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The mechanical behavior of granular materials is gov- erned primarily by their structure and the applied ef- fective stresses. Structure depends on the arrangement of particles, density, and anisotropy. Particle sizes, shapes, and distributions, along with the arrangement of grains and grain contacts comprise the soil fabric. The packing characteristics of granular materials are discussed further in Chapter 5.

Particle Size and Distribution

Figure 4.3 illustrates the tremendous range in particle sizes that may be found in a soil, where different sizes are shown to the same scale. The largest size shown represents fine sand. It may be recalled that particles finer than about 0.06 mm cannot be seen by the naked eye. The orders of magnitude difference in particle sizes found in any one soil is often better appreciated from a representation such as that in Fig. 4.3 than by the usual size distribution (or grading) curve where particle diameters are shown to a logarithmic scale.

The origin of a cohesionless soil can be reflected by its grading. Alluvial terrace deposits and aeolian de- posits tend to be poorly graded or sorted. Glacial de- posits such as Boulder clays and tills are often well graded, containing a wide variety of particle sizes. Small particles in a well-graded soil fit into the voids between larger particles. Well-graded cohesionless soils are relatively easy to compact to a high density by vibration. The loss of fine fraction by internal ero- sion can lead to large changes in engineering proper- ties. Uniformly graded soils are usually used for controlled drainage applications because they are not susceptible to loss of fines by internal erosion and their

Figure 4.2 Interactions between clay minerals as indicated by liquid limit (data from Seed et al., 1964).

hydraulic conductivity can be maintained within defin- able and narrow limits.

The slope of the grain size distribution curve is char- acterized by the coefficient of uniformity Cu:

d60

Cu⫽ (4.1)

d10

where d60and d10correspond to the sieve sizes that 60

and 10 percent of the particles by weight pass through. A soil with Cu ⬎ 5 to 10 is considered well-graded.

The possible range of packing of soil particles is often related to the maximum and minimum void ratios (or minimum and maximum densities) reflecting the loosest and densest states, respectively. Uniformly graded soils tend to have a narrower range of possible densities compared to well-graded soils. Soils contain- ing angular particles tend to be less dense than soils with rounded particles, as discussed later in this sec- tion. However, angular and weak materials may crush significantly more during compression, compaction, or deformation. Figure 4.4 shows how the maximum and

minimum void ratios change by mixing sand and silt in different proportions. At low silt contents, silt par- ticles fit into the voids between larger sand particles, so the void ratio of sand–silt mixtures decreases with increase in silt content. However, at a certain silt con- tent, the silt fully occupies the voids, and the increase in silt content results in sand particles floating inside the silt matrix. Then, the void ratios increase with fur- ther increase in silt content.

The relative density, DR, a measure of the current

void ratio in relation to the maximum and minimum void ratios, and applied effective stresses controls the mechanical behavior of cohesionless soils. Relative density is defined by

emax⫺ e

DR ⫽ ⫻ 100% (4.2)

emax⫺ emin

in which emax, emin, and e are the maximum, minimum,

and actual void ratios.

The relative density correlates well with other prop- erties of granular soils. As different standard test meth-

ENGINEERING PROPERTIES OF GRANULAR SOILS 87

Figure 4.3 Different grain sizes in soil.

0 10 20 30 40 50 60 70 80 90 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Silt content (%) Maximum void ratio Minimum void ratio

Void ratio

Figure 4.4 Maximum and minimum void ratios of Monterey sand–silt mixtures (from Polito and Martin, 2001).

Morphology (large scale) _{Roundness Texture} (intermediate scale)

Surface Texture (small scale) Roundness Texture

(intermediate scale)

Figure 4.5 Scale-dependent particle shape characterization. The solid line gives the particle outline. Morphology de- scribes overall shape of the particle as given by the heavy dotted line. Texture reflects the smaller scale local features of the particles as identified by light dotted circles. The ex- amples are surface smoothness, roundness of edges and cor- ners, and asperities.

ods can give different limiting void ratios, the use of the relative density is sometimes criticized, especially when considered in relation to the random in situ var- iations of the density of most sand and gravel deposits. Nonetheless, if properly interpreted, relative density can provide a very useful measure of cohesionless soil properties.

Particle Shape

Particle shape is an inherent soil characteristic that plays a major role in mechanical behavior of soils. Characterization of particle shape is scale dependent, as shown in Fig. 4.5. At larger scales, that is, that of the particle itself, the particle morphology might be described as spherical, rounded, blocky, bulky, platy,

elliptical, elongated, and so forth. At smaller scales, the texture, which reflects the local roughness features such as surface smoothness, roundness of edges and corners, and asperities, is important.

With the exception of mica, most nonclay minerals in soils occur as bulky particles.2 _{Most particles are}

2_{Quartz particles become flatter with decreasing size and may have}

a platy morphology when subdivided to a fineness approaching clay size (Krinsley and Smalley, 1973).

Figure 4.6 Grain shape distribution of Monterey No. 0 sand. Results are based on study of

277 particles, d_{50 ⫽}0.43 mm, Cu⫽1.4 (Mahmood, 1973).

not equidimensional, however, and are at least slightly elongate or tabular. A frequency histogram of particle length-to-width ratio (L /W ) for Monterey No. 0 sand is shown in Fig. 4.6. This well-sorted beach sand is composed mainly of quartz with some feldspar. The mean of all the particle measurements is an L /W ratio of 1.39. This distribution is typical of that for many sands and silty sands.

Particle morphology in soil mechanics has histori- cally been described using standard charts against which individual grains may be compared. A typical chart and some examples are shown in Fig. 4.7 (Krum- bein, 1941; Krumbein and Sloss, 1963; Powers, 1953).

Sphericity is defined as the ratio of the diameter of a

sphere of equal volume to the particle to the diameter of the circumscribing sphere. Roundness is defined as the ratio of the average radius of curvature of the cor- ners and edges of the particle to the radius of the max- imum sphere that can be inscribed (Wadell, 1932). Sphericity and roundness are measures of two very dif- ferent morphological properties. Sphericity is most de- pendent on elongation, whereas roundness is largely dependent on the sharpness of angular protrusions from the particle. Different definitions of sphericity and roundness are available, as shown in Table 4.1. Due to the variety of definitions available, the quanti- fication of particle shape requires accurate specifica- tion of their definition.

In recent years, techniques for computer analysis of shape data by digital imaging have improved greatly, and standard software applications include determina-

tion of aspect ratio and roundness. A convenient way to characterize particle shapes in more detail is by a Fourier mathematical technique. For instance, the (R,

) Fourier method is in the following form:

N

R() ⫽a₀ ⫹

冘

(a cos nn  ⫹b sin nn ) (4.3)

n⫽1

where R() is the radius at angle, N is the total num- ber of harmonics, n is the harmonic number, and a and

b are coefficients giving the magnitude and phase for

each harmonic. The lower harmonic numbers give the overall shape; for instance, the sphericity is expressed by the first and second harmonics. The coefficient val- ues for higher-order descriptors generally decay with increasing descriptor or harmonic number, which ex- presses smaller features (i.e., texture) (Meloy, 1977). Other mathematical methods to curve-fit particle shapes are listed in Table 4.1. Further discussion on particle shape characterization is given by Barrett (1980), Hawkins (1993), Santamarina et al. (2001), and Bowman et al. (2001).

In an assembly of uniform size spherical particles, the loosest stable arrangement is the simple cubic packing giving a void ratio of 0.91. The densest pack- ing is the tetrahedral arrangement giving a void ratio of 0.34. Particle shape affects minimum and maximum void ratios as shown in Fig. 4.8 (Youd, 1973). The values increase as particles become more angular or the roundness (defined as roundness 1 in Table 4.1)

ENGINEERING PROPERTIES OF GRANULAR SOILS 89

Very Angular

Angular Rounded Well

Rounded (b) Roundness (a) Sphericity High Sphericity Low Sphericity Subangular Subrounded 0.9 0.7 0.5 0.3 0.9 0.7 0.5 0.3 0.1

Figure 4.7 Particle shape characterization: (a) Chart for visual estimation of roundness and sphericity (from Krumbein and Sloss, 1963). (b) Examples of particle shape characterization (from Powers, 1953).

decreases. When R ⫽ 1, the particle is a sphere. As particles become more angular, R decreases to zero. Void ratios are also a function of particle size distri- bution; the values decrease as the range of particle sizes increases (increase in the coefficient of unifor- mity Cu).

The friction angle increases with increase in particle angularity, possibly as a result of an increase in coor- dination number. For example, values of the angle of repose3 _{are plotted against roundness in Fig. 4.9 and}

3_{Angle of repose can be determined by pouring soil in a graduated}

cylinder filled with water. Tilt the cylinder more than 60 and bring it back slowly to the vertical position. The angle of the residual sand slope is the angle of repose. Further details of the method can be found in Santamarina and Cho (2004).

the following linear fit to the relationship is proposed (Santamarina and Cho, 2004);

repose ⫽42 ⫺17R (4.4) where R is the coefficient of roundness defined as roundness 1 in Table 4.1. Similar data relating friction angle from drained triaxial tests and particle shape is presented by Sukumaran and Ashmawy (2001). Particle Stiffness

Soil mass deformation at very small strains originates from the elastic deformations at points of contact be- tween particles. Contact mechanics shows that the elas- tic properties of particles control the deformations at particle contacts (Johnson, 1985), and these deforma-

Table 4.1 Methods for Particle Shape Characterization

Method Definition

Morphology—Sphere

Sphericity 1 Diameter of a sphere of equal volume Diameter of circumscribing sphere

Sphericity 2 Particle volume

Volume of circumscribing sphere Sphericity 3

Projection sphericity Area of particle outline

Area of a circle with diameter equal to the longest length of outline Inscribed circle sphericity Diameter of the largest inscribed circle

Diameter of the smallest inscribed circle

Morphology—Ellipse

Eccentricity p/ Rap, where the ellipse is characterized by Rp ⫹pcos 2 in polar coordinates

Elongation Smallest diameter

Diameter perpendicular to the smallest diameter

Slenderness Maximum dimension

Minimum dimension

Texture—Roundness

Roundness 1 Average of radius of curvature of surface features, (兺r ) / Ni

Radius of the maximum sphere that can be inscribed, rmax Roundness 2 Radius of curvature of the most convex part

0.5 (longest diameter through the most convex part) Roundness 3 Radius of curveture of the most convex part

Mean radius

Morphology—Texture

Fourier method Eq. (4.3), first and second harmonics, characterize sphericity, whereas higher harmonics (around 10th) characterizes roundness. Surface texture is characterized by much higher harmonics.

Fourier descriptor method

More flexible than the Fourier method by using the complex plane (Bowman et al., 2001). Lower harmonics give shape characteristics such as elongation, triangularity, squareness, and asymmetry. Higher harmonics (larger than 8th) give textural features. Fractal analysis Use as a measure of texture (Vallejo, 1995; Santamarina, et al. 2001).

From Hawkins (1993), Santamarina et al. (2001), and Bowman et al. (2001).

ENGINEERING PROPERTIES OF GRANULAR SOILS 91 0.6 0.6 0.4 0.2 0.8 0.8 1.0 1.2 1.4 1 2 3 4 6 10 15 Subrounded Rounded Coefficient of Uniformity, Cu 0.35 0.49 0.70 0.30 0.25 0.20 R = 0.17 Subangular Subangular Angular R = 0.20 Angular R = 0.20 Angular Subrounded Rounded 0.70 0.49 0.35 0.30 0.25 0.20

Maximum Void ratio,

emax

Minimum Void Ratio,

emin

Figure 4.8 Maximum and minimum void ratios of sands as a function of roundness and the coefficient of uniformity (from Youd, 1973). 50 40 30 20 10 0.0 0.2 0.4 0.6 0.8 1.0 Roundness R φrepose = 42 – 17R Angle of repose φrepose

Figure 4.9 Angle of repose as a function of roundness (from Santamarina and Cho, 2004).

Table 4.2 Elastic Properties of Geomaterials at Room Temperature Material Young’s Modulus (GPa) Shear Modulus (GPa) Poisson’s Ratio Quartz 76 29 0.31 Limestone 2–97 1.6–38 0.01–0.32 Basalt 25–183 3–27 0.09–0.35 Granite 10–86 7–70 0.00–0.30 Hematite 67–200 27–78 — Magnetite 31 19 — Shale 0.4–68 5–30 0.01–0.34

After Santamarina et al. (2001).

tions in turn influence the stiffness of particle assem- blages. Elastic properties of different minerals and rocks are listed in Table 4.2. The modulus of a single grain, which determines the particle contact stiffness, is at least an order of magnitude greater than that of the particle assembly. Further details on the relation between particle stiffness and particle assemblage stiff- ness are given in Chapter 11.

Particle Strength

The crushability of soil particles has large effects on the mechanical behavior of granular materials. At high stresses, the compressibility of sand becomes large as a result of particle crushing, and the shape of an e–log

p compression curve becomes similar to that of nor-

mally consolidated clay (Miura et al., 1984; Coop, 1990; Yasufuku et al., 1991). Under constant states of stress, the amount of particle breakage increases with time, contributing to creep of the soil (Lade et al., 1996). The amount of crushing in a soil mass depends both on the stiffness and strength of the individual grains and how applied stresses are transmitted through the assemblage of soil particles.

Particle strength or hardness is characterized by crushing at contacts or particle tensile splitting. There is a statistical variation in grain strength for particles of a specified material and of a given size (Moroto and Ishii, 1990; McDowell, 2001). Random variation in grain strengths leads to distributions of particle sizes when large stress is applied to a soil assembly. Table 4.3 lists the characteristic tensile strengths of some soil particles. The values are smaller than the yield strength of the material itself. The strength also depends on the particle shape. For example, Hagerty et al. (1993) show that angular glass beads were more susceptible to breakage than round glass beads.

Table 4.3 Strength of Soil Particles

Sand Name Size (mm)

37% Tensilea

Strength (MPa)

Mean Strengthb

(MPa) Reference

Quartz

Leighton Buzzard silica sand 1.18 — 29.8 Lee (1992)

2.0 — 24.7

3.36 — 20.5

Toyoura sand 0.2 147.4 136.6 Nakata et al. (2001)

Aio quartz sand 0.85 51.2 52.1 Nakata et al. (1999)

1.0 47.7 46.6

1.18 37.9 35.6

1.4 46.7 42.4

1.7 39.6 38.5

Silica sand 0.5 147.4 132.5 McDowell (2001)

1 66.7 59.0

2 41.7 37.3

Silica sand 0.28 110.9 147.3 Nakata et al. (2001)

0.66 72.9 73.1

1.55 31.0 29.7

Feldspar

Aio feldspar sand 0.85 20.9 24.6 Nakata et al. (1999)

1.0 24.3 22.8

1.18 18.1 18.2

1.4 23.1 21.4

1.7 18.9 18.3

Calcareous Sand

Oolitic limestone particle 5 — 2.4 Lee (1992)

8 — 2.1 12 — 1.8 20 — 1.5 30 — 1.3 40 — 1.2 50 — 1.1 Carboniferous limestone 5 — 14.9 particle 8 — 12.2 Lee (1992) 12 — 10.3 20 — 8.3 30 — 7.0 40 — 6.2 50 — 5.7

Quiou sand 1 109.3 96.19 McDowell and Amon (2000)

2 41.4 36.20

4 4.2 3.87

8 0.73 0.63

16 0.61 0.54

Others

Masado decomposed granite

soil 1.55 24.2 22.1 Nakata et al. (2001)

Glass beads 0.93 365.8 339.6 Nakata et al. (2001)

Angular glass 0.93 62.1 60.0 Nakata et al. (2001)

a_{Stress below which 37% of the particles do not fracture.} b_{Force / d}2_{at which particle of size d is crushed.}

ENGINEERING PROPERTIES OF GRANULAR SOILS 93 Grain size (mm) 100 80 60 40 20 0 0.01 0.1 1.0 20.7 MPa 41.4 MPa 62.1 MPa 103 MPa 345 MPa 517 MPa 689 MPa Maximum stress Uncrushed

Percent Finer by Weight

Figure 4.11 Evolution of particle size distribution curve upon crushing (from Hagerty et al., 1993).

1 5 10 50 100 0.2 1.0 0.5 5.0 10 50

Average Particle Size (mm) Leighton Buzzard Sand

Carboniferous Limestone Angular River Gravel

Rounded River Gravel

Oolitic Limestone

Particle Strength (MPa)

Figure 4.10 Relationship between tensile strength and particle size (from Lee, 1992).

The breakage potential of a single soil particle in- creases with its size as illustrated in Table 4.3. This is because larger particles tend to contain more and larger internal flaws and hence have lower tensile strength. Fig. 4.10 shows that oolitic limestone, carboniferous limestone, and quartz sand exhibit near linear declines in strength with increasing particle size on a log–log plot (Lee, 1992).

The amount of particle crushing in an assemblage of particles depends not only on particle strength, but also on the distribution of contact forces and arrange- ment of different size particles. It can be argued that larger size particles are more likely to break because the normal contact forces in a soil element increase with particle size and the probability of a defect in a given particle increases with its size as shown in Fig 4.10 (Hardin, 1985). However, if a larger particle has contacts with neighboring particles (i.e., larger coor- dination number), the load on it is distributed, and the probability of facture is less than for a condition with fewer contacts. Experimental evidences suggest that fines increase as particles break by increase in applied pressure. For example, the evolution of particle size distribution curves for Ottawa sand in one-dimensional compression is shown in Fig. 4.11 (Hagerty et al., 1993). Hence, the coordination number dominates over size-dependent particle strength. Larger particles have higher coordination numbers because they are in con-

tact with many smaller particles. The very smallest particles have a lower coordination number because there are fewer smaller particles available for contact. Hence, the largest particles in the aggregate become protected by the surrounding newly formed smaller particles, and smaller particles are more likely to break

Figure 4.12 Weight–volume relationships for a saturated clay-granular soil mixture.

or move. Further details on particle breakage effects on compression behavior of sands are given in Chap- ter 10.

4.4 DOMINATING INFLUENCE OF THE CLAY

PHASE

In general, the more clay in a soil, the higher the plas- ticity, the greater the potential shrinkage and swell, the lower the hydraulic conductivity, the higher the com- pressibility, the higher the cohesion, and the lower the internal angle of friction. Whereas surface forces and their range of influence are small relative to the weight and size of silt sand particles, the behavior of small and flaky clay mineral particles is strongly influenced by surface forces, as discussed in Chapter 6. Water is strongly attracted to clay particle surfaces, also dis- cussed in Chapter 6, and results in plasticity, whereas nonclay particles have much smaller specific surface and less affinity for water and do not develop signifi- cant plasticity, even when in finely ground form.

If it is assumed as a first approximation that all of the water in a soil is associated with the clay phase, the amount of clay required to fill the voids of the granular phase and prevent direct contact between granular particles can be estimated for any water con- tent. The weight and volume relationships for the dif- ferent phases of a saturated soil are shown in Fig. 4.12. In this figure W represents weight, V is volume, C is the percent clay by weight, GSCis the specific gravity

of clay particles, w is the water content in percent, w

is the unit weight of water, and GSG is the specific

gravity of the granular particles. The volume of voids in the granular phase is eGVGS, where eG is the void

ratio of the granular phase and VGS is the volume of

granular solids, given by

C Ws

e VG GS⫽

冉 冊

1⫺ eG (4.5)

100 GSG w

The volume of water plus volume of clay is given by

w WS C WS

VW⫹ VC⫽ ⫹ (4.6)

100 w 100 GSC w

If clay and water completely fill the voids in the gran- ular phase, then

w Ws C W C Ws ⫹ ⫽

冉 冊

1⫺ eG (4.7) 100 w 100 GSC w 100 GSG w which simplifies to w C C eG ⫹ ⫽

冉 冊

1⫺ (4.8) 100 100GSC 100 GSG

The void ratio of a granular material composed of bulky particles is of the order of 0.9 in its loosest pos- sible state. The specific gravity of the nonclay fraction in most soils is about 2.67, and that of the clay fraction is about 2.75. Inserting these values in Eq. (4.8) gives

C⫽48.4 ⫺1.42w (4.9)

This relationship indicates that for water contents typ- ically encountered in practice, say 15 to 40 percent, only a maximum of about one-third of the soil solids need be clay in order to dominate the behavior by pre- venting direct interparticle contact of the granular par- ticles. In fact, since there is a tendency for clay particles to coat granular particles, the clay can signif- icantly influence properties. For example, just 1 or 2 percent of highly plastic clay present in gravel used as a fill or aggregate may be sufficient to clog handling and batching equipment.

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