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In order to make it easier to investigate some problems such as the mechanism of Janssen effect and the stress distribution in granular medium, we simplify a granular column into a lattice system, in which a lattice point represents a small lump of granular medium and only neighbor interactions are considered. To study the disordered granular columns, a force propagation lattice model determined by the absorption coefficient p and the lateral transfer coefficient q is proposed, and this model is analyzed from the theoretical view. Firstly, the equation of force propagation in the matrix form is given, and this equation is determined by a tridiagonal matrix A(p,q) that is called transfer coefficient matrix. Based on the force transfer equation, the bottom force distribution varying with the top force distribution and the layer of lattice system is deduced, and its analytical solution refers to the similarity diagonalization of matrix A(p, q). Then, a method based on the second order difference equations is proposed to obtain the eigenvalues and eigenvectors of the transfer coefficient matrix. The eigenvalues and eigenvectors of A(p, q) can be rigorously deduced for a typical case, and with these results the pressure distribution relationship between the top and the bottom of the container is given. Based on these theoretical expressions, the relationship between the effective mass and the total mass of granular medium is deduced, and it means that the force propagation model and the Janssen model can lead to similar results. Moreover, the bottom stress distribution is calculated without the top load. Calculations show that the stress distribution reaches a maximum at the center bottom and drops down to either side. Finally, numerical calculations are performed to investigate the effects of parameters p and q on the relation between bottom pressure and packing height. Numerical results show that the saturated value of pressure decreases while parameter p or q increases.
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Keywords:
- Janssen effect /
- stress distribution /
- lattice systems /
- tridiagonal matrix
[1] Lu K Q, Liu J X 2004 Physics 33 629 (in Chinese)[陆坤权, 刘寄星2004物理33 629]
[2] de Gennes P G 1999 Rev. Mod. Phys. 71 374
[3] Vanel L, Claudin P, Bouchaud J P, Cates M E, Wittmer J P 2000 Phys. Rev. Lett. 84 1439
[4] Sun Q C, Hou M Y, Jin F 2011 The 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)]
[5] Bertho Y, Frdrique G D, Hulin J P 2003 Phys. Rev. Lett. 90 144301
[6] Peng Z, Li X Q, Jiang L, Fu L P, Jiang Y M 2009 Acta Phys. Sin. 58 2090 (in Chinese)[彭政, 李湘群, 蒋礼, 符力平, 蒋亦民2009物理学报58 2090]
[7] Wambaugh J F, Hartley R R, Behringer R P 2010 Eur. Phys. J. E 32 135
[8] Li X Q, Jiang Y M, Peng Z 2010 J. Shandong Univ. (Natural Science) 45 101 (in Chinese)[李湘群, 蒋亦民, 彭政2010山东大学学报45 101]
[9] Cambau T, Hure J, Marthelot J 2013 Phys. Rev. E 88 022204
[10] Li Z F, Peng Z, Jiang Y M 2014 Acta Phys. Sin. 63 104503 (in Chinese)[李智峰, 彭政, 蒋亦民2014物理学报63 104503]
[11] Landry J W, Grest G S, Silbert L E, Plimpton S J 2003 Phys. Rev. E 67 274
[12] Marconi U M B, Petri A, Vulpiani A 2000 Physica A 280 279
[13] Zhang X G, Hu L, Long Z W 2006 Chin. J. Comput. Phys. 23 642 (in Chinese)[张兴刚, 胡林, 隆正文2006计算物理23 642]
[14] Jiang Y M, Zheng H P 2008 Acta Phys. Sin. 57 7360 (in Chinese)[蒋亦民, 郑鹤鹏2008物理学报57 7360]
[15] Gendelman O, Pollack Y G, Procaccia I 2016 Phys. Rev. Lett. 116 078001
[16] Liu C, Nagel S R, Schecter D A, Coppersmith S N, Majumdar S, Narayan O, Witten T A 1995 Science 269 513
[17] Coppersmith S N, Liu C, Majumdar S, Narayan O, Witten T 1996 Phys. Rev. E 53 4673
[18] Yang S L 2010 Math in Practice and Theory 40 155 (in Chinese)[杨胜良2010数学的实践与认识40 155]
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[1] Lu K Q, Liu J X 2004 Physics 33 629 (in Chinese)[陆坤权, 刘寄星2004物理33 629]
[2] de Gennes P G 1999 Rev. Mod. Phys. 71 374
[3] Vanel L, Claudin P, Bouchaud J P, Cates M E, Wittmer J P 2000 Phys. Rev. Lett. 84 1439
[4] Sun Q C, Hou M Y, Jin F 2011 The 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)]
[5] Bertho Y, Frdrique G D, Hulin J P 2003 Phys. Rev. Lett. 90 144301
[6] Peng Z, Li X Q, Jiang L, Fu L P, Jiang Y M 2009 Acta Phys. Sin. 58 2090 (in Chinese)[彭政, 李湘群, 蒋礼, 符力平, 蒋亦民2009物理学报58 2090]
[7] Wambaugh J F, Hartley R R, Behringer R P 2010 Eur. Phys. J. E 32 135
[8] Li X Q, Jiang Y M, Peng Z 2010 J. Shandong Univ. (Natural Science) 45 101 (in Chinese)[李湘群, 蒋亦民, 彭政2010山东大学学报45 101]
[9] Cambau T, Hure J, Marthelot J 2013 Phys. Rev. E 88 022204
[10] Li Z F, Peng Z, Jiang Y M 2014 Acta Phys. Sin. 63 104503 (in Chinese)[李智峰, 彭政, 蒋亦民2014物理学报63 104503]
[11] Landry J W, Grest G S, Silbert L E, Plimpton S J 2003 Phys. Rev. E 67 274
[12] Marconi U M B, Petri A, Vulpiani A 2000 Physica A 280 279
[13] Zhang X G, Hu L, Long Z W 2006 Chin. J. Comput. Phys. 23 642 (in Chinese)[张兴刚, 胡林, 隆正文2006计算物理23 642]
[14] Jiang Y M, Zheng H P 2008 Acta Phys. Sin. 57 7360 (in Chinese)[蒋亦民, 郑鹤鹏2008物理学报57 7360]
[15] Gendelman O, Pollack Y G, Procaccia I 2016 Phys. Rev. Lett. 116 078001
[16] Liu C, Nagel S R, Schecter D A, Coppersmith S N, Majumdar S, Narayan O, Witten T A 1995 Science 269 513
[17] Coppersmith S N, Liu C, Majumdar S, Narayan O, Witten T 1996 Phys. Rev. E 53 4673
[18] Yang S L 2010 Math in Practice and Theory 40 155 (in Chinese)[杨胜良2010数学的实践与认识40 155]
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