Linear viscoelastic models from creep and complex modulus testing in soil-emulsion

Abstract

This article describes the development and validation of a linear viscoelastic model 2S2P1D for silty sand mixed with 16% asphalt emulsion. Experimental data from creep tests were employed to propose and validate the model. The methodology involved soil collection and characterization, after which it was mixed with asphalt emulsion, and exposed to air for 24 hours, followed by the Superpave rotary compaction of the soil-emulsion mixture. Test samples were then exposed to air until a constant density loss was detected, followed by conducting complex modulus tests at thirty different temperature-frequency pairs. Based on these results and the smooth construction of the master curve of the mixture, it was confirmed that the material studied is thermorheologically simple, and the parameters of the linear 2S2P1D model were defined. After defining the constitutive model, a creep test was conducted on the material at a temperature of 0°C for 7200 seconds. The model exhibited satisfactory adjustments to the creep test results, even in the nonlinearity domain. The differences between the strains predicted by the model and those defined experimentally were considered within an acceptability limit for geotechnical materials, and the introduction of a modulus decrease coefficient of approximately 65% was sufficient to simulate the nonlinearity effect in the 2S2P1D model.

Keywords:
soil; asphalt emulsion; complex modulus; creep.

1. Introduction

Soil stabilization with asphalt emulsion has already been a topic known and studied for quite some time (Nassar et al., 2016; Andavan and Maneeshkumar, 2020). However, in recent years, its application as an earth-dam core is being studied due to its cohesive and waterproofing properties that derive from the asphalt binding agent (Dantas Neto et al., 2020; Brito et al., 2022). Such studies evidence that, unlike soil stabilization for paving (B. D. Oluyemi-Ayibiowu, 2019; Fernandes et al., 2022), the results of the use of soil-emulsion in watertight cores have proven more promising with an emulsion content over 16% in density (Pereira, 2018; Dantas Neto et al., 2020).

The fact that engineering structures, such as earth dams, undergo different types and intensities of loadings throughout their construction and use, and particularly the possibility of soil-emulsion mixtures behaving time-dependently in terms of strains under permanent static loading, make it essential to conduct studies of viscoelastic characterization and to model such materials to ensure its safety. To date, there are a few studies investigating the rheological characteristic of soil mixtures with asphalt binders (Collop and Khanzada, 2001; Kuvat and Sadoglu, 2020; Sarajpoor et al., 2021). However, none of them specifically focus on mixtures with high binder levels, particularly for their application in impermeable cores, being impossible to simulate the strain behavior of the dam body and evaluate the risk of material flow. For that, rheological models are needed.

Pereira (2018) and Dantas Neto et al. (2020) have demonstrated that the most promising results regarding shear resistance and permeability coefficient for potential dam core applications were achieved with emulsion contents exceeding 16% by mass in their studied mixture of silty sand and emulsion. Similarly, Brito et al. (2022) investigated the influence of curing time on the shear strength parameters of mixtures of silty sand (SM) with asphalt emulsion contents ranging from 10% to 25% by mass, compacted using Marshall energy. These authors found that an increase in the emulsion content led to an increase in the cohesive intercept and a decrease in the friction angle. There was no influence of curing time on the shear strength parameter values obtained.

Rodrigues et al. (2023) studied the impact of emulsion content on the strength parameters of the same soil-emulsion mixture. It was observed that at exceptionally high content levels, around 28% by mass, the internal structure of aggregate particles underwent significant changes due to the substantial amount of binder present. This alteration reduced the soil's grain-to-grain contact, leading to a decline in internal friction and cohesive forces, consequently affecting the mixture's strength properties. Table 1 provides a summary of studies conducted on soil-asphalt emulsion mixtures, further elucidating their characteristics and behaviors under various conditions.

Table 1
Summary of studies on soil-asphalt emulsion mixtures.

Authors	Studied Aspects	Emulsion Contente Evaluated in Study	Main conclusions
Jacintho (2010)	Shear strength	0 - 8%	Not observed gains in shear strength due to macropores formed by the low asphalt emulsion contents in the mixtures
	Permeability		No reduction in permeability due to macropores formed by the low asphalt emulsion contents in the mixtures
Lima and Dantas Neto (2019)	Compaction process	13% - 31%	Methodologies with higher compaction efforts yielded more homogeneous test specimens, and the higher the initial viscosity of the asphalt binder, the greater the formation of asphalt film between the fine fraction of the soil and the asphalt emulsion
Pereira (2018)	Permeability	13% - 28%	The addition of high levels of asphalt emulsion provided a better distribution of the binder within the soil mass, and the higher the emulsion content used, the lower the coefficient of permeability of the mixture
Dantas Neto et al., (2020)	Shear strength	13% - 28%	The addition of high levels of asphalt emulsion provided an increase in cohesion and reduction of the friction angle for low levels of applied normal stress
Brito et al. (2020)	Influence of curing time on shear strength	10% - 25%	The curing time did not exert a significant influence on the strength parameters of soil mixtures with high levels of asphalt emulsion under saturated conditions
Rodrigues et al. (2023)	Influence of emulsion content on shear strength	16% - 28%	Adding emulsion below the optimum content increases grain adhesion, while above this limit tends to decrease cohesion due to excess binder. Increasing emulsion content results in lower friction angles, with higher cohesion values for mixtures with 16% and 22%, but with 28% emulsion, there is a reduction in cohesive intercept due to excess binder

One potential concern associated with this material is its tendency to accumulate strains over time under continuous loading conditions, a phenomenon commonly referred to as creep (2018). For conventional asphalt mixes (often consisting of different sizes of crushed mineral aggregates and asphalt binder with 5% approximate content in density), the mechanical behavior is influenced by factors, such as the properties of the binding agent, the rate and intensity of mechanical load application, temperature variations, as well as time and load history (Nur et al., 2013).

Despite the absence of large aggregates, the behavior of a soil-emulsion, as well as asphalt mixes, tends to acquire the viscoelastic properties of the asphalt binder (Di Benedetto et al., 2004a; Poueget et al., 2012; Li et al., 2018), just as occurs for asphalt fillers and mortars (Di Benedetto et al., 2004b; Babadopulos et al., 2019). Meanwhile, there are still few characterization studies and viscoelastic modelling of soil-emulsion mixes, and it is important to understand their behavior at different timescales for applications, such as dams which are designed for long periods of continuous loading.

The mechanical behavior of linear viscoelastic materials can be investigated by two different kinds of tests: time domain tests and frequency domain tests (Ferry, 1980). In the time domain, a persistent load is applied, allowing for the evaluation of strain increments to determine the creep function. Conversely, by applying constant strain and assessing decreasing stress, the relaxation modulus can be established. Within the frequency domain, significant attention is directed towards the complex modulus, characterized by both magnitude and phase angle. The range of the complex modulus is very often known as “dynamic modulus”, although there is no inert effect in this property, and that must not be confused with the stiffness modulus obtained in wave spread tests (Huang and Di Benedetto, 2015). There are interconversion methods between the linear viscoelastic properties for a given material (Park and Schapery, 1999; Schapery and Park, 1999; Tiouajni et al., 2011).

The mechanical model 2S2P1D (two springs, two parabolic dashpots, one dashpot) (Olard and Di Benedetto, 2003), derived from an enhancement of the Huet-Sayegh mode (Sayegh, 1965), has the advantage of effectively representing the rheological behavior of both binding agents and asphalt mixes across a wide range of frequencies and temperatures (Olard and Di Benedetto, 2003; Di Benedetto et al., 2004b; Mounier et al., 2012; Nur et al., 2013). While this model has been successfully tested for various types of mixes, mastics, mortars, and conventional asphalt mixes (Olard and Di Benedetto, 2003; Nur et al., 2013; Babadopulos et al., 2019), there is currently no literature characterizing soil-emulsion asphalt mixes. Nor is it known if there is an effective capacity to represent the behavior of this material with linear viscoelastic models. This type of study is not yet available, nor is there validation for soil-emulsions. Moreover, validation of this kind of model could be a challenge, especially considering that when in time domain testing, it is difficult to guarantee linear conditions. Also, granular materials have non-linear behavior that will need to be considered (Santos, 1999; Gomes Correia et al., 2001; Ruiz et al., 2022).

Once the applicability of the linear viscoelastic model 2S2P1D to soil-asphalt emulsion mixtures is confirmed, future numerical simulations of deformation behavior over extended periods in dams with soil-emulsion cores will become feasible. This will enable enhanced design of dams with soil-emulsion cores. So, the purpose of this article is to propose a linear viscoelastic model for a soil-emulsion consisting of silty sand mixed with 16% asphalt density emulsion, investigating test results in the time and frequency domains. The response of this soil-emulsion for loads were examined over a wide range of temperatures (-10.0ºC to 54.4ºC) and frequencies (0.1 to 25Hz), in order to characterize the material analyzed in the linear domain, restricted to minor stresses and strains, adopting the criterion of limiting sine strains to a peak-to-peak range of 68μm/m (Carpenter, Ghuzlan, and Shen, 2003). The model is based on model 252P1D, seeking validation with the prediction of strain behavior for a creep test not used in the adjustment of the original model.

2. Materials and methods

2.1 Soil sample

To accomplish this study, two soil samples were collected within the limits of the Pici Campus of the Federal University of Ceará (UFC), on the site of coordinates UTM 3°45’7.5”S and 38°34’21.9W. The soil in question was identified as silty sand (SM), according to the Unified Soil Classification System (SUCS). Table 2 provides a summary of the average values of the geotechnical parameters obtained for both samples, as well as the standards used in characterization.

Table 2
Summary of the geotechnical parameters of the collected soil.

Geotechnical parameters of soil
NBR 7182 (2016a)	Maximum apparent dry specific weight (g/cm3) - PN	1.87
NBR 7182 (2016a)	Optimum moisture (%)	9.74
NBR 6459 (2016)	Liquid limit (%)	NL*
NBR 7180 (2016b)	Plastic limit (%)	NP*
DNER - ME 093 (1994)	Soil particle density	2.64
NBR 7181 (2016c)	Particle size analysis	-

• ^*
  The soil did not present liquid or plasticity (non-liquid, NL, non-plastic, NP).

2.2 Asphalt emulsion

The asphalt emulsion chosen in this study was a slow setting cationic emulsion (CSS). In nominal terms, this emulsion takes up to four hours to complete the breaking process, in order to have ample time for completing the mixture with the soil.

The characterization of the asphalt emulsion was tested by the manufacturer, having followed the procedures provided in standards NBR 14950 (ABNT, 2003), NBR 14393 (ABNT, 2012) and NBR 6568 (ABNT, 2005). Table 3 gives a summary of the results from each of the aforementioned tests.

Table 3
Summary of the parameters of the used slow setting cationic emulsion (CSS).

Asphalt emulsion parameters
NBR 14950 (2003)	Saybolt-Furol viscosity, sSF, at 50ºC	44
NBR 14393 (2012)	Screening, 0.84 mm, % in density	0.01
NBR 6568 (2005)	Residue, % in density	63.2

2.3 Determining asphalt emulsion contents

The 16% density content of asphalt emulsion was selected, since significant gains in cohesion and impermeability are observed for the mixture studied in this article. Starting with this emulsion content, enables the use of this material in impermeable dam cores (Pereira, 2018; Brito, 2020; Brito et al., 2022). At the same time an emulsion content was achieved that would enable shaping the TSs with enough stiffness to permit application of load levels respecting the resolution limits of the hydraulic press used, UTM25 (Universal Testing Machine), in the characterization of the material.

2.4 Mixing soil and asphalt emulsion

The mixing procedure of the materials was the same used in the studies by Pereira (2018) and Brito (2020), and obeyed the following stages: (i) Soil de-clumping - carried out using mortar and pestle, until there was no clumps in the soil; (ii) Use of a sample splitter until the material is reduced to a representative form; (iii) Oven drying at 100ºC for 24 hours - a non-aerated unventilated oven was used; (iv) Soil cooling at ambient temperature of around 25ºC - carried out in a laboratory on a site with approximately 70% moisture (average for the city of Fortaleza, Brazil); (v) Weighing and mixing by hand the soil with asphalt emulsion for approximately five (5) minutes, using a spatula and metal receptacle, continuously mixing; (vi) The mixed material exposed to the air for 24 hours (aeration) on an uncompacted layer around 10 cm thick; (vii) Compaction: using a gyratory compactor with gyration angle of 1.16o at continuous gyratory speed of 30 gyrations a minute and compaction pressure of 600 kPa.

2.5 Making test specimens

The TSs were shaped in a Superpave gyratory compactor, producing test specimens (TS) 100 mm in diameter, and 150 mm in height. The compaction procedure adopted the standard AASHTO T312 (2019), for several types of asphalt mixes.

The Superpave compaction method was selected over the Marshall and Proctor methodologies, since the dynamic modulus test (AASHTO T 342, 2022) requires a 1.5 height diameter ratio (H/D) for the test specimens in order to minimize the edge effects, whereby the Proctor and Marshall methodologies would not meet this ratio. In addition, investigations in mixes with bituminous binders have demonstrated that this type of compaction by crushing produces materials more similar to those obtained from field compaction (Chen et al., 2012).

The standard AASHTO T 342 (2022) states that the test specimens are molded in the gyratory compactor with a diameter of 100 mm and a height of 170 mm and later drilled for further testing. However, this procedure was not undertaken, since handling a core drill is difficult to cut into the soil-emulsion test specimens, which have less initial stiffness in an ambient temperature, compared to an asphalt concrete specimen. Also, the procedure aims to regularize the contact surfaces of the material, which are already quite regular in the soil-emulsion test specimens due to the small particle size. After molding, the test specimens were exposed to the open air for the dilution water of the emulsion to evaporate. To guarantee completion of the breaking process, the test specimens exposed to the open air were weighed daily, and remained there for at least two consecutive days until verifying a maximum variation of 0.3g in the values of their mass.

2.6 Complex modulus characterization

The dynamic modulus was tested according to the standard AASHTO T 342 (2022) Three TSs were tested at different frequencies and temperatures, taking the dynamic modulus value of the material as the average of the five (5) final cycles. Next, for each test condition, the dynamic modulus of the material was determined by the average of those three values. The equipment used for this test was the UTM-25 press, programmed to undertake a sine load at predetermined temperatures and frequencies, as stated in AASHTO T 342 (2022). This is a servo-controlled hydraulic press fitted with a temperature-controlled heat chamber and its results are recorded using a data collection system and computer. Then the data can be processed to obtain the values of the complex modulus and of the phase angle. This processing is addressed in more detail below.

During the complex modulus test, a uniaxial load of harmonic sine compression is applied to the test specimens, recording the strain result, which also has a permanent sine format. Unlike elastic materials, for viscoelastic materials the stress and strain peaks do not coincide at the same moment in time, and this gap is associated with the phase angle (δ), as occurs for asphalt mixes (Papazian, 1962; Huet, 1963). This sine load application process is repeated and recorded for the different temperatures and frequencies when performing test characterization. The ratio between complex notional stress (σ*) and complex notional strain (ε*) of a material under a harmonic load is defined as complex modulus E^* (Equation 1). This property contains both phase angle information and the ratio between the stress and strain ranges, defining the absolute value of this parameter, which is the property commonly referred to as “dynamic modulus” |E* | (Equation 2) (Papazian, 1962), although there are no inert effects.

(1) $$E^{\ast }=\frac{{\sigma }^{\ast }}{{\epsilon }^{\ast }}=\frac{{\sigma }_0}{{\epsilon }_0}\cos \delta +i\frac{{\sigma }_0}{{\epsilon }_0}\sin \delta$$

(2) $$\left|E^{\ast }\right|=\frac{{\sigma }_0}{{\epsilon }_0}$$

Where: σ₀ is the range of load stress; ε₀ is the range of strain response; δ is the phase angle.

For a variety of viscoelastic materials, a superposition of test points in different frequency-temperature pairs is noted in representations containing both the modulus and phase information (such as Black Space curves - dynamic modulus on a logarithmic scale vs. phase angle - or Cole-Cole - an imaginary part of the complex modulus vs. real part of the complex modulus). This means that there is an equivalence between frequency variation effects (or load application time) and temperature variation effects. In this case, the materials with this characteristic are referred to as thermorheologically simple. Accordingly, after obtaining the dynamic modulus for different temperatures and frequencies, regarding the time-temperature superposition principle, the dynamic modulus could be converted into a single plotted curve in function of an equivalent frequency, also known as reduced frequency (f_r), which is obtained through Equation 3. The resulting curve is known as the master curve. Each test point will have a horizontal translation factor (α_T), which is the function of the temperature, and when the test frequency is divided by this factor, it is referred to as reduced frequency (Equation 3).

(3) $$f_r=f^{\ast }{\alpha }_T$$

Where: α_T is the translation factor; f is the data-obtaining frequency; f_r is the reduced frequency.

The translation factors were calculated based on the Williams-Landel-Ferry (WLF) equation (Williams et al., 1955) as per Equation 4. The C₁ and C₂ coefficients are optimized so that the test points and modeled points to have the least possible error.

(4) $$\log \left({\alpha }_r\right)=\frac{-C_1(T-T_r)}{-C_2+T-T_r}$$

Where: α_T is the translation factor (also known as shift factor); C₁ and C₂ are coefficients of the WLF equation; T is the data collection temperature; T_r is the reference temperature for building the master curve.

The strains of the cylindrical test specimens are based on the average of the values collected by three (3) devices of the Linear Variable Differential Transformers (LVDT) type, positioned equidistant to each 120° in the test specimens. The positioning is made with targets (small metal hexagonal elements glued to the TS with epoxy adhesive), then coupled to the devices in which the LVDTs are fitted. The distance between the gluing points used to calculate the strains is 75mm.

The frequencies, temperatures and minimum conditioning times of test specimens were recommended by AASHTO T 342 (2022). The stress levels adopted in the test were selected so that the test specimens would not have excessive strains and would leave the linear domain of viscoelasticity considered typically for asphalt materials (Mangiafico et al., 2018). Therefore, to ensure this condition, it was sought to apply strain amplitudes no greater than 67.5μm/m in the test specimens. Since the test control is loaded, the limits chosen must be in stress, and determined by trial and error with a few loading cycles, in a procedure commonly known as fingerprinting (AASHTO T 342, 2022). Table 4 shows the maximum stress levels applied to the tests for all three (3) test specimens. It should be mentioned that a contact stress equal to 5% was applied as the minimum stress during loading.

Table 4
Stress levels in the Complex Modulus test.

Maximum stress levels applied in the Complex Modulus test
Frequency (Hz)	25	25	10	5	1	0.5	0.1
Number of cycles	200	200	200	100	20	15	15
Maximum stress, T=-10°C (kPa)	620	620	540	510	450	420	380
Maximum stress, T=4.4°C (kPa)	480	480	320	240	140	120	100
Maximum stress, T=21.1°C (kPa)	210	210	145	105	58	49	28
Maximum stress, T=37.8°C (kPa)	90	90	60	45	20	16	10
Maximum stress, T=54.4°C (kPa)	50	50	40	30	10	8	8

2.7 Creep test

The equipment used in the creep test was the same as that used in the Dynamic Modulus test, the UTM25. The strains of this test were measured based on two LVDTs fixed at the top end of the TS. This procedure to measure dislocations needs to be different due to the fact that over time, the strains accumulate continuously, so that the use of the LVDTs fitted between the targets is not possible.

The test temperature was 0°C, in order to enable application of strength levels above the minimum resolution band of the equipment, which is 0.35 kN, without causing excessive strains on the TS as soon as it began. Therefore, the creep test was performed at two stress levels to check if the proposed model is valid for different loads. In CP1, a stress of 343 kPa was applied and in TS2 and TS3, the stress of 150 kPa. The creep tests were used to validate the linear viscoelastic linear model.

The maximum testing time was determined at 7200 seconds in order to test the level of reliability of the proposed model in the linear domain, bearing in mind that this was modeled in the linearity domain, and at very high testing times, this domain is no longer respected and the comparison between the idealized predicted values in 2S2P1D tend to be decreasingly representative with the development of the strains. It is expected in the creep test that, after sufficient strain accumulation, the behavior deviates from the linear.

2.8 The 2S2P1D rheological model

The 2S2P1D model stands as a robust methodology for characterizing the behavior of both asphalt binders and mixes (Olard and Di Benedetto, 2003; Di Benedetto et al., 2004; Mounier et al., 2012). Originating from an enhancement of the Huet- Sayegh model, this model was developed by Di Benedetto (2003) and comprises a configuration composed of two springs, two parabolic creep elements, and one dashpot. Eq. 5 represents how the complex modulus is expressed by this model. Models like the 2S2P1D, employing parabolic dashpots, are regarded as continuous spectrum models. These models resemble those with a discrete spectrum of infinite elements, allowing for a far more accurate representation of the behavior of a viscoelastic material (Olard and Di Benedetto, 2003).

(5) $$E^{\prime }(i\omega )-E_0+\frac{E_{00}-E_0}{1+\delta (i\omega \tau {)}^{-k}+(i\omega \tau {)}^n+(i\omega \beta \tau {)}^{-1}}$$

Where: E₀ is the glassy modulus when ω → ∞; E₀₀ is the static modulus when ω → 0; ω is the solicitation pulsation; h, k are exponents with 0 < k < h < 1; δ is a dimensionless constant which works as a shape factor; τ is a characteristic of time; β is viscous related constant.

3. Results and discussions

3.1 Dynamic modulus

The dynamic modulus test results of the soil mix with 16% asphalt emulsion are expressed in Figure 1 by isotherms. The dynamic modulus is found dependent on temperature, since its value drops drastically as the temperature rises, as well as the dependence that this material’s property has with the test frequency. The investigated material can not only present stiffness around 104MPa (54.5°C and 0.1Hz), a figure compatible with some clays (Sultanov et al., 2019), but also around 7.49GPa (-10°C and 0.1Hz), compatible with Portland cement mortars (Nunes et al., 2019). This finding occurs in a similar way to the other analyzed frequencies and temperatures; and this reinforces the need to investigate the linear viscoelastic behavior even in unexpected field temperatures, since they are a part of the material’s behavior spectrum, and, as will be seen, interfere in the rheological model to be adopted.

Figure 1
Isotherms of soil with 16% asphalt emulsion.

Figure 2 presents the master curve of the material constructed from the text results and compares with a curve obtained from the proposed model. Table 5 provides the coefficients of the 2S2P1D model, obtained from a manual adjustment of the parameters, and translation coefficients according to the WLF equation (Williams et al., 1955).

Table 5
Parameters of model 2S2P1D and WLF equation.

Parameters of model 2S2P1D
E00 (Mpa)	E0 (Mpa)	k	h	δ	τΕ (Tref)	β	C1	C2
90	11500	0.25	0.5	2.3	0.01	250	12.11	103.16

Figure 2
Soil-emulsion master curve with 16%.

Figure 3 presents a graphical illustration of the fitting outcomes obtained by comparing 90 distinct dynamic modulus values. These values correspond to the experimental data points collected for each frequency-temperature combination across three test specimens. These values were then compared with the optimized parameters derived from the proposed model.

Figure 3
Comparison of experimental results with proposed models on a logarithmic scale.

Overall, the fitting results indicate a significant agreement between the experimental data points and the corresponding 2S2P1D model. This agreement primarily concerns the |E*| values, suggesting that the model effectively captures the behavior of soil-emulsion mixtures within the linear domain. As shown in Figure 3, the maximum experimental error relative to the proposed modeling is below 20% for 91% of the data points. Notably, higher experimental errors are observed at elevated temperatures, while for temperatures below 21.1°C, all experimental data showed errors below 20% compared to the corresponding value of the 2S2P1D model. Additionally, the overall trend of the model reveals a high coefficient of determination (R²) of 0.987.

The good adhesion of the test results to the 2S2P1D model regarding the value |E*| indicates that this model can satisfactorily represent the soil-emulsion behavior in the linear domain. Outside the linear region, as the strains develop, the material’s modulus tends to change significantly in value (Santos, 1999; Gomes Correia et al., 2001). Due to this modulus change in the soils outside the linear regime, Santos (1999) and Gomes Correia et al. (2001) drew equations to predict this variation in the strain properties, based on test results of a wide variety of soils, including clays, sands, residual, laterites and saprolites. These equations are defined according to a key parameter (γ_0.7), calculated as the shear strain for a stiffness degradation factor G/G₀ = 0.7, where G₀ is the initial shear modulus for very minor strains γ ≈ 10^-6.

From the results of this study and considering the soils as isotropic materials having a constant Poisson coefficient, Figure 4 was created to present intervals of variation of the soil elasticity modulus as the standard strain values evolve Ζ = ε  ε_0.7 .

Figure 4
Degradation of the sand and clay modulus as a function of Z.

Knowing that the strain modulus of soils is reduced as the strain evolves, more models were created with different parameters of the 2S2P1D model that had their E₀ and E₀₀ values reduced by 50%, 65% and 80%, since the creep test was conducted for strains up to 20 times greater than those observed in the dynamic modulus test. Such variations of the model were produced to approximate their results to the behavior in the non-linear regime of the mix and soil-emulsion.

The initial 2S2P1D model, and its variations, were discretized for the generalized Kelvin -Voigt (KVG) model that consisted of a serial association of a sufficient number of viscoelastic elements, each represented by a spring parallel to a linear dashpot. This study involved the use of 40 viscoelastic linear elements and a free spring. Such discretization, obtained in accordance with the procedures proposed by Di Benedetto et al. (2007), was due to the high computational effort in calculating the strains from continuous spectrum models, such as 2S2P1D (Tiouajni et al., 2011). Moreover, the KVG model explicitly provides the creep curve of the material, enabling a direct interconversion between the complex modulus and creep function, while the 2S2P1D model does not have an explicit creep form. Figure 5 compares between the 2S2P1D model and KVG discrete model with 40 viscoelastic elements and a spring, evidencing the proximity in the results of both.

Figure 5
Comparing the 2S2P1D and KVG models.

Having obtained the parameters of the KVG models, it was calculated what the estimated behavior of the soil-emulsion mix would be in a creep test. Next, this behavior was compared to the test results for the creep test, according to the results in Figure 6.

Figure 6
Predicted strain vs. creep test results.

First, it is found that the creep functions (D(t)) measured in the TSs, even subjected to different normal stress levels, had very similar results in D(t) terms. The observed differences (30%) are compatible with the variability in the TS soil properties (Wang et al., 2016). These differences are not enough to evidence non-linearity in the behavior during the creep tests, which achieved strains of up to 780μm/m, for the tested TSs (TS2 andTS3) with less stress, and 1600μm/m, for the TS with greater test stress (TS1). However, the 2S2P1D linear model on its own did not sufficiently represent the behavior of the asphalt soil-emulsion mix in absolute terms, since it predicted just around half the actual strain found in the tests. Yet, it should be noted that in the dynamic modulus adjustment tests, the maximum strains were 67.5 μm/m, while during the creep test, after 7200 seconds, a strain of 780 μm/m was found for the TSs tested with less stress, and 1600 μm for the test specimens with the highest test stress. The error, at the end of the test, when predicting the strain, varied between 44% and 61%, using a calibrated model to determine the behavior at strain levels of 20 times less than the actual occurrence. These differences, resulting from the load amplitudes, are compatible with typical soil non-linearities (Santos, 1999; Gomes Correia et al., 2001), as shown in Figure 4.

Accordingly, adjustments were made to the linear model 2S2P1D to consider the effect of non-linearity. The behavior of another three (3) new 2S2P1D models (varying only the asymptotic moduli) was determined bearing in mind different diminishing levels of the modulus with the strain level. The new models were made considering 50%, 65% and 80% variations in the determined stiffness value. The strain results of those new models were also provided in Figure 6.

It was found that, in the model considering 65% stiffness, the results were much closer to those tested, since the errors in the strains dropped to values with a 6% difference at the end of the test. Also, for the test’s high strain levels, all the analyzed test results stood between the models with 80% and 50% of the determined stiffness value.

Therefore, it is suggested that the idealized linear model 2S2P1D may potentially be useful for predicting the behavior of soil-emulsion, even under high deformations where nonlinear influences begin to emerge, if information on deformation levels is available. It may also be used in the future to predict the behavior of dams with a soil-emulsion core. The suggested contribution lies in the idea that incorporating a dynamic modulus reduction coefficient may be suitable for predicting deformations in the nonlinear regime. However, the results of this article alone cannot conclusively determine this coefficient.

Later studies require more tests on this type of material, so that with enough test data it may be possible to establish a rule for applying the modulus reduction coefficient or a suitable safety factor for this type of estimation, compatible with geotechnical work. For the results obtained so far, it would be necessary to use a 1.53 safety coefficient, which would be equal to this modulus reduction coefficient of 65%. This safety factor’s order of magnitude is, in fact, already used in many geotechnical projects to assess material properties, which yet again evidences the good approximation based on the proposed model.

4. Conclusions

This study investigated the mechanical behavior of soil-emulsion regarding the strain in time domain testing (creep function) and frequency domain (complex modulus). While the modulus test was used to adjust the linear viscoelastic models (2S2P1D and KVG), creep testing was used to validate the proposed models. Model 2S2P1D was considered satisfactory, since a test was run to determine dynamic modulus in minor strains (below 100μm/m) and when predicting the creep behavior in strains of up to 20 times greater, prediction errors were found varying between 44% and 61% difference, rectifiable when considering non-linearity coefficients compatible with granular materials. From the results, it was possible to conclude that:

• • The studied soil-emulsion mix is a thermorheologically simple material, and therefore, a parameter for the linear constitutive model 2S2P1D could be determined;

• • For minor strains, the idealized linear model 2S2P1D is able to satisfactorily represent the behavior of the studied soil-emulsion mix;

• • Model 2S2P1D provided good adjustments to test results for the creep tests, even at different stress levels, as long as the expected non-linearity in the creep tests is considered;

• • Even at high strain levels in the non-linear domain, the differences between the predicted and test behavior were considered within an acceptable limit for geotechnical applications;

• • The insertion of a modulus reduction coefficient of 65% was sufficient, in this case study, to simulate the effect of non-linearity at the material’s high strain levels.

Acknowledgments

The authors would like to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the financial support and the Graduate Program in Civil Engineering at the Federal University of Ceará.

References

• AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORTATION OFFICIALS. AASHTO T 312 Preparing and determining the density of asphalt mixture specimens by means of the superpave gyratory compactor, 2019.
• AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORTATION OFFICIALS. AASHTO T 342 Standard method of test for determining dynamic modulus of hot mix asphalt (HMA), 2022.
• ANDAVAN, S.; MANEESHKUMAR, B. Case study on soil stabilization by using bitumen emulsions - a review. Materials Today: Proceedings, v. 22, n. 3, p. 1200-1202, jan. 2020.
• ASSOCIAÇÃO BRASILEIRA DAS NORMAS TÉCNICAS - ABNT. NBR 6568: emulsões asfálticas - determinação do resíduo de destilação. Rio de Janeiro: ABNT, 2005.
• ASSOCIAÇÃO BRASILEIRA DAS NORMAS TÉCNICAS - ABNT. NBR 7182: solo - ensaio de compactação. Rio de Janeiro: ABNT, 2016a.
• ASSOCIAÇÃO BRASILEIRA DAS NORMAS TÉCNICAS - ABNT. NBR 7180: solo - determinação do limite de plasticidade. Rio de Janeiro: ABNT, 2016b.
• ASSOCIAÇÃO BRASILEIRA DAS NORMAS TÉCNICAS - ABNT. NBR 7181: solo - análise granulométrica. Rio de Janeiro: ABNT, 2016c.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS - ABNT. NBR 6459: solo - determinação do limite de liquidez. Rio de Janeiro: ABNT, 2016.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS - ABNT. NBR 14950: materiais betuminosos - determinação da viscosidade Saybolt Furol, 2003.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS - ABNT. NBR 14393: emulsões asfálticas - determinação de peneiração, 2012.
• BABADOPULOS, L. et al. Influence of loading amplitude on viscoelastic properties of bitumen, mastic and bituminous mixtures. Road Materials and Pavement Design, v. 20, n. sup2, p. S780-S796, 31 jul. 2019.
• BRITO, N. J. C. de O. et al. Avaliação do tempo de cura na resistência ao cisalhamento de misturas solo-emulsão com teores de emulsão superiores a 10%. Revista Matéria, v. 27, n. 2, jun. 2022.
• BRITO, N. J. C. O. Estudo da influência do tempo de cura no comportamento mecânico das misturas entre solos e altos teores de emulsão asfáltica. Dissertação (Mestrado) - Universidade Federal do Ceará, Fortaleza, 2020.
• CARPENTER, S. H.; GHUZLAN, K. A.; SHEN, S. Fatigue endurance limit for highway and airport pavements. Transportation Research Record, v. 1830, p. 131-138, 2003.
• CHEN, J. et al. Application of discrete element method to Superpave gyratory compaction. Road Materials and Pavement Design, v. 13, n. 3, p. 480-500, 2012.
• COLLOP, A. C.; KHANZADA, S. Permanent deformation In idealized “Sand Asphalt” Bituminous Mixtures. Road Materials and Pavement Design, v. 2, n. 1, p. 7-28, 2001.
• DANTAS NETO, S. A.; PEREIRA, C. G. F.; ABREU, A. A. Stabilization of sandy soil with high content of asphalt emulsion. REM - International Engineering Journal, v. 73, n. 2, p. 163-169, 2020.
• DEPARTAMENTO NACIONAL DE ESTRADAS DE RODAGEM - DNER. ME 093: solos - determinação da densidade real. Rio de Janeiro: DNER, 1994.
• DI BENEDETTO, H. et al. Fatigue of bituminous mixtures. Materials and Structures, v. 37, p. 202-216, 2004a.
• DI BENEDETTO, H. et al. Linear viscoelastic behaviour of bituminous materials: from binders to mixes. Road Materials and Pavement Design, v. 5, p. 163-202, 1 jan. 2004b.
• DI BENEDETTO, H. et al. Three-dimensional thermo-viscoplastic behaviour of bituminous materials: The DBN model. Road Materials and Pavement Design, v. 8, n. 2, p. 285-315, abr. 2007.
• FERNANDES, L. F. et al. Sandy soil stabilization with asphalt emulsion for paving purposes. Research, Society and Development, v. 11, n. 1, p. e0711124274, jun. 2022.
• FERRY, J. D. Viscoelastic properties of polymers. 3. ed. New York: John Wiley & Sons, 1980.
• GOMES CORREIA, A. et al. An approach to predict shear modulus of soils in the range of 10^-6 to 10^-2 strain levels. In: INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN GEOTECHNICAL EARTHQUAKE ENGINEERING AND SOIL DYNAMICS, 4. Anais [...]. San Diego, 2001.
• HUANG, S. C.; DI BENEDETTO, H. Advances in asphalt materials. Cambridge: Elsevier, 2015.
• HUET, C. Etude par une méthode d’impédance du comportement viscoélastique des matériaux hydrocarbonés. Tese (Doutorado) - Faculté des Sciences de l’université de Paris, Paris, 1963.
• JACINTHO, E. C. Estudos de propriedades e comportamentos de misturas solo - Emulsão aplicado a barragens. Brasília: Universidade de Brasília, 2010.
• KUVAT, A.; SADOGLU, E. Dynamic properties of sand-bitumen mixtures as a geotechnical seismic isolation material. Soil Dynamics and Earthquake Engineering, v. 132, 1 maio 2020.
• LI, P. et al. Analysis of viscoelastic response and creep deformation mechanism of asphalt mixture. Construction and Building Materials, v. 171, p. 22-32, 20 maio 2018.
• MANGIAFICO, S. et al. Nonlinearity of bituminous mixtures. Mechanics of Time-Dependent Materials, v. 22, n. 1, p. 29-49, 1 fev. 2018.
• MOUNIER, D.; DI BENEDETTO, H.; SAUZÉAT, C. Determination of bituminous mixtures linear properties using ultrasonic wave propagation. Construction and Building Materials, v. 36, p. 638-647, nov. 2012.
• NASSAR, A. I. et al. Mechanical, durability and microstructure properties of Cold Asphalt Emulsion Mixtures with different types of filler. Construction and Building Materials, v. 114, p. 352-363, 1 jul. 2016.
• NUNES, V. A.; BORGES, P. H. R.; ZANOTTI, C. Mechanical compatibility and adhesion between alkali-activated repair mortars and Portland cement concrete substrate. Construction and Building Materials, v. 215, p. 569-581, 10 ago. 2019.
• NUR, N. I. et al. Modeling the rheological properties of bituminous binders using the 2S2P1D Model. Construction and Building Materials, v. 38, p. 395-406, jan. 2013.
• OLARD, F.; DI BENEDETTO, H. General “2S2P1D” model and relation between the linear viscoelastic Behaviours of Bituminous Binders and Mixes. Road Materials and Pavement Design, v. 4, n. 2, p. 185-224, 1 jan. 2003.
• OLUYEMI-AYIBIOWU, B. D. STABILIZATION OF LATERITIC SOILS WITH ASPHALT- EMULSION. Nigerian Journal of Technology, v. 38, n. 3, p. 603-608, jul. 2019.
• PAPAZIAN, H. S. The response of linear viscoelastic materials in the frequency domain with emphasis on asphaltic concrete. In: INTERNATIONAL CONFERENCE ON THE STRUCTURAL DESIGN OF ASPHALT PAVEMENTS. Anais [...]. Michigan, 1962.
• PARK, S. W.; SCHAPERY, R. A. Methods of interconversion between linear viscoelastic material functions. Part I - a numerical method based on Prony series. International Journal of Solids and Structures, v. 36, p. 1653-1675, 1999.
• PEREIRA, C. G. F. Avaliação do comportamento hidráulico e da resistência ao cisalhamento de misturas entre solo arenoso e elevados teores de emulsão asfáltica para aplicações em barragens. Dissertação (Mestrado) - Universidade Federal do Ceará, Fortaleza, 2018.
• POUEGET, S. et al. Prediction of isotropic linear viscoelastic behavior for bituminous materials - forward and inverse problems. EURASPHALT AND EUROBITUME CONGRESS -EEC, 5, 2012, Istanbul. Proceedings [...]. Istanbul: EEC, 2012.
• RODRIGUES, P. M. B.; DANTAS NETO, S. A.; BABADOPULOS, L. F. de A. L. Avaliação do comportamento cisalhante de misturas solo-emulsão com teores de emulsão variando de 16% a 28% em massa. Revista Matéria, v. 28, n. 2, 2023.
• RUIZ, J. F. et al. Influence of non-linear soil properties on railway critical speed. Construction and Building Materials, v. 335, n. 13, p. 127485, jun. 2022.
• SANTOS, J. A. Soil characterisation by dynamic and cyclic torsional shear tests. Application to the study of piles under lateral static and dynamic loadings. Tese (Doutorado) - Technical University of Lisbon, Lisboa, Portugal, 1999.
• SARAJPOOR, S.; GHALANDARZADEH, A.; KAVAND, A. Dynamic behavior of sand-bitumen mixtures using large-size dynamic hollow cylinder tests. Soil Dynamics and Earthquake Engineering, v. 147, 1 ago. 2021.
• SAYEGH, G. Variation des modules de quelques bitumes purs et bétons bitumineux. Paris: Faculté des Sciences de l’université de Paris, 1965.
• SCHAPERY, R. A.; PARK, S. W. Methods of interconversion between linear viscoelastic material functions. Part II an approximate analytical method. International Journal of Solids and Structures, v. 36, p. 1677-1699, 1999.
• SULTANOV, K. et al. Variable moduli of soil strain. In: E3S WEB OF CONFERENCES. Anais [...]. EDP Sciences, 29 maio 2019.
• TIOUAJNI, S. et al. Approximation of Linear Viscoelastic Model in the 3 Dimensional Case with Mechanical Analogues of Finite Size: application to Bituminous Materials. Road Materials and Pavement Design, v. 12, n. 4, p. 897-930, 1 jan. 2011.
• WANG, Y.; CAO, Z.; LI, D. Bayesian perspective on geotechnical variability and site characterization. Engineering Geology, v. 203, p. 117-125, 25 mar. 2016.
• WILLIAMS, M. L.; LANDEL, R. F.; FERRY, J. D. The temperature dependence relaxation mechanism in amorphous polymers and other glass forming liquids. Journal of ACS, v. 77, p. 3701, 1955.

Publication Dates

• Publication in this collection
  10 Jan 2025
• Date of issue
  2025

History

• Received
  16 Jan 2023
• Accepted
  24 Apr 2024