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CRUSHING OF GRANULAR BASES: FRACTAL AND LABORATORY ANALYSES

Luis E. Vallejo

Associate Professor Department of Civil and Environmental Engineering

University of Pittsburgh, USA E-Mail: vallejo@civ.pitt.edu

ABSTRACT

Granular materials forming part of the base of flexible pavements experience crushing as a result of static and dynamic loads. Very little research has been conducted to date on how to evaluate crushing and what effect varying levels of crushing have on the engineering properties of these granular materials (for example, shear strength and hydraulic conductivity). In this study crushing of granular materials is evaluated using fractals. Crushing of a particle can be the result of either its abrasion or its total fragmentation. Abrasion takes place when the sharp corners of the particle are removed as a result of shear, compression or both. Thus, as a result of abrasion, the particle changes in shape. In this study, the fractal dimension of the particle profile evaluates the changes in shape of a particle before and after abrasion. It was determined that the rougher a particle profile is; the higher is the fractal dimension. Thus, the fractal dimension concept is an excellent tool to measure abrasion in particles. When crushing is the result of fragmentation of a particle, the structure of the particle changes from a single solid element to a mixture of many small particles of varying sizes. When a granular base experience fragmentation, the resulting granular mass will be composed of a granular mixture that has a fractal distribution in particle sizes (large, medium and small). In this study, crushing as a result of fragmentation was evaluated using the fragmentation fractal dimension of the size distribution of the particles before and after crushing. This study presents the results of a ring shear tests on sand. This study was conducted to evaluate the crushing experienced by the sand using fractals. The effect of crushing on the shear strength and the hydraulic conductivity of the sand as a result of a sustained application of normal and shear stresses were also evaluated. The hydraulic conductivity, K, of the sand was calculated using a relationship developed by Hansen. This relationship relates K to the D10 obtained from the grain size distribution curve. The hydraulic conductivity of the sand was found to decrease with an increase in its level of crushing. The friction angle measured in the ring shear tests was found to decrease with an increase in the fractal dimension values. High values of fractal dimension are associated with high normal and shear stresses in the ring shear test. These high normal and shear stresses cause the grains to change from rough to smooth with the resulting decrease in shear strength These changes in the roughness of the sand grains seem to be the controlling factor for the decrease in shear strength measured in the ring shear tests.

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1.0 INTRODUCTION Granular materials form part of engineering structures such as the base of flexible pavements, highway embankments, and foundations. The granular materials forming part of these structures are subjected during their engineering lives to either static or dynamic loads. Very little research has been conducted to date on the effect that varying levels of crushing have on the engineering properties of granular materials (i.e. hydraulic conductivity, shear strength). Because of sustained crushing, the original engineering properties with which a structure (i.e. a pavement or a highway embankment) was designed will change during its engineering life. Changes in the original engineering properties could affect the stability of the structure and could make it unsafe. Thus, there is a need to understand the evolution of crushing in granular materials. In this study, the evaluation of crushing of granular materials is conducted using fractal theory. Also, laboratory experiments in the form of ring shear strength and static compression tests are used to induce crushing in granular materials. Grain size distribution analysis of the granular materials was conducted before and after crushing. The grain size distribution curves are used to evaluate crushing levels and to calculate changes in the hydraulic conductivity and the shear strength of the sample. 1.1 The Crushing of Granular Materials

Granular materials form part of engineering structures such the base of flexible pavements, highway embankments, and foundations. The granular materials forming part of these structures are subjected during their engineering lives to either static or dynamic loads. As a result of these loads, particle breakage occurs (Hendron, 1963; Vesic and Clough, 1968; Lee and Farhoomand, 1967; Miura and Ohara, 1979; Hardin, 1985; Hagerty et al., 1993; Lade et al., 1996; Coop, 1999; Bolton, 1999; and Feda, 2002). According to Lee and Farhoomand (1967) and Coop (1999), particle breakage or crushing seems to be a general feature for all granular materials. Grain crushing is influenced by grain angularity, grain size, uniformity of gradation, low particle strength, high porosity, and by the stress level and anisotropy (Bohac et al., 2001). According to Lee and Farhoomand (1967), one of the most important factors influencing the crushing of a mass of granular materials is the crushing resistance of the grains. Coarse granitic sand particles with an average diameter of 2.8 mm experienced breakage at pressures equal to 2 MPa, while calcareous shells begin crushing at 0.05 to 0.2 MPa (Lade and Farhoomand, 1967, Bohac et al, 2001). Angular particles of freshly quarried materials undergo fragmentation under ordinary pressures (about 0.98 MPa) due to breakdown of sharp angularities (Ramamurthy, 1968). When a granular mass is subjected to a compressive load, the particles resist the load through a series of contacts between the grains (Oda and Konishi, 1974; Radjai, 1995)(Fig. 1). As shown in Fig.1, particles do not share equally in the bearing of the applied load. Some particles carry more load than others. In fact, some particles can actually be removed without affecting the mechanical equilibrium of the packing. The particles with highly loaded contacts are usually aligned in chains (Cundall and Strack, 1979).. Crushing starts when these highly loaded particles fail and break into smaller pieces that

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move into the voids of the original material. These load chains change in intensity and direction as the crushing develops in the particle assemblage On crushing, fines are produced and the grain size distribution curve becomes less steep. Consequently, with continuing crushing, the soil becomes less permeable and more resistant to crushing. Grain size distribution is a suitable measure of the extent of crushing (Hardin, 1985; Lade et al., 1996).

Figure 1. Force chains in two dimensional static assembly of discs in a horizontal container with three fixed walls and one piston, on which a fixed vertical force is applied (Radjai, 1995). As mentioned before, compression induced crushing of granular material (as in the unbound granular base under an asphalt pavement) causes a decrease in volume of the original assembly of grains (settlement if the grains are laterally confined). This decrease in volume is the result of the breakage of some of the grains. The grains that break could consist of pieces that are large enough to occupy the space of the original particles, or they can break into multiple small pieces. If the breakage produces few large pieces, this type of breakage will not cause a substantial volume decrease of the original granular structure. On the other hand, if the grains that break produce multiple small pieces that are small enough to migrate to the adjacent voids in the granular structure, then the decrease in volume of the original structure will be substantial. The compressive pressure acting on the small broken grains aids in their migration inside the granular mass. Also, the number of broken grains will be a function of the level of compressive force acting on the granular assembly. The larger this compressive force, the larger will be the number of broken grains. Lade et al. (1996) found that if a uniform granular material is crushed, the resulting grain size distribution approaches that of a well graded soil for very large compressive loads. Bolton (1999) established that the grain size distribution of a granular assembly

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that has been crushed under large compressive loads is a fractal distribution. A well graded particle distribution or a fractal distribution represents a granular structure that is made of grains of all sizes including the original unbroken grains. These original large grains did not break based on the fact that with more small size particles surrounding them, the average contact stress acting on these large grains tends to decrease (Lade et al., 1996). However, before the granular structure reaches a well graded or a fractal particle size distribution, the granular structure will experience gradual changes in particle sizes depending on the magnitude of the compressive load applied to it. A simplified representation of the crushing process of a laterally confined granular assembly (i.e. the unbound granular base under an asphalt pavement) that is subjected to a vertical load, P, is shown in Fig. 2. This first figure shows an unbroken granular structure represented by discs which are stacked in a cubical arrangement [Fig. 2(a)]. This granular arrangement represents a loose granular packing. At small magnitudes of the compressive load, P, few discs (particles) break. The particles that break form small isolated dense zones that are surrounded by connected zones of loose granular packings [Fig. (2(b)]. In Fig. 2(b), vo id space which has been filled by crushed material is equal to 31% of the total void area. As the compressive force producing crushing increases, the number of dense zones will also increase because additional particles break [Figs. 2(c) and 2(d)]. Fig. 2(c) is a simplified representation of a high level of crushing. In Fig. 2(c) the dense packing zones are interconnected and the loose packing zones become isolated. In Fig. 2 (c), the percentage of the void area covered with crushed material has increased to 64.5%. In Fig. 2(d), the percentage of void area covered by crushed material has reached a value of 82%. Pavements are the most unusual structures designed by civil engineers. Water enters through their tops, bottoms, and sides, but because pavements are relatively flat, the water flows out again very slowly unless they are well drained under their full width (Cedergreen, 1994). Most serious problems are caused to asphalt pavements when their granular bases are unable to remove the water that enters the pavement. Fig. 2(a) represents a well drained granular base assuming drainage goes vertically or laterally. In Fig. 2(b) the loose zones that drains the water are interconnected. Thus, drainage in the vertical or horizontal direction is still possible. In Fig. 2(b) the dense zones, which are zones that prevent drainage, are isolated and not continuous; for that reason drainage is still possible. In Figs. 2(c) and 2(d), the loose zones that drain the granular base in either the vertical or horizontal direction are no longer connected. These loose zones must be interconnected in order for water to drain from underneath the pavement. In Figs. 2(c) and 2(d), the dense zones made of crushed material are the ones that are interconnected. These dense zones made of crushed granular material surround and isolate the loose zones that promoted drainage.

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Figure 2. Evolution of crushing in a confined granular material under compression

Thus, when the granular base reaches the conditions of Figs. 2(c) and 2(d) as a result of crushing, serious problems will develop in pavements. Due to traffic loads, the material in the loose isolated zones will act as closed hydraulic systems that will develop excess pore water pressures, producing the failure of the granular base as well as the pavement (Cedergreen, 1994). The granular base for flexible pavement structures requires very durable rock particles that do not crush under traffic loads. Because this type of uncrushable material is sometimes difficult to obtain, the choice of a granular material that crushes somewhat and allows water to drain is warranted [Fig. 2(b)]. To avoid the detrimental effect of the settlement reported in Fig. 2(b) on a flexible pavement, the granular base could be designed with a grain size distribution acquired by this base after some moderate crushing that produces some fine material [Fig. 2(b)]. The finer material in the mixture will help reduce point-to-point contact between the large particles, prevent ing their rotation and reorientations, which is one of the causes of the long-term settlement of rockfills (Vallejo, 2001). Thus, the grain size distribution effective in Fig. 2(b) will produce a structure with an improved resistance to crushing (since resistance to crushing increases with crushing) and will also allow water to drain. In the field, the size distribution after some crushing [Fig. 2(b)] could be achieved using heavier compaction equipment. After this distribution has been achieved, the flexible pavement could be laid on this granular material. In this case, a certain amount of crushing could have positive effects on the performance of flexible pavement structures. Very little is known about the effect that varying levels of crushing have on the engineering properties of granular materials (i.e. hydraulic conductivity, and shear strength). Because of sustained crushing, the original engineering properties with which a structure (i.e. a pavement or a highway embankment) was designed will change during its engineering life. Changes in the original engineering properties could affect the stability of the structure and could make it unsafe.

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1.2 Levels of Crushing in Granular Bases The granular base makes up the greatest thickness in a flexible pavement structure, and provides both bearing strength and drainage for the pavement structure. Hence, proper size, grading, shape and durability are important attributes to the overall performance of the pavement structure. Granular base aggregates may consist of durable particles of crushed stone, gravel or slag capable of withstanding the effects of handling, spreading, and compacting without the generation of fines. The use of angular, nearly equidimensional aggregate with rough surface texture is preferred over rounded, smooth aggregate particles. Granular bases with this shape and texture give granular bases greater stability and proper drainage (Barksdale, 1991; AASHTO, 1993). However, as a result of static and dynamic loads, the aggregates could produce fines as a result of particle abrasion or complete particle disintegration. The particle abrasion can change particles from rough to round, and the complete disintegration of the particles can produce a granular base with a size distribution that is fractal in nature (a mixture of large, small, and fine particles). If the granular base experience abrasion only, its shear strength, stability and hydraulic conductivity could decrease. If large portions of the granular base experience complete fragmentation, the granular base will experience large settlements (Fig. 2) that could be detrimental to the pavement layer located on top of it. Fragmentation of the granular base will result in a decrease of its hydraulic conductivity. The shear strength and stability could be enhanced by the fractal size distribution of the grain structure. Next, a theoretical method based on fractal theory to evaluate crushing (abrasion and complete fragmentation) in granular materials is presented. This fractal methodology will be used to measure crushing in granular materials subjected to a combination of normal and shear stresses. 2.0 FRACTALS AND THE CONCEPT OF FRACTAL DIMENSION The shape of forms in nature is usually analyzed using Euclidean geometry. According to this kind of geometry, straight lines are perfectly straight lines and curves are arcs of perfect circles. However such perfection is seldom found in natural forms. Most of the time, the shapes of natural forms are irregular. Fractals are a relative ly new mathematical concept to describe the geometry of irregularly shaped objects in terms of fractional numbers (fractal dimension) rather than an integer. In this study the fractal dimension concept from fractal theory will be used to measure the degree of irregularity of closed profiles forming part of granular materials. Fractals will also be used to evaluate the size distribution in a granular material subjected to varying crushing levels. The key parameter for fractal analysis is the fractal dimension. This value is a real number, which differs from the more familiar Euclidean or topological dimension. The latter is an integer, with a value of one for a line of any shape and two for a surface. The fractal dimension for a line of any shape varies between one and two, and for a surface

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between two and three. The difference between the fractal and the Euclidean dimensions can be explained by what happens when a thin ink line of any shape is drawn on a sheet of paper. This line has an Euclidean or topological dimension equal to one. However, if the line is drawn in such a way as to increase it wiggliness, the paper will appear to be almost covered with ink, giving the line a dimension better represented by that of the area of the sheet of paper. The area of the sheet of paper has a topological dimension equal to two. Thus, the real dimension of irregular complex lines lie somewhere between that for the ideal Euclidean line with a fine width, and the extreme case of a complete surface cover. The fractal dimension measures the surface filling properties of wiggling or irregular lines and its value approaches that of the Euclidean dimension of the surface that encloses them. The fractal dimension for lines or profiles of any shape is a real number that varies between one and two. The rougher or more irregular the line, the larger is its fractal dimension (Vallejo, 1995, 1996). 2.1 The Fractal Dimension of Open Profiles Many methods have been developed to measure the fractal dimension of open and closed form profiles such as those forming part rock joints, geomembranes, pavements, sands, gravels, and voids in soils (Vallejo and Zhou, 1995). The most commonly used methods are: (a) the divider method, (b) the box method, and (c) the spectral method (Cox and Wang, 1993). In the present study, the divider method will be used to measure the fractal dimension of open and closed form profiles. The divider method used to obtain the fractal dimension, D, can be explained using Figs.3 and 4. Fig. 3 represent s two profiles, one very complex (profile ab in Figs. 3 and 4), and the other a very simple one (profile a’b’ in Fig. 3). Suppose we wish to measure The divider method used to obtain the fractal dimension, D, can be explained using Figs.3 and 4. Fig. 1 represents two profiles, one very complex (profile ab in Figs. 3 and 4), and the other a very simple one (profile a’b’ in Fig. 3). Suppose we wish to measure the length L of the of the profiles shown in Fig. 1 using a ruler or yardstick of fixed length, r. We may begin by setting two arms of a divider to a known distance (step or segment length r) and step off the outline of the profiles as shown in Fig. 4. The length of the profiles, L, is obtained from the product of the number of segments, n, and the chosen segment length, r. Three different segment lengths, r, were used to measure both the complex and the simple profiles. The scales for the length of these segments are shown in Fig. 2. Table 1 shows the length of the profiles using different segments r. An analysis of the results shown in Table 1 indicates that the length of the profile ab increased as the length of the segments used to measure its length decreased. The reason for this is that as the segment length r is decreased, the smaller details in the profile which were previously stepped over add their contribution to the length of the profile (Fig. 4). As smaller segments are used, the length of the profile ab continue to increase. Opposite results were obtained for the case of the simple profile a’b’. Table 1 indicates that the length of this

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Fig. 3. Plots of a complex and simple open form profiles Figure 4. Segment scale r and the number of segments n to cover the complex profile ab.

Table 1 Data for fractal dimension calculation for profiles shown in Figs. 1 and 2

____________________________________________________________________ Profile Segment Length Number of Segments Length of Profile r n L = nr _____________________________________________________________________ 1 79.3 79.3 ab 4 17.8 71.2 8 8.5 68.0 _____________________________________________________________________ 1 52.8 52.8 a’b’ 4 13.2 52.8 8 6.6 52.8 ______________________________________________________________________

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profile remained constant regardless of the length of the segment r used to measure the length of the simple profile. According to Mandelbrot (1977) and Turcotte (1992), if a linear relationship develops between the values n and r when plotted on a log- log paper, the profiles analyzed are fractal profiles. The absolute value of the slope of the linear relationship between n and r values represents the fractal dimension, D, of the profiles. The values of n and r for the profiles ab and a’b’ (Table 1 and Fig. 4) were plotted on a log- log paper and the results are shown in Fig. 3. An examination of Fig. 5 indicates that the n and r values for both profiles plotted on straight lines. The fractal dimension, D , of the rough, complex profile ab was equal to 1.0739. The fractal dimension for the smooth, simple profile a’b’ was equal to 1.0000. Thus, the fractal dimension concept can be used to evaluate the degree of roughness of closed profiles (particle profiles).

Figure 5 Plot of number of segments n versus segment length r to obtain the fractal dimension, D , of profiles in Fig. 3.

2.2 The Fractal Dimension of Closed (Particle) Profiles: Abrasion Measurement Fig. 6 represents the profile of two particles having the same cross sectional area but different profiles. Fig. 6(A) shows the two dimensional profile of a smooth, ellipsoidal particle repeated twice. Fig. 4(B) shows the profile of a rough, ellipsoidal particle, also repeated twice. The profiles of the particles in Fig. 6(A) and 6(B) have been covered using segments of different lengths, r. The number of segments, N, of each length, r, to cover the profile of the particles are shown in Fig. 6. The number of segments, N, and the corresponding length of the segments, r, are plotted in a log- log paper (Fig. 7). The slope of the best fit line passing through the points relating N and r represents the fractal dimension D of the profiles. As expected, the fractal dimension, D, of the rough profile [Fig. 6(B)] is greater than the fractal dimension, D, for the smooth profile [Fig. 6A). The

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fractal dimension of the rough profile equal to 1.1036, and the fractal dimension of the smooth profile is equal to 1.0498 (Fig. 7). Fig. 6(B) can represent the profile of a particle before abrasion occurs. Fig. 6(A) can represent the profile of a particle after abrasion occurs. Thus, the fractal dimension concept is very useful to measure abrasion in the particles forming part of granular bases under flexible pavements.

Fig. 6. Smooth and rough particle for fractal analysis

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Fig. 7. Fractal dimension, D, for particles shown in Fig. 6 2.3 Fractal Dimension of the Grain Size Distribution: Fragmentation Measurement Grain size distribution of naturally occurring soils have been found by Tyler and Wheatcraft (1992) and Hyslip and Vallejo(1997) to be fractal. According to Mandelbrot (1977) and Tyler and Wheatcraft, the distribution of grains by size in a natural soil can be obtained using the following equation:

>r) = (1)

where N(R>r) is the total number of particles with linear dimension R (radius of the particle) which is greater than a given size r; k is a proportionality constant; and DF is the fractal dimension of the size distribution of grains. As a result of shear stresses, or a combination of compression and shear stresses, the size distribution in a granular soil will change. Changes in the size distribution of the grains will be reflected in the values of DF. Thus, grain fragmentation in soils subjected to shear stresses or a combination of compressive and shear stresses can be evaluated by the changes in their fragmentation fractal dimension, DF. To apply the number-based relationship expressed by Eq. (1), is very time consuming. Another relationship that uses the results of a standard sieve analysis test was developed by Tyler and Wheatcraft (1992) to calculate the fragmentation fractal dimension, DF , of natural soils. This relationship is:

(2) where M(R<r) is the cumulative mass (weight) of particles with size R smaller (finer) than a given comparative size r;

MT is the total mass (weight) of particles; r is the sieve size opening; rL is the maximum particle size as defined by the largest sieve size opening used in the sieve analysis; and DF is the fragmentation fractal dimension. The results of a sieve analysis tests using Eq. (2) can be plotted on log- log paper. The slope, m , of the best fitting line through data obtained using Eq. (2) and the fractal dimension, DF , are related as follows: DF = 3 – m (3) Eqs. ( 2) and (3) will be used to obtain the fractal dimension of the size distribution in a sand subjected to crushing in a ring shear apparatus. 2.4 The Fragmentation Fractal Dimension of a Sand Subjected to Ring Shear Test

RN ( FDkr−

FD

LT rr

MrRM

−

=

<3

)(

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An oven dried sand with a specific gravity equal to 2.6 containing grains with a diameter that passed sieve No. 10 (2 mm) and grains that were retained in No. 16 sieve (1.18 mm) was subjected to fragmentation in a Bromhead’s ring shear apparatus. The ring shear tests were carried out to investigate crushing of the sand as a result of sustained normal and shearing stresses. The sand was subjected first to one normal constant stress after which shearing was induced in the sample for the completion of one 3600 rotation. After this rotation was completed, the normal stress was increased and the sample was sheared again for another 3600 rotation. These rotations were carried out for various normal stresses that vary between 15 and 1,374.3 kPa. The combination of the normal and shear stresses caused some of the sand grains to crush. The crushing of some of the grains caused the original size distribution to change from a uniform sand to a well graded or fractal sand. The grain size distribution of the particles after some of the fifteen 3600 rotations is shown in Fig. 8. The fractal dimension of the grain size distribution was calculated using Eqs. (2) and (3) and the results shown in Figs. 9 and 10. An analysis of Fig. 10 indicates that the fragmentation fractal dimension, DF, changed gradually from a value of 1.4 to a value of 2.3. This change in the fractal dimension represents a sand that is gradually being crushed or fragmented in the ring shear test.

Fig. 8. Grain size distribution of sand crushed in the ring shear apparatus.

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Fig. 9. Typical plot of sieve analysis results to obtain DF at a normal stress = 1374.3 kPa.

Fig. 10.

Fragmentation fractal dimension values for sand subjected to crushing in ring shear test. 2.5 Effects of Sand Fragmentation on the Hydraulic Conductivity There are a number of empirical correlations relating grain size with hydraulic conductivity. One of the most widely used correlations relating hydraulic conductivity and grain size is that of Hansen (1911) formula K = 100 (D10)2 (4) in which K is the hydraulic conductivity (cm/s), and D10 = grain diameter (cm) corresponding to 10% of the material being smaller by weight (also called the effective grain size). Eq. (4) was used in conjunction with the grain size distribution curves shown in Fig. 8 to evaluate the hydraulic conductivity during the crushing of the sand in the ring shear apparatus. The results of this analysis is shown in Fig. 11.

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Fig. 11. Relationship between hydraulic conductivity and fractal dimension DF.

An analysis of Fig. 11 shows the hydraulic conductivity, K, decreases as the value of the fractal dimension, DF, increases. At the beginning of the ring shear test, the normal stress is low, the sample is loose, and the profile of the sand particles is rough. These conditions make that the hydraulic conductivity of the sand be large. As the normal and shear stresses acting on the sample increases, the sand experience crushing. The crushed material fills the void spaces located within the grains that have not crushed. This results in an overall decrease in the hydraulic conductivity of the sand. The changes experienced by the sand under a combination of normal and shear stresses in the ring shear apparatus and the influence that these changes have on the hydraulic conductivity of the crushed sand can be best explained using Fig. 12. This figure shows what happens to a pore located within three large particles when it is gradually filled by smaller and smaller grains resulting from the gradual crushing of the larger grains. The grains shown in Fig. 12 are self-similar with respect to their sizes and represent a sand with a fractal size distribution. Fig. 12(a) shows the pore between the three grains when it is filled by one small grain. The same pore continues to be filled by smaller and smaller grains as one goes from Fig. 12(b) to Fig. 12(d). The pore space decreases gradually until it becomes completely blocked [Fig. 12(d)]. Thus, when the pore reaches the condition shown in Fig. 12(d), water can not move through the pore. Thus, the filling of the pore space by a soil with a fractal size distribution will influence the hydraulic conductivity of the pore and the sand that contains it.

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Fig. 12 . A pore being filled by a fractal soil

2.6 Effect of Sand Fragmentation on the Shear Strength Fig. 13 shows the relationship between the friction angle, φ , measured in the ring shear apparatus and the fractal dimension, DF. This figure shows that the friction angle decreases with an increase in the fractal dimension values. From the ring shear tests it was determined that an increase in fractal dimension is associated with an increase in normal and shear stresses as well as in the levels of crushing (Figs. 8 and 10). At low normal and shear stress levels, the san particles are in a loose arrangement and their profile are rough. This roughness of the partic les causes them interlock when subjected to shear. This high level of interlocking makes the friction angle of shearing resistance to be high. At high normal and shear stress levels, the roughness of the particles is somewhat eliminated due to the abrasion that takes place during the shearing of the sand. This abrasion makes the particles to change from rough to smooth, producing as a result a decrease in the friction angle of shearing resistance. This decreases in shearing resistance due to particles loosing their roughness seem to be a dominant factor over the increase in shear resistance that should take place as a result of the sand sample changing from uniform to a well graded sand (Figs. 8 and 12).

Fig. 13. Relationship between friction , φ , angle and fractal dimension, DF.

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2.7 Some Comments on the Application of Fractals to Granular Base In the previous paragraphs the methodology of using fractals to analyze the abrasion and fragmentation of granular materials and their effect on the mechanical properties (shear strength and hydraulic conductivity) was applied to only a sand sample. This methodology, however, is general and can be used to analyze the effect that the partial and complete fragmentation has on the engineering properties of the aggregates that form part of a pavement’s granular base. The testing equipment to measure crushing, shear strength, and hydraulic conductivity of these aggregates has to be large enough in order accommodate the large size of their particles. 3. CONCLUSIONS The fractal dimension concept from fractal theory has been presented to evaluate partial crushing (abrasion) and fragmentation of granular materials. Crushing was produced by conducting ring shear tests on a sand sample. The ring shear tests cause the size distribution of the sand to change. This change was the result of sustained normal and shear stresses during the tests. The size distribution of the sand changed from non-fractal to a fractal one. The changes in the particles size distribution in the sand had a large influence on the hydraulic conductivity and the shear strength. The hydraulic conductivity decreased as the particle size distribution changed from non-fractal to a fractal one. The opposite took place when the shear strength of the sand was considered.. The shear strength of the sand measured by its friction angle decreased as the sand changed from non-fractal to a fractal one. 4. ACKNOWLEDGEMENTS The work described in this study was sponsored by Grant CMS: 0301815 to the University of Pittsburgh from the National Science Foundation, Washington, D.C. This support is gratefully acknowledged. The author gives special thanks to Drs. Bernardo Caicedo and Arcesio Lizcano from the Universidad de los Andes for helpful discussions related to the subject of this study. Thanks are also given to Mr. Zamri Chik, Ph.D. student in the Department of Civil and Environmental Engineering at the University of Pittsburgh for conducting some of the tests described in this paper. 5. REFERENCES

AASHTO Guide for the design of pavement structures (1993). American Association of State Highway and Transportation Officials, Washington, D,C. Barksdale, R. (1991). The aggregates handbook. National Stone Association, Washington, D.C.

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