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Small-strain shear modulus (Gmax) and microscopic pore structure of calcareous sand with different grain size distributions

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Abstract

The maximum shear modulus (Gmax) is a key material characteristic that is incorporated in advanced soil constitutive models. Numerous experimental studies have been conducted to describe the effects of particle sizes and packing characteristics on Gmax. However, most of these studies were conducted on quartz-based sands. A review of the literature revealed that few studies have described the effects of grain size distribution (GSD) on Gmax in calcareous sands. Therefore, bender element (BE) tests were performed on calcareous sands with different mean grain sizes (d50), uniformity coefficients (Cu), and void ratios to obtain Gmax. The BE results revealed that the Gmax of calcareous sand increases slightly with increasing d50 but decreases significantly with increasing Cu. A modified model of Gmax incorporating the effects Cu and d50 was therefore developed for calcareous sand. Moreover, microscopic observations of pore size distributions (PSD) obtained from nuclear magnetic resonance (NMR) tests were presented to demonstrate the effect of GSD on PSD and its correlation with Gmax. The NMR results revealed that the interaggregate pore structure proportion and uniformity of the PSD decreased significantly with increasing Cu but increased slightly with increasing d50. The underlying mechanism for the effect of GSD on Gmax was related to its substantial impact on microstructure. The significant decrease in Gmax with increasing Cu can be attributed to the remarkable reduction in the ratio of the interaggregate void ratio to the intraaggregate void ratio. Additionally, Gmax was enhanced as the heterogeneity of the microporosity structure distribution decreased.

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Data availability statement

All data and models generated or used during the study appear in the published article.

Abbreviations

G max :

Small strain shear modulus

d 50 :

Mean grain size

C u :

Uniformity coefficient

e :

Void ratio

e t :

Target void ratio after consolidation

e c :

Actual void ratio after consolidation

A, n :

Model parameters in Eq. (1)

pʹ:

Mean effective stress

p a :

A standard atmospheric pressure

S v :

Percentage passing

R, R max :

Pore size and maximum pore size

D a :

Fractal dimension

ζ :

Ratio of interaggregate void ratio to intraaggregate void ratio

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Acknowledgements

This work was financially supported by the Chinese National Natural Science Foundation (Grant No. 51508506), Joint Fund of Zhejiang Provincial Natural Science Foundation (Grant No. LHZ20E080001), Hangzhou Science Technology Plan Project (Grant No. 20172016A06, 20180533B06, 20180533B12, 20191203B44), Scientific Research Cultivation Fund of Zhejiang University City College (J-202112), China Scholarship Council, and Alexander von Humboldt-Stiftung.

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Authors and Affiliations

  1. Department of Civil Engineering, Zhejiang University City College, Hangzhou, 310015, China

    Zhi Ding

  2. Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou, 310058, China

    Shao-Heng He & Zhi Ding

  3. Chair of Soil Mechanics, Foundation Engineering and Environmental Geotechnics, Ruhr-Universität Bochum, 44801, Bochum, Germany

    Shao-Heng He, Meisam Goudarzy & Yifei Sun

  4. State Key Laboratory of Internet of Things for Smart City and Department of Civil and Environmental Engineering, University of Macau, Macau S.A.R., 999078, China

    Tao Xu

  5. Engineering Research Center of Smart Rail Transportation of Zhejiang Province, Hangzhou, 310014, China

    Qiong-Fang Zhang

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  1. Shao-Heng He
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Appendix

Appendix

NMR is a specific physical phenomenon of protons with spin characteristics. The process of returning the magnetization vector to the equilibrium state is called relaxation. According to the relaxation mechanism for low-field NMR, there are three different relaxation mechanisms available for fluids in porous media. Therefore, the relaxation time of the pore fluid can be expressed as follows:

$$\frac{1}{{T_{2} }} = \frac{1}{{T_{2B} }} + \frac{1}{{T_{2S} }} + \frac{1}{{T_{2D} }}$$
(14)

where T2 is the transverse relaxation time of the pore fluid collected by the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence; T2B is the free transverse relaxation time of the liquid, which is determined by the physical properties of the liquid (such as viscosity and chemical composition); T2S is the transverse relaxation time caused by surface relaxation; and T2D is the transverse relaxation time of pore fluid caused by diffusion under a magnetic field gradient. For water in porous media, the influence of the first and third terms on the equation can be ignored, and surface relaxation plays the main role as follows:

$$\frac{1}{{T_{2} }} = \frac{1}{{T_{2S} }} = \rho_{2} \left( \frac{S}{V} \right)$$
(15)

where ρ2 is the surface relaxation coefficient, which is a constant and is not affected by temperature and pressure, and S/V is the ratio between the pore surface area (S) and the pore volume (V), which is related to the pore shape. The shapes of pores in the soil can be approximated as cylindrical tubes. Therefore, Eq. (16) can be written as follows:

$$R = 2\rho_{2} T_{2}$$
(16)

where R is the pore size.

Fractal theory is a mathematical method that quantifies the degree of self-similarity, complex geometry and heterogeneity of porous media with fractal dimensions. According to previous studies, the pore structures of porous media exhibit fractal characteristics and can be studied using fractal theory [79, 80]. Fractal theory has been widely used in quantitatively describing and studying the fractal characteristics of PSDs. According to fractal theory, the fractal dimension D in the PSD corresponding to pore size R obtained by NMR can be provided as follows:

$$S_{v} = \left( {\frac{R}{{R_{\max } }}} \right)^{3 - D}$$
(17)

Therefore:

$$\lg S_{v} = (3 - D)\log R + (D - 3)\log R_{\max }$$
(18)

where R is the pore size, Rmax is the maximum pore size, D is the fractal dimension, and Sv is the percentage passing.

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He, SH., Goudarzy, M., Ding, Z. et al. Small-strain shear modulus (Gmax) and microscopic pore structure of calcareous sand with different grain size distributions. Granular Matter 24, 112 (2022). https://doi.org/10.1007/s10035-022-01270-2

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