Damage and fracture characterization of asphalt concrete mixtures using the equivalent micro-crack stress approach

Construction and Building Materials 148 (2017) 521–530

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Damage and fracture characterization of asphalt concrete mixtures using the equivalent micro-crack stress approach Ibrahim Onifade a,⇑, Björn Birgisson b a b

KTH – Royal Institute of Technology, Department of Civil and Architecture Engineering, Brinellvägen 23, 114 28 Stockholm, Sweden Zachry Dept. of Civil Engineering, 115F CVLB, Texas A&M University, College Station, TX 77843, USA

h i g h l i g h t s
Equivalent micro-crack stress (EMCS) proposed for asphalt damage characterization.
The EMCS is obtained from fundamental material properties.
Provides good agreement in both cyclic and monotonic loading modes.
The higher the effective micro-crack stress, the better the fracture performance.

a r t i c l e

i n f o

Article history: Received 5 September 2016 Received in revised form 26 April 2017 Accepted 6 May 2017

Keywords: Equivalent micro-crack stress Micro-crack initiation threshold MCIT Damage and fracture characterization Asphalt concrete Energy-based criterion

a b s t r a c t In this paper, a new parameter termed ‘‘equivalent micro-crack stress” ðrmc Þ is proposed for the evaluation of the cracking performance of asphalt mixtures with respect to their resistance to the initiation of micro-crack. The ‘‘equivalent micro-crack stress” ðrmc Þ is a function of the material stiffness and the ‘‘micro-crack initiation threshold” (MCIT). The MCIT is a critical strain energy density at the instance of initiation of micro-crack. Experimental testing is carried out for the evaluation of the cracking performance of unmodified and wax modified asphalt mixtures using the Superpave IDT tests at 20 °C, 10 °C and 0 °C. The low temperature range is used in the study to minimize the effect of viscoplastic dissipation on the material cracking behaviour. The result shows that the ‘‘equivalent micro-crack stress” ðrmc Þ gives a good indication of the material cracking performance of the unmodified and wax modified mixtures. A Finite Element Analysis is performed to assess the validity of the proposed approach under cyclic loading condition in the controlled-stress mode. The result shows that there is a good agreement between the material cracking performance in both monotonic and cyclic loading conditions using the proposed approach. The higher the ‘‘effective micro-crack stress” ðrmc Þ , the better the fracture performance of the mixture. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The mechanisms of damage and fracture in asphalt concrete are complex phenomena that require a full understanding of the constitutive behaviour of the material and the constitutive laws that govern the different stages in the damage and fracture process. Generally, the mechanism of damage and fracture in asphalt concrete mixtures can be categorized into a micro-crack initiation stage, followed by a micro-crack propagation stage which results in the coalescence and accumulation of micro-cracks. The coalescence and accumulation of micro-cracks results in the formation of macro-cracks which consequently propagate and result in the ⇑ Corresponding author. E-mail address: [email protected] (I. Onifade). http://dx.doi.org/10.1016/j.conbuildmat.2017.05.076 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.

failure of the material. Damage and fracture characterization in asphalt concrete mixtures is quite challenging considering the complexities in asphalt material behaviour due to the heterogeneity of the mixture, the presence of air voids, the differences in the behaviour of the constituent materials, and the complexities in the distinction and description of the full material constitutive behaviour in the damage, plastic and the fracture domains for practical applications. The fracture performance of asphalt mixtures is dependent on a number of contributing factors which include temperature and the rate of loading (frequency). Epps and Monismith [1] carried out a study to evaluate the influence of different factors that affect the fracture performance of asphalt concrete mixtures under a cyclic loading condition. The different factors considered in the study include the asphalt penetration, asphalt content, aggregate type,

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aggregate gradation, air void content and temperature. The fatigue loading was also conducted in both controlled stress-mode and controlled strain-mode to evaluate the influence of the different loading modes on the fracture and fatigue performance. The result showed that there is a significant influence of the loading mode (i.e. controlled stress-mode or strain-mode) on the fracture performance of the mixtures at different temperatures. It was observed that under the controlled stress mode, the fracture performance increases with decrease in temperature while for the controlled strain mode, the fracture performance showed a reverse tendency as the fracture performance decreases with decrease in temperature. The loading speed also have a significant influence on the fracture performance of asphalt mixtures due to their viscoelastic nature. Generally, asphalt mixtures tend to withstand more number of load cycles before failure under fatigue loading at increasing loading rates or frequencies e.g., [2]. Different criteria have been used for the evaluation of the crack performance of asphalt mixtures by researchers in monotonic and cyclic loading conditions. A 50% reduction in the stiffness of the material was used as a failure criterion e.g., [3]. Some researchers have use a deformation-based criterion where the total deformation under loading condition was used a criterion to characterize the cracking resistance of the mixture e.g., [4,5]. Other researchers have used criteria based on the ratio of the dissipated energy and the change in the phase angle e.g. [6,7]. Kim et al. [8] proposed three different damage indicators (i.e. change in dynamic modulus; change in pseudo stiffness; and change in dissipated strain energy) for the characterization of fatigue performance of asphalt mixtures using Dynamic Mechanical Analysis (DMA) testing. Research efforts in the SHRP project A-003A resulted in the development of a model for the determination of the total number of load repetition before crack initiation [9]. The regression-based model was developed by correlating the number of cycles to crack initiation in a fatigue beam test and in a pavement layer system. The Miner’s law has been used for fatigue characterization in the Mechanistic Empirical Pavement Design Guide (MEPDG) [10]. The Miner’s law have certain limitations with respect to it’s application to viscoelastic asphalt mixtures since it does not capture the fundamental mechanisms of damage initiation and propagation in asphalt mixtures, as well as the rate-dependent damage behaviour observed in asphalt concrete mixtures. Based on the limitations of regression-based, deformationbased and phenomenological approaches, efforts have been made to develop more fundamental understanding of the fracture behaviour of asphalt concrete mixtures and to relate the mixture cracking behaviour in both monotonic and cyclic loading conditions. These research efforts rely on the mechanics-based continuum approach to study the behaviour of asphalt mixtures with emphasis on the characterization of the effect of non-linearity due to damage initiation and propagation on the stress-states in the material. The continuum approach is applicable up to the limit of the formation of a macro-crack, where the displacement fields are discontinuous and a discrete crack is necessary to model the propagation of the macro-crack. Various researchers have used the continuum approach based on energy consideration to characterize the damage and fracture behaviour of asphalt concrete mixtures. The Hot-mix asphalt Fracture Mechanics (HMA-FM) e.g., [11–15] identifies the existence of fundamental energy thresholds, which have been associated with the formation of micro cracks and macro crack in asphalt concrete mixtures i.e. the Dissipated Creep Strain Energy (DCSE) and Fracture Energy (FE) limits. In an effort to characterize the cracking performance of asphalt mixtures using parameters obtained from the HMA-FM, it was found there is no single mixture property or characteristic obtained from experimental testing that can reliably predict cracking performance of hot-mix asphalt mixtures [16]. For this reason, the Energy Ratio

(ER) parameter is introduced into the HMA-FM to characterize the cracking performance of asphalt mixtures [16]. However, the ER approach is a semi-empirical approach which requires assumptions of a pavement layer system in order to obtain bending stress in the asphalt concrete layer, the bending stress together with the mixture properties obtained from the Superpave IDT test is used for the determination of the ER value. Mixtures with better cracking performance exhibit higher ER values. The viscoelastic continuum damage (VECD) model e.g., [17–22] is developed based on the work potential theory [23] and the elasticviscoelastic correspondence principle [24]. The mechanics-based discrete crack approach extends the application of fracture mechanics principle for viscoelastic asphalt concrete material e.g., [25–29]. However, the consideration of a crack size in the energy balance equation for the mechanics-based discrete crack approach complicates the analysis procedure and limits their use for simple mixture performance evaluation. Onifade et al. [30,31] proposed an energy-based damage and fracture model which is based on the continuum damage mechanics and the thermodynamics of irreversible processes. The damage model identifies the existence of a critical micro-crack energy limit below which no micro-crack is formed and above which microcrack propagation occurs. The existence of the energy threshold was investigated in [32] using micromechanical analysis techniques presented in [33,34]. The result shows that there exist an energy threshold that is invariant of the rate of loading for the initiation of micro-cracks in asphalt mixtures. The computational procedure in the damage model can be used to identify the fundamental energy threshold for micro-crack initiation and to characterize the damage accumulation that accompanies the fracture process. In this paper, an equivalent micro-crack stress approach is proposed for the characterization of the cracking performance of asphalt mixtures. The approach is derived from the energy-based viscoelastic damage model by Onifade et al. [30,31]. The equivalent micro-crack stress is derived as a function of the material stiffness and a critical strain energy density threshold value at the instance of micro-crack initiation. The Superpave (indirect tension) IDT resilient modulus and creep test are used to obtain the material stiffness and viscoelastic material properties, while the Superpave IDT strength test is used to obtain the critical strain energy density at the instance of micro-crack initiation. The Superpave IDT tests are performed at 20  C;  10  C and 0  C to minimize the influence of plastic and viscoplastic dissipation on the material response.

2. Energy-based viscoelastic damage model Recently Onifade et al. [31] proposed a new viscoelastic damage model based on energy balance with potentials for the identification of the critical threshold for micro-crack initiation and its consequent evolution based on thermodynamics of irreversible processes and continuum damage mechanics. The main components of the energy-based viscoelastic damage model are presented as follows: The generalized Maxwell model of n-terms with inclusion of a viscous strain ðvi Þ from each of the dashpot components is used in the model formulation. The generalized Maxwell model is categorized into a long term equilibrium (time-independent) part with long term equilibrium stiffness ðE1 Þ and a series of nonequilibrium parts (time-dependent) with different stiffness ðEi Þ and dashpot viscosity ðgi Þ. The total strain is additively decomposed into an elastic strain ðei Þ and a viscous strain ðvi Þ on each Maxwell branch i.e.  ¼ ei +vi .

I. Onifade, B. Birgisson / Construction and Building Materials 148 (2017) 521–530

The viscoelastic damage formulation is based on the energy equivalence principle with anisotropic damage representation to capture the behaviour of asphalt concrete in tension and in compression. The anisotropic damage variable takes the form of a second-order tensor ðDÞ and the integrity tensor / is identified and expressed as / ¼ I  D, where I is a second-order identity matrix and principal values of both / and D varies from 0 to 1. Considering the energy equivalence principle, the effective stress and strain are related to their nominal counterpart and expressed as shown in Eqs. (1) and (2):

r
¼ M : r;

ð1Þ


¼ MT : ;

ð2Þ


is the effective stress,  is the where r is the nominal stress and r
is the effective strain, and M is a fourth-order nominal strain and  damage tensor. The fourth-order damage tensor M is defined in terms of the integrity tensor / (where / ¼ I  DÞ according to [35] as:

1 M ¼ ðI  /1 þ /1  IÞ; 2

ð3Þ

The matrix representation of M 1 tensor exhibits the diagonal form shown in Eq. (4) in the principal axis of damage. A similar representation of the fourth-order damage tensor was used by [36,37]:

M 1

  /ð1Þ       ¼      

/ð2Þ /ð3Þ

        pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  /ð1Þ /ð2Þ   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  /ð2Þ /ð3Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  / / ð3Þ

ð4Þ

523

negative component of the stress tensor is obtained by subtracting the positive component ðrþ Þ obtained using the spectral decomposition technique from the total stress tensor ðrÞ. The negative part of the stress tensor can be obtained using the expression in Eq. (9)

r ¼ r  rþ ;

ð9Þ

Once the positive and negative components of the stress and strain tensors have been obtained, the damage conjugate ðYÞ can then be decomposed to obtain the positive ðY þ Þ and negative ðY  Þ parts as well. ðY þ Þ is the damage driving force under tensile conditions while ðY  Þ is the driving force under compressive conditions. The damage conjugate ðYÞ is written in terms of the total
and the total viscoelastic strain  as: effective stress r

Y ¼

1  
þr
   ; ½  r 2

ð10Þ

where  designates the positive and the negative components of the stress and strain tensors. 2.1. Damage initiation and evolution A non-associative damage formulation is used to derive different criteria for damage initiation and evolution. The criteria is considered only on the positive part of damage conjugate Y þ . The micro-crack initiation criteria is expressed in terms of the micro-crack initiation potential ð} 1 Þ, the critical viscoelastic micro-crack energy limit ð} 1;c Þ, and the damage softening parameter (R). The micro-crack initiation criterion is expressed as: d

f ¼ } 1 ðY þ Þ  } 1;c ðSo Þ  RðrÞ ¼ 0;

ð11Þ

where:

ð1Þ

The relationship between the nominal stiffness E1 and the effective stiffness E1 for the long term part is shown in Eq. (5). In a similar manner, the nominal and effective stiffness relationship for the non-equilibrium part is given in Eq. (6).

E1 ¼ M 1 : E1 : M T ;

ð5Þ

Ei ¼ M 1 : Ei : M T ;

ð6Þ

} 1 ðY þ Þ: is the micro-crack initiation potential } 1;c ðSo Þ: is the critical micro-crack energy limit Y þ : is a measure of the strain energy density So : is an energy term obtained from a strength test The micro-crack initiation potential } 1 ðY þ Þ is driven by the thermodynamic conjugate of the damage variable ðY þ Þ and expressed as:

So  k2 þ 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!k2 þ1 Yþ : Yþ 2  S0

The nominal non-equilibrium term stiffness Ei can be written in terms of the effective non-equilibrium Lame’s constants
ki and Gi

} 1 ðY  Þ ¼

and Eq. (6) is expressed as:

where k2 and So are material parameters. The critical viscoelastic micro-crack energy limit } 1;c ðSo Þ is expressed in terms of k2 and So as:

Ei ¼ M 1 : M T ð
ki þ 2Gi Þ

ð7Þ

The matrix representation of the stiffness matrix for the nonequilibrium part showing the influence of damage on the principal damage directions is given as:  2  / ð
ki þ 2Gi Þ
ki
ki  ð1Þ  2


ki / ð k þ 2G Þ k i i  ð2Þ i  
ki
ki /2ð3Þ ð
ki þ 2Gi Þ  Ei ¼   /ð1Þ /ð2Þ Gi      

/ð2Þ /ð3Þ Gi

                /ð3Þ /ð1Þ Gi 

} 1;c ðSo Þ ¼

ð13Þ

The micro-crack propagation criterion F D which is used to derive the evolution of damage is expressed as:

F D ¼ } 2 ðY þ Þ  a  } 1;c ðSo Þ  RðrÞ ¼ 0;

ð14Þ

where a is k1 =k2 . } 2 ðY þ Þ is the micro-crack propagation potential expressed as:

ð8Þ Similar representation used in Eqs. (7) and (8) are used for the stiffness matrix of the long-term part E1 . The spectral decomposition technique introduced by [38,39] is used to decompose the stress and strain tensors into positive and negative components so as to distinguish between the behaviour of asphalt concrete in tension and compression. For example, the

So k2 þ 1

ð12Þ

} 2 ðY þ Þ

So ¼a  k2 þ 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!k2 þ1 Yþ : Yþ 2  S0

ð15Þ

The evolution of micro-crack is obtained with respect to the dissipative micro-crack potential ðF D Þ by taking the derivative of the dissipation potential. The resulting power-law type damage evolution law is given as:

524

@F D k1 D_ ¼ k_  ¼  ›Y þ k2

I. Onifade, B. Birgisson / Construction and Building Materials 148 (2017) 521–530

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!k2 Yþ : Yþ Yþ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  r_ 2  S0 Yþ : Yþ

ð16Þ

where, k1 ; k2 and So are material parameters. 3. Materials and testing In this paper, the asphalt concrete test results used were obtained and re-analyzed from an earlier study on the evaluation of fracture and moisture damage performance of wax modified asphalt mixtures [40]. Unmodified 70/100 penetration grade binder was used as the base binder. The base binder was modified by adding 4% by weight of two types of commercial waxes, FTparaffin and Asphaltan-B. The wax modified bitumen mixtures were prepared by adding 4% of wax by weight of base binder. The mixture is heated for 30 min at 155  C and then placed in preheated blocks in a paint shaker and homogenized by shaking for 90s. 3.1. Specimen fabrication A dense graded crushed granite aggregate mixture with a maximum aggregate size of 11 cm was used for the preparation of the mixtures used in the study. A total of three different mixtures were produced with the combination of three different types of binder i.e. unmodified 70/100 binder, Asphaltan-B modified and FTparaffin wax modified binders. The mixture with the base binder and crushed granite aggregate is denoted as ‘‘AG1-O” while the Asphaltan-B modified and FT-paraffin wax modified mixtures are designated as ‘‘AG1-AB” and ‘‘AG1-FT” respectively. The aggregate gradation used for the mixture fabrication is presented in Table 1. The binder content for both unmodified and modified mixtures is 6.2% by weight of mixture and the target air void content is 7  1% by volume. A total of nine samples were fabricated for each mixture for testing at 20 degrC;  10  C and 0  C using the Superpave IDT test. 3.2. Binder testing The rheological effect of the modification of the unmodified 70/100 bitumen using Asphaltan-B and FT-praffin wax is investigated using standard test methods. The methods used are the softening point test (EN 1427), penetration test at 25  C (EN 1426), Brookfield viscosity test at 135  C and 165  C (EN 13302), and the force ductility test at 10  C (EN 13589, EN 13703). 3.3. Asphalt mixture testing The Superpave IDT test is used to characterize the performance of the asphalt concrete mixtures used in this study. The Superpave

Table 1 Crushed granite aggregate gradation used for mixture fabrication Sieve size (mm)

% passing

Specification limits

22.4 16 11.2 8 5.6 4 2 1 0.5 0.25 0.125 0.063

100 99.1 92.2 78.6 67.2 57 43.3 29.7 21.8 15.7 10.6 7.8

100 98 85 70 58 48 33 23 16 11 8 6

100 100 99 88 75 66 52 42 31 22 15 9

IDT test procedure consists of three different tests i.e. resilient modulus test, creep test and strength test. The resilient modulus and creep compliance tests are used to obtain the linear viscoelastic properties of the asphalt mixtures. In the Superpave IDT strength test, a cylindrical disc-shaped sample of asphalt mixture is loaded at a fast constant deformation rate of 50 mm/min until failure to obtain the tensile strength and fracture properties. In the Superpave IDT tests, the material properties are characterized using the observed stress and strain response at the center of the specimen. The horizontal tensile stress ðrx Þ is obtained using Eq. (17), where P is the applied load, t is the thickness of the test specimen and d is the diameter of the test specimen. The horizontal tensile strain ðx Þ is obtained using the relationship in Eq. (18), where MH is the horizontal deformation, and GL is the gauge length. The expression for the vertical compressive stress in the direction of the applied load in the Superpave IDT test is given in Eq. (19). C sx and C ex are Correction factors introduced to correct for the effects of biaxial stress state and bulging on the test results. The correction factor C sx and C ex are functions of the Poissons ratio and the horizontal and vertical deformations, see [41] for details.

rx ¼

2P  C sx ; ptd

ð17Þ

x ¼

MH  C ex ; GL

ð18Þ

ry ¼ 

6P  C sx ¼ 3rx ; ptd

ð19Þ

It was observed that the expression used for the determination of the Poisson’s ratio ðlÞ in [41] results in an over-estimation of the resilient modulus of the asphalt concrete mixtures used in this study. This was verified by comparing the experimental response in the Superpave IDT strength test with the predicted undamaged response using effective material properties from the resilient modulus and creep test as well as the load history from the Superpave IDT strength test. It was observed from the comparison that the over-estimation of the resilient modulus results in a mismatch between the experimental strain response and the predicted linear viscoelastic strain response. For this reason, a correction factor ðcl Þ was introduced to correct the Poisson’s ratio ðlÞ to eliminate the error in the material property characterization. The corrected Poisson’s ratio ðlc Þ is given as:

lc ¼ cl  l;

ð20Þ 

It was also observed at high strain levels at 0 C, the Poisson’s ratio does not have a significant influence on the material response during the Superpave IDT strength test. However, at 10  C and 20  C, the strain level at the instance of material failure is much lower due to the increase in the stiffness of the material and the influence of the Poisson’s ratio on the material response is significant. The observation of the influence of the Poisson’s ratio on the material response was similar for all 27 mixtures used in the study, including the modified and unmodified asphalt concrete samples. 4. Results and discussion 4.1. Binder characterization The results obtained from the binder tests are provided in Table 2. The results show that wax modification of the bitumen produced a stiffening effect on the binder with a decrease in the binder penetration and an increase in the softening point temperature. The wax modification also resulted in a general reduction in the viscosity of the binder as observed in the Brookfield viscosity test results. This decrease in the viscosity is a major advantage of

525

I. Onifade, B. Birgisson / Construction and Building Materials 148 (2017) 521–530 Table 2 Bitumen characteristic properties. Bitumen

Penetration

Softening point ð CÞ

Brookfield viscosity (mPa s) 135  C

Brookfield viscosity (mPa s) 165  C

Forced ductility (Nm) 10  C

Penetration index

Unmodified 70/100 +4% wax AB +4% wax FT

81 52 45

46 85 89

345 263 270

101 82 80

1.38 3.54 4.03

1.1 5.1 5.2

the use of wax modification of the binder, as mixtures can be mixed and compacted at lower temperatures without a significant loss in mixture workability for adequate compaction. The FTparaffin exhibits the highest stiffening effect as seen from the test results. 4.2. Superpave IDT test

at which the micro-crack initiation occurs. Below the MCIT, the material response is purely linear viscoelastic for the range of temperature and binder viscosity studied in this paper. Above the MCIT value, further micro-crack propagation and accumulation occur which then results in the formation of a macro-crack and eventual fracture. The expression for the MCIT is given in Eq. (13) and reexpressed in Eq. (21).

MCIT ¼ } 1;c ðSo Þ ¼

So k2 þ 1

ð21Þ

The results obtained from the Superpave IDT resilient modulus test, creep compliance test and strength test are presented in Table 3, where Mr is the resilient modulus, D1 and m are powerlaw creep compliance parameters, EE; DCSE and FE are the elastic energy, dissipated creep strain energy and the fracture energy densities respectively. The results from the resilient modulus test show that the wax modification results in an increase in the overall stiffness of the asphalt mixtures. However, at 20  C, there is no significant difference in stiffness of the unmodified and Asphaltan-B wax modified asphalt mixtures. Significant influence of the wax modification on the mixture stiffness can be observed at 10  C and 0  C with the FT-paraffin wax modified mixtures showing the highest stiffness properties. The Superpave IDT resilient modulus test result shows that there is a stiffening effect due to the wax modification of the asphalt binder which can be quite pronounced at higher temperatures. The result from the creep compliance test ðD1 and m-value) in Table 3 show that the wax modified mixtures have a lower mvalue than the unmodified mixture. The wax modified mixtures exhibit lower creep rate values with the FT-paraffin wax modified mixtures having the lowest creep rate at the temperatures considered. The elastic energy (EE), dissipated creep strain energy (DCSE) and fracture energy (FE) are energy thresholds obtained from the Superapve IDT strength test. The FE corresponds to the total amount of energy the material can sustain before the formation of a macro-crack. The DCSE has been used as a fundamental energy threshold to evaluate the resistance of asphalt mixtures to macrocrack formation. The values of the EE, DCSE and FE obtained are presented in Table 3.

Eq. (24) is then used to evaluate the strain in the effective (undam
ðtÞ is the effective strain, CðtÞ is the aged) configuration, where 
ðtÞ is the stress applied on the undamaged creep compliance and r test specimen during the Superpave IDT strength test.

4.3. Damage and fracture characterization


ðt Þ ¼ C ðt Þ  r
ðt Þ;

The viscoelastic damage model proposed by Onifade et al. [30,31] is used to determine the micro-crack initiation threshold (MCIT). The MCIT is the critical strain-energy density threshold

For simplification of the interpretation of the Superpave IDT strength test results, a one-dimensional representation of the stress–strain relationship in Eq. (24) in the horizontal tensile direction can be expressed as:

To evaluate the expression in Eq. (21), the material damage parameters k2 and So are needed. The material damage parameters can be obtained using a simplified and improved version of the procedure in Onifade et al. [31], briefly presented and discussed below. The test results from the resilient modulus and creep compliance tests can be used to obtain the linear viscoelastic properties of the different mixtures. Assuming a plane stress condition in the Superpave IDT test specimen i.e. rz ¼ 0, the resilient modulus from the Superpave IDT resilient modulus test can be obtained based on the biaxial stress–strain relationship in the horizontal tensile direction given as:

Mr ¼

rx  lc ry ; x

ð22Þ

The creep compliance CðtÞ is then fitted with a Prony series model in Eq. (23) to obtain the Prony series model parameters where C o is the inverse of the resilient modulus M r ; C i is the spring th

compliance of the i prony series term, t is the time in seconds and si is the retardation time.

CðtÞ ¼ C o þ C i 1 

n X

!

st

exp

i

;

ð23Þ

i

ð24Þ

Table 3 Summary of Superpave IDT test results. Temp

Mr. (GPa)

D1 (1/GPa)

m

EE ðkJ=m3 Þ

DCSE ðkJ=m3 Þ

FE ðkJ=m3 Þ

AG1-O

20 10 0

20.96 16.70 10.88

0.021 0.028 0.443

0.623 0.817 0.753

0.21 0.21 0.26

0.31 0.77 3.21

0.52 0.97 3.47

AG1-AB

20 10 0

20.68 17.24 12.30

0.024 0.057 0.264

0.598 0.653 0.693

0.23 0.27 0.3

0.41 0.8 2.69

0.63 1.07 2.99

AG1-FT

20 10 0

22.26 17.40 13.5

0.014 0.041 0.135

0.65 0.685 0.758

0.22 0.26 0.35

0.39 0.89 2.63

0.61 1.15 2.98

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x ðtÞ ¼ CðtÞ  r
y ðtÞ ;
x ðtÞ  lc r

ð25Þ

where, lc is the corrected Poisson’s ratio given in Eq. (20). The recommended correction factor cl for the Superpave IDT resilient modulus and strength tests is 2/3. However, for the Superpave IDT strength test at 0  C, a correction factor cl of 0 was found to

be suitable due to the reduced influence of the Poisson’s ratio on the material response at high strain levels. Fig. 1 shows a plot of the strain for the AG1-O in the effective (undamaged) and nominal (damaged) configurations at 10  C. It can be observed that there is a good match between the initial portion of the predicted effective strain and the observed experimental nominal strain curves. The deviation of the nominal strain curve from the effective strain path is as a result of damage (micro-crack) initiation and accumulation in the material. The rate of change of both the effective and nominal strain is used to obtain the point of deviation from linear viscoelastic response, which corresponds to the instance of initiation of damage (micro-crack). The damage parameter So is the viscoelastic strain energy density at the instance of damage initiation. Based on the principles of energy equivalence, the damage state variable D in the principal damage
ðtÞ is the effective direction is estimated using Eq. (26), where  (undamaged) strain and ðtÞ is the nominal (damaged) strain in the principal direction of damage. In the absence of damage, the nominal and effective strain are homogeneous in time and the damage state variable D will be zero.

D¼1

Fig. 1. Plot of strain in the nominal and effective configuration for AG1-O at 10  C.


ðtÞ ; ðtÞ

ð26Þ

The resulting damage evolution D is then fitted with Eq. (16) using the least squares means method to obtain the damage parameters k1 and k2 . Table 4 shows the result of the damage parameters obtained from the material strength test.

Table 4 Damage parameters for unmodified (AG1-O) and wax modified (AG1-AB, AG1-FT) mixtures. Tempð CÞ

k1

k2

So ðkJ=m3 Þ

MCIT ðkJ=m3 Þ

AG1-O

20 10 0

253.5 116.2 24.6

1.57 0.95 0.52

0.177 0.169 0.264

0.0689 0.0864 0.1740

AG1-AB

20 10 0

300.5 105.4 54.6

1.18 0.85 0.67

0.172 0.253 0.524

0.0788 0.1368 0.3136

AG1-FT

20 10 0

386.6 93.9 64.2

1.29 1.06 0.71

0.206 0.231 0.409

0.0900 0.1125 0.2395

Fig. 2. Plot of micro-crack initiation threshold for the unmodified (AG1-O) and wax modified (AG1-AB, AG1-FT) mixtures.

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Fig. 2 shows the plot of the micro-crack initiation threshold (MCIT) obtained using Eq. (21) for both unmodified and wax modified mixtures. It can be observed that the MCIT value increases as the temperature increases due to the increase in the strain energy release rate at higher temperature range. The wax modified mixtures generally have higher MCIT values than the unmodified mixture. AG1-FT exhibits a higher MCIT value than the AG1-AB mixture at 20  C. However at 10  C and 0  C, AG1-AB has the highest MCIT values. It can be quite tempting to conclude that AG1-AB exhibits the highest resistance to micro-crack formation based on the value of the MCIT obtained, a conclusion that can be erroneous. It is important to note that the mixtures have different viscoelastic material properties and the evolution of the strain energy density which is the driving force for damage will follow different paths for the different mixtures. To illustrate this phenomenon, Fig. 3 shows the schematic plot of the stress–strain curve of two materials with similar MCIT values but different stiffness E1 and E2 , where material stiffness E1 is greater than E2 . It can be observed that at the same MCIT value, material with stiffness E1 can take more stress at the instance of micro-crack initiation than the material with stiffness E2 . It is clear from this analogy that the MCIT value itself cannot be used as measure of the resistance of the mixtures to micro-crack initiation as the evolution of the strain energy density path is different for materials with different material stiffness. Therefore, to characterize the mixture crack performance, a parameter called ”the equivalent micro-crack stress ðrmc Þ” is proposed and expressed as a function of the MCIT value and the material stiffness. For viscoelastic material with time-dependent stiffness properties, a relationship between the reference stiffness ðEref Þ and the MCIT value was envisaged to obtain the equivalent micro-crack stress ðrmc Þ expressed as:

rmc

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Eref  MCIT ;

Eref is given as:

Eref ¼

Eo 1 þ ðEo  D1  t m Þ

ð28Þ

where Eo is the instantaneous (elastic) modulus, D1 and m are the power-law creep compliance parameters and t is the reference evaluation time in seconds. A reference evaluation time of 100s is used in this study. It can be observed that for a purely elastic material (i.e. D1 and m = 0), the reference stiffness ðEref Þ reduces to the modulus of elasticity of the material. The influence of the loading frequency can be taken into consideration in the equivalent microcrack stress ðrmc Þ equation through the reference evaluation time i.e. high loading frequency correspond to a low reference evaluation time. It can be seen from Eq. (28) that at very high loading frequencies, the reference time tends to zero ðt ! 0Þ and the elastic modulus is once more recovered. It can be observed that Eq. (27) takes a similar form as the failure stress obtained from the Griffith energy balance equation (in Eq. (29)), where E is a measure of the material stiffness, wf is the fracture work, and a is the crack radius. The fracture work is usually a combination of plastic, viscoelastic and viscoplastic effects depending on the behaviour of the material [42]. By comparing Eqs. (27) and (29), it can be seen that the MCIT is a global mixture parameter that incorporates a measure of the combination of the surface energy and the viscoelastic dissipation as well as the volumetric content and spatial distribution of air voids in the material. One would expect that the higher the amount of flaws in the material (air voids), the lower will be the MCIT value. In a similar manner, an increased tendency of viscoplastic and plastic dissipation at higher temperatures will result in higher MCIT values.

rf ¼

Fig. 3. Schematic plot of the stress–strain curve of two materials with similar MCIT values but different stiffness E1 and E2 , where material stiffness E1 is greater than E2 .

ð27Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  E  wf ; pa

ð29Þ

Table 5 shows the result of the equivalent micro-crack stress ðrmc Þ obtained for the unmodified and wax modified mixtures. Fig. 4 shows a plot of the equivalent micro-crack stress ðrmc Þ for unmodified (AG1-O) and wax modified (AG1-AB and AG1-FT) mixtures as a function of temperature. The wax modified mixtures possess higher equivalent micro-crack stress ðrmc Þ with FTparaffin wax modified mixture exhibiting the highest equivalent micro-crack stress ðrmc Þ values. It can be observed that as the temperature increases, the equivalent micro-crack stress ðrmc Þ decreases and the overall mixture resistance to micro-crack formation decreases in the range of temperatures considered in this study. A Finite Element Analysis (FEA) is carried out to assess the fatigue behaviour of the mixtures and to evaluate the validity of the equivalent micro-crack stress ðrmc Þ in Eq. (27) for repetitive load conditions. The linear viscoleastic and damage parameters

Table 5 Equivalent micro-crack initiation stress ðrmc Þ. Temp ð CÞ

Ecreep (Gpa)

MCIT (kPa)

rmc (MPa)

AG1-O

20 10 0

2.394 0.790 0.070

0.0689 0.0864 0.1740

0.406 0.261 0.110

AG1-AB

20 10 0

2.352 0.826 0.154

0.0788 0.1368 0.3136

0.431 0.336 0.220

AG1-FT

20 10 0

3.084 0.982 0.222

0.0900 0.1125 0.2395

0.527 0.332 0.231

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Fig. 4. Plot of the equivalent micro-crack initiation stress ðrmc Þ for unmodified (AG1-O) and wax modified (AG1-AB and AG1-FT) mixtures at different temperatures.

Fig. 5. Plot of evolution of the damage variable ðD33 Þ in the principal damage axis for unmodified (AG1-O) and wax modified (AG1-AB and AG1-FT) mixture at 0  C.

obtained from the Superpave IDT tests are used to model the material behaviour in the linear viscoelastic and damage regime. A three-dimensional uniaxial condition is considered with the material subjected to a haversine load function in Eq. (30) expressed as:

r ¼ ½sinð3ptÞsinð3ptÞ þ 0:2  600 kPa;

ð30Þ

where r is the applied pressure, and t is time. The load function in Eq. (30) is repeated after every 1 second period. From Eq. (30), the applied maximum and minimum pressure are 720 kPa and 120 kPa respectively. The evolution of the damage variable ðD33 Þ in the principal damage axis under repetitive load condition is obtained and presented in Fig. 5. It can be observed from Fig. 5 that AG1-FT can withstand the highest number of load repetitions before micro-crack initiation at 0  C. AG1-Ft can withstand 133 load repetitions, while AG1-AB and AG1-O can withstand 118 and 26 load repetitions respectively before micro-crack initiation. In general, the wax modified mixtures show a better resistance to micro-crack initiation than the unmodified mixture. Table 6 shows a summary of the number of cycles to micro-crack initiation for the unmodified (AG1-O) and modified (AG1-AB and AG1-FT) mixtures at 10  C and 0  C. Fig. 6 shows a plot of the number of cycles to micro-crack initiation ðN mc Þ and the equivalent micro-crack stress ðrmc Þ for unmod-

Table 6 Number of cycles to micro-crack initiation ðN mc Þ at 10  C and 0  C.

10  C 0 C

AG1-O

AG1-AB

AG1-FT

252 26

440 118

442 133

ified (AG1-O) and wax modified (AG1-AB and AG1-FT) mixtures. It can be observed from Fig. 6 that there is a good agreement between the number of cycles to micro-crack initiation ðN mc Þ from the fatigue simulation results and the equivalent micro-crack stress ðrmc Þ values obtained from Eq. (27). The number of cycles to micro-crack initiation ðN mc Þ increases as the equivalent microcrack stress ðrmc Þ increases. It can be seen from Fig. 6 that the wax modified mixtures can be easily distinguished from the unmodified mixture using the N mc - rmc relationship. It can be observed that the wax-modified mixtures show a rapid increase in micro-crack resistance with respect to changes in the equivalent micro-crack stress ðrmc Þ. The wax-modified mixtures exhibit a higher N mc - rmc slope value between 2775 and 3059, while the unmodified mixture exhibit a slope of 1496 for the range of temperatures used in the relationship.

I. Onifade, B. Birgisson / Construction and Building Materials 148 (2017) 521–530

529

Fig. 6. Plot of Number of cycles to micro-crack initiation ðN mc Þ and the equivalent micro-crack initiation stress ðrmc Þ for unmodified (AG1-O) and wax modified (AG1-AB and AG1-FT) mixtures.

5. Summary and conclusions In this paper, a new mixture cracking performance parameter termed ‘‘the equivalent micro-crack stress” ðrmc Þ, obtained from fundamental material properties is proposed for fracture characterization of asphalt concrete mixtures with respect to resistance to micro-crack initiation. The equivalent micro-crack stress ðrmc Þ takes a similar form as the Griffith’s failure stress criterion and accounts for the influence of the material stiffness and a critical strain energy density at the instance of micro-crack initiation on the material fracture behaviour. The critical strain energy density at the instance of initiation of micro-crack is termed the ‘‘microcrack initiation threshold” (MCIT). It was observed that there is a good agreement between the effective micro-crack stress obtained from monotonic loading mode and the fracture behaviour using a cyclic loading condition (without rest periods) under controlled stress mode. The higher the effective micro-crack stress parameter, the better the fracture performance of the mixture. The equivalent micro-crack stress approach is used to characterize the fracture behaviour of unmodified and wax modified asphalt concrete mixtures. The result shows that the wax modified mixtures exhibit better resistance to micro-crack formation at the temperatures considered. At higher temperatures, the increased tendency of the material to undergo plastic deformation can have a significant influence on the MCIT parameter, as the increased tendency of the material to withstand more deformation before the initiation of micro-crack results in an increase in the MCIT, which invariably reflects in the effective micro-crack stress value. The FTparaffin wax modified mixtures exhibit the best resistance to micro-crack formation among the mixtures considered in this study. The observation is in agreement with the results of an earlier investigation of the fracture characterization of wax modified mixtures using the Energy Ratio (ER) approach in [40]. The effective micro-crack stress is a valuable material cracking performance criterion that can be used as the basis for the selection of materials for specific engineering purpose. The viscoelastic energy-based damage model provides a mechanics basis for the characterization of asphalt damage behaviour which presents the possibility of developing further insight into the fundamentals of the material behaviour. The influence of aging and moisture on asphalt cracking performance can also be studied using the equivalent micro-stress approach presented in this paper. Pending

further validation, it can be seen that the equivalent micro-crack stress approach presented may be applicable to a wide range of materials, and it offers a certain advantage in the fracture characterization of porous and composite materials. The approach has been validated based on experimental results from the Superpave IDT test results, the same approach may be applicable using other material characterization tests such as the uniaxial tensile test and test results from actual fatigue testing. References [1] J.A. Epps, C.L. Monismith, Fatigue of Asphalt Concrete Mixtures—Summary of Existing Information, in: B. Gallaway (Ed.), Fatigue of Compacted Bituminous Aggregate Mixtures, STP38807S, ASTM International, 1972, pp. 19–45. [2] G.G. Al-Khateeb, K.A. Ghuzlan, The combined effect of loading frequency, temperature, and stress level on the fatigue life of asphalt paving mixtures using the IDT test configuration, Int. J. Fatigue 59 (2014) 254–261, http://dx. doi.org/10.1016/j.ijfatigue.2013.08.011. [3] R.G. Hicks, F.N. Finn, C.L. Monismith, R.B. Leahy, Validation of SHRP binder specification through mix testing (with discussion), J. Assoc. Asphalt Paving Technol. 62 (1993). [4] Y.R. Kim, N.P. Khosla, N. Kim, Effect of temperature and mixture variables on fatigue life predicted by diametral fatigue testing, Transp. Res. Record 1317 (1991) 128–138. [5] G.D. Airey, A. Collop, N.H. Thom, A. Shiratori, S.E. Zoorob, A laboratory evaluation of secondary aggregates in bituminous mixtures, J. Assoc. Asphalt Paving Technol. 73 (2004) 731–770. [6] R. Reese, Properties of aged asphalt binder related to asphalt concrete fatigue life, Journal of the Association of Asphalt Paving Technologists; vol. 66. Association of Asphalt Paving Technologists (AAPT), 1997. [7] G.M., Rowe, M.G. Bouldin, Improved techniques to evaluate the fatigue resistance of asphaltic mixtures. ISBN 90-802884-3-8, 2000. [8] Y.R. Kim, D.N. Little, R.L. Lytton, Fatigue and Healing Characterization of Asphalt Mixtures, Journal of Materials in Civil Engineering 15 (2003). doi: 10.1061/(ASCE)0899-1561(2003)15:1(75)#sthash.K8dNiele.dpuf. [9] R.L. Lytton, J. Uzan, E.G. Fernando, R. Roque, D. Hiltunen, S.M. Stoffels. Development and Validation of Performance Prediction Models and Specifications for Asphalt Binders and Paving Mixes. SHRP-A-357, 1993. ISBN 0-309-05617-9. [10] Inc. A. Guide for mechanistic-empirical design of new and rehabilitated pavement structures. Tech. Rep.; Champaign, 2004. [11] R. Roque, B. Birgisson, B. Sangpetngam, Z. Zhang, Hot mix asphalt fracture mechanics: a fundamental crack growth law for asphalt mixtures. In: Journal of the Association of Asphalt Paving Technologists; vol. 71. Association of Asphalt Paving Technologists (AAPT), 2002. [12] B. Sangpetngam, B. Birgisson, R. Roque, Development of an efficient hot mix asphalt fracture mechanics-based crack growth simulator, in: Proc., 82nd Transportation Research Board Annual Meeting, Transportation Research Board (TRB), Washington, D.C, 2003, http://dx.doi.org/10.3141/1832-13. [13] B. Birgisson, J. Wang, R. Roque, B. Sangpetngam, Numerical Implementation of a Strain Energy-based Fracture Model for HMA Materials, Road Mater.

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