The marble and granite industries in Egypt produce a vast amount of by-product slurry waste that could be used in green mortar production suitable for construction purposes. This research highlights the effect of the chemical constituents of marble and granite waste powders on the compressive strength of the green concrete produced. A chemical analysis of the constituents of granite and marble wastes was compared with those of the cement to study the effect of these components on the hydration reaction inside the mixture. The experiment was based on replacing the same proportions of sand and cement in the green concrete mixes with each of granite and marble waste powders after dissolving it in the water content.

The study revealed that by replacing 5% of cement in (NC5) mix, 10% of sand in (NF10) mix, as well as 5% of cement and 10% of sand in (NC5 + NF10) mix, by granite waste powder, the compressive strength values increased by 33%, 39%, and 41%, respectively. This was due to the presence of more than 26% fine free silica particles in granite which undergo pozzolanic reaction with calcium hydroxide present in mortar pores producing calcium silicate hydrate (CSH) crystals resulting in high strength to the cement mortar. For the same mixes containing marble powder, the compressive strength showed less values by 14%, 10%, and 0% for NC5, NF10, and NC5 + NF10 mixes, respectively, when compared to the control mix values.

Ornamental stones, specifically marble and granite, are common building materials being widely used in construction processes. Handling as well as disposal of these stones wastes is considered one of the major environmental and land pollution problems due to both its highly alkaline nature, and its manufacturing and processing techniques, which impose a health threat to the surroundings (Hamza et al. 2011).

Several researches worldwide attempted to use these industrial by-products efficiently in producing mortar and concrete on an experimental scale. Marble and granite wastes (MGWs) do not require any processing before its use in mortar and concrete production. Moreover, its physical and chemical properties are suitable for making concrete products. Also, due to its finesse, MGW is a promising material which acts as a micro-filler in cement aggregate matrix (Ramos et al. 2013; Bacarji et al. 2013).

Bahar (2010) studied the effect of using marble dust waste as a fine material on the mechanical properties of the concrete mix. The study revealed that substituting the very fine aggregate passing through a 0.25-mm sieve by marble waste performed better than the control mix in terms of compressive strength. Marble dust had a filler effect at an early age and played a noticeable role in the hydration process. The SEM investigations indicated a difference between the appearance of CH crystals with and without marble dust addition, verifying the fact that the marble dust played a noticeable role during the hydration process.

Manpreet et al. (2017a, 2017b) indicated in his study that replacing 15% of the cement content in the concrete mix at a water to cement ratio of 0.35 with marble slurry would decrease the water permeability and abrasion of the mix. Higher replacement ratios increased the density of the mix and reduced water penetration in the long term.

Similarly, Sarbjeet et al. (2015) observed that the optimum replacement percentage of marble waste for both cement and sand was found 15% to produce paver concrete blocks. Increasing this replacement percentage affected negatively the compressive strength of the produced blocks. The percentage increment in the compressive strength for the optimum mix is greater in case of sand replacement as compared to the cement replacement.

Additional studies were conducted to evaluate the effect of using granite waste as a replacement for cement and sand in the concrete and mortar mixes. Shehdeh (2016) observed that substitution of 10% of sand by weight with granite powder in concrete resulted in a maximum increase in compressive strength to approximately 500 kg/cm2 compared to 365 kg/cm2 of control concrete and an increase in splitting tensile strength to 30 kg/cm2 compared to 26 kg/cm2 of control concrete.

A distinctive study was undertaken by Abhishek and Pradeep (2015) to examine the effect of partial replacement of sand with granite quarry dust and cement with marble powder in concrete. The results indicated that the compressive strength increased gradually by adding up to 10% replacement of cement with marble powder and 20% replacement of fine aggregate with granite dust. Beyond these proportions of waste addition, the compressive strength of the concrete mix decreased. This was referred to the lack of combination between the CSH gel and the granite marble wastes during the hydration process leading to a weak micro-structure of concrete.

Bakhoum et al. (2017) presented a study on the use of nano-granite waste particles as a partial replacement of cement and fine aggregate in mortar production. The research concluded that replacing 5% cement and 10% sand with nano-granite waste in the mortar mix increased the compressive strength of the green mortar by 41% compared to that of the control mix (CM). SEM images reinforced this result as the green mortar mix showed maximum density and minimum micro-cracks and a number of pores. The research was extended to study the social, environmental, and economic effects of using granite waste as a partial replacement for both cement and sand simultaneously. Savings in energy consumption and CO2 emissions reached 5%.

The materials used in this study were obtained from the local Egyptian market. Normal Portland cement used was CEM I 42.5N, confirming the Egyptian standards ES 47561. The specific gravity of cement used in this study was 3.15 gm/cm3, and the percentage of fine particles passing from sieve 170 was 9% which presents the fineness of cement used. The initial and final setting times were 2h and 3.2h, respectively.

Natural sand composed of siliceous materials was used as fine aggregate in this study. The nominal maximum size of the sand was 4.75mm. The marble and granite waste powder were obtained from Shaqu-Elteban area, Egypt. This fine powder was used to substitute conventional cement and fine aggregates in the green mortar mixes produced. The fine waste material is obtained in the form of slurry material containing various percentage of water. In order to ensure a constant W/C ratio for mortar mixes, the natural waste material was dried up using an oven at a temperature of 200C for 6h. The waste material powder was weighed before and after the drying process, and the difference of weight proved to be less than 10% to ensure proper drying state of this waste material. The water was clean tap water with a temperature ranging 2030C.

The waste materials were then sieved and the fine particles passing through sieve 300m were used as a partial replacement of cement. This waste was dissolved in water resulting in a consistent solution which in turn was added to the other mix components to produce the resulting green mortar mixes.

X-ray fluorescence spectrometry (XRFS) is a method of elemental analysis that assesses the presence and concentration of various elements by measurement of secondary X-radiation from the sample that has been excited by an X-ray source.

Classically, elements from the heaviest down to atomic number 9 (F) can be determined at levels of a few milligram/kilogram (ppm). Newer developments with wavelength dispersive spectrometers (WDXRF) allow the determination of some of the ultralow atomic number elements including (O).

The design mixes used in this study to produce green mortar were prepared by partially replacing cement, sand, and both of them with different percentages by weight of waste material. It should be stated that the W/C ratio was 0.5 for all mixes produced.

The first mortar mix prepared (CM) was a control mix with a 0% replacement ratio of waste. The first mortar green mix (NC5) was containing 5% of marble waste as a partial replacement of cement, while (NF10) was a green mortar containing 10% of marble waste as a partial replacement of sand. The last green mix (NC5 + NF10) was prepared using 5% cement replacement and 10% sand replacement together. Table 1 presents the mixes components produced for this study.

It should be noted that the mix proportions used in this study were selected similar to those done on granite waste powder in Bakhoum et al. (2017) in order to conclude a complete understanding of the behavior of the chemical constituents of marble and granite in mortar mixes.

A compressive strength test was carried out for mortar mixes after curing for 28days. The Shumadsu 1000 KN universal compression machine was used in testing the mortar samples. The machine is equipped with a data analyzing output for data recording.

Table 2 shows the chemical analyses of the raw materials as obtained from XRF analyses, whereas Table 3 presents the percent of free silica and organic matter of the two wastes. The loss on ignition obtained is significantly higher in case of marble waste than that of granite waste. This is mainly due to the loss of carbon dioxide from the decomposition of marble waste (consisting mainly of calcium carbonate), in addition to the organic matter content as revealed in Tables 2 and 3.

The compression tests were carried out to investigate the mechanical behavior of green mortar mixes prepared using marble waste powder. Table 4 presents the compression test results for mortar mixes at a curing time of 28days.

Test results show that for the control mix (CM), the compression strength was 251kg/cm2. By replacing 5% of cement with marble powder in NC5, the compressive strength value dropped by 14%. For sand replacement, and by replacing 10% of sand with marble powder in (NF10), the compressive strength value rises up to 5% above the control CM. Replacing 5% of cement and 10% of sand with marble waste powder in NC5 + NF10 mix gives almost the same compressive strength value obtained from CM.

Comparing these results of marble replacement to those of granite with the same mixing proportions cited in Bakhoum et al. (2017), the results of green mortar using granite waste powder in Table 5 showed an increase in the compressive strength for all mixes NC5, NF10, and NC5 + NF10, of 33%, 39%, and 41%, respectively.

The probable reason for that behavior is the presence of a relatively large amount of fine free silica particles in granite as evidenced from Table 3. The table shows that granite waste contains more than 26% fine free silica particles. These particles will undergo pozzolanic reaction with calcium hydroxide present in mortar pores producing CSH crystals which result in high strength to the cement mortar. On the other hand, marble waste is largely composed of limestone (Table 2) which possesses no pozzolanic properties.

This result concedes with those presented by Manpreet et al. (2017a, 2017b), which revealed that the increase in compressive strength is observed only due to micro-filler effect of marble powder whereas a decrease in strength begins to appear at 10% substitution as the amount of C3A and C2S required for hydration process reduces.

Marble waste has a high percentage of fines but does not have a considerable amount of silica and alumina. Calcite and dolomite are the main constituents present in the marble slurry. Replacing cement and sand by marble waste did not result in a significant increase in the cement mortar compressive strength when compared to the control mix.

Granite waste contains more than 26% fine free silica particles which undergo pozzolanic reaction with calcium hydroxide present in mortar pores. This pozzolanic reaction produces CSH crystals which increase the strength of the cement mortar.

This result is corroborating with the chemical composition analysis conducted by Bruna et al. (2018), which revealed that the main components of the granite waste were silica (42.80%), calcium oxide (19.00%), and aluminum trioxide (8.07%) together with small amounts of other oxides.

For future work, it is highly recommended to study the effect of using both marble and granite wastes together as a partial replacement to cement and sand on the cement aggregate matrix with different mixing proportions to reach the optimum green mortar compression strength results.

Bruna SA, Fernanda GP, Luciane SCM, White JS, Maria TPA (2018) Study of Portland cement composites replacing cement for waste from the cutting and polishing of ornamental rocks. Int J Sci Eng Invest 7(76):120124

Sarbjeet S, Anshuman T, Ravindra N (2015) Comparative assessment of effects of sand & cement replacement in concrete by marble dust & in turn deriving an optimum design mix for concrete paver blocks. Conference Paper: Global Stone Technology Forum (GSTF)

SA performed the chemical laboratory analysis tests for raw materials, and SA, HE, and MA performed the compression mechanical test. MA, SA, and GG analyzed and interpreted the results of chemical analysis and compression test values obtained, and all authors read and approved the final manuscript.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Amin, S.K., Allam, M.E., Garas, G.L. et al. A study of the chemical effect of marble and granite slurry on green mortar compressive strength. Bull Natl Res Cent 44, 19 (2020). https://doi.org/10.1186/s42269-020-0274-8

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The tensile strength and fracturing behavior of Carrara marble subjected to the dynamic Brazilian disc test using the split Hopkinson pressure bar technique are determined and compared with those obtained by the conventional quasistatic Brazilian disc test. Detailed observation of the cracking processes is aided by high-speed video footage captured at a frame rate of 100,000 frames per second. The dynamic increase factor is computed, revealing a strong strain rate dependence of the Carrara marble when subjected to strain rates above 1s1. Similar to the quasistatic loading tests, conspicuous white zones/patches commonly appear prior to the initiation of visible cracks in the dynamic loading tests. Identification of the white patch initiation and evolution is aided by image comparison software. Comparing the cracking and failure processes under quasistatic and dynamic loading, some distinct differences in the white patch geometry and initiation load are observed. In addition, the extent of the compressive failure zones around the contact points between the loading platens and specimens is found to increase with the strain rate.

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Huang J, Chen G, Zhao Y, Wang R (1990) An experimental study of the strain field development prior to failure of a marble plate under compression. Tectonophysics 175(13):269284. doi:10.1016/0040-1951(90)90142-u

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Today we are faced with an important consumption and a growing need for aggregates because of the growth in industrial production, this situation has led to a fast decrease of available resources. On the other hand, a high volume of marble production has generated a considerable amount of waste materials; almost 70% of this mineral gets wasted in the mining, processing and polishing stages which have a serious impact on the environment. The processing waste is dumped and threatening the aquifer. Therefore, it has become necessary to reuse these wastes particularly in the manufacture of concrete products for construction purposes. The main goal of this study is to demonstrate the possibility of using marble wastes as a substitute rather than natural aggregates in concrete production. The paper presents the study methodology, the characterization of waste marble aggregates and various practical formulations of concrete. This experimental investigation was carried out on three series of concrete mixtures: sand substitution mixture, gravel substitution mixture and a mixture of both aggregates (sand and gravel). The concrete formulations were produced with a constant water/cement ratio. The results obtained show that the mechanical properties of concrete specimens produced using the marble wastes were found to conform with the concrete production standards and the substitution of natural aggregates by waste marble aggregates up to 75% of any formulation is beneficial for the concrete resistance.

Yidong Wang, Wenjiao Dan, Yongfu Xu, Yue Xi, "Fractal and Morphological Characteristics of Single Marble Particle Crushing in Uniaxial Compression Tests", Advances in Materials Science and Engineering, vol. 2015, Article ID 537692, 10 pages, 2015. https://doi.org/10.1155/2015/537692

Crushing of rock particles is a phenomenon commonly encountered in geotechnical engineering practice. It is however difficult to study the crushing of rock particles using classical theory because the physical structure of the particles is complex and irregular. This paper aims at evaluating fractal and morphological characteristics of single rock particle. A large number of particle crushing tests are conducted on single rock particle. The force-displacement curves and the particle size distributions (PSD) of crushed particles are analysed based on particle crushing tests. Particle shape plays an important role in both the micro- and macroscale responses of a granular assembly. The PSD of an assortment of rocks are analysed by fractal methods, and the fractal dimension is obtained. A theoretical formula for particle crushing strength is derived, utilising the fractal model, and a simple method is proposed for predicting the probability of particle survival based on the Weibull statistics. Based on a few physical assumptions, simple equations are derived for determining particle crushing energy. The results of applying these equations are tested against the actual experimental data and prove to be very consistent. Fractal theory is therefore applicable for analysis of particle crushing.

Granular materials are widely used in rock fill dams, highways, railways, and dykes, due to their engineering properties such as high hydraulic permeability, high density, high shear strength, and low settlement. Coarse particles are susceptible to particle crushing at a high compressive strength, which directly modifies their structure, influencing dilatancy, friction angle, strength, and permeability [1]. The strength of granular material decreases during particle crushing as its compression increases, which may eventually lead to significant deformations and ultimately to structural instability [2]. Determining the exact mechanics of particle crushing, or particle breakage of granular material, is one of the most intractable problems in the geosciences. This topic is of interest to many subfields of research including powder technology, minerals and mining engineering, geology, geophysics, and geomechanics [3].

The problems associated with particle crushing in geomechanics began to attract attention in the early 1960s and from then on gradually developed into a topic of significant study. Research on particle crushing has since been carried out via four approaches. Firstly, researchers have experimented with various artificial materials. Takei et al. [4] carried out single particle compressive strength testing and one-dimensional compression testing with plaster and talc sticks, glass beads, and quartz, discussing the fragmentation mechanism of these four different materials in detail. Secondly, studies have attempted to mathematically describe the crushing characteristics of particles, which is of particular interest to the field of geomechanics. They linked the crushing behaviour of particles and their mechanical response through the use of behavioural constitutive models. Existing constitutive models are based on simple curve-fitting parameters, which are determined in isolation by discrete stress-strain tests. Nakata et al. [5] conducted single particle crushing tests on three kinds of particles and analysed the results using particle survival probability curves. Matsui et al. [6] investigated methods for estimating the ratio of net work input to crushing and specific surface area produced based on load conditions and material properties. Rozenblat et al. [7] expressed particle strength distributions based on an extended logistic function of the crushing force of a number of individual particles. Thirdly, the mechanical properties of crushed particles on the macroscopic level were determined. The macroscopic mechanical properties of particle crushing have been studied through direct shear tests, ring shear tests, uniaxial compression tests, and triaxial shear tests, which mainly characterised the strength and the deformation [8] characteristics of the particles. Lastly, mesoexperiments on particle crushing were carried out through the use of some advanced instruments and methods such as electron microscopy and X-ray [9] in order to obtain detailed structural information.

The majority of analyses focused on experimentally determining the physical characteristics of single particle crushing. Russell et al. [1012] analysed breakage behaviour of characteristic elastic-plastic granules using compression tests. The study describes the influences of granule size, moisture content, and loading intensity on the energy absorption and recovery at stressing. Mader-Arndt et al. [13] investigated the particle contacts by means of atomic force microscopy (AFM), nanoindentation, and shear tests. Ribas et al. [14] and Portnikov et al. [15] designed compression testers to accurately measure the force-displacement curves, the distribution of strengths, and the fracture energies of single particles.

Mandelbrot [16] has however observed that several natural phenomena can be accurately described by fractal theory. The fractal dimension, , is a fraction with a value between 0 and 3.0. Fractals are self-similar objects, allowing the fractal fragment size distribution at any scale to be predicted [17]. If the shape of some rock fragments is fractal, then their fractal dimension may hence be estimated from their size distribution [18]. Prior studies have generally been aimed towards characterising the particle size distribution (PSD). Many statistical methods have been proposed for describing the PSD of comminuted materials, and the most significant of such PSD functions, including normal, log-normal, Gates-Gaudin-Schumann, and Rosin-Rammler distribution functions, have been reviewed in detail by Allen [19]. The use of fractal size distribution analysis is another approach for linearizing the size distribution curve [2022]. The concept of utilising probability for studying particle crushing was introduced by McDowell et al. [23], who described a method whereby the compression behaviour of particles could be expressed using a probabilistic approach based around the mathematics of fractals. Xu et al. [24] expressed a significant size effect in ice failure strength and used the fractal model for studying ice particle fragmentation by employing modified Weibull [25] statistics. Combining traditional experimental methods with insights from fractal theory is hence an effective method for determining the morphology and mechanical characteristics of particle crushing.

This paper aims at exploring fractal crushing characteristics of single particle. Crushing tests on various sized particles are carried out to evaluate the crushing characteristics of each individual particle. Particle shape plays an important role in both the micro- and macroscales responses of a granular assembly [26]. The relationships between failure modes and particle shapes are analysed, and the fractal model is employed to describe the particle crushing behaviour. The fractal dimension of the particle size distribution is determined and found to be equal to 2.48 for marble particles. The crushing strength is then connected to the particle size combining with the fractal fragmentation of marble particles. The Weibull statistics was modified to estimate the probability of fracture for marble particles, and the Weibull modulus was determined using the fractal dimension of particle fragmentation. The formula of the size effect on the crushing energy is also analysed using fractal model for particle fragmentation.

A fractal is a shape consisting of parts similar to the whole in some manner [16]. A certain value is usually used to express the similarity between the parts and the entirety, which is termed the fractal dimension. The fractal dimension attempts to objectively represent how densely a fractal occupies the metric space in which it lies. Fractal dimensions are important because they can be defined in relation to real-world material behaviour and can hence be measured experimentally.

A fractal particularly suited to the analysis of particle crushing is that defined as a box-counting measure , the number of particles having diameters greater than or equal to , which displays scale invariance with a noninteger exponent [27]:In (1), the fractal dimension can be estimated from the slope of a straight line in the log-log plot. The number of particles is very difficult to count accurately, but the mass of the particles may easily be measured. A simple approach to calculating the fractal dimension of particle crushing is the following algorithm [18]:where is the total mass of the particles.

The particle density is a constant value, so the mass of the particles of diameter less than can be expressed as where is a shape factor. Equation (3) normalized by the total mass of all the particles givesFollowing (4), the fractal dimension of the particle-size distribution can be determined from the slope of versus .

The material used for the experiment is marble pebble, an actual rock, the main components of which are CaCO3, MgCO3, and SiO2, with impurities of Al2O3 and Fe2O3. The characteristic diameters of the particles used for the experiment range from 6.0mm to 32.0mm. The shapes of the particles are not perfectly spherical, so it has no way to using diameter in the strict sense to describe particle size. Then characteristic diameter is employed to express the size of the particles, which is defined as , where , , and are the lengths of each particle measured along three different axes.

Figure 1 contains an apparatus of particle crushing test. The test is carried out by transducer placing the particle between two hardened platens and then bringing the upper platen downwards at a constant velocity in order to crush the particle. The transducer diameter of the removable hardened platens is 150mm, and the load-measuring capacity of the apparatus is 50kN, with a resolution of 0.01kN. The transducers measuring the displacement and the force are arranged on the instrument, which can collect and record data automatically.

The purpose of experiments is to get fractal crushing and mechanical properties of marble particles using single particle crushing tests. The fractal dimension of particle crushing is measured from the particle size distribution of particle fragmentation. A total of 600 single particle crushing tests are carried out on marble particles.

The rock particles are divided into six groups according to their characteristic particle size: 6.08.0mm, 8.010.0mm, 10.013.0mm, 13.016.0mm, 16.025.0mm, and 25.033.0mm, with each group containing 100 particles. This will ensure not only the accuracy of the experimental results, but also the controllability of time cost. Before the test, single particle is placed on the centre of the bottom platen for preparation. The test commences with a loading velocity of 1.0mm/min that continues until the particles are crushed, with the crushed particles then being collected for the sieve tests. The instrument is able to automatically capture force and displacement data in real time during tests. Since the loading velocity is 1mm/min and the characteristic particle sizes are in range of 6.033.0mm, the strain rates range from 5.05 104 to 2.78 103s1. The loading velocity has been chosen to guarantee quasistatic loading of the particles.

Figure 2 shows the typical force-displacement behaviour of single marble particle in uniaxial compression tests. Prior to diametrical loading, a contact arises between the particle surface and the punch surfaces where negligible deformation and yielding of the surface asperities occur. This phenomenon is the same as that presented by Russell et al. [10, 11]. On diametrical loading, the particle undergoes elastic contact deformation until reaching the crushing point. The breakages are brittle; no obvious plastic behavior occurs in Figure 2.

When the force reaches the crushing point as the load increases, the particle will be primarily crushed. After the primary crushing, force and displacement curve is different for different particles. Figure 3 shows three typical and complete processes of force-displacement relationships obtained from the particle crushing tests. Type I compression curve is shown in Figure 3(a). The primary crushing occurs when the stress reaches a maximum of around 1.8kN at the displacement of 0.35mm. Then the stress suddenly decreases and after that always remains at low level. This type of particles tested tends to form only two or three fragments after crushing, as shown in Figure 4(a), and hence may be described as incompletely crushed. Type II compression curve is shown in Figure 3(b), and while the force still decreases after the first peak is reached, but compared with type I, this curve is slowly and stepwisely reduced. This type of particle breaks into several small fragments when crushed, as seen in Figure 4(b), which is also an incompletely crushed particle type. In type III compression curve, as shown in Figure 3(c), the force-displacement curve may reach a second peak that is even higher than the first peak. The particles seen in Figure 4(c) form into many tiny fragments and hence are completely crushed. It is noted that the different types of particle crushing are a result of the irregular shapes and flaws of particles.

The diameters of each particle in three directions are, in descending order, , as shown in Figure 5. Also, they are perpendicular to each other. As the data for each particle are 3D, two aspect ratios are calculated, the elongation index (EI) and the flatness index (FI), which are defined as EI = and FI = . The shape of the particles is lanker when EI is smaller, like a rod, while the bottom of the particles is round when EI is large. The smaller the FI value, the more flat the particles are, and when both values of EI and FI are close to 1.0, the particles are approximately spherical.

The connection between the crushing type and the shape of the particles is shown in Figure 6. The particles in area III of Figure 6 are inerratic in shape and mostly fit type I force-displacement compression curve shown in Figure 3(a). The anomalistic particles fall in the shaded region III and mostly fit type III compression curve, as shown in Figure 3(c). The particles in region II are less uniformly distributed and fit into all three compression curves. The peak forces in Figure 3 each represent the partial crushing of a particle. For inerratic particles, the crushing will be strongest at the first peak as the mother particle is broken into two or three pieces, and after that there will be no further significant crushing. This leads them to possessing type I force-displacement compression curve. For flat particles, the panel of the instrument and the mother particle are in sufficient contact for crushing at the initial stage of loading. This leads to many tiny fragments formation after crushing, manifesting in type III compression curve. Type II curve is an intermediate stage of the other two types.

The first peak value on the force-displacement compression curve in the single particle crushing tests is defined as the particle crushing force. The relationship between the particle crushing force and the particle size is shown in Figure 7. It can be seen that the particle crushing force grows as particle size increases.

The fractal dimension of the particle crushing may be determined by analysing data from the sieve tests according to (4). The apertures of the sieve tests used are 0.1mm, 0.5mm, 1.0mm, 2.0mm, 5.0mm, 10.0mm, 14.0mm, and 20.0mm. There are some fragments generated by friction and internal defects. These fragments will also be counted in the sieve tests for determining the fractal dimension.

The mass-size distribution of the crushed particles is shown in Figure 8(a). The abscissa represents the particle diameter after crushing, and the ordinate represents the ratio of the cumulative mass under the sieve to the total mass of crushed particles. The minimum diameter of the marble particles is 6.0mm, and particles with diameter greater than 6.0mm are not crushed. It should be noted that the crushed particles with diameter less than 5.0mm (mm) demonstrate a linear distribution in log-log that is approximately 0.52. The mass-size distribution of the particles with diameter greater than 5.0mm deviates the log-log coordinates in Figure 8(a). The deviation in Figure 8(a) is due to the fact that particles are not completely crushed. All particles with the diameter less than 5.0mm are collected and the mass size distribution of crushed particles is shown in Figure 8(b). It is seen from Figure 8(b) that the mass-size distributions of crushed particles in a log-log plot can be described by a linear relationship, whose slope is approximately 0.52. According to (4), the fractal dimension of crushed particles is 2.48.

It should be noted that the result is only suitable for the specific material and the specific failure mode here in this paper. It means that the result might not be applicable to comminution operations where strain rates are very high (likely to be 104106s1). However, this method here is still suitable because the different loading rates will lead to the change of the particle size distribution; then the fractal dimension will also change. In that case, an updated can be obtained for comminution operations with different strain rates. This also proves that, at higher displacement controlled loading rates, higher compressive loads are required to realize equivalent deformation in comparison to those achieved by loads at lower displacement-controlled loading rates, which has been addressed by Russell et al. [10, 11].

The fractal dimension obtained in the previous section will play an important role in predicting the mechanical properties of the particles. The apparent crushing strength of particles is given bywhere is the apparent crushing strength of particles, is the ultimate fracture force, and is the apparent area of the particle in the section perpendicular to . The apparent area of the particle is written aswhere is the particle diameter and is a shape factor; when the contact surface between the particle and the platens is square, .

The fractal dimension of the particle size distribution is in three dimensions; therefore the fractal dimension of the particles size distribution is in two-dimensional cross-sectional planes according to the rules of thumb [16, 28]. The real particle area in any section with size is written aswhere is the real section area of the particles. The intrinsic tensile strength of particles is a constant value that does not vary with particle size, which is written asCombining (5), (6), (7), and (8),The particle diameter is difficult to measure, so it is replaced by the characteristic diameter in this section. Figure 9 shows the relationship between crushing strength and characteristic diameter of marble particle in a log-log plot. The solid line in Figure 9 is the prediction of (9). The fractal dimension of crushing particles is 2.48 obtained from the mass size distribution in Figure 8. According to (10), the value of is 0.52. It is seen from Figure 9 that the prediction of (9) matches well with the experimental data. It is seen however from Figure 9 that the discrete value of the failure strength is quite high even when the size of the particles is the same, which is mainly due to the influence of the irregularity of the particles. Since even particles are of the same size, their shape may not be exactly the same. Thus, their contact surfaces may differ, and their crushing strength is not necessarily the same.

The probability of survival at a given crushing stress is defined asFigure 10 shows the curves representing the probabilities of survival under a given characteristic stress . Curves are presented for the particle size ranges of 6.08.0mm, 8.010.0mm, 10.013.0mm, 13.016.0mm, 16.025.0mm, and 25.033.0mm, respectively.

The survival probability of granular material can be calculated from [25]where is the probability of survival of any particle subjected to a stress , is the characteristic stress at which 37% of particles survive, is the Weibull modulus, and . In the fractal model, , so (13) can be expressed asIf the tensile strength is defined as tensile stress at a standard probability of survival for various batches of particles with diameter , it may be expressed asCombining with (9) givesIt can be seen that increases linearly as increases, as shown in Figure 11, and the slope is . Taking the experiment result in (16), it can be calculated that . The predictions of (14) based on the use of a fractal dimension of 2.48 are also plotted in Figure 11. The predictions of (14) match satisfactorily with the experimental data.

The experimental system is set up to detect the force at which particle crushing occurs. The crushing energy is the integration of the crushing force with respect to displacement of the particle as shown inIn Figure 3, a linear relationship may be observed between the crushing force and displacement before the crushing point; hence the crushing energy isFigure 12 shows the relationship between particle displacement and the characteristic diameter . The experiment reveals that the displacement and diameter are directly proportional:where is the proportionality coefficient between and .

Substitute (7), (8), and (19) into (18),where is a constant and . So it is a linear relationship between the crushing energy and the diameter of the particles in log-log coordinates, and the slope should equal the fractal dimension .

Figure 13 contains all the experimental data of crushing energy. The relationship between crushing energy and characteristic diameter is shown in a log-log plot. The relationship between crushing energy and characteristic diameter is an approximate line, with the slope of 2.48, which is equal to the fractal dimension of crushing particles.

(1)Uniaxial compression tests on single particle reveal that the force-displacement curves of marble particles are divided into three types depending on the particle shape.(2)Particle crushing occurs in uniaxial compression tests on single particle. The particle size distribution of crushed particles obeys fractal model with the fractal dimension of 2.48 for marble particles. The fractal dimension of crushed particles should be calculated from the particle size distribution of completely crushed particles.(3)The relationship between the crushing strength and particle size is presented in the fractal model for the particle crushing. The proposed relationship between the crushing strength and particle size is verified by experiments of marble particles.(4)Based on the fractal model for the particle crushing, the probability of survival and crushing energy of marble particles is also derived and verified by experiments.(5)In this paper, the marble particles are selected to conduct the particle crushing tests and to derive the fractal model for crushing. However, the fractal model is not limited to marble particles and can be conveniently applied to the other particles. In addition, a simple and goodness-of-fit method is proposed to predict the crushing strength, the crushing energy, and the probability of survival of particles based on the fractal fragmentation of marble particles.

Copyright 2015 Yidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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