Research Papers

Contact Law and Coefficient of Restitution in Elastoplastic Spheres

[+] Author and Article Information
Daolin Ma

State Key Laboratory for Turbulence
and Complex Systems,
College of Engineering,
Peking University,
Beijing, China 100871

Caishan Liu

State Key Laboratory for Turbulence
and Complex Systems,
College of Engineering,
Peking University,
Beijing, China 100871
e-mail: liucs@pku.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 14, 2015; final manuscript received August 30, 2015; published online September 18, 2015. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 82(12), 121006 (Sep 18, 2015) (9 pages) Paper No: JAM-15-1245; doi: 10.1115/1.4031483 History: Received May 14, 2015; Revised August 30, 2015

A complete contact cycle of an elastoplastic sphere consists of loading and unloading phases. The loading phase may fall into three sequential regimes: elastic, mixed elastic–plastic, and fully plastic. In this paper, we distinguish the transition points among the three regimes via the material hardness and a dimensionless geometric parameter corresponding to the onset of the fully plastic regime. Based on Johnson’s simplified spherical expansion model, together with the well-supported force–indentation relationships in the elastic and fully plastic regimes, we build an analytical approximation for the mixed elastic–plastic regime by enforcing the C1 continuity of a loading force–indentation curve. Unloading responses of the elastoplastic sphere are characterized by an elastic force–indentation relation, which has a Hertzian-type form but takes into account the effects of the strain hardening that occurs in the mixed elastic–plastic regime. We validate the model by comparing with existing quasi-static and impact experiments and show that the model can precisely capture the force–indentation responses. Further validation is performed by employing the proposed compliance model to investigate the coefficient of restitution (COR). We achieve agreement between our numerical results and the experimental data reported in other studies. Particularly, we find that the COR is inversely proportional to the impacting velocity with an exponent equal to 1/6, instead of 1/4 reported by many other models.

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Grahic Jump Location
Fig. 1

A sketch of a conical indentation

Grahic Jump Location
Fig. 4

Experiment data reported by Wang et al. [21] for a bead pair with identical diameters (2R=9.525 mm) and made of brass alloy material. Green circle (color version online) represents the transition from a mixed elastic–plastic stage into a fully plastic stage. Simulations are performed under ψ=2.6 and ε=18.

Grahic Jump Location
Fig. 3

Parameter ψ extracted from the experiment curves shown in Ref. [34] for brass beads with different sizes. Inset details the sizes of bead pairs

Grahic Jump Location
Fig. 7

Relationship between COR and impacting velocity according to the experimental data given in Refs. [3033] and [35]

Grahic Jump Location
Fig. 5

Experiment data reported in Ref. [21] for bead pairs with identical diameters (2R=9.525 mm) and made of stainless steel 440C. Green circle (color version online) represents the transition from a mixed elastic–plastic stage into a fully plastic stage. Simulations are performed under ψ=2.8 and ε=18.

Grahic Jump Location
Fig. 6

COR obtained from our model (solid lines) and the experiment data (symbols) reported by Minamoto and Kawamura [31,35], Brake et al. [32], Wong et al. [30], and Kharaz and Gorham [33]. The blue dotted line curve (color version online) corresponds to the simulated results by always changing the curvature of the contact surface during an unloading phase.

Grahic Jump Location
Fig. 2

Red squares (color version online) represent the experimental curves reported in Ref. [20] for a tungsten carbidesphere with radius 0.5 mm pressed into a steel plate. Solid lines stand for the simulated curves under parameters ψ=2.9 and ε=38. The green circle is the position of δp at the transition from a mixed elastic–plastic stage to a fully plastic stage. Inset details the critical point at the transition from an elastic stage to a mixed elastic–plastic stage.



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