Highly nonlinear solitary waves in chains of ellipsoidal particles

We study the dynamic response of a one-dimensional chain of ellipsoidal particles excited by a single compressive impulse. We detail the Hertzian contact theory describing the interaction between two ellipsoidal particles under compression, and use it to model the dynamic response of the system. We observe the formation of highly nonlinear solitary waves in the chain, and we also study their propagation properties. We measure experimentally the traveling pulse amplitude (force), the solitary wave speed, and the solitary wave width. We compare these results with theoretical predictions in the long wavelength approximation, and with numerical results obtained with a discrete particle model and with finite element simulations. We also study the propagation of highly nonlinear solitary waves in the chain with particles arranged in different configurations to show the effects of the particle's geometry on the wave propagation characteristics and dissipation. We find very good agreement between experiment, theory, and simulations for all the ranges of impact velocity and particle arrangements investigated.

Figure 1

(a) Digital image showing the ellipsoidal particles used in experiments. (b) Schematic diagram showing the front and side views for the contact between two ellipsoidal particles and the dimensions and the radii of the maximum and minimum principal of curvatures at the contact points between ellipsoidal particles. (c) Schematic diagram for assembly of a 20 particle chain of vertically stacked stainless steel ellipsoidal particles. Piezoelectric sensors were embedded in particles 7 and 12. (d) Schematic diagram representing the assembly of the piezogauges embedded inside selected ellipsoidal particles.

Figure 2

(a) Finite element mesh of (i) two adjacent ellipsoidal particles arranged in the minor axis direction, (ii) two adjacent ellipsoidal particles arranged in the major axis direction, and (iii) two adjacent equivalent spherical particles. (b) Comparison of the contact force-displacement relations between two ellipsoidal particles arranged in minor [curve group (i)] and major [curve group (ii)] axes directions, and also between two equivalent spherical particles [curve group (iii)] obtained from both finite element simulations [dotted (blue) curves] and Hertzian elliptical contact law [solid (black) curves].

Figure 3

(a) Comparison of experiments and numerical results on the formation and propagation of a solitary wave in a chain of 20 stainless steel ellipsoidal particles excited by impacting a stainless steel ellipsoidal striker of mass $m$ $=$ 0.925 g, with an initial velocity of 0.626 m/s. The curves in group 1 show the results for particle 7 in the chain from the top, and similarly, the curves in group 2 show the results for particle 12 from the top. Experimental results are shown by solid (green) curves. (b) Dependence of the wave speed on the maximum contact dynamic force in the chains of ellipsoidal particles arranged in both minor [curve group (i)] and major [curve group (ii)] axes directions and in the chain of equivalent spherical particles [curve group (iii)] under gravitational loading. Experimental data for a chain of ellipsoidal particles arranged in the minor axis direction are shown by solid (green) diamonds in curve group (iii). The solid (black) curves represent the theoretical predictions. In both panels the dashed (red) curves represent the discrete particle results and the dotted (blue) curves represent the finite element results.

Figure 4

Results obtained in a chain composed of 50 ellipsoidal particles, excited by a spherical striker with an impact velocity $v$ $=$ 0.37 m/s. The (green) solid curves represent experimental force-time signals obtained from instrumented particles positioned in locations 8, 16, 28, and 41. The (red) dashed curves represent numerical results obtained from a modified discrete particle model with linear damping. γ is the relaxation coefficient and has a value of −11.67.