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BRIEF RESEARCH REPORT article

Front. Phys., 25 November 2021
Sec. Interdisciplinary Physics
This article is part of the Research Topic Microscopic Structure Effect on the Macroscopic Property of Geomaterials View all 63 articles

Empirical Formulas of Shear Modulus and Damping Ratio for Geopolymer-Stabilized Coarse-Grained Soils

Shengnian WangShengnian Wang1Xinqun GaoXinqun Gao1Wei MaWei Ma2Guoyu Li
Guoyu Li2*Chong ShiChong Shi3Peng ZhangPeng Zhang1
  • 1College of Transportation Science and Engineering, Nanjing Tech University, Nanjing, China
  • 2State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou, China
  • 3Institute of Geotechnical Engineering, Hohai University, Nanjing, China

The contribution of gravel fraction on the maximum shear modulus (Gmax), dynamic shear modulus ratio (G/Gmax), and damping ratio (λ) of cementitious coarse-grained soils has not been fully understood yet. Large-scale triaxial cyclic tests for geopolymer-stabilized coarse-grained soils (GSCGSs) were conducted with different volumetric block proportions (VBPs) under various confining pressures (CPs) for investigating their dynamic behaviors and energy dissipation mechanisms. Results indicate that the Gmax of GSCGS increases linearly with VBPs but nonlinearly with CP. High VBPs will probably result in a gentle decrease in G/Gmax and a rapid increase in normalized λ (λnor), while the opposite is the case for a high CP. With the shear strain amplitude being normalized, the G/Gmax and λnor are distributed in a narrow band with low dispersion and thus can be well-described by empirical functions of the normalized shear strain amplitude.

Introduction

Cementitious coarse-grained soils (CCGSs) are widely used as filling materials in infrastructure projects such as high-speed railway subgrades, earth dams, and highways [1,2]. However, the design and construction of engineering structures on CCGS are always challenging for engineers due to parameter determination difficulties. Dynamic soil properties including the maximum shear modulus (Gmax), dynamic shear modulus ratio (G/Gmax), and damping ratio (λ) from small to large shear strain amplitude (γ) are crucial indices for the seismic design and stability evaluation of geotechnical structures subjected to periodic random loads. Previous studies showed that CCGS was inhomogeneous and heterogeneous geotechnical materials [1,2]. Their cyclic shear behaviors were affected by gravel fraction, cementation, interparticle contact stiffness, void ratio, curing period, and deformation within individual particles [36]. Of these factors, the gravel fraction and cementation played a particularly significant role in the shear behavior of CCGS. However, no consensus exists on their effects up to now. Geopolymer binders (GBs) are alkali-activated aluminosilicate gel materials with enormous advantages in high strength, fast hardness, weak shrinkage, etc. Their primary raw materials are solid wastes, such as fly ash, glass waste, red mud, metakaolin (MK), and combinations of two or more of these materials [7]. The coarse-grained soil stabilized with GBs (GSCGS) thus can also be a better choice for engineering practices, regardless of safety performance in seismic resistance and durability or feasibilities in resource acquisition and cost control. This study conducted large-scale undrained triaxial cyclic tests on GSCGS with different volumetric block proportions (VBPs) under various confining pressures (CPs). The evolution of Gmax, G/Gmax, and λ was investigated, and their relationships with γ were discussed.

Experiments

The dynamic behaviors of GSCGS in this study were investigated via a large-scale triaxial cyclic shear instrument (HCA300) developed by the American company GCTS. Each GSCGS cylindrical specimen was 100 mm in diameter and 200 mm in height. For the convenience of sample preparation, coarse-grained soils were considered a mixture of the soil matrix and rock blocks. The soil matrix was fine-grained residual soil, with a maximum grain size of 2 mm. The natural dry density was 1.64 g/cm3. The maximum dry density and optimum water content were 1.72 g/cm3 and 18.3%, respectively. The rock blocks mainly comprised crushed stones with a dry density of 2.42 g/cm3. The maximum rock block size was limited to be 0.2 times the diameter of the specimen to avoid the grain size effect, namely, the rock block size used in sample preparation was 2–20 mm.

Considering that the VBP greater than 60% may result in considerable hollow phenomena among rock blocks and significant difficulties in packing GSCGS samples in the mold, only five VBPs (0/15/30/45/60, %) combined with four CPs (0.05/0.10/0.20/0.40, MPa) were considered in this study. The previous study showed that GBs could synthesize from MK, CaO, and NaHCO3 with a mass ratio of 4:1:1, and their optimal mixing ratio in fine-grained soil was 15 wt% [7]. Therefore, the dosage of GBs in the coarse-grained soil samples was determined by the relative content of the soil matrix because of the cementation of GB functions primarily in the fine-grained soil. In other words, once the VBP is selected, the dosage of fine-grained soil in a GSCGS specimen is known, and the dosage of GBs can be determined. The water consumption for sample preparation was the sum of the amount of water required for the fine-grained soil to reach its maximum dry density and an extra water compensation of 5% for rock blocks’ water absorption. All the specimens were cured in a humid environment at room temperature for 7 days and saturated by a vacuum extractor on GCTS until the B-value reached 0.95 at least before loading. The axial strain amplitude was increased from 1 × 10–5 to 1 × 10–2 in a level-by-level manner. The number of cyclic loadings for each strain amplitude was 5. The loading frequency was 0.5 Hz.

Results and Discussion

Dynamic soil properties, including G and λ, were achieved by following the calculation methods for symmetrical and asymmetric hysteresis loops suggested by Kumar et al. [8]. Figure 1A presents the relationship between the Gmax of GSCGS and the VBP. The Gmax always increases linearly with the VBP, despite GSCGS being subjected to tensile or compressive stress. The increasing gradient of fitting curves suggests that there is a positive correlation between the Gmax and CP. Hence, the relationship of the Gmax and VBP can be described as follows:

Gmax=kpVBP+Gmatrix,(1)

where kp is the gradient of fitting curves and Gmatrix is the intercept denoting the fundamental stiffness of the soil matrix under a specified CP. The fitting results based on Eq. 1 illustrate that the kp increases with the CP, namely, high CP will result in larger values in Gmax. Figure 1B presents the relationship between the Gmax of GSCGS and the CP. The Gmax increases nonlinearly with the CP at the same VBP. Seed et al. [9] proposed a simplified relationship between the Gmax and CP for gravelly soil as follows:

Gmax=K2(CP)0.5,(2)

where K2 is a regression coefficient. Rollins et al. [10] reported that K2 was a function of relative density for soils. Since the GSCGS is regarded as the soil matrix and rock blocks, the density of GSCGS can be summarized as a function of the VBP. Therefore, K2 is related to the VBP of GSCGS. Figure 1C illustrates an excellent linear correlation between K2 and VBP. Thus, a new empirical formula for the Gmax of GSCGS is defined as follows:

Gmax=(k0VBP+C)(CP)0.5,(3)

where k0 and C are regression coefficients. Figure 1D presents the measured and predicted Gmax of GSCGS. Both are close to the bisecting line with a high correlation coefficient (R2) of 0.9741, which indicates that the proposed empirical formula can predict the Gmax of GSCGS well.

FIGURE 1
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FIGURE 1. Relationships of the Gmax of GSCGS with the VBP and CP.

Figure 2A presents the G/Gmax envelope curves of GSCGS with different VBPs under various CPs. The G/Gmax is distributed within a band on the whole. The shape of the curves is very close as γ is less than the order of 10−4%. When γ lies between 10–4% and 0.01%, the G/Gmax is scattered. When γ lies between 0.01 and 1.0%, the G/Gmax decreases significantly. The reduction rate of G/Gmax slows down once γ is higher than 1.0%. As a whole, the G/Gmax of GSCGS is more likely to be characterized following a hyperbolic G/Gmax function proposed by Hardin and Drnevich [11], which is given in the following equation:

G/Gmax=1/(1+γ/γr)n,(4)

where γr is the reference shear strain and n is the curvature coefficient. It can be observed that the envelope region of G/Gmax overlaps with the bounds proposed by Rollins et al. [10] when the VBP of GSCGS is higher than 45%. However, when the VBP is less than 45%, they have not overlapped anymore, especially when γ ranges between 0.01 and 1.0%. Seed et al. [9] pointed out that the G/Gmax of sands always decreased faster than gravelly soils as γ increased, namely, high VBP would result in a gentle decrease in G/Gmax of gravelly soils. This discovery explains why the G/Gmax envelope curves of GSCGS are relatively higher than those of gravelly soils used in studies by Seed et al. [9] and Rollins et al. [10].

FIGURE 2
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FIGURE 2. G/Gmax and λ envelope curves of GSCGS with different VBPs under various CPs.

Figure 2B shows the normalized λ (λnor) envelope curves of GSCGS with different VBPs under various CPs, wherein the empirical model proposed by Chen et al. [12] is applied.

λnor=λ0(1G/Gmax)n/(λmaxλmin),(5)

where λmin and λmax are the minimum and maximum λ, respectively, and λ0 and n are regression parameters related to soil properties. It can be observed that λnor is distributed in a narrower band overall. The shape of the curves becomes unanimous when γ is less than the order of 10−3%. This result implies that the VBP and CP might have a minimal impact on λnor. The reason why the λnor envelope curves of GSCGS are lower than those of gravelly soils examined by Seed et al. [9] and Rollins et al. [10] maybe that a high VBP is more likely to result in significant difficulties in compaction of coarse-grained soils, while cementation improves the integrity of CGS significantly, and thereby results in relatively low λnor when subjected to cyclic loadings.

Figure 3 presents the relationship of the G/Gmax and λnor of GSCGS vs. normalized γ (γnor = γ/γr). It can be observed that both G/Gmax and λnor are distributed within a narrow band, namely, both of them are insensitive to the VBP and CP via γnor. Martin and Seed [13] had summarized a nonlinear elastic model for gravel soils with γnor, which is

G/Gmax=1[γnor2β/(1+γnor2β)]α,(6)

where α and β are regression parameters. The fitting results of G/Gmax show that this nonlinear model is also available to GSCGS with an excellent correlation coefficient of 0.9870 and can be simplified as follows:

G/Gmax=1/(1+γnor).(7)

Substituting Eqs 4, 6 into Eq. 5 yields

λnor=[γnor2β/(1+γnor2β)]αn.(8)

The fitting results of λnor show a perfect correlation of 0.9757 with γnor, and can be rewritten as follows:

λnor=γnor/(1+γnor).(9)

This empirical formula thus can characterize λ of GSCGS under cyclic loadings.

FIGURE 3
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FIGURE 3. Relationships of the G/Gmax and λnor of GSCGS with γnor.

Conclusion

The dynamic properties of GSCGS were investigated via large-scale triaxial cyclic tests in this study. Outcomes illustrate that the Gmax of GSCGS increases linearly with the VBP but nonlinearly with CP. Thus, new empirical formulas of Gmax referring to the VBP and CP are proposed. A high VBP may result in a gentle decrease in G/Gmax and a rapid increase in λnor, while the opposite is the case for a high CP. G/Gmax and λnor are insensitive to VBP and CP via γnor so that they can be described by empirical formulas of γnor. The proposed empirical formulas can provide a reference to understand the dynamic behaviors of GSCGS and other similar cementitious geomaterials.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author Contributions

Funding acquisition and formal writing of the work, SW; investigation and data analysis of the work, XG; review and editing of the work, WM, GL, CS, and PZ. All authors have read and agreed to the published version of the manuscript.

Funding

This study was financially supported by the National Natural Science Foundation of China (41902282) and State Key Laboratory of Frozen Soil Engineering (SKLFSE201809).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: coarse-grained soils, geopolymer, shear modulus, damping ratio, empirical formulas

Citation: Wang S, Gao X, Ma W, Li G, Shi C and Zhang P (2021) Empirical Formulas of Shear Modulus and Damping Ratio for Geopolymer-Stabilized Coarse-Grained Soils. Front. Phys. 9:754377. doi: 10.3389/fphy.2021.754377

Received: 06 August 2021; Accepted: 01 September 2021;
Published: 25 November 2021.

Edited by:

Wanqing Shen, Université de Lille, France

Reviewed by:

Lanlan Yang, Jiangnan University, China
Jun Yu, Nantong University, China

Copyright © 2021 Wang, Gao, Ma, Li, Shi and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Guoyu Li, guoyuli@lzb.ac.cn

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.