Propagação do momento linear e energia cinética em cadeias granulares afiladas. - Machado, Luis Paulo Silveira (2011)
Recently, tapered granular chains have been studied for the development of impact-absorbing devices (shock waves). The characteristic of this system to mitigate shock waves (pulses of linear momentum), makes this material an excellent candidate for such absorbers. Therefore, many studies have been conducted in this direction, but most are numeric character, and some analytical. However, the analytical approximations presented do not show good fits with numerical solutions. In this work we study analytically the propagation of linear momentum and kinetic energy in one-dimensional tapered chains of spherical grains that interact according to the Hertz potential. We apply the binary approximation, based on the supposition that the energy transferred along the chain occurs through successive collisions between two particles. Four settings taper were studied: forward linear, forward exponential, backward linear and backward exponential. With the Binary Theory, we correctly anticipate the trends of increase and decrease the propagation of linear momentum and kinetic energy. However, to capture the correct values of the amplitudes of the pulses of these quantities we developed a numericalanalytic correction to the velocities of particles. Compared the results with numerical solutions of the equations of motion, where we use the Runge-Kutta fourth order. The results show: good agreement between the analytical and numerical theories; forward exponentially tapered chains are the most suitable in the design of absorbers (present exponential decay of linear momentum); a weaker scenario for the propagation of kinetic energy (due to quadratic dependence with the velocity), but the linear momentum is the quantity relevant in the context of impacts.Furthermore, we found good agreement with experimental results presented by F. Melo et al (2006). Thus, we present analytical expressions that capture correctly the propagation of linear momentum and kinetic energy in tapered granular chains. These systems actually demonstrate be excellent for absorbing shock waves, attenuating pulses of linear momentum and saturating kinetic energy. An extension of this work is to study tapered decorated chains and with precompression, allowing the absorption in chains composed of a number smaller grains (short chains).