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Research Papers

# Impact of Viscoplastic Bodies: Dissipation and Restitution

[+] Author and Article Information
K. A. Ismail

Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK

W. J. Stronge

Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UKwjs@eng.cam.ac.uk

For rate-independent compliance where $c→∞$ (i.e., the dashpot behaves as a rigid link), the plastic loss factor γ equals the $COR e∗$.

Although the equation of motion (Eq. 2) is conveniently written in a form similar to that of the Kelvin–Voight model (3), the physics are completely different (Eq. 2) since the damping factor $ζ$ is inversely proportional to the damping constant $c$.

J. Appl. Mech 75(6), 061011 (Aug 20, 2008) (5 pages) doi:10.1115/1.2965371 History: Received November 22, 2007; Revised March 11, 2008; Published August 20, 2008

## Abstract

A viscoplastic coefficient of restitution (COR) that accounts for nonfrictional sources of energy dissipation is determined for direct collision between hard compliant bodies. This COR incorporates effects of both irreversible elastic-plastic material (rate-independent) and viscoelastic (rate-dependent) behaviors. The COR is calculated based on a modified Maxwell model for compliance of the bodies in the small deforming region around the initial contact point. Modifications to the Maxwell model incorporate the effects of plasticity and viscoelasticity, so the calculated COR gives a value that considers both hysteresis due to the plastic deformation and viscoelastic (rate-dependent) sources of energy dissipation during collision.

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## Figures

Figure 1

Viscoplastic compliance for impact between hard bodies with local compliance

Figure 2

Force-deflection relations for bilinear elastic-plastic element representing viscoplastic compliance of the contact region. Stiffness of the elastic-plastic element increases from κ to κγ−2 at the instant of maximum compression giving a ratio of energy dissipation from hysteresis to maximum energy stored in elastic-plastic element γ2.

Figure 3

Contact force for impact with viscoelastic compliance γ=1 for different values of the damping factor ζ. Note that γ=1, ζ=0 (i.e., damping constant c→∞) represents a perfectly elastic collision with no energy dissipation.

Figure 4

Contact force profile for viscoplastic compliance with a damping factor ζ=1/4

Figure 5

Distribution of the work done during compression by viscous and elastic-plastic elements as a function of damping factor ζ. The total work equals the initial kinetic energy of normal relative motion.

Figure 6

Distribution of the work done during restitution by viscous and elastic-plastic elements as a function of damping factor ζ for the viscoplastic compliance (Case I: ζ<γ). Also shown are curves for the total work by the system.

Figure 7

COR for bilinear Maxwell model with both plastic deformation and rate-dependent energy losses (viscoplastic compliance). Also plotted is the result of COR for viscoelastic compliance (6), which is identical to the viscoplastic compliance if internal dissipation is negligible (γ=1).

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