Jammed Packings of Soft Grains in Two Dimensions: Mechanical and Statistical Mechanical Properties.

摘要

Granular Materials present challenging problems to the physics community. Such systems are out of equilibrium and dissipative in nature. Since they are not susceptible to thermal fluctuations, mechanical agitation such as shearing and vibration is necessary to explore different states of the granular system. Granular systems share many challenges with glassy systems in general. A complex energy landscape, with enormous relaxation times, and a disordered structure pose problems to theorists as well as experimentalists who are interested in the critical properties of such systems.;This thesis will focus on the problem of unjamming of mechanically stable granular packings. In a model granular system, a correlation function is constructed which identifies a diverging correlation length, established through the use of finite size scaling, as the granular packing loses rigidity (unjams). The exponent associated with this correlation length is found to deviate from the mean-field prediction of (Wyart et al, PRE 72:051306, 2005). This deviation is attributed to novel structures resulting from boundary conditions that emerge and propate in from the boundary as the packing unjams. An entropic description of unjamming is presented. The correlation function is applied to jammed packings of frictionless disks formed in computer simulations.;Anisotropic elliptical grains are studied as well. The phenomenon of hypostaticity, the onset of rigidity at surpisingly low contact number, is addressed. Hypostaticity is shown to result from the onset of quadratically stable vibrational modes that become quartically stable in just-touching packings. An analysis of the vibrational spectrum shows unjamming in ellipse packings to be qualitively different from unjamming in disk packings. The dynamical matrix can be expressed in terms of two contributions, one from the packing geometry and another from the grain curvature (Donev et al, PRE 75, 2007). The contribution from packing geometry is shown to stabilize disk packings, while the contribution from grain curvature is shown to stabilize ellipse packings.