The aim of this work is to present a new procedure for modelling industrial processes that involve granular material flows, using a numerical model based on the Particle Finite Element Method (PFEM). The numerical results herein presented show the potential of this methodology when applied to different branches of industry. Due to the phenomenological richness exhibited by granular materials, the present work will exclusively focus on the modelling of cohesionless dense granular flows.

The numerical model is based on a continuum approach in the framework of large-deformation plasticity theory. For the constitutive model, the yield function is defined in the stress space by a Drucker-Prager yield surface characterized by two constitutive parameters, the cohesion and the internal friction coefficient, and equipped with a non-associative deviatoric flow rule. This plastic flow condition is considered nearly incompressible, so the proposal is integrated in a u p - mixed formulation with a stabilization of the pressure term via the Polynomial Pressure Projection (PPP). In order to characterize the non-linear dependency on the shear rate when flowing a visco-plastic regularization is proposed.

The numerical integration is developed within the Impl-Ex technique, which increases the robustness and reduces the iteration number, compared with a typical implicit integration scheme. The spatial discretization is addressed within the framework of the PFEM which allows treating the large deformations and motions associated to granular flows with minimal distortion of the involved finite element meshes. Since the Delaunay triangulation and the reconnection process minimize such distortion but do not ensure its elimination, a dynamic particle discretization of the domain is proposed, regularizing, in this manner, the smoothness and particle density of the mesh. Likewise, it is proposed a method that ensures conservation of material or Lagrangian surfaces by means of a boundary constraint, avoiding in this way, the geometric definition of the boundary through the classic -shape method.

For modelling the interaction between the confinement boundaries and granular material, it is advocated for a method, based on the Contact Domain Method (CDM) that allows coupling of both domains in terms of an intermediate region connecting the potential contact surfaces by a domain of the same dimension than the contacting bodies. The constitutive model for the contact domain is posed similarly to that for the granular material, defining a correct representation of the wall friction angle.

In order to validate the numerical model, a comparison between experimental results of the spreading of a granular mass on a horizontal plane tests, and finite element predictions, is carried out. These sets of examples allow us validating the model according to the prediction of the different kinematics conditions of granular materials while spreading – from a stagnant condition, while the material is at rest, to a transition to a granular flow, and back to a deposit profile.

The potential of the numerical method for the solution and optimization of industrial granular flows problems is achieved by focusing on two specific industrial applications in mining industry and pellet manufacturing: the silo discharge and the calculation of the power draw in tumbling mills. Both examples are representative when dealing with granular flows due to the presence of variations on the granular material mechanical response.

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[1] M. R. Aboutorabi, An Endochronic Plasticity Model for the Coupled Nonlinear Mechanical Response of Porous and Granular Materials: University of Iowa, 1986.

[2] I. Babuska and A. K. Aziz, "On the Angle Condition in the Finite Element Method," SIAM Journal on Numerical Analysis, vol. 13, pp. 214-226, 1976.

[3] T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures: John Wiley & Sons, 2000.

[4] E. J. Benink, Flow and Stress Analysis of Cohesionless Bulk Materials in Silos Related to Codes, 1989.

[5] R. Boer and W. Brauns, "Kinematic hardening of granular materials," Ingenieur-Archiv, vol. 60, pp. 463-480, 1990/01/01 1990.

[6] J. W. Bonet, R. D., Nonlinear Continuum Mechanics for Finite Element Analysis: Cambridge University Press, 1997.

[7] T. Borzsonyi and R. Stannarius, "Granular materials composed of shapeanisotropic grains," Soft Matter, vol. 9, pp. 7401-7418, 2013.

[8] R. W. Boulanger and K. Ziotopoulou, "Formulation of a sand plasticity plane-strain model for earthquake engineering applications," Soil Dynamics and Earthquake Engineering, vol. 53, pp. 254-267, 2013.

[9] C. E. Brennen, Fundamentals of Multiphase Flow: Cambridge University Press, 2005.

[10] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods: Springer, 2007.

[11] H. H. Bui, R. Fukagawa, K. Sako, and J. C. Wells, "Numerical Simulation of Granular Materials Based on Smoothed Particle Hydrodynamics (SPH)," AIP Conference Proceedings, vol. 1145, pp. 575-578, 2009.

[12] J. Cante, M. D. Riera, J. Oliver, J. Prado, A. Isturiz, and C. Gonzalez, "Flow regime analyses during the filling stage in powder metallurgy processes: experimental study and numerical modelling," Granular Matter, vol. 13, pp. 79-92, 2011/02/01 2011.

[13] J. C. Cante, J. Oliver, and S. Oller, "Simulación numérica de procesos de compactación de pulvimateriales: aplicación de técnicas de cálculo paralelo," Escola Tècnica Superior d'Enginyers de Camins, Canals i Ports de Barcelona, Universitat Politècnica de Catalunya, 1995.

[14] J. Carbonell, E. Oñate, and B. Suárez, "Modeling of Ground Excavation with the Particle Finite-Element Method," Journal of Engineering Mechanics, vol. 136, pp. 455-463, 2010/04/01 2009.

[15] M. E. Cates, J. P. Wittmer, J. P. Bouchaud, and P. Claudin, "Jamming and static stress transmission in granular materials," Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 9, pp. 511-522, 1999.

[16] P. W. Cleary, "Charge behaviour and power consumption in ball mills: sensitivity to mill operating conditions, liner geometry and charge composition," International Journal of Mineral Processing, vol. 63, pp. 79-114, 2001.

[17] B. Clermont and B. de Haas, "Optimization of mill performance by using online ball and pulp measurements," The Journal of the Southern African Institute of Mining and Metallurgy, vol. 110, p. 8, March 2010 2010.

[18] P. Coussot, Rheometry of Pastes, Suspensions, and Granular Materials: Applications in Industry and Environment: Wiley, 2005.

[19] P. Coussot and M. Meunier, "Recognition, classification and mechanical description of debris flows," Earth-Science Reviews, vol. 40, pp. 209-227, 1996.

[20] S. C. Cowin and M. A. Goodman, "A Variational Principle for Granular Materials," ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, vol. 56, pp. 281-286, 1976.

[21] P. A. Cundall and O. D. L. Strack, "A discrete numerical model for granular assemblies," Géotechnique, vol. 29, pp. 47-65, 1979.

[22] H. W. Chandler, "A variational principle for granular materials," International Journal for Numerical and Analytical Methods in Geomechanics, vol. 12, pp. 371-378, 1988.

[23] D. J. Chen, D. K. Shah, and W. S. Chan, "Interfacial stress estimation using least-square extrapolation and local stress smoothing in laminated composites," Computers & Structures, vol. 58, pp. 765-774, 1996.

[24] J. F. Chen, J. M. Rotter, J. Y. Ooi, and Z. Zhong, "Flow pattern measurement in a full scale silo containing iron ore," Chemical Engineering Science, vol. 60, pp. 3029-3041, 2005.

[25] J. F. Chen, J. M. Rotter, J. Y. Ooi, and Z. Zhong, "Correlation between the flow pattern and wall pressures in a full scale experimental silo," Engineering Structures, vol. 29, pp. 2308-2320, 2007.

[26] J. Chung and G. M. Hulbert, "A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-alpha Method," Journal of Applied Mechanics, vol. 60, pp. 371-375, 1993.

[27] Y. F. Dafalias and L. R. Herrmann, "Bounding surface formulation of soil plasticity," in Soil Mechanics - Transient and Cyclic Loads, G. N. Pande and O. C. Zienkiewicz, Eds., ed Chichester, U.K.: John Wiley and Sons, 1982, pp. 253-282.

[28] C. R. Dohrmann and P. B. Bochev, "A stabilized finite element method for the Stokes problem based on polynomial pressure projections," International Journal for Numerical Methods in Fluids, vol. 46, pp. 183-201, 2004.

[29] J. Duran, Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials (Partially Ordered Systems): Springer, 1999.

[30] H. Edelsbrunner, D. Kirkpatrick, and R. Seidel, "On the shape of a set of points in the plane," Information Theory, IEEE Transactions on, vol. 29, pp. 551-559, 1983.

[31] S. A. Elaskar, L. A. Godoy, D. D. Gray, and J. M. Stiles, "A viscoplastic approach to model the flow of granular solids," International Journal of Solids and Structures, vol. 37, pp. 2185-2214, 2000.

[32] B. Erskine. (2013). Sale key to N.S. mine’s fate. Available: http://thechronicleherald.ca/business/1119942-sale-key-to-ns-mine-s-fate

[33] P. J. Frey and P. L. George, Mesh Generation: Application to Finite Elements: Hermes Science, 2000.

[34] P. L. George and H. Borouchaki, Delaunay Triangulation and Meshing: Application to Finite Elements: Hermès, 1998.

[35] C. González-Montellano, F. Ayuga, and J. Y. Ooi, "Discrete element modelling of grain flow in a planar silo: influence of simulation parameters," Granular Matter, vol. 13, pp. 149-158, 2011/04/01 2011.

[36] C. González, "El Método PFEM: aplicación a problemas industriales de pulvimetalurgia," Doctoral, Escola Tècnica Superior d'Enginyers de Camins, Canals i Ports de Barcelona, Universitat Politècnica de Catalunya. Departament de Resistència de Materials i Estructures a l'Enginyeria, Barcelona, Spain, 2009.

[37] G. Gustafsson, "Mechanical characterization and modelling of iron ore pellets," Doctoral, Division of Mechanics of Solid Materials, Luleå University of Technology, Sweden, 2012.

[38] M. Harr, "Stress Distribution," in The Civil Engineering Handbook, Second Edition, ed: CRC Press, 2002.

[39] S. Hartmann, J. Oliver, R. Weyler, J. C. Cante, and J. A. Hernández, "A contact domain method for large deformation frictional contact problems. Part 2: Numerical aspects," Computer Methods in Applied Mechanics and Engineering, vol. 198, pp. 2607-2631, 2009.

[40] J. M. Hill and A. P. S. Selvadurai, "Mathematics and mechanics of granular materials," in Mathematics and Mechanics of Granular Materials, J. Hill and A. P. S. Selvadurai, Eds., ed: Springer Netherlands, 2005, pp. 1-9.

[41] D. Hirshfeld and D. C. Rapaport, "Granular flow from a silo: Discreteparticle simulations in three dimensions," The European Physical Journal E: Soft Matter and Biological Physics, vol. 4, pp. 193-199, 2001.

[42] G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering: John Wiley & Sons, 2000.

[43] D. Humphrey, "Strength and Deformation," in The Civil Engineering Handbook, Second Edition, ed: CRC Press, 002.

[44] M. Hürlimann, D. Rickenmann, and C. Graf, "Field and monitoring data of debris-flow events in the Swiss Alps," Canadian Geotechnical Journal, vol. 40, pp. 161-175, 2003/02/01 2003.

[45] S. Idelsohn, E. Onate, F. Pin, and N. Calvo, "Fluid–structure interaction using the particle finite element method," Computer Methods in Applied Mechanics and Engineering, vol. 195, pp. 2100-2123, 2006.

[46] S. R. Idelsohn, E. Oñate, and F. Del Pin, "A Lagrangian meshless finite element method applied to fluid–structure interaction problems," Computers & Structures, vol. 81, pp. 655-671, 2003.

[47] S. R. Idelsohn, E. Oñate, and F. D. Pin, "The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves," International Journal for Numerical Methods in Engineering, vol. 61, pp. 964-989, 2004.

[48] H. M. Jaeger, S. R. Nagel, and R. P. Behringer, "Granular solids, liquids, and gases," Reviews of Modern Physics, vol. 68, pp. 1259-1273, 1996.

[49] H. A. Janssen, "Versuche über Getreidedruck in Silozellen," Vereines Deutscher Ingenieure, vol. 39, pp. 1045-1049, 1895.

[50] A. W. Jenike, Storage and Flow of Solids: University of Utah, 1964.

[51] H. Jiang and Y. Xie, "A note on the Mohr–Coulomb and Drucker–Prager strength criteria," Mechanics Research Communications, vol. 38, pp. 309-314, 2011.

[52] P. Jonsén, B. I. Pålsson, and H.-Å. Häggblad, "A novel method for fullbody modelling of grinding charges in tumbling mills," Minerals Engineering, vol. 33, pp. 2-12, 2012.

[53] P. Jonsén, B. I. Pålsson, K. Tano, and A. Berggren, "Prediction of mill structure behaviour in a tumbling mill," Minerals Engineering, vol. 24, pp. 236-244, 2011.

[54] M. Jung and U. Rüde, "Implicit Extrapolation Methods for Multilevel Finite Element Computations," SIAM Journal on Scientific Computing, vol. 17, pp. 156-179, 1996/01/01 1996.

[55] K. Kamran, R. Rossi, and E. Oñate, "A contact algorithm for shell problems via Delaunay-based meshing of the contact domain," Computational Mechanics, vol. 52, pp. 1-16, 2013/07/01 2013.

[56] T. Karlsson, M. Klisinski, and K. Runesson, "Finite element simulation of granular material flow in plane silos with complicated geometry," Powder Technology, vol. 99, pp. 29-39, 1998.

[57] A. S. Khan and S. Huang, Continuum Theory of Plasticity: Wiley, 1995.

[58] J. B. Knight, C. G. Fandrich, C. N. Lau, H. M. Jaeger, and S. R. Nagel, "Density relaxation in a vibrated granular material," Physical Review E, vol. 51, pp. 3957-3963, 1995.

[59] P. M. Knupp, "Algebraic mesh quality metrics for unstructured initial meshes," Finite Elements in Analysis and Design, vol. 39, pp. 217-241, 2003.

[60] R. Kobyłka and M. Molenda, "DEM modelling of silo load asymmetry due to eccentric filling and discharge," Powder Technology, vol. 233, pp. 65-71, 2013.

[61] K. Krabbenhøft, "A variational principle of elastoplasticity and its application to the modeling of frictional materials," International Journal of Solids and Structures, vol. 46, pp. 464-479, 2009.

[62] E. Lajeunesse, A. Mangeney-Castelnau, and J. P. Vilotte, "Spreading of a granular mass on a horizontal plane," Physics of Fluids, vol. 16, pp. 2371-2381, 2004.

[63] E. Lajeunesse, J. B. Monnier, and G. M. Homsy, "Granular slumping on a horizontal surface," Physics of Fluids, vol. 17, pp. -, 2005.

[64] A. O. Larese, E.; Ross, R., A Coupled Eulerian-PFEM Model for the Simulation of overtopping in Rockfill Dams vol. 133. Barcelona, Spain: International Center for Numerical Methods in Engineering, 2012.

[65] H. Ling, D. Yue, V. Kaliakin, and N. Themelis, "Anisotropic Elastoplastic Bounding Surface Model for Cohesive Soils," Journal of Engineering Mechanics, vol. 128, pp. 748-758, 2002/07/01 2002.

[66] G. R. Liu and B. Liu, Smoothed Particle Hydrodynamics: A Meshfree Particle Method: World Scientific, 2003.

[67] J. Lubliner, Plasticity Theory: Macmillan Publishing, 1990.

[68] A. Mangeney-Castelnau, F. Bouchut, J. P. Vilotte, E. Lajeunesse, A. Aubertin, and M. Pirulli, "On the use of Saint Venant equations to simulate the spreading of a granular mass," Journal of Geophysical Research: Solid Earth, vol. 110, p. B09103, 2005.

[69] M. Mier-Torrecilla, S. R. Idelsohn, and E. Oñate, "Advances in the simulation of multi-fluid flows with the particle finite element method. Application to bubble dynamics," International Journal for Numerical Methods in Fluids, vol. 67, pp. 1516-1539, 2011.

[70] B. K. Mishra and C. Thornton, "An improved contact model for ball mill simulation by the discrete element method," Advanced Powder Technology, vol. 13, pp. 25-41, 2002.

[71] N. Mitarai and F. Nori, "Wet granular materials," Advances in Physics, vol. 55, pp. 1-45, 2006.

[72] S. Moriguchi, R. Borja, A. Yashima, and K. Sawada, "Estimating the impact force generated by granular flow on a rigid obstruction," Acta Geotechnica, vol. 4, pp. 57-71, 2009/03/01 2009.

[73] A. Mota, W. Sun, J. Ostien, J. Foulk, III, and K. Long, "Lie-group interpolation and variational recovery for internal variables," Computational Mechanics, pp. 1-19, 2013/06/14 2013.

[74] Z. Mróz and S. Pietruszczak, "A constitutive model for sand with anisotropic hardening rule," International Journal for Numerical and Analytical Methods in Geomechanics, vol. 7, pp. 305-320, 1983.

[75] R. M. Nedderman, Statics and Kinematics of Granular Materials: Cambridge University Press, 2005.

[76] J. Oliver, J. C. Cante, R. Weyler, C. González, and J. Hernandez, "Particle Finite Element Methods in Solid Mechanics Problems," in Computational Plasticity. vol. 7, E. Oñate and R. Owen, Eds., ed: Springer Netherlands, 2007, pp. 87-103.

[77] J. Oliver, S. Hartmann, J. C. Cante, R. Weyler, and J. A. Hernández, "A contact domain method for large deformation frictional contact problems. Part 1: Theoretical basis," Computer Methods in Applied Mechanics and Engineering, vol. 198, pp. 2591-2606, 2009.

[78] J. Oliver, A. E. Huespe, and J. C. Cante, "An implicit/explicit integration scheme to increase computability of non-linear material and contact/friction problems," Computer Methods in Applied Mechanics and Engineering, vol. 197, pp. 1865-1889, 2008.

[79] E. Oñate, M. Celigueta, S. Idelsohn, F. Salazar, and B. Suárez, "Possibilities of the particle finite element method for fluid–soil–structure interaction problems," Computational Mechanics, vol. 48, pp. 307-318, 2011/09/01 2011.

[80] E. Oñate, S. R. Idelsohn, M. A. Celigueta, and R. Rossi, "Advances in the particle finite element method for the analysis of fluid–multibody interaction and bed erosion in free surface flows," Computer Methods in Applied Mechanics and Engineering, vol. 197, pp. 1777-1800, 2008.

[81] J. Y. Ooi and K. M. She, "Finite element analysis of wall pressure in imperfect silos," International Journal of Solids and Structures, vol. 34, pp. 2061-2072, 1997.

[82] A. Piccolroaz, D. Bigoni, and A. Gajo, "An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part I – small strain formulation," European Journal of Mechanics -A/Solids, vol. 25, pp. 334-357, 2006.

[83] J. Ravenet, "Silos problems," Bulk Solids Handling, vol. 1, p. 14, 1981.

[84] P. Richard, M. Nicodemi, R. Delannay, P. Ribiere, and D. Bideau, "Slow relaxation and compaction of granular systems," Nat Mater, vol. 4, pp. 121-128, 2005.

[85] M. W. Richman, "The Source of Second Moment in Dilute Granular Flows of Highly Inelastic Spheres," Journal of Rheology, vol. 33, pp. 1293-1306, 1989.

[86] I. Roberts, "Determination of the Vertical and Lateral Pressures of Granular Substances," Proceedings of the Royal Society of London, vol. 36, pp. 225-240, 1883.

[87] J. M. Rodríguez Prieto, "Numerical modeling of metal cutting processes using the particle finite element method (PFEM)," Doctoral Degree, Resistència de Materials i Estructures a l'Enginyeria, Universitat Politècnica de Catalunya, 2014.

[88] K. H. Roscoe, A. N. Schofield, and C. P. Wroth, "On The Yielding of Soils," 8, Géotechnique, 1958.

[89] J. M. Rotter, J. Y. Ooi, J. F. Chen, P. J. Tiley, and I. Mackintosh, "Flow Pattern Measurement in Full Scale Silos," The University of Edinburgh, Edinburgh, Scotland, UK R95-008, 1995.

[90] C. Rycroft, K. Kamrin, and M. Bazant, "Assessing continuum postulates in simulations of granular flow," Journal of the Mechanics and Physics of Solids, vol. 57, pp. 828-839, 2009.

[91] S. K. Saxena, R. K. Reddy, and A. Sengupta, "Verification of a constitutive model for granular materials," Proceedings of International Workshop on Constitutive Equations for Granular Non-cohesive soils, pp. 629-645, July 1987 1987.

[92] G. Si, H. Cao, Y. Zhang, and L. Jia, "Experimental investigation of load behaviour of an industrial scale tumbling mill using noise and vibration signature techniques," Minerals Engineering, vol. 22, pp. 1289-1298, 2009.

[93] J. C. Simo and T. J. R. Hughes, Computational Inelasticity: Springer, 1998.

[94] J. Stener, "Development of Measurement System for Laboratory Scale Ball Mill," Master Thesis, Luleå University of Technology, Luleå, Sweden, 2011.

[95] Q. Sun, F. Jin, J. Liu, and G. Zhang, "Understanding force chains in dense granular materials," International Journal of Modern Physics B, vol. 24, pp. 5743-5759, 2010/11/20 2010.

[96] Q. Sun, F. Jin, and G. Zhang, "Mesoscopic properties of dense granular materials: An overview," Frontiers of Structural and Civil Engineering, vol. 7, pp. 1-12, 2013/03/01 2013.

[97] T. Sundström, "An Autogenous Mill Application in the Mining Industry," SPM Instrument, Strängnäs, Sweden2013.

[98] B. Tang, J. F. Li, and T. S. Wang, "Some improvements on free surface simulation by the particle finite element method," International Journal for Numerical Methods in Fluids, vol. 60, pp. 1032-1054, 2009.

[99] K. Tano, Continuous Monitoring of Mineral Processes with Special Focus on Tumbling Mills: A Multivariate Approach: Luleå University of Technology, 2005.

[100] U. S. G. S. USGS. (2010). Landslides in Central America. Available: http://landslides.usgs.gov/research/other/centralamerica.php

[101] K. C. Valanis and J. F. Peters, "An endochronic plasticity theory with shear-volumetric coupling," International Journal for Numerical and Analytical Methods in Geomechanics, vol. 15, pp. 77-102, 1991.

[102] R. Weyler, J. Oliver, T. Sain, and J. C. Cante, "On the contact domain method: A comparison of penalty and Lagrange multiplier implementations," Computer Methods in Applied Mechanics and Engineering, vol. 205–208, pp. 68-82, 2012.

[103] M. Wójcik and J. Tejchman, "Modeling of shear localization during confined granular flow in silos within non-local hypoplasticity," Powder Technology, vol. 192, pp. 298-310, 2009.

[104] T. F. Wolff, "Soil Relationships and Classification," in The Civil Engineering Handbook, Second Edition, ed: CRC Press, 2002.

[105] W. Wu, E. Bauer, and D. Kolymbas, "Hypoplastic constitutive model with critical state for granular materials," Mechanics of Materials, vol. 23, pp. 45-69, 1996.

[106] Y.-H. Wu, J. M. Hill, and A. Yu, "A finite element method for granular flow through a frictional boundary," Communications in Nonlinear Science and Numerical Simulation, vol. 12, pp. 486-495, 2007.

[107] H. S. Yu, Plasticity and Geotechnics: Springer, 2007.

[108] O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, "Finite Element Method - Its Basis and Fundamentals (6th Edition)," ed: Elsevier.

[109] O. C. Zienkiewicz and J. Z. Zhu, "The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique," International Journal for Numerical Methods in Engineering, vol. 33, pp. 1331-1364, 1992.

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