#

aðT o  T i Þ 1  l 2ð1  lÞ 2ð1  lÞðT o  T R Þ : þ þ 2ð1  lÞ lnðr i =ro Þ ðro =r i Þ2  1 ðT o  T i Þ

ðer Þr¼ro

#

ð6aÞ



þ

ez ¼

#

aðT o  T i Þ 1  l ð1 þ lÞ ln ðr=r o Þ  2ð1  lÞ ln ðr i =ro Þ ln ðr i =r o Þ

ð4aÞ "

ðrh Þr¼ro



#

aEðT o  T i Þ 1 2 ln ðr=r o Þ 2 þ þ : 2ð1  lÞ ln ðri =ro Þ ln ðr i =r o Þ ðr o =r i Þ2  1



aðT o  T i Þ 2l ð1 þ lÞ ln ðr=r o Þ  2ð1  lÞ ln ðr i =ro Þ ln ðr i =r o Þ

!

ðez Þr¼ro ¼ a

In Eq. (5), er, eh, and ez are the radial, circumferential, and longitudinal strains, respectively. By substituting Eq. (3) into (5), the thermal strains of the hollow cylinder can be expressed as:

1 1 ðT o  T i Þ þ aðT o  T R Þ: þ 2lnðr i =ro Þ ðro =r i Þ2  1 ð7cÞ

According to Eq. (7), ðeh Þr¼ro ¼ ðez Þr¼ro , and these terms are independent of Poisson’s ratio, l. Thus, they can be further simplified as:

ðeh Þr¼ro ¼ ðez Þr¼ro ¼ aT 

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! 1 1 ðT o  T i Þ þ ðT o  T R Þ: þ 2 ln ðr i =r o Þ ðro =r i Þ2  1

It can be observed that T  is simply a function of the geometry of the circular hollow cylinder and is thus defined as the geometric temperature in this study.

To exclude the effects of ag and bg from the strain measurements, an identical set of dummy strain gauges can be employed in a reference material with known CTE aR. The unknown CTE of the studied material can then be obtained using the thermal output measurements of the two specimens:

2.3. Thermal strain measurements using strain gauges

a ¼ aR þ

Strain measurement by using metal foil strain gauges involves employing the following formula:

Consequently, when mechanical strain and thermal output are the only terms considered, Eq. (13) can be reexpressed as

T ¼

R¼q

L A

ð8Þ

ð9Þ

where R is the resistance, q is the resistivity, L is the length, and A is the area of the strain gauge. By differentiating Eq. (9) and comparing it with the parameters before stretching (R0, q0, L0, A0), the unit resistance change is obtained as follows:

dR=R0 ¼ dq=q0 þ dL=L0  dA=A0 :

ð10Þ

For strain gauges composed of circular metal wires, the relationship between strain (dL=L0 ) and unit resistance change can be expressed as

e¼

dL dR ¼ L0 R0



  dq=q0 dR FG þ ð1 þ 2lÞ ¼ R0 dL=L0

ð11Þ

where FG is termed the gauge factor of a strain gauge. However, when a strain gauge is employed and the unit resistance change measured, FG in Eq. (11) is termed the instrument gauge factor FI [20] because the deformation dL=L0 in this case is induced not only by mechanical (dLM) but also thermal loadings (dLT), amongst others (dLO), as expressed in Eqs. (12) and (13).

dL=L0 ¼

dLM þ dLT þ dLO L0

ð12Þ

e ¼ eM þ eT þ eO :

ð13Þ

In Eq. (13), eM is the mechanical strain, and eT is the temperature-induced strain, which is also referred to as apparent strain eapp or thermal output eTO [11,14,18,20].



eT ¼ eapp ¼ eTO ¼ ðas  ag Þ þ

 bg DT s : FG

ð14Þ

In Eq. (14), as is the CTE of the substrate material, ag is the CTE of the strain gauge element, bg is the thermal coefficient of resistivity of the grid material, and DTs is the temperature change of the substrate. In this study, as ¼ a and DT s ¼ DT are used, accordingly. The term eO can be attributed to the transverse sensitivity, bondable resistance, and misalignment of the strain gauge [19] as well as errors in the data acquisition system such as nonlinearity of the Wheatstone bridge and electric heating of the lead wires. This study simply ignores the effect of eO to place focus on the feasibility of the derived strain gauge apparatus in measuring the CTE of AC. Although the influence of eO is not significant and can be manipulatively averted or alleviated from the strain measurements, considering eO is beyond the scope of this study. The investigation of it may require more sophisticated experimental apparatus.

eTO  ðeTO ÞR DT

:

ð15Þ

eI ¼ eM þ eTO ¼ eM þ ðeTO ÞR þ ða  aR ÞDT

ð16aÞ

eITO ¼ eM þ aDT ¼ eI  ðeTO ÞR þ aR DT

ð16bÞ

eITO ¼ aT  ; T  ¼ eM =a þ ðT  T R Þ

ð16cÞ

where eI is the instrument strain and eITO is the instrument strain after thermal output calibration. The effects of ag and bg can be considered by using a dummy strain gauge set as depicted in Eq. (15), or by simply referring to the nominal values provided by the strain gauge suppliers. 3. Experimental procedures Based on the calibrated thermoelasticity model of circular hollow cylinders described previously, heating and cooling experiments were performed using Marshall AC specimens to obtain the corresponding CTE and CTC. Fig. 2 shows the instrumentations of the thermal probe, strain gauges, and thermocouples on the AC specimens. Fig. 3 illustrates the overall experimental setup, in which heating, measuring, and data acquisition systems were assembled accordingly. The heating system was composed of heating coil wrapping on a magnesium oxide tube with a K-type thermocouple embedded. The tube was then wrapped with stainless steel to form a U10 mm  H100 mm thermal probe. The nominal resistance of the thermal probe was 43 X, and the maximum driving power was 25 W. The measuring system consisted of eight K-type thermocouples and eight strain gauges (gauge length = 20 mm; four for active strain gauges and four for dummy strain gauges; R0 = 120 X and ag = 11.7 ppm/°C). The data acquisition system was composed of ADAM 4017 and 4018 with sixteen 16-bit channels. To eliminate the effect of room temperature variation on the strain measurements, dummy gauges were added to the system with similar connection wires to those of the active gauges. Dense-graded AC with a 19-mm (3/4 in.) nominal maximum aggregate size containing slag was designed and mixed. The AC-20 asphalt binder was acquired from a local supplier and the consensus properties were tested. The penetration at 25 °C, viscosity at 60 °C, and viscosity at 135 °C of the asphalt binder were 63 (0.1 mm), 1940 poises, and 357 poises, respectively. The mixing and compacting temperatures were thereafter determined between 149–154 °C and 138–143 °C. The job mix formula was designed to achieve the maximum density according to

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Fig. 2. Strain gauge and thermocouple instrumentation on AC specimens.

Fig. 3. Illustration of the self-assembled measurement and acquisition system.

the AI MS-2 method [24]. The formula was 4.6% of the optimum asphalt binder content, and air voids, density, VMA, VFA, stability, and flow were 4.6%, 2476 kg/m3, 13.1%, 71%, 1400 kgf, and 9.6 (0.25 mm), respectively. The standard U100  H60-mm specimen was compacted and a U10  H60-mm hole was drilled in the centre of the cross-sectional surface. Before the strain gauges were implemented, specimen surfaces where the gauges (g1–g4) were located, as shown in Fig. 2, were carefully polished with sandpaper. In addition, all lead wires were of the same type and had an equal length for both active and dummy gauges to remove the possible effect of room temperature fluctuation and, consequently, improve the measuring accuracy, because eO was excluded from this study. After the thermal probe was placed, the remaining space in the hole was filled with heat transfer cement (TG-S606C) with a satisfactory workability at a high temperature, a thermal conductivity coefficient of

5 W/m K, a specific weight of 2.3, and operating temperatures ranging from 40 to 180 °C. It was also recognised that the accuracy of temperature measurement would affect the output results of the thermoelasticity model. Accordingly, all K-type thermocouples were calibrated with a mercurial thermometer before use. The calibration tasks were repeated three times and an R-square higher than 0.998 was obtained. In addition to the studied AC specimen having eight thermocouples (Fig. 1); one extra thermocouple was placed on the dummy AC specimen (not shown) to provide the TR for a calibration analysis of the thermoelasticity model. The heating procedures were mainly controlled by the driving power of the thermal probe. The six scenarios of thermal flux (q1–q6) generated by the thermal probe were 4.59, 18.34, 41.27, 73.38, 114.27, and 164.64 W/m2. Each heating procedure was performed for more than 2 h to achieve thermal equilibrium. Similarly, the cooling

T.-C. Hou et al. / Measurement 85 (2016) 222–231

227

procedures were performed by switching off the power of the thermal probe for 6 h. Generally, the outer part of the AC specimen returned to room temperature after 2 h of cooling, whereas the inner part required longer to achieve thermal equilibrium, upon which the next heating procedure could be performed. The room temperature was kept at 27 °C throughout the testing procedures to eliminate the influences of ambient temperature fluctuation.

strain gauges is represented by Ra ¼ Rd þ DR, Eq. (17) can be rewritten as

4. Results and discussion

Furthermore, by considering the amplifier gain Vo = G  eo, where G = 50 in this study, and Vo is the amplified voltage, the instrument strain eI can then be calculated according to

As stated previously, the thermal fields of the circular hollow cylinder were assumed to be radially distributed. Fig. 4 shows the measured values and theoretical curves (according to Eq. (2)) of the thermal fields. The two observations were quite close to each other with a root mean square error of 9.2% (TRMSE = 9.2%TR). The recorded reference temperature, TR, varied from 28.11 to 31.49 °C with an average of 29.55 °C throughout the testing procedures. Therefore there was a measuring error of ±2.72 °C. The measured values meet the theoretical curves at r = 5 and 50 mm simply because the measurements serve as Ti and To for the model. 4.1. Approach I (neither eM nor eT are compensated, eO = 0) For a Wheatstone bridge circuit composed of one active gauge installed on the studied specimen, one dummy gauge on the dummy specimen, and one half-bridge with two constant gauges R, the following relationship can be built:

eo Ra R ¼  V i Ra þ Rd R þ R

ð17Þ

where eo =V i is the ratio of output to input voltages, Ra is the resistance of the active strain gauge, and Rd the dummy. If the temperature-induced resistance change of

eo R d þ DR 1 ¼  : V i 2Rd þ DR 2

ð18Þ

By assuming DR  Rd , the result is

eo DR  : V i 4Rd

eI ¼

4 Vo : G  FI V i

ð19Þ

ð20Þ

Fig. 5 shows the experimental results of instrument strain eI versus temperature at location g1/T1. The strain generally increases with temperature in each heating scenario, but slightly decreases as the thermal equilibrium is approached (2 h later). Similarly, cooling procedures cause a decrease of the strain reading as expected, but also result in a negative strain measurement at the end. The strain measurements obtained here include the thermal strain of the strain gauge itself, as depicted in Eq. (13), and this strain must be excluded to improve the interpretation of the results. Nevertheless, the CTE and CTC can still be interpreted simply by applying linear regression to the ascending and descending sections of Fig. 3. The measurements of heat flow q1 (4.59 W/m2) provide insufficient temperature change, leading to a relatively small value of R2 compared with that of other measurements; thus, the results are excluded for interpreting the average CTE. As listed in Table 1, the mean and standard deviation are 13.64 ppm/°C and 1.82 ppm/°C for the coefficient of thermal expansion, and 16.07 ppm/°C and 2.72 ppm/°C for thermal contraction, respectively.

Fig. 4. Theoretical and measured thermal fields.

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Fig. 5. Instrument strain versus temperature obtained by g1/T1.

Table 1 Linear regression (y = Ax + B) of CTE and CTC measured by the g1/T1 pair. g1/T1 I

Heating

Cooling

II

Heating

Cooling

III

Heating

Cooling

q1

q2

q3

q4

q5

q6

Avg.

Std.

A B R2 A B R2

27.39 3.18 0.08 7.00 4.64 0.77

14.62 1.45 0.70 17.16 93.90 0.79

10.56 2.74 0.94 18.37 177.45 0.97

14.93 0.48 0.99 18.53 218.34 0.98

14.67 0.60 0.99 13.15 363.00 0.99

13.43 2.40 0.98 13.16 347.51 0.98

13.64 0.01 0.92 16.07 240.04 0.94

1.82 2.00 0.12 2.72 114.47 0.08

A B R2 A B R2

22.62 4.24 0.21 13.79 8.66 0.38

26.57 1.45 0.88 29.47 95.02 0.92

22.55 2.77 0.99 31.01 181.78 0.99

26.92 0.51 1.00 28.97 228.44 0.99

26.69 0.73 1.00 26.37 379.08 1.00

25.75 1.96 1.00 26.48 377.46 0.98

25.70 0.12 0.97 28.46 252.35 0.97

1.81 1.89 0.05 2.00 124.52 0.032

A B R2 A B R2

9.85 8.52 0.35 11.62 11.11 0.39

18.58 3.63 0.96 24.54 65.45 0.91

18.67 17.47 0.99 24.79 116.46 0.98

21.983 34.555 0.99 27.16 200.94 0.99

22.65 66.03 0.99 23.71 328.89 1.00

21.10 74.04 1.00 24.988 354.55 0.98

20.60 39.14 0.98 25.04 213.26 0.97

1.88 30.38 0.02 1.28 127.18 0.04

4.2. Approach II (only eT is compensated, eO = 0) The effect of the thermal strain of the strain gauge eT can be modified using Eqs. (14) and (16) when interpreting the CTE values [11,14,18,20]. In this study, ag = 11.7 ppm/ °C and eTO, as provided by the strain gauge suppliers, were both adopted:

eTo ffi 0:29  102 þ 0:22  101 T  0:41  101 T 2  0:38  105 T 3 þ 0:11  105 T 4 :

ð21Þ

Fig. 6 shows the modified CTE obtained using Approach II from g1/T1. The slight decreases observed in the strain readings upon thermal equilibrium shown in Fig. 5 do not occur in this scenario. In addition, the modified strains

are approximately twice the unmodified strains. As in Approach I, the heat flow q1 provides an insignificant temperature change, and therefore, the results were ignored. The resulting mean and standard deviation, as listed in Table 1, are 25.70 ppm/°C and 1.81 ppm/°C for the CTE, and 28.46 ppm/°C and 2.00 ppm/°C for the CTC, respectively. Both the mean and standard deviation of the CTE and CTC are larger than those obtained using Approach I. 4.3. Approach III (both eM and eT are compensated, eO = 0) The studied AC hollow cylinder was heated using a thermal probe from the centre of the specimen, resulting in an inhomogeneous thermal field along the radial coordinates. Thus, the geometric temperature T  defined

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229

Fig. 6. Modified instrument strain versus temperature obtained by g1/T1.

in Eq. (8) should be used for interpreting the CTE. The known parameters used are listed as follows:

r i =r o ¼ 5 mm=50 mm To ¼ T1;

Ti ¼ T5;

T R;mean ¼ 29:55 C:

ð22Þ

T R ¼ 28:11—31:49 C; ð23Þ

By employing Eqs. (8) and (16), the relationship between the thermal strains ðeh Þr¼ro ¼ ðez Þr¼ro ¼ eITO and the geometric temperature T  can be obtained, as shown in Fig. 7. The results in Fig. 7 are similar to those in Fig. 6, with even slighter hysteresis occurring in this scenario (compared with Approach I, Fig. 5). The resulting

mean and standard deviation are 20.60 ppm/°C and 1.88 ppm/°C for the CTE, and 25.04 ppm/°C and 1.28 ppm/°C for the CTC, respectively. The resulting CTE/ CTC values are larger than those obtained using Approach I, but smaller than those obtained using Approach II, as listed in Table 1. The results associated with the heat flow q1 were neglected in this scenario because its applicability has been discussed previously. Theoretically, the CTE approaches can be ranked according to accuracy as III > II > I. The averaged thermal coefficient (means of expansion and contraction) obtained using the three approaches are 23.61 ± 2.91 (III), 26.91 ± 2.50 (II), and 15.51 ± 4.26 (I) ppm/°C, respectively. Because Approach III is considered more rational than the

Fig. 7. Thermal strain versus geometric temperature obtained by g1/T1 and T5.

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Fig. 8. Comparison of CTE and CTC obtained using Approaches I–III.

Table 2 Comparison of the CTEs with reported values in literature. Investigated

Reported

CTE

I

II

III

[25]

[26]

[27]

[28]

[29]

(ppm/°C)

15.5 ± 4.3

26.9 ± 2.5

23.6 ± 2.9

20.5–63.2

25–100

22.3–28.0

22.5–27.7

22.5–26.9

other two from a thermoelasticity standpoint, its CTE values were used as the basis for addressing the validity of the other approaches, as shown in Fig. 8. Fig. 8 also provides all readings of strain gauge and thermocouple pairs, namely g1/T1–g4/T4, as illustrated in Fig. 2. A result closer to the diagonal with less scattered values indicates a more accurate interpretation of the CTE of a specific composition of AC. For Approach III, the CTE varies from 18.37 to 27.35 ppm/°C with X  r = 21.93 ± 2.85 ppm/°C, and the CTC varies from 23.35 to 28.74 ppm/°C with X  r = 25.29 ± 1.84 ppm/°C. Compared with the previously reported values of the AC mixtures, which are 20.46–63.21 ppm/°C [25], 25–100 ppm/°C [26], 22.3–28.0 ppm/°C [27], and 22.5–27.7 ppm/°C [28], 22.5–26.9 ppm/°C [29], it can be concluded that the strain gauge apparatus coupled with the thermoelasticity model proposed in this study for measuring the CTE of AC is rational and applicable. The comparisons of CTEs are listed in Table 2. Furthermore, the standard deviations of ±2.85 and ±1.84 ppm/°C are equal to 13% and 7.3% if represented

by the coefficient of variation. This is a relatively scattered value for a general measurement technique. The factors that can be considered and improved in future study include the effect of eO in Eq. (13), the quality and homogeneity of the AC specimens, the potential porous interfaces between strain gauges and AC specimen surfaces, strain gauges with larger gauge lengths (considering the maximum AC aggregate size = 19 mm), and preventable thermal drainage from the outer surfaces of the cylinder specimens. It should still be noted that bonding strain gauges to nonhomogenous materials such as AC is an experience-required task. Careful selection of the adhesive and sample surface polishing would also help obtaining reliable strain gauge measurements. 5. Conclusions The CTE is a physical parameter for assessing the quality of AC pavement. The changes in the mechanical properties of AC with respect to current climate changes are now

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more urgent than ever because they govern the safety and durability of transportation infrastructures. This study developed a simple strain gauge apparatus with a hollow cylinder thermoelasticity model for measuring the CTE of AC materials. The CTE results obtained from Approach I, in which eM and eT were not compensated, exhibit lower values than expected and were thus considered inappropriate. The values interpreted from Approach II, in which eT was compensated, are the highest amongst the three approaches; however, theoretically, the accuracy is not sufficient. The CTEs with eM and eT compensation (Approach III) are the most accurate, with the results being comparable to several reported values. Thus, the proposed strain gauge apparatus with a hollow cylinder thermoelasticity model for measuring the CTE of AC is rational and applicable. The CTE measured using Approach III varies from 18.37 to 27.35 ppm/°C with a mean value of 21.93 ppm/°C, and the CTC varies from 23.35 to 28.74 ppm/°C with a mean value of 25.29 ppm/°C, respectively. Future efforts suggested for improving the accuracy and applicability are summarised as follows: Considering the effect of eO. Improving control of the quality and homogeneity of the AC specimens. Improving the potential porous interfaces between strain gauges and AC specimen surfaces. Applying strain gauges with larger gauge lengths. Preventing thermal drainage from the outer surfaces of the cylinder specimens. Acknowledgement The author would like to thank the Ministry of Science and Technology of the Republic of China, Taiwan for financially supporting this research under Contract No. MOST 102-2221-E-006-169. References [1] C. Tong, Q. Wu, The effect of climate warming on the Qinghai-Tibet Highway, China, Cold Reg. Sci. Technol. 24 (1996) 101–106. [2] B.N. Mills, S.L. Tighe, J. Andrey, J.T. Smith, K. Huen, Climate change implications for flexible pavement design and performance in Southern Canada, J. Transp. Eng. 135 (2009) 773–782. [3] W. Meagher, J. Daniel, J. Jacobs, E. Linder, Method for evaluating implications of climate change for design and performance of flexible pavements, Transp. Res. Rec. 2305 (2012) 111–120. [4] Y. Qiao, G. Flintsch, A. Dawson, T. Parry, Examining effects of climatic factors on flexible pavement performance and service life, Transp. Res. Rec. 2349 (2013) 100–107. [5] M. Anyala, J.B. Odoki, C.J. Baker, Hierarchical asphalt pavement deterioration model for climate impact studies, Int. J. Pavement Eng. 15 (2014) 251–266. [6] J.D. James, J.A. Spittle, S.G.R. Brown, R.W. Evans, A review of measurement techniques for the thermal expansion coefficient of metals and alloys at elevated temperatures, Meas. Sci. Technol. 22 (2001) R1.

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