A Numerical Study of Compaction of Dry Granular Material
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A Numerical Study of Compaction of Dry Granular Material
Deborah Sulsky Department of Mathematics and Statistics Department of Mechanical Engineering University of New Mexico Albuquerque, NM 87131, U.S.A.
ABSTRACT The material-point method is used to examine numerically the macroscopic stress-strain response of a granular sample under compression. The simulations reproduce experimental observations of the large stiffening that occurs as the granular bed becomes packed. We also show the network of force chains that forms and how the character of contacts between grains changes for large deformations. Finally, we examine the probability distribution of forces and observe exponential distributions above the mean with a small peak at the mean for small deformations, and a transition to a larger peak at larger deformations. INTRODUCTION A striking feature of granular material under weak compression is that forces are carried in a network of force chains. Thus, under weak compression, only a fraction of the grains, those contained in the network, support the applied force. These chains have been visualized using photoelastic materials in three-dimensional packings of beads [1] and in two-dimensional arrays of cylinders [2]. The distribution of forces in the medium is not uniform and can be locally large within the chain. Experiments [1] quantify the inhomogeneous nature of the forces by establishing the probability distribution of normal forces between neighboring grains. This distribution is shown to decay exponentially for forces above the mean. This relationship holds for a wide array of parameters, including grain geometry and interparticle friction. For large deformations, the relationship is not as clear [3]. Another measure of the inhomogeneity in the force distribution is that the macroscopic force, for example under uniaxial compression, is a power law function of the applied deflection. The exponent in this macroscopic relation is, in general, different than that in the microscopic Hertzian law governing compression of individual grains in contact [4]. An exponent m with a value larger than one in the macroscopic law indicates that the number of paths of contacts increases as the deflection to the m − 1 power [5]. Elban and Chiarito [6] studied compaction of the energetic material HMX (cyclotetramethylenetetranitramine). In one set of experiments, a narrow, #20 sieve cut of HMX provided a narrow distribution of grain sizes around the mean value of about 900 µm. Elban and Chiarito measured applied and transmitted forces, as well as the
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porous bed displacement. These data were plotted to show the dramatic increase in applied stress as a function of percent theoretical maximum density (% T M D) that occurs above about 80% T M D. The change in stiffness of the granular sample under compression has been attributed to a consolidation phase at small forces and to a consolidated phase at larger forces [7]. At small forces, the deformation of the sample is due to local displacements like sliding and
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