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Estimating forces during ploughing of a granular bed

Published online by Cambridge University Press:  19 July 2019

Prasad Sonar ,
Sachin Modi  and
Ishan Sharma Open the ORCID record for Ishan Sharma [Opens in a new window]
Prasad Sonar*
Affiliation:
Mechanics and Applied Mathematics Group, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, UP 208016, India
Sachin Modi
Affiliation:
Mechanics and Applied Mathematics Group, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, UP 208016, India
Ishan Sharma
Affiliation:
Mechanics and Applied Mathematics Group, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, UP 208016, India
*
†Email address for correspondence: prasads@iitk.ac.in
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Abstract

We present a method for predicting forces on a plough – modelled as a flat, rigid plate inclined in the direction of motion – as it moves through a granular bed. Our method combines coarse, but representative, discrete element (DE) simulations with continuum mechanics. We first homogenize the kinematic information obtained from DE simulations to obtain a continuum strain field. The strain field is then combined with an appropriate continuum constitutive law for the granular material being ploughed and linear momentum balance to obtain forces acting on the plough. Our method has the advantage that it does not require (i) detailed DE simulations nor (ii) extensive calibration of grain parameters to match experiments which, in turn, requires significant effort and may be system dependent. Both (i) and (ii) are necessary if forces are to be estimated directly from simulations. We confirm the effectiveness of our approach by comparing our predictions with results from calibrated DE simulations and experiments.

JFM classification

Granular media: Granular media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 875 , 25 September 2019 , pp. 376 - 410
Copyright
© 2019 Cambridge University Press 

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References

Aguilar, J., Zhang, T., Qian, F., Kingsbury, M., McInroe, B., Mazouchova, N., Li, C., Maladen, R., Gong, C., Travers, M. et al. 2016 A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems. Rep. Prog. Phys. 79 (11), 110001.Google Scholar
Alekseevskii, V. P. 1966 Penetration of a rod into a target at high velocity. Fizika Goreniya I Vzryva (translated Combustion, Explosion, and Shock Waves, pp. 63–66) 2, 99–106.Google Scholar
Asaf, Z., Rubinstein, D. & Shmulevich, I. 2007 Determination of discrete element model parameters required for soil tillage. Soil Till. Res. 92 (1), 227–242.10.1016/j.still.2006.03.006Google Scholar
Askari, H. & Kamrin, K. 2016 Intrusion rheology in grains and other flowable materials. Nat. Mater. 15 (12), 1274–1279.10.1038/nmat4727Google Scholar
Baligh, M. M. 1985 Strain path method. J. Geotech. Engng 111 (9), 1108–1136.10.1061/(ASCE)0733-9410(1985)111:9(1108)Google Scholar
Barker, T., Schaeffer, D. G., Bohorquez, P. & Gray, J. M. N. T. 2015 Well-posed and ill-posed behaviour of the mu-rheology for granular flow. J. Fluid Mech. 779, 794–818.10.1017/jfm.2015.412Google Scholar
Bharadwaj, R. L., Wassgren, C. & Zenit, R. 2006 The unsteady drag force on a cylinder immersed in a dilute granular flow. Phys. Fluids 18 (4), 043301.10.1063/1.2191907Google Scholar
Bhateja, A., Sharma, I. & Singh, J. K. 2016 Scaling of granular temperature in vibro-fluidized grains. Phys. Fluids 28 (4), 043301.10.1063/1.4944795Google Scholar
Bhateja, A., Sharma, I. & Singh, J. K. 2017 Segregation physics of a macroscale granular ratchet. Phys. Rev. Fluids 2 (5), 052301.10.1103/PhysRevFluids.2.052301Google Scholar
Birkhoff, G., MacDougall, D. P., Pugh, E. M. & Taylor, G. 1948 Explosives with lined cavities. J. Appl. Phys. 19, 563–582.10.1063/1.1698173Google Scholar
Coetzee, C. J. 2014 Discrete and continuum modelling of soil cutting. Comput. Part. Mech. 1 (4), 409–423.10.1007/s40571-014-0014-7Google Scholar
Coetzee, C. J. 2017 Calibration of the discrete element method. Powder Technol. 310, 104–142.10.1016/j.powtec.2017.01.015Google Scholar
Coetzee, C. J. & Els, D. N. J. 2009 Calibration of granular material parameters for DEM modelling and numerical verification by blade–granular material interaction. J. Terramechanics 46 (1), 15–26.10.1016/j.jterra.2008.12.004Google Scholar
Crandall, S. H., Dahl, N. C. & Dill, E. H. 1960 An Introduction to the Mechanics of Solids. McGraw-Hill.10.1063/1.3057081Google Scholar
Cundall, P. A. & Strack, O. D. L. 1979 A discrete numerical model for granular assemblies. Géotechnique 29 (1), 47–65.10.1680/geot.1979.29.1.47Google Scholar
Ding, Y., Gravish, N. & Goldman, D. I. 2011 Drag induced lift in granular media. Phys. Rev. Lett. 106 (2), 028001.10.1103/PhysRevLett.106.028001Google Scholar
Goldrein, H. T., Grantham, S. G., Proud, W. G. & Field, J. E. 2002 The study of internal deformation fields in granular materials using 3D digital speckle x-ray flash photography. AIP Conf. Proc. 620, 1105–1108.10.1063/1.1483731Google Scholar
Goldrein, H. T., Palmer, S. J. P. & Huntley, J. M. 1995 Automated fine grid technique for measurement of large-strain deformation maps. Opt. Lasers Engng 23 (5), 305–318.10.1016/0143-8166(95)00036-NGoogle Scholar
Grantham, S. G., Proud, W. G., Goldrein, H. T. & Field, J. E. 2006 Study of internal deformation fields in granular materials using 3D digital speckle x-ray flash photography. In Laser Interferometry X: Techniques and Analysis, vol. 4101, pp. 319–327. International Society for Optics and Photonics.10.1117/12.498392Google Scholar
Gravish, N., Umbanhowar, P. B. & Goldman, D. I. 2010 Force and flow transition in plowed granular media. Phys. Rev. Lett. 105 (12), 128301.10.1103/PhysRevLett.105.128301Google Scholar
Gravish, N., Umbanhowar, P. B. & Goldman, D. I. 2014 Force and flow at the onset of drag in plowed granular media. Phys. Rev. E 89 (4), 042202.Google Scholar
Guo, H., Goldsmith, J., Delacruz, I., Tao, M., Luo, Y. & Koehler, S. A. 2012 Semi-infinite plates dragged through granular beds. J. Stat. Mech. Theory Exp. 2012 (07), P07013.10.1088/1742-5468/2012/07/P07013Google Scholar
Hettiaratchi, D. R. P. & Reece, A. R. 1967 Symmetrical three-dimensional soil failure. J. Terramechanics 4 (3), 45–67.10.1016/0022-4898(67)90126-7Google Scholar
Hettiaratchi, D. R. P., Witney, B. D. & Reece, A. R. 1966 The calculation of passive pressure in two-dimensional soil failure. J. Agric. Engng Res. 11 (2), 89–107.10.1016/S0021-8634(66)80045-8Google Scholar
Jop, P., Forterre, Y. & Pouliquen, O. 2006 A constitutive law for dense granular flows. Nature 441 (7094), 727–730.10.1038/nature04801Google Scholar
Katsuragi, H. & Durian, D. J. 2013 Drag force scaling for penetration into granular media. Phys. Rev. E 87 (5), 052208.Google Scholar
Li, C., Umbanhowar, P. B., Komsuoglu, H., Koditschek, D. E. & Goldman, D. I. 2009 Sensitive dependence of the motion of a legged robot on granular media. Proc. Natl Acad. Sci. USA 106 (9), 3029–3034.10.1073/pnas.0809095106Google Scholar
Li, C., Zhang, T. & Goldman, D. I. 2013 A terradynamics of legged locomotion on granular media. Science 339 (6126), 1408–1412.10.1126/science.1229163Google Scholar
Liu, C. H., Nagel, S. R., Schecter, D. A., Coppersmith, S. N., Majumdar, S., Narayan, O. & Witten, T. A. 1995 Force fluctuations in bead packs. Science 269 (5223), 513–515.10.1126/science.269.5223.513Google Scholar
Loret de Mola Lemus, D., Kohanbash, D., Moreland, S. & Wettergreen, D. 2014 Slope descent using plowing to minimize slip for planetary rovers. J. Field Rob. 31 (5), 803–819.10.1002/rob.21518Google Scholar
McKyes, E. 1985 Soil Cutting and Tillage, vol. 7. Elsevier.Google Scholar
McKyes, E. & Ali, O. S. 1977 The cutting of soil by narrow blades. J. Terramechanics 14 (2), 43–58.10.1016/0022-4898(77)90001-5Google Scholar
Metcalfe, G., Tennakoon, S. G. K., Kondic, L., Schaeffer, D. G. & Behringer, R. P. 2002 Granular friction, coulomb failure, and the fluid-solid transition for horizontally shaken granular materials. Phys. Rev. E 65 (3), 031302.Google Scholar
MiDi, G. D. R. 2004 On dense granular flows. Eur. Phys. J. E 14 (4), 341–365.Google Scholar
Moreland, S., Skonieczny, K., Inotsume, H. & Wettergreen, D. 2012 Soil behavior of wheels with grousers for planetary rovers. In 2012 IEEE Aerospace Conference, pp. 1–8. IEEE.Google Scholar
Murthy, T. G., Gnanamanickam, E. & Chandrasekar, S. 2012 Deformation field in indentation of a granular ensemble. Phys. Rev. E 85 (6), 061306.Google Scholar
Murthy, T. G., Saldana, C., Yadav, S. & Du, F. 2013 Experimental studies on the kinematics of cutting in granular materials. AIP Conf. Proc. 1542, 919–922.10.1063/1.4812082Google Scholar
Nedderman, R. M. 1992 Statics and Kinematics of Granular Materials. Cambridge University Press.10.1017/CBO9780511600043Google Scholar
Obermayr, M., Dressler, K., Vrettos, C. & Eberhard, P. 2011 Prediction of draft forces in cohesionless soil with the discrete element method. J. Terramechanics 48 (5), 347–358.10.1016/j.jterra.2011.08.003Google Scholar
Oldroyd, J. G. 1947 A rational formulation of the equations of plastic flow for a Bingham solid. Math. Proc. Camb. Phil. Soc. 43, 100–105.10.1017/S0305004100023239Google Scholar
Ono, I., Nakashima, H., Shimizu, H., Miyasaka, J. & Ohdoi, K. 2013 Investigation of elemental shape for 3D DEM modeling of interaction between soil and a narrow cutting tool. J. Terramechanics 50 (4), 265–276.Google Scholar
Palmer, A. C. 1999 Speed effects in cutting and ploughing. Gèotechnique 49 (3), 285–294.10.1680/geot.1999.49.3.285Google Scholar
Pawar, H.2013 Interaction laws in discrete element method. MTech thesis, Indian Institute of Technology, Kanpur, India.Google Scholar
Percier, B., Manneville, S., McElwaine, J. N., Morris, S. W. & Taberlet, N. 2011 Lift and drag forces on an inclined plow moving over a granular surface. Phys. Rev. E 84 (5), 051302.Google Scholar
Pitman, E. B. & Schaeffer, D. G. 1987 Stability of time dependent compressible granular flow in two dimensions. Commun. Pure Appl. Maths 40 (4), 421–447.10.1002/cpa.3160400403Google Scholar
Pouliquen, O. & Forterre, Y. 2009 A non-local rheology for dense granular flows. Proc. R. Soc. Lond. A 367 (1909), 5091–5107.Google Scholar
Prager, W. 1961 Introduction to Mechanics of Continua. Ginn and Co.Google Scholar
Qiong, G., Pitt, R. E. & Ruina, A. 1986 A model to predict soil forces on the plough mouldboard. J. Agric. Engng Res. 35 (3), 141–155.10.1016/S0021-8634(86)80053-1Google Scholar
R Core Team 2013 R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing.Google Scholar
Rubin, M. B. 2012 Analytical formulas for penetration of a long rigid projectile including the effect of cavitation. Intl J. Impact Engng 40, 1–9.Google Scholar
Sauret, A., Balmforth, N. J., Caulfield, C. P. & McElwaine, J. N. 2014 Bulldozing of granular material. J. Fluid Mech. 748, 143–174.10.1017/jfm.2014.181Google Scholar
Schaeffer, D. G. 1987 Instability in the evolution equations describing incompressible granular flow. J. Differ. Equ. 66 (1), 19–50.10.1016/0022-0396(87)90038-6Google Scholar
Schaeffer, D. G. 1990 Instability and ill-posedness in the deformation of granular materials. Intl J. Num. Anal. Meth. Geomech. 14 (4), 253–278.10.1002/nag.1610140403Google Scholar
Schaeffer, D. G. & Pitman, E. B. 1988 Ill-posedness in three-dimensional plastic flow. Commun. Pure Appl. Maths 41 (7), 879–890.10.1002/cpa.3160410703Google Scholar
Seguin, A., Bertho, Y., Gondret, P. & Crassous, J. 2011 Dense granular flow around a penetrating object: experiment and hydrodynamic model. Phys. Rev. Lett. 107 (4), 048001.10.1103/PhysRevLett.107.048001Google Scholar
Sevenhuijsen, P. J., Sirkis, J. S. & Bremand, F. 1993 Current trends in obtaining deformation data from grids. Expl Techn. 17 (3), 22–26.10.1111/j.1747-1567.1993.tb00747.xGoogle Scholar
Sharma, I. 2017 Shapes and Dynamics of Granular Minor Planets: The Dynamics of Deformable Bodies Applied to Granular Objects in the Solar System. Springer International Publishing.10.1007/978-3-319-40490-5Google Scholar
Sharma, I. 2019 High-speed impacts of slender bodies into non-smooth, complex fluids. J. Fluid Mech. 861, R1.10.1017/jfm.2018.938Google Scholar
Shmulevich, I., Asaf, Z. & Rubinstein, D. 2007 Interaction between soil and a wide cutting blade using the discrete element method. Soil Till. Res. 97 (1), 37–50.10.1016/j.still.2007.08.009Google Scholar
Silbert, L. E., Ertaş, D., Grest, G. S., Halsey, T. C., Levine, D. & Plimpton, S. J. 2001 Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64 (5), 051302.Google Scholar
Slonaker, J., Motley, D. C., Zhang, Q., Townsend, S., Senatore, C., Iagnemma, K. & Kamrin, K. 2017 General scaling relations for locomotion in granular media. Phys. Rev. E 95 (5), 052901.Google Scholar
Sokolovski, V. V. 1960 Statics of Solid Media. Butterworths Scientific Publications.Google Scholar
Tanaka, H., Momozu, M., Oida, A. & Yamazaki, M. 2000 Simulation of soil deformation and resistance at bar penetration by the distinct element method. J. Terramechanics 37 (1), 41–56.10.1016/S0022-4898(99)00013-0Google Scholar
Tate, A. 1967 A theory for the deceleration of long rods after impact. J. Mech. Phys. Solids 15, 387–399.10.1016/0022-5096(67)90010-5Google Scholar
Tate, A. 1969 Further results in the theory of long rod penetration. J. Mech. Phys. Solids 17, 141–150.10.1016/0022-5096(69)90028-3Google Scholar
Tate, A. 1978 A simple hydrodynamic model for the strain field produced in a target by the penetration of a high speed long rod projectile. Intl J. Engng Sci. 16, 845–858.Google Scholar
Tate, A. 1986a Long rod penetration models. Part I. A flow field model for high speed long rod penetration. Intl J. Mech. Sci. 28, 535–548.10.1016/0020-7403(86)90051-2Google Scholar
Tate, A. 1986b Long rod penetration models. Part II. Extensions to the hydrodynamic theory of penetration. Intl J. Mech. Sci. 28, 599–612.10.1016/0020-7403(86)90075-5Google Scholar
Thomas, B. J. & Schaeffer, D. G. 1988 Nonlinear behavior of model equations which are linearly ill-posed. Commun. Partial Diff. Eq. 13 (4), 423–467.10.1080/03605308808820548Google Scholar
Tripathi, A. & Khakhar, D. V. 2010 Steady flow of smooth, inelastic particles on a bumpy inclined plane: hard and soft particle simulations. Phys. Rev. E 81 (4), 041307.Google Scholar
Tsuji, T., Nakagawa, Y., Matsumoto, N., Kadono, Y., Takayama, T. & Tanaka, T. 2012 3-D DEM simulation of cohesive soil-pushing behavior by bulldozer blade. J. Terramechanics 49 (1), 37–47.10.1016/j.jterra.2011.11.003Google Scholar
Wettergreen, D., Moreland, S., Skonieczny, K., Jonak, D., Kohanbash, D. & Teza, J. 2010 Design and field experimentation of a prototype lunar prospector. Intl J. Rob. Res. 29 (12), 1550–1564.10.1177/0278364910370217Google Scholar
Wieghardt, K. 1975 Experiments in granular flow. Annu. Rev. Fluid Mech. 7 (1), 89–114.10.1146/annurev.fl.07.010175.000513Google Scholar
Yadav, S., Saldana, C. & Murthy, T. G. 2015 Deformation field evolution in indentation of a porous brittle solid. Intl J. Solids Struct. 66, 35–45.10.1016/j.ijsolstr.2015.04.009Google Scholar
Yarin, A. L., Rubin, M. B. & Roisman, I. V. 1995 Penetration of a rigid projectile into an elastic-plastic target of finite thickness. Intl J. Impact Engng 16, 801–831.10.1016/0734-743X(95)00019-7Google Scholar