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.

References

References
1.
Ma
,
W.
,
Liu
,
C.
,
Chen
,
B.
, and
Huang
,
L.
,
2006
, “
Theoretical Model for the Pulse Dynamics in a Long Granular Chain
,”
Phys. Rev. E
,
74
(
4
), p.
046602
.
2.
Daraio
,
C.
,
Nesterenko
,
V.
,
Herbold
,
E.
, and
Jin
,
S.
,
2006
, “
Energy Trapping and Shock Disintegration in a Composite Granular Medium
,”
Phys. Rev. Lett.
,
96
(
5
), p.
058002
.
3.
Ayesh
,
A.
,
Brown
,
S.
,
Awasthi
,
A.
,
Hendy
,
S.
,
Convers
,
P.
, and
Nichol
,
K.
,
2010
, “
Coefficient of Restitution for Bouncing Nanoparticles
,”
Phys. Rev. B
,
81
(
19
), p.
195422
.
4.
Glaubitz
,
M.
,
Medvedev
,
N.
,
Pussak
,
D.
,
Hartmann
,
L.
,
Schmidt
,
S.
,
Helmd
,
C. A.
, and
Delcea
,
M.
,
2014
, “
A Novel Contact Model for AFM Indentation Experiments on Soft Spherical Cell-Like Particles
,”
Soft Matter
,
10
(
35
), pp.
6732
6741
.
5.
Zhang
,
H.
,
Brogliato
,
B.
, and
Liu
,
C.
,
2014
, “
Dynamics of Planar Rocking-Blocks With Coulomb Friction and Unilateral Constraints: Comparisons Between Experimental and Numerical Data
,”
Multibody Syst. Dyn.
,
32
(
1
), pp.
1
25
.
6.
Gonzalez
,
M.
, and
Cuitino
,
A. M.
,
2012
, “
A Nonlocal Contact Formulation for Confined Granular Systems
,”
J. Mech. Phys. Solids
,
60
(
2
), pp.
333
350
.
7.
Frenning
,
G.
,
2013
, “
Towards a Mechanistic Model for the Interaction Between Plastically Deforming Particles Under Confined Conditions: A Numerical and Analytical Analysis
,”
Mater. Lett.
,
92
(
1
), pp.
365
368
.
8.
He
,
A.
, and
Wettlaufer
,
J. S.
,
2014
, “
Hertz Beyond Belief
,”
Soft Matter
,
10
(
13
), pp.
2264
2269
.
9.
Johnson
,
K. L.
,
1987
,
Contact Mechanics
,
Cambridge University Press
, Cambridge.
10.
Stronge
,
W. J.
,
2004
,
Impact Mechanics
,
Cambridge University Press
, Cambridge.
11.
Pal
,
R. K.
,
Awasthi
,
A. P.
, and
Geubelle
,
P. H.
,
2013
, “
Wave Propagation in Elasto-Plastic Granular Systems
,”
Granular Matter
,
15
(
6
), pp.
747
758
.
12.
Yan
,
S.
, and
Li
,
L.
,
2003
, “
Finite Element Analysis of Cyclic Indentation of an Elastic-Perfectly Plastic Half-Space by a Rigid Sphere
,”
Proc. Inst. Mech. Eng., Part C
,
217
(
5
), pp.
505
514
.
13.
Pei
,
L.
,
Hyun
,
S.
,
Molinari
,
J.
, and
Robbins
,
M. O.
,
2005
, “
Finite Element Modeling of Elasto-Plastic Contact Between Rough Surfaces
,”
J. Mech. Phys. Solids
,
53
(
11
), pp.
2385
2409
.
14.
Wu
,
C.-Y.
,
Li
,
L.-Y.
, and
Thornton
,
C.
,
2005
, “
Energy Dissipation During Normal Impact of Elastic and Elastic-Perfectly Plastic Spheres
,”
Int. J. Impact Eng.
,
32
(
1–4
), pp.
593
604
.
15.
Tabor
,
D.
,
1948
, “
A Simple Theory of Static and Dynamic Hardness
,”
Proc. R. Soc. London, Ser. A
,
192
(
1029
), pp.
247
274
.
16.
Goldsmith
,
W.
,
1960
,
Impact: The Theory and Physical Behaviour of Colliding Solids
,
Edward Arnold
,
London
.
17.
Aryaei
,
A.
,
Hashemnia
,
K.
, and
Jafarpur
,
K.
,
2010
, “
Experimental and Numerical Study of Ball Size Effect on Restitution Coefficient in Low Velocity Impacts
,”
Int. J. Impact Eng.
,
37
(
10
), pp.
1037
1044
.
18.
Mesarovic
,
S. D.
, and
Fleck
,
N. A.
,
1999
, “
Spherical Indentation of Elastic–Plastic Solids
,”
Proc. R. Soc. London, Ser. A
,
455
(
1987
), pp.
2707
2728
.
19.
Mesarovic
,
S. D.
, and
Fleck
,
N. A.
,
2000
, “
Frictionless Indentation of Dissimilar Elastic–Plastic Spheres
,”
Int. J. Solids Struct.
,
37
(
46
), pp.
7071
7091
.
20.
Bartier
,
O.
,
Hernot
,
X.
, and
Mauvoisin
,
G.
,
2010
, “
Theoretical and Experimental Analysis of Contact Radius for Spherical Indentation
,”
Mech. Mater.
,
42
(
6
), pp.
640
656
.
21.
Wang
,
E.
,
Geubelle
,
P. H.
, and
Lambros
,
J.
,
2013
, “
An Experimental Study of the Dynamic Elasto-Plastic Contact Behavior of Metallic Granules
,”
ASME J. Appl. Mech.
,
80
(
2
), p.
021009
.
22.
Thornton
,
C.
,
1997
, “
Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres
,”
ASME J. Appl. Mech.
,
64
(
2
), pp.
383
386
.
23.
Li
,
L.
,
Wu
,
C.
, and
Thornton
,
C.
,
2002
, “
A Theoretical Model for the Contact of Elastoplastic Bodies
,”
Proc. Inst. Mech. Eng., Part C
,
216
(
4
), pp.
421
431
.
24.
Burgoyne
,
H. A.
, and
Daraio
,
C.
,
2014
, “
Strain-Rate-Dependent Model for the Dynamic Compression of Elastoplastic Spheres
,”
Phys. Rev. E
,
89
(
3
), p.
032203
.
25.
Brake
,
M.
,
2012
, “
An Analytical Elastic-Perfectly Plastic Contact Model
,”
Int. J. Solids Struct.
,
49
(
22
), pp.
3129
3141
.
26.
Mesarovic
,
S. D.
, and
Johnson
,
K. L.
,
2000
, “
Adhesive Contact of Elastic–Plastic Spheres
,”
J. Mech. Phys. Solids
,
48
(
10
), pp.
2009
2033
.
27.
Etsion
,
I.
,
Kligerman
,
Y.
, and
Kadin
,
Y.
,
2005
, “
Unloading of an Elastic Plastic Loaded Spherical Contact
,”
Int. J. Solids Struct.
,
42
(
13
), pp.
3716
3729
.
28.
Li
,
L.
, and
Gu
,
J.
,
2009
, “
An Analytical Solution for the Unloading in Spherical Indentation of Elastic–Plastic Solids
,”
Int. J. Eng. Sci.
,
47
(
3
), pp.
452
462
.
29.
Hunter
,
S.
,
1957
, “
Energy Absorbed by Elastic Waves During Impact
,”
J. Mech. Phys. Solids
,
5
(
3
), pp.
162
171
.
30.
Wong
,
C. X.
,
Daniel
,
M. C.
, and
Rongong
,
J. A.
,
2009
, “
Energy Dissipation Prediction of Particle Dampers
,”
J. Sound Vib.
,
319
(
1–2
), pp.
91
118
.
31.
Minamoto
,
H.
, and
Kawamura
,
S.
,
2011
, “
Moderately High Speed Impact of Two Identical Spheres
,”
Int. J. Impact Eng.
,
38
(
2
), pp.
123
129
.
32.
Brake
,
M. R.
,
Reu
,
P. L.
,
VanGoethem
,
D. J.
,
Bejarano
,
M. V.
, and
Sumali
,
A.
,
2011
, “
Experimental Validation of an Elastic–Plastic Contact Model
,”
ASME
Paper No. IMECE2011-65736.
33.
Kharaz
,
A.
, and
Gorham
,
D.
,
2000
, “
A Study of the Restitution Coefficient in Elastic–Plastic Impact
,”
Philos. Mag. Lett.
,
80
(
8
), pp.
549
559
.
34.
Wang
,
E.
,
On
,
T.
, and
Lambros
,
J.
,
2013
, “
An Experimental Study of the Dynamic Elasto-Plastic Contact Behavior of Dimer Metallic Granules
,”
Exp. Mech.
,
53
(
5
), pp.
883
892
.
35.
Minamoto
,
H.
, and
Kawamura
,
S.
,
2009
, “
Effects of Material Strain Rate Sensitivity in Low Speed Impact Between Two Identical Spheres
,”
Int. J. Impact Eng.
,
36
(
5
), pp.
680
686
.
36.
Weir
,
G.
, and
Tallon
,
S.
,
2005
, “
The Coefficient of Restitution for Normal Incident, Low Velocity Particle Impacts
,”
Chem. Eng. Sci.
,
60
(
13
), pp.
3637
3647
.
You do not currently have access to this content.