Dynamical behaviors related to the Hertz contact:
Collective processes of collision and solitary waves propagation in granular media

Eric Falcon

Laboratoire de Physique, École Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France

PhD Thesis Université Claude Bernard Lyon I (1997) (in french)

French title:

Comportements dynamiques associés au contact de Hertz :
processus collectifs de collision et propagation d'ondes solitaires dans les milieux granulaires

This work is both an experimental and theoretical study of dynamical behaviors linked to the Hertz contact. At first, we quickly remind the Hertz theory of elastic contact, the theory of the normal impact between two elastic bodies as well as the different mechanisms of energy loss occuring during a collision. Afterwards, we analyse the behavior of one bead bouncing repeatedly off a stationary flat surface. At the end of the bouncing process, we observe that the bead oscillates on the surface, with a characteristic period, before coming to rest. Since the impact velocity tends towards zero for the last bounces, the gravity effect becomes prominent and leads to a strong decrease of the effectif restitution coefficient. In the third part, we analyse the collective processes occuring during the collision of a column of beads with a horizontal fixed plane. Our work shows that this collision is governed by two basic mechanisms: The energy dispersion through the whole column and the energy dissipation due to the inelastic collision. The fourth part is devoted to the propagation of moderate or high amplitude compression waves in a chain of beads, submitted or not to a static force. We observe the propagation of supersonic nonlinear solitary waves in the chain, both the shape and the velocity of these waves being in agreement with the theoretical predictions. Finally, we observe the appearance of new free-surface instabilities in a deep bed of granular material under vertical vibration: At low frequency of vibration, the corrugation of the heap surface and the formation of hexagons (at higher amplitude of vibration); at high frequency, standing waves patterns (stripes) with a vanishing frequency.


Hertz contact - waves propagation - non linear waves - contact dynamics - inelastic collision - impact - restitution coefficient - granular medium - instabilities - bounce.

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