Analysis of Shear Bands in Sand Under Reduced Gravity Conditions

Transcription

1 Analysis of Shear Bands in Sand Under Reduced Gravity Conditions Jason P. Marshall, Ryan C. Hurley, Dan Arthur, Ivan Vlahinic, Carmine Senatore, Karl Iagnemma, Brian Trease, and José E. Andrade Abstract The strength of granular material, specifically sand is of pivotal importance for understanding physical phenomena on other celestial bodies. However, relatively few experiments have been conducted to determine the dependence of strength properties on gravity. In this work, we experimentally investigated three measures of strength (peak, confined flow, and unconfined flow friction angle) in Earth, Martian, Lunar, and near-zero gravity. The angles were captured in a passive Earth pressure experiment conducted on a reduced gravity flight. The results showed no dependence of the peak friction angle on gravity, a weak dependence of the confined flow friction angle on gravity, and no dependence of the unconfined flow friction angle on gravity. These results highlight the importance of understanding strength and deformation mechanisms of granular material at different levels of gravity. Jason P. Marshall Ryan C. Hurley Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Dan Arthur Ivan Vlahinic Carmine Senatore Massachusetts Institute of Technology, Cambridge, MA, USA Karl Iagnemma Massachusetts Institute of Technology, Cambridge, MA, USA Brian Trease University of Toledo, Toledo, OH, USA José E. Andrade 1

2 2 Marshall, et. al. 1 Introduction and Background Information Granular materials are ubiquitous in nature encompassing everything from foods to soils. These particulate materials play a pivotal role in human society on Earth and an increasingly prominent role on other celestial bodies that humanity is exploring. Understanding granular materials is important for future off-world scientific endeavors; these materials impact all functional aspects of missions including penetrator experiments, drilling events, and robotic mobility. Research on granular materials has been conducted for centuries, but almost exclusively under Earth gravity. While many of the conclusions from existing research will remain true in non-earth gravity environments, it is possible that many will not. We provide experimental evidence of internal friction angles, hereafter called friction angles [11], corresponding to peak strength, confined flow strength, and unconfined flow strength in Earth, Martian, Lunar, and near-zero gravity. These results add data to an under-researched regime of granular materials, namely strengths at low-confinement levels and their dependence on gravity. In this work, we assumed a Mohr-Coulomb failure envelope [5] given by τ = σtanφ + c, where τ is the shear stress, σ is the normal stress, and c is the cohesion. We deal exclusively with a drained granular material and thus assume it is cohesion-less. The Mohr-Coulomb failure envelope defines failure in terms of confining stresses and a friction angle, which is best understood graphically as seen in FIG. 1. Many experimental measurements of friction angles have been made at high confining pressures [6, 2, 3, 8, 9, 7], but few studies have been conducted with low confining pressures [4, 1] and even fewer with varying gravity levels [12, 1]. In addition to the friction angle, the shape of the failure pattern in passive Earth pressure experiments is important, for which we assume a log-spiral pattern [11]. 2 Experimental Setup The above properties were investigated at different gravity levels with a passive Earth pressure experiment [11]. The experiment was conducted on a reduced gravity flight with granular material deformed via a push block in a box with a glass face on the front. As the block was pushed in, the sand failed and ruptured in a clearly defined shear band. Data was recorded on 3 Martian, 2 Lunar, and 20 near-zero gravity parabolas in addition to 5 tests at Earth gravity. The sand was fluidized by vibrating and injecting air into the sample between each test, resulting in a relative density of nearly 100%. The free surface was mostly flat with a slight amount of buildup of material on the left side, however, the surface was the same for all tests conducted. A pixel video of the deformation on the front face was recorded at 60 frames-per-second. Individual images were then extracted and analyzed using digital image correlation (DIC) [10]. The resulting strain field was used to calculate the log-spiral failure surface and the friction angles for each test.

3 Analysis of Shear Bands in Sand Under Reduced Gravity Conditions 3 Fig. 1 a) The Mohr-Coulomb failure envelope is shown. The angle φ is used to describe the strength. b) Strength in granular material can be defined by a peak strength, φ p ; confined flow strength, φ f ; and an unconfined flow strength, φ r. c) Failure in granular material along a shear band with an angle of 45 φ 2. 3 Results 3.1 Peak Friction Angle The peak friction angle was captured by assuming a 2% max shear strain in the sand as the point of failure. While we chose 2%, other reasonably small values will result in nearly the same failure surface. Color plots of all regions of the sand above this max shear strain value resulted in a clear log-spiral failure surface as shown in FIG. 2(a). This failure pattern was repeatedly seen at Earth, Martian, and Lunar gravity levels and occurred within a fraction of a second of initial loading and with a translation of the push block less then 0.02 inches. We calculated the

4 4 Marshall, et. al. friction angle corresponding to this failure surface and found that a value of around 20 was consistently seen across these gravity levels. FIG. 2(b) shows the friction angle values at the different gravity levels with standard error bars. We found that there was essentially no dependence of the peak friction angle on gravity. Fig. 2 a) Log-spiral failure surface for the peak friction angle at Lunar gravity. b) Peak friction angles at different gravity levels with standard error bars. 3.2 Confined Flow Friction Angle Almost immediately after the log-spiral failure surface formed, flow began in a localized shear band. The shear band shape was a planar surface at a lower friction angle between 0 and 10, as shown in FIG. 3. The angle was clearly lower than the peak values at all gravity levels. Additionally, there was a weak dependence of the confined flow friction angle on gravity level, as shown in FIG. 3(b). At near-zero and lunar gravity, the confined flow friction angle essentially dropped to 0, while angles at the higher gravity levels for Mars and Earth were around We note that there were a few negative friction angle values in the data counter to theoretical lower bounds at 0. We attributed these values to random variations of granular contacts within the sample and errors in establishing the angle of the failure surface that led to slight deviations from the theoretically bounded values. 3.3 Unconfined Flow Friction Angle As the push block progressed into the mass of sand, a pile formed. Eventually, the slope of the pile increased beyond the unconfined flow friction angle and failure occurred. Individual grains rolled or slid down the pile causing a reduction of the

5 Analysis of Shear Bands in Sand Under Reduced Gravity Conditions 5 Fig. 3 a) Planar flow surface for the confined flow friction angle at Martian gravity. b) Confined flow friction angles at different gravity levels with standard error bars. pile slope to a stable configuration. We captured this angle at all non-zero gravity levels on the left side of the pile, as shown in FIG. 4(a). The calculated unconfined flow friction angle at Earth, Martian, and Lunar gravities was around 33 and we found no dependence of this value on the gravity level, as shown in FIG. 4(b). This value corresponded closely with values calculated at Earth gravity with a standard test. Fig. 4 a) Example of unconfined flow friction angle at Martian gravity. b) Unconfined flow friction angles at different gravity levels with standard error bars.

6 6 Marshall, et. al. 4 Conclusions These results have a few important consequences with specific application to space exploration. The lack of dependence of the peak and unconfined flow friction angles on gravity suggests that soil experiments conducted at Earth gravity with representative granular materials will be valid with respect to these angles. However, the weak dependence of the confined flow friction angle on gravity suggests that any extrapolation needs be done carefully as different levels of gravity can affect the soil properties. In general careful consideration must be taken when designing components and experiments for non-earth celestial bodies that include granular material interactions, as the material strength and deformation mechanisms may be significantly different. These results also suggest that further experiments should be pursued to refine the trends found. Acknowledgements This research was supported in part by the Keck Institute for Space Studies (KISS) at the California Institute of Technology under the program x-terramechanics - Integrated Simulation of Planetary Surface Missions. This support is gratefully acknowledged. References 1. Khalid A Alshibli, Susan N Batiste, and Stein Sture. Strain localization in sand: plane strain versus triaxial compression. Journal of Geotechnical and Geoenvironmental Engineering, 129(6): , Khalid A Alshibli and Stein Sture. Shear band formation in plane strain experiments of sand. Journal of Geotechnical and Geoenvironmental Engineering, 126(6): , K Been, MG Jefferies, and J Hachey. The critical state of sands. Geotechnique, 41(3): , T Chakraborty and R Salgado. Dilatancy and shear strength of sand at low confining pressures. Journal of geotechnical and geoenvironmental engineering, 136(3): , Robert D Holtz and William D Kovacs. An introduction to geotechnical engineering. Prentice Hall, D Negussey, WKD Wijewickreme, and YP Vaid. Constant-volume friction angle of granular materials. Canadian Geotechnical Journal, 25(1):50 55, A Sadrekarimi and SM Olson. Critical state friction angle of sands. Géotechnique, 61(9): , T Schanz and PA Vermeer. Angles of friction and dilatancy of sand. Geotechnique, 46: , Alessandro Simoni and Guy T Houlsby. The direct shear strength and dilatancy of sand gravel mixtures. Geotechnical & Geological Engineering, 24(3): , MA Sutton, WJ Wolters, WH Peters, WF Ranson, and SR McNeill. Determination of displacements using an improved digital correlation method. Image and vision computing, 1(3): , Karl Terzaghi, Ralph B Peck, and Gholamreza Mesri. Soil mechanics in engineering practice. John Wiley & Sons, Meng Zou, Shichao Fan, Ruiyang Shi, Yanjing Yang, and Jianqiao Li. Effect of gravity on the mechanical properties of lunar regolith tested using a low gravity simulation device. Journal of Terramechanics, 60:11 22, 2015.