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The starting premise of any soft discrete element method simulation, widely used in granular physics and granular mechanics, is the modelling of grain-grain contact force. Most of models often used in the literature including the famous ones by Hertz-Mindlin and Luding, do not present the algorigthy of total elastic potential, or the rate of dissipation which is mainly due to the partially frictional character of the forces. This renders the question of thermodynamic consistency unsettled. A model that possesses explicit expressions for both is proposed here. It is conceptually closely related to the continuum-mechanical theory of granular solid hydrodynarmics (GSH). This theory contains expressions for the total elastic potential and the thermal energy, it accounts for energy conservation and the positivity of entropy production, and it clarifies the equilibrium properties of granular media. All these are lacking (or hidden) in the contact models widely used in the literature. A preliminary calculation shows that the restitution coefficient varies with the impact velocity, which is an added bonus, and demonstrates the model's increased realism. For simplicity, the equations presented in this work are limited to the 2D-case and neglect granular rotations. Nevertheless, the generalization to the 3D-case and the inclusion of granular rotations are carefully discussed, clarifying how to treat rolling and the torsional forces in a thermodynamically consistent fashion. A key point of the present approach, and the major difference to other force models, is the fact that, starting from the characteristic thermodynamic potential, we employ the Onsager reciprocity relation to set up the transport coefficients. The contact forces (usually postulated) are then derived from them. This difference is both conceptually and methodologically relevant. We discussed in detail off-diagonal transport coefficients, especially the so called gear ratio that is particular to granular matter. It reflects the difference between the elastic and the total strain, and is closely related to the slip movement of contact surface, which occur during shear, rolling and torsional deformations. It is relevant to both the macroscopic GSH scales, and the mesoscopic granular scale.
[1] Hertz H 1881 J. Reine Angew. Math. 92 156
[2] Thornton C 2015 Granular Dynamics, Contact Mechanics and Particle System Simulations–A DEM Study (Particle Technology Series, Volume 24) (eBook DOI 10.1007/978-3-319-18711-2) (Switzerland:Springer International Publishing AG)
[3] Laughlin R B, Pines D 2000 Proc. Natl. Acad. Sci. USA 97 28
[4] Laughlin R B 2004 A Different Universe (Changsha:Hunan Science and Technology Press) (in Chinese)[王文浩 译 2008 不同的宇宙(长沙:湖南科学技术出版社)]
[5] Luding S 2008 Granular Matter 10 235
[6] Truesdell C 1972 Rational Thermodynamics (Berlin:Springer-Verlag)
[7] Boussinesq J 1873 C. R. Hebd. Seances Acad. Sci. 77 1521
[8] Bonneau L, Andreotti B, Clément E 2007 Phys. Rev. E 75 016602
[9] Sun Q C, Jin F, Zhou G D 2013 Granular Matter 15 119
[10] Kumar N 2014 Ph. D. Dissertation (Enschede:University of Twente. The Netherlands ISBN:978-90-365-3634-9, DOI:10.3990/1.9789036536349)
[11] Jiang Y M, Liu M 2009 Granular Matter 11 139
[12] Sun Q C, Hou M Y, Jin F 2011 Physics and Mechanics of Granular Matter (Beijing:Science Press) (in Chinese)[孙其诚, 厚美瑛, 金峰 2011 颗粒物质物理与力学(北京:科学出版社)]
[13] Jiang Y M, Liu M 2014 Acta Mech. 225 2363
[14] Jiang Y M, Liu M 2015 Eur. Phys. J. E 38 15
[15] Jiang Y M, Liu M 2003 Phys. Rev. Lett. 91 144301
[16] Jiang Y M, Liu M 2004 Phys. Rev. Lett. 93 148001
[17] Jiang Y M, Liu M 2007 Phys. Rev. Lett. 99 105501
[18] Torquato S, Stillinger F H 2010 Rev. Mod. Phys. 82 2633
[19] Edwards S F, Mounfield C C 1996 Physica A 226 1
[20] Edwards S F, Mounfield C C 1996 Physica A 226 12
[21] Edwards S F, Mounfield C C 1996 Physica A 226 25
[22] Parisi G, Zamponi F 2010 Rev. Mod. Phys. 82 789
[23] Luding S 2009 Nonlinearity 22 R101
[24] Hayakawa H, Otsuki M 2008 Prog. Theor. Phys. 119 381
[25] Kuang S B, Zou R P, Pan R H, Yu A B 2012 Ind. Eng. Chem. Res. 51 14289
[26] Hou Q F, Kuang S B, Yu A B 2017 Chem. Engineer. Sci. 161 67
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[1] Hertz H 1881 J. Reine Angew. Math. 92 156
[2] Thornton C 2015 Granular Dynamics, Contact Mechanics and Particle System Simulations–A DEM Study (Particle Technology Series, Volume 24) (eBook DOI 10.1007/978-3-319-18711-2) (Switzerland:Springer International Publishing AG)
[3] Laughlin R B, Pines D 2000 Proc. Natl. Acad. Sci. USA 97 28
[4] Laughlin R B 2004 A Different Universe (Changsha:Hunan Science and Technology Press) (in Chinese)[王文浩 译 2008 不同的宇宙(长沙:湖南科学技术出版社)]
[5] Luding S 2008 Granular Matter 10 235
[6] Truesdell C 1972 Rational Thermodynamics (Berlin:Springer-Verlag)
[7] Boussinesq J 1873 C. R. Hebd. Seances Acad. Sci. 77 1521
[8] Bonneau L, Andreotti B, Clément E 2007 Phys. Rev. E 75 016602
[9] Sun Q C, Jin F, Zhou G D 2013 Granular Matter 15 119
[10] Kumar N 2014 Ph. D. Dissertation (Enschede:University of Twente. The Netherlands ISBN:978-90-365-3634-9, DOI:10.3990/1.9789036536349)
[11] Jiang Y M, Liu M 2009 Granular Matter 11 139
[12] Sun Q C, Hou M Y, Jin F 2011 Physics and Mechanics of Granular Matter (Beijing:Science Press) (in Chinese)[孙其诚, 厚美瑛, 金峰 2011 颗粒物质物理与力学(北京:科学出版社)]
[13] Jiang Y M, Liu M 2014 Acta Mech. 225 2363
[14] Jiang Y M, Liu M 2015 Eur. Phys. J. E 38 15
[15] Jiang Y M, Liu M 2003 Phys. Rev. Lett. 91 144301
[16] Jiang Y M, Liu M 2004 Phys. Rev. Lett. 93 148001
[17] Jiang Y M, Liu M 2007 Phys. Rev. Lett. 99 105501
[18] Torquato S, Stillinger F H 2010 Rev. Mod. Phys. 82 2633
[19] Edwards S F, Mounfield C C 1996 Physica A 226 1
[20] Edwards S F, Mounfield C C 1996 Physica A 226 12
[21] Edwards S F, Mounfield C C 1996 Physica A 226 25
[22] Parisi G, Zamponi F 2010 Rev. Mod. Phys. 82 789
[23] Luding S 2009 Nonlinearity 22 R101
[24] Hayakawa H, Otsuki M 2008 Prog. Theor. Phys. 119 381
[25] Kuang S B, Zou R P, Pan R H, Yu A B 2012 Ind. Eng. Chem. Res. 51 14289
[26] Hou Q F, Kuang S B, Yu A B 2017 Chem. Engineer. Sci. 161 67
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