Comparison of Inter-conversion and Analytical-numerical Approaches on Thermal Stress Computation of Asphalt binder

KOREAN ASPHALT INSTITUTE Vol. 8, No. 1, 14-20, 2018

*Corresponding author : Ki-Hoon Moon [Tel: 82-10-9464-1665,

E-mail: [email protected], [email protected]]

Comparison of Inter-conversion and Analytical-numerical Approaches on Thermal Stress Computation of Asphalt binder

Ki-Hoon Moon ^{1, *} , Augusto Cannone Falchetto ² , and Oh Sun Kwon ³

1 Senior Researcher, Korea Expressway Corporation (KEC) pavement research division, 208-96, Dong Budaero 922, Dong Tan, Hwa Sung city, Gyung Gi do, South Korea 18489, Tel:+82-31-8098-6294

2 Technische Universität Braunschweig, Department of Civil Engineering (ISBS), Germany

3 Senior Researcher, Korea Expressway Corporation (KEC) pavement research division, 208-96, Dong Budaero 922, Dong Tan, Hwa Sung city, Gyung Gi do, South Korea 18489

(Received April 14, 2018; Revised April 30, 2018; Accepted May 3, 2018)

Abstract: In the northern U.S. and Canada, low temperature cracking represents one of crucial distresses on asphalt pavements. As temperature drops thermal stress develops in asphalt layer then cracking occurs when it reaches at crucial level. Because of that reason, thermal stress is recognized as an important parameter for evaluating the low temperature performance of asphalt pavement. Conventionally, thermal stress is computed by converting creep compliance into its corresponding relaxation modulus based on different techniques. In this paper, analytical and approximate inter-conversion methods are applied to compute the asphalt binder thermal stress and then the results are compared. Clear differences in thermal stress are found when using power-law based approach compared to the numerical solution of the convolution integral according to Hopkins and Hamming’s algorithm. Based on the findings it can be said that both two different method can successfully be applied for providing lower and upper boundaries on estimating thermal stress of given asphalt binders.

Keywords: Thermal stress, Analytical numerical method, Inter conversion method, Hopkins and Hamming’s algorithm

1. Introduction

Thermal cracking is a serious distress for asphalt pavements built in northern U.S. and Canada. As temperature drops below 0°C, an increase in thermal stress can be observed in the restricted asphalt layers and when the temperature drops below a limit value known as critical temperature, cracking occurs. Low temperature cracking is source of significant negative effects on asphalt pavement performance; this is due the development of surface cracks and, then, to the infiltration of water in the pavement structures, potentially leading to a rapid deterioration process when associated to severe freeze-thaw cycles and traffic

loading (Marasteanu et al., 2007; Moon, 2010;

Moon et al., 2013; Moon et al., 2014; Cannone Falchetto et al., 2014).

Currently, many road agencies are devoting considerable efforts to maintain asphalt pavement performances and network systems at acceptable service levels. Along with these actions, various experimental methods have been developed during the past years to characterize the low temperature behavior of bituminous materials (Marasteanu et al., 2009; Moon, 2010; Moon et al., 2014; Cannone Falchetto et al., 2014). One fundamental test method to characterize viscoelastic materials is the creep test (Marasteanu et al., 2009; Moon et al., 2010;

AASHTO T313-12, 2012). Through this test, creep

compliance: D(t), can be obtained then from D(t),

relaxation modulus: E(t), can be calculated based

on different mathematical methods. Finally,

thermal stress: s(T), can be calculated with

different temperature level (Marasteanu et al., 2007; Moon et al., 2010; Moon et al., 2014). Even though various thermal stress computation approaches were introduced in many studies, not many studies considered and compared the effectiveness of each different thermal stress computation approach.

In the present paper, two different methods (i.e.

numerical method for computing the exact solution and approximate computation techniques) are used to convert BBR creep compliance data:

D(t), to relaxation modulus: E(t). Christensen, Anderson and Marasteanu (CAM) model (Marasteanu and Anderson, 1999) and Gauss quadrature method (Marasteanu et al., 2007; Moon et al.

2013) are then applied to compute thermal stress and the results are graphically compared. At the end of this paper, the recommendations for thermal stress computing approach are identified.

2. Inter-conversion Techniques for Converting Creep Compliance Into

Relaxation Modulus

2.1. Inter-conversion based on exact solutions - numerical algorithm

The creep compliance, D(t), and the relaxation modulus, E(t) of a viscoelastic material are correlated by the convolution integral as (Findley et al., 1976; Ferry, 1980):

(1)

(2)

where t > 0

Based on Laplace transformation, Equations (1) or (2) can be rewritten as:

(3)

where s > 0 is the Laplace transformation parameter.

However, in some cases the expressions for E(t) and D(t) cannot be easily manipulated in the Laplace domain. Therefore, a numerical approach can be used as an alternative and convenient method for solving the integral in Equation (1) and (2).

In this paper a simple and widely used

numerical algorithm developed by Hopkins and Hamming (1967) is applied to compute E(t) from D(t). The main computational steps for obtaining E(t) are as follows:

a. Select a time interval as:

(4) b. Express the integral form of D(t) as:

(5)

c. From Equation (5), f(t) can be calculated with the trapezoid integration rule:

(6)

d. Equation (1~2) can be modified using Equation (6) as:

(7)

From Equation (7), each element of the integral can be approximated as:

(8)

Where, (9)

Equation (8) can be then expressed as:

(10)

Solving for , Equation (10) becomes:

0

( ) ( )

t

E t − ⋅ τ D τ τ d = t

∫

0

( ) ( )

t

E τ ⋅ D t − τ τ d = t

∫

2

( ) ( ) 1 E s D s

⋅ = s

0 0, 1 1, 2 2,... _n t = t = t = t = n

0

( ) ^t ( ) f t = ∫ D t dt

1 1 1 1

( ) ( ) ( ( ) ( )) ( )

n n 2 n n n n

f t ₊ = f t + ⋅ D t ₊ + D t ⋅ t ₊ − t

1

1 1

0

( ) ( )

t n

n n

t ₊ = ∫ ⁺ E τ ⋅ D t ₊ − τ τ d

1

0 ⁱ ( ) ( 1 )

i n t i t n

E D t d

τ ⁺ τ ₊ τ τ

=

= ∑ ∫ ⋅ −

1

( ) ( 1 )

i i t

t n

E τ D t τ τ d

+

⋅ + − =

∫

1/2 1 1 1

( _i ) [ ( _{n i} ) ( _{n i} )]

E t ₊ f t ₊ t ₊ f t ₊ t

− ⋅ − − −

1 1

2

1 ( )

2 ^{i i}

t i t ₊ t

+ = ⋅ +

1

1 1/2 1 1 1

0 ( ) [ ( ) ( )]

n

n i n i n i

t ₊ i ⁻ E t ₊ f t ₊ t ₊ f t ₊ t

= − ∑ = ⋅ − − −

1/2 1

( _n ) ( _{n n} ) E t ₊ f t ₊ t

+ ⋅ −

) ( t _{n + 1 / 2} E

1

1 1/2 1 1 1

1/2 0

1

( ) [ ( ) ( )]

( )

( )

n

n i n i n i

n i

n n

t E t f t t f t t

E t f t t

−

+ + + + +

+ =

+

− ⋅ − − −

= −

∑

(11)

where, , ,

Then, the converted E(t) can be used to compute thermal stress based on the CAM model (Marasteanu and Anderson, 1999) and on Gaussian quadrature.

2.2. Approximated inter-conversion techniques 2.2.1. A simple Power-law based inter-conversion technique

In previous studies, it was found that values of D(t) and E(t) can be interrelated through a simple power-law functions as (Park and Kim, 1999;

Ebrahimi et al., 2014):

(12)

(13)

where E

1

, D

1

and n are positive constants. By taking the Laplace transformation of Equations (12) and (13), the following result can be derived as can be seen in Equation (14):

(14)

where parameter n can be expressed as:

or (15)

These power-law approximation methods provide very good results when D(t), and/or E(t), is close to a straight line in a log-log scale plot against time.

3. Materials and Testing

A set of two regular asphalt binders for HMA

pavement construction were prepared in this research. The given Performance Grade (PG) (AASHTO M 320-10-UL, 2010) was PG58-28 (unmodified binder), PG58-34 (Styrene Butadiene Styrene - SBS modified binder), respectively; both asphalt binders were provided from Korea Expressway Corporation Research Division (KECRD).

To compute creep stiffness, relaxation modulus and thermal stress, BBR tests (AASHTO T313-12, 2012) were performed. All the binders were tested in long term aged condition according to the Pressurized Aging Vessel (PAV) (AASHTO R028-12, 2012) procedure, respectively. Table 1 presents a summary of the prepared asphalt binders used in this study.

BBR creep tests; as can be seen in Figure 1, were performed on thin asphalt binder beams (102.0 ±5 mm×12.7±0.5 mm×6.25±0.5 mm) according to the current AASHTO specification (AASHTO T 313-12, 2012).

In this test, the mid-span deflection, d(t), of an asphalt binder specimen is measured for the entire duration of the test (total = 240 s), and then used to calculate the creep stiffness, S(t) and the m-value (slope of logarithmic S(t) vs time curve) as:

(16)

(17)

where S(t) is the flexural creep stiffness; D(t) is the creep compliance; σ is the maximum bending stress in the beam; ε(t) is the bending strain; P is the constant applied load (980±50 mN); l is the length of specimen (102±5 mm); b is the width of specimen (12.7 ±0.5 mm); h is height of specimen (6.25±0.5 mm); δ(t) is the deflection at the mid-span of the beam, and t is testing time (0~240 s).

1

1 1/2 1

0 1

( ) [ ( ) ( )]

( ) ( )

n

n i i i

i

n n

t E t f t f t

f t f t

−

+ + +

= +

− ⋅ −

= −

∑

( ) 0

0

f t = E t =( ) 0

₀ E t ( ) ₁ = t f t ₁ / ( ) ₁

( ) 1 ⁿ

E t = E t ⋅ ( ) 1 ⁿ

D t = D t ⋅ ⁻

sin( ) ( ) ( ) n E t D t

n π π

⋅ = ⋅

⋅

log ( )

log _t

d D t n

τ τ ₌

= log ( )

log _t

d E t n

τ τ ₌

=

3

3 1

( ) ( ) 4 ( ) ( )

S t P l

t b h t D t

σ

ε δ

= = ⋅ =

⋅ ⋅ ⋅ log ( ) ( ) log

d S t m t = d t Table 1. Prepared asphalt binders

ID Binder Description Material Source PG Aging Condition

1 Asphalt binder No.1

Plain KECRD 58-28 Long term (PAV)

Test temp: -18,-24ºC 2 Asphalt binder No.2

SBS modified KECRD 58-34 Long term (PAV)

Test temp: -24,-30ºC

*KECRD: Korea Expressway Corporation Research Division

To compute creep stiffness and relaxation modulus, generate master curves, and to finally calculate thermal stress and critical cracking temperature, BBR creep tests were performed at two different temperatures (i.e. low PG + 10°C and low PG + 10 - 6ºC). Moreover, two specimens were prepared at each temperature. Therefore, total four asphalt binder specimens were used (i.e. two for lowPG + 10ºC and other two for lowPG + 10 - 6ºC) for one set of binder testing.

4. Thermal Stress Computation Process

Thermal stress represents a critical component of the computation module used for predicting the low-temperature pavement performance in the current AASHTO Mechanistic-Empirical Pavement Design Guide (MEPDG) (AASHTO, 2008). Thermal stress can be calculated based on the following convolution integral expression:

(18)

where is the time dependent stress; is the relaxation modulus and . The strain, , can be expressed as:

(19) where, is the temperature change rate and a is the coefficient of thermal expansion or contraction;

based on previous research effort it is assumed α=0.00017 (Moon et al., 2010; Moon et al., 2013).

Thermal stress, , was numerically estimated in the temperature range +22ºC~-40ºC based on the one dimensional hereditary integral and using 24 Gauss points quadrature (points used for Gauss-Legendre numerical integration) as (Moon

et al., 2013):

⁽²⁰⁾

where is the reduced time; a

T

is the shift factor; is the thermal stress (MPa);

is the relaxation modulus; t is time (s);

and T is temperature (ºC) in the range +22 to -40ºC with cooling rate of 2ºC/hour as proposed in different studies (Marasteanu et al., 2007; Moon, 2010; Moon et al., 2013). In this research, the following steps were followed to obtain :

1. Compute creep compliance, D(t), from the experimental BBR creep stiffness, S(t) as D(t)= 1/

S(t);

2. Convert D(t) to the corresponding relaxation modulus, E(t), by using the five different interconversion methods described in section 2 of this paper.

3. Generate E(t) master curve using the BBR experimental data obtained at two different temperatures and the CAM model (Marasteanu and Anderson, 1999) which can be expressed as:

(21) ( ) t ^t ( ) ( t E t t dt ') '

σ ε

−∞

= ∫ ^ ⋅ −

σ t() E t()

( ) t d t ( ') / ' dt ε  = ε ε t()

( ) t T ε = ⋅Δ α

ΔT

σ ξ ( )

( ) ( ') ( ') '

'

d E d

d

ξ ε ξ

σ ξ ξ ξ ξ

−∞ ξ

= ∫ ⋅ −

( ) ( ( ) '( )) ' '

t d T

E t t dt

dt

α ξ ξ

−∞

= ∫ ⋅ Δ ⋅ −

ξ t a = ⁄

T

σ ξ ( ) E ξ ξ′ ( – )

σ ξ ( )

( ^{( ) ( ) ( )} )

( ) 1

( ) 1

1 10 ^{T c}

v w v

g c

v

g c

Log t Log a Log t v g

E t E t t

w t

LogE t LogE Log

v t

LogE w Log v

−

+ −

⎡ ⎛ ⎞ ⎤

⎢ ⎥

= ⋅ + ⎜ ⎟ ⎢ ⎣ ⎝ ⎠ ⎥ ⎦

⎡ ⎛ ⎞ ⎤

⎢ ⎥

⇒ = − ⋅ ⎢ ⎣ + ⎜ ⎟ ⎝ ⎠ ⎥ ⎦

⎡ ⎤

⇒ − ⋅ ⎢ ⎣ + ⎥ ⎦

Fig. 1. BBR asphalt binder specimen (left) and BBR testing equipment setup (right)

where E

g

is the glassy modulus, assumed equal to 3GPa for asphalt binders (Moon et al., 2010) and t

c

, v and w are fitting parameters. a

T

is the horizontal shift factor, which is expressed as:

(22)

where C

1

and C

2

are constant parameters and T

s

is the reference temperature (ºC).

Calculate thermal stress using Equations (20~22) and 24 Gauss point quadrature (Moon et al., 2013).

5. Data Analysis

5.1. Comparison of relaxation modulus

The E(t) curves obtained from the two different

computation methods are presented in Figures 2 and 3. In Figures 2 and 3, Method 1 and 2 means simple power-law computation method and approximated inter-conversion computation approach, respectively.

From a visual comparison of Figures 2 to 3, it can be observed that in all cases lower values of E(t) were computed when using Method 2 (i.e.

Hopkin and Hamming’s algorithm, 1967) compared to using Method 1 (i.e. Simple power- law approach). Therefore, in can be said that by using two different computation approaches, upper and lower boundaries of relaxation modulus can be successfully computed from given creep stiffness: S(t), data from current BBR low temperature binder creep test.

1 2

10 ^{C C T s} a T = ^{+ ⋅}

Fig. 2. Comparison of relaxation modulus (PG 58-28 binder)

Fig. 3. Comparison of relaxation modulus (PG 58-34 binder)

5.2. Comparison of thermal stress results

Thermal stress results were calculated based on the computed results from previous section (Section 5.1) and on the computation approach explained in Section 4. Figure 4 to 5 presents the thermal stress curves. Similar to the previous section, graphical comparisons were performed.

Similarly to the analysis performed for the relaxation modulus computation (i.e. see Section 5.1), higher values of thermal stresses were observed when using Method 1 (i.e. Power-law based model), compared to using Method 2 (i.e.

Hopkins and Hamming’s algorithm). Based on the results, it can be said that by using different computation approach, range of thermal stress of given asphalt binder can be successfully set.

Eventually, these ranged results can provide positive data base for predicting several asphalt pavement performance with several simulation pavement analysis software (e.g. MEPDG or AASHTOWare-Pavement analysis tool).

6. Summary and Conclusion

In this paper, two different computation approaches: inter-conversion and approximated numerical approaches, for thermal stress computation were performed, analyzed then compared. The following conclusions can be drawn:

The power-law based approximated models provide relatively higher values of the relaxation modulus compared to the numerical estimation of the exact solution method.

Approximately 20~30% higher thermal stress values were computed when using power-law based models compared to the exact solution method.

Thermal stress takes an important role in managing asphalt pavement conditions especially for road build in cold and severe climates.

Therefore, reliable and easy to compute results of thermal stress are essential for pavement agencies and road authorities even when used as input in sophisticated calculation systems such the current MEPDG (AASHTO, 2008).

The two introduced computation methods investigated in this paper provides sufficiently close upper and lower bounds for the computations of relaxation modulus and thermal stress. Therefore, the use of two different model approaches is highly recommended for obtaining the low temperature properties of the asphalt materials, since it provides to practitioners a reasonable data range including averaged values of thermal stress of given asphalt binder.

Acknowledgements

The author would like to gratefully acknowledge the Korea Expressway Corporation Research Division (KECRD) for technical support during the experimental phase.

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