An improved method of characterizing fracture resistance of asphalt mixtures using modified Paris’ law: Part II—Establishment of index for fracture resistance

Mechanics of Materials 138 (2019) 103168

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An improved method of characterizing fracture resistance of asphalt mixtures using modified Paris’ law: Part II—Establishment of index for fracture resistance

T

⁎

Rong Luo , Hui Chen School of Transportation, Wuhan University of Technology, Hubei Highway Engineering Research Center, 1178 Heping Avenue, Wuhan, Hubei Province 430063, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Asphalt mixture Fracture resistance Modified Paris’ law Damage density

The modified Paris’ law has been employed to characterize the fracture resistance of asphalt mixtures. However, the correlation between the coefficients of the modified Paris’ law has lacked theoretical justification. This study formulated the coefficients of the original Paris’ law in terms of the stress intensity factor for a long crack whose failure zones were much smaller than the crack size. These formulations were then utilized to derive the formulas of the coefficients of the modified Paris’ law in terms of the pseudo J-integral for the multitude of short cracks in a viscoelastic material such as an asphalt mixture. A theoretical model was developed between the two coefficients of the modified Paris’ law. This theoretical relation provided solid theoretical support to the empirical relation reported in the literature that was identified through regression analysis based on the results of hundreds of tests. The experimental data in this study further justified the theoretical relation. An index for fracture resistance of asphalt mixtures was established through rigorous theoretical derivation and experimental justification. The established index for fracture resistance serves as an excellent indicator of fatigue properties of asphalt mixtures. This index can be utilized in mix designs for comparing different mixtures, in field core evaluations for quantifying the fracture resistance of in-service asphalt pavements, in multiscale analyses of asphalt mixtures for linking mechanical properties at a macro level to interface adhesion at a micro level, and in other potential applications.

1. Introduction Fracture resistance is one of the most important properties of asphalt mixtures for asphalt pavements. The modified Paris’ law shown in Equation 1 has been employed to characterize the fracture resistance of asphalt mixtures (Si et al., 2002, Masad et al., 2008, Zhang et al., 2012, Luo et al., 2013):

dϕ = A JR (JR)n JR dN

(1)

where: ϕ = damage density, dimensionless; N = number of load cycles, cycle; A JR = modified Paris’ law coefficient, m2/(J•cycle); n JR = modified Paris’ law coefficient, dimensionless; and JR = average pseudo Jintegral, J/m2. However, the current methods reported in the literature are associated with deficiencies, including (Luo et al., 2013, 2012; Gu et al., 2015; Luo et al., 2012; Tong et al., 2015; Park et al., 1996): (1) the application of incompatible energy balances equations

⁎

simultaneously; (2) abandonment of the relationship between the dissipated pseudo strain energy (DPSE) and the damage density; and (3) lack of theoretical justification of the correlation between the coefficients of the modified Paris’ law. In order to address these deficiencies, this investigation developed an improved method to better characterize the fracture resistance of asphalt mixtures, which is presented in a set of two companion papers. The first companion paper detailed the development of the improved method of determining the fracture parameters including the modified Paris’ law coefficients ( A JR and n JR ) and the damage density (ϕ). This developed method had the merits of averting the application of incompatible energy balance equations and of applying the relationship between the DPSE and ϕ in model construction. Therefore, the first two deficiencies listed above were explicitly addressed. As the second of the two companion papers, this paper focuses on addressing the third deficiency. Specifically, this paper elaborates on

Corresponding author. E-mail addresses: [email protected] (R. Luo), [email protected] (H. Chen).

https://doi.org/10.1016/j.mechmat.2019.103168 Received 24 May 2019; Received in revised form 28 August 2019; Accepted 1 September 2019 Available online 10 September 2019 0167-6636/ © 2019 Elsevier Ltd. All rights reserved.

Mechanics of Materials 138 (2019) 103168

R. Luo and H. Chen

coordinate perpendicular to x1. The formulation of KI was valid for both linearly elastic and viscoelastic materials. Let:

the theoretical justification of the correlation between the two coefficients of the modified Paris’ law, which supports the establishment of a legitimate index for the fracture index of asphalt mixtures. The elaboration starts from the formulation of the coefficients of the original Paris’ law for a long crack whose failure zones were much smaller than the crack size, as detailed in the next section. The subsequent section describes the derivation of the theoretical model for the relationship between the two coefficients of the modified Paris’ law for a multitude of short cracks in a viscoelastic material. Next, the following section illustrates the development of the relationship between the modified Paris’ law coefficient and the damage density as well as the establishment of the index for fracture resistance of asphalt mixtures. The final section summarizes the major conclusions of the two companion papers.

x1 α

η=

(5)

σf (x1)

f=

(6)

σm

where: σm = maximum of σf(x1) with respect to x1, Pa. The formulation of KI shown in Eq. (4) was then re-written as: 1

2α 2 KI = ⎛ ⎞ σm ⎝π ⎠

∫0

1

1

η− 2 f (αη) dη

(7)

Let:

I1 =

2. Formulation of coefficients of original Paris’ law (A and n)

∫0

1

1

η− 2 f (αη) dη

(8)

Eq. (7) was simplified as:

2.1. Derivation of failure zone length

1

2α 2 KI = ⎛ ⎞ σm I1 ⎝π ⎠

The original Paris’ law was established to predict the growth of a single fatigue crack in a linearly elastic material. Specifically, the stress intensity factor range was related to the crack growth rate under a fatigue stress regime, as shown in Eq. (2) (Paris and Erdogan, 1963):

da = A (ΔKI )n dN

Rearranging Eq. (9) produced the formulation of α:

α=

(2)

KI w (t )

For viscoelastic materials, the energy criterion of failure was established by relating the fracture energy for crack propagation to motion of the surrounding continuum, as shown in Eq. (11) (Schapery, 1975):

(3)

G=

KImin = minimum where: KImax = maximum KI in a load cycle, 1 KI in a load cycle, Pa·m 2 ; t = loading time, s; and w(t) = wave function of KI in terms of loading time, dimensionless. If the crack had a failure zone with a length of α, as illustrated in Fig. 1, KI could be formulated in terms of the failure zone length (Schapery, 1975):

2 KI = ⎛ ⎞ ⎝π ⎠

∫0

α −1 σf (x1) x1 2 dx1

(10)

2.2. Derivation of crack tip velocity

1 Pa·m 2 ;

1 2

πKI2 2σm2 I12

This formulation of α was utilized to derive the crack tip velocity, which is presented in the following subsection.

where: a = crack length, m; A = coefficient of the original Paris’ law, 1 m 2 /(Pa·cycle) ; n = coefficient of the original Paris’ law, dimensionless; and ΔKI = range of the stress intensity factor for the opening mode 1 (Mode I) during a fatigue cycle, Pa·m 2 . If the material was subjected to a cyclic tensile load, the stress intensity factor for the opening mode, KI, would vary cyclically following the variation of the tensile stress since KI is a function of the tensile stress. As a result, ΔKI could be written as:

ΔKI = KI max − KI min =

(9)

1 Cv (t˜α ) KI2 8

(11) 2

where: G = fracture energy, J/m , which was a constant material property that was independent of crack growth history and external loading; Cv(t) = plane-strain creep compliance, Pa−1; and t˜α had a formulation shown in Eq. (12): 1 α t˜α = λmm da

(12)

dt

(4)

in which m = model parameter, dimensionless, and

where: α = length of the failure zone, m; σf(x1) = tensile stress applied to the failure zone, Pa; and x1 = variable of the coordinate along the failure zone, m. The parameter x2 in Fig. 1 was the variable of the

1

λm =

3π 2 Γ(m + 1)

(

4 m+

3 2

)(

Γ m+

)

3 2

, Γ(m) =

∫0

∞

t m − 1e−t dt (13)

The linear viscoelastic creep compliance Cv(t) was represented using the generalized power law approximation (Schapery, 1975):

Cv (t ) = C0 + C2 t m

(14) −1

where: C0 = model parameter, Pa , C0 = Cv (0) ; and C2 = model parameter, Pa−1•s. As a result, Cv (t˜α ) was written as: m

⎛α⎞ Cv (t˜α ) = C0 + C2 λm ⎜ da ⎟ ⎝

dt

(15)

⎠

Substituting Eq. (10) into Eq. (15) produced:

Cv (t˜α ) = C0 + C2 λm

π m 2

KI 2m σm I1 da m

()( ) ( ) dt

(16)

Incorporating Eq. (16) into the energy criterion of failure (see Eq. (11)) resulted in the formulation of crack tip velocity:

Fig. 1. Half of a mode I crack with a failure zone length of α. 2

Mechanics of Materials 138 (2019) 103168

R. Luo and H. Chen 1

2+ 2

πKI m ⎛ C2 λm ⎞ m da = ⎜ ⎟ dt 2σm2 I12 ⎝ 8G − C0 KI2 ⎠

The formulations of the crack growth rate and the coefficients of the original Paris’ law applied to a long crack whose failure zones were much smaller than the crack size. These formulations were utilized to formulate the coefficients of the modified Paris’ law, as detailed in the following section.

(17)

Since G was a constant and Cv (t˜α ) was a continuous and monotonically increasing function of t˜α , the value of KI had a range of (KIe, KIg), in which KIe was named as the equilibrium critical stress intensity factor as defined in Eq. (17), and KIg was the glassy critical stress intensity factor, as formulated in Eq. (18) (Schapery, 1975; Schapery, 1 1975). Both KIe and KIg had a unit of Pa·m 2 .

KIe =

8G Cv (∞)

KIg =

8G = Cv (0)

3. Derivation of relationship between coefficients of modified Paris’ law 3.1. Formulation of coefficients of Paris’ law in terms of J-integral (AJ and nJ)

(18)

8G C0

For asphalt mixtures, fatigue cracks started from small air voids that were considered short cracks. Their failure zone lengths were not much smaller than the crack lengths, and the failure zones had nonlinear and rate-dependent mechanical behavior. As a result, it was more appropriate to present the Paris’ law by means of the cyclic J-integral instead of ΔK for large-scale yielding at short crack tips in asphalt mixtures, as shown in Eq. (28) (Dowling and Begley, 1976; Ngoula et al., 2018):

(19)

Rearranging Eq. (19) yielded:

C0 =

8G KIg2

(20)

Substituting Eq. (20) into Eq. (17) produced:

da = AJ (ΔJ )nJ dN

1 m

⎤ ⎡ 2+ 2 ⎥ πKI m ⎢ C2 λm da = ⎥ ⎢ dt KI2 ⎥ 2σm2 I12 ⎢ ⎛ 8G 1 − K 2 ⎞ ⎢ ⎝ Ig ⎠ ⎥ ⎦ ⎣

where: AJ = coefficient of the Paris’ law in terms of the J-integral, 1/ (Pa•cycle); nJ = coefficient of the Paris’ law in terms of the J-integral, dimensionless; and ΔJ = cyclic J-integral, J/m2. For an elastic material, the cyclic J-integral was defined as (Ngoula et al., 2018; Lamba, 1975; Tanaka, 1983; Banks-Sills and Volpert, 1991; Vormwald, 2014; Ngoula and Wvormwald, 2018; Metzger et al., 2015):

(21)

For typical asphalt pavements under traffic loading, it was reasonable to assume that a Mode I fatigue crack inside the asphalt layer would propagate at a low velocity. This assumption indicated that KI2

KI ≪ KIg, which suggested

2 KIg

fied as: 2+ 2

≪ 1. Consequently, Eq. (21) was simpli-

ΔW =

(22)

= ∫

(

)

1

C2 λm m dt 8G

(23)

da dN

where: = crack growth per load cycle, or crack growth rate, m/ cycle. According to Eq. (3), KI was a function of t:

KI = ΔKI w (t )

(24)

Substituting Eq. (24) into 23 led to: 1

da C λ m ⎡ π = ⎢ 2 2 ⎛ 2 m⎞ dN 2σm I1 ⎝ 8G ⎠ ⎣

∫0

Δt

2 2+ 2 ⎤ w (t )2 + m dt ⎥·ΔKI m ⎦

A=

π ⎛ C2 λm ⎞ 2σm2 I12 ⎝ 8G ⎠

2 n=2+ m

∫0

Δσij d Δ˜εij

(30) (31)

Δεij = εij − εij0

(33)

Δui = ui − ui0

(34)

(25)

Je =

∂u e

∫Γ ⎛W edx2 − Ti ∂x1i ⎞ ds ⎜

⎟

⎝

⎠

∫Γ ⎛ΔW edx2 − ΔTi ⎜

⎝

2

w (t )2 + m dt

Δεij

(32)

ΔJ e = Δt

∫0

Δσij = σij − σij0

By simply comparing Eq. (25) with Eq. (2), it was inferred that the coefficients of the original Paris’ law were formulated as: 1 m

(29)

where: Γ = contour path enclosing the crack tip; W = strain energy density, J/m3; σij = stress tensor, Pa; ɛij = strain tensor, dimensionless; Ti = surface traction vector exerted on the material within the contour, Pa; nj = outward normal vector, dimensionless; ui = displacement vector, m; Δ indicates the relative changes of the parameters between two states corresponding to the external applied stresses σm, i and σm, j, respectively; and the superscript 0 designates the initial state at the beginning of a load cycle. Based on the correspondence principle, the pseudo J-integral was established for viscoelastic materials (Shapery, 1984; Anderson, 2005), as presented in Eq. (35). Accordingly, the cyclic pseudo J-integral was formulated in Eq. (36).

Δt da dt 0 dt

2 2+ m

⎠

and with

For a viscoelastic material subjected to cyclic loading, the crack growth per load cycle was formulated by integrating the crack tip velocity (see Eq. (22)) over the interval [0, Δt], in which Δt was the duration of a load cycle:

2 I2 2σm 1

⎟

⎝

ΔTi = Δσij nj

2.3. Derivation of crack growth rate subjected to cyclic load

Δt πKI

⎜

with

This formulation of the crack tip velocity was employed to derive the crack growth rate in typical asphalt pavements under traffic loading, as detailed in the next subsection.

= ∫0

∫Γ ⎛ΔWdx2 − ΔTi ∂∂Δxu1 i ⎞ ds

ΔJ =

1

πKI m C2 λm m da ⎛ ⎞ = dt 2σm2 I12 ⎝ 8G ⎠

da dN

(28)

(26)

with

(27)

We = 3

∫0

εije

σij dε˜ije

(35)

∂Δuie ⎞ ds ∂x1 ⎠ ⎟

(36)

(37)

Mechanics of Materials 138 (2019) 103168

R. Luo and H. Chen

ΔW e =

∫0

Δuie

uie

=

Δεije

−

Δσij d Δ˜εije

(38)

uie0

(39)

e

2

e

where: J = pseudo J-integral, J/m ; W = pseudo strain energy density, J/m3; uie = pseudo displacement vector, m; ΔJe = cyclic pseudo Jintegral, J/m2; εije = pseudo strain tensor, dimensionless; the notation Δ and the superscript 0 hold the same indications as specified above. According to the specific cyclic loading pattern applied to the asphalt mixture specimens in this study, every load cycle applied started from a time point when the external stress was zero (σm, i = 0) , as illustrated in Figure 6 of the first companion paper. This fact indicated that σij0 = 0 , which suggested ΔW e = W e and ΔTi = Ti . In addition, since ∂Δuie ∂x 1

=

ΔJ e =

∂uie ∂x 1

, Eq. (29) was re-written as:

∂u e

∫Γ ⎛W edx2 − Ti ∂x1i ⎞ ds = J e ⎜

⎟

⎝

⎠

(40)

In other words, the cyclic pseudo J-integral was equal to the pseudo J-integral in every load cycle for the specific cyclic loading protocol developed in this study. Consequently, ΔJe could be replaced by Je for this specific cyclic loading protocol that had an external load of zero at the beginning of every load cycle. Thus, the Paris’ law for asphalt mixtures subjected to this loading protocol was presented in terms of Je, as follows:

da = AJ (J e )nJ dN

Fig. 2. Elliptical-shaped cracks randomly distributed in an asphalt mixture.

(41) where: AC = average area of the crack opening, m2. The growth of AC per load cycle was calculated to be:

Assuming that the asphalt mixtures were linearly viscoelastic materials in plane strain, the pseudo J-integral was related to KI for the opening mode of crack tip deformation, as follows (Shapery, 1984; Anderson, 2005):

K 2 (1 − ν 2) Je = I ER

d (πab) dAC db da ⎞ = = π ⎛a +b dN dN dN ⎠ ⎝ dN

Since the crack was growing along the major axis, the minor axis db was considered to be constant, which suggested dN = 0 . Thus Eq. (47) was simplified as:

(42)

where: ν = Poisson's ratio, dimensionless; and ER = reference modulus, Pa, which was the magnitude of the complex modulus of the asphalt mixture in the linear viscoelastic stage. Incorporating Eq. (24) and Eqs. (42) into Eq. (41) produced:

d (πab) dAC da = = πb dN dN dN

⎟

1

1 ER nJ π ⎛ C2 λm ⎞ m ⎞ ⎛ AJ = 2 n J w (t ) ⎝ 1 − ν 2 ⎠ 2σm2 I12 ⎝ 8G ⎠

∫0

Δt

2 w (t )2 + m dt

1 m

ϕ=

(44)

where: AL = cracking area in the cross section of the asphalt mixture, m2; and A0 = total area of the cross section, m2. Since k cracks with an average opening area of AC formed at the cross section of the asphalt mixture, AL was equal to the product of k and AC. As a result, the damage density ϕ was presented as:

(45)

ϕ=

The formulations of the coefficients of the Paris’ law presented by means of the pseudo J-integral were applied to a single crack propagating from an air void in an asphalt mixture. These formulations were employed to derive the relationship between the coefficients of the modified Paris’ law for the growth of multiple cracks inside an asphalt mixture, which is detailed in the next subsection.

(49)

kAC A0

(50)

Based on Eqs. (48) and (50), the growth of the damage density per load cycle was formulated as:

( )= kA

d AC dϕ 0 = dN dN

3.2. Formulations of coefficients of modified Paris’ law ( A JR and n JR )

k dAC πkb da = A0 dN A0 dN

(51)

Rearranging Eq. (51) produced:

Air voids serving as initial cracks were randomly distributed inside a typical asphalt mixture, as illustrated in Fig. 2(a). The cross section of the asphalt mixture was assumed to possess k cracks, and each crack was assumed to be elliptical-shaped, with an average semi-major axis of a and an average semi-minor axis of b, as illustrated in Fig. 2(b). Therefore, the average area of the crack opening approximated the area of the ellipse:

AC ≈ πab

AL A0

(43)

Comparing Eq. (43) with Eq. (25) led to:

nJ = 1 +

(48)

The damage density in Eq. (1) was defined as:

n

da 1 − ν2 ⎞ J = AJ ⎛ w (t )2nJ ΔKI2nJ dN ⎝ ER ⎠ ⎜

(47)

da A dϕ = 0 dN πkb dN

(52)

Substituting Eq. (1) into Eq. (52) led to:

da A = 0 A JR (JR)n JR dN πkb

(53)

where: A JR and n JR = coefficients of the Paris’ law in terms of the average pseudo J-integral. The parameter JR was in fact the average

(46) 4

Mechanics of Materials 138 (2019) 103168

R. Luo and H. Chen

coefficients of the modified Paris’ law in terms of the average pseudo Jintegral, as presented in Eq. (61). This fact suggested that either A JR or n JR was sufficient to characterize the fracture resistance of the asphalt mixture. As a result, the modified Paris’ law coefficient n JR was taken as the index for fracture resistance of the asphalt mixture. The relationship between n JR and the damage density ϕ was then analyzed, which is detailed in the following subsection.

pseudo strain energy release rate for the multitude of cracks per unit volume. The value of JR was considered as γ times the value of Je, which was the pseudo strain energy release rate for a specific crack tip. Sub1 stituting J e = γ JR into Eq. (41) produced: n

da J J A = AJ ⎛⎜ R ⎞⎟ = nJ (JR)nJ dN γJ ⎝γ⎠

(54)

Comparing Eqs. (53) and (54), A JR and n JR were formulated as:

A JR =

πkb AJ A0 γ nJ

4.2. Analysis of relationship between modified Paris’ law coefficient n JR and damage density ϕ

(55)

n JR = nJ

The formulation of the damage density ϕ was already derived in the first companion paper, as shown Eq. (62):

(56)

Incorporating Eqs. (44) and (45) into Eqs. (55) and (56) led to:

ϕ = aN b + ϕ0

A JR =

π 2kb ER n JR ⎛ C2 λm ⎞n JR − 1 1 1 ⎞ · · n · ·⎛ · 2 2 2 n J J 2A0 σm I1 γ R w (t ) R ⎝ 1 − ν 2 ⎠ ⎝ 8G ⎠

∫0

Δt

where: N = number of load cycles; ϕ0 = initial damage density that equaled the air void content of the asphalt mixture; and a and b = model parameters that were determined in the first companion paper. Consequently, the incremental damage density due to repeated loading, Δϕ, were formulated as:

w (t )2n JR dt (57)

n JR = 1 +

1 m

(58)

Δϕ = ϕ − ϕ0 = aN b

In order to simplify the formulation of A JR , the integral in Eq. (57) was approximated as:

∫0

Δt

w (t )2n JR dt ≈ w (t )2n JR ·Δt

(59)

π 2kb 1 1 ER n JR ⎛ C2 λm ⎞n JR − 1 ⎞ · · · ·⎛ ·w (t )2n JR ·Δt 2A0 σm2 I12 γ n JR w (t )2n JR ⎝ 1 − ν 2 ⎠ ⎝ 8G ⎠ (60)

Taking the logarithm with base 10 of both sides of Eq. (60) produced:

log A JR ≈ n JR log

ER C2 λm 4π 2kbG Δt + log 8Gγ (1 − ν 2) A0 σm2 I12 C2 λm

(63)

By applying Eq. (63), Δϕ was calculated for each replicate specimen that was subjected to 103, 104 and 105 destructive load cycles, respectively, in the second controlled-strain repeated direct-tension test, which is illustrated in Fig. 2 in the first companion paper. The calculated values of Δϕ are listed in Table 2. The average values of Δϕ (see Table 2) were plotted against the corresponding average values of n JR (see Table 1), as shown in Fig. 4. A linear relationship was identified between Δϕ and n JR at any number of load cycles. The R2 value of every model was larger than 0.98, which demonstrated the goodness of model fit. This finding indicated that a larger value of n JR was associated with more cracking damage in the asphalt mixture at a specific number of load cycles. In other words, an asphalt mixture with a smaller n JR possessed better capability of fracture resistance. Therefore, the parameter n JR was proven to be an excellent index for fracture resistance of asphalt mixtures. Another observation from Table 2 and Fig. 4 was that the specimens were close to complete failure after 105 destructive load cycles since the total damage density ϕ (ϕ = ϕ0 + Δϕ ) was approaching its maximum value of 1. The incremental damage density Δϕ of the third replicate specimen of Type I mixture even exceeded 1 at N = 105 , which demonstrated that this replicate specimen broke into two pieces completely before the application of 105 load cycles. The inference from this observation was that Eq. (61) could be utilized to predict the fatigue life of the asphalt mixture at a given level of controlled strain. Since a, b, and ϕ0 were known parameters, the fatigue life N of the asphalt mixture could be determined at ϕ = 1 (or ϕ at any other number that was considered to be corresponding to the complete failure).

Accordingly, an approximation of A JR was obtained, as follows:

A JR ≈

(62)

(61)

Equation 61 indicated that the logarithm of A JR had an approximate linear relationship with n JR . This finding was obtained through rigorous theoretical derivation, as detailed above. The established theoretical relation coincided with the statistical relation between log A JR and n JR that was reported in the literature (Molenaar, 1983; Jacobs, 1995; Medani and Molenaar, 2000; Medani and Molenaar, 2000; Uzan, 1997; Lytton et al., 1993). In other words, the derivation in this study provided solid theoretical support to the empirical relation that was identified through regression analysis based on the results of hundreds of tests. Based on the theoretical relation shown in Eq. (61), models were constructed for the determined values of log A JR and n JR of the four types of asphalt mixtures investigated in this study, which is detailed in the next section.

4.3. Potential applications of modified Paris’ law coefficient n JR 4. Development of relationship between damage density and coefficient of modified Paris's law

The modified Paris’ law coefficient n JR can certainly be utilized in mix design to quantify and compare the fatigue properties among asphalt mixtures subjected to repeated tensile loading. In addition, n JR can also be used to evaluate the fracture resistance of field cores taken from asphalt pavements in service. Specifically, a series of cores may be taken from the same asphalt pavement at different service time points; by comparing their n JR values, the reduction of fracture resistance of the asphalt pavement can be investigated as its serving time increases. The modified Paris’ law coefficient n JR is in fact an index of the mechanical property at a macro level, which can be further linked to the micro-adhesion at the asphalt-aggregate interface. For example, it was found from Table 1 that the asphalt mixtures made of the SBSmodified binder had smaller values of n JR compared with the mixtures

4.1. Construction of model for modified Paris’ law coefficients A JR and n JR The first companion paper detailed the determination of the modified Paris’ law coefficients A JR and n JR of the four types of asphalt mixtures. The determined values of A JR and n JR were listed in Table 6 of the first companion paper, and the values of log A JR and n JR are reproduced in Table 1. Applying a linear model to fit the data of log A JR and n JR , the R2 value reached 0.9760, as shown in Fig. 3. The high R2 value demonstrated the goodness of mode fit. The excellent linear relationship between log A JR and n JR in turn justified the theoretical derivation of the relationship between the two 5

Mechanics of Materials 138 (2019) 103168

R. Luo and H. Chen

Table 1. Determined coefficients of modified Paris’ law. Mixture Type

Type of Aggregates

Type of Asphalt

I

Diabase

#70

II

Diabase

SBS-modified binder

III

Limestone

#70

IV

Limestone

SBS-modified binder

Replicate No.

n JR

logA JR

−43.0397 −46.1084 −51.9658 −33.9442 −27.2367 −34.4488 −44.0836 −53.0653 −29.4384 −34.2956 −34.8864 −35.3399

1 2 3 1 2 3 1 2 3 1 2 3

Individual

Average

12.3787 12.7282 14.0733 9.8749 8.6682 9.4361 12.3898 15.1863 8.1607 9.9093 9.9417 10.2317

13.0601

9.3264

11.9122

10.0276

Fig. 4. Relationship between incremental damage density Δϕ and modified Paris’ law coefficient n JR .

Fig. 3. Relationship between modified Paris’ law coefficients.

made of the #70 asphalt regardless of the type and gradation of the aggregates. This fact indicated that using the SBS-modified binder improved the fracture resistance of the asphalt mixtures, which very likely resulted from better adhesion between the SBS-modified binder and the aggregates.

formulations were then utilized to derive the formulas of the two coefficients ( A JR and n JR ) of the modified Paris’ law in terms of the pseudo J-integral for the multitude of short cracks in a viscoelastic material such as an asphalt mixture. Based on the formulas of A JR and n JR , a theoretical model was developed between the two coefficients, and this model indicated an approximate linear relationship between log A JR and n JR , which coincided with the statistical relation between log A JR and n JR that was reported in the literature. Specifically, the derivation in this study provides solid theoretical support to the empirical relationship that was identified through regression analysis based on the results of hundreds of tests. Linear models were constructed between log A JR and n JR of all replicated specimens of the four types of asphalt mixtures tested in this study, and the R2 value reached 0.9760. This finding demonstrated that it was absolutely unnecessary to make

5. Conclusion This is the second of a series of two companion papers. This paper presented the establishment of the index for fracture resistance of asphalt mixtures through rigorous theoretical derivation and experimental justification. The coefficients A and n were formulated for the original Paris’ law in terms of the stress intensity factor (KI) for a long crack whose failure zones were much smaller than the crack size. These Table 2. Incremental damage density due to repeated loading. Mixture Type

I

II

III

IV

Replicate No.

1 2 3 1 2 3 1 2 3 1 2 3

ϕ0

0.040

0.043

Δϕ (N = 103) Individual 0.4556 0.3947 0.5343 0.3527 0.3145 0.4451 0.4585 0.4508 0.4101 0.4306 0.3099 0.3928

Average 0.4615

0.3708

0.4398

0.3778

6

Δϕ (N = 104) Individual 0.6519 0.5444 0.7861 0.4835 0.4233 0.6342 0.6567 0.6439 0.5763 0.6101 0.4162 0.5480

Average 0.6608

0.5137

0.6256

0.5248

Δϕ (N = 105) Individual 0.9326 0.7511 1.1567 0.6627 0.5697 0.9038 0.9406 0.9196 0.8099 0.8645 0.5588 0.7646

Average 0.9468

0.7121

0.8900

0.7293

Mechanics of Materials 138 (2019) 103168

R. Luo and H. Chen

References

use of both A JR and n JR to evaluate the fracture resistance of the asphalt mixture. Either A JR or n JR was sufficient to characterize the fracture resistance. The parameter n JR was selected as the index for fracture resistance. The incremental damage density due to repeated tensile load (Δϕ) was calculated at 103, 104, and 105 destructive load cycles, respectively. It was found that the average values of n JR had an approximate linear relationship with the corresponding average values of Δϕ at each number of load cycles. The high R2 values (> 0.98) demonstrated the goodness of model fit. This finding illustrated that an asphalt mixture with a larger value of n JR would have more cracking damage at a specific number of load cycles. As a result, an asphalt mixture with a smaller n JR value is preferred in mix design because it has better capability of fracture resistance. The major contribution of the entire study is the development of an improved method to characterize the fracture resistance of asphalt mixtures using the modified Paris’ law. This contribution consists of two major components:

Anderson, T.L., 2005. Fracture Mechanics: Fundamentals and Applications. Taylor & Francis Group, Boca Raton, Florida, USA. Banks-Sills, L., Volpert, Y., 1991. Application of the cyclic J-integral to fatigue crack propagation of Al 2024-T351. Eng. Fract. Mech. 40 (2), 355–370. Dowling, N.E., Begley, J.A., 1976. Fatigue crack growth during gross plasticity and the Jintegral. Mechanics of Crack Growth, ASTM STP 590. American Society for Testing and Materials, pp. 82–103. Gu, F., Luo, X., Zhang, Y., Lytton, R.L., 2015. Using overlay test to evaluate fracture properties of field-aged asphalt concrete. Construct. Build. Mater. 101, 1059–1068. Jacobs, M.M.J., 1995. Ph.D. Thesis, Delft University of Technology, The Netherlands. Lamba, H.S., 1975. The J-integral applied to cyclic loading. Eng. Fract. Mech. 7, 693–703. Luo, X., Luo, R., Lytton, R., 2013. Modified Paris's law to predict entire crack growth in asphalt mixtures. Transp. Res. Rec. J. Transp. Res. Board 2373, 54–62. Luo, X., Luo, R., Lytton, R.L., 2012a. Characterization of fatigue damage in asphalt mixtures using pseudostrain energy. J. Mater. Civ. Eng. 25 (2), 208–218. Luo, X., Luo, R., Lytton, R.L., 2012b. Energy-based mechanistic approach to characterize crack growth of asphalt mixtures. J. Mater. Civ. Eng. 25 (9), 1198–1208. Lytton, R.L., Uzan, J., Fernando, E.G., Roque, R., Hiltunen, D., Stoffels, S., 1993. Development and Validation of Performance Prediction Models and Specifications for Asphalt Binders and Paving Mixes. Strategic Highway Research Program, Project Report SHRP-A-357, USA. Masad, E., Castelo Branco, V.T.F., Little, D.N., Lytton, R., 2008. A unified method for the analysis of controlled-strain and controlled-stress fatigue testing. Int. J. Pavement Eng. 9 (4), 233–246. Medani, T.O., Molenaar, A.A.A., 2000a. A simplified practical procedure for estimation of fatigue and crack growth characteristics of asphaltic mixes. Road Mater. Pavement Des. 1 (4), 451–465. Medani, T.O., Molenaar, A.A.A., 2000b. Estimation of fatigue characteristics of asphaltic mixes using simple tests. Heron 45 (3), 155–165. Metzger, M., Seifert, T., Schweizer, C., 2015. Does the cyclic J-integral ΔJ describe the crack-tip opening displacement in the presence of crack closure. Eng. Fract. Mech. 134, 459–473. Molenaar, A.A.A., 1983. Ph.D. Thesis, Delft University of Technology, The Netherlands. Ngoula, D.T., Madia, M., Beier, H.T., Vormwald, M., Zerbst, U., 2018. Cyclic J-integral: numerical and analytical investigations for surface cracks in weldments. Eng. Fract. Mech. 198, 24–44. Ngoula, D.T., Wvormwald, M., 2018. Short fatigue crack growth in welded joints described by the effective cyclic J-integral. MATEC Web of Conferences, Fatigue 165. pp. 09002 2018. Paris, P., Erdogan, F., 1963. A critical analysis of crack propagation laws. J. Basic Eng. 85 (4), 528–533. Park, S.W., Kim, Y.R., Schapery, R.A., 1996. A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete. Mech. Mater. 24 (4), 241–255. Schapery, R.A., 1975a. A theory of crack initiation and growth in viscoelastic media: II. Approximate methods of analysis. Int. J. Fract. 11 (3), 369–388. Schapery, R.A., 1975b. A theory of crack initiation and growth in viscoelastic media: II. Analysis of continuous growth. Int. J. Fract. 11 (4), 549–562. Schapery, R.A., 1975c. A theory of crack initiation and growth in viscoelastic media: I. Theoretical development. Int. J. Fract. 11 (1), 141–159. Shapery, R.A., 1984. Correspondence principles and a generalized J integral for large deformation and fracture analysis of viscoelastic media. Int. J. Fract. 25, 195–223. Si, Z., Little, D.N., Lytton, R.L., 2002. Characterization of microdamage and healing of asphalt concrete mixtures. J. Mater. Civ. Eng. 14 (6), 461–470. Tanaka, K., 1983. The cyclic J-integral as a criterion for fatigue crack growth. Internal J. Fract. 22, 91–104. Tong, Y., Luo, R., Lytton, R.L., 2015. Moisture and aging damage evaluation of asphalt mixtures using the repeated direct tensional test method. Int. J. Pavement Eng. 16 (5), 397–410. Uzan, J., 1997. Evaluation of fatigue cracking. Transp. Res. Rec. No. 1570, 89–95. Vormwald, M., 2014. Fatigue crack propagation under large cyclic plastic strain conditions. Procedia Mater. Sci. 3, 301–306. Zhang, Y., Luo, R., Lytton, R.L., 2012. Mechanistic modeling of fracture in asphalt mixtures under compressive loading. J. Mater. Civ. Eng. 25 (9), 1189–1197.

(1) determining fracture parameters, including the modified Paris’ law coefficients ( A JR and n JR ) and the damage density (ϕ), which is detailed in the first companion paper; and (2) establishing an index for fracture resistance, n JR , which has an approximate linear relationship with the incremental damage density (Δϕ), as presented in the second companion paper. The merits of the new method developed in this study include: (1) averting the application of incompatible energy balance equations simultaneously; (2) applying the relationship between the DPSE and the damage density in model construction; and (3) developing the theoretical relation between the two coefficients of the modified Paris’ law. The established index for fracture resistance serves as an excellent indicator of fatigue properties of asphalt mixtures. This index can be utilized in mix design for comparing different mixtures, in field core evaluation for quantifying the fracture resistance of in-service asphalt pavements, in multiscale analysis of asphalt mixtures for linking mechanical properties at a macro level to interface adhesion at a micro level, and in other potential applications. Declaration of Competing Interest None. Acknowledgements The authors acknowledge the financial support of the “973 Program” of the Ministry of Science and Technology of China (Project no. 2015CB060100). Special thanks to the 1,000-Youth Elite Program of China for the start-up funds used for purchasing the laboratory equipment that was crucial to this research.

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