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Klug, A.; , . Studies on Application of Viscoelastic Continuum Damage Theory. Encyclopedia. Available online: (accessed on 14 June 2024).
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Studies on Application of Viscoelastic Continuum Damage Theory

According to the viscoelastic continuum damage (VECD) theory, the time dependency of viscoelastic materials can be eliminated by means of correspondence principles, which transform physical variables (stress, strain, and stiffness) in pseudo variables (pseudo stress, pseudo strain, and pseudo stiffness). According to this theory, a damaged body presenting internal microcracks is assumed to be an undamaged body with a reduced pseudo stiffness (C), with the microcracks uniformly distributed within the body. The material damage evolution is described as the function C(S), in which a reduction in pseudo stiffness (C) is related to a material internal state variable (S).

viscoelastic continuum damage theory

1. Introduction

An accurate prediction of the fatigue behavior of an asphalt mixture has been the goal of many studies focused on improving the asphalt mixture’s design and the flexible pavement’s performance [1][2][3][4][5][6][7][8][9]. Phenomenological models, which relate the stress or strain in the specimen with the number of cycles to failure, are simple tools to determine the fatigue behavior of an asphalt mixture in the laboratory [1][10][11][12][13]. However, this approach does not account for the complexity of the fatigue phenomenon [3]. More recently, mechanistic approaches have been employed rather than the phenomenological ones, as mechanistic models account for how damage evolves throughout the fatigue life at different loading and environmental conditions, leading to a better estimation of the fatigue behavior of the asphalt mixture [2][3][5][7].
Studies on the response of asphalt mixtures to fatigue cracking are divided into two categories: (1) the full asphalt mixture, which comprised asphalt binder, coarse aggregate, fine aggregate, and mineral filler; and (2) the fine aggregate matrix (FAM), composed of fine aggregate, mineral filler, and asphalt binder [14]. Based on the premise that the changes in the material microstructure are the beginning of the fatigue process [4][14][15], many researchers have studied the fatigue behavior of asphalt mixtures using the FAM approach [5][6][14][16][17][18]. The fine aggregate matrix represents an intermediate scale between the asphalt mastic and the full asphalt mixture, presenting an internal structure that is not affected by the coarse aggregate particles [19]. Studies at the FAM scale might be capable of providing a more realistic characterization of the fatigue response of full asphalt mixtures than the one provided by tests performed on the mastic [14]. Another advantage of using FAMs in fatigue testing is that the reduced size of the specimens requires a smaller amount of material, as compared with the amount needed to produce specimens of full asphalt mixtures, which also reduces the laboratory work needed in the preparation of specimens [20].
Asphalt concrete is a material with high viscoelasticity imparted by the binder matrix [21]. The data obtained in tests performed with both FAMs and full asphalt mixtures can be analyzed by means of the continuum mechanics theory. According to the viscoelastic continuum damage (VECD) theory, the time dependency of viscoelastic materials can be eliminated by means of correspondence principles, which transform physical variables (stress, strain, and stiffness) in pseudo variables (pseudo stress, pseudo strain, and pseudo stiffness) [15][22][23]. According to this theory, a damaged body presenting internal microcracks is assumed to be an undamaged body with a reduced pseudo stiffness (C), with the microcracks uniformly distributed within the body [22][23]. The material damage evolution is described as the function C(S), in which a reduction in pseudo stiffness (C) is related to a material internal state variable (S) [15][21][22][23].

2. Theory of Viscoelastic Continuum Damage

The work potential theory, which is based on the methods of thermodynamics of irreversible process, was developed by Schapery [15][16][17][18][19][20][21][22][23] in order to describe the mechanical behavior of elastic materials with growing damage. The theory characterizes the material using macroscale observations, quantifying the changes of the material microstructure by means of internal state variables (S). The elastic model was extended to describe the mechanical behavior of viscoelastic media, by means of elastic–viscoelastic correspondence principles that eliminate the time dependence of the viscoelastic material.
Park et al. [21], Lee [24], and Lee and Kim [25] applied Schapery’s theory to develop a constitutive model that describes the damage evolution process of asphalt mixtures for different materials, as well as loading and environmental conditions. This constitutive model was simplified by Lee et al. [7] to build a practical fatigue prediction model for specimens of asphalt mixtures under uniaxial cyclic loading. Such a model was later adapted by Kim and Little [5] for specimens of sand asphalt mixtures under torsional shear without rest periods. This improved model is considered capable of providing a reasonable representation of the fatigue life of asphalt mastics and fine aggregate matrices [5].

3. Studies on the Application of the VECD Theory

3.1. Full Asphalt Mixture Approach

Based on previous studies developed to characterize the damage behavior of asphalt mixtures [21][24][25][26][27], Daniel and Kim [28] proposed a testing procedure for fatigue characterization of asphalt concrete specimens under monotonic (constant-crosshead rate tests to failure) and cyclic (controlled-crosshead strain amplitude cyclic fatigue testing in tension) loading, which consists of two steps: (i) to perform a frequency sweep test on at least three replicate specimens at different frequencies and temperatures, for linear-viscoelastic material characterization (phase angle and relaxation modulus); and (ii) to perform a constant strain rate test to failure on all replicate specimens at a single rate at the desired temperature, for damage characterization. Pseudo strains are calculated according to Equation (15). Normalized pseudo stiffness, C, and damage parameter, S, for all times are calculated by Equation (34) and Equation (39), respectively. The values are cross-plotted to construct the characteristic curve that describes the reduction in material integrity as damage grows in the specimen (CxS) and to determine the functional coefficients C1 and C2 (Equation (40)). Daniel and Kim [28] showed that a single CxS curve can be obtained for each material, regardless of the applied loading conditions (cyclic vs. monotonic, amplitude/rate, frequency). However, Lundström and Isacsson [29] indicated that it was difficult to generally predict fatigue results based on characteristic curves obtained from monotonic tests. A later study conducted by Keshavarzi and Kim [30] applied the viscoelastic continuum damage (VECD) theory to simulate asphalt concrete behavior under monotonic loading. In that study, direct tension monotonic testing that incorporated a constant crosshead displacement rate and four temperatures was used to simulate thermal cracking of asphalt concrete prepared with four reclaimed asphalt pavement (RAP) proportions. The predictions of monotonic simulation matched the measured data of the monotonic tests very well up to the point of maximum stress. More recently, Cheng et al. [31] observed that the asphalt mixture CxS curves were independent of the strain level, but affected by the loading waveform.
Daniel and Kim [28] also verified that the characteristic curve at any temperature below 20 °C can be found by utilizing the time–temperature superposition principle (t-TS) and the concept of reduced time. Later, Chehab et al. [32] demonstrated that t-TS can be extended from material’s linear-viscoelastic range to high damage levels. Findings from compression tests [33][34] complemented the findings from Chehab et al. [33] for tension tests. Underwood et al. [35] also found that the t-TS principle with growing damage was applicable to mixtures with modified asphalt binders. The analysis of the fatigue behavior using the constant-crosshead rate test method and the t-TS principle [36] has been successfully employed to evaluate mixtures prepared with RAS, mixtures prepared with RAP, warm-mix asphalt (WMA) mixtures, mixtures prepared with modified binders, and other factors such as aggregate gradation, air voids, moisture, and aging [9][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55].
In the studies carried out by Kim et al. [56] and Underwood et al. [57][58], a simplified VECD model was implemented in a finite element package (FEP++) to predict the fatigue performance of asphalt mixtures tested at test road project sites. In this VECD-FEP++ approach, the viscoelastic nature of asphalt concrete (AC) mixtures with growing damage is addressed using the VECD model, whereas the finite element program (FEP++) accounts for other important characteristics, such as temperature, layer thickness, stiffness gradient, etc. Comparisons between the field fatigue performance of the test road pavements to those predicted by the VECD-FEP++ simulations showed a generally positive relationship between model predictions and field observations. For a quick fatigue assessment, Underwood et al. [57][58][59] implemented the simplified VECD (S-VECD) model in the FEP++, instead of the original VECD model. The fatigue life of the pavements predicted using the S-VECD-FEP++ was found to agree well with the measured field response (R² = 0.8473% for the no-Terpolymer scenario, and R² = 0.9932 for the with-Terpolymer scenario). The S-VECD-FEP++ model was shown to be able to capture the effects of structure, climatic region, unbound layer modulus, and asphalt mixture properties, and to distinguish between top-down and bottom-up cracking.
The flexural bending test, also known as the beam fatigue test [60], is another testing method used to characterize the damage behavior of asphalt mixtures. This test measures the fatigue life of a compacted asphalt beam subjected to repeated flexural bending. According to De Mello et al. [61], what most likely happens in the field, and which is common to flexural fatigue tests, is a stress/strain field varying through the section from maximum compression to maximum tension in opposite sides instead of a homogeneous state of stress/strain throughout the sample section in cylindrical specimens subjected to uniaxial loading. The VECD approach was applied to flexural fatigue tests in studies with different goals, for example: (i) simplifying the calculation of damage parameters in the VECD model by considering the peak-to-peak values of stress and strain [62], (ii) evaluating which factors can influence the fatigue behavior more significantly [61], (iii) comparing fatigue cracking characteristics of a fine mix and a coarse mix [55], and (iv) studying the influence of RAP content on fine aggregate matrix (FAM) mixes [63]. Zhang et al. [63] also concluded that the linear amplitude sweep (LAS) test of FAM mixes under flexural bending mode can provide acceptable data with good repeatability as an alternative test method for tests with cylindrical samples in the dynamic shear rheometer (DSR).

3.2. Fine Aggregate Matrix Approach

Microstructural discontinuities, such as air voids and microcracks, coalesce and propagate due to repeated dynamic loading from the traffic of heavy vehicles and environmental loads, giving rise to the fatigue cracking process. The fatigue cracking reduces the structural performance of the pavement, with a negative impact on its service life. This process develops under two circumstances: (i) after adhesive failure, when the crack occurs at the aggregate–mortar interface; and/or (ii) after cohesive failure, when the crack develops within the mortar. Based on such an interpretation of the cracking phenomenon, Kim et al. [8] began to study the fatigue behavior of asphalt mixtures using the fine aggregate matrix (FAM) approach, and developed a protocol to evaluate the FAM’s properties.
The FAM is the matrix phase of the asphalt concrete composed of fine aggregates, filler, binder, and air voids, and represents the intermediate scale between the asphalt mastic and the full asphalt mixture. In studies with FAM mixtures, the primary assumption is that it reproduces the internal structure of the fine portion of the aggregate gradation of a full hot-mix asphalt (HMA) mixture. Another important assumption is that the physicochemical interactions between the aggregate and binder are replicated in the FAM specimen. Studies with a FAM gained notoriety for having a good agreement between the FAM and asphalt concrete (AC) properties, which was observed for the moisture characterization [64], fatigue cracking, and permanent deformation characterization [65][66][67][68][69].
Caro et al. [64] carried out surface energy measurements and dynamic mechanical analyzer tests on specimens of four FAM mixtures, and compared the results with the ones obtained by assessing the moisture susceptibility of the corresponding full asphalt mixtures by means of the saturation aging tensile stiffness (SATS) test [70]. A good agreement was observed between the results obtained for the fine-graded asphalt mixtures and the ones obtained for the dense asphalt mixtures. In both approaches, the granite mixture treated with 2% (by weight) hydrated lime was the most resistant to moisture damage, and the mixture containing only granite was the most susceptible to damage. However, the authors reported some differences between the FAM and HMA results for the sample containing crushed granite and limestone dust.
In order to investigate the inherent fatigue cracking resistance of modified asphalt binders (PPA, SBS, PPA + SBS, and PPA + Elvaloy), Motamed et al. [65] submitted FAM samples produced with modified asphalt binders and glass beads to torsional loading (controlled strain mode–275 kPa) at 10 Hz and 16 °C. The rationale for the use of glass beads in substitution for the mineral aggregate was to simulate the same stress state to which the binder was submitted in the asphalt concrete structure. The FAMs and the asphalt concrete mixtures fatigue lives (number of cycles to achieve 50% of the initial modulus) were compared, and it was observed that the FAM mixtures presented the same rank order for fatigue life of the asphalt concrete mixtures produced with the same modified asphalt binders [65].
Gudipudi and Underwood [18] observed a good agreement for the damage characteristic curves (C vs. S) between FAM and asphalt concrete for the tests carried out at 10 and 19 °C, but the C-values at failure for the FAMs were lower as compared to those of the asphalt concrete. It was not possible to compare results from tests with FAM and asphalt concrete carried out at 25 °C once the damage curve for both materials (FAM and asphalt concrete) presented a significant variation that could be related to viscoplasticity or another mechanism [18]. Coutinho [66] found the same rank order between the fatigue resistance of AC mixtures and the fatigue resistance of FAM mixtures when the FAM mixtures were subjected to stress-controlled time-sweep tests. Im et al. [67] observed a strong correlation between the linear and nonlinear viscoelastic and viscoplastic deformation characteristics of the asphalt concrete and its corresponding FAM. Underwood and Kim [71] evaluated the effect of different compositions for the four material scales (binder, mastic, FAM, and asphalt concrete) using linear-viscoelastic properties, such as dynamic shear modulus (|G*|) and phase angle (δ). The authors concluded that the dynamic modulus and the phase angle for the FAM materials were much more similar to the full mixture data than were the mastic materials. The study showed that the materials at different scales presented differing levels of sensitivity to changes in the blending parameters. The materials at the FAM scale presented a sensitivity that was more in line with that observed for asphalt concrete mixtures under all of the tested conditions.
Palvadi et al. [17] validated the VECD theory to characterize damage in FAM specimens based on the similarity of the characteristic curves (CxS) for a given FAM for both monotonic and cyclic loading modes and different amplitudes. Palvadi et al. [17] also proposed a test procedure to investigate the healing characteristics of FAM specimens. This test procedure consists of four rest periods (5, 10, 20, and 40 min) and three levels of stiffness (20%, 30%, and 40% reduction in C). In this method, four specimens of each FAM were tested in order to apply a specific rest period in a specific specimen. Palvadi et al. [17] concluded that the healing percentage of each FAM is a material characteristic, once that the values for this parameter were similar, regardless of both the sequence of application of the rest period and the damage level. In an attempt to improve the procedure proposed by Palvadi et al. [17], Karki et al. [16][72] developed an integrated testing procedure that was capable of quantifying damage and healing characteristics using a single specimen without separating the damage and healing tests. Karki et al. [16][72] were the first to apply the simplified viscoelastic continuum damage (S-VECD) theory to characterize FAMs. The researchers highlighted that the characteristic curve (CxS) is a unique material property due to the similarity of the curves for a given material, regardless of the loading conditions (different amplitudes and frequencies) and the introduction of rest periods during the test.
Gudipudi and Underwood [18] analyzed the fundamental similarities or differences between FAM and AC scales by means of the S-VECD theory. They observed similarities in material properties between the two material scales, and the CxS curves for a particular FAM and its corresponding AC mix were very similar. However, the C-value when failure occurred was generally lower in the FAM as compared to the mix. This result suggested that the FAM can reach a greater damage accumulation before failure occurs. The use of FAM testing for material characterization and ranking of AC mixtures has a great potential, if the material fabrication protocols are accurately followed. Freire et al. [73] applied the S-VECD theory to evaluate the effect of different maximum nominal aggregate sizes (MNS) of the mineral aggregate particles on the FAM fatigue resistance to identify which one best represented the damage characteristics of the asphalt mixture. The FAM mixtures were prepared with three different MNS (4.00, 2.00, and 1.18 mm), and their fatigue characteristics were evaluated and compared to the ones of a hot-mix asphalt (HMA) mixture prepared with an MNS of 12.5 mm. The main finding was that the Wöhler curves created for the FAMs produced with mineral aggregate particles of 2.00 mm and the corresponding asphalt mixture presented similar trends. The authors pointed out that a direct comparison of the absolute results, instead of the trends observed, could not be done directly because the parameters for the HMA mixture were obtained by axial loading, whereas the FAM parameters were obtained by shear loading.
Some researchers adapted the linear amplitude sweep test (LAS) method proposed by Johnson [74] to characterize the fatigue resistance of the FAMs using the VECD approach. The investigations evaluated the effect of (i) different particle size distributions [66], (ii) different nominal maximum aggregate size [75][76], (iii) the thermal and photochemical aging [77][78], and (iv) the effect of RAP content and rejuvenating agents (RAs) [79][80]. However, Freire et al. [76] did not recommend the use of the LAS test to analyze the fatigue resistance of the FAM mixes due to the difficulty to achieve the failure, once the torque capacity of the DSR was low and unable to take the sample to failure. The authors observed that for the highest strain amplitudes of the LAS test, the equipment needed to work near its capacity due to the high stiffness of the FAM specimens.
Regarding the FAM mixes containing RAP and recycled asphalt shingle (RAS), researchers [6][63][68][79][80][81][82] concluded that the use of these materials decreased the fatigue life of the mixture due to the hard binder present in the RAS and RAP. The use of RAs (petroleum tech, green tech, and agriculture tech) in the FAM mixes containing RAS and RAP was investigated by Zhang et al. [63], Nabizadeh [68], Nabizadeh et al. [82], and Zhu et al. [6] as an alternative to increase the fatigue life of the FAMs. Nabizadeh [68] and Zhang et al. [63] concluded that the RAs resulted in softer mixtures with improved fatigue life (especially for the FAMs with high RAP contents). Zhu et al. [6] observed the same behavior in the case of the FAM with RAS mixed with another petroleum-based RA. The combination of the WMA additive with the petroleum tech rejuvenator was evaluated by Nabizadeh [68], and this combination resulted in the softest FAM compared with other rejuvenators (green tech and agriculture tech).
With the aim of investigating the fatigue cracking of the asphalt binders at the FAM scale without the physicochemical interaction with the mineral aggregate, Motamed et al. [65] used rigid particles, such as glass beads, in substitution for the mineral aggregate to produce the FAM specimens. This new technique resulted in similar fatigue cracking characteristics between the FAMs and the asphalt mixtures produced with the same asphalt binder. The researchers concluded that the glass beads could be used in substitution for the mineral aggregate when the binder properties were the main issue of interest. More recently, Li et al. [83] used a combined fatigue–healing method based on the VECD model to evaluate fatigue and self-healing properties of three rock asphalt composites. Li et al. [83] indicated that the replacement of a portion of the virgin asphalt binder by the rock asphalts enhanced the fatigue cracking resistance of the FAM mixtures, and the influence on fatigue life was dependent on both the type and concentration of the rock asphalt.
Warm fine aggregate mixtures (W-FAM) fabricated using different WMA additives were compared with an HMA (control mixture) in a study by Sadeq et al. [84]. The control mixture presented higher dissipated pseudo-strain energy (DPSE) than the W-FAM, and the fatigue life in the VECD analysis approach was not statistically significant among the control mixture and the W-FAM mixtures, indicating that the WMA mixtures had fatigue resistance comparable to the hot-mix asphalt mixtures. In studies by Sadek et al. [85] and Sharma and Swamy [86], a probabilistic analysis approach was applied to the fatigue life prediction model deduced from the VECD theory. The inherent variability of asphaltic materials exhibited in the fatigue test results led the researchers to develop a new probabilistic approach. Probabilistic approaches present the ability to account for uncertainties associated with fatigue tests, models, and model parameters. In a study conducted by Sadek et al. [87], the efficacy of using the probabilistic approach in the analysis of the viscoelastic continuum damage (VECD) and fatigue life was examined for hot and warm fine aggregate mixtures (H-FAM and W-FAM). The probabilistic VECD approach had the advantage of providing more reliable predictions of fatigue life that accounted for uncertainty in determining the model parameters, instead of the deterministic approach. The probabilistic analysis results showed that the W-FAMs had shorter fatigue lives than the one obtained by the control H-FAM mixture; however, their fatigue lives presented more consistency and less uncertainty.


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