Introduction

Sand has many applications for use such as construction materials and as a raw material for glass and silicon. Sand is often used to provide ballistic protection for the military structures. Understanding the mechanical behavior of sand is also important for investigations of the interaction of a blast wave induced by an improvised explosive device with military vehicles and personnel, as well as earthquakes and landslides. Numerous publications have focused on the mechanical behavior of bulk sand for geological and civil engineering applications, at low stress levels under quasi-static compression [1–4]. For applications involving high-speed impact, the mechanical behavior of sand under high stress needs to be determined. Under such a condition, a fundamental understanding of the underlying mechanisms of the deformation, flow, and fracture of granular materials under load, is a prerequisite to investigate the mechanical behavior at mesoscale and continuum scale. Ultimately, it can be used to explain the large scale phenomena such as penetration.

The mesoscale behavior of sand depends strongly on the mechanical behavior of individual sand grains. The inherent complexity of mesoscale phenomena, including the interactions of solitary waves in granular materials with different particle sizes, requires the understanding of statistical variation of the sizes and mechanical properties of those individual grains. The measurement of individual particle properties can provide accurate information for the simulations of granular materials especially in particulate mechanics. For example, the accuracy in mesoscale simulations of sand behavior using discrete element method (DEM) depends on the mechanical properties of individual sand grains used as input [5–7]. The mechanical properties of sand at grain level are necessary input for the constitutive law in meso/multiscale simulations and grain fracture process such as sand penetration [8–12]. For this purpose it is particularly important to determine the mechanical properties such as the Young’s modulus, hardness, fracture toughness, and constitutive law for individual sand grains. Moreover, the mechanical properties of the grains are affected by the mineral composition, size, crystal structure, and defects in the form of voids, ridges and cracks [13]. Many efforts have been made on characterizing the mechanical behavior of bulk sand, through experiments such as compression [14–18], shear [19, 20], and stress wave propagation [21, 22]. Recently, results have been reported for uniform strain extension tests on sand and high strain rate behavior of sand [23, 24]. For understanding the mechanical behavior at mesoscale it is necessary to determine the grain-level mechanical properties of sand. However, due to the small dimensions in sand grains, typically in the range of 0.1–1.5 mm with irregular shapes, it is difficult to carry out a conventional tensile, compressive or shear experiment on individual sand grains to determine the mechanical properties. Hence, nanoindentation becomes an important and effective tool to characterize the mechanical properties of individual sand grains.

In the past three decades, nanoindentation has been used for measurement of mechanical properties of different types of materials [25–27]. The instrumented nanoindentation has been widely accepted as an effective testing method for the characterization of mechanical properties of elastic–plastic materials at submicron scale [28]. The resolution can reach a fraction of a nanometer in displacement, and μN in load. In addition, continuous stiffness measurement can provide hardness and Young’s modulus as a function of depth. As the depth increases, the hardness and Young’s modulus trend to approach steady values [29–31]. On nanoindentation on granular materials, results have been reported for 0.6 mm Ottawa sand grains [32], 25–100 μm size ceramic powders [33], and 0.155–0.476 μm pharmaceutical powders [34]. Recently, nanoindentation has been used to characterize the mechanical behavior of Stillwater sand grains of 0.7–1.5 mm [13].

In this investigation, nanoindentation was conducted to determine the mechanical properties for six sizes of Eglin sand grains, sorted from the as-received dry sand (assorted) using a sieving technique. The energy dispersive X-ray spectroscopy (EDS), X-ray diffraction (XRD) and transmission electron microscopy (TEM) were used to determine the chemical constituents and structures (crystalline vs. amorphous). The Young’s modulus and the hardness of sand grains were determined by a Berkovich tip, using the nanoindentation method developed by Oliver and Pharr [28]. A cube-corner indenter tip was used to induce cracks in a sand grain to measure the fracture toughness. Statistical analysis was conducted on the experimental data. An inverse problem solving approach was used to determine the yield strength and stress–strain relationships for the sand grains.

Experiment and Simulation

Preparation of Sand Grains of Different Sizes

All dry sand used in this study was silica based fine grain Quikrete #1961 sand, quarried in Pensacola, FL, washed and kiln-dried at Eglin Air Force Base. Eglin sand has been used in several previous publications [21, 22, 35] by groups in the US and Europe. The as-received Eglin sand was washed and subsequently dried in a temperature chamber at 55 °C for 24 h before characterization. A multi-directional shaker was used to sort out the sand grains [35]. The shaker (D-4326, DUAL Manufacturing Co, Franklin Park, IL) used a series of metal wire sieves following the ASTM E-11-09e1 standard (Specification for Wire Cloth and Sieves for Testing Purposes). To separate different sizes of sand, the sand agglomerate was agitated through a series of sieves stacked together, with the larger open-hole sizes at the top and gradually reducing to the smaller ones at the bottom. Following the ASTM D2487 standard, the sand grains were sorted into grain sizes of 0.70 mm, 0.60 mm, 0.50 mm, 0.42 mm, 0.30 mm, 0.212 mm, and 0.15 mm through shaking on a set of stacked sieves of #25, #30, #35, #40, #50, #70, and #100, respectively. After shaking, the sand that fell through a mesh was given the designation of passed weight and the sand that remained on the sieve was designated as the remaining weight.

Microscopic Characterization

There are various factors that may affect the mechanical properties of sand grains, including the micro/macro structure, chemical constituents and sand grain size. The as-received sand grains were coated with 10 nm thin Pt/Au for observation of surface morphology under a JEOL JSM 6400 scanning electron microscope (SEM). The transmission electron microscopy (JEOL 2100F TEM/STEM operating at 200 kV) was used to characterize the microstructure of an Eglin sand grain. TEM samples were prepared using a focused ion beam apparatus (FEI Nova NanoLab 200). X-ray diffraction (Rigaku Ultima III X-Ray Diffractometer) was conducted on sand grains of different sizes, to assess their crystalline structures.

Measurements of Modulus and Hardness by Nanoindentation

The sorted sand grains of each size were embedded in an epoxy matrix in a sample holder. Each sample was cured in a temperature chamber at 50 °C for 24 h, resulting in a composite of sand grains in a hardened epoxy matrix. Each sample was wet-polished using alumina abrasive slurry (Buehler Inc., Minneapolis, MN) on a rotating polishing wheel covered with a cloth pad. To obtain a smooth surface suitable for nanoindentation, the minimum abrasive size used in the final polishing was 50 nm. After polishing, the sand sample surfaces were cleaned by acetone followed by alcohol. The polished sand/epoxy surface was examined by a Zeiss OPMI optical microscope.

An MTS Nano Indenter XP system was used for nanoindentation measurements. This nanoindenter can reach a maximum indentation depth of 500 μm (resolution at 0.2 nm) and a maximum load of 500 mN (resolution at 50 nN). A diamond Berkovich indenter tip was used for the measurements of the Young’s modulus and hardness. Nanoindentations were made under a constant rate loading. The nanoindentation load was increased until it reached a set value, followed by unloading. The load–displacement curves were analyzed to determine the mechanical properties of the sand grains based on the contact mechanics analysis of the nanoindentation.

The hardness (H) is obtained using

$$ H=\frac{P_{\max }}{A_c} $$
(1)

where P max is the maximum indentation load, A c is the contact area corresponding to the maximum contact depth (h c ) at the maximum load, which is calculated based on the tip area function [28]. The reduced modulus of the specimen (E r ) is calculated using

$$ \frac{1}{E_r}=\frac{1-{\nu}_s^2}{E_s}+\frac{1-{\nu}_i^2}{E_i} $$
(2)

where E s and ν s are the Young’s modulus and Poisson’s ratio of the specimen, respectively, and E i and ν i are the Young’s modulus and the Poisson’s ratio of the indenter tip (diamond), respectively. Due to the finite stiffness of the indenter tip, its modulus is considered in the calculation of sand grain modulus from the contact stiffness. The contact stiffness S is calculated from the slope of the initial unloading curve,

$$ S=\frac{ dP}{ dh}=\beta \frac{2}{\sqrt{\pi }}{E}_r\sqrt{A_c\left({h}_c\right)} $$
(3)

where β is the indenter shape factor [36]. Equations (2) and (3) were used to determine the Young’s modulus for a sand grain. The hardness was calculated from the peak load and the corresponding contact area using equation (1). These were the direct outputs from the nanoindentation software, based on equations (1)–(3).

Fracture Toughness Measurement

The sorted sand grains of each size were also embedded in an epoxy matrix in a sample holder, following the same preparation procedure as a sample for measurement of modulus. A diamond cube-corner indenter tip was used in this investigation. When a brittle material is loaded by a sharp nanoindenter tip under a sufficiently high load, radial cracks will emanate on the sand surface from the contacting points with the nanoindenter corners. In such case, the fracture toughness can be estimated by measuring the lengths of the radial cracks produced under a given indentation load.

In order to estimate the fracture toughness, a formula for the mode-Ι fracture toughness K C derived by Pharr et al. [37] was used,

$$ {K}_C=\beta {\left(\frac{E}{H}\right)}^{0.5}\left(\frac{P_{\max }}{c^{3/2}}\right) $$
(4)

where β is an empirical constant which depends on the geometry of the indenter tip (for a cube-corner tip β = 0.032).

Statistical Analysis of Mechanical Properties

The Weibull distribution is one of the most widely used distribution functions and was found appropriate to describe the distribution of the mechanical properties for materials with defects [38, 39]. For Eglin sand, the Weibull distribution is found to be suitable to describe the size distributions for both intact and impacted sand after high-strain rate compression; it can describe the statistical distribution better than the Gaussian distribution [40]. The Weibull probability density function is given as

$$ f(x)=\frac{k}{\lambda }{\left(\frac{x}{\lambda}\right)}^{k-1}{e}^{-{\left(\frac{x}{\lambda}\right)}^k},0\le x\le +\infty $$
(5)

where k > 0 is the shape parameter, and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function given as

$$ F(x)=1-{e}^{-{\left(\frac{x}{\lambda}\right)}^k},x\ge 0 $$
(6)

and F(x) = 0 for x < 0. The Weibull plot was used to examine whether the data follows a Weibull distribution [40, 41]. The Weibull plot is a diagram of the empirical cumulative distribution function of data plotted using ln(x) as the abscissa and ln(-ln(1-F(x))) as the ordinate. If the relationship follows a straight line, i.e., ln(−ln(1 − F(x))) = k ln x − k ln λ, then the data follows the Weibull distribution. The mechanical properties including modulus, hardness and fracture toughness were examined using the Weibull plots.

FEM Simulations to Determine Constitutive Model

Eglin sand grains are polycrystalline materials, modeled as isotropic materials. The Ramberg-Osgood model [42, 43] is used. The strain components ε ij are calculated through the stress components σ ij

$$ {\varepsilon}_{ij}=\frac{1}{E}\left[\left(1+\nu \right){\sigma}_{ij}-\nu {\delta}_{ij}{\sigma}_{kk}\right]+\frac{3}{2}\alpha {\left(\frac{\sigma_e}{\sigma_y}\right)}^{n-1}\frac{s_{ij}}{E} $$
(7)

where α is the yield offset, n is the hardening exponent, σy is the yield stress, ν is the Poisson’s ratio, E is the Young’s modulus, δ ij is the Kronecker delta, σ e is the von Mises stress defined as \( {\sigma}_e=\sqrt{{\scriptscriptstyle \frac{3}{2}}{s}_{ij}{s}_{ij}} \), and s ij are the deviatoric stress components, given as \( {s}_{ij}={\sigma}_{ij}-{\scriptscriptstyle \frac{1}{3}}{\sigma}_{kk} \), where σ kk is the first stress invariant. In equation (7), the first term represents the linear elastic response, and the second term is the result of the power-law plastic flow normal to the von Mises stress potential. In the uniaxial stress state, the Ramberg-Osgood model becomes

$$ \varepsilon =\frac{\sigma }{E}+\alpha \frac{\sigma_y}{E}{\left(\frac{\sigma }{\sigma {}_y}\right)}^n $$
(8)

For modeling individual sand grains, the stress at 0.2 % offset σ0.2 is considered as the yield stress σy, defined as σ y  = 0.002E/α. In the Ramberg-Osgood model, only three parameters (α, n, σy) are independent.

The inverse problem solving methodology proposed by Shim et al. [44] was used to extract the stress–strain curve of a single grain from FEM simulation of the nanoindentation data. Their work is based on a relationship between the hardness and the flow stress, first provided by Tabor [45]

$$ H={C}_{\varphi }{\sigma}_f $$
(9)

where C φ is the constraint factor, depending on the angle φ of the indenter, and the flow stress is calculated from a characteristic value of the plastic strain. The characteristic strain, ε c , can be obtained for a given indenter angle. Several approaches have been proposed to obtain the stress–strain curves by relating the hardness to the stress, and the indenter angle to the characteristic strain [46–49]. Combining equations (8) and (9), the hardness can be expressed in terms of E, σ y , C φ and ε c ,

$$ \log H=n \log \left(\frac{E}{\sigma_y}{\varepsilon}_c\right)+ \log \left({C}_{\varphi }{\sigma}_y\right) $$
(10)

where the unknown model parameters are the exponent n and the yield stress σ y for a given nanoindenter angle. Johnson [50] showed that C φ follows a relationship with E/σ y and the nanoindenter angle given by

$$ {C}_{\varphi }=\frac{2}{3}\left[1+ \ln \left(\frac{E}{\sigma_y} \cot \varphi \right)\right] $$
(11)

Johnson also developed a simple equation to calculate the characteristic strain in equation (12) which is used in this simulation.

$$ {\varepsilon}_c=0.2 \cot \varphi $$
(12)

In this paper, finite element simulations were conducted to solve the inverse problem to obtain the values of n and σ y for different sizes of Eglin sand grains. The Young’s modulus E, and hardness H, have been measured from nanoindentation experiments. In this case, two parameters were calculated from equations (9)–(12), and there was only one parameter remaining unknown.

ABAQUS V6.9 standard [51] was used for simulations with finite deformation considered. The model of the nanoindentation is shown in Fig. 1. Due to the three-fold symmetry of the pyramidal Berkovich indenter, only one sixth of the entire model was used in simulations to reduce the computational time. The mesh size selected was examined for the convergence of the load–displacement curve. In this work, there were 1,646,381 C3D4 (brick) elements used to mesh the sand. The displacement history from the experiment was given as input to the FEM analysis. The Poisson’s ratio (for silica quartz) was assumed as 0.18 for all sand grains. The hardness values were substituted into equation (10) to determine parameters n and σ y . The different yield stress values were used to calculate the exponent value for the power law equation, which is used as the input for the plastic behavior of sand. The output of the FEM analysis was the resulting nanoindentation load under the prescribed displacement. This numerical load was plotted versus the displacement into the surface, to generate the simulated load–displacement curve that is consistent with the nanoindentation experiments.

Fig. 1
figure 1

Finite Element Analysis model for nanoindentation on 1/6 spherical sand grain model using a Berkovich tip. (a) Side view of the model. (b) Top view of the model

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Results

Particle Size Distribution

The curve of passed weight versus grain size is plotted in Fig. 2, to show the sand size distribution. The six major sizes of sand grains were 0.6 mm, 0.5 mm, 0.425 mm, 0.3 mm, 0.212 mm and 0.15 mm. The median grain size was determined as 0.425 mm. The sieve opening size of d60 allows 60 % of the sand by the weight to pass, determined as 0.41 mm, close to the median size 0.425 mm. Similarly, the effective grain size for d10 was determined as 0.18 mm; and the uniformity U (defined as d60/d10) was 2.28, indicating that Eglin sand grains have a narrow size distribution, close to the values reported previously [35].

Fig. 2
figure 2

Size distribution curve of sand grains

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Microscopic Analysis

Figure 3(a) shows the SEM images of typical sand grains, which exhibit different shapes, sizes, and surface characteristics. Eglin sand grains with a median size of 0.425 mm are shown in Fig. 3(b), indicating irregular shapes with surface ridges. The bright field TEM images, including the selected area electron diffraction patterns (Fig. 4(a)), clearly show two layers of core-shell structure in an Eglin sand grain: an amorphous outer layer and a crystalline interior layer. Dislocations are also observed in the interior layer as shown in the transmission electron microscopy image (Fig. 4(b)). The EDS spectra in Fig. 5 show the presence of silicon and oxygen in their oxide compounds. It is noted that the Ga signal was from the focused Ga ion beam, the Pt signal was from the Pt coating used to prepare transmission electron microscopy samples, and the Cu signal was from the grid used to support the transmission electron microscopy sample. There are some slight differences in the chemical constituents between the amorphous outer surface (Fig. 5(a)) and the crystalline interior (Fig. 5(b)). It appears that the outer amorphous layer contains a little more oxygen than the crystalline interior, in consideration of the oxygen/silicon ratio. EDS results show that the Eglin sand consisted primarily of silica quartz with the presence of small amounts of other chemical elements, which may contribute to the variation of mechanical properties.

Fig. 3
figure 3

SEM images of Eglin sand grains. (a) Assorted sand grains; (b) A single sand grain of 0.425 mm (in the center)

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Fig. 4
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TEM images of Eglin sand grain. (a) Bright field image showing two layers with different diffraction patterns; (b) Bright field image showing dislocations in the interior layer

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Fig. 5
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EDS and electron diffraction patterns of Eglin sand grains. (a) Amorphous outer layer; (b) Crystalline interior layer

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The XRD patterns (intensity vs. 2 theta, 2θ) of six sizes of sand grain samples are shown in Fig. 6. Defects such as voids in the sand grains can also contribute to the property variation. Most of the diffraction peak angles (2θ) are the same, confirming the EDS results that the sand is primarily silica. There are, however, some differences between these six patterns, which indicate that sand grains with different sizes may have slightly different structures, different material constituents, and defects. In comparison of sand diffraction data with the database maintained by the International Centre for Diffraction Data, it is found that most of the sand grain patterns match reasonably well with the α-quartz diffraction pattern.

Fig. 6
figure 6

X-ray Diffraction patterns of Eglin sand grains

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Young’s Modulus and Hardness

Since the structures and chemical constituents of the Eglin sand vary with the grain size, nanoindentation was conducted on different sizes of the sand to determine how the properties change with the grain size. Figure 7 shows an optical micrograph of a polished sand surface with several grains. In addition to a few polishing ridges, small pits are observed, indicating the presence of defects inside the sand grains. Each nanoindentation location was chosen in an area with a smooth polished surface, to avoid the surface region with the ridges and pits. However, there might be defects underneath a nanoindenter which could contribute to variation in mechanical properties. Nanoindentation sites were near the center of a sand grain, to minimize the edge effect.

Fig. 7
figure 7

Optical image of polished sand grains embedded in an epoxy matrix

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We first conducted nanoindentation on a relatively large (~0.7 mm) sand grain. The Young’s modulus of 72.4 ± 2.8 GPa and a Hardness of 11.2 ± 0.7 GPa were obtained from ten tests on several grains. When nanoindentation was carried out on the same grain, a distance of at least 50 μm between neighboring indents was maintained. For the same grain, the Young’s modulus values were consistent when nanoindentation was made on the same surface, indicating homogeneous behavior on the same grain. For nanoindentation conducted on different grains the mechanical properties for nearly 0.7 mm sand grains have very small deviation (within an error range of 5 %). The percent weight of 0.7 mm grains or larger is less than 5 % as seen in Fig. 2. For the next six sizes of sand grains, one nanoindentation was conducted on each sand grain, and nanoindentation was made on more than one hundred grains; results are summarized in Tables 1 and 2. Figure 8(a) shows a typical nanoindentation residual impression and Fig. 8(b) shows a typical inverted image (3D) obtained using NanoVision. The inverted image (Fig. 8(b)) enables the convenient determination of the indentation depth and ease of observation of its topographical features.

Table 1 Properties and parameters of Weibull distribution of Young’s modulus of Eglin sand grains
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Table 2 Properties and parameters of Weibull distribution of hardness of Eglin sand grains
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Fig. 8
figure 8

Typical indent image on a sand grain using Berkovich tip. (a) 2D residual indent impression; (b) 3D inverted indent image (dimensions in μm)

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The Weibull plots for the Young’s modulus and the hardness are shown in Figs. 9 and 10, respectively. The curves show that most data points fall on a straight line, indicating that these properties follow the Weibull distribution. The actual median values, or P50 values, defined as 50 % cumulative value in equation (6) for Young’s modulus and hardness for the six sizes of sand grains are listed in Table 1. The P50 values for hardness are listed in Table 2. The P30 values (defined as 30 % cumulative probability in equation (6)), often used in geosciences, are also given in comparison with the P50 values in Tables 1 and 2. The Weibull parameters (the shape k, and the scale λ) are also given in Tables 1 and 2, with 95 % confidence intervals for six sizes of sand grains. A close examination indicates that when sand grain size changes from 0.6, 0.5 to 0.425 mm, the P30 Young’s modulus values increase from 97.4 GPa, 102.1 GPa to 108.9 GPa, respectively. As the grain size reduces further, the P30 Young’s modulus values become 80.8 GPa, 77.5 GPa, and 71.5 GPa for grain sizes 0.3 mm, 0.21 mm and 0.15 mm, respectively. The P50 values of Young’s modulus show the similar trend. The maximum Young’s modulus is attained at sand size of 0.425 mm, which is the median grain size of the Eglin sand. Similarly for hardness of the six grain sizes, the P50 hardness value reaches the highest at the median grain size (0.425 mm). Nature has natural selection to allow grains with the highest modulus and hardness to remain.

Fig. 9
figure 9

The Weibull plots of Young’s modulus from nanoindentation on six different sizes of sand grains. (a) 0.6 mm; (b) 0.5 mm; (c) 0.425 mm; (d) 0.3 mm; (e) 0.212 mm; (f) 0.15 mm

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Fig. 10
figure 10

The Weibull plots of nanoindentation hardness from nanoindentation on six sizes of sand grains. (a) 0.6 mm; (b) 0.5 mm; (c) 0.425 mm; (d) 0.3 mm; (e) 0.212 mm; (f) 0.15 mm

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Fracture Toughness

Before proceeding with the nanoindentation on sand grains, a sample test was also conducted on a standard fused silica sample (results were not included here) for evaluation of the method. The average value for fracture toughness obtained was 0.60 MPa⋅m0.5, which is in a reasonable agreement with the value (0.58 MPa⋅m0.5) reported by Harding [52] for fused silica.

For sand, the values of the elastic modulus (E) and hardness (H) were obtained using the Berkovich tip (see Microscopic Analysis section) prior to indenting with a cube-corner tip. The average values for E and H were used to determine E/H for all sizes of sand grains. The E/H was determined as 8.5 for the assorted sand grains. In equation (4), the crack length c was determined from the surface profile scanning (inverted images); a typical image is shown in Fig. 11. Both indent impression (Fig. 11(a)) and inverted indent (Fig. 11(b)) are shown to illustrate the cracks. Due to potentially slight misalignment, inhomogeneity of the sand grains, and instability of the crack propagation, it is rather difficult to allow all three cracks to propagate to equal lengths. When there are differences between the crack lengths at the same indent, the average crack length was used for the calculation of K C . The Weibull plot for K C is shown in Fig. 12; it follows a straight line, indicating that the fracture toughness also follows the Weibull distribution. The distribution parameters were also determined. The histogram of the fracture toughness values shows the Weibull distribution (Fig. 12). Two parameters in the Weibull distribution of the fracture toughness are given in Table 3. From the Weibull distribution, the median value for the fracture toughness was determined as 2.32 MPa⋅m0.5, with a range from 1.3 to 4.0 MPa⋅m0.5.

Fig. 11
figure 11

Typical nanoindentation crack image under a cube corner tip at maximum load of 60 mN for determination of the fracture toughness. (a) 2D surface indent image; (b) 3D inverted image (dimension in μm)

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Fig. 12
figure 12

The Weibull plot of fracture toughness from nanoindentation of assorted sand grains in all sizes

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Table 3 Fracture toughness and Weibull parameters for Eglin sand grains
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FEM Simulations to Determine Constitutive Law

It is noted that the Young’s modulus and hardness values obtained from the nanoindentation follow the Weibull distribution. For each grain size, the P50 values in nanoindentation load–displacement were used for simulation. In this work, the maximum strain (up to which the stress–strain curve is valid) is limited by the strain produced by the nanoindentation test [49]. Examination of the indent impressions (for the Berkovich nanoindenter tip) obtained from NanoVision indicates that no cracks were formed in nanoindentation using the Berkovich tip. Thus, the continuum plasticity model is appropriate to model nanoindentation under the Berkovich tip. An inverse problem solving approach was used to determine the stress–strain relationship of sand at granular level by correlating the FEM simulated nanoindentation load–displacement data with the measured results. Figure 13 shows the simulation results obtained using the inverse problem solving approach on the nanoindentation P50 data for 0.3 mm sand grains. Similar simulation results were found for other sizes of sand grains. The results indicate that the simulation data agrees reasonably well with the experimental results. For the six sizes of samples, the values for yield stress and exponent of power law equation are about 4.5–7 GPa and 2.44–4.69, respectively, using P50 load–displacement curves. The model parameters are given in Table 4. These best-fit parameters can be used to determine the uniaxial stress–strain relationships for the Eglin sand grains under compression. These values, together with the measured Young’s modulus values can be used as the material parameters for the elastic–plastic constitutive model for Eglin sand grains.

Fig. 13
figure 13

Comparison of experimental and simulation results for 0.3 mm sand grain

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Table 4 Simulation parameters of Eglin sand grains
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Discussion

Sand Crystal Structure

The method developed by Oliver and Pharr [28], in general, is not suitable for materials with anisotropic properties [53]. However, the XRD results (Fig. 6) show that the Eglin sand grains have the same characteristic feature as single crystal α-quartz, which has a rhombohedral structure. To investigate the effect of anisotropy on indentation, a series of nanoindentation experiments were conducted on single crystal quartz along different crystal orientations. The single crystal α-quartz samples (Boston Piezo-Optics Inc., MA, USA.) were indented on Z-cut plane [(0001) direction] and X-cut plane [(\( 2\overline{1}10 \)) plane]. The Young’s moduli, if assuming isotropy and using Oliver and Pharr’s analysis, are determined as 331 GPa on Z-cut surface, and 252 GPa on the X-cut surface. They are significantly higher than the elastic properties of the quartz (103 GPa for the Z-cut surface and 79 GPa for the X-cut surface, respectively) [54]. The results on single crystal quartz also confirmed that the Oliver and Pharr approach does not work for single crystal quartz. The Young’s moduli are also significantly higher than that of Eglin sand grains (with an average value of 90.4 GPa for assorted sand grains). These results indicate that the individual Eglin sand grains are polycrystalline with polycrystalline grain sizes much smaller than the indent size so that nanoindentation will give the effective-isotropic properties of an ensemble of polycrystalline grains. The effective behavior of an ensemble of polycrystalline grains is then considered isotropic, hence the Oliver and Pharr approach is still valid to determine the Young’s modulus of individual Eglin sand grains.

Statistical Analysis of Modulus and Hardness

It is noted that the Young’s modulus values for Eglin sand grains are not coincident with those in different crystallographic orientations of single crystal quartz. From XRD results (Fig. 6), differences among the diffraction patterns indicate differences in structures of sand grains of various sizes. The Young’s moduli of X-cut, Y-cut and Z-cut of single crystal quartz reported are 79 GPa, 79 GPa and 103 GPa respectively [54]. The nanoindentation on different 0.6 mm grains shows 97.4 GPa modulus, indicating orientation independent properties. It is noted that the amorphous layer of sand grains should have Young’s modulus around 70 GPa (fused silica). The amorphous layer constitutes a higher volume fraction in small Eglin sand grains than in larger ones.

A possible reason for the small modulus observed in small grains is the effect of underfill epoxy. To examine this, we conducted FEM simulations for a small sand grain (0.15 mm) under nanoindentation with and without an epoxy substrate. The results indicate that the epoxy effect is negligible. However, after polishing, some sand grains might become very thin, especially for smaller grains, because a portion of the grain was removed during the polishing. In this case, the underfill epoxy could affect the nanoindentation results on a few small sand grains. As shown in Fig. 10, for smaller sand grains there are several small Young’s modulus values (less than 40 GPa) which could be induced by the effect of the epoxy layer, with a Young’s modulus on the order of ~3 GPa, close to epoxy’s modulus.

Further, the material constituents and defects can contribute to differences in properties. To ensure the reliability of statistical calculation, a large number of nanoindentation tests (more than 100 for each sand size) have been conducted. Combining the nanoindentation results of all sand grains with all six sizes together, the Weibull distributions for Young’s moduli and hardness for assorted Eglin sand are determined, as shown in Figs. 14 and 15, respectively. The Weibull parameters (the shape k, and the scale λ) are also given in Tables 1 and 2, with 95 % confidence intervals for the assorted sand grains. From Figs. 14 and 15, the overall Young’s modulus for the Eglin sand grains are found to be 90.4 GPa (with a range from 33.4 to 119.8 GPa), hardness to be 12.8 GPa (range 4.2–20.3 GPa).

Fig. 14
figure 14

The Weibull plot of Young’s modulus from nanoindentation for assorted sand grains of all sizes

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Fig. 15
figure 15

The Weibull plot of hardness from nanoindentation for assorted sand grains of all sizes

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Parameter Sensitivity in Simulation

Although the nanoindentation methods used for extracting Young’s modulus have been well established [55–61], the constitutive law of materials cannot be directly determined from nanoindentation data. To determine the elastic–plastic properties, von Mises yield stress was used along with isotropic hardening to simulate the deformation characteristics of an individual sand grain. The plastic behavior under compression was assumed to follow the Ramberg-Osgood relationship [13]. In simulations, by changing the yield stress values for different grain size of the samples and comparing the simulation data with the experimental results until a best agreement reached, the two unknown material parameters, n and σ y were adjusted until reaching the best-fitting. Several parameters (Young’s modulus, hardness and yield stress) were adjusted to fit the simulated load–displacement curves with the experiment data. As shown in equation (10), the power law index n, is related with Young’s modulus, hardness and yield stress. In order to determine the parameter sensitivity of the simulation model, several sets of parameters were assumed and the corresponding simulations were conducted.

The Young’s modulus values were changed by 2–10 % of the experimental results while other parameters were fixed at first. The corresponding load–displacement curves did not change very much, except for 2–10 % of change at initial unloading portion. However, the corresponding stress–strain curves changed by 24 % at 40 % compressive strain level. Next, when the hardness values changed by 2–10 %, the stress–strain curves calculated changed by almost 36 % at 40 % compressive strain level while the load–displacement curves changed 2.3 % at the initial unloading portion, which indicate that the nanoindentation simulation model is very sensitive to Young’s modulus and hardness. However, nanoindentation can measure both Young’s modulus and hardness very accurately using the Oliver-Pharr approach, typically with 95 % of accuracy, so that the effects of Young’s modulus and hardness can be minimized. At last, in addition to Young’s modulus and hardness, the effect of yield stress was analyzed as well. If the yield stress values changed by 20 % while other parameters remained the same, the stress–strain curves changed by 5 % at 40 % compressive strain level, indicating that the model is not very sensitive to yield stress. In this case, the Young’s modulus and hardness value adjustment is limited to within 5 % of the experimental results and the yield stress is the main parameter for fitting the model.

Conclusions

In order to assess the granular level mechanical behavior of sand, nanoindentation was conducted on individual sand grains in different sizes. Mechanical properties of six main sizes of sand grains, namely in 0.6 mm, 0.5 mm, 0.425 mm, 0.3 mm, 0.212 mm and 0.15 mm, were measured to investigate the particle size effect. The mechanical properties of different size sands, Young’s modulus, 0.2 % offset yield strength, hardness, stress–strain relationship and fracture toughness were determined from more than seven hundreds nanoindentation experiments. The properties of sand at the granular level have a wide variation, and parameters of the Weibull distribution were determined. Larger sand grains were found in general to have higher Young’s modulus and hardness. From statistical aspects, the overall median Young’s modulus for the Eglin sand grains was found to be ~90.4 GPa (range 33.4–119.8 GPa), the median hardness to be 12.8 GPa (range 4.2–20.3 GPa) and the median fracture toughness to be ~2.32 MPa⋅m0.5 (range 1.3–4.0 MPa⋅m0.5). The maximum Young’s modulus and hardness were attained at the median grain size of 0.425 mm, indicating that the nature has optimized the sand grain size by allowing the sand grains with the highest mechanical properties to remain. The Ramberg-Osgood power-law relationship was used to describe the homogenous and isotropic stress–strain behavior for sand at the granular level through FEM simulation of the nanoindentation, and the material parameters were determined. These data can be used for mesoscale simulations of sand with the Weibull distributions through such methods as the stochastic mechanics in the future.