Kuwabara-Kono numerical dissipation: a new method to simulate granular matter

A new method is introduced for the simulation of multiple impacts in granular media using the Kuwabara-Kono (KK) contact model, a nonsmooth (not Lipschitz continuous) extension of Hertz contact that accounts for viscoelastic damping. We use the technique of modified equations to construct time-discretizations of the nondissipative Hertz law matching numerical dissipation with KK dissipation at different consistency orders. This allows us to simulate dissipative impacts with good accuracy without including the nonsmooth KK viscoelastic component in the contact force. This tailored numerical dissipation is developed in a general framework, for Newtonian dynamical systems subject to dissipative forces proportional to the time-derivative of conservative forces. Numerical tests are performed for the simulation of impacts in Newton’s cradle and on alignments of alternating large and small balls. Resulting wave phenomena (oscillator synchronization, propagation of dissipative solitary waves, oscillatory tails) are accurately captured by implicit schemes with tailored numerical dissipation, even for relatively large time steps.

A Numerical Method for the Dynamic Analysis of Mechanical Systems in Impact

This paper develops a general procedure for solving mechanism problems where intermittent separations and impacts can occur between mating parts. The numerical technique employed to solve the problem identifies the onset of separation and gives the behavior of the mechanism during separation and impact. The Hertz law is used to find force displacement relationships in impact. Equations of motion are generated by using Hamilton’s principle. A key contribution of this paper is the development of a general approach to the handling of constraint conditions that occur during the separation and impact phase of the motion. Example problems are solved to illustrate the generality and breadth of the method of solution.

The Penetration of Hard-Steel Spheres Into Plane Metal Surfaces

An experimental investigation was undertaken to determine the force-indentation relations governing the contact of hard-steel balls and plane surfaces of various metals both under static and dynamic conditions, the latter involving the Hopkinson-bar technique, with maximum elastic strain rates of 500 sec−1. Excellent correlation was obtained between the measured permanent crater diameter at the contact point and that calculated from strain-gage data by means of an equation treating the bar as a one-dimensional member. A comparison also was effected between the static and dynamic force-indentation curves, the Hertz law of contact, and a relation based upon the concept of a constant flow pressure in the plastic regime.