Macro- and Micromechanical Assessment of the Influence of Non-Plastic Fines and Stress Anisotropy on the Dynamic Shear Modulus of Binary Mixtures

Abstract

Resonant column tests were carried out on Hostun sand mixed with 5%, 10% and 20% non-plastic fines (defined as grains smaller than 0.075 mm) in order to quantify the combined influence of the void ratio (e), anisotropic stress state (defined as σ_v′/σ_h′) and fines content (f_c) on the maximum small-strain shear modulus G_max. A significant reduction in the G_max with increasing f_c was observed. Using the empirical model forwarded by Roesler, the influence of e and σ_v′/σ_h′ on G_max was captured, although the model was unable to capture the influence of varying fines content using a single equation. From the micro-CT images, a qualitative observation of the initial skeletal structure of the ‘fines-in-sand’ grains was performed and the equivalent granular void ratio e* was determined. The e was henceforth replaced by e* in Roesler’s equation in order to capture the variation in f_c. The new modification was quantified in terms of the mean square error R². Furthermore, the G_max of Hostun sand–fine mixtures was predicted with good accuracy by replacing e with e*. Additionally, a micromechanical interpretation based on the experimental observation was developed.

1. Introduction

The determination of soil stiffness is of primary importance, considering its application in practical fields of geotechnical engineering, including foundation settlements, deformations caused by excavations or wave propagation in the ground due to vibration. In addition, all geotechnical sub/superstructure designs require the soil stiffness parameters in order to estimate the resistance of a structure against dynamic motions, such as earthquakes. In recent years, the development of elasto-plastic and advanced constitutive models (e.g., hardening soil model with small-strain stiffness, SANISAND) exclusively requires the small-strain soil stiffness parameter G_max as an input, underlining its importance in terms of research and practice in the field of soil dynamics.

Soil stiffness typically has a maximum value at low strains (strain < 10⁻⁶), denoted either G_max or G₀, and decreases by increasing the strain amplitude. Since the 1960s, several researchers ([1,2,3]) identified the influence of the void ratio and the effective stress on the small-strain shear modulus G_max. The authors of [1] were the earliest to formulate an empirical equation to capture the influence of the void ratio and effective stress under isotropic stress conditions (Equation (1)):

$$G_{max}=A p_a f\left(e\right){\left(\frac{p\prime }{p_a}\right)}^n$$

(1)

where A is a material constant depending on the soil type, p_a is the atmospheric pressure (≈100 kPa), p’ is the effective stress, n is a stress exponent and f(e) is the void ratio function, which popularly takes the form of Equation (2) ([1]) or Equation (3) ([4]):

$$f\left(e\right)=\frac{{\left(c-e\right)}^2}{1+e}$$

(2)

$$f\left(e\right)=e^{-d}$$

(3)

c and d are fitting parameters. The authors of [5,6] studied the influence of the grain size distribution on G_max, where they reported a significant influence of the uniformity coefficient C_u and the mean grain size D₅₀, which they characterized via empirical relations. Other than grain size, the fines content (i.e., grains passing through a standard US 200 sieve and smaller than 0.075 mm) can also influence the mechanical behavior of soils, as was evidenced in previous studies, e.g., refs. [7,8,9,10,11]. Systematic studies on the effect of fines content and isotropic stress on G_max have been published by [8,9,10,11,12,13,14,15,16,17,18]. These studies showed a strong influence of the non-cohesive fines fraction on G_max—a decrease in G_max was observed with increasing fines content. Therefore, ref. [12] proposed a reduction factor for Equation (1), as shown in Figure 1.

It must be noted that Equation (1), in conjunction with the reduction factor, is not very suitable since it assumes different values for each soil type, as demonstrated by [17,18,19]. The authors of [9,11] reported that the fitting parameters of Equations (1)–(3) are influenced by fines content, which means for every binary mixture, the fitting parameters must be separately determined.

It is possible to capture the influence of the fines content on the mechanical behavior of granular soils by utilizing the concept of equivalent granular void ratio, as defined by [7]. Furthermore, ref [19] defined that a certain portion of fines actively participate in the granular structure contributing to force chains, based on which Equation (4) was suggested, where a boundary was formulated, below which the soil behavior would be dominated by sand (defined as ‘fines-in-sand’) and above which by fines (‘sand-in-fines) ([19,20]).

$$e^{\ast }=\frac{e+\left(1-b\right)f_c}{1-\left(1-b\right)f_c} f_c<f_{thr}$$

(4)

The b-value holds a value between 0 and 1 and is responsible for denoting the active proportion of fines in the soil structure; therefore, it is a function of the fine content—the higher the value the higher the percentage of fines contributing to force chains in the mixture. In [20,21], the authors developed a semi-empirical equation (Equation 5) to calculate b-parameter.

$$b=\left[1-exp\left\{-0.3\frac{\left(f_c/f_{thr}\right)}{k}\right\}\right]\times {\left(r\frac{f_c}{f_{thr}}\right)}^r$$

(5)

where r = (D₁₀/d₅₀)⁻¹, D₁₀ = sand grain size corresponding to 10% finer materials by weight passing through, d₅₀ = fine grain size corresponding to 50% finer materials by weight passing through, k = (1 − r^0.25). The authors of [17,19] had previously used this formulation in their work on binary mixtures.

Ref. [10] carried out resonant column (RC) tests on Hostun sand mixed with a fine fraction. They reported the significant influence of the fines content on the maximum shear modulus (Figure 2a). The concept of equivalent granular void ratio e* [22,23] was successfully used to predict the maximum shear modulus in Equation (1) under isotropic stress conditions. However, no investigation was done for anisotropic stress states.

In order to evaluate the effect of anisotropy, numerous laboratory tests using bender elements ([11,24,25,26,27,28,29,30]), RC and torsional shear tests ([1,11,31,32,33,34,35]) have already been carried out. Furthermore, [1] and [24] extended the empirical relations to account for anisotropy. [1] believed that shear stress plays an insignificant role on shear modulus and suggested replacing p′ in Equation (1) with the average of vertical and horizontal stresses Equation (6).

$$G_{max}=Af\left(e\right)p_a{}^{\left(1-n\right)}{\left(\frac{{\sigma }_v+{\sigma }_h}{2}\right)}^n$$

(6)

${\sigma }_v$ and ${\sigma }_h$ denotes the vertical and horizontal stresses.

Ref. [24] conducted experimental studies on cubic soil samples to investigate the effects of stress components on shear wave velocity under anisotropic loading, where a significant influence of vertical stress on shear wave velocity and, consequently, on G_max was reported. Therefore, Hardin’s relationship was modified accordingly by using ${\sigma }_v^{\prime }$ and ${\sigma }_h^{\prime }$ instead of p′, as shown in Equation (7):

$$G_{max}=Af\left(e\right)p_a{\left(\frac{{\sigma }_v^{\prime }}{p_a}\right)}^{n_v}{\left(\frac{{\sigma }_h^{\prime }}{p_a}\right)}^{n_h}{\left(\frac{{\sigma }_c}{p_a}\right)}^{n_c}$$

(7)

$n_v$, $n_h$ and $n_c$ are the stress exponents, while ${\sigma }_c$ represents the out-of-plane stress.

Furthermore, it was also found out that the placing technique of the sand does not influence the shear wave velocities. Reference [11] performed tests on glass beads, concluding that reference shear strain was affected by anisotropy, and an empirical relationship was developed, considering confining pressure and anisotropic stress components. Performing tests on Ticino sand under both horizontal and vertical directions using geophones equipped with bender and compression elements, ref. [25] noted stiffnesses in the horizontal plane were larger than in the vertical plane. Using a similar procedure, ref. [26] investigated the interdependence of the active earth pressure coefficient K₀ on stress-induced anisotropy. The work of [24] was further extended by [27], in which the author attempted to establish a difference between the effects of fabric anisotropy and effective stress on soil stiffness. A similar approach using a multiaxial triaxial cell equipped with bender/extender elements was adopted by [28] to assess the evolution of elastic anisotropy under axially symmetric stress conditions. Using results backed by experiments, ref. [29] carried out discrete element method simulations and stated that more contact normal tensors preferred to distribute along the horizontal direction. Reference [30] used three different sample preparation methods to investigate the fabric anisotropy of Hostun sand, where air-pluviated and tamped samples exhibited higher stiffness in the horizontal and vertical directions, respectively.

Most of the previous studies have focused on either of the two aspects: either anisotropy of clean sands or sands with varying fines content. In nature, sands are often mixed with fines, and subjected to anisotropic stress states, possibly due to an existing sub-structure or repetitive loading over time. Therefore, studies combining both fines content under anisotropic stress conditions are necessary considering their practical importance. Therefore, in the present work, the influence of fines content on G_max under anisotropic stress conditions was investigated in detail. In addition, the application of e* in Equation (7) in specimens under anisotropic stress was inspected.

2. Experimental Procedure and Sample Preparation

The following sub-sections describe the experimental device and procedure followed by the tested materials and the method of sample preparation.

2.1. The RC Device

The RC and Bender element tests are common methods to determine the small-strain dynamic properties of soils (e.g., shear wave velocity, shear stiffness and damping). To determine the shear modulus and damping, the RC device available at the Ruhr-Universität Bochum was used. The device is capable of investigating the influence of small (shear strain γ < 10⁻⁵) and medium shear strains (10⁻³ < γ < 10⁻⁵). This free-free mode of vibration, i.e., the bottom and top plates are freely movable ([36]), was employed. This specimen is enclosed by a latex membrane of thickness 0.4 mm. The inserted cylindrical soil sample is subjected to harmonic torsional vibrations about the longitudinal axis by two electromagnetic excitation heads. The generated acceleration is measured with the help of two accelerometers, which then results in the linearly distributed torsion γ over the specimen height. This recorded acceleration is displayed on an oscilloscope as a sinusoidal excitation signal, the frequency of which is varied using a function generator until the resonant frequency f_R is determined. The dynamic shear modulus for the specific load amplitude can then be calculated from f_R ([36]):

$$G={\left(\frac{2\pi f_R}{a}\right)}^2$$

(8)

$$a tan\left(a\right)-\frac{J^2}{J_0 J_L} \frac{tan\left(a\right)}{a}=\frac{J}{J_0}+\frac{J}{J_L}$$

(9)

The polar mass moment of inertia for the bottom and top parts of the device, as well as the sample, are denoted by $J_0$, $J_L$ and $J$ respectively. $\rho$ denotes the sample density, while $a=\frac{\omega L}{v_s}$, $\omega$ is the rotational frequency, L is the sample height and $v_s$ the shear wave velocity. The schematic figure of the RC device is shown in Figure 3.

The device was modified for performing tests under anisotropic stress states ([37]). A pneumatic pressure cylinder was mounted on the top of the cell, as shown in Figure 3, for applying vertical pressure on the specimen. The calibration and validation of the device were performed following different methods, further details of which are in [11,37]. At the beginning, the specimen was isotropically loaded with a cell pressure ${\sigma }_h^{\prime }$ = 50 kPa. From a cell pressure of 200 kPa onwards, the vertical pressure ensures anisotropy in the specimen. The results on isotropically loaded specimens were shown previously in [17]. This paper aimed to describe the results on the same soil mixtures, but under anisotropic loading. The stress path is currently divided into two areas: isotropic condition at ${\sigma }_h^{\prime }$ = 200 kPa ([11,17]) and anisotropic condition where ${\sigma }_h^{\prime }$ is constant 200 kPa and ${\sigma }_v^{\prime }$ is increased up to 600 kPa (i.e., effective vertical to horizontal stress ratio ${\sigma }_v^{\prime }/{\sigma }_h^{\prime }$ = 3).

2.2. Tested Materials

In the present work, four series of tests were carried out with a mixture of Hostun sand (a quartz sand originally mined in France) and non-plastic quartz (obtained locally) as fines with different proportions by weight—0%, 5%, 10% and 20%. The material mixtures of Hostun sand and the different proportions of fines are shown in Figure 4. The white opaque parts show the quartz powder and the light-colored elements denote sand particles. Hostun sand has been used previously in numerous tests ([37]). Silica is the predominant chemical that constitutes the sand grains (SiO₂ > 98%). The sand is angular with a grain density of 2.65 g/cm³. The mean grain size of Hostun sand and quartz powder (also consisting of SiO₂ as the main component) is 0.375 mm and 0.004 mm, respectively. Figure 5 shows the grain size distribution of the tested materials obtained following [38].

3. Experimental Results

From Figure 6a–d, the influence of the anisotropic stress on the variation in void ratio e over different fines content is shown. The results show that with an increase in the effective anisotropic deviatoric stress ${\sigma }_v^{\prime }/{\sigma }_h^{\prime }$, the e decreases with increasing fines, whereas the decrease is minimal for the clean sand (Figure 6a). This implies a higher compressibility of the mixtures with higher fines.

Furthermore, Figure 7 shows the relationship between G_max, e and ${\sigma }_v^{\prime }/{\sigma }_h^{\prime }$. It is clear that the G_max increases with decreasing e and increasing $\frac{{\sigma }_v^{\prime }}{{\sigma }_h^{\prime }}.$ It is interesting to note the relative magnitudes of the G_max for differing fines content—it is higher for the clean Hostun sand (Figure 7a) but reduces with increasing fines content. In addition, the reduction in G_max is much larger at higher fines content, although the specimens are prepared at an initially dense state (see Figure 7d). Practically, the influence of e is diminished at f_c = 20%, while it is largest at f_c = 0%.

In Figure 8, the variation of the G_max with different e and f_c values is shown along with isotropic (Figure 8a) and anisotropic (Figure 8b) states. For all specimens with 0%, 5%, 10% and 20% fines content, G_max decreases with the increase in e and increase of vertical stress, ${\sigma }_v^{\prime }$. In comparison to the isotropic specimen with initial e = 0.80 and containing 5% fines, the 20% fines specimen under the same boundary conditions shows a lower G_max- for 5% f_c, which is close to 100 MPa, while for the other, it measures around 75 MPa. The same trend can be noted for the anisotropic specimen in Figure 8b, which implies an increase in anisotropic stress load causes an increase in the maximum shear modulus, whereas the increase in f_c causes a reduction in the maximum shear modulus.

For ${\sigma }_v^{\prime }/{\sigma }_h^{\prime }$ = 2, the G_max for clean sand measures approximately 160 MPa at e = 0.70 (Figure 8b), while under isotropic conditions (Figure 8a), it measures around 135 MPa. This reinforces the observation that at anisotropic stress conditions, an increase in the e as well as fines content results in a significant decrease in the G_max.

4. Analysis of Results

Using the popular models of [1,23], the variation in the G_max with e, p′ and ${\sigma }_v^{\prime }/{\sigma }_h^{\prime }=1$ (isotropic) and 2 (anisotropic) was quantified as shown in Figure 9. In general, a good fit with the experimental data can be seen. Later, the G_max was normalized with the void ratio function employing an average value of the fitting parameter c (= 2.12, 2.15, 2.37 and 3.25 for 0%, 5%, 10% and 20% fines, respectively). To capture the variation in the anisotropic stress, the G_max was initially normalized with the Hardin’s void ratio function, considering the average value of the fitting parameter c (= 2.12, 2.15, 2.37 and 3.25 for 0%, 5%, 10% and 20% fines, respectively), following which the normalized G_max was further plotted against the ${\sigma }_v^{\prime }/{\sigma }_h^{\prime }$ (shown only for the clean sand case in Figure 10a) to obtain the best fit magnitudes of the fitting parameters $n_v$ and $n_h$. Figure 10b presents a 3D overview of the variation of G_max with regards to e,${\sigma }_v^{\prime }$ and ${\sigma }_h^{\prime }$ (Equation (10) in the case of the clean sand only, shown in Figure 10b below, where the proposed model shows a good fit with the experimental data with R² = 0.95 ($K_0$ denotes the ratio ${\sigma }_h^{\prime }/{\sigma }_v^{\prime }$).

$$\frac{G_{max}}{f\left(e\right)}=0.836p_a{\left(\frac{{\sigma }_v^{\prime }}{p_a}\right)}^{0.21}{\left(\frac{{\sigma }_h^{\prime }}{p_a}\right)}^{0.23}$$

(10)

However, it is not possible to capture in a single curve the influence of varying fines content. Therefore, as suggested by various studies, the global void ratio can also be replaced by the equivalent granular void ratio e*, which would have the same magnitude as e for clean sand but would depend on the parameter b for fine content below the threshold fines. Based on the suggestion of [20] from their experiments with 10 sands, a value of 30% was deemed suitable for the f_thr since it was able to capture many characteristic responses of undrained soil behavior regardless of host sands. The b value was calculated accordingly in order to convert the e to e* up to a fines content of 20%. The resulting G_max − e* curves for both isotropic and anisotropic stress states are shown in Figure 11 below. The data at higher e* and lower G_max show relatively greater scatter than at lower e* and higher G_max. Hardin’s relationship also shows a slight deviation to higher e* and lower G_max, which may be attributed to the inherent variability of G_max. Figure 12 shows a comparison of the measured and predicted G_max with the proposed Equation (10), where a good fit is evident (R² = 0.80) for different f_c values ranging between 0–20% as well as different degrees of anisotropy achieved by loading in the vertical direction ($1\le {\sigma }_v^{\prime }/{\sigma }_h^{\prime }\le 3$).

5. Discussion

As shown in the previous section, by using e* instead of e in the Hardin’s model, it was possible to capture the influence of varying fines lower than the threshold value. The biggest advantage afforded by using e* is that one can avoid the numerous back calculations to determine the fitting parameters for each fines content. In addition, a solid micromechanical interpretation based on certain existing studies employing similar concepts [40,41,42,43,44] asserts a more logical approach, which, presently, is achieved via the b parameter—a micromechanical parameter specially designed to account for the positive or negative effects of f_c or the micromechanical influence of f_c in fine sand mixtures, and can be roughly estimated if the grain size distribution is available.

Micromechanical Interpretation

The stiffness of soil samples depends on microstructural properties, mainly grain-to-grain contacts that dictate the stiffness of the material. According to the theory of Hertz–Mindlin, the tangential or shear stiffness K_T between two grains is a function of the normal stiffness K_N, contact forces f_T and f_N and the elastic properties of the grains ([45,46]), and is mathematically represented as:

$$K_T=C_2 K_N {\left[1-\frac{f_T}{f_N tan\phi }\right]}^{\eta }$$

(11)

$$K_N=C_1{f}_N^n$$

(12)

$$\eta =n=\frac{1}{3}$$

(13)

n and $\eta$ are fitting parameters. With the increase in the vertical load f_N, the normal stiffness increases, which results in increments in the friction and, consequently, the tangential stiffness and shear modulus G_max between particles. An additional load applied in the vertical direction results in an increase in f_T. However, according to Equation (11), an increase in f_T should result in a decrease in K_T, which explains the slower increase in G_max for anisotropic stresses compared to G_max for isotropic cases.

Presently, to reinforce our observations, a micro-CT scan was performed on the sample having 10% fines content in Kumamoto University in Japan (Figure 13a). The sample height and diameter for the CT scan was 40 mm and 7 mm, respectively (Figure 13b). Three densities were selected: loose ($\mathsf{\rho }=1.207$ g/cm³, Figure 14a), medium-dense ($\mathsf{\rho }=1.312$ g/cm³, Figure 14b) and dense ($\mathsf{\rho }=1.437$ g/cm³, Figure 14c), details of which are in

Table 1. Schedules of the various samples for micro-CT analysis for different densities.

Case	Material	Weight Ratio (%)		Sample Height (mm)	Weight (g)	Dry Density (g/cm3)
		Quartz	Hostun
1	Quartz/Hostun	10	90	37.5	1.742	1.207
2				34.5		1.312
3				31.5		1.437

. The white particles denote sand grains, while those in gray denote fines, i.e., the quartz powder. From Figure 14a, the fines cover some of the coarse particles merely like a coating. Some of the fine grains arrange themselves between the coarse particles, i.e., these fine grains actively interact with the sand grains. The remaining fine grains distribute themselves in the voids between the coarse grains, i.e., these parts of the fine grains act inactively ([17]). An external load causes friction between the individual grains. For large grains, the friction surface is correspondingly large, making the grain-to-grain contact system relatively stable. Adding more fine grains (Figure 14b,c) reduces the friction area between the individual grains, causing the system to lose stability. Mathematically, this results in a lower contact force K_T and thus results in a lower maximum shear stiffness ([10]).

6. Conclusions

The aim of the study was to demonstrate the variation of the maximum shear modulus in binary mixtures of sand–non-plastic fines considering stress anisotropy due to additional vertical loading. As already mentioned, individual studies considering stress anisotropy or mixtures were conducted in previous research, however this work presents one of the first considering both these influences. This paper briefly summarizes the results and interpretation of the tests with respect to the influence of stress anisotropy and fine grain content on the G_max. The major findings are summed up below:

• Based on a series of resonant column tests with different initial global void ratios and confining pressures, it was found that G_max is lower for increasing fines content as well as higher void ratios. Using a popular empirical model, the variation of G_max with void ratio, confining pressure and stress state was captured with a good accuracy.
• Furthermore, to include fines content, the concept of equivalent intergranular void ratio was used where the global void ratio e was replaced by e* in the various formulations with a higher degree of accuracy, particularly for lower magnitudes of e*.
• Micro-CT scans of the binary mixtures were additionally made to enable a closer look into the microstructure, e.g., grain-to-grain contacts. From the skeletal structure, the contact forces between the individual grains were evident. In the case of anisotropic stress, a shear force acts in addition to the normal force for isotropically loaded specimens. An increase in fines results in a reduction in the friction between sand grains which leads to lower contact stiffness.

The results of the present work will provide practical knowledge for design engineers to predict the shear modulus in binary mixtures considering the designed overburden vertical stress leading to a stress anisotropy, and therefore, allow them to reliably calculate ground settlements/factor of safety of structures under dynamic loads. Furthermore, advanced constitutive models can also incorporate both these influencing factors in complex multi-variable numerical simulations.

For future work, the influence of stress anisotropy and grain distribution on elastic modulus E, grain shape or size and grain minerals can be investigated. In addition, the influence of other stress paths can be checked. Furthermore, a higher percentage of fines greater than 20% can be investigated. For a more accurate interpretation of the results, various methods, such as the discrete element method (DEM), to determine the number of grain contacts or the force distribution on the grains can be applied.