# American Institute of Mathematical Sciences

March  2008, 3(1): 125-148. doi: 10.3934/nhm.2008.3.125

## A network model of geometrically constrained deformations of granular materials

 1 Department of Mathematics, Washington State University, Pullman, WA 99164, United States, United States 2 Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802

Received  March 2007 Revised  November 2007 Published  January 2008

We study quasi-static deformation of dense granular packings. In the reference configuration, a granular material is under confining stress (pre-stress). Then the packing is deformed by imposing external boundary conditions, which model engineering experiments such as shear and compression. The deformation is assumed to preserve the local structure of neighbors for each particle, which is a realistic assumption for highly compacted packings driven by small boundary displacements. We propose a two-dimensional network model of such deformations. The model takes into account elastic interparticle interactions and incorporates geometric impenetrability constraints. The effects of friction are neglected. In our model, a granular packing is represented by a spring-lattice network, whereby the particle centers correspond to vertices of the network, and interparticle contacts correspond to the edges. We work with general network geometries: periodicity is not assumed. For the springs, we use a quadratic elastic energy function. Combined with the linearized impenetrability constraints, this function provides a regularization of the hard-sphere potential for small displacements.
When the network deforms, each spring either preserves its length (this corresponds to a solid-like contact), or expands (this represents a broken contact). Our goal is to study distribution of solid-like contacts in the energy-minimizing configuration. We prove that under certain geometric conditions on the network, there are at least two non-stretched springs attached to each node, which means that every particle has at least two solid-like contacts. The result implies that a particle cannot loose contact with all of its neighbors. This eliminates micro-avalanches as a mechanism for structural weakening in small shear deformation.
Citation: K. A. Ariyawansa, Leonid Berlyand, Alexander Panchenko. A network model of geometrically constrained deformations of granular materials. Networks & Heterogeneous Media, 2008, 3 (1) : 125-148. doi: 10.3934/nhm.2008.3.125
 [1] Qun Lin, Antoinette Tordesillas. Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions. Journal of Industrial & Management Optimization, 2014, 10 (1) : 337-362. doi: 10.3934/jimo.2014.10.337 [2] Shige Peng, Mingyu Xu. Constrained BSDEs, viscosity solutions of variational inequalities and their applications. Mathematical Control & Related Fields, 2013, 3 (2) : 233-244. doi: 10.3934/mcrf.2013.3.233 [3] Jean Dolbeault, Maria J. Esteban, Gaspard Jankowiak. Onofri inequalities and rigidity results. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3059-3078. doi: 10.3934/dcds.2017131 [4] Haolei Wang, Lei Zhang. Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions. Electronic Research Archive, 2020, 28 (1) : 405-421. doi: 10.3934/era.2020023 [5] Rong Hu, Ya-Ping Fang, Nan-Jing Huang. Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequality constraints. Journal of Industrial & Management Optimization, 2010, 6 (3) : 465-481. doi: 10.3934/jimo.2010.6.465 [6] Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567 [7] Na Zhao, Zheng-Hai Huang. A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function. Journal of Industrial & Management Optimization, 2011, 7 (2) : 467-482. doi: 10.3934/jimo.2011.7.467 [8] Mariano Giaquinta, Paolo Maria Mariano, Giuseppe Modica. A variational problem in the mechanics of complex materials. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 519-537. doi: 10.3934/dcds.2010.28.519 [9] Andrew D. Lewis. Nonholonomic and constrained variational mechanics. Journal of Geometric Mechanics, 2020, 12 (2) : 165-308. doi: 10.3934/jgm.2020013 [10] Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105 [11] Serena Dipierro. Geometric inequalities and symmetry results for elliptic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3473-3496. doi: 10.3934/dcds.2013.33.3473 [12] Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201 [13] Danijela Damjanović, James Tanis. Cocycle rigidity and splitting for some discrete parabolic actions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5211-5227. doi: 10.3934/dcds.2014.34.5211 [14] James Tanis, Zhenqi Jenny Wang. Cohomological equation and cocycle rigidity of discrete parabolic actions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3969-4000. doi: 10.3934/dcds.2019160 [15] Xiaodi Bai, Xiaojin Zheng, Xiaoling Sun. A survey on probabilistically constrained optimization problems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 767-778. doi: 10.3934/naco.2012.2.767 [16] Dmitri E. Kvasov, Yaroslav D. Sergeyev. Univariate geometric Lipschitz global optimization algorithms. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 69-90. doi: 10.3934/naco.2012.2.69 [17] Christiane Pöschl, Jan Modersitzki, Otmar Scherzer. A variational setting for volume constrained image registration. Inverse Problems & Imaging, 2010, 4 (3) : 505-522. doi: 10.3934/ipi.2010.4.505 [18] Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063 [19] Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433 [20] Michel Chipot, Karen Yeressian. On the asymptotic behavior of variational inequalities set in cylinders. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4875-4890. doi: 10.3934/dcds.2013.33.4875

2019 Impact Factor: 1.053