1 Introduction

Crushing of soil grains occurs due to extremely high stresses that may generate during pile driving, construction of high earth or rock fill dams, laying foundations of the offshore gravity structures, impact of projectiles, drilling at great depths for petroleum extraction etc. (e.g., Miura et al. 1984; Hardin 1985; Zheng et al. 1990; Coop and Lee 1993; Hagerty et al. 1993). Extensive laboratory investigations have been conducted by researchers to assess compression and crushing behaviour of the granular material (e.g., Miura et al. 1984; Hagerty et al. 1993; Schanz and Kitamura 1993; Mcdowell et al. 1996; Yamamuro et al. 1996; Nakata et al. 2001; Feda 2002; Chuhan et al. 2003). Also, triaxial compression and one-dimensional compression have been carried out by researchers to study the bulk and microscopic response of a variety of granular materials by varying the initial voids ratio e and the aspect ratio H/d, which is the ratio of the height H and diameter d of the sample (e.g., Hagerty et al. 1993; Morland et al. 1993; Ghosal and Self 1995; Anand and Gu 2000; Chuhan et al. 2003; Gupta et al. 2004; Bartake and Singh 2005). Based on these experiments, parameters such as total breakage factor (Hardin 1985), aggregate stability (Oztas et al. 1999), and probability of crushing and particle breakage factor (Nakata et al. 1999) have been defined. These parameters are found to be of immense help for understanding the crushing behaviour of the granular material in a very comprehensive manner. Also, these studies indicate that the factors which primarily influence the crushing of grains are particle-size distribution, grain shape, hardness, density or the voids ratio, the aspect ratio etc. (e.g., Miura et al. 1984; Hardin 1985; Feda 2002; Chuhan et al. 2003).

In a recent study, based on the extensive experimental investigations on different sands, correlation between the yield stress and the grain size, shape and mineralogy of the granular material has been developed (Chuhan et al. 2003). However, this study does not yield empirical or semi-empirical relationships that can be employed for determining crushing strength of the sand, directly. In addition to this, researchers have tried to simulate “single particle crushing” by employing a simple test setup that consists of two parallel plates and a loading arrangement (e.g., Nakata et al. 1999; McDowell and Bolton 1998; McDowell 2001). However, these studies do not simulate the in situ behaviour of the granular material and results are found to be sample specific.

In such a situation, the potential of the methodology proposed by Bartake and Singh (2005) for determining crushing strength of cenospheres, based on the specific gravity G and the load-deformation characteristics, can be extended for determining crushing strength of granular materials, in general. This methodology can also be utilized for developing a generalized mathematical model for determining crushing strength of the granular material, based on its G and the particle-size distribution characteristics. Such a model would also be useful in overcoming the difficulties associated with the experimental investigations, which are quite cumbersome, time consuming and are dependent on the experimental and sample conditions, to a certain extent.

With this in view, experiments were conducted on different types of sands, cenospheres and glass beads and the results are used for developing a generalized model that can be employed for determining crushing strength of the granular material. Validation of the model has been done by comparing the computed results vis-à-vis experimental findings and those available in the literature.

2 Experimental investigations

Three grades of sands and cenospheres (denoted as S1, S2, S3 and CS1, CS2, CS3, respectively), were used in this study. These samples were characterized for their mineralogical composition by X-ray diffraction (XRD) spectrometer (Phillips, Holland), which is fitted with a graphite monochromator and Cu-kα radiation source. For identifying various minerals in these samples JCPDS (1994) files were used. Sand samples exhibit presence of quartz as the most predominant mineral while cenospheres are noticed to consist of quartz (as the most predominant), mullite, sillimanite and hematite minerals. These sands were mixed in different proportions, as listed in Table 1, to create 12 different samples of sand.

Table 1 Designation of the sand mixtures used in the study
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For establishing particle-size distribution characteristics of these samples, dry sieving was conducted (ASTM D 422-63). Glass beads used in this study are of uniform diameter D GB (= 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 mm).

G of sands, cenospheres and glass beads was obtained by using an Ultra-Pycnometer (Quantachrome, USA), which employs Helium gas (ASTM D 5550).

2.1 Test setup

The test setup depicted in Fig. 1 was fabricated to study the crushing characteristics of the granular material (sands, cenospheres and glass beads). As depicted in the figure, the test setup consists of a stainless steel (SS) mold of inner diameter 33 mm, outer diameter 51 mm and height 110 mm, in which granular material can be filled to achieve a sample of desired aspect ratio H/d and density ρ. Two stainless steel pistons (top and bottom) of 32 mm diameter were used for crushing the sample. A collar was provided at the bottom of the mold on which two removable stainless steel clips of thickness 10 mm can be fixed. These clips were removed before commencing the test to achieve loading of the sample from both the ends. With this setup, influence of H/d on the measured crushing strength σCR of the samples can also be studied.

Fig. 1
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The test setup

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2.2 Testing methodology

A 5 T capacity servo hydraulic universal testing machine UTM (AIMIL, India) was employed for establishing crushing characteristics of the sample. These tests were performed at a strain rate of 1.25 mm/min (Karner et al. 2004). A 5000 kg load cell, attached to a digital readout unit was used for recording the load P transmitted to the sample. Prior to the loading of the sample, silicon grease was applied on the two pistons to minimize the side friction. The sample was poured into the mold by adopting the rainfall technique (i.e., pouring the sample using a funnel by maintaining a constant height of fall) to achieve a certain H/d and ρ equal to 1.74 g/cm3. The P transmitted to the sample was recorded, at an interval of 10 s, until it reaches 3500 kg.

3 Results and discussion

Figure 2 depicts the particle-size distribution characteristics of the sand and cenosphere samples considered in this study. Gradational characteristics of the samples of cenospheres CS1, CS2 and CS3 are also superimposed in the figure to highlight the range of particle-size of the granular materials considered in this study. From the figure, particle sizes D x corresponding to x percent finer, uniformity coefficient C u, and the coefficient of curvature C c, were computed and are listed in Table 2. The value of G for these samples is also presented in the table. G for all types of glass beads is found to be 2.46.

Fig. 2
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Particle-size distribution characteristics of the granular materials

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Table 2 Physical characteristics of the granular materials used in the study
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Load-deformation (P-δ) characteristics of different sands and cenospheres corresponding to different H/d values, were developed. However, for the sake of brevity, results pertaining to Sample S1 and CS3 (with H/d = 1.0) only are presented in Figs. 3 and 4, respectively. It can be noted from Fig. 3 that followed by the initial elastic compression of the grains (depicted by the portion AB of the P-δ characteristics), the sample transforms to a new state of the material. Hence, intersection of the tangents drawn to the initial (AB) and final (DE) portions of the P-δ characteristics would intersect at C (= 703 kg), which would yield σCR (= 87.4 kg/cm2) of the sample. Incidentally, the results obtained are found to be consistent with those reported in the literature (e.g., Hardin 1985; Yamamuro et al. 1996; Chuhan et al. 2003).

Fig. 3
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Load-deformation characteristics of Sample S1

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Fig. 4
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Load-deformation characteristics of Sample CS3

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It can be noted from Fig. 4 that after an initial elastic response (depicted by the portion AB), rearranging and repacking of the particles starts (depicted by the portion BC). Incidentally, point C corresponds to the point on the P-δ characteristics with the maximum curvature. Beyond point C, cenospheres lose their integrity and alter to a new state of the material. Hence, a tangent drawn to the P-δ characteristics, at point C, would intersect its linear portion (represented as DE) at F, which would yield σCR of the sample. As depicted in Fig. 4, the crushing load of the sample is 1280 kg, which would yield σCR equal to 159.2 kg/cm2. This crushing strength falls in the range of the values reported in literature (Kruger 1996, http://www.sinosi.com), which does not present details of the testing methodology adopted.

As a practice, researchers have used e versus logarithm of effective stress σ′ relationship, rather than P-δ characteristics, to study the crushing characteristics of the granular materials (e.g., Hagerty et al. 1993; Nakata et al. 2001). It must be noted that e-log(σ′) relationship would yield a point of maximum curvature, which has been termed as the “yield point” (DeSouza 1958) and the “break point” (Roberts 1964) and is representative of the stress required to initiate crushing of the grains. However, determination of the yield point involves uncertainty and error, mainly, due to improper identification of the point of maximum curvature on e-log (σ′) relationship. In such a situation, the methodology described in this paper, for determining crushing strength of granular materials, seems to be quite useful.

Further, the influence of H/d on σCR of different samples of sands was investigated. For the sake of brevity, P-δ characteristics of the Sample S1, corresponding to different H/d, are depicted in Fig. 5. It can be noted that the variation in crushing strength values of the sample with respect to H/d is almost insignificant. In addition to this, for checking the reproducibility of the P-δ characteristics, tests were performed on Sample S1 by maintaining H/d = 1.0 and the results are depicted in Fig. 6. It can be noted from the figure that, the three trials yield practically same response.

Fig. 5
figure 5

Load-deformation characteristics of Sample S1 for different H/d values

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Fig. 6
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Reproducibility of the load-deformation characteristics of sand (Sample S1, H/d = 1.0)

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Similarly, the influence of H/d on σCRof different cenosphere samples was investigated. For the sake of brevity, P-δ characteristics of Sample CS3, corresponding to different H/d, are depicted in Fig. 7. Using the above-mentioned methodology, σCR of Samples CS1, CS2 and CS3, corresponding to different H/d, was computed and is depicted in Fig. 8. From the figure, it can be noticed that initially, σCR increases non-linearly with H/d and thereafter attains a constant value. The transition point between these two trends corresponds to H/d = 1. Hence, for determining σCR of the cenosphere sample, tests must be conducted with H/d = 1. For checking the reproducibility of the P-δ characteristics, tests were performed on Sample CS3 by maintaining H/d = 1 and the results are depicted in Fig. 9. It can be noticed from the figure that the four trials yield practically same trends.

Fig. 7
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Load-deformation characteristics of cenospheres (Sample CS3)

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Fig. 8
figure 8

Variation of σCR with H/d for different cenospheres

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Fig. 9
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Reproducibility of the load-deformation characteristics of cenospheres (Sample CS3, H/d = 1)

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Further, σCR for different samples of sands and cenospheres, corresponding to H/d = 1.0, was determined and is tabulated in Table 3.

Table 3 Experimental results
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As samples used for this study exhibit a very wide range of the particle-sizes (refer Fig. 2) and specific gravity, attempts were made to correlate their σCR values with a parameter σCR/G, which is indicative of crushing strength per unit mass of the material. Hence, by using the optimization software Mathematica 4 (Mathematica 2000) and the results of 11 samples (S1, S2, S3, E, F, G, I, K, L, CS1 and CS3), relationships represented by Eqs. 1–6 were developed. As stated earlier, D x appearing in these relationships corresponds to x percent finer fraction of the granular material. Mathematica 4 has a built-in ‘Nonlinear-fit’ function, which is based on the least-square fit concept. It fits the data to the fitting function and yields the best-fit parameters.

$$ \sigma_{\rm CR}/G=\frac{25\cdot D_{45} \cdot D_{80} }{\left({D_{10} \cdot D_{30}}\right)^{0.7}\cdot \left( {D_{90}}\right)^{1.5}} $$
(1)
$$ \sigma _{\rm CR}/G=\frac{20\cdot D_{15}\cdot D_{45} \cdot D_{50} \cdot D_{80} }{\left({D_{10}}\right)^{1.6}\cdot D_{30}\cdot D_{60} \cdot \left( {D_{90}}\right)^{1.2}} $$
(2)
$$ \sigma _{\rm CR} /G=\frac{25\cdot \left( {D_{15} } \right)\cdot \left( {D_{45} } \right)^{1.5}\cdot \left( {D_{60} } \right)^{0.5}\cdot \left( {D_{85} } \right)^{2.5}}{\left( {D_{10} \cdot D_{30} } \right)^{1.5}\cdot \left( {D_{80} } \right)^{0.3}\cdot \left( {D_{90} } \right)^3} $$
(3)
$$ \sigma _{\rm CR} /G=15\cdot \left[ {\left( {\frac{\left( {D_{15} } \right)^{0.5}}{\left( {D_{30} } \right)^{1.55}}} \right)+\left( {\frac{\left( {D_{60} } \right)^2}{D_{90} }} \right)} \right] $$
(4)
$$ \sigma _{\rm CR} /G=\frac{15}{\left( {D_{10} } \right)^{0.8}\cdot D_{60} } $$
(5)
$$ \sigma _{\rm CR} /G=\frac{\hbox{25}\cdot \left( {D_{10} } \right)^{0.15}}{D_{30}} $$
(6)

Using Eqs. 1–6, σCR/G for these 11 samples was computed, as listed in Table 4, and its percentage error with respect to the experimentally obtained σCR/G, as depicted in italics, was also determined. The average percentage error for the relationships presented by Eqs. 1–6 is found to vary from 4 to 20%. However, the minimum value of the average percentage error (= 4.31) is obtained by using the relationship presented by Eq. 2. This indicates that the crushing strength of the granular material can be estimated by using Eq. 2, with the least amount of error. Further, for validating this equation, parameter σCR/G was computed; (i) using Eq. 2 and (ii) from the experimental results available in the literature (Been et al. 1991; Pestana and Whittle 1995; Yamamuro et al. 1996; Bopp and Lade 1997; Nakata et al. 2001), as listed in Table 5. For most of the soils, the percentage difference between these two values of σCR/G, with respect to the experimental value (depicted in italics), is found to be less than 10. This demonstrates the usefulness and efficiency of Eq. 2 for computing crushing strength of the granular material, in general.

Table 4 A comparison of the experimental and computed value of σCR/G (in kg/cm2) for different samples
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Table 5 Validation of the Eq. 2
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However, the preceding discussion is applicable for the granular material with a wide range of particle sizes and hence can not be employed for determining σCR of the material which is poorly-graded, such as glass beads, which are of uniform diameter size D GB. Hence, in order to prove the applicability of Eq. 2 even for the poorly-graded granular material, the load-deformation characteristics of the glass beads (with different D GB) were established, by maintaining H/d = 1. For the sake of brevity, results of only three glass beads (D GB = 1, 2 and 3 mm) are depicted in Fig. 10. σCR for the glass beads was computed (corresponding to the peak failure load), normalized with specific gravity G and is plotted with respect to D GB, as depicted in Fig. 11.

Fig. 10
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Load-deformation characteristics of the samples of glass beads

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Fig. 11
figure 11

Variation of crushing strength of glass beads with their diameter

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The trend presented in Fig. 11 can be represented as:

$$ \sigma_{\rm CR}/G=47\cdot (D_{\rm GB})^{-0.54}. $$
(7)

Further, using Eq. 2, σCR/G for glass beads was computed by assuming D x   =  D PG, where D PG corresponds to the diameter of poorly-graded granular material. The results obtained from this analysis are compared with those obtained from Eq. 7, as depicted in Fig. 12.

Fig. 12
figure 12

Comparison of the results obtained from Eq. 2 and Eq. 7

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From Fig. 12, it can be noted that, results obtained from Eqs. 2 and 7 exhibit almost similar trends. However, with respect to Eq. 2, Eq. 7 is found to over-predict values of σCR/G for the same D PG. With this in view, a multiplication factor, MF, which is the ratio of σCR/G obtained from Eq. 7 and Eq. 2 was computed and its variation with D PG was established:

$${\rm MF}=2.5\cdot (D_{\rm PG})^{0.3}. $$
(8)

As such, σCR of any granular material can be obtained by employing the following guidelines:

  1. (a)

    For uniformly-graded material, Eq. 2 should be employed, and

  2. (b)

    For poorly-graded material, results from Eq. 2 should be multiplied by MF.

4 Concluding remarks

A methodology for determining crushing strength of the granular material, based on its load-deformation characteristics has been developed and reported in this paper. Based on the rigorous experimentation on different samples of sands, cenospheres and glass beads, a mathematical relationship between the crushing strength, particle-size characteristics and specific gravity of the granular material has been proposed. This relationship is found to be quite efficient and useful for determining crushing strength of granular materials.