# Application of the Viscoelastic Continuum Damage Theory to Study the Fatigue Performance of Asphalt Mixtures—A Literature Review

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## Abstract

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_{FAM}) indicated that the increase in RAP from 20% to 40% decreased the fatigue life of the mixtures. A strict control of the mixture variables is required, since the intrinsic heterogeneity of asphalt mixtures can lead to different characteristic curves for the same material.

## 1. Introduction

## 2. Theory of Viscoelastic Continuum Damage

#### 2.1. The Work Potential Theory

_{j}, and independent generalized displacements, q

_{j}, as shown in Equation (1), where δq

_{j}is the virtual displacement and δW’ is the virtual work. For different physical situations, q

_{j}can represent strain, displacement, or rotation; and Q

_{j}can be stress, force, or moment. For any process of interest, the existence of a strain energy function, W = W(q

_{j}, S

_{m}), was assumed, where S

_{m}(m = 1, 2, 3, M) refers to the increase in the value of the internal state variable, S. The relationship between the work done on a body during the process in which damage occurs and the strain energy function is expressed by Equation (2), where f

_{m}is the thermodynamic force (Equation (3)).

_{1}–t

_{2}, and assuming that a state function W

_{S}= W

_{S}(S

_{m}) exists such that the thermodynamic force, f

_{m}, is given by Equation (4), the work to vary the internal state of the material from a state 1 to a state 2 is expressed by Equation (5). The variable $\dot{{\mathrm{S}}_{\mathrm{m}}}$ is the damage evolution rate. By solving the integral in Equation (5), the work is given by Equation (6). Assuming that the time t

_{1}= 0, the total work from t = 0 to the current time t

_{2}is given by Equation (7).

_{j}is given by Equation (8), where j = 1, 2, …, J, and W

_{T}is the total work done on the body while considering that S

_{m}is variable in time. The elements of Schapery’s theory, expressed in terms of stress–strain relationships, can be represented as follows in Equations (9) and (10), where σ

_{ij}is the stress tensor, ε

_{ij}is the strain tensor, and S

_{m}is the internal state variable. By considering Equations (3) and (4), the damage evolution law for the elastic media is represented by Equation (11), in which W

_{S}= W

_{S}(S

_{m}) is the dissipated energy due to damage growth. The right-hand side of the law represents the required force for damage growth, while the left-hand side of the damage evolution law represents the available thermodynamic force to produce damage growth.

#### 2.2. Elastic–Viscoelastic Correspondence Principle

^{R}) and pseudo strain (ε

^{R}). According to the second correspondence principle (CP II), σ

^{R}= σ, where σ is the time-dependent stress applied to a viscoelastic material; and the pseudo strain is given by Equation (15), where ε is the time-dependent strain in a viscoelastic material, G(t) is the linear-viscoelastic relaxation modulus of the material, and E

^{R}is the modulus of hypothetical elastic material [22,23].

^{R}) function, where the physical strain, ε, is substituted by the pseudo strain, ε

^{R}, and S

_{m}is the internal state variable. The stress-pseudo strain relationship is given by Equation (17), where $\mathsf{\sigma}$ is the stress, W

^{R}is the pseudo strain energy density, and ε

^{R}is the pseudo strain [23]. For most viscoelastic media, the available force for damage growth and the resistance against the growth are rate-dependent. For this reason, the damage evolution law for elastic materials (Equation (11)) cannot just be transformed into a damage evolution law for viscoelastic materials by the use of correspondence principles without further modification. The new damage evolution law for viscoelastic materials is given by Equation (18), where $\dot{{\mathrm{S}}_{\mathrm{m}}}$ is the damage evolution rate, W

^{R}is the pseudo strain energy density, S

_{m}is the internal state variable, and α

_{m}is a material-dependent constant [26].

#### 2.3. Viscoelastic Continuum Damage Model Applied to Asphalt Mixtures

^{R}= strains in a body (Equation (15)); ${\mathsf{\epsilon}}_{\mathrm{L}}^{\mathrm{R}}$ = maximum pseudo strain in the past history; and S

_{p}= damage parameter [28]. The form of the damage parameter based on pseudo strain is shown as follows in Equation (20), where p = (1 + N)k; N = the exponent of the power law between stress and strain, σ~|ε

^{R}|

^{N}; k = 2(1 + 1/m); m = the exponent of the power law between creep compliance and time; and D(t) = D

_{1}t

^{m}. When repetitive loading is applied, numerical integration can be used to obtain S

_{p}, as shown in Equation (21), assuming that dε

^{R}/dt is constant within the range of the experimental data points. A uniaxial tensile testing was employed to generate all data for the construction of the constitutive equation, which satisfactorily predicted effects due to multilevel loading, the sequence of multilevel loading, and various durations of rest periods. From the tests, it was proved that the history dependence of asphalt concrete with negligible damage growth can successfully be eliminated by the CP II, σ

^{R}= σ.

^{R}) function is given by Equation (22), where C is a function of the damage parameter S, and ε

^{R}is the pseudo strain. For linear-viscoelastic behavior and fixed damage, the stress, σ, can be written as in Equation (23). The damage evolution law (Equation (18)) is reduced to the single equation for S (Equation (24)), where S is the damage parameter, α is a material-dependent constant, and the over dot denotes a time derivative.

^{R}is the pseudo strain, and C is a function of the damage parameter S.

^{R}and strain rate, but not on S. In order to find the dependence of the C modulus on S, the use of the damage evolution law (Equation (18)) is required. However, the evolution law itself requires prior knowledge of C(S), making this procedure inefficient to find C and its dependence on S. The method proposed by Park et al. [21] to overcome this problem was to determine a transformed damage variable, Ŝ (Equation (28)), that may be obtained from the numerical scheme presented in Equation (29), where ${\mathsf{\epsilon}}_{\mathrm{i}}^{\mathrm{R}}$(i = 1, 2, 3, N) denotes pseudo strain levels, C’ ≡ dC/dŜ, and Ŝ(0) = 0. This method allows one to obtain the function C(Ŝ) from experimental stress–pseudo strain curves, and then the function C(S) can be obtained from Equation (28), by replacing the transformed damage variable Ŝ for the original damage variable, S, where α is a material-dependent constant.

^{R}cycle as the number of loading increased, and they found it necessary to define the secant pseudo stiffness, S

^{R}, to represent this change in the slope of stress–pseudo-strain loops (Equation (33)), where ${\mathsf{\epsilon}}_{\mathrm{m}}^{\mathrm{R}}$ is the peak pseudo strain in each stress–pseudo-strain cycle and σ

_{m}is the stress that corresponds to ${\mathsf{\epsilon}}_{\mathrm{m}}^{\mathrm{R}}$. To minimize the sample-to-sample variability, the pseudo stiffness was divided by the initial pseudo stiffness, I, resulting in the normalized pseudo stiffness, C, represented by Equation (34). By considering Equation (33) and Equation (34), the constitutive equation for viscoelastic materials with growing damage is expressed by Equation (35). The normalized pseudo stiffness, C(S

_{m}), is a function of the internal state variables, S

_{m}, and represents the microstructural changes of the body. The researchers assumed an internal state variable, S

_{1}, to determine the change in pseudo stiffness due to growing damage, and the work function (W

^{R}) for viscoelastic materials is given by Equation (36), where C

_{1}(S

_{1}) is a function that represents S

^{R}.

_{1}can be determined by using experimental data and the damage evolution law (Equation (18)), this procedure is not convenient to find C

_{1}and its dependence on S

_{1}, because the evolution law requires prior knowledge of C

_{1}(S

_{1}). The method presented to overcome this problem was to use a chain rule (Equation (37)) to eliminate the S on the right-hand side of the evolution law, and by means of mathematical substitutions (Equation (38)) the numerical approximation given by Equation (39) was obtained, where ε

^{R}is the pseudo strain, t is the time, I is a factor to normalize the pseudo stiffness, and α is a material-dependent constant (Equations (30) and (31)). The function C

_{1}(S

_{1}) can be obtained by cross-plotting the C values obtained from Equation (35) against the S values obtained from Equation (39), and by performing a regression on the data (Equation (40)), where C

_{10}, C

_{11}, and C

_{12}are regression coefficients; or an exponential function (Equation (41)), where a and b are the calibration constants.

#### 2.4. Pseudo-Strain Calculation

_{0}is the shear strain amplitude, ω is the angular velocity, θ is a regression constant, and H(t) is the Heaviside step function. Substituting Equation (42) within the definition of pseudo strain (Equation (15)) and assuming that E

_{R}is 1, the pseudo strain at the current time can be analytically represented by Equation (43), where ϕ is the phase angle and |G*| is the linear-viscoelastic dynamic shear modulus. As can be seen, the pseudo strain at any time can be predicted with a well-defined strain history as a function of time and two material properties: dynamic modulus and phase angle. In the study of fatigue, only the peak pseudo strain within each cycle is used, and the pseudo strain reaches the peak pseudo strain in each cycle when the sine function in Equation (43)) becomes 1, as represented by Equation (44)), where ${\mathsf{\epsilon}}_{\mathrm{m}}^{\mathrm{R}}$ is the peak pseudo strain at each cycle [8,30].

#### 2.5. Simplified Viscoelastic Continuum Damage Model

^{th}load cycle for a stress-controlled cyclic test were simplified, as shown in Equations (46) and (47). In this FHWA S-VECD method, damage is expected to grow according to Equation (48).

_{k}, and |G*|

_{LVE}| represent, respectively, the shear stress amplitude, the shear pseudo-stress amplitude, the shear pseudo-strain amplitude, the dynamic shear modulus at the k

^{th}loading cycle, and the linear-viscoelastic dynamic shear modulus. The comma subscript, pp, denotes the pseudo strain computed based on peak-to-peak values. The I factor is used as a correction factor in order to normalize test results while considering the sample-to-sample variability of the initial dynamic shear modulus. Hou [9] conducted a study in order to verify the S-VECD model proposed by Underwood et al. [33] by applying it to various types of asphalt concrete mixtures under various conditions. Hou [9] observed that the model could be applied to accurately predict the fatigue life of asphalt concrete under cyclic loading at multiple temperatures and strain levels. Lee et al. [36] recommended that the power law model (Equation (40)) should be fitted to the characteristic curves obtained from the S-VECD model after filtering the data to produce damage (S) increments of 5000. Lee et al. [36] also proposed a shape factor, which is calculated as the ratio of the area above the damage characteristic curve for an individual test replicate (i.e., A

_{measured}) to the area above the fitted model damage characteristic curve obtained by Equation (40) (i.e., A

_{predicted}). Replicates with shape factors greater than 1.1 or lower than 0.9 are considered outliers and should not be considered for analysis.

#### 2.6. Mechanistic Fatigue Life Prediction Model

_{f}, to a certain pseudo-stiffness level, C, or to reach a certain amount of damage, S

_{f}, at an arbitrary frequency, f, and controlled pseudo-strain amplitude, ε

^{R}. The parameters C

_{11}and C

_{12}are obtained from Equation (40) [5,7]. Kim and Little [5] performed torsional shear cyclic tests in sand asphalt mixtures, and compared the measured fatigue lives of the materials with the values predicted from the fatigue model (Equations (49)–(51)). They concluded that the model parameters might provide a reasonable representation of the fatigue response.

#### 2.7. Fatigue Failure Criterion

^{R}, and the number of cycles to fatigue failure. The criterion was able to predict the fatigue life of asphalt concrete mixtures across different temperatures and strain amplitudes [45,46], and has been applied to predict fatigue behavior of reclaimed asphalt pavement (RAP) mixtures and non-RAP mixtures, mixtures prepared with modified and unmodified binders, warm-mix asphalt mixtures, and long-term aged mixtures [51,52,53,54,55,56,57,58,59,60]. Keshavarzi and Kim [49] extended the G

^{R}criterion concepts to determine when fracture occurs in monotonic failure tests, such as thermal stress restrained specimen tests (TSRSTs).

^{R}index that is more representative of field conditions, because there was not a clear trend between RAP increase and a change in the mixture fatigue performance when the damage characteristic curves and G

^{R}failure criterion were used. More recently, Wang and Kim [48] developed and validated a new energy-based failure criterion based on the S-VECD model (D

^{R}). The advantages of the D

^{R}failure criterion in comparison to the previous G

^{R}failure criterion are: (i) the D

^{R}can be computed for each fatigue test and then used to check the sample-to-sample variability for each test, (ii) D

^{R}is obtained in arithmetic scale rather than in log–log scale, and (iii) the number of tests needed to obtain D

^{R}is fewer than for the G

^{R}failure criterion [48]. In a study by Wang et al. [62], a three-dimensional finite element program (FlexPAVETM) was used to simulate the fatigue performance of field test sections. The fatigue damage of the sections was predicted using the FlexPAVETM software by considering both G

^{R}and D

^{R}criteria. The D

^{R}failure criterion was found to yield more realistic fatigue-cracking performance predictions than the G

^{R}failure criterion [62], and has been checked for RAP and aged mixtures [63,64].

^{R}, an index parameter referred to as apparent damage capacity (S

_{app}) [65], and mix design factors such as nominal maximum aggregate size (NMAS), asphalt binder type, and binder content. The S

_{app}parameter was shown to be able to predict the fatigue-cracking propensity of asphalt mixtures [47,65,66], and it was found to have a strong relationship with polymer modification and the NMAS of asphalt mixtures [47]. Based on experimental data and the Georgia Department of Transportation’s practical guidelines for specific mixtures, a study by Etheridge et al. [47] developed S

_{app}threshold values for different traffic levels. In a study by Zhang et al. [67], the D

^{R}parameter showed good correlation with three new performance indices from the linear amplitude sweep (LAS) test: (i) strain tolerance (ε

_{T}), (ii) strain energy tolerance (ε

_{E}), and (iii) average reduction in integrity to failure (I

^{R}).

#### 2.8. Linear Viscoelasticity

_{max}is the maximum shear stress at each cycle and γ

_{max}is the applied cyclic shear strain amplitude; and in Equation (54) for the complex shear modulus, G*(ω), where i is equal to √(−1).

_{e}is the equilibrium modulus, G

_{i}is the elastic modulus, ρ

_{i}is the relaxation time, ω is the angular frequency, and n is the number of elements of the Prony series needed to fit the analytical representation to the experimental data.

_{i}) are defined by means of the collocation method, a matching process between the analytical representation and the experimental data for a certain number of points. Considering the Prony series parameters found by the collocation method, the static relaxation shear modulus as a function of time (time domain) can be predicted from the dynamic shear modulus as a function of frequency (frequency domain) by Equation (58). The relaxation property (m-value) is determined as the slope of the relaxation modulus curve, in logarithm scale, and is used in the VECD approach to determine the damage evolution rate of the material. This material property can be obtained by adjusting a power law function (Equation (59)) to the relaxation curve predicted by the Prony series, where G

_{0}and G

_{1}are material constants, t is the time, and m is the slope of the relaxation curve in the time domain.

_{LVE}, is equal to the dynamic shear modulus value that is measured by applying a small level of strain or stress within the linear-viscoelastic range of the material [16].

## 3. Studies on the Application of the VECD Theory

#### 3.1. Full Asphalt Mixture Approach

_{1}and C

_{2}(Equation (40)). Daniel and Kim [75] 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 [76] 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 [77] 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. [66] observed that the asphalt mixture CxS curves were independent of the strain level, but affected by the loading waveform.

#### 3.2. Fine Aggregate Matrix Approach

## 4. Analysis Protocol of Tests with FAM Using the S-VECD Approach

## 5. Application of the S-VECD Theory: Laboratory Tests and Discussion of Results

#### 5.1. Experimental Method—Materials and Preparation of the FAM Specimens

#### 5.2. Experimental Method: Fingerprint Test

_{LVE}) was the averaged |G*| values measured at 1 Hz in the fingerprint.

#### 5.3. Experimental Method: Damage Test

_{f}, which is the number of axle load repetitions capable of leading the material to failure, was predicted by employing the mechanistic fatigue life prediction model (Equation (51)) developed by Lee et al. [7] and Kim and Little [5]. The fatigue models were adjusted for a 50% reduction in the material’s pseudo stiffness. Figure 4a,b shows an example of the CxS curve and the fatigue curve obtained for a single specimen. The detailed description of the procedure devised to deal with replicates of the same FAM and to build the CxS and fatigue curves is presented in the next section. Figure 4a presents the CxS curve built from laboratory data and the power law fitted to the data for the specimen 1-FAM1. Figure 4b presents the fatigue curve for the specimen 1-FAM1.

#### 5.4. Analysis and Discussion of Results

_{LVE}, and the material relaxation rate, m) for each FAM specimen. Table 4 also shows the average linear-viscoelastic properties of each FAM and the coefficient of variation of each specimen. The specimen shape factor and the D

^{R}criterion were tools that were able to check the specimen-to-specimen variability, and they are also presented in Table 4. As recommended by Lee et al. [7], the specimen-to-specimen variability can be quantified and minimized by means of a shape factor. By following such an idea, a shape factor was defined in this study as the ratio of the area above a particular specimen’s CxS curve to the area above the average CxS curve of the FAM. A shape factor close to 1 indicated a valid specimen; i.e., the results of a particular specimen were very close to the FAM average curve. In the study by Lee at al. [7], replicates with shape factors greater than 1.1 or lower than 0.9 were considered outliers. By following this premise, curves with shape factors of 1.0 ± 0.1 were accepted and used to build the characteristic curve of the FAMs. In the calculation of the shape factors, a decay until a pseudo stiffness equal to 0.3 was considered. The shape factors of all specimens are presented in Table 4, where it is possible to observe shape factors between 0.92 and 1.08, except for FAM2 s1, which presented a shape factor of 0.8. The other tool applied in this study to check the specimen-to-specimen variability was the D

^{R}criterion, which is defined as the average reduction in pseudo stiffness up to failure [48]. Wang and Kim [48] found a variation of ±0.04 in the D

^{R}value of specimens of the same mixture. The authors also carried out sensitivity studies of the pavement performance analysis using the S-VECD model with the D

^{R}failure criterion, and they found that with this variation of ±0.04 in the D

^{R}value, the predicted fatigue damage for the pavements did not differ significantly. The variation between the specimens of the FAMs evaluated in this study was ±0.05.

_{LVE}values as compared to the mixtures containing 20% of RAP (1 and 3). For the same RAP proportion (20% or 40%), the highest complex modulus was observed for the FAMs containing the binder PG 64-22. Regarding the relaxation rate, m, the increase in the RAP content from 20 to 40% led to a decrease in this rate, with FAMs 1 and 3 (20% of RAP) presenting higher m values as compared to FAMs 2 and 4 (40% of RAP). Among the FAMs prepared with 20% of RAP, FAM1 (prepared with binder PG 64-22) and FAM3 (prepared with binder PG 58-16) presented equivalent m values (FAM3 presented a slight 0.2% reduction as compared to the FAM1 rate). Among the FAMs prepared with 40% of RAP, FAM4 (prepared with binder PG 58-16) presented an increase of 8.5% in the m value as compared to FAM2 (prepared with binder PG 64-22).

_{FAM}) and the rank order of the FAMs according to this factor, by assuming that the higher the factor, the higher the fatigue performance of the material.

^{R}values presented in Table 4, it can be observed that FAM4, prepared with 40% RAP and the binder content of the RAP material adjusted with the binder PG 58-16, presented the higher D

^{R}values, followed by FAM3, prepared with 20% RAP and the binder content of the RAP material adjusted with the binder PG 58-16. It can also be observed that the lower values of the D

^{R}parameter were the ones of FAM2, which was prepared with 40% RAP and the binder content of the RAP material adjusted with the binder PG 64-22. The trend observed by Wang and Kim (2019) was that the D

^{R}value decreased as the RAP content increased, which was the same trend observed in this study for the two mixtures prepared with the binder PG 64-22. However, for the two mixtures prepared with the binder PG 58-16, the D

^{R}values increased with the increase in RAP. Wang and Kim (2019) emphasized that the D

^{R}value alone cannot be used to compare the fatigue performance of different asphalt mixtures.

## 6. Conclusions

- The findings of the experimental study with RAP and binders PG 64-22 and PG 58-16 indicated that the FAMs containing 40% of RAP (2 and 4) presented higher |G*|
_{LVE}values and higher damage evolution ratios as compared to the FAMs containing 20% of RAP (1 and 3). Out of the FAMs prepared with 20% of RAP (FAM1 and FAM3), the highest |G*|_{LVE}was observed for the FAM containing binder PG 64-22 (FAM1), and the damage evolution ratios were the same for both FAMs, which was an expected result, once the presence of the softest binder (PG 58-16) was supposed to reduce the stiffness of the FAMs (3 and 4). - Regarding the prediction of the fatigue lives of the materials evaluated in the experimental study, the addition of RAP increased the parameter A of the fatigue model (related to the initial stiffness of the material and how the stiffness changed with the evolution of the damage) and the parameter B (related to the damage evolution rate)—the resulting fatigue lives of the FAMs prepared with 20% RAP were longer than the ones obtained for the FAMs prepared with 40% of RAP. The fatigue performance was directly related to the specimen stiffness: the higher the stiffness, the higher its susceptibility to damage and the lower the relaxation rates (which resulted in higher damage accumulation rates). The best solution to adjust the binder content of FAMs produced with 20% and 40% of RAP was the use of the binder PG 58-16. The FAM tests combined with the S-VECD theory as a tool to analyze the results was a practical approach, and is widely used to evaluate all sorts of variables of an asphalt mixture. However, some variables, such as low temperatures and/or high percentages of RAP, turn the mixtures into overly stiff materials, and the tests can be unpractical due to the limits of the rheometer torque. Equipment with a higher torque capability could accelerate the test duration.
- The improvement of computational simulations of the test protocols is an important subject for future works, and could contribute to a better understanding of the mechanisms and variables involved in the fatigue process, and could also help overcome the rheometer limitations.
- Comparisons between fractures mechanics and continuum mechanics results could also be an interesting topic to improve the VECD model in order to account for the different types of damage: adhesive or cohesive.
- Regarding materials science and development of advanced/new materials, the FAM approach combined with the S-VECD approach offered several new possibilities in terms of material performance evaluation and material development. Some examples can be mentioned concerning the fatigue performance: (i) the evaluation of the impact of higher RAP contents added to new AC mixtures; (ii) the evaluation of the impact of recycling agents at different contents, including petroleum-based materials, vegetable-based oils, and recycled oils; (iii) the assessment of the aging impact on fatigue; (iv) the assessment of moisture damage on fatigue resistance; (v) the assessment of new asphalt modifiers, including hybrid modification using virgin and recycled materials; and (vi) the evaluation of the effect of distinct aggregate types and aggregate gradations, among others. Several doubts related to these subjects can be countered by carrying out tests at the FAM scale and using the S-VECD approach. However, one must keep in mind that such a development also depends on a larger number of experiments on the correlation between the fatigue performance at the two scales (FAM and full asphalt mixtures). Such experiments are essential for the development and popularization of these very promising techniques.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Generalized Maxwell model [68].

**Figure 3.**Fingerprint: (

**a**) relaxation modulus vs. frequency; and (

**b**) relaxation modulus vs. time (log).

**Figure 4.**(

**a**) CxS curve of the specimen 1-FAM1 and power law fitted to data; and (

**b**) fatigue model for the specimen 1-FAM1.

**Figure 6.**Individual CxS curves obtained by using the average viscoelastic properties, and average CxS curves. (

**a**) FAM1, (

**b**) FAM2, (

**c**) FAM3, (

**d**) FAM4.

Fingerprint Test | FAM1 Sample 1 | ||
---|---|---|---|

1. Data obtained from the fingerprint | Dynamic shear modulus within the linear-viscoelastic region (|G*_{LVE}|), relaxation rate (m) (Figure 3a,b) | - | |G*|_{LVE} = 9.93 × 10^{8}m = 0.476 |

2. Prony series fitted to the storage modulus values to obtain ${\mathrm{G}}_{\mathrm{e}}$, ρ_{i}, and ${\mathrm{G}}_{\mathrm{i}}$. | ${{\mathrm{G}}^{\prime}\left(\mathsf{\omega}\right)=\mathrm{G}}_{\mathrm{e}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}\frac{{\mathrm{G}}_{\mathrm{i}}{\mathsf{\omega}}^{2}{\mathsf{\rho}}_{\mathrm{i}}{}^{2}}{{\mathsf{\omega}}^{2}{\mathsf{\rho}}_{\mathrm{i}}{}^{2}+1}$ | Equation (56) | ${\mathrm{G}}_{\mathrm{e}}$, ρ_{i}, and ${\mathrm{G}}_{\mathrm{i}}$ are calculated by using Solver |

3. Laplace transform to convert data from frequency to time domain | ${\mathrm{G}\left(\mathrm{t}\right)=\mathrm{G}}_{\infty}+{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{G}}_{\mathrm{i}}{\mathrm{e}}^{-\mathrm{t}/{\mathsf{\rho}}_{\mathrm{i}}}$ | Equation (58) | $\mathrm{G}\left(\mathrm{t}\right)$ values obtained by using the parameter of the Prony series |

4. Model adjusted to data to obtain the material parameter, m | G(t) = G_{0}+G_{1}t^{−m} (Figure 3b) | Equation (59) | G_{0} = 1G _{1} = 2.65 × 10^{8}m = 0.476 |

5. Equation to obtain the material parameter, α | α = (1 + 1/m) for strain-controlled tests α = 1/m for stress-controlled tests | Equation (30) Equation (31) | α = 2.10 |

Damage Test | FAM1 sample 1 k = 5 | ||

1. Data obtained from the damage tests | Complex modulus (|G*|), phase angle (φ), and strain (ε), at each cycle k | - | |G*|_{k = 1} = 7.87 × 10^{8}|G*| _{k = 5} = 6.91 × 10^{8}φ _{k = 5} = 49.5ε _{k = 5} = 0.051% |

2. Peak pseudo strain during each cycle k, ${\mathsf{\epsilon}}_{\mathrm{m}}^{\mathrm{R}}\left(\mathrm{t}\right)$ | ${\mathsf{\epsilon}}_{0}^{\mathrm{R}}{\left(\mathrm{t}\right)=\mathsf{\epsilon}}_{0}{|\mathrm{G}}^{\ast}{|}_{\mathrm{LVE}}$ | Equation (45) | ${\mathsf{\epsilon}}_{0}^{\mathrm{R}}{\left(\mathrm{t}\right)=\mathsf{\epsilon}}_{0\mathrm{k}=5}=5.03\text{}\times \text{}10$^{5} |

3. Pseudo stress, σ^{R} | σ^{R} = σ | Elastic-viscoelastic correspondence principle | σ^{R} = 350 kPa |

4. Initial pseudo stiffness, I | $\mathrm{I}=\frac{{|\mathrm{G}}^{\ast}{|}_{\mathrm{k}=1}}{{|\mathrm{G}}^{\ast}{|}_{\mathrm{LVE}}}$ | Dynamic shear modulus at the first cycle | $\mathrm{I}=\frac{7{.87\text{}\times \text{}10}^{8}}{9{.93\text{}\times \text{}10}^{8}}=0.792$ |

5. Normalized pseudo stiffness at each cycle k, C | ${\mathrm{C}}_{\mathrm{k}}\left(\mathrm{S}\right)=\frac{{|\mathrm{G}}^{\ast}{|}_{\mathrm{k}=\mathrm{n}}}{{\mathrm{I}\text{}|\mathrm{G}}^{\ast}{|}_{\mathrm{LVE}}}$ | Equation (47) | ${\mathrm{C}}_{5}=\frac{6.91}{0.792\text{}\times \text{}7.87}=0.88$ |

6. Damage parameter, S | $\mathrm{dS}\text{}\equiv {\left[\frac{\mathrm{I}}{2}{\left({\mathsf{\epsilon}}_{0,\mathrm{k},\mathrm{pp}}^{\mathrm{R}}\right)}^{2}\left({\mathrm{C}}_{\mathrm{k}-1}-{\mathrm{C}}_{\mathrm{k}}\right)\right]}^{\mathsf{\alpha}/(1+\mathsf{\alpha})}{\left({\mathrm{t}}_{\mathrm{k}}-{\mathrm{t}}_{\mathrm{k}-1}\right)}^{1/(1+\mathsf{\alpha})}$ | Equation (48) | ${\mathrm{S}}_{5}\text{}\equiv \text{}4{.71\text{}\times \text{}10}^{7}$ |

7. Characteristic curve, CxS | The characteristic curves, CxS, are obtained by cross-plotting the C values against the S values at each cycle k | Figure 4a | C_{5} = 0.88 vs. S_{5} = 4.71 × 10^{7} at cycle k = 5 |

Fatigue Life Prediction Model | |||
---|---|---|---|

1. Power law fitted to the CxS curve to obtain C_{10}, C_{11}, and C_{12} | ${\mathrm{C}}_{1}\left({\mathrm{S}}_{1}\right){=\mathrm{C}}_{10}-{\mathrm{C}}_{11}{{(\mathrm{S}}_{1})}^{{\mathrm{C}}_{12}}$ | Equation (40) | Figure 4a |

2. Calculation of the parameter A | $\mathrm{A}=\mathrm{f}{\left\{\frac{1}{2}{\mathrm{C}}_{1}{\mathrm{C}}_{2}\right\}}^{\mathsf{\alpha}}{\left\{1+\mathsf{\alpha}\left(1-{\mathrm{C}}_{2}\right)\right\}}^{-1}{\mathrm{S}}_{\mathrm{f}}^{\left[1+\mathsf{\alpha}(1-{\mathrm{C}}_{2})\right]}$ | Equation (49) | A = 7.35 × 10^{29} |

3. Calculation of the parameter B | B = 2α | Equation (50) | B = 4.35 |

4. Prediction fatigue life curve | N_{f} = A[ε^{R}]^{−B} | Equation (51) | Figure 4b |

5. N_{f} (strain = 0.005%) | N_{f(0.005%)} = 7.35 × 10^{29} [4.46 × 10^{4}]^{−4.35} | Equation (51) | 4.59 × 10^{9} |

6. N_{f} (strain = 0.2%) | N_{f(0.2%)} = 7.35 × 10^{29} [1.78 × 10^{6}]^{−4.35} | Equation (51) | 500.25 |

Basalt Rock | |||
---|---|---|---|

Quarry identification | Bandeirantes | ||

Specific gravity of coarse aggregates (g/cm³) | 2.904 | AASHTO T 85 | |

Specific gravity of fine aggregates (g/cm³) | 2.999 | AASHTO T 84 | |

Specific gravity of filler (g/cm³) | 2.769 | ASTM D7928 | |

Absorption (%) | 0.6 | ASTM C128 | |

RAP Material | |||

Quarry location | São Carlos/SP | ||

Maximum specific gravity (g/cm³) | 2.596 | AASHTO T209 | |

Asphalt Binders | |||

Performance grade (PG) | PG 58-16 | PG 64-22 | ASTM D6373 |

Specific gravity (g/dm³) | 1.015 | 1.004 | ASTM D70 |

Continuous grade—virgin (°C) | 61.07 | 66.84 | ASTM D7175 |

Continuous grade—short-term aged (°C) | 65.52 | 66.94 | ASTM D7175 |

Continuous grade—long-term aged (S [60]^{1}) | −20.7 | −26.9 | ASTM D6648 |

Continuous grade—long-term aged (m [60]^{2}) | −20.2 | −27.4 | ASTM D6648 |

^{1}Creep stiffness at 60 s (MPa);

^{2}slope at 60 s.

Material | Sample | Air Voids | Linear-Viscoelastic Properties | CxS Parameters | ||||||
---|---|---|---|---|---|---|---|---|---|---|

m | M _{(av)} | cv (%) | |G*|_{LVE} (kPa) | |G*|_{LVE (av)} (kPa) | cv (%) | Shape Factor |
D^{R} | |||

FAM1 | s1 | 5.15 | 0.476 | 0.461 | 3.2 | 9.93 × 10^{8} | 8.92 × 10^{8} | 11.3 | 1.05 | 0.382 |

s2 | 5.28 | 0.490 | 6.2 | 7.53 × 10^{8} | −15.6 | 0.92 | 0.373 | |||

s3 | 4.96 | 0.422 | −8.6 | 9.46 × 10^{8} | 6.1 | 0.97 | 0.350 | |||

s4 | 4.99 | 0.467 | 1.2 | 8.82 × 10^{8} | −1.1 | 0.94 | 0.345 | |||

s5 | 4.99 | 0.452 | −2.0 | 8.85 × 10^{8} | −0.8 | 1.08 | 0.394 | |||

FAM2 | s1 | 5.19 | 0.342 | 0.375 | −8.7 | 1.57 × 10^{9} | 1.70 × 10^{9} | −7.6 | 0.8 | - |

s2 | 5.05 | 0.386 | 2.9 | 1.61 × 10^{9} | −5.3 | 0.99 | 0.332 | |||

s3 | 5.09 | 0.380 | 1.3 | 1.96 × 10^{8} | 15.3 | 1.03 | 0.332 | |||

s4 | 5.05 | 0.392 | 4.5 | 1.66 × 10^{9} | −2.4 | 0.98 | 0.325 | |||

FAM3 | s1 | 5.16 | 0.463 | 0.460 | 0.7 | 5.04 × 10^{8} | 5.95 × 10^{8} | −15.3 | 1.04 | 0.395 |

s2 | 4.67 | 0.465 | 1.1 | 5.90 × 10^{8} | −0.8 | 0.96 | 0.415 | |||

s3 | 5.47 | 0.448 | −2.5 | 6.61 × 10^{8} | 11.1 | 0.99 | 0.406 | |||

s4 | 5.09 | 0.463 | 0.7 | 6.24 × 10^{8} | 4.9 | 0.91 | 0.360 | |||

FAM4 | s1 | 4.57 | 0.398 | 0.406 | −1.8 | 1.54 × 10^{9} | 1.57 × 10^{9} | −2.1 | 1.03 | 0.455 |

s2 | 4.68 | 0.402 | −1.0 | 1.63 × 10^{9} | 3.6 | 1.00 | 0.448 | |||

s3 | 4.89 | 0.417 | 2.7 | 1.55 × 10^{9} | −1.5 | 0.97 | 0.423 |

FAM | A | B | N_{f}(0.005%) | Rank Order | N_{f}(0.20%) | Rank Order | FF_{FAM} | Rank Order |
---|---|---|---|---|---|---|---|---|

FAM1 | 7.35 × 10^{29} | 4.35 | 4.59 × 10^{9} | 4 | 500.25 | 2 | 2.32 | 3 |

FAM2 | 2.68 × 10^{36} | 5.35 | 1.17 × 10^{10} | 3 | 31.80 | 4 | 2.30 | 4 |

FAM3 | 5.25 × 10^{29} | 4.35 | 1.77 × 10^{10} | 1 | 1885.85 | 1 | 2.49 | 1 |

FAM4 | 2.16 × 10^{34} | 4.93 | 1.52 × 10^{10} | 2 | 189.63 | 3 | 2.39 | 2 |

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**MDPI and ACS Style**

Klug, A.; Ng, A.; Faxina, A.
Application of the Viscoelastic Continuum Damage Theory to Study the Fatigue Performance of Asphalt Mixtures—A Literature Review. *Sustainability* **2022**, *14*, 4973.
https://doi.org/10.3390/su14094973

**AMA Style**

Klug A, Ng A, Faxina A.
Application of the Viscoelastic Continuum Damage Theory to Study the Fatigue Performance of Asphalt Mixtures—A Literature Review. *Sustainability*. 2022; 14(9):4973.
https://doi.org/10.3390/su14094973

**Chicago/Turabian Style**

Klug, Andrise, Andressa Ng, and Adalberto Faxina.
2022. "Application of the Viscoelastic Continuum Damage Theory to Study the Fatigue Performance of Asphalt Mixtures—A Literature Review" *Sustainability* 14, no. 9: 4973.
https://doi.org/10.3390/su14094973