Impulse propagation in dissipative and disordered chains with power-law repulsive potentials

Abstract

We report particle dynamics based studies of impulse propagation in a chain of elastic beads with dissipative contacts and with randomly distributed masses. The interaction between the beads is characterized by the potential V( δ)∼ δ n , δ≥0 being grain overlap, n>2 and at zero external loading, i.e., under conditions of “sonic vacuum” in which sound cannot propagate through the chain [J. Appl. Mech. Technol. Phys. 5 (1983) 733]. In the earlier work, we have confirmed the studies of Nesterenko and coworkers and have reported that impulses propagate as solitary waves in the system of interest in the absence of dissipation and disorder [Physica A 268 (1999) 644]. In the present study, we first discuss the effects of restitution and velocity dependent friction on the propagation of the impulse. We next report that the maximum energy E max of the solitary wave as it propagates from a chain of monodisperse grains of mass m to a chain with masses m(1+ r( z) ϵ), where −1≤ r( z)≤1 and ϵ=const. that measures the degree of randomness, decays with linear distance traveled z as exp(− α E z), α E ∼ ϵ 2+ f( n) , f( n) being some n dependent constant for 2< n<∞. In monodisperse chains, the velocity of the solitary wave c∼ E max ( n−2)/2 n . In polydisperse chains, we show that the propagation speed of a non-dispersive solitary wave decays with distance as exp(− α c z), where α c = α E ( n−2)/2 n.