Behavior of Rigid-Soft Particle Mixtures

Behavior of Rigid-Soft Particle Mixtures Jong-Sub Lee1; Jake Dodds2; and J. Carlos Santamarina3 Abstract: Mixtures of rigid sand particles and soft fine-grained rubber particles are tested to investigate their small and large-strain responses. Mixtures are prepared with different volumetric sand fraction sf to identify the transition from a rigid to a soft granular skeleton using wave propagation, k0 loading, and triaxial testing. Deformation moduli at small, middle, and large strains do not change linearly with the volume fraction of rigid particles; instead, deformation moduli increase dramatically when the sand fraction exceeds a threshold value between sf= 0.6–0.8 that marks the formation of a percolating network of stiff particles. The friction angle increases with the volume fraction of rigid particles. Conversely, the axial strain at peak strength increases with the content of soft particles, and no apparent peak strength is observed in specimens with low sand fraction 共sf艋 0.6兲. The presence of soft particles alters the formation of force chains. Although soft particles are not part of high-load carrying chains, they play the important role of preventing the buckling of stiff particle chains. DOI: 10.1061/共ASCE兲0899-1561共2007兲19:2共179兲 CE Database subject headings: Elasticity; Mixtures; Porosity; Shear waves; Velocity. Introduction The number of disposed tires in the United States exceeds 500 million per year. Therefore, there is a very large stock of rubber particles from recycled tires. Previous studies have explored their use in soil mixtures for highway construction, lightweight backfills, and highway embankments 共Ahmed and Lovell 1993; Bosscher et al. 1997; Lee et al. 1999; Garga and O’Shaughnessy 2000; Feng and Sutter 2000; Zornberg et al. 2004兲. Typically, these studies involve rubber particles larger than sand particles 共Drubber / Dsand ⬇ 5–10兲. It has been reported that the addition of rubber particles to sand causes a decrease in permeability, a reduction in the minimum void ratio but an increase in the maximum void ratio, a decrease in stiffness, and a reduction in friction angle 共Masad et al. 1996; Feng and Sutter 2000兲. When used as backfill material in retaining walls, the tire shred settles more and produces less horizontal pressure than gravel backfill 共Lee et al. 1999兲. Two mechanisms are involved in the deformation of granular materials: Distortion of individual particles and relative motion between particles as the result of sliding or rolling 共Lambe and Whitman 1979兲. These are seldom independent of one another: A small particle distortion allows particles to move past one another and may cause a previously stable chain of particles to collapse. Chains made of “primary” particles carry most of the load transferred through granular materials; the role of “secondary” particles is to prevent the buckling of these chains 共Radjai et al. 1998兲. Because rubber particles have much lower stiffness than mineral sand particles, it is hypothesized that mixtures of rigid and soft particles may show surprising performance due to the different roles particles may assume, as either load carriers or buckling preventers. The purpose of this study is to explore particle-level mechanisms that are responsible for the macroscale small-to-large strain behavior of rigid-soft granular mixtures, for the special case where Drubber ⬍ Dsand 关a complementary study for Drubber ⬎ Dsand is presented by Kim 共2005兲兴. The choice of Drubber ⬍ Dsand reflects our interest to explore particle-level pore filling and chain distortion effects rather than relative stiffness and arching effects that develop when Drubber Ⰷ Dsand. Mixtures are tested in oedometer and triaxial devices; shear wave velocity is measured during k0 loading in the oedometer cell. From these measurements, the evolution of elastic modulus 共strain ⬃10−2兲, constrained modulus 共strain ⬃10−2 – 10−4兲 and small-strain shear modulus 共strain 艋10−6兲 are investigated at different stress levels and for different sand volume fractions. Test procedures and results follow. Experimental Study 1 Assistant Professor, Dept. of Civil and Environmental Engineering, Korea Univ., Seoul 136-701, Korea. 2 Civil Engineer, Utah National Resource Conservation Service 共NRCS兲, Price, UT 84501. 3 Professor, Dept. of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332. Note. Associate Editor: John S. Popovics. Discussion open until July 1, 2007. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on May 23, 2005; approved on November 4, 2005. This paper is part of the Journal of Materials in Civil Engineering, Vol. 19, No. 2, February 1, 2007. ©ASCE, ISSN 08991561/2007/2-179–184/$25.00. The rigid-soft granular mixtures are prepared using shredded rubber tires and Ottawa 50/70 sand. Rubber and sand properties are summarized in Table 1. The selected grain size of the sand 共D50 = 0.35 mm兲 is about 4 times larger than the mean grain size of the rubber particles 共D50 = 0.09 mm兲. Photographic images are shown in Fig. 1. The sand-rubber mixtures are carefully placed in the oedometer and triaxial cells to prevent segregation. Densification is attained by tamping. Specimens are prepared at the following sand volume fractions sf= Vsand / Vtotal: 0, 0.2, 0.4, 0.6, 0.7, 0.8, 0.9, and 1.0. The mass density of the mixtures increases with sand fraction as shown in Fig. 2. A linear mixing model ␳mix JOURNAL OF MATERIALS IN CIVIL ENGINEERING © ASCE / FEBRUARY 2007 / 179 V total of solids Table 1. Properties of Rubber from Pneumatic Tires and Quartz Used material properties Rubber Quartz sand 共Ottawa 50/70 sand兲 1.08–1.15 2.65 Specific gravitya 1 29⫻ 103 Shear modulus 共MPa兲a Poisson’s ratioa 0.49 0.31 0.09 0.35 D50 共mm兲 Sphericity ⬃0.25 0.9 Roundness ⬃0.60 0.5 — 0.85 emax — 0.50 emin Mass density 共g / cm3兲 0.58 1.68 a Relevant to the material that makes the particles 共data adapted from Beatty 1980兲. = 共sf兲␳sand + 共1 − sf兲␳rubber is superimposed on the data. At high sand fraction 共sf⬎ 0.6兲, the mass density of the mixture is higher than the linear model because small rubber particles fill the porosity between the large sand grains. Specimens are incrementally loaded in the oedometer cell until the effective vertical stress reaches 556 kPa 共cell diameter: 100 mm, specimen height: 30– 40 mm; load increment ratio is 2兲. The shear wave propagation velocity is measured at each stress level using bender elements installed on the top and bottom plates of the oedometer cell. Consolidated drained triaxial tests are conducted on 35 mm diameter and 70 mm height specimens subjected to a confining pressure of ␴c⬘ = 80 kPa. Experimental Results The measured load-deformation response of the different mixtures is summarized next, starting with the small-strain shear modulus Gmax, followed by mid-strain k0 compressibility and large strain triaxial response. Small Strain Shear Modulus Gmax Fig. 3 shows two sequences of measured S-wave time series gathered during oedometic loading and unloading 共sf= 0.2 and sf= 0.8兲. The low frequency content and long travel time in signals gathered for low sand fraction mixtures point to the low stiffness of these mixtures. The evolution of S-wave velocity versus vertical effective stress ␴⬘ during oedometer loading and unloading is plotted in Fig. 4 for the same two mixtures 共sf= 0.2 and sf= 0.8兲. Clearly, the higher the sand fraction, the higher the Fig. 2. Measured mass density as a function of volumetric sand fraction for rigid-soft mixtures 共Dsand / Drubber ⬇ 4兲 S-wave velocity VS at the same stress level. The hysteretic VS–␴⬘ behavior in both mixtures is characteristically observed in k0 loading and reflects the evolution of k0 during unloading, that is, the contribution of the locked-in horizontal stress on shear stiffness in the vertical plane. Fig. 5 is a summary plot of Gmax versus vertical effective stress during loading for all tested rigid-soft mixtures. The measured responses fall into either of two well-defined groups: Specimens with sf艌 0.7 are “sand-like” and exhibit high shear modulus, however, specimens with sf艋 0.4 are “rubber-like” and exhibit low shear moduli. The transition mixture sf= 0.6 displays very high stress sensitivity: it behaves rubber-like at low confinement, but it turns sand-like at high stress. Zero Lateral Strain Loading „k0 Condition… The oedometric stress–strain response is shown in Fig. 6 for selected specimens 共sf= 0, 0.6, 0.8, and 1.0兲. The vertical strain decreases with increasing sand fraction for any given load. All mixtures retain residual deformation after unloading 共in agreement with the hysteresis in VS–␴⬘v observed in Fig. 4兲. The slope of the stress–strain curve is calculated at each load increment to determine the evolution of the constrained modulus M with vertical stress. The summary plot in Fig. 7 shows the increase in stiffness M with increasing sand fraction; there is a marked increase observed for mixtures with sf艌 0.9 共notice loglog scale兲. Strength at Constant Confining Stress Fig. 1. 共a兲 Rubber; 共b兲 sand particles The stress difference ␴1 − ␴3 versus axial strain response is shown in Fig. 8共a兲 for all rigid-soft mixtures. The stress–strain response of the rubber specimen 共sf= 0.0兲 is quasilinear. Peak strengths are observed in high sand fraction specimens for the tested densities and applied low confinement 共refer to Fig. 2兲. The strain at peak strength increases with decreasing sand fraction, the peak strength decreases, and no peak strength is apparent in specimens with low sand fraction 共sf艋 0.6兲. Similar stress-strain response for rubbersand mixtures was observed by other researchers 共Lee et al. 1999; Youwai and Bergado 2003兲. Peak and large-strain 共␧z = 15% axial strain兲 friction angles are plotted in Fig. 8共b兲. Both the peak and large-strain friction angles increase with increasing sand fraction. 180 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING © ASCE / FEBRUARY 2007 Fig. 3. Shear wave signals collected at different confining pressures: signals for 共a兲 sf= 0.8; 共b兲 sf= 0.2. Amplitudes are normalized with respect to the peak amplitude of each signal. Analyses and Discussion The packing of binary mixtures of two-size spherical rigid particles of relative size Dlarge / Dsmall ⬇ 4 is schematically drawn in Fig. 9共a兲. As the volume fraction of small particles increases 关right-to-left in Fig. 9共a兲兴, large particles become increasingly disconnected. Eventually, small particles form the percolating granular skeleton and large particles “float” within. The packing is densest when there are enough small particles to fill the voids left between large particles; this occurs when the ratio between the volume of large particles relative to the total volume is approximately 0.6 关Fig. 9共b兲—Guyon et al. 1987兴. Consequently, the measured porosity of rubber-sand mixtures is minimum near the sand fraction sf⬇ 0.6 关Fig. 9共c兲兴. The excess of soft rubber particles when sf⬍ 0.6 separates sand particles and forms a soft “rubber fabric.” Stiffness–stress Fig. 4. Shear wave velocity as a function of vertical effective stress for sf= 0.2 and sf= 0.8 data show that a mixture with some excess rubber fraction may behave rubber-like at low confinement, but become sand-like at high confining stress when rubber particles deform allowing sand particles to come in contact to form a sand skeleton 共Figs. 5 and 7兲. Fig. 10 shows 共1兲 the large-strain elastic Modulus E from triaxial tests; 共2兲 the middle-strain constrained Modulus M from oedometer tests; and 共3兲 the small-strain shear Modulus Gmax from shear wave velocity measurements. Middle- and large-strain moduli gathered in oedometer and triaxial cells reflect contact deformation and skeletal changes. The transition sand fraction from rubber-like behavior to sand-like behavior is similar in E and M and it is observed at sf⬇ 0.8. An earlier rise is apparent in Gmax; further, the sand fraction that distinguishes rubber-like from Fig. 5. Gmax as a function of vertical effective stress during loading for all tested rigid-soft mixtures. The numbers in the figure denote the volumetric sand fraction. Extrapolated trends are based on results by Santamarina and Aloufi 共1999兲. JOURNAL OF MATERIALS IN CIVIL ENGINEERING © ASCE / FEBRUARY 2007 / 181 Fig. 6. Vertical strain as a function of vertical effective stress in oedometer cell. Numbers denote the volumetric sand fraction. sand-like behaviors varies with stress level: The rigid particle percolation threshold decreases at high stress. Trends in Gmax can be further analyzed by recognizing that the small strain shear stiffness Gmax is a “measure of state” 共constant fabric measurement兲, which is controlled by the nature of interparticle contacts and interparticle coordination. The effective stress governs the shear stiffness Gmax of uncemented particulate materials when capillary effects are negligible, as predicted by the semiempirical power relation Gmax = ␳V2S = A 冉 冊 ␴⬘0 kPa Fig. 8. Drained triaxial tests at 80 kPa confining stress: 共a兲 deviatoric stress versus axial strain: numbers denote the volumetric sand fraction sf in the different rigid-soft mixtures; 共b兲 peak ␾peak and large-strain ␾LS friction angle 共␧z = 15% axial strain兲 b 共1兲 where ␴⬘0⫽average effective stress on the polarization plane; A and b⫽experimentally determined parameters; and ␳ is the mass density. The A factor is the value of Gmax when ␴0⬘ = 1 kPa and it is related to packing 共porosity and coordination number which is the average number of contacts per particle兲, the properties of the particles, contact behavior, and fabric changes; the A factor is low for rubber-like mixtures, and high for sand-like mixtures 关Fig. 10共d兲兴. Fig. 7. Constrained modulus M as a function of vertical effective stress for all rigid-soft mixtures. M is calculated at each stress increment by dividing the average stress by the strain measured during the loading increment. Numbers denote the volumetric sand fraction. Fig. 9. Packing and porosity of binary mixtures: 共a兲 packing; 共b兲 porosity of a binary mixture of spherical particles 共Guyon et al. 1987兲; and 共c兲 measured porosity for the rubber-sand mixtures 182 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING © ASCE / FEBRUARY 2007 Fig. 11. The number of contacts between sand particles in transition mixtures increases as the effective stress increases. This results in a high b exponent in the stiffness-stress power relation 关Fig. 9共d兲兴. Conclusions and Recommendations Fig. 10. Moduli variation with sand fraction: 共a兲 large-strain elastic modulus; 共b兲 middle-strain constrained modulus; 共c兲 small-strain shear modulus; and 共d兲 variation of the A factor and b exponent used to model Gmax = A共␴⬘0 / kPa兲b The b exponent captures the sensitivity of Gmax to stress changes. General guidelines for b values are: b ⬇ 0 for ideal solid or a cemented soil; b = 1 / 3 for Hertzian contact; b = 0.33–0.40 for rounded and dense sands; b ⬇ 0.5 for loose or angular sands; and b 艌 0.6 for soft clays. As the shear wave velocity is measured at different stress states, the b exponent reflects not only contact behavior but fabric changes as well 共Santamarina et al. 2001兲. These guidelines help analyze the evolution of the b exponent with sand fraction: rubber-like mixtures 共sf⬍ 0.5兲 resemble soft systems such as clays, sand-like mixtures 共sf⬎ 0.7兲 are like sand 共b → 0.5兲, and the transition mixture experiences a large increase in sand-to-sand coordination that causes a very high b value 共b ⬃ 1.0兲. The observed behavior is summarized in Fig. 11. Photoelasticity is used to gain further insight into the interaction between small-soft and large-stiff particles, and the role of this interaction in force propagation and stiffness. Rigid particles are modeled with 12.6 mm diameter stiff photoelastic disks, whereas soft particles are represented by 9.3 mm rubber cylinders. Both particle types are 12.6 mm thick. A typical force percolation chain observed in photoelastic studies is shown in Fig. 12 共volumetric rigid particle fraction is ⬃0.8兲. The rubber particles are generally not members of primary force chains; in fact, the prevalence of force chains and the amount of load each chain carries depends on the number of viable rigid particle paths. When there are few viable rigid particle paths, the resulting force chains carry high load. Soft particles often play the secondary yet important role of preventing the buckling of stiff particle chains. The load-deformation behavior of rigid-soft granular mixtures is studied using specimens prepared with uniform sand and small rubber particles 共Dsand / Drubber ⬇ 4兲 at different volume fractions. The main observations from this study follow: • Small-, middle-, and large-strain deformation moduli are not linear functions of the volume fraction of rigid particles. A threshold volume fraction separates soft from rigid skeleton conditions. The threshold volume fraction is confining stress dependent. • In transition mixtures, the coordination among stiff particles increases with confinement so that the mixture behaves rubberlike at low confinement and sand-like at high confinement. The highest value of the b exponent in Gmax = A共␴⬘0 / kPa兲b is observed for sf⬇ 0.6 and it reflects the high increase in coordination number among stiff particles during this transition. • The friction angle increases with sand fraction and no peak strength is apparent in specimens with low sand fraction 共sf艋 0.6兲. Fig. 12. Force chains interpreted from typical photoelastic results obtained with rigid-soft mixtures 共photoelastic large particles D = 12.6 mm and small soft particles D = 9.3 mm兲. High load carrying chains do not involve soft particles. The vertical stress is applied under zero lateral strain conditions, i.e., fixed lateral walls. The solid area fraction of rigid particles is ⬃0.8. JOURNAL OF MATERIALS IN CIVIL ENGINEERING © ASCE / FEBRUARY 2007 / 183 • In most cases, load carrying particle chains do not involve soft particles. However, soft particles do participate in preventing the buckling of load carrying chains. Acknowledgments This study is part of a research initiative on engineered soils and was supported by Vulcan Materials and other Georgia mining companies, The Goizueta Foundation, and the National Science Foundation. References Ahmed, I., and Lovell, C. W. 共1993兲. “Rubber soils as light weight geomaterials.” Transportation Research Record. 1422, Transportation Research Board, Washington, D.C., 61–70. Beatty, J. R. 共1980兲. “Physical properties of rubber compounds.” Mechanics of pneumatic tires, Chap. 10, U.S. Dept. of Transportation, Washington, D.C. Bosscher, P. J., Edil, T. B., and Kuraoka, S. 共1997兲. “Design of highway embankments using tire chips.” J. Geotech. Geoenviron. Eng., 123共4兲, 295–304. Feng, Z. Y., and Sutter, K. G. 共2000兲. “Dynamic properties of granulated rubber sand mixtures.” Geotech. Test. J., 23共3兲, 338–344. Garga, V. K., and O’Shaughnessy, V. 共2000兲. “Tire-reinforced earthfill. Part 1: Construction of a test fill, performance, and retaining wall design.” Can. Geotech. J., 37共1兲, 75–96. Guyon, E., Oger, L., and Plona, T. J. 共1987兲. “Transport properties in sintered porous media composed of two particle sizes.” J. Physics D: Applied Physics, 20共12兲, 1637–1644. Kim, H. K. 共2005兲. “The influences of spatial variability on soil mechanics—Stiffness and strength.” Ph.D. thesis, Georgia Institute of Technology, Atlanta. Lambe, T. W., and Whitman, R. V. 共1979兲. Soil mechanics—SI version, Wiley, New York. Lee, J. H., Salgado, R., Bernal, A., and Lovell, C. W. 共1999兲. “Shredded tires and rubber-sand as lightweight backfill.” J. Geotech. Geoenviron. Eng., 125共2兲, 132–141. Masad, E., Taha, R., Ho., C., and Papagionnakis, T. 共1996兲. “Engineering properties of tire/soil mixtures as a lightweight fill material.” Geotech. Test. J., 19共3兲, 297–304. Radjai, F., Wolf, D. E., Jean, M., and Moreau, J. J. 共1998兲. “Bimodal character of stress transmission in granular packings.” Phys. Rev. Lett., 80共1兲, 61–64. Santamarina, J. C., and Aloufi, M. 共1999兲. “Small strain stiffness: A micromechanical experimental study.” Proc., Pre-failure Deformation Characteristics of Geomaterials, M. Jamiolkowski, R. Lancellotta, and D. Lo Presti, eds., 451–458. Santamarina, J. C., Klein, K. A., and Fam, M. A. 共2001兲. Soils and waves—Particulate materials behavior, characterization, and process monitoring, Wiley, New York. Youwai, S., and Bergado, D. 共2003兲. “Strength and deformation characteristics of shredded rubber tire–sand mixtures.” Can. Geotech. J., 40共2兲, 254–264. Zornberg, J. G., Cabral, A., and Viratjandr, C. 共2004兲. “Behaviour of tire shred-soil mixtures.” Can. Geotech. J., 41共2兲, 227–241. 184 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING © ASCE / FEBRUARY 2007