1 Introduction

The mechanical properties of sandy soils depend on the characteristics of the individual sand particles that compose them. Sand particles tend to fragment into smaller particles even at low-stress level [4, 8,9,10,11, 35, 40], generating soils with different gradation and mechanical responses. Therefore, to understand how soil properties change due to crushing, it is essential to quantify changes in gradation and particle size produced by loading in laboratory experiments. Sand particle crushability depends on particle mineralogy, size, shape, tensile strength, initial gradation, and stress state [11, 35, 40, 51, 58]. Both silica and carbonate sand particles undergo crushing when the stresses imposed on the particles are greater than the particle crushing strength [4, 11, 13, 31, 35, 37, 42, 43, 62]. From a macroscopic point of view, particle crushing causes shifting and broadening of the original particle-size distribution curve [11, 17, 35]. At particle scale level, different modes of particle crushing have been observed [13, 42, 58]:

  1. 1.

    minor abrasion or breakage of one asperity;

  2. 2.

    chipping [58], attrition [13], or mixed crushing modes [58], consisting of the breakage of several asperities without significant change of particle shape;

  3. 3.

    particle splitting into two or more particles of similar size.

Recent research shows that, in addition to the stress state, the size and shape (morphology) of sand particles are key factors that control the crushing mode [5, 39, 51, 58]. For example, Wang and Coop [58] observed that angular particles tend to crush at the asperities. In contrast, rounded particles tend to split into smaller particles (as compared to the original particle) of similar size. Nakata et al. [42] and Sohn et al. [51] indicated that, although small particles tend to be stronger than large particles because they have fewer internal flaws, large particles undergo less crushing than small particles in compressive loading when the sand is well-graded due to a greater number of inter-particle contacts between it and the surrounding particles, leading to less stress concentration at the surface of the large particles [55].

Various types of laboratory tests (e.g., triaxial, direct shear, one-dimensional compression, ring shear, impact loading, and compaction tests) have been used to study crushing [5, 12, 11, 15, 17, 21, 25, 27, 29, 34, 35, 41, 42, 51, 57, 60]. However, particle crushing produced by a pile driven or jacked in sand has been investigated only in a few experimental studies [6, 30, 34, 50, 54, 59, 62]. Pile driving or jacking in sand creates complex stress and strain fields below the pile base, and crushing of a soil element under these conditions cannot be easily studied using standard element tests. In addition, recovering samples for quantification of crushing around full-scale field piles, especially at great depths, would not be feasible; hence, calibration chamber testing is the most suitable and economical alternative to studying crushing produced under conditions associated with great depths. Additionally, it is possible to recover disturbed samples from calibration chamber samples for crushing assessment and quantification with conventional techniques.

The degree of crushing a sand experiences can be quantified by assessing changes in its particle-size distribution with loading and calculating a crushing index or a breakage parameter [14, 24, 31, 32, 36, 61]. Quantification of particle crushing produced by penetration or jacking of piles is important (1) for the development of constitutive models that consider crushing in their formulation and that can be used to simulate more accurately pile response to loading and (2) to improve pile design methods using such simulations [8, 9, 13, 28, 44, 45, 53, 63,64,65].

In this paper, we investigate particle crushing produced by jacking and loading of a model pile in dense and medium-dense silica sand samples prepared by air pluviation in a half-cylindrical calibration chamber with imaging capabilities. The effects of the number of jacking strokes used to install the model pile, the relative density of the sand, and the surcharge applied on the sand samples are considered. In addition to the detailed characterization of the original sand, particle-size distribution curves and breakage and morphology parameters were determined from sand samples recovered along the shaft and base of the model pile after testing. Relationships between the measured base resistance and three breakage parameters are proposed. The displacement of individual soil particles in the path of the model pile during loading was tracked using digital images to better understand the displacement patterns in the sand domain next to the model pile. The experimental results are discussed considering the effects of the sand relative density, particle morphology, and mode of particle crushing on the degree of crushing produced by model pile installation and loading in uniform silica sand.

2 Materials and methods

2.1 Test equipment

The model pile experiments were carried out in a half-cylindrical steel calibration chamber with three observations windows made of transparent polymethyl methacrylate (PMMA) reinforced with a steel frame. Digital images of the sand domain were captured during installation and loading of the half-cylindrical model pile. Table 1 summarizes the dimensions and capabilities of the calibration chamber testing facility. Further details can be found in [7, 6, 18, 52, 54].

Table 1 Details of the half-cylindrical calibration chamber with image capabilities
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Figure 1 shows the brass model pile used in the experiments. It consists of an 800-mm-long, half-circular rod with a diameter B = 38.1 mm, to which a half-section conical tip with a 60°-apex angle is attached. A specially designed tension/compression load cell (Model WMC, Interface Inc., Scottsdale, Arizona) with a rated capacity of 42 kN was used to measure the load at the pile head. Four electrical-resistance strain gages (Kyowa Americas Inc. model KFG-2-350-C1-23) installed diametrically opposed to each other in a cylindrical brass rod (load-transfer bar) with a diameter of 12.7 mm were used to measure the axial deformation of the load-transfer bar and to obtain the load at the base of the model pile (see Fig. 1).

Fig. 1
figure 1

Schematic of the model pile used in the experiments

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2.2 Test sand

The test sand is a poorly graded silica sand known as Ohio Gold Frac (SP according to [1]) with sub-angular to sub-rounded particles of high sphericity (see Fig. 2). The sand has a mean particle size D50 of 0.62 mm. Further details on the Ohio Gold Frac sand are provided in [23].

Fig. 2
figure 2

Ohio Gold Frac sand particles

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3 Experimental program

3.1 Test matrix

Twelve tests were carried out in the calibration chamber (see Table 2). The tests in Table 2 are identified by a code that specifies the sand density (D denotes “dense,” and MD denotes “medium dense”), the installation type (M denotes “monotonic,” MS denotes “multi-stroke”), the number of jacking strokes in the installation (the number after the acronym MS), and finally the test number (from 1 to 12). Out of these twelve tests, ten were stopped 1B (38.1 mm) above the target installation depth h* (= 408.1 mm) to perform a static load test (for a pile head displacement w of 1B) and to locate the digital cameras closer to the observation windows for imaging of individual sand particles near the base the model piles.

Table 2 Test program
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Tests D-M-T2 and MD-M-T6 were performed in dense and medium-dense sand to obtain the unit base resistances at the target installation depth (= 408.1 mm); these resistances were compared with the base resistances measured in the other tests in which the model pile was installed using multiple jacking strokes and then tested statically in compression.

3.2 Test procedure

The sand samples were prepared by air pluviation using a large pluviator [33] positioned at the top of the calibration chamber. The target relative density of the sample (dense or medium dense) was obtained by controlling the sand flow during pluviation through a removable diffuser sieve below the pluviator [6, 18, 52, 54].

The model pile was pushed in the sand sample to an initial depth of 50 mm (without the surcharge at the top of the sample) to prevent measurement of loads that may be affected by stress non-uniformity at the top of the sample where the top bladder pressure is applied. With this initial model pile embedment, a check was made to ensure perfect contact between the flat side of the pile and the glass on the observation windows. Then, the model pile was unloaded, and the surcharge was gradually applied on the top of the sample using an air-rubber bladder placed between the sand surface and a reaction steel lid bolted at the top of the chamber. The initial vertical pressure in the sand at the final installation depth was measured in different tests (not reported in this paper) using a pressure cell (Model No. 4800, GEOKON, New Hampshire, USA) placed in the sand sample. The measurements of the pressure cell indicated that the tests reported in this research were performed under an initial vertical effective stress at the final installation depth of the model piles of 17, 33, and 57 kPa (see Table 3). The installation depth indicated in Table 2 was achieved using either monotonic or multi-stroke jacking. The jacking rate was 1.0 mm/s for all the tests. When the pile was jacked using multiple strokes, there was a pause of 5 s after each stroke before unloading the head of the pile (the load at the top load cell was close to zero). Then, a new pause of 2 s preceded the next stroke. This procedure was repeated until the pile reached the desired installation depth. In those tests in which a static load test was performed (see Table 2), the pile head was disconnected from the jacking system to simulate the end of the installation. Then, the pile and the jacking system were connected again, and the model pile was pushed into the sand for an additional distance equal to 1.0B (= 38.1 mm) at a reduced speed of 0.1 mm/s. In this last stage, the cameras were placed in front of the chamber to take digital images of a region in the sand domain of 70 mm × 80 mm. These images were used for tracking individual particles. Finally, sand samples from a 3-to-4 mm-thick annular zone around the shaft of the model pile were carefully recovered by applying water to the sand after testing as the sand was removed from the calibration chamber [62].

Table 3 Values of qbL from static load tests performed after pile installation
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4 Experimental results

Careful examination of the digital images taken during installation and loading shows that crushing of particles occurs below the base [54] as the model pile penetrates deeper in the sand sample. The digital images and videos also show that particle crushing is negligible along the shaft of the model pile as installation proceeds. This is because the stresses normal to the smooth surface of the model pile shaft (as back-calculated from shaft resistance) are not large enough to produce crushing of the sand particles.

4.1 Base resistance

Figure 3 shows unit base resistance qb versus installation depth h* for all the tests performed. In Fig. 3, for those tests in which the model pile was installed with multiple jacking strokes, the readings taken during pauses in loading or during unloading between jacking strokes are not shown to facilitate visualization.

Fig. 3
figure 3

Unit base resistance qb mobilized in dense and medium-dense sand samples with different surcharges: a during pile installation by jacking, b during static load tests after installation

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The results in Fig. 3 show that, for tests D-M-T2 and MD-M-T6, which were performed for an initial effective stress level at the pile base \(\sigma_{0}^{\prime }\) of 33 kPa and with the model pile installed to a final depth of 408 mm, an increase in the sand relative density DR from ≈ 63% to ≈ 90% produces an increase in the unit base resistance qb from ≈ 5.5 MPa to ≈ 13.7 MPa (at h* = 408 mm = 10.70 B). The unit base resistance qb of tests D-M-T2 and MD-M-T6 at an installation depth h* of 370 mm (= 9.71 B), the depth to which the model pile was installed for the remaining tests, is 13.6 MPa and 5.3 MPa in dense and medium-dense samples, respectively. Figure 3 shows that the unit base resistance qb at a depth of 370 mm for the tests performed with \(\sigma_{0}^{\prime }\) equal to 33 kPa and with different numbers of jacking strokes (1, 16, and 32) is 13.3 in dense sand and 5.1 MPa in medium-dense sand. The qb values at a depth of 370 mm for the tests performed with \(\sigma_{0}^{\prime }\) equal to 17 kPa is 10.6 MPa and 2.7 MPa in dense and medium-dense sand, respectively, while the corresponding qb values for the tests performed with \(\sigma_{0}^{\prime }\) equal to 57 kPa are 18.3 MPa and 6.4 MPa.

The results of all the static load tests (SLT) performed after model pile installation (see Table 3) showed that the limit unit base resistance qbL is mobilized right after a model pile head displacement w of 4 mm (≈ 0.1 B). Table 3 provides the qbL values measured in the SLTs at large pile head displacements (w = 38 mm). For the tests performed in dense sand with σ′0 equal to 33 kPa, the results in Table 3 show that, when the model pile was installed with 16 or 32 jacking strokes, the measured qbL values differ by less than 2% with respect to the value mobilized by the monotonically jacked model pile. In medium-dense sand, when the model pile was installed with 32 and 16 jacking strokes, qbL is 7.1% and 9.2% greater than the qbL measured for monotonic installation. A higher qbL value for the model pile installed in medium-dense sand with 16 jacking strokes compared with the qbL for the model pile installed with 32 jacking strokes might be related to the method of sample preparation (medium-dense sand samples are not as uniform as dense sand samples). In medium-dense sand, changes in qbL values of ± 5% from test to test are typical.

The results in Table 3 suggest that, during the installation of a model pile in medium-dense sand, the jacking strokes may increase the density of the sand below the base, producing a slight increase in qbL. Nevertheless, differences smaller than 9% in the measured value of qbL in medium-dense samples are small. Figure 3b and Table 3 show that qbL at the end of installation (h* = 370 mm) is on average 2.2% and 9.8% smaller than at the end of the static load tests (h* = 406 mm) in dense and medium-dense sand, respectively.

4.2 Particle-size distribution after pile testing

Figure 4 shows images of the sand in the neighborhood of the model pile after installation in a typical test performed in a dense sand sample. In dense sand samples, particle crushing is more intense than in medium-dense sand samples. Crushed particles can be easily identified through the observation windows of the calibration chamber and in the digital images by the characteristic change in color of the crushed particles, which contrasts with the color of the original sand (dark color for uncrushed, light color for crushed particles). The average thickness (directly measured on the observation windows and in the digital images) of the annular zone around the model pile where the crushed particles accumulate along the shaft is ≈ 3.0 mm.

Fig. 4
figure 4

View of the model pile and particles crushed during installation of the model piles in dense sand with \(\sigma_{0}^{\prime }\) equal to 33 kPa

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To recover samples of crushed sand from the ≈ 3 mm-thick annular zone around the shaft of the model pile after testing (postmortem sampling), water was carefully dropped on top of the sample first and then, as sand was excavated from the chamber, water was also sprayed around the pile shaft. The annular samples with the desired dimensions were collected using curved and sharp tools. The average length of these samples was about 80 mm, which provided sufficient sand for sieve analyses [2]. The samples were air-dried and sieved to obtain the particle-size distribution curves. After sieving, the gradation of the particles smaller than 0.075 mm (passing the sieve # 200) was obtained from sedimentation analyses [3].

The particle-size distribution curves tend toward a self-similar distribution (approximately parallel) with an increase in the degree of particle crushing [14, 26, 37, 38, 56]. An ultimate particle-size distribution curve would result if a sand sample were subjected to large mean stresses and shear strains [14, 42, 49]. This ultimate particle-size distribution [14, 56] can be described using a fractal law:

$$F_{{\text{u}}} (d) = \left( {\frac{d}{{d_{{\text{M}}} }}} \right)^{{3 - D_{{\text{f}}} }}$$
(1)

where Fu(d) is the ultimate particle-size distribution (cumulative mass of particles with size finer than d), d is the current particle size, dM is the largest particle size, and Df is the fractal dimension [14]. Df can be back-calculated from Eq. (1) by plotting the results of the sieve analyses in a log–log space in which the slope m of the line in this space is equal to (3 − Df). A fractal dimension of 2.6 was selected as the fractal dimension corresponding to the ultimate particle-size distribution of the Ohio Gold Frac sand; this value is in the range of values suggested for sand by Samis et al. [49], Einav [14], and Buscarnera and Einav [9]. The ultimate particle-size distribution curve is used to calculate the breakage parameters proposed by Einav [14] and Xiao et al. [61].

The particle-size distribution curves of the sand samples collected at different depths for all the tests performed are shown in Fig. 5, h is the vertical distance from the base of the model pile to the center of the samples of crushed sand (≈ 3 mm-thick and 80 mm high) collected at different depths along the model pile shaft after testing, and rp is the model pile radius. As shown in Fig. 5, the particle-size distribution curves shift to the left and broaden because of particle crushing. Also, the degree of crushing decreases as the vertical distance h from the pile base increases because crushing happens below the model pile base. The changes in gradation are greater for the samples recovered from the tests performed in dense sand than in medium-dense sand. Based on the results in Fig. 5 and the unit base resistance measurements, when the stresses at the base of the model pile are greater than approximately 3 MPa, Ohio Gold Frac sand particles crush.

Fig. 5
figure 5figure 5

Particle-size distribution curves of samples from a 3 mm-thick annular zone around the shaft and the base of the jacked model piles after installation and loading for: a monotonic installation in dense sand with \(\sigma_{0}^{\prime }\) = 33 kPa; b multi-stroke installation (32 strokes) in dense sand with \(\sigma_{0}^{\prime }\) = 33 kPa; c multi-stroke installation (16 strokes) in dense sand with \(\sigma_{0}^{\prime }\) = 33 kPa; d monotonic installation in medium-dense with \(\sigma_{0}^{\prime }\) = 33 kPa; e multi-stroke installation (32 strokes) in medium-dense sand with \(\sigma_{0}^{\prime }\) = 33 kPa; f multi-stroke installation (16 strokes) in medium-dense sand with \(\sigma_{0}^{\prime }\) = 33 kPa; g, h model pile installed monotonically in dense and medium-dense sand with \(\sigma_{0}^{\prime }\) = 17 kPa; i, j model pile installed in dense and medium-dense sand with \(\sigma_{0}^{\prime }\) = 57 kPa

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Table 4 presents the measured values of the coefficient of uniformity Cu, the coefficient of curvature Cc, and the mean particle size D50 of the crushed sand samples recovered after testing. Cu and Cc increase as the distance from the samples to the pile base decreases, indicating that the particle-size distribution is less uniform near the pile base than elsewhere. This trend is more evident in dense sand (Cu increases from 1.46 to 10.41 and Cc from 0.93 to 3.34) than in medium-dense sand (Cu increases from 1.46 to 2.08 and Cc from 0.93 to 1.25). The mean particle size D50 of the crushed samples varies between 0.46 and 0.59 mm in dense sand and between 0.56 and 0.60 mm in medium-dense sand.

Table 4 Gradation parameters of the crushed samples recovered after testing
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4.3 Morphology of particles

Particle morphology influences the mode in which particles tend to crush [42, 51, 58]. To obtain the morphology parameters for the Ohio Gold Frac sand particles and how these changed (due to crushing) after testing, 25 particles retained in each one of the sieves used to construct the particle-size distribution curves presented in Fig. 5 were randomly selected and observed with an optical microscope. A charge-coupled camera attached to the microscope was then used to capture digital images of the particles (see Fig. 6). The digital images captured have a resolution of 8.0 megapixels. To guarantee the accuracy of the results, different combinations of lenses were used to capture images in which each particle is composed of at least 150 pixels per circumscribed circle diameter [66]. The digital images were processed with the MATLAB ® code developed by Zheng and Hryciw [66] and the open source software ImageJ ® [16] to obtain the roundness R, the convexity CX, the aspect ratio AR, and the five most used definitions of sphericity: area sphericity SA, diameter sphericity SD, circle ratio sphericity SC, perimeter sphericity SP, and width-to-length ratio sphericity SWL [66]. Definitions of these parameters are presented in Table 5. The morphology parameters of particles passing the sieve No. 100 (< 0.15 mm) were not calculated due to lack of resolution of the digital images.

Fig. 6
figure 6

Digital images of a set of crushed particles retained on different sieves as observed in an optical microscope: a sieve #20 (mesh opening size = 0.841 mm), b sieve #40 (mesh opening size = 0.420 mm), c sieve #60 (mesh opening size = 0.250 mm), d sieve #80 (mesh opening size = 0.180 mm), and e sieve #100 (mesh opening size = 0.150 mm)

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Table 5 Morphology parameters of particles before and after testing (based on the definitions of Zheng and Hryciw [66] and Xiao et al. [60])
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The particles used in the morphology analyses were selected from different tests (with \(\sigma_{0}^{\prime }\) = 33 kPa) and from samples recovered at positions closer to the base of the model pile (h/rp < 4.2), where greater changes in the particle-size distribution curves occurred. Crushed particles recovered after testing in both dense and medium-dense tests were analyzed.

Table 5 summarizes the morphological parameters (for an average of 25 particles per sieve) of untested (original) and tested (crushed) sand particles. A comparison of the average values of the morphological parameters is presented in Fig. 7.

Fig. 7
figure 7

Mean values of morphological parameters of crushed and uncrushed samples of Ohio Gold Frac sand (R = roundness, CX = convexity, AR = aspect ratio, SA = area sphericity, SD = diameter sphericity, SC = circle ratio sphericity, SP = perimeter sphericity, SWL = width-to-length ratio sphericity)

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Figure 7 shows that the roundness of particles increases from 0.51 (original sand) to 0.60 and 0.65 for crushed samples of medium-dense and dense sand, respectively. This indicates that the surface of the sand particles becomes less angular as a result of crushing. Furthermore, crushed particles from dense sand samples are more rounded (R = 0.65) than particles from medium-dense sand samples (R = 0.60), confirming that dense sand undergoes more crushing than medium-dense sand under the same test conditions. Also, the dominant mode of crushing is the breakdown of particle surface irregularities and sharp edges (attrition or chipping) rather than particle splitting in several fragments (particle sizes reduce only slightly after testing). This is in agreement with the particle-size distribution curves (see Table 4), which showed a maximum reduction of 20.7% in the value of D50 for dense sand and 9.6% for medium-dense sand.

As shown in Fig. 7, the convexity parameter Cx has similar values for samples collected before and after testing, indicating that Cx is not greatly influenced by particle breakage, as also observed by [60]. Moreover, the aspect ratio AR (shortest Feret diameter Fmin divided by the longest Feret diameter Fmax) of the particles obtained from dense sand samples is greater for the crushed sand than for the original sand. A greater AR after the particles undergo crushing may suggest that particle breakage occurs mainly in the direction of the longest Feret diameter, reducing the value of Fmax and thus, increasing AR. The results also show that, on average, the AR of the sand particles takes the same value before and after testing when analyzing medium-dense sand samples. The results of CX and AR for dense and medium-dense samples are in accordance with the dominant mode of crushing explained in the previous paragraph.

Figure 7 also shows that all five sphericity parameters of the crushed sand increase (with respect to the parameters of the original sand) between 2.2 and 11.7% for dense sand and between 0 and 3.3% for medium-dense sand. Among these parameters, the area sphericity SA is the parameter that increases the most for crushed particles from the dense sand tests (from 0.68 to 0.76) and the medium-dense sand tests (from 0.68 to 0.70). These results suggest that the changes in particle shape are small and confirm previous observations indicating that crushing affects mainly the surface of the particles (surface edges).

4.4 Breakage parameter

Several parameters have been proposed in the literature that can be used to quantify the change in particle size and gradation due to crushing [14, 24, 31, 32, 36, 41]. Figure 8a, b shows the definitions of the relative breakage parameter Br proposed by Hardin [24] and the breakage parameter B* proposed by Einav [14], respectively. Einav [14] proposed that an ultimate particle-size distribution be used as reference to define the boundary of the breakage potential Bp (the maximum breakage state that the soil can achieve), while Hardin [24] proposed a cutoff boundary at a particle size equal to silt-size threshold (0.074 mm). Xiao et al. [61] analyzed the Einav [14] and Hardin [24] breakage parameters and proposed a new breakage parameter B*N that addresses some of the issues with the previous expressions proposed to quantify crushing. B*N is calculated as follows:

$$B_{N}^{*} (\% ) = \frac{{\left( {\sum\nolimits_{k = 0}^{n} {p_{k}^{c} } - \sum\nolimits_{k = 0}^{n} {p_{k}^{i} } } \right)}}{{\left( {\sum\nolimits_{k = 0}^{n} {p_{k}^{u} } - \sum\nolimits_{k = 0}^{n} {p_{k}^{i} } } \right)}}$$
(2)

where p indicates the cumulative passing-sieve percentage for each superscript state i (initial), c (current), and u (ultimate). The index can be calculated by replacing the superscript “*” in the B*N parameter with either H or E. In the case of BHN, the breakage parameter is calculated based on Hardin’s Br, thus disregarding particle sizes smaller than 0.074 mm. In the case of BEN (see Fig. 8c), the breakage parameter is calculated based on Einav’s ultimate distribution. This study uses BEN since the fractal dimension (equal to 2.6) needed to calculate Einav’s [14] ultimate distribution was obtained using Eq. (1).

Fig. 8
figure 8

Breakage parameter definitions: a relative breakage Br [24], b breakage parameter B* [14], and c new breakage parameter BEN [61]

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Table 6 summarizes the calculated Br, B*, and BEN values of the crushed sand samples whose particle-size distribution curves are presented in Fig. 5. From Table 6, it is observed that the definition of the breakage potential leads to values of B* that are on average 1.18 times the values of Br (coefficient of correlation R2 = 0.98). The new breakage index BEN has a strong correlation with B* (R2 = 0.99) and Br (R2 = 0.98), with BEN being on an average 1.06 times smaller than Br and 1.26 times smaller than B*. The stronger correlation with B* is probably due to the fact that these parameters share the use of the same ultimate distribution. Furthermore, values of Br, B*, and BEN increase as the density of the sand increases and the vertical distance h from the position at which the sample was recovered to the pile base decreases. The maximum values of the breakage parameter are for the samples recovered closer to the base of the model pile after testing and from tests carried out with \(\sigma_{0}^{\prime }\) = 57 kPa in dense sand (Br = 32.44%, B* = 41.74%, and BEN = 40.77%). The minimum value of the breakage parameters are for the samples recovered from the test carried out in medium-dense sand and \(\sigma_{0}^{\prime }\) = 17 kPa (Br = 5.44%, B* = 5.68%, and BEN = 3.87%). Since the stress level developed at the base of the model pile is linked to qb, which is a function of the relative density of the sand and the surcharge used [47], these results suggest that crushing of particles, quantified through breakage parameters, can be related to the cone or base resistance mobilized in the model pile tests.

Table 6 Breakage parameters of the crushed sand samples obtained in the tests
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4.5 Base resistance and breakage parameter

Yang et al. [62] indicated that, during model pile installation in a cylindrical calibration chamber, crushing of silica sand particles (NE34 Fontainebleau) starts below the base of the model piles when the cone resistance qc is around 5 MPa, suggesting a relationship between qc and the crushing of particles. At the particle level, the stress required to break or crush a particle can be much higher. The average fracture stress σf,avg of individual silica sand particles is greater than 5 MPa [19] and develops at inter-particle contacts in a loading process. Ganju [19] found an average fracture stress σf,avg of 75.2 MPa for individual Ohio Gold Frac sand particles. In this research, the normalized position h/rp of every sample of crushed sand obtained along the shaft of the model piles is known (measured from the pile base to the center of the sample). Moreover, the value of the unit base resistance qb was recorded continuously during the installation and loading of the model pile. Thus, it is possible to relate the breakage parameters of the crushed sand with the value of the limit unit base resistance qbL measured at depths where a plateau in the curve qb versus h* is observed (see Fig. 3). Figure 9a–c plots the breakage parameters Br, B*, and BEN versus the limit unit base resistance qbL normalized by the reference stress pA. Figure 9 shows that the degree of crushing of the sand, as quantified by Br, B*, and BEN increases with increasing qbL.

Fig. 9
figure 9

Breakage parameters of sand samples next to the model pile as a function of the measured limit unit base resistance normalized by the reference stress pA: a qbL/pA versus Br, b qbL/pA versus B*, and c qbL/pA versus BEN

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Based on the data in Fig. 9, the relationships between Br, B*, or BEN and the normalized limit unit base resistance qbL/pA for Ohio Gold Frac can be written as:

$$B_{{\text{r}}} (\% ) = 0.1725 \times \left( {\frac{{q_{{{\text{bL}}}} }}{{p_{{\text{A}}} }}} \right)^{0.9936}$$
(3)
$$B^{*}(\% ) = 0.1138 \times \left( {\frac{{q_{{{\text{bL}}}} }}{{p_{{\text{A}}} }}} \right)^{1.1219}$$
(4)
$$B_{N}^{E} (\% ) = 0.0382 \times \left( {\frac{{q_{{{\text{bL}}}} }}{{p_{{\text{A}}} }}} \right)^{1.3208}$$
(5)

where pA is the reference stress (= 100 kPa = 2000 psf ≈ 1 tsf) and qbL has the same units as pA.

Based on the experimental results presented in Fig. 9, Eqs. (3), (4), and (5) predict breakage parameters within ± 5% for the observed range of base resistance, 0 ≤ qb ≤ 18.5 MPa. Equations (3), (4), and (5) are valid for Ohio Gold Frac sand and can be used as an approximation for other sands with similar crushability.

5 Image analyses

5.1 Displacement paths of particles below the pile base

To observe and track individual particles of Ohio Gold Frac sand (D50 = 0.62 mm) in the half-cylindrical calibration chamber, a 12-megapixel camera (Basler beat beA400-62 km) was located close to the observation window before the static load tests. The camera can take images of an area of 70 mm × 80 mm in the sand domain near the base of the model pile (at h* = 370 mm). The rate of capture of the digital images during the SLTs was twenty pictures per second (1 picture per each 0.005 mm of pile settlement). Videos built from the digital images were used to track particles using the free software Tracker (Open Source Physics ®). Figure 10 shows a typical picture and the resulting displacement paths of 50 particles next to the inclined faces of the conical base after a pile base displacement of 36 mm (in the static load test).

Fig. 10
figure 10

Displacement paths of individual sand particles after a pile base displacement of 36 mm (in the static load test)

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Figure 11 shows the normalized radial and vertical displacements u/rp and v/rp of four sand particles (A, B, C, and D) located at different radial positions below the model pile base in dense (D-MS16-T4) and medium-dense sand (MD-MS16-T8) samples. The values of u/rp and v/rp are plotted against the normalized pile base displacement wb/rp as the model pile penetrates 36 mm (0.94 B = 1.88 rp) in the sand sample (from h* = 370 mm to h* = 406 mm). Figure 11 shows that the vertical displacement dominates for particles closer to the pile centerline. It also shows that the vertical displacement of particles located further away from the pile centerline (e.g., particle D located at r > 1.0 rp) decreases with respect to the vertical displacement of particles closer to the pile centerline and that the horizontal displacement dominates. The displacement of the particle closer to the pile centerline (particle A) could not be tracked after a pile base displacement of ≈ 1.1rp because white crushed particles from it and other neighboring particles covered its surface, making it impossible to continue tracking its position. The mixture of crushed, uncrushed, and partially crushed particles covered by the smallest white particles form a band of soil that moves parallel to the conical faces of the pile base and end their movement next to the shaft at horizontal distances less than about 3 mm measured radially from the pile shaft. Additional measurements not presented in Fig. 11 showed that, at radial distances greater than r = 27.3 mm (1.43rp), the normalized displacements u/rp and v/rp of the particles are less than 0.05 (u and v are less than 1.0 mm).

Fig. 11
figure 11

Cumulative normalized radial and vertical displacement of particles below the pile base for a pile base displacement wb of 36 mm: a particles in dense sand and b particles in medium-dense sand

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Figure 12 shows the displacement paths of 50 particles in dense and medium-dense sand samples relative to the initial position of the pile base (only the relative movement of the particles with respect to the pile base is plotted). The results in Fig. 12 show that particles below the pile centerline move vertically until they reach the sharp edge of the conical base. Then, their movement aligns with the direction of either of the conical faces of the base (which make a 60° with respect to a horizontal plane). In addition, particles located initially below the pile base (except those at the pile centerline) start changing the direction of their movement from a vertical path toward a path aligned with the conical faces of the base when the vertical position h (measured vertically from the tip of the conical base to the particle centroid) is about 5 mm (0.26 rp = 8 D50). As seen in Fig. 12, the inclination of the particle paths (measured from a horizontal plane) in dense and medium-dense sand samples increases as the radial distance from the particle to the pile centerline increases (from approximately 60° at the pile centerline to 81° at r = 1 rp). The inclination of the particle trajectories plotted in Fig. 12 increases at a rate of 1.1 degrees per millimeter (measured from the pile centerline). This indicates that particles initially located at r > 27.3 mm (r = 1.43 rp) or further away would follow a vertical path as the pile is pushed deeper into the sand. Furthermore, particles follow a vertical path as soon as they reach the level of the cone shoulder (because the particles barely move after the shoulder has passed its position).

Fig. 12
figure 12

Displacement paths of individual particles relative to the initial position of the pile base: a particles in dense sand and b particles in medium-dense sand

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Based on the measurements presented in Fig. 12, Fig. 13 shows a schematic representation of the displacement paths of individual particles around a penetrating model pile with a conical base in dense sand (left-hand side) and medium-dense sand (right-hand side). The streamlines in Fig. 13 for dense and medium-dense sand differ in their relative radial position with respect to the pile centerline by 10% or less. Figure 13 also shows the original position (0 < |r| < ≈ 15 mm = 0.78 rp) and path of particles that crushed below the base of the model pile and that, with continuous pile penetration, are located within the annular zone of crushed particles (about 3.0 mm thick) around the shaft at the end of penetration (the zone from which samples were recovered for sieve analyses).

Fig. 13
figure 13

Schematic representation of displacement paths of individual particles under continuous pile settlement in dense sand (left side from pile centerline) and medium-dense sand (right side from pile centerline)

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5.2 Crushing modes and zones of crushing below model piles in the DIC calibration chamber

From the digital images of the particles taken with a 12-megapixel camera (Basler beat beA400-62 km), a high-speed camera capturing up to 270 pictures per second (iSight, Apple, Cupertino, California), and a portable polarizing microscope (Dino-Lite AM7013MZT4 digital handheld microscope, Dunwell Tech, Torrance, California), it was determined that particles of Ohio Gold Frac sand crush mainly due to breakage of one or several asperities (the particles are not fragmented into several small pieces). Due to the speed at which these asperities break from the surface of the particles, the fractured sand fragments below the base of the model piles separate from the original sand particle very suddenly, with a visual effect that resembles the popping of popcorn in a closed container. This crushing mode was called “mixed mode” by Wang and Coop [58] and “attrition” by Daouadji et al. [13].

Figure 14 shows schematically the process of crushing (as observed from multiple digital images and videos) of the particles in the path of the model pile during penetration in dense sand. The original sand particles start to develop fissures on their surface as the model pile base approaches their location. This occurs as the base resistance reaches a value of about 3 MPa. The 2D digital images show that, as the distance between the base of the model pile and the particles decreases and the base resistance increases to values above 3 MPa, small fragments of sizes less than 10% of the original size of the sand particles break away from the particle surface and move into inter-particle voids. This mode of crushing occurs more frequently not along inter-particle contacts, but on the free surfaces of the particles (at particle-void interfaces). This observation suggests that the sand particles crush mostly due to tensile stresses, as also observed by McDowell et al. [38], McDowell and Bolton [37], and Nakata et al. [41], among others. The particles closer to the path of the model pile experience more attrition than the particles located at radial distances r > rp. It was observed, especially in dense sand samples, that attrition is very intense and that large sand particles become “cushioned” by small sand particles (see Fig. 14), with crushing of the smaller fragments occurring more frequently afterward. This explains why the value of D50 decreases by 25% or less after particles crush.

Fig. 14
figure 14

Schematic of the evolution of crushing during continuous model pile settlement

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As also observed by Yang et al. [62], the degree of crushing varies with the distance to the surface of the model pile. Figure 15 shows three crushing zones observed in a test performed in a dense sand sample (D-M-T1, DR = 89.9%) when the tip of the conical base was at a depth of 445 mm below the sand surface and qb was 13.6 MPa. The boundaries between zones were initially drawn using Otsu’s threshold clustering algorithm implemented in the software ImageJ ® [16] and then smoothed and connected manually based on the true colors of the pictures by using the software AutoCAD®.

Fig. 15
figure 15

Schematic delineation of zones of crushing for model pile installation in a dense sand sample (test D-M-T1, DR = 90.7%, \(\sigma_{0}^{\prime }\) = 33 kPa, h.* = 407 mm)

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In Fig. 15, zone I defines the area of crushed particles closer to the model pile shaft. It differs from the other two zones next to it by the presence of more fines (fragments of white color) around and on the surface of larger particles. The thickness of this zone ranges from 1.5 to 3.0 mm. Yang et al. [62] observed a very well-defined zone of gray color with a thickness ranging from 0.5 to 1.8 mm immediately next to the surface of their model pile after performing static and cyclic tests. However, in our experiments, we only observe a small variation in the color of the sand in contact with the pile without a clear boundary that could be measured.

Zone II contains fewer white particles than zone I (suggesting a smaller content of fine particles) and has a thickness ranging from 2.5 to 5.0 mm. In contrast to zone I, in this zone, particles of size similar to that of uncrushed sand can be identified in the digital images. Zone III is composed mainly of uncrushed sand and fragments that have migrated through the inter-particle spaces from zones I and II. The thickness of zone III ranges from 2.0 to 5.0 mm.

The digital images and videos indicate that crushing occurs initially mostly in zones I and II, with minimal activity (particles popping up) in zone III. As installation continues, the smaller fragments from the crushed particles start clogging the inter-particle spaces. At installation depths greater than 7B, it was observed that most of the crushing (attrition, as shown by images and videos) occurs in zone II rather than in zone I. In zone I, below the inclined faces of the conical base, the small sand fragments produced by crushing of larger particles continue to crush further with increased pile penetration. This process creates fine particles (of white color) that surround the large particles and prevent them from further crushing due to the “cushioning” effect [55].

For medium-dense sand samples, only one crushing zone with thickness ranging from 2.0 to 4.0 mm was observed; however, the degree of particle crushing in this zone, as indicated by the presence of fine particles of white color, was similar to that observed in zone II in Fig. 15.

6 Summary and conclusions

This paper presented the results of a series of tests performed to investigate and quantify the crushing of sand particles produced by model piles jacked in dense and medium-dense samples prepared by air pluviation in a half-cylindrical calibration chamber with digital image capabilities. The experimental program included measurement of loads at the base of the model piles during pile installation and loading, the exhumation of samples from different locations along the shaft of the model pile for particle-size analyses, and determination of morphology parameters of sand particles before and after testing. These results were combined with analyses of digital images of the pile and the surrounding sand particles during pile installation and loading to characterize the crushing process that takes place in the path of model piles penetrating in uniform sand.

The measurement of the loads during pile installation indicates that an increase in the sand relative density and the surcharge at the top of the sample lead to an increase in the unit base resistance and the crushing of particles below the base of the model piles.

Samples of crushed sand from an annular zone 3.0 mm-thick around the shaft of the model piles show that the degree of particle crushing, as indicated by the widening of the particle-size distribution curve, increases with sand relative density and the surcharge used in the tests. In the tests with an initial maximum stress level equal to 57 kPa, the mean particle size D50 of the crushed sand dropped as much as 25% and 9% when the model pile was installed in dense and medium-dense sand samples, respectively.

The results of the morphology analyses show that crushing produced by pile installation produces particles that are more rounded and spherical than those of the original sand. This is consistent with detailed observations of individual particles in digital images captured during the pile installation and loading that showed that the predominant crushing mode of the sand particles is the breakage of asperities and sharp edges of the original particles. The finer particles produced by crushing particles surround the larger particles, protecting them from further crushing, as also observed by previous researchers using different laboratory techniques.

Based on the experimental results of this research, equations to predict the breakage parameters Br, B*, and BEN from the measured limit unit base resistance of model piles penetrating in silica sand (Ohio Gold Frac) were proposed.

The measurement of limit unit base resistance qbL of piles installed with multiple strokes changed by less than 3% in dense sand and 9% in medium-dense sand with respect to qbL of piles installed monotonically. These results and the particle-size distribution curves of sand from the crushed zone next to the shaft of the model pile indicate that the jacking strokes have a negligible effect on the degree of crushing produced during pile jacking.