Author's personal copy International Journal of Plasticity...

Author's personal copy

A modified viscoplastic model to predict the permanent deformation ofasphaltic materials under cyclic-compression loading at high temperatures

Masoud K. Darabi a, Rashid K. Abu Al-Rub a,⇑, Eyad A. Masad a,b, Chien-Wei Huang b,Dallas N. Little a

a Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843, USAb Mechanical Engineering Program, Texas A&M University at Qatar, Doha, Qatar

a r t i c l e i n f o

Article history:Received 23 October 2011Received in final revised form 28 January 2012Available online 8 March 2012

Keywords:Hardening-relaxationViscoplastic-softeningViscoelasticityAsphalt concreteConstitutive modeling

a b s t r a c t

When subjected to cyclic creep (ratcheting) loading with rest periods between the loadingcycles, the viscoplastic behavior of asphaltic materials changes such the rate of accumula-tion of the viscoplastic strain at the beginning of the subsequent loading cycle increasescomparing to that at the end of the preceding loading cycle. This phenomenon is referredto as the hardening-relaxation (or viscoplastic-softening) and is a key element in predictingthe permanent deformation (rutting) of asphalt pavements which is one of the most impor-tant distresses in asphalt pavements. This paper presents a phenomenological-basedrate-dependent hardening-relaxation model to significantly enhance the prediction ofthe permanent deformation in asphaltic materials subjected to cyclic-compression load-ings at high temperatures. A hardening-relaxation memory surface is defined in the visco-plastic strain space as the general condition for the initiation and evolution of thehardening-relaxation (or viscoplastic-softening). The memory surface is formulated to bea function of an internal state variable memorizing the maximum viscoplastic strain forwhich the softening has been occurred during the deformation history. The evolution func-tion for the hardening-relaxation model is then defined as a function of the hardening-relaxation internal state variable. The proposed viscoplastic-softening model is coupledto the nonlinear Schapery’s viscoelastic and Perzyna’s viscoplastic models. The numericalalgorithms for the proposed model are implemented in the well-known finite element codeAbaqus via the user material subroutine UMAT. The model is then calibrated and verifiedby comparing the model predictions and experimental data that includes cyclic creep-recovery loadings at different stress levels, loading times, rest periods, and confinementlevels. Model predictions show that the proposed approach provides a promising tool forconstitutive modeling of cyclic hardening-relaxation in asphaltic materials and in generalin time- and rate-dependent materials.

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1. Introduction

Asphalt concrete, as one of the main constituents of flexible pavements, are subjected to repeated actions of traffic load-ing. These repetitive loadings cause the pavements to experience several major distresses such as fatigue damage and rutting(i.e. permanent deformation) where the contribution of each depends strongly on the pavement’s material, structure, tem-perature, moisture level, and age. It is well-known that the rutting is one of the most important pavement’s distresses at high

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⇑ Corresponding author. Tel.: +1 979 862 6603; fax: +1 979 845 6554.E-mail addresses: [email protected], [email protected] (R.K. Abu Al-Rub).

International Journal of Plasticity 35 (2012) 100–134

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temperatures, whereas, fatigue damage is more pronounced at intermediate and low temperatures (e.g. Morrison et al.,1994; Uzan, 1996; Ramadhan et al., 1998; Tashman et al., 2005; Kim et al., 2008). In this paper, we focus on predictingthe permanent deformation of asphalt concrete. The fatigue damage response of asphalt concrete has already been discussedin the authors’ previous works (e.g. Abu Al-Rub et al., 2010; Darabi et al., 2011a).

Several authors have used different approaches and proposed constitutive models to predict the plastic/viscoplastic re-sponse of time- and rate-dependent materials such as bituminous materials and polymers (e.g. Perzyna, 1971; Hashiguchi,1980; Schapery, 1999; Kaneko and Oyamada, 2000; Seibi et al., 2001; Chehab et al., 2003; Gibson et al., 2003; Hashiguchi,2005; Masad et al., 2005; Hashiguchi and Mase, 2007; Masad et al., 2007; Anandarajah, 2008; Hashiguchi and Ozaki, 2008; LaRagione et al., 2008; Vorobiev, 2008; Abu Al-Rub et al., 2009, 2010; Berbenni et al., 2010; Zhu et al., 2010; Abu Al-Rub et al.,2012; Tehrani and Abu Al-Rub, 2011). To name few works on constitutive modeling of asphalt concrete, Chehab et al. (2003),Gibson et al. (2003), and Underwood et al. (2006) used Schpaery viscoelastic-damage model (Schapery, 1975a–c) which hasbeen coupled to viscoplasticity (Schapery, 1999) to model the permanent deformation of asphalt concrete. However, theviscoplastic component of these models can only be used to predict the viscoplasticity due to tensile stresses. To modelthe viscoplastic response of asphalt concrete in both tension and compression, Seibi et al. (2001) and Masad et al. (2005,2007) used a Perzyna-type viscoplasticity model (Perzyna, 1971). Later, Saadeh et al. (2007) and Huang et al. (2011a) coupledthe Schapery (1969a) nonlinear-viscoelastic model to the Perzyna (1971) viscoplastic model in order to accurately predictthe permanent deformation of asphalt concrete. Recently, Abu Al-Rub et al. (2010) and Darabi et al. (2011a,b) coupledSchapery (1969a) nonlinear-viscoelastic and Perzyna (1971) viscoplastic models to time-dependent damage and micro-damage healing (Darabi et al., 2012a) models in order to accurately predict the mechanical response of asphalt concreteunder various loading and environmental conditions.

Regardless of the specific viscoplastic constitutive model used for asphalt concrete, all the above mentioned works em-ploy the classical hardening viscoplasticity theories to predict the permanent deformation of asphalt concrete materials.Fig. 1(b) shows experimental measurements for an asphalt concrete subjected to the repeated constant stress creep-recovery

(a)

(b)

Deviatoric stress (kPa)

827(kPa)

LT UT Time (sec)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12

Stra

in(%

)

Time (Sec)

Experimental measurements

Model prediction using the calibrated VE-VP model

Fig. 1. (a) Stress history for a compression repeated creep-recovery test at 55 �C when the loading time (LT) is 1.6 s and unloading time (UT) is 0.9 s; (b)experimental measurements and prediction of a calibrated classical Perzyna-type viscoplastic model which is coupled to viscoelasticity. Model predictionsshow that the strain response saturates after two cycles. Therefore, no more viscoplastic strain is predicted after the second cycle. However, experimentalmeasurements show that the material builds up more viscoplastic strain in each loading cycle.

M.K. Darabi et al. / International Journal of Plasticity 35 (2012) 100–134 101


loading shown in Fig. 1(a). Fig. 1(b) shows the model predictions when using a calibrated classical hardening Perzyna-typeviscoplastic model that is coupled to Schapery’s nonlinear-viscoelastic model (e.g. Huang et al., 2011a). Fig. 1(b) shows thatthe strain response predicted by the classical hardening Perzyna-type viscoplastic model saturates and reaches its maximumvalue after the second loading cycle. Therefore, no further viscoplastic strain is expected during the next loading/unloadingcycles unless the applied stress level increases. However, the experimental data in Fig. 1(b) show that if the material is put inrest before the next loading cycle is applied, then it keeps accumulating viscoplastic strain during each loading cycle andafter the saturation of the viscoplastic strain. Obviously, the model predictions significantly deviate from experimental mea-surements if the number of loading cycles increases. This discrepancy between the experimental measurements and predic-tion of classical hardening viscoplasticity models for asphalt concrete has been briefly reported by Masad et al. (2005) andSaadeh et al. (2007). They explained this distinct behavior of asphalt concrete based on the anisotropic nature of deformationin asphalt concrete that is induced by aggregate reorientation during deformation.

This discrepancy between theory predictions and experimental measurements is due to an inherent assumption in theclassical hardening viscoplasticity theories. The classical viscoplastic hardening theories assume that the viscoplastic prop-erties of the material do not change during the unloading (or the rest period), such that the viscoplastic properties at the endof the preceding loading cycle is the same as the viscoplastic properties at the beginning of the subsequent loading cycle.This inherent assumption of the classical hardening viscoplasticity theories can be reasonably applied to homogenous mate-rials with no microstructure or to materials where the change in the microstructure during the unloading is negligible. How-ever, this is not the case for asphalt concrete where the microstructure changes significantly during the unloading and restperiods, specially at high temperatures.

The microstructure of asphalt concrete can be considered as the coarse aggregates surrounded by the fine aggregate mix-ture (FAM) which consists of fine fillers, fine aggregates, air voids, and the asphalt binder. The overall mechanical response ofan asphalt concrete is controlled by the properties of its constituents. For example, the viscoelastic response of asphalt con-crete is mainly controlled by the viscoelastic properties of the binder (or more precisely by the viscoelastic properties of fineaggregate matrix, FAM). On the other hand, aggregate positions, aggregate orientations, friction between the aggregates, andaggregate interlocking mainly control the viscoplastic response of asphalt concrete, especially in compression. Therefore, theviscoplastic state and the viscoplastic properties of asphalt concrete are strongly related to their microstructure.

During the loading cycle, aggregates are pushed against each other, reorient, change position, and deform the binder inbetween. This change in the microstructure brings the material to a specific viscoplastic and hardening state at the end of theloading cycle. However, during the rest period, the confined binder between the aggregates apply residual stresses to thesurrounding aggregates causing them to redistribute and change position and orientation, such that the viscoplastic stateof the material (which is related to the material’s microstructure) at the end of the rest period is different from its stateat the end of the preceding loading cycle. Therefore, it seems quite natural that some of the viscoplastic properties of asphaltconcrete materials evolve during the rest period, such that it increases the potential of asphalt concrete to develop moreviscoplastic deformation during the next loading cycle. In this work, we relate this distinct behavior to the cyclicviscoplastic-softening (or hardening-relaxation) behavior of asphalt concrete at high temperatures. However, it should benoted that the softening in the viscoplastic response investigated in this study is not that due to damage (cracks and voids)evolution or temperature increase. This cyclic viscoplastic-softening behavior can be explained through using viscoplasticitytheory as a recovery in the material’s ability to harden. Therefore, during the rest period, the change in the asphalt concrete’smicrostructure allows the asphalt composite to recover a previous viscoplastic hardening state and consequently to exhibitmore viscoplastic deformation. In fact, the authors feel that the term hardening-relaxation is more suitable to describe thisviscoplastic-softening, which is a characteristic of time-dependent materials. In other words, the hardening level inducedin the material during the loading stage starts relaxing during the rest periods.

Numerous works are available in the literature to model the plastic/viscoplastic softening response of materials duringthe monotonic loading conditions. These works mainly relate the plastic/viscoplastic softening response of materials duringthe loading to the strain localization, damage evolution, and temperature change during the deformation (e.g. Kachanov,1958; Rabotnov, 1969; Cordebois and Sidoroff, 1982; Lemaitre, 1985; Ortiz, 1985; Ohtani and Chen, 1989; Voyiadjis andKattan, 1990; Needleman and Tvergaard, 1991; Ord, 1991; Aifantis, 1992; deBorst and Muhlhaus, 1992; Zhu and Cescotto,1995; Krajcinovic, 1996; Zhou and Clode, 1998; Schapery, 1999; Borja and Regueiro, 2001; Semiatin and Bieler, 2001; AbuAl-Rub and Voyiadjis, 2003; Engelen et al., 2003; Voyiadjis et al., 2003; Wei et al., 2008; Ling and Belytschko, 2009). How-ever, few studies are available in the literature on modeling the viscoplastic-softening effect (or hardening-relaxation in thiswork) on the mechanical response of time-dependent materials subjected to cyclic creep loadings when rest periods areintroduced between the loading cycles. Hashiguchi and Ozaki (2008) explained the plastic-softening response of slidingbodies under cyclic loading using the subloading-friction model (Hashiguchi et al., 2005). However, their model can onlybe applied to predict the plastic-softening response for very simple cases, such as two discrete bodies sliding over together,and cannot be applied to model the plastic/viscoplastic softening response of materials with complex microstructure such asasphalt concrete. Their model and approach will be discussed in more details in the subsequent sections.

Surprisingly, very few works are available in the literature on modeling the hardening-relaxation behavior of asphalt con-crete under cyclic loading despite the high importance of hardening-relaxation (or viscoplastic-softening) effect on the pre-diction of rutting performance of asphalt pavements at high temperatures, which is one of the most important distresses inasphalt pavements. Masad et al. (2005) and Saadeh et al. (2007) stated that the deviation of their model predictions fromexperimental measurements for the repeated creep-recovery tests is related to the anisotropic deformation of asphalt

102 M.K. Darabi et al. / International Journal of Plasticity 35 (2012) 100–134


concrete. However, they did not modify their model to capture the experimental measurements. Recently, Yun and Kim(2011) modeled the viscoplastic-softening of asphalt concrete by defining a softening function in the yield surface duringthe unloading which is different from the hardening function during the loading stage. However, this model is augmentedwith many model parameters that significantly increase the number of required tests to identify these parameters. More-over, this model can only be used in uniaxial compression and for specific loading conditions. Huang et al. (in preparation)also proposed a phenomenological model to capture the viscoplastic response of asphalt concrete at high temperatures andin uniaxial compression. However, their model is also developed for specific loading conditions and does not possess a gen-eral initiation criterion that can be used for general cases. This work tries to propose a robust approach for constitutive mod-eling of the phenomenon reported by Masad et al. (2005) and Yun and Kim (2011). However, the term ‘‘viscoplastic-softening’’ is renamed here ‘‘hardening-relaxation’’ in order to avoid the confusion for the viscoplastic softening due to dam-age and strain localization. Moreover, we believe that this term describes the observed cyclic viscoplastic response of time-dependent materials in a more natural way.

Based on the experimental evidences and motivated by the arguments on the physical mechanism of the hardening-relaxation phenomenon in asphalt concrete, this study introduces a surface in the viscoplastic strain space (referred to asthe hardening-relaxation memory surface). This surface can be used as the general criterion for the initiation and evolutionof the hardening-relaxation during the cyclic loading. The concept of the proposed hardening-relaxation memory surfaceis similar to the pioneering works of Ohno and his co-workers (e.g. Ohno, 1982; Ohno and Wang, 1993) on the cyclic plas-ticity of metals. Moreover, Hassan and Kyriakides (1994a,b) have presented a systematic experimental and modeling studyof cyclic hardening and softening metals and modified Ohno (1982) plastic memory surface. In the current study, a differentmemory surface is defined as a function of an internal state variable which memorizes the maximum experienced viscoplas-tic strain during the loading/unloading history. An evolution function is then proposed to capture the cyclic viscoplastic re-sponse of asphalt concrete. The model predictions show that the hardening-relaxation memory surface approach provides apromising tool for constitutive modeling of cyclic viscoplastic deformation in asphalt concrete and ultimately the rutting inasphalt pavements.

The outline of this paper is as follows: Section 2 discusses the mechanism of hardening-relaxation in asphaltic materialsfollowed by the proposed hardening-relaxation model. The proposed hardening-relaxation model is coupled to theSchapery’s nonlinear-viscoelastic model in Section 3. Determination of the viscoelastic, viscoplastic, and hardening-relaxation model parameters is discussed in Section 4. In Section 5, the implemented hardening-relaxation model isvalidated against a set of experimental measurements on asphalt concrete. The numerical implementation techniques forthe proposed constitutive model are presented in Appendix I. Final conclusions are drawn in Section 6.

2. Hardening-relaxation mechanism and proposed hardening-relaxation model

2.1. Classical hardening Perzyna-type viscoplastic constitutive model

In this section, the Perzyna-type viscoplastic model which has been used previously to model the viscoplastic response ofasphalt concrete (as presented in Masad et al. (2005)) is summarized. We will show that the classical Perzyna-type visco-plastic model with either classical isotropic or kinematic hardening rules fails to predict the permanent deformation of as-phalt concrete materials subjected to cyclic loading at high temperatures when the rest period is introduced between theloading cycles. We will show that the classical isotropic/kinematic hardening rules cannot capture the effects of pulse load-ing time and/or unloading time (i.e. rest period between the loading cycles) on permanent deformation response of asphaltconcrete at high temperatures. Specifically, we will show that while the incorporation of kinematic hardening partially cap-tures the effect of different pulse loading times, it fails to capture different viscoplastic strain responses for different unload-ing (resting) times and does not capture of the pulse (loading) time effect properly. A modeling technique will be proposed inthe subsequent sections to remedy this problem and to enhance the classical Perzyna-type viscoplastic model in capturingboth pulse loading time and unloading time effects of viscoplastic response of asphalt concrete materials subjected to cyclicloading at high temperatures.

Based on the classical viscoplastic theories, the rate of the viscoplastic strain tensor _evp is defined through the followingclassical non-associative viscoplastic flow rule:

_evpij ¼ _cvp @F@rij

ð1Þ

where the superimposed ‘‘�’’ designates the derivative with respect to time; _cvp and F are the viscoplastic multiplier and theviscoplastic potential function, respectively; and r represents the stress tensor. Physically, oF/orij determines the direction ofthe viscoplastic strain tensor, whereas _cvp is a positive scalar representing the magnitude of the viscoplastic strain which canbe expressed in terms of an overstress function and a fluidity parameter, such that (Perzyna, 1971):

_cvp ¼ CvphUðf ÞiN ð2Þ

where Cvp is the viscoplastic fluidity parameter such that 1/Cvp has the units of time and represents the viscoplasticity relax-ation time, N is the viscoplastic rate sensitivity exponent, and U is the overstress function which is expressed in terms of the



yield function f. Moreover, hi in Eq. (2) is the Macaulay brackets defined by hUi = (U + |U|)/2. The following simple expres-sion is assumed here for the overstress function U:

Uðf Þ ¼ fr0y

ð3Þ

where r0y is a yield stress quantity used to normalize the overstress function and can be assumed unity. Eqs. (1)–(3) indicatethat the viscoplasticity occurs only if the overstress function U is positive.

A Drucker–Prager-type yield surface function has been used by several researchers for describing the viscoplastic flowbehavior of asphalt concrete since it takes into consideration the confinement, aggregate friction, aggregate interlocking,and dilative behavior of asphalt concrete (e.g. Seibi et al., 2001; Cela, 2002; Dessouky, 2005; Tashman et al., 2005; Masadet al., 2007; Saadeh et al., 2007; Huang, 2008). In this study, we use a modified Drucker–Prager yield function that distin-guishes between the distinct behavior of asphalt concrete in contraction and extension and also takes into considerationthe pressure sensitivity (e.g. Dessouky, 2005; Masad et al., 2007). This modified Drucker–Prager yield function f can be ex-pressed as follows:

f ¼ s� aI1 � jðpÞ ð4Þ

where a is the pressure-sensitivity material parameter; j(p) is the isotropic hardening function associated with the cohesivecharacteristics of the material, which depends on the microstructure of asphalt concrete, and is a function of the effective(equivalent) viscoplastic strain p; I1 = rkk is the first stress invariant; and s is the deviatoric effective shear stress modifiedto distinguish between the mechanical response of asphalt concrete in contraction and extension loading modes, such that:

s ¼ffiffiffiffiffiffiffi3J2

p2

1þ 1dvpþ 1� 1

dvp

� �3J3ffiffiffiffiffiffiffi

3J32q

264375 ð5Þ

where J2 ¼ 12 ðSij � XijÞðSij � XijÞ and J3 ¼ 12 ðSij � XijÞðSjk � XjkÞðSki � XkiÞ are the second and third deviatoric stress invariants,respectively. The parameter dvp is a material constant that gives the distinction of the viscoplastic response of asphalt con-crete in contraction and extension loading conditions and can be characterized as the ratio of the material’s yield strength intension to that in compression. Also, S (Sij ¼ rij � 13 rkkdij with dij is the Kronecker delta) and X are the stress and backstressdeviator tensors, respectively. The kinematic hardening rule of Armstrong and Frederick (1966) is used in this work for theevolution of X, such that:

_Xij ¼23

C _evpij � cXij _p ð6Þ

where C and c are material constants.Following the work of Lemaître and Chaboche (1990), the isotropic hardening function j(p) is expressed as an exponen-

tial function of the effective viscoplastic strain p, such that:

jðpÞ ¼ j0 þ j1½1� expð�j2pÞ� ð7Þ

where j0, j1, and j2 are material parameters; j0 defines the initial yield strength; j0 + j1 determines the saturated yieldstress; and j2 is the strain hardening rate. It should be noted that other isotropic hardening functions can be assumed with-out changing the overall conclusions of the current study.

Several studies have shown that the viscoplastic deformation of asphalt concrete and geomaterials in general is non-asso-ciative such that the plastic potential function F should be different from the yield surface function f (e.g. Zienkiewicz et al.,1975; Oda and Nakayama, 1989; Cristescu, 1994; Florea, 1994; Bousshine et al., 2001). Hence, the direction of the viscoplas-tic strain increment tensor is no longer normal to the yield surface function f, but to the plastic potential F. To have non-asso-ciative viscoplasticity and to avoid the unnecessary increase in the model parameters, this study uses a viscoplastic potentialfunction F which has a form similar to the yield surface function f (Eq. (4)) but with a pressure sensitivity model parameter bwhich is different to that of the a in the yield surface function, such that:

F ¼ s� bI1 ð8Þ

where b is a material parameter describing the dilation or contraction behavior of the asphalt concrete. The effective visco-plastic strain rate _p can, therefore, be defined as:

_p ¼ A�1ffiffiffiffiffiffiffiffiffiffiffiffiffi_evpij _e

vpij

q; A ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2 0:5þ b=3

1� b=3

� �2sð9Þ

We will show in Section 2.2 that the Perzyna-type viscoplastic model as presented in this section is not capable of effec-tively capturing the effects of rest time and pulse time on the viscoplastic response of asphalt concrete subjected to cycliccompressive loading at high temperatures.



2.2. Rest time and pulse time effects

In this section, we show that the classical hardening viscoplasticity theories cannot effectively capture the effects of boththe rest and the pulse times on the viscoplastic response of asphaltic materials. The isotropic hardening model presented inEq. (7) and the kinematic hardening model presented in Eq. (6) are used in this section to investigate the capabilities of theclassical hardening viscoplasticity model in capturing the viscoplastic response at different loading (pulse) and unloading(rest) times. The material parameters for the isotropic hardening model used in obtaining the results are listed in Table 3.The values of C = 250 MPa and c = 10 are also used for the kinematic hardening model. The identification procedure of thesematerial parameters will be presented in Section 4. It should be noted that the viscoelasticity model that will be presented inSection 3 has not been used in obtaining the results in this section in order to investigate the effect of the viscoplasticitymodel more clearly. However, the conclusion of this section remains intact even if the viscoelastic model is also consideredas will be shown later.

The loading history presented in Fig. 1(a) is recalled here. To investigate the effect of the unloading time or rest period onviscoplastic response, the simulations are conducted at the fixed loading time of 1.6 sec but at three different unloadingtimes of 0.05, 1, and 200 s. The simulations are repeated for the viscoplastic model with isotropic hardening only, kinematichardening only, and combined isotropic and kinematic hardening models to investigate the effect of each hardening compo-nent on capturing the rest period effect. Moreover, in all these simulations, the accumulated loading time is set to be thesame in order to present more meaningful results.

Fig. 2 clearly shows that the classical hardening Perzyna-type viscoplastic model that is presented in the previous sectionwith either isotropic hardening or kinematic hardening rules does not distinguish between different unloading times andpredicts identical responses for the viscoplastic strain regardless of the duration of the rest time. In fact, once the stress iszero, neither the viscoplastic strain nor the kinematic/isotropic hardening functions evolve anymore which makes the clas-sical hardening viscoplasticity theories incapable of capturing the effect of rest period on viscoplastic response of asphalticmaterials. However, the experimental measurements in Fig. 2 clearly show that the material builds up more viscoplasticstrain at the same accumulated loading time as the rest period increases.

The model predictions using isotropic hardening only (Fig. 2) saturates soon after the accumulated loading time increases.The isotropic hardening model causes the initial yield surface to expand during the loading time, such that the yield surfacereaches to the stress state soon after the loading time increases. During the unloading and rest period, however, the stressfield will always remain within the yield surface since the yield surface does not shrink or move during the unloading andrest periods. Therefore, no extra viscoplastic strain will be predicted after several loading cycles unless the stress level ex-ceeds its previous level.

However, as shown in Fig. 2, the model with kinematic hardening only shows that the viscoplastic strain accumulateslinearly with the accumulated loading time after the first loading cycle. The reason is that the yield surface moves towardthe stress point due to kinematic hardening, such that the stress point will be located on the yield surface at the end of

0

0.25

0.5

0.75

0 3 6 9 12 15

Vi

VVsc

opla

stic

stra

in(%

)

Accumulated loading time (sec)

Prerr diction: kinematic hardening only, UT=0.05, 1, and 200 secPrerr diction: isotropic haraa dening only, UT=0.05, 1, and 200 secPrerr diction: isotropic and kinematic haraa dening, UT=0.05, 1, and 200 secExperirr memm ntal memm asurerr menmm ts: UT=0.05 secExperirr memm ntal memm asurerr menmm ts: UT=1 secExperirr memm ntal memm asurerr menmm ts: UT=200 sec

Fig. 2. Model prediction using calibrated classical hardening Perzyna-type viscoplastic model as compared to experimental measurements for different restperiods for the loading condition in Fig. 1(a). The Perzyna-type viscoplastic model with either isotropic or kinematic hardening components does notcapture the effect of unloading time (UT) or rest period on viscoplastic response of asphalt concrete and predicts the same viscoplastic strain response fordifferent rest periods. However, experimental data show that the duration of the resting time has a significant effect on the viscoplastic strain response,such that the longer the rest period the higher the viscoplastic strain.



the first loading cycle. During the unloading, the yield surface moves back, such that zero stress state will be located on theyield surface during the rest period. For the next loading cycles, the yield surface keeps moving forward and backward to thesame stress point and the zero stress level, such that it predicts the same increment of the viscoplastic strain during a load-ing-unloading cycle. Therefore, after the first loading cycle, the viscoplastic strain increases linearly with the accumulatedloading time.

Fig. 3 schematically presents the effect of the combined isotropic and kinematic hardening models on the viscoplasticstrain response for the loading history shown in Fig. 1(a). This Figure shows that the yield surface expands and also its centermoves by DX toward the stress state ‘‘A’’ as the results of isotropic and kinematic hardening models, respectively. The rate ofthe viscoplastic strain at time ‘‘t’’ is governed by the distance of stress point ‘‘A’’ from the yield surface at time ‘‘t’’ (i.e. UKI).However, as the accumulated loading time increases, the yield surface eventually reaches the stress point and the overstressbecomes zero. After this time, no further viscoplastic strain accumulates. It should be noted that the predicted viscoplasticstrain when both isotropic and kinematic hardening models are used has its lowest value comparing to the cases when eitherthe isotropic or the kinematic hardening models are used since the yield surface expands and translates at the same time(see Fig. 2). For the same reason, the viscoplastic strain also saturates faster comparing to the case with isotropic hardeningonly (see Fig. 2).

Moreover, to investigate the effect of the pulse time on the viscoplastic response of asphalt concrete, the simulations areconducted for the loading history shown in Fig. 1(a) at a fixed unloading time of 0.9 s but at different pulse times (loadingtimes) of 0.1, 0.4, and 1.6 s. Fig. 4 shows that the Perzyna-type viscoplastic model with isotropic hardening only does notcapture the effect of different pulse times on the viscoplastic strain response of asphalt concrete, such that it predicts iden-tical viscoplastic strain values at the same accumulated loading time regardless of the pulse load duration. The model withboth isotropic and kinematic hardening components saturates very fast and therefore does not distinguish between differentpulse times. However, the model with kinematic hardening only predicts partially the proper trend of the pulse time effect,such that at a fixed accumulated loading time, the shorter the pulse time the higher the predicted viscoplastic strain. How-ever, as shown in Fig. 4, the model with the kinematic hardening only builds up the same viscoplastic strain increment dur-ing each loading cycle regardless of the duration of the pulse time, such that the predictions for different pulse times lie onthe same line. However, the experimental measurements in Fig. 4 show clear differences for the viscoplastic strain versusnumber of loading cycle as the pulse time changes. As shown in Fig. 4, at the same number of loading cycles, the value ofthe viscoplastic strain increases as the pulse time duration increases. The reason is that at the same number of loading cycles,the material has experienced the stress for a longer time and therefore builds up more viscoplastic strain as the pulse timeincreases. Figs. 2 and 4 clearly shows that the Perzyna-type viscoplasticity model with isotropic and/or kinematic hardeningcomponents are not capable of capturing either the effect of pulse load time or rest time on the viscoplastic response of as-phalt concrete. Therefore, the focus of this paper is to propose a remedy to this problem.

This section clearly shows that the classical hardening viscoplasticity theories as presented in the previous section andconsistently used in the literature cannot effectively capture the effect of both pulse time and rest period on the viscoplastic

Fig. 3. Schematic representation of the effect of the combined isotropic and kinematic hardening on the viscoplastic strain response. The yield surface is atits initial location at the beginning of the first loading cycle. During the loading cycles, the yield surface expands as a result of the isotropic hardening modeland also moves toward the stress point ‘‘A’’ by DX as a result of the kinematic hardening model. The rate of the viscoplastic strain at time ‘‘t’’ is governed bythe overstress UKI, distance from the stress point ‘‘A’’ to the yield surface, which decreases as the accumulated loading time increases until the yield surfacereaches the stress state ‘‘A’’ and the viscoplastic strain response saturates. After this point, no further viscoplastic strain will be accumulated.



response of asphalt concrete which motivates the necessity for modifying them in order to capture the viscoplastic responseof asphalt concrete. This shortcoming of the classical hardening viscoplasticity theories is the main motivation in conductingthis work in order to enhance these theories in predicting the permanent deformation of asphalt concrete accurately. Themicrostructural mechanism and the modeling techniques associated with this distinct behavior of asphalt concrete willbe discussed in more details in the subsequent sections.

2.3. Micromechanical arguments for hardening-relaxation mechanism

Asphalt concrete materials are subjected to repeated loading conditions during the pavement service life. The repetitiveaction of the loading/unloading cycles specially at high temperatures causes the asphalt concrete to experience permanentdeformations. However, the material’s microstructure changes during the rest period and becomes different from its micro-structure before at the end of the preceding loading stage. These changes in the microstructure specifically at high temper-atures progressively change the viscoplastic properties of the asphalt concrete which pose a challenge in modeling of theviscoplastic response of asphalt concrete.

Fig. 5(b)–(d) schematically represent intuitive arguments concerning the changes in the microstructure of an asphalt con-crete during a compressive loading/unloading stress history shown in Fig. 5(a). A simple schematic of the asphalt concretemicrostructure before loading is shown in Fig. 5(b). As shown in Fig. 5(c) aggregates are pushed against each other during thecompressive loading stage. The compressive stress reorients the aggregates such that the angle between the longest dimen-sions of aggregate and normal to the stress direction decreases compared to that before the loading is applied. During thisstage, asphalt binder confines amongst the aggregates and starts to creep as the aggregates are pushed together. Clearly, theconfinement of the asphalt binder induces time-dependent residual stresses that tend to push the aggregates back in order torelease these induced residual stresses. However, as shown in Fig. 5(d), during the rest period, the induced time-dependentresidual stresses rearrange and rotate the aggregates to positions that are not the same as their locations and orientations atthe end of the loading stage (i.e. position and orientation of the aggregates in Fig. 5(d)). This time-dependent rearrangementin the microstructure due to the viscoelastic response of the asphalt binder allows the asphalt composite to recover part of itsviscoplastic hardening (i.e. the asphalt concrete’s ability to harden) during the rest period. As temperature increases, the dif-ference between the orientation and position of the aggregates at the end of the loading and during the unloading increases.The reason is that the binder gets softer and deforms more easily as temperature increases which provides more mobility forthe aggregates. Moreover, since the binder and the residual stresses relax faster during the rest period as temperature in-creases, the time-dependent residual stresses cause faster rearrangement of the microstructure and in turn more hardeningrecovery. Obviously, those properties of the asphalt concrete which are correlated to the relative positions, orientations, and

0

0.5

1

1.5

0 20 40 60 80 100

Vis

copl

asti

c st

rain

(%)

Number of loading cycles (N)

Prediction: kinematic hardening only, LT=0.1 secPrediction: kinematic hardening only, LT=0.4 secPrediction: kinematic hardening only: LT=1.6 secPrediction: isotropic hardening only, LT=0.1, 0.4, and 1.6 sec Prediction: kinematic and isotropic hardening, LT=0.1, 0.4, and 1.6 secExperimental measurements: LT=0.1 secExperimental measurements: LT=0.4 secExperimental measurements: LT=1.6 sec

Fig. 4. Model prediction using Perzyna-type viscoplastic model with kinematic hardening only, isotropic hardening only, and both isotropic and kinematichardening as compared to experimental measurements for the stress history shown in Fig. 1(a) at fixed unloading time of 0.9 s and for different pulse loadtimes or loading times (LT) of 0.1, 0.4, and 1.6 s. The kinematic hardening model predicts proper trend for the effect of the LT. However, the viscoplasticstrain increases linearly when only the kinematic hardening is considered. The viscoplastic model with isotropic hardening does not capture the effect ofloading time (LT) on viscoplastic response of asphalt concrete and predicts the same viscoplastic strain response for different rest periods. The model withboth isotropic and kinematic hardening models predicts the lowest values for the viscoplastic strain comparing to the other cases. However, experimentaldata show that the duration of the pulse load has a significant effect on the viscoplastic strain response, such that the shorter the pulse load time the higherthe viscoplastic strain. In conclusion, the classical viscoplastic theory with isotropic and kinematic hardening components cannot effectively capture theeffect of pulse loading time on viscoplastic strain response of asphalt concrete at high temperatures.



degree of interlocking of the aggregates change more substantially after the loading as compared to those properties beforethe loading. Specifically, the viscoplastic behavior of the asphalt concrete in compression, which is mainly governed by thefriction between the aggregates and aggregate interlocking, is more prone to change. This change makes asphalt concretemore prone to permanent deformation such that they can undergo more viscoplastic deformation during the next loadingcycle. This behavior is referred to as the hardening-relaxation or viscoplastic-softening.

2.4. Hardening-relaxation model

Changes in the viscoplastic response of asphalt concrete as a result of rearrangement of the microstructure during the restperiod (i.e. hardening-relaxation response of asphalt concrete) can be modeled using several approaches, three of which arebriefly presented here. The first approach is to define an isotropic softening function in the yield surface during the unloadingwhich is different from the isotropic hardening function during the loading. Yun and Kim (2011) used this approach and pro-posed an empirical model to explain the viscoplastic response of asphalt concrete in compression. However, their model isaugmented with many model parameters, which significantly increases the number of required tests to identify theseparameters. The second approach is to additively decompose the hardening/softening function j (see Eq. (4)) into hardeningand softening components (i.e. j = jh + js; jh and js being the hardening and softening components of the function j,respectively). The hardening component is then allowed to evolve during the loading and the softening component is al-lowed to evolve during the unloading. The third approach, which is used here, is to capture the hardening-relaxation effectthrough the saturation limit, j1, of the viscoplastic hardening function, Eq. (7). The saturation limit of the hardening functionused in this work is j0 + j1 where j0 is the initial yield strength. Therefore, the saturation limit of the hardening function iscontrolled by the parameter j1. However, as explained in Fig. 5, material’s microstructure changes progressively if unloadingtime (or the rest period) is introduced between the loading cycles. This change in the microstructure shows itself in the form

(a)

(b) (c) (d)

Aggregates Asphalt binder

Stress

Time (sec) A

B

C

Point “A” ”C“tnioP”B“tnioP

1θ

2θ 3θ

2 3 1θ θ θ< <

Fig. 5. Schematic representation of the changes in the microstructure of asphalt concrete during a loading/unloading cycle. (a) Compressive stress historyfor a creep-recovery test; (b) microstructure before applying the load; (c) during the loading, aggregates are pushed together and confine the trappedbinder. Aggregates reorient, such that the angle between the aggregates’ longest dimension and the normal to the loading direction decreases; (d) duringthe rest period, the induced pressure inside the confined binder pushes aggregates changing position and orientation of the aggregates such that the anglebetween the aggregates’ longest dimension and the normal to the loading direction lies between that before the loading and that at the end of the loading.



of additional softening (or, equivalently, recovery in hardening ability), such that the material after the rest period becomesmore prone to development of the viscoplastic strain comparing to its state at the end of the preceding loading. This hard-ening-relaxation effect can therefore be captured effectively through the model parameter j1 which controls the saturationlimit of the isotropic hardening function. Therefore, it is argued here that the material’s ability to accommodate more hard-ening changes during the rest period such that j1 is an internal state variable that evolves during the rest period.

To explain this approach in more details, a loading scenario is presented in Fig. 6(a). Fig. 6(a) shows a repeated creep-recovery test with a constant stress level and with rest periods between the loading cycles. As shown in Fig. 6(b), the classicalhardening plasticity assumes that the j1 parameter is constant through the whole loading/unloading history. Therefore, the

(a)

(b)

A

1κ( )1Cκ

( )1Eκ

κ

p

B

C

D

EF

A

B,C

D,E

F

Classical hardening viscoplasticity

When hardening-relaxation is considered

σ

TimeA B C D E F

(c)

A

B,C

D,E

F

D,E

F

p

Accumulative loading time

Classical hardening viscoplasticity

When hardening-relaxation is considered

Fig. 6. Schematic representation of the effect of the hardening-relaxation on the mechanical response of asphalt concrete. (a) A repeated creep-recoveryloading history; (b) isotropic hardening versus the effective viscoplastic strain j–p. Classical hardening viscoplasticity predicts constant hardeningparameters, such that the response does not depend on the rest period. The hardening-relaxation, on the other hand, shows the drop in the hardeningfunction during the rest period, such that the response follows a different path after the rest period; (c) effective viscoplastic strain versus accumulativeloading time p–t. The response is independent of the rest period for the classical hardening viscoplasticity, such that the rate of the viscoplastic straindecreases progressively as the accumulative loading time increases. The hardening-relaxation model predicts higher values for the rate of the viscoplasticstrain after the rest period comparing to its value before the rest period, such that rate of the viscoplastic strain is no longer a decreasing function.



isotropic hardening j versus the effective viscoplastic strain p (i.e. j–p curve) follows a single path (i.e. the path with thesolid line in Fig. 6(b)). Obviously, the unloading time does not affect the viscoplastic hardening/softening of the materialin case of classical hardening viscoplasticity. Therefore, as shown in Fig. 6(c), effective plastic strain versus time is a contin-uous curve which does not depend on the introduced rest period. Clearly, Fig. 6(c) shows that the rate of the viscoplasticstrain decreases as the number of loading cycles increases for the classical hardening viscoplasticity.

However, when the hardening-relaxation is introduced through decreasing j1 parameter, the response will strongly de-pend on the introduced rest period. Fig. 6(b) shows that the saturation limit of the viscoplastic hardening function decreasesduring the rest period when considering hardening-relaxation, e.g. when moving from point ‘‘B’’ to point ‘‘C’’, such that thehardening function will have a new saturation limit at point ‘‘C’’ (i.e. j0 þ jðCÞ1 in Fig. 6(b)). Therefore, as shown in Fig. 6(b), j–p curve follows a different path after the rest period. As a result the material softens during the rest period (i.e. decrease ofthe hardening function from point ‘‘B’’ to point ‘‘C’’ or from point ‘‘D’’ to point ‘‘E’’ in Fig. 6(b)). Hence, for the same stress levelduring the next loading cycle (e.g. moving from point ‘‘C’’ to point ‘‘D’’ in the loading history), the rate of the viscoplasticstrain increases. Fig. 6(c) demonstrates this distinct response of asphalt concrete and shows that the rate of the effectiveviscoplastic strain is no longer a decreasing function, but has a higher rate after the rest period comparing to its rate atthe end of the preceding loading cycle.

We will use the third approach in this work. However, defining a condition that could be used for the general multi-axialstate of stress in order to decide when to activate the hardening-relaxation model is a challenging task. In other words, theaforementioned three approaches cannot be applied to general cases unless several other conditions are defined. For exam-ple, how the hardening-relaxation model evolves during a continuous loading history for which a well-defined rest perioddoes not exist? how the hardening-relaxation evolves at very small viscoplastic strain values? or what is the maximumallowable recovery in the hardening-relaxation models? These questions can be answered by postulating several constraintson the hardening-relaxation model that still will not guarantee the applicability of these phenomenological models for thegeneral state of stress. A robust phenomenological approach is proposed here to remedy these issues without the need toincrease the model parameters required for postulating extra conditions. In this approach, we define a surface in the visco-plastic strain space which we refer to as the hardening-relaxation memory surface. The memory surface can be imagined asthe counterpart of the yield surface function in the viscoplastic strain space. This memory surface is expressed in terms of aninternal state variable which memorizes the maximum experienced viscoplastic strain for which the softening (or hardeningrecovery) is occurred up to the last unloading stage. The evolution of the hardening-relaxation model will then be based onthis surface and the defined hardening-relaxation internal state variable. This approach is inspired by the pioneering worksof several researchers on the cyclic plasticity of metals (e.g. Murakami and Ohno, 1982; Ohno, 1982; Chaboche, 1989; Hassanand Kyriakides, 1994a,b).

In this study, the static hardening-relaxation memory surface is defined as follows:

f vp;s ¼ p� qvp 6 0 ð10Þ

where fvp,s is the static hardening-relaxation (viscoplastic-softening) memory surface, p is the effective viscoplastic strain,and qvp is the hardening-relaxation internal state variable that memorizes the maximum experienced viscoplastic strainfor which the hardening recovery has occurred. Eq. (10) shows that the material does not undergo hardening-relaxation iffvp,s < 0. However, similar to the rate-dependent plasticity (i.e. viscoplasticity), the value of the static hardening-relaxationmemory surface can have a positive value if p exceeds qvp. The difference between p and qvp controls the evolution rate ofqvp, such that, analogous to classical viscoplasticity theory, a dynamic hardening-relaxation memory surface can be definedas follows:

vvp;s ¼ p� qvp �_qvp

Cvp;s

� � 1S16 0 ð11Þ

where vvp,s is the dynamic hardening-relaxation memory surface; Cvp,s is the hardening-relaxation fluidity parameter, suchthat 1/Cvp,s can be assumed as the hardening-relaxation relaxation time controlling the rate of the hardening-relaxation; andS1 is the hardening-relaxation rate-sensitivity parameter.

The rate of the hardening-relaxation internal state variable qvp can be determined based on Eq. (11), such that:

_qvp ¼ Cvp;shf vp;siS1 ð12Þ

which clearly shows that _qvp depends on the difference between the effective viscoplastic strain and the maximum value ofthe effective viscoplastic strain for which the hardening-relaxation has occurred (i.e. the hardening-relaxation internal statevariable). In other words, fvp,s = p � qvp is the amount of the viscoplastic strain for which the material has not yet had the timeto rearrange its microstructure completely for a specific value of the effective viscoplastic strain (or equivalently the amountof the viscoplastic strain for which the hardening-relaxation has not occurred). In fact, this difference is the driving force forrearranging the materials’ microstructure and for the hardening-relaxation phenomenon.

It should be noted that the hardening-relaxation internal state variable, qvp, evolves during the unloading stage (i.e. dur-ing the deformation recovery). Fig. 7 schematically shows the evolution of the effective viscoplastic strain, p, and qvp during ahypothetical loading-unloading cycle. As shown in Fig. 7, p evolves during the loading. During this stage, no hardening-relax-ation occurs. However, during the unloading stage, p remains constant and qvp evolves. The difference between p and qvp



(i.e. fvp,s = p � qvp) is the driving force for recovery in the hardening ability of the material. The hardening-relaxation contin-ues with a decreasing rate as qvp evolves and stops once qvp reaches p.

For simplicity, we can assume that the evolution function for the rate of the saturation limit of the hardening j1 to be alinear function of the rate of the hardening-relaxation internal state variable, such that:

_j1 ¼ �S2 _qvp ð13Þ

However, Eq. (13) is modified to consider the history effect in the hardening-relaxation model since the experimental mea-surements show that the rate of the hardening-relaxation decreases as the saturation limit of the hardening function j1 de-creases, such that:

_j1 ¼ �S2 _qvpj1

jinitial1

� �S3ð14Þ

where jinitial1 is the initial value of the saturation limit of the hardening function j before the hardening-relaxation phenom-enon (or equivalently, the saturation limit of the hardening function in monotonic loading without rest periods). Fig. 8 sche-matically represents the concept of the hardening-relaxation static and dynamic memory surfaces.

This approach provides a promising framework for constitutive modeling of the cyclic hardening-relaxation phenomenonin the materials. The hardening-relaxation model parameters can be identified using repeated creep-recovery tests at hightemperatures and for different loading/unloading times.

The simulations presented in Figs. 2 and 4 are repeated for the model with the hardening-relaxation component. Fig. 9(a)and (b) show the simulation results for different resting times and pulse times, respectively. The hardening-relaxation modelparameters used in these simulations are listed in Table 3. Fig. 9(a) clearly shows that, unlike the classical hardening visco-plasticity theories, the hardening-relaxation model captures the effect of different resting times, such that longer resting

p

t

vpq

p

Loading stage Unloading stage

(the driving force for viscoplastic

softening or recovery in the viscoplastic hardening)

vpp q−

Fig. 7. Schematic representation of the evolution of the effective viscoplastic strain and the hardening-relaxation internal state variable during ahypothetical loading–unloading cycle.

Static hardening-relaxation memory surface

Determines rate of the hardening-relaxation internal state variable

Dynamic hardening relaxation memory surface

vpq

1vpε

2vpε 3

vpε

p

Fig. 8. Schematic representation of the concept of the hardening-relaxation memory surface in the principal viscoplastic strain space.



times yield larger values for the viscoplastic strain. Fig. 9(b) also shows that the predicted values for the viscoplastic strainincreases as the pulse time decreases. Fig. 9(a) and (b) are in agreement with the experimental data presented in Figs. 2 and4, respectively.

The hardening-relaxation model possesses the hardening-relaxation fluidity parameter Cvp,s, such that 1/Cvp,s has theunits of time and can be regarded as a measure of the hardening relaxation time. This parameter controls the rate of changeof the hardening-relaxation internal state variable (i.e. _qvp), such that q evolves faster and reaches the maximum experiencedaccumulated viscoplastic strain faster as the hardening-relaxation fluidity parameter (i.e. Cvp,s) increases. Therefore, thematerial undergoes the hardening-relaxation in a shorter period of time as Cvp,s increases. It should be noted that this param-eter does not explicitly affect the evolution rate of j1, but it affects the evolution rate of qvp which implicitly affects the evo-lution rate of j1. The rate-sensitivity model parameter S1 exponentially affects the rate of increase in the viscoplastic strain.The effect of this parameter is very similar to the effect of the rate-sensitivity parameter N in the Perzyna-type viscoplasticmodel (i.e. Eq. (2)).

The model parameter S2 also affects the rate of hardening-relaxation. Substituting Eq. (12) into Eq. (14) shows that Cvp,sS2has an interesting physical meaning. It has the units of stress/time and shows that Cvp,sS2 is the measure of the material’s

(a)

(b)

0

0.25

0.5

0.75

1

0 5 10 15

Eff

ecti

ve v

isco

plas

tic

stra

in (%

)


Prediction using hardening-relaxation model: UT=0.05 sec

Prediction using hardening-relaxation model: UT=1 sec

Prediction using hardening-relaxation model: UT=200 sec

0

0.25

0.5

0.75

1

0 3 6 9 12 15

Eff

ecti

ve v

isco

plas

tic

stra

in (%

)


Prediction using hardening-relaxation model: LT=0.1 sec



Fig. 9. Effect of unloading (rest) time and loading (pulse) time on the viscoplastic strain response when the hardening-relaxation model is used.(a) Different resting times; (b) different loading times.



capacity in decreasing its yield strength with resting time. In other words, this parameter (i.e. Cvp,sS2) can be used in clas-sification of different asphalt concrete materials to measure their propensity for permanent deformation under cyclic loadingconditions, such that larger values for Cvp,sS2 means that the material is more prone to the permanent deformation duringcyclic loading conditions. Therefore, the effect of S2 parameter on the viscoplastic response is similar to the effect of Cvp,s.However, unlike the parameter Cvp,s, the S2 parameter directly affects the evolution rate of the j1 parameter. Finally, the his-tory model parameter S3 affects the rate of the hardening-relaxation as the loading cycles increases, such that different val-ues of the S3 does not significantly affect the viscoplastic strain response at initial loading cycles. However, the viscoplasticresponses for different S3 parameters deviate significantly as the number of loading cycles or the viscoplastic strainincreases.

It should be mentioned that several researchers have argued that the classical kinematic hardening models, such asArmstrong–Frederick model as in Eq. (6), are not capable of effectively describing the cyclic viscoplastic response of frictionalmaterials and have proposed various cyclic plasticity models (e.g. Dafalias and Popov, 1975; Krieg, 1975; Hashiguchi, 1980;Mroz et al., 1981; Hashiguchi, 1989). In these works, two or multiple yield surfaces are usually defined. Therefore, a subload-ing surface within the bounding surface which can be regarded as the conventional yield surface is defined, such that thestress point always remains on the surface of the subloading surface even during the unloading (e.g. Hashiguchi and Ueno,1977; Dafalias and Herrmann, 1980). Sekiguchi and Ohta (1977) assumed that the yield surface rotates around the origin ofthe stress space in order to describe the inherent anisotropy of soils. Hashiguchi (1979) renamed this concept as the rota-tional hardening and proposed several rotational hardening models. Hashiguchi (1989) introduced the subloading surfacewhich expands/contracts as a function of the rate of the plastic deformation, such that the stress point always remains onthe subloading surface during both loading and unloading stages. Therefore, the subloading surface keeps the shape ofthe bounding surface and only contracts/expands and rotates around the origin of the stress point. Obviously, the plastic/viscoplastic response during the next loading stage is governed by the current subloading surface. In fact, the underlying fun-damental mechanisms of the proposed hardening-relaxation model is similar to the incorporation of the subloading surfacein cyclic plasticity. However, the approach used in this work is different from the previous works in order to simplify theformulations and implementation techniques and provide a more physically-sound approach. In this work, we model thehardening-relaxation phenomenon by introducing a hardening-relaxation memory surface in the viscoplastic strain spaceand the hardening-relaxation internal state variable qvp that memorizes the history of the loading. During the unloadingas well as during the rest period, the hardening-relaxation internal state variable qvp evolves leading to the evolution ofthe saturation limit (i.e. j1) of the isotropic hardening function (i.e. j). Therefore, the induced hardening in the materialstarts relaxing during the unloading (rest) period. This phenomenon is precisely equivalent to the contraction of the yieldsurface during the unloading. Therefore, the current yield surface (which has been contracted during the unloading stage)can be regarded similar to the subloading surface concept. It should be noted that the current yield surface can never exceedsthe boundaries of the yield surface at the end of preceding loading stage which can be regarded similar to the bounding sur-face (normal-yield surface according to the notion of Hashiguchi and Chen (1998)). However, Hashiguchi and Chen (1998)expressed the evolution function for the expansion/contraction of the subloading surface as a function of the rate of plasticdeformation which makes their model time-independent. In other words, their evolution function for the contraction of thesubloading surface does not evolve during the rest period (at the stress free configuration), such that it cannot capture theeffect of different rest periods on the (visco)plastic response of materials. Obviously, the two approaches (i.e. the currenthardening-relaxation concept and the subloading surface concept) follow very similar underlying physics with differentmodeling approaches. The current work uses the concept of the memory surface, originally proposed by Ohno and hisco-workers (Ohno, 1982; Ohno and Wang, 1993), in order to explain this distinct behavior of time-dependent materialsand specifically, in this work, the asphalt concrete. It should be noted that this work does not consider the rotation of theyield surface since the current experimental data does not include the tests required for determination of the model param-eters associated with this model.

3. Coupling viscoelasticity to the hardening-relaxation viscoplastic model

The total deformation of the asphalt concrete includes both recoverable and irrecoverable components, where the con-tribution of each component strongly depends on temperature, stress level, and the loading rate. In this study, we assumesmall deformations since the deformations are very small during each loading cycle in asphalt concrete. Therefore, the totalstrain tensor e can be additively decomposed into the viscoelastic strain tensor eve (i.e. the recoverable component) and theviscoplastic strain tensor evp (i.e. the irrecoverable component), such that:

e ¼ eve þ evp ð15Þ

The Schapery’s (1969b) nonlinear viscoelastic model will be used to predict the recoverable component of the strain tensor,whereas, the Perzyna-type viscoplastic model (Section 2.1) and the proposed hardening-relaxation model (Section 2.3) willbe used to predict the irrecoverable component of the strain tensor.



3.1. Schapery’s nonlinear viscoelastic constitutive model

The key equations for the Schapery’s nonlinear viscoelastic model is presented in this section. However, we will not pres-ent detailed formulation and numerical implementation of the constitutive models. The readers are referred to previousworks for detailed explanation of the numerical implementation techniques of the nonlinear Schapery’s viscoelasticity(e.g. Haj-Ali and Muliana, 2004; Huang, 2008; Levesque et al., 2008; Abu Al-Rub et al., 2010; Huang et al., 2011b; Darabiet al., 2012a) and Perzyna’s viscoplasticity (e.g. Heeres et al., 2002; Huang, 2008; Abu Al-Rub et al., 2010; Darabi et al.,2011b; Huang et al., 2011b).

The Schapery’s one-dimensional single-integral nonlinear viscoelastic model can be written as follows:

eve;t ¼ g0D0rt þ g1Z t

0DDðw

t�wsÞ dðg2rsÞds

ds ð16Þ

where superscripts t and s designate the response at a specific time; g0, g1, and g2 are nonlinear parameters which are a func-tion of the stress tensor r and the temperature T; D0 and DD are the instantaneous and transient creep compliances, respec-tively; and wt is a reduced time, which is given by:

wt ¼Z t

0

dnaT

ð17Þ

where aT is the time-temperature shift factor which will be used to capture the viscoelastic response of asphalt concrete atdifferent temperatures. The transient creep compliance DD can be expressed in the form of Prony series, for numerical pur-poses, such that:

DDwt¼XNn¼1

Dn½1� expð�knwtÞ� ð18Þ

where N is the number of terms; and Dn and kn are the nth coefficients of the Prony series.Lai and Bakker (1996) generalized Eq. (18) to the general three-dimensional problems by decomposing the total nonlinear

viscoelastic strain tensor eve into the deviatoric viscoelastic strain tensor eve and the volumetric viscoelastic strain evekk , suchthat:

eveij ¼ eveij þ13evekkdij ¼

J2

Sij þB3rkkdij ð19Þ

where J and B are the shear and bulk compliances, respectively; Sij = rij � rkkdij/3 are the components of the deviatoric stresstensor; and dij is the Kronecker delta. The deviatoric nonlinear viscoelastic strain tensor and the volumetric viscoelastic straincan be expressed in terms of the shear and bulk compliances as follows:

eveij ¼12

g0J0Stij þ

12

g1

Z t0

DJðwt�wsÞ dðg2S

sijÞ

dsds ð20Þ

evekk ¼13

g0B0rtkk þ13

g1

Z t0

DBðwt�wsÞ dðg2rskkÞ

dsds ð21Þ

where J0 and B0 are the instantaneous shear and bulk compliances, respectively; and DJ and DB are the transient shear andbulk compliances, respectively. It should be noted that the instantaneous and transient shear and bulk compliances are re-lated to the instantaneous and transient creep compliances as follows:

J0 ¼ 2ð1þ mÞD0; B0 ¼ 3ð1� 2mÞD0 ð22Þ

DJ ¼ 2ð1þ mÞDD; DB ¼ 3ð1� 2mÞDD ð23Þ

where m is the Poisson’s ratio, which is assumed to be time-independent in this work.The numerical algorithms for the proposed coupled viscoelastic, viscoplastic, hardening-relaxation constitutive model are

implemented in the well-known finite element code Abaqus via the user material subroutine UMAT. The nonlinear viscoelas-tic model is implemented using a recursive-iterative algorithm, whereas an extension of the classical rate-independent re-turn mapping algorithm to the rate-dependent problems is used for numerical implementation of the hardening-relaxationviscoplastic model. Appendix I briefly presents the numerical algorithms and implementation procedures for such constitu-tive models. Readers are referred to the authors previous works (e.g. Abu Al-Rub et al., 2010; Huang et al., 2011b; Darabiet al., 2012b) for a detailed description of the numerical algorithms and implementation techniques of the presentedconstitutive model. The implemented numerical algorithms are used to obtain the numerical results that will be presentedin the next sections.



4. Identification of the model parameters

The presented viscoelastic–viscoplastic constitutive model is calibrated against the US Federal Highway (FHWA)Accelerated Load Facility (ALF) data. The experimental measurements on ALF materials are conducted at North Carolina StateUniversity (NCSU) (e.g. Kim et al., 2008). The material used in this study is compacted using the Superpave gyratorycompactor to the dimension of 178 mm in height and 150 mm in diameter. To obtain the uniform air void distribution,the specimens are cored and cut to a height of 150 mm with a diameter of 100 mm. The asphalt concretes have 5.3% bindercontent with approximately 4% air voids and the asphalt binder is specified as unmodified PG 70-22. The dynamic modulustest in compression at different frequencies and temperatures, and the repeated creep-recovery test in compression withvariable loading (VL) at 55 �C are used to calibrate the viscoelastic and viscoplastic model parameters, respectively. Table1 lists the tests which are used to identify the model parameters.

As explained in Section 2.2, the kinematic hardening model is not capable of effectively capturing the effects of pulseloading time and rest period on the viscoplastic response of asphalt concrete materials under compressive cyclic loading con-ditions at high temperatures. Therefore, it should be noted that the kinematic hardening model is not considered in the sub-sequent calibration and validation processes. The reasons for this are: (1) the conventional kinematic hardening model isproved not to be capable of capturing the desired viscoplastic response of asphalt concrete under cyclic loading conditionsat high temperatures (see Section 2.2); (2) several researchers have argued that the classical kinematic hardening models arenot capable of effectively describing the cyclic viscoplastic response of frictional materials and have proposed various cyclicplasticity models without kinematic hardening (e.g. Dafalias and Popov, 1975; Krieg, 1975; Hashiguchi, 1980, 1989; Mrozet al., 1981); (3) as will be shown in the next section, the proposed model with isotropic hardening and isotropic harden-ing-relaxation is sufficient to capture the current experimental measurements; and (4) the current available experimentaldata does not include proper experimental tests to fully calibrate the kinematic hardening model. Therefore, because ofthe aforementioned reasons and also in order to avoid the unnecessary increase in the number of model parameters, thekinematic hardening model is not considered for the calibration and validation of the experimental tests presented in thiswork. However, this does not imply that kinematic hardening is not needed for describing the cyclic viscoplastic behaviorof asphalt concrete. In fact, future work should focus on also incorporating the proposed hardening-relaxation concept inthe kinematic hardening rules of time-dependent materials such as bituminous materials, polymers, and biomaterials.

Moreover, it should be emphasized that most of the tests used in this work are conducted in compression and at 55 �C. Athigh temperatures, the failure mechanism of asphalt concrete is primarily related to the flow of the material. Therefore, dam-age (micro-cracks and micro-voids) modeling has not been incorporated here to predict the degradation of the asphalt con-crete. Instead, the flow-type response of asphalt concrete at high temperatures in compression is captured using theproposed hardening-relaxation model.

4.1. Identification of the viscoelastic model parameters

The dynamic modulus test is used to identify the linear viscoelastic model parameters as well as the temperature cou-pling term model parameters (i.e. the time-temperature shift factors). This test is conducted at four temperatures (�10,10, 35, and 55 �C) and eight frequencies (0.01, 0.05, 0.1, 0.5, 1, 5, 10, and 25 Hz). The strain amplitude is controlled to below enough (50–70le) such that the material does not get damaged. The standard procedure is used to identify the linearviscoelastic model parameters and time-temperature shift factors using the complex compliance D⁄ and the phase angleh. In other words, the master curve is first constructed for the complex compliance D⁄ from which the time-temperature shiftfactors are also identified. The next step is to calculate the storage compliance D00 ¼ kD�k sin h and the loss complianceD00 ¼ kD�k cos h versus the reduced frequency xR (i.e. the natural circular frequency normalized by the time-temperatureshift factor). The Prony series coefficients (Dn and kn) in Eq. (18) are related to D0 and D

00 as follows (Park and Schapery, 1999):

D0 ¼ D0 þXNn¼1

Dn1þ ðx=knÞ2

; D00 ¼XNn¼1

1kn

Dnx1þ ðx=knÞ2

" #ð24Þ

where Dn and kn can be identified from the above expressions by minimizing the error between the experimental andcalculated D0 and D00, such that:

Table 1Summary of the tests used for identification of the viscoelastic–viscoplastic and viscoplastic–softening model parameters.

Test Temperature(�C)

Stress level(kPa)

Confinement(kPa)

Loading time(s)

Rest period(s)

Complex modulus test �10, 10, 35, 55 – 140 – –Repeated creep-recovery test with variable loading

(VL)55 Varies 140 0.4 200



err ¼ D0

D0Exp� 1

!2þ D

00

D00Exp� 1

!2ð25Þ

Fig. 10 presents the dynamic modulus test data before and after the time-temperature shifting.The variable loading (VL) test at 55 �C is used to identify the nonlinear viscoelastic model parameters (i.e. g0, g1, and g2)

and the viscoplastic model parameters (see Fig. 11). VL is a repeated creep-recovery test for which the loading and unloadingtimes remain constant through the entire test (i.e. loading time of 0.4 s and unloading time of 200 s). This test consists ofseveral loading blocks. Each loading block consists of eight creep-recovery cycles with increasing applied deviatoric stresslevel. The deviatoric stress level starts from 137.9 kPa in the beginning of the first loading block and increases with the factorof 1.2 for the next deviatoric stress level until it reaches the last creep-recovery within that block. For the next loading block,however, the first deviatoric stress level equals to the third stress level in the previous block. Fig. 11 schematically shows theapplied stress history for the VL test.

(a)

(b)

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-1.5 -0.5 0.5 1.5 2.5

Log

(D*)

[kP

a-1 ]

Log(Angular frequency) [Rad/Sec]

T=-10C

T=10C

T=35C

T=55C

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-2 0 2 4 6 8 10 12

Log

(D*)

[kP

a-1 ]

Log(Reduced angular frequency) [Rad/Sec]

T=-10C Shifted

T=10C Shifted

T=35C Shifted

T=55C Shifted

Fig. 10. Complex compliance data in compression at different temperatures. (a) Before time-temperature shift and (b) after time-temperature shift.



The recovery part of the VL test is used to extract the viscoelastic response from the total measured strain during therecovery part of the VL test. The nonlinear model parameters are then identified by comparing the prediction of the linearviscoelastic model using the identified linear viscoelastic model parameters and to the extracted viscoelastic response duringthe recovery part of the VL test. More details on the procedure for extracting the viscoelastic strain response during therecovery part of a repeated creep-recovery test are presented by Abu Al-Rub et al. (2009) and Masad et al. (2009). The advan-tage of using the VL test for identifying the nonlinear viscoelastic model parameters is that the VL test is conducted at severalstress levels, and therefore, it can be used to identify a relation for the nonlinear viscoelastic model parameters as a functionof the stress level. The analysis has shown that the instantaneous creep compliance (i.e. D0) does not change with the stresslevel, such that the nonlinear viscoelastic model parameter g0 is unity. Moreover, for simplicity, the nonlinear parameter g1 isassumed to be one. The deviation of the extracted viscoelastic strain in the VL test from the viscoelastic strain response usingthe linear viscoelastic model is used to identify the nonlinear parameter g2 as a function of stress. However, as we will showin the next sections, most of the tests in this work are conducted at a constant stress level of 827 kPa. Therefore, a singlevalue for the nonlinear model parameter g2 that represents the value of this parameter at the stress level of 827 kPa is

Block 1

Block 2

0

200

400

600

800

1 3 5 7 9 11 13 15

Dev

iato

ric

stre

ss le

vel (

kPa)

Loading Number

Fig. 11. Stress history for the variable loading (VL) test.

Table 2Identified viscoelastic model parameters at T0 = 55 �C and the time-temperature shift factors.

n 1 2 3 4 5 6 7

Viscoelastic model parameterskn (s�1) 107 105 103 102 10 1 10�1

Dn (kPa�1) 1.8 � 10�8 5.6 � 10�8 2.3 � 10�7 4.4 � 10�7 5.6 � 10�7 3.3 � 10�7 3.5 � 10�7

D0 (kPa�1) 3 � 10�8

g0 g1 g21 1 2.03

Time-temperature shift factors (T0 = 10oC)

T (oC) �10 10 35 551.26 � 108 105 63 1

Table 3Identified viscoplastic and viscoplastic–softening model parameters at T0 = 55 �C.

Viscoplastic model parametersa b r0y (kPa) C

vp (s�1) N j0 (kPa) j1 (kPa) j20.25 0.2 100 2.4 � 10�3 1 50 1800 135

Viscoplastic–softening model parametersCvp,s (s�1) S1 S2 (kPa) S310�3 0.3 2 � 105 2.5



(b)

(c)

0

0.1

0.2

0.3

0 500 1000 1500 2000 2500 3000 3500

Vis

copl

asti

c st

rain

(%)

Time (sec)


Model prediction without memory surface

Model prediction with memory surface

0

600

1200

1800

0 500 1000 1500 2000 2500 3000 3500

k 1(k

Pa)

Time (sec)

(a)

0

0.1

0.2

0.3

0.4

0.5

0 500 1000 1500 2000 2500 3000 3500

Stra

in (%

)

Time (sec)




Fig. 12. Experimental measurements and model predictions of the variable loading test (VL) at 55 �C with and without using the hardening-relaxationmemory surface. (a) Total strain response; (b) viscoplastic strain response; (c) evolution of j1 parameter with time.



reported in this work. The identified viscoelastic model parameters using the aforementioned procedure at 55 �C are listed inTable 2.

4.2. Identification of the hardening-relaxation viscoplastic model parameters

The creep part of the VL test at 55 �C is used to identify the viscoplastic model parameters (see Fig. 11). Once the linearand nonlinear viscoelastic model parameters are identified, the viscoelastic strain response during the creep part of the VLtest is known. The viscoplastic strain response during the creep part of VL test can be easily calculated by subtracting theviscoelastic strain response from the total measured strain using Eq. (15).

The extracted viscoplastic strain response during the initial loading cycles of the VL test are used to identify the viscoplas-tic model parameters (the parameters for the model presented in Section 2.1). The detailed procedure for identifying theseviscoplastic model parameters can be found in the previous works (please refer to Huang et al. (2008, 2011b) and Abu Al-Rubet al. (2009) for more details on identification of the classical viscoplastic model parameters). Table 3 lists the hardening-relaxation viscoplastic model parameters identified at 55 �C in compression.

Fig. 12(a) shows the comparison between the experimental data and model predictions for the VL test at 55 �C. Fig. 12(a)shows that the model predictions agree well with the experimental data within the first loading block, the loading blockagainst which the viscoplastic model is calibrated. The model does not show more viscoplastic strain accumulation withinthe second block until the applied stress level exceeds its maximum value experienced at the end of the preceding loadingblock. However, experimental measurements in Fig. 12(a) show asphalt concretes progressively develop permanent defor-mations even when the applied stress is less than its maximum value during the previous loading cycle (more viscoplasticstrain is accumulated for the first four loading cycles in the second block even though the applied stress is still less than theapplied stress at the end of the first block).

Again, this distinct behavior of asphalt concrete subjected to the repeated loading at high temperatures cannot be ex-plained by hardening plasticity and viscoplasticity theories. In other words, the hardening plasticity/viscoplasticity theoriesstate that the material point remains inside the yield surface (i.e. elastic/viscoelastic) when the applied stress in the subse-quent loading cycle is less than the stress level in the preceding loading cycle such that no further permanent deformation isexpected. However, as it is obvious from Fig. 12, once yielded, asphalt concretes show accumulation of the plastic/viscoplas-tic strain in each loading cycle even though the applied stress level decreases. This behavior is related, in this work, to thehardening-relaxation effect. The second block of the VL test is used to identify the hardening-relaxation model parameterswhich are listed in Table 3.

The variable loading test (VL) is re-simulated using the hardening-relaxation memory surface concept as shown inFig. 12. Fig. 12 shows that the hardening-relaxation model improves the model predictions of both viscoplastic strainand total strain responses for the VL test. Evolution of the saturation limit of the viscoplastic model (i.e. j1) is also

Table 4Summary of the tests used for validation of the viscoplastic–softening model.a

Test Stress level (kPa) Loading time (s) Rest period (s)

Repeated creep-recovery test with constant loading level and time (CLT) 827 0.1, 0.4, 1.6, 6.4 0.9Repeated creep-recovery test with variable loading time (VT) 827 Sequence of (0.05, 0.1, 0.4, 1.6, 6.4) 0.05, 1, 200Repeated creep-recovery test with reversed variable loading time (RVT) 827 Sequence of (6.4, 1.6, 0.4, 0.1, 0.05) 200

a All tests have done at 55 �C and 140 kPa confinement level in compression.

……..

Time (sec)

Deviatoric stress (kPa)

827(kPa)

LT UT=0.9sec

……..

200 sec rest between loading blocks

Block 1 Block 2

Fig. 13. Schematic representation of the stress input for the constant loading time test (CLT). Both loading time (LT) and unloading time (UT) are constantthrough the entire test. CLT test data is available for four different loading times (LT) of 0.1, 0.4, 1.6, and 6.4 s.



(a)

-1

-0.5

0

0.5

1

0 200 400 600 800 1000

Axi

al st

rain

(%

)

Time (sec)




Rad

ial s

trai

n (%

)

(b)

(c)

-1

-0.5

0

0.5

1

0 200 400 600 800 1000

Axi

al V

isco

plas

tic

stra

in (%

)

Time (sec)




Rad

ial v

isco

plas

tic

stra

in (%

)

0

600

1200

1800

0 200 400 600 800 1000

k 1(k

Pa)

Time (Sec)

Fig. 14. Experimental measurements and model prediction with and without the hardening-relaxation memory surface for the constant loading and timetest (CLT) at 55 �C in compression when the loading time is 0.1 s. This test consists of four loading blocks and 240 loading cycles. (a) Total strain response;(b) viscoplastic strain response; AND (c) evolution of the j1 parameter versus time.



(a)

(b)

-0.8

-0.4

0

0.4

0.8

0 100 200 300 400 500

Axi

al st

rain

(%

)

Time (sec)




Rad

ial s

trai

n (%

)

-0.8

-0.4

0

0.4

0.8

0 100 200 300 400 500

Axi

al v

isco

plas

tic

stra

in (%

)

Time (sec)




Rad

ial v

isco

plas

tic

stra

in (%

)

(c)

0

600

1200

1800

0 100 200 300 400 500

k1(k

Pa)

Time (Sec)

Fig. 15. Experimental measurements and model prediction with and without the hardening-relaxation memory surface for the constant loading and timetest (CLT) at 55 �C in compression when the loading time is 0.4 s. (a) Total strain response and (b) viscoplastic strain response; (c) evolution of the j1parameter versus time.



plotted in Fig. 12(c). Fig. 12(c) shows that j1 relaxes during the rest period, such that the model with memory surfacepredicts higher increment of the viscoplastic strain during the next loading cycle as compared to the model predictionswithout using the memory surface, Fig. 12(b). It should be noted that the hardening-relaxation continues until the rateof the viscoelastic strain reaches a negligible value. After that point, no more softening occurs since the hardening-relax-ation is physically related to the rearrangement of the microstructure during the unloading and deformation recovery

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