本研究使用離散元素法(Discrete Element Method, DEM)模擬顆粒體在一圓柱容器內受到單向度束制壓縮時的力學行為，探討的內容分為三部分：(1)不同的顆粒間與顆粒-牆壁間摩擦係數、顆粒間恢復係數對一般顆粒體受到壓縮時力學傳遞性質的影響。(2)不同堆積結構對一般顆粒體受到壓縮時力學傳遞性質的影響。(3)鍵結強度對於可破裂顆粒體受到壓縮時力學傳遞性質的影響，以及探討其粉碎的機制。研究結果顯示：(1)當顆粒-牆壁間摩擦係數增加時，靠近壁面的顆粒體越來越不易移動，故靠近上壓板區域的平均配位數逐漸減少，且範圍往下蔓延，而顆粒間摩擦係數增加時，顆粒發生排列運動的機會減少，造成上壓板區域的顆粒堆發生硬化現象，且硬化的範圍相較於增加顆粒-牆壁間摩擦係數的情況來得大，所以造成靠近上壓板邊壁區域平均配位數降低的範圍較大。(2)在水平面接觸力數目所佔比例與方位關係圖中，無論堆積結構為何，分佈面積皆隨著高度的增加，逐漸減少，除了Random堆積結構在每一高度的分佈大致呈現均向性分佈，其餘堆積結構每一高度的分佈則呈現異向性分佈。(3)當壓縮載重達到4000N時，增加單方向或是兩方向鍵結強度時，皆使得顆粒體的支撐能力增加，進而造成顆粒體的勁度增加。靠近上壓板區域的接觸力強度逐漸增加是由於顆粒體隨著鍵結強度的增加，發生破裂的機會減少，此一現象使得能量不易消散造成接觸力容易向下傳遞，影響範圍也往下蔓延。此外，垂直方向的接觸數目所佔比例與方位關係，在分佈狀況並沒有隨著鍵結強度的增加有明顯變化，不同高度也沒有分佈上的差異，有趣的是分佈均在低接觸角度的分佈較高，呈現異向性分佈。

The purpose of this study is to investigate the mechanical response of a granular assembly under confined compression by using discrete element modelling. The effects of inter-particle friction coefficient, particle-wall friction coefficient, inter-particle restitution coefficient, and packing structure on the internal physical properties of the unbroken particles were extensively examined. The packing structures studied here include FCC, BCC, HCP and random configurations. In addition, the effect of bond strength on the mechanical behaviour and the pulverization mechanism of the crushable particles was also examined. Several key findings are highlighted as follows: (1) As the particle-wall friction increases, the average coordination number of the region near the upper circumference gradually reduces. Similarly, the average coordination number of the upper region decreases with the increase of the inter-particle friction. This also makes the hardening area extend from top to bottom; (2) For the four packing structures studied here, the contact number generally decreases with the increase of height. Only random packed structure exhibits uniform distribution of the contact orientation in the horizontal plane, whereas BCC, FCC and HCP packed structures show anisotropic distributions. The contact forces in all the four packing structures show a strongly anisotropic distribution in the vertical plane; (3) As the bond strength increases in one direction or both directions, the self-supporting capacity of granular assemblies is enhanced, leading to strong loading stiffness. In addition, the contact force intensity increases with bond strength. The enhancement of bond strength does not tend to dissipate the system energy, so the contact forces can transmit into the deeper depth. However, the orientation distribution of contact forces in the vertical plane does not change significantly with the increase of bond strength, and exhibits strongly anisotropic characteristics, especially for a large portion with low contact angles.

[1] P.A. Cundall, O.D. L.Strack, “A discrete numerical model for granular assemblies”, Géotechnique, 29, 47-65, 1979.
[2] D. Markauskas, R. Kačianauskas, “Compacting of particles for biaxial compression test by the discrete element method”, Journal of Civil Engineering and Management, XII, 153-161, 2006.
[3] K. Odagi, T. Tanaka, K. Yamane, “DEM simulation of compression test of particles”, Australia, 127, 1-8, 2002.
[4] C.L. Martin, D. Bouvard, “Study of the cold compaction of composite powders by the
discrete element method”, Acta Materialia, 51, 373-386, 2003.
[5] J.R. Williams, N. Rege, “The development of circulation cell structures in granular
materials undergoing compression”, Powder Technology, 90, 187-194, 1997.
[6] C. O'Sullivan, L. Cui, “Micromechanics of granular material response during load
Reversals: Combined DEM and experimental study”, Powder Technology, 193,
289-302, 2009.
[7] P. Redanz, N.A. Fleck, “The compaction of a random distribution of metal cylinders by the discrete element method”, Acta Materialia, 49, 4325-4335, 2001.
[8] K. Iwashita, M. Oda, “Micro-deformation mechanism of shear banding process based on modified distinct element method” , Powder Technology, 109, 192-205 , 2000.
[9] C.Y. Wu, O.M. Ruddy, A.C. Bentham, B.C. Hancock, S.M. Best, “Modelling the mechanical behaviour of pharmaceutical powders during compaction”, Powder Technology, 152, 107-117, 2005.
[10] Z. Bi, Q. Sun, F. Jin, M. Zhang, “Numerical study on energy transformation in granular matter”, Granular Matter, 13, 503-510, 2011.
[11] T. Ueda, T. Matsushima, Y. Yamada, “Micro structures of granular materials with various grain size distributions”, Powder Technology, 217, 533-539, 2012.
[12] F. Radjai, S. Roux, J.J. Moreau, “Contact forces in a granular packing”, American Institute of Physics Journals, 9, 544-550, 1999.
[13] T.S. Majmuder, R.P. Behringer, “Contact force measurements and stress-induced anisotropy in granular materials”, Nature, 435, 1079-1082, 2005.
[14] C. Thornton, “Force Transmission in Granular Media”, KONA Power and Particle Journal, 15, 81-90, 1997.
[15] J.F. Peters, M. Muthuswamy, J. Wibowo, A. Tordesillas, “Characterization of force chains in granular material”, Physical Review E, 72, 041307, 2005.
[16] H.A. Makse, D.L. Johnson, L.M. Schwartz, “Packing of Compressible Granular
Materials”, Physical Review Letters, 84, 4160-4163, 2000.
[17] D.M. Mueth, H.M. Jaeger, S.R. Nagel, “Force distribution in a granular medium”, Physical Review E, 57, 3164-3169, 1998.
[18] D.L. Blair, N.W. Mueggenburg, A. H.Marshall, H. M.Jaeger, S. R. Nagel, “Force distributions in three-dimensional granular assemblies:Effects of packing order and interparticle friction”, Physical Review E, 63, 041304, 2001.
[19] J.M. Erikson, N.W. Mueggenburg, H.M. Jaeger, S.R.Nagel, “Force distributions in three-dimensional compressible granular packs”, Physical Review E, 63, 041301, 2001.
[20] G. Løvoll, K.J. Måløy, E.G. Flekkøy “Force measurements on static granular materials”, Physical Review E, 60, 5872-5878, 1999.
[21] L.E. Silbert, G.S. Grest, J.W. Landry, “Statistics of the contact network in frictional and frictionless granular packings”, Physical Review E, 66, 061303, 2002.
[22] S.J. Antony, “Evolution of force distribution in three-dimensional granular media”, Physical Review E, 63, 011302, 2000.
[23] C.H. Liu, S.R. Nagel, D.A. Schecter, S.N. Coppersmitj, S. Majumdar, O. Narayan, T.A. Witten, “Force Fluctuations in Bead packs”, Science, 269, 513-515, 1995.
[24] F. Radjai, M. Jean, J.J. Moreau, S. Roux, “Force Distributions in Dense Two-
Dimensional Granular System”, Physical Review Letters, 77, 274-277, 1996.
[25] 邱國豪，「可破裂顆粒在單向度壓力及膨脹收縮之力學行為」，國立中央大學，碩士論文，民國105。
[26] Y.C. Chung, H.H. Liao, S.S. Hsiau, “Convection behaxior of non-spherical particles in a vibrating bed: Discrete element modeling and experimental validation”, Powder Technology, 237, 53-66, 2013.
[27] Y. Tsuji, T. Kawaguchi, T. Tanaka, “Discrete particle simulation of two-dimensional fluidized bed”, Powder Technology, 77, 79-87, 1993.
[28] Y. Tsuji, T. Tanaka, T. Ishida, “Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe”, 71, 239-250, 1992.
[29] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985.
[30] C. O’Sullivan, J.D. Bray. “Selecting a suitable time step for discrete element simulation that use the central difference time integration scheme”, Engineering Computation, 21, 278-303, 2004.
[31] Y.C. Chung, J.Y. Ooi, “Influence of discrete element model parameters on bulk behavior of a granular solid under confined compression”, Particulate Science and Technology, 26, 83-96, 2008.
[32] C. Kittel, (2014)，「伯克利物理學教程(SI版)第1卷：力學」(陳秉乾 等譯)，機械工業出版社，2016。
[33] http://www.materialhouse.co.jp/tech/kesshou.html
[34] Y.C. Chung, J.Y. Ooi, “Benchmark tests for verifying discrete element modelling codes
at particle impact level”, Granular Matter, 13, 643-656, 2011.
 
[35] Y.C. Chung, C.K. Lin, P.H. Chou, S.S. Hsiau, “Mechanical behaviour of a granular solid and its contacting deformable structure under uni-axial compression – Part I: Joint DEM–FEM modelling and experimental validation”, Chemical Engineering Science, 144, 404-420, 2016.
[36] D.O. Potyondy, P.A. Cundall, “A bonded-particle model for rock”, International Journal of Rock Mechanics & Mining Sciences, 41, 1329-1364, 2004.
[37] Y.C. Chung, C.K. Lin, J. Ai, “Mechanical behaviour of a granular solid and its contacting deformable structure under uni-axial compression – Part II: Multi-scale exploration of internal physical properties”, Chemical Engineering Science, 144, 421-443, 2016.
[38] M.D. Silva, J. Rajchenbach, “Stress transmission through a model system of cohesionless elastic grains”, Nature, 406, 708-710, 2000.
[39] C. Goldenberg, I. Goldhirsch, “Force Chains, Microelasticity, and Macroelasticity”, Physical Review Letters, 89, 084302, 2002.
[40] J. Geng, D. Howell, E. Longhi, E. Clément, “Footprints in Sand: The Response of a Granular Material to Local Perturbations”, Physical Review Letters, 87, 035506-1–035506-4, 2001.