Thermal stress computation of asphalt mixture using simple power law transfer function approach

Ki Hoon Moon

doi:10.22702/jkai.2024.14.2.27

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Scientific Paper

Journal of the Korean Asphalt Institute. 31 December 2024. 297-305
https://doi.org/10.22702/jkai.2024.14.2.27

Thermal stress computation of asphalt mixture using simple power law transfer function approach

Ki Hoon Moon¹^*

¹Principal Researcher, Korea Expressway Corporation Research Division

^{*Corresponding Author}

License (open-access, https://creativecommons.org/licenses/by-nc/4.0/):

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

It is well recognized that low temperature cracking is one of the crucial distresses on asphalt pavement. When low temperature cracking happens, easy infiltration of water and moisture can be happened. Then this can lead significant pavement distresses such as huge macro cracks, pot hole issues and pavement deflections. It is well known that Thermal Stress (TS) can be a successful and reliable parameter on low temperature performance evaluation of given asphalt material however, the computation process is complex. In this paper, a simple computation method of TS was introduced. Based on the recommendations in this paper, many pavement agencies can easily compute TS results from the current asphalt material mechanical testing method.

Keywords

Low temperature cracking

Thermal Stress (TS)

Power-law function Transformation Approach (PTA)

In-Direct Tensile (IDT) test

MAIN

1. Introduction
2. Experimental work
2.1 Part 1: Material preparation
2.2 Part 2: In-Direct Tensile (IDT) mixture creep test
3. Computation of Thermal Stress (TS) with simple Power law function Transformation Approach (PTA) procedure
4. Data analysis
5. Summary and conclusion

1. Introduction

The low temperature cracking: shows similarity to the transverse cracking on the pavement surface, is one of the serious distresses on asphalt pavement especially for northern U.S., Canada and South Korea (Moon, 2010, 2012; Cannone Falchetto et al., 2014; Moon et al., 2014). After low temperature cracking happens water and moisture can easily be infiltrated into restricted asphalt pavement layer which leads to further serious pavement distress such as macro cracking, fatigue cracking, pavement deformation and pot hole issues (Marasteanu et al., 2009; Moon, 2010, 2012).

Due to this reason, many pavement agencies and universities in U.S. and Canada put lots of effort to maintain asphalt pavement at certain level especially at cold weather session (Moon, 2010, 2012; Cannone Falchetto et al., 2014; Moon et al., 2014). To estimate (and/or predict) the severity of low temperature cracking on asphalt pavement, computation of Thermal Stress (TS) is essential (Moon, 2010, 2012; Cannone Falchetto et al., 2014; Moon et al., 2014). Generally, low temperature creep testing work with complicated mathematical approaches are needed for TS computation (Moon, 2010, 2012).

A simple performance test: named Bending Beam Rheometer (BBR) test is used for measuring low temperature creep properties for asphalt binder based on current AASHTO specification (AASHTO, 2012). For asphalt mixture, In-Direct Tensile (IDT) test (Buttlar and Roque, 1994; AASHTO, 2003) is widely used for measuring low temperature creep properties rather than using BBR creep test.

In asphalt material, TS can be computed as follows:

1) The deflection: $δ$ (t), under constant loading condition (e.g. for BBR creep test: 240 seconds and for IDT creep test: 1,000 seconds) is measured for the certain duration of testing procedure.

2) The creep compliance: D(t) and corresponding creep stiffness: S(t) results are generated using Euler-Bernoulli beam theory based on measured $δ$ (t) results (Ferry, 1980).

3) Then relaxation modulus: E(t), is computed by applying several inter-conversion techniques such as Hopkins and Hamming’s algorithm (1967) (Park and Kim, 1999).

4) Finally, Thermal Stress (TS): $σ$ (Tº), is computed by using various mathematical (and/or numerical) approaches with viscoelastic theory (Marasteanu et al., 2009; Moon, 2010, 2012; Moon et al., 2014).

The most complicated computation part on TS computation is the inter-conversion procedure between D(t) and E(t). Because of this computation complexity, not many pavement agencies and universities could derive and apply TS results of asphalt material easily before.

In this paper, a simple Power-law function Transformation Approach (PTA) procedure for TS computation was developed and introduced. Two asphalt mixtures with In-Direct Tensile (IDT) testing was considered for experimental work. The final computation results (i.e. generation of TS results trend) were visually inspected then the findings and future research topics were discussed.

2. Experimental work

2.1 Part 1: Material preparation

Two different asphalt mixture sets were prepared in this paper. More detailed information is provided in Table 1.

Table 1.

Prepared asphalt materials

Mixture Number	Mixture design information	Binder Grade	Other Materials
A (Regular)	- WC-1, NMAS = 13 mm (Wearing Course, type-1) - Passing sieve (%) 13 mm: 100%, 10 mm: 87% 5 mm: 52%, 2.5 mm: 41% 0.6 mm: 19%, 0.075 mm: 5%	Performance-Grade PG 64-28 (Un modified binder) (AASHTO, 2010)	-
B (RAP 25%)	- WC-1, NMAS = 13 mm - Passing sieve (%) 13 mm: 100%, 10 mm: 84% 5 mm: 53%, 2.5 mm: 40% 0.6 mm: 18%, 0.075 mm: 6%	Performance-Grade PG 64-28 (Unmodified binder) (AASHTO, 2010)	RAP (Reclaimed Asphalt Pavement), 25%

WC = Wearing Course, NMAS = Nominal Maximum Aggregate Size.

2.2 Part 2: In-Direct Tensile (IDT) mixture creep test

In-Direct Tensile (IDT) test was considered for measuring low temperature creep performance of asphalt material (Buttlar and Roque, 1994; Zhang et al. 1997; AASHTO, 2003). In IDT test, cylinder shaped asphalt mixture specimen with dimension of 150 mm ± 3 (diameter) × 40 ± 3 mm (thickness) is prepared. Then vertical and horizontal deflections are measured with total four Linear Variable Differential Transducers (LVDTs) for 1000 ± 2.5 seconds with constant load application (e.g. equal to 7.0~8.5 kN). Schematic information on IDT testing is shown in Fig. 1.

https://cdn.apub.kr/journalsite/sites/jkai/2024-014-02/N0850140219/images/jkai_2024_142_297_F1.jpg

Fig. 1.

IDT testing set up (with LVDT sensors) and specimen sample

Table 2 shows detailed information of IDT testing preformed in this paper.

In IDT test, the creep compliance: D(t), and corresponding creep stiffness: S(t), can be calculated as (Zhang et al., 1997):

(1)

\{\begin{cases} D (t) = \frac{1}{S (t)} = \frac{{H_{m}}^{(t)} ∙ d ∙ t}{P ∙ G L} ∙ C_{c o m p l i a n c e} \\ C_{c o m p l i a n c e} = 0.6354 \times {(\frac{x}{y})}^{- 1} - 0.332 \\ \frac{x}{y} = \frac{ε_{x i n 500 s e c o n d s}}{ε_{y i n 500 s e c o n d s}} \end{cases}

Where D(t) = Creep compliance, 1/MPa;

S(t) = Creep stiffness, MPa (= 1/D(t));

$H_{m}^{(t)}$ = Measured horizontal deflection at time t, mm;

$d$ = Diameter of specimen (= 150 mm, prepared from SGC);

$t$ = Thickness of specimen (= 40 mm, prepared from SGC);

$G L$ = Gauge Length (= 38 mm);

$P$ = Constant applied load, kN (= 7.0~8.5);

$ε_{x}$ = strain in horizontal direction, mm/mm;

$ε_{y}$ = strain in vertical direction, mm/mm;

Then the computed D(t) (and/or S(t)) needs to be converted into relaxation modulus: E(t), for further thermal stress computation. More detailed information is presented in the next section of this paper.

Table 2.

Information of IDT (In-Direct Tensile) mixture creep test

Mixture ID	Testing Temperature (°C)	Number of specimens	Applied load (kN)
Mixture A (Regular)	-18	3 specimens (-18°C)	7.0~8.5 (for 1,000 sec)
Mixture B (RAP 30%)	-18	3 specimens (-18°C)	7.0~8.5 (for 1,000 sec)

3. Computation of Thermal Stress (TS) with simple Power law function Transformation Approach (PTA) procedure

Generally, creep compliance: D(t), and relaxation modulus: E(t), are inter-related through convolution integral as (see Equation (2)):

(2)

\int_{0}^{t} E (t - τ) ∙ D (t) d τ = \int_{0}^{t} E (t - τ) ∙ \frac{1}{S (t)} d τ = t \int_{0}^{t} E (t) ∙ D (t - τ) d τ = \int_{0}^{t} E (t) ∙ \frac{1}{S (t - τ)} d τ = t

Additionally, D(t) (= 1/S(t), see Equation (1)) and corresponding E(t) can be estimated well with Power-law function (see Equation (3), Ferry, 1980; Park and Kim, 1999; Moon et al., 2019):

(3)

E (t) ≅ E_{1} ∙ \frac{1}{t^{n}} = E_{1} ∙ t^{- n}, D (t) ≅ D_{1} ∙ t^{n}

A method for Thermal Stress (TS) computation with simple Power law function Transformation Approach (PTA) can be explained as the following steps:

1) Perform Laplace transformation on Equation (3) then Equation (4) can be derived.

(4)

\{\begin{cases} L (E (t) = E_{1} ∙ t^{- n}) = E (s) = E_{1} ∙ \frac{Γ (1 - n)}{s^{1 - n}} \\ L (D (t) = D_{1} ∙ t^{n}) = D (s) = D_{1} ∙ \frac{n!}{s^{1 + n}} = D_{1} ∙ \frac{Γ (1 + n)}{s^{1 + n}} \end{cases}

In Equation (4), $Γ$ is the Gamma function which can be written as:

(5)

Γ (n) = \int_{0}^{\infty} t^{n - 1} ∙ e^{- t} d t

From Equations (2) to (5) with Euler reflection formula, relationship between D(t) and E(t) can be derived as (see Equation (6)):

(6)

D (s) = \frac{1}{s^{2}} ∙ \frac{1}{E (s)} = \frac{1}{s^{2}} ∙ \frac{s^{1 - n}}{E_{1} ∙ Γ (1 - n)} = \frac{1}{E_{1} ∙ Γ (1 - n)} ∙ \frac{1}{s^{n + 1}} \Rightarrow E (t) ∙ D (t) = \frac{\sin (n ∙ π)}{n ∙ π}

The parameter n in Equation (6) can be written as (see Equation (7)):

(7)

n = | \frac{d \log D (t)}{\log τ} |_{τ = t} or n = | \frac{d \log E (t)}{\log τ} |_{τ = t}

From Equation (7), E(t) can be derived as:

(8)

E (t) = \frac{1}{D (t)} ∙ \frac{\sin (n ∙ π)}{n ∙ π}

From the results in Equation (8), E(t) master curve can be constructed based on experimental results from Table 2 as:

(9)

E (t) = E_{g} ∙ [1 + (\frac{t}{t_{c}})^{v}]^{- \frac{w}{v}} \Rightarrow L o g E (t) = L o g E_{g} - \frac{w}{v} ∙ L o g [1 + (\frac{t}{t_{c}})^{v}]

Where E_g = glassy modulus, assumed equal to 30~40GPa for asphalt mixtures (Moon et al., 2014),

t_c, v and w = fitting parameters.

a_T = horizontal shift factor which can be expressed as:

(10)

a_{T} = 10^{C_{1} + C_{2} ∙ T_{s}} \Rightarrow L o g a_{T} = C_{1} + C_{2} ∙ T_{s}

C₁, C₂ = constant parameters, and

T_s = reference temperature (°C, in this paper lowPG+10°C was set as reference temperature).

Finally, TS: $σ$ (T°C), can be computed by solving Equation (11) with 24 Gauss points integration approach at temperature (T°C) ranging from 22°C(= T_i°C) to -40°C. An asphalt binder cooling rate of 2°C/hour was applied in this paper (Moon, 2010, 2012; Moon et al., 2014).

(11)

σ (t) = \int_{- \infty}^{t} \dot{ε} (t) ∙ E (t - t') d t' \Rightarrow σ (ξ) = \int_{- \infty}^{ξ} \frac{d ε (ξ')}{d ξ'} ∙ E (ξ - ξ') d ξ' = \int_{- \infty}^{t} \frac{d (a ∙ ∆ T)}{d t'} ∙ E (ξ (t) - ξ' (t)) d t'

(12)

where ξ = \frac{t}{a_{T}} reduced time

$\dot{ε} (t) = d ε (t') / d t'$ = strain rate which can also be expressed as:

(13)

ε (t) = α ∙ ∆ T

$α$ = Coefficient of thermal expansion or contraction on asphalt mixture; in this study, it is assumed $α$ = 0.00003 (Moon, 2010, 2012; Moon et al., 2014),

$∆ T$ = temperature cooling rate.

4. Data analysis

Based on the theoretical analysis procedure (see Section 2 and 3), TS results were derived. The results are shown in Figs. 2 and 3.

It can be seen that TS can easily be computed by means of simple mathematical approach. Moreover, higher values of TS were found for recycled asphalt material (e.g. Mixtuer B) compared to conventional asphalt mixture (e.g. Mixture A). The major reason is: generally RAP mixture contains more oxidyzed asphalt binder which relates into more brittle characteristics at low temperature. Based on the theoretical approach, low temperature performance of asphalt material can easily be evaluated which means more detailed investigation on low temperature characteristics is available.

https://cdn.apub.kr/journalsite/sites/jkai/2024-014-02/N0850140219/images/jkai_2024_142_297_F2.jpg

Fig. 2.

Results of TS: $σ$ (T°C), Mixture A (left) & Mixture B (right)

https://cdn.apub.kr/journalsite/sites/jkai/2024-014-02/N0850140219/images/jkai_2024_142_297_F3.jpg

Fig. 3.

Results of TS: $σ$ (T°C), of two sample tested mixtures (Mixtuees A and B)

5. Summary and conclusion

For viscoelastic material such as asphalt material, TS computation process was not an easy task. Therefore, only limited number of pavement research agencies were able to compute and analyze TS. In this paper, simple mathematical approach for TS computation (with PTA method) was introduced. Two different types of asphalt material including RAP were prepared and tested by means of IDT experimental work. It was found that a simple computation method introduced in this paper can successfully provide TS computation results. However, more extensive experimental works and corresponding numerical analysis efforts are needed as future research topic.

References

AASHTO (2003). Determining the creep compliance and strength of Hot Mix Asphalt using the Indirect Tensile Test (IDT) device, AASHTO T322-03, American Association of State Highway and Transportation Officials, Washington D.C.

AASHTO (2010). Standard Specification for Performance-Graded Asphalt Binder, AASHOTO M320-10, American Association of State Highway and Transportation Officials, Washington D.C.

AASHTO (2012). Determining the Flexural Creep Stiffness of Asphalt Binder Using the Bending Beam Rheometer (BBR), AASHTO T313-12, American Association of State Highway and Transportation Officials, Washington D.C.

Buttlar, W.G. and Roque, R. (1994). "Development and evaluation of the strategic highway research program measurements and analysis for indirect tensile testing at low temperature", Transportation Research Record, 1454, pp. 163-171.

Cannone Falchetto, A., Marasteanu, M.O., Balmurugan, S. and Negulescu, I.I. (2014). "Investigation of Asphalt Mixture Strength at Low Temperatures with the Bending Beam Rheometer", Road Materials and Pavement Design, 15(1), pp. 28-44.

10.1080/14680629.2014.926618

Ferry, J.D. (1980). Viscoelastic properties of polymers 3^rd edition, Wiley, New York, N. Y.

Marasteanu, M.O., Velasquez, R.A., Cannone Falchetto A. and Zofka A. (2009). Development of a Simple Test to Determine the Low Temperature Creep Compliance of Asphalt Mixtures. IDEA Program Final Report, NCHRP-133.

Moon, K.H. (2010). Comparison of thermal stresses calculated from asphalt binder and asphalt mixture creep compliance data, M.S. thesis, University of Minnesota, Twin Cities, U.S.

Moon, K.H. (2012). Investigation of asphalt binder and asphalt mixture low temperature properties using analogical models, Ph.D. thesis, University of Minnesota, Twin Cities, U.S.

Moon, K.H., Cannone Falchetto, A. and Yeom, W.S. (2014). "Low-Temperature Performance of Asphalt Mixtures Under Static and Oscillatory Loading", Arabian Journal for Science and Engineering, 39(11), pp. 7577-7590.

10.1007/s13369-014-1321-2

Moon, K.H., Kwon, O.S., Cho, M.J. and Cannone Falchetto, A. (2019). "An alternative one-step computation approach for computing thermal stress of asphalt mixture: the Laplace transform", Journal of Korean Society of Civil Engineers, 39(1), pp. 219-225.

Park, S.W. and Kim, Y.R. (1999). "Interconversion between Relaxation Modulus and Creep Compliance for Viscoelastic Solids", Journal of Materials in Civil Engineering, ASCE, 11(1), pp. 76-82.

10.1061/(ASCE)0899-1561(1999)11:1(76)

Zhang, W., Drescher, A. and Newcomb, D.E. (1997). "Viscoelastic behavior of asphalt concrete in diametral compression", Journal of Transportation Engineering, 123(6), pp. 495-502.

10.1061/(ASCE)0733-947X(1997)123:6(495)

Journal of the Korean Asphalt Institute ISSN:2234-0785(Print) 2635-9553(Online) 한국아스팔트학회지

Preview

Thermal stress computation of asphalt mixture using simple power law transfer function approach

ABSTRACT

MAIN

Table 1.

Prepared asphalt materials

Fig. 1.

IDT testing set up (with LVDT sensors) and specimen sample

(1)

Table 2.

Information of IDT (In-Direct Tensile) mixture creep test

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

Fig. 2.

Results of TS: σ (T°C), Mixture A (left) & Mixture B (right)

Fig. 3.

Results of TS: σ(T°C), of two sample tested mixtures (Mixtuees A and B)

References

Results of TS: $σ$ (T°C), Mixture A (left) & Mixture B (right)

Results of TS: $σ$ (T°C), of two sample tested mixtures (Mixtuees A and B)