Graphical Abstract Figure

Schematic of the behavior of the elastoplastic contact during the compression phase: (a) elastoplastic contact of rigid sphere indenting a flat surface to a penetration depth of h and contact radius a, (b) plastic contact of the sphere with penetration depth hp and contact radius a, and (c) elastic deformation around the contact region caused by an equivalent flat punch of radius a

Graphical Abstract Figure

Schematic of the behavior of the elastoplastic contact during the compression phase: (a) elastoplastic contact of rigid sphere indenting a flat surface to a penetration depth of h and contact radius a, (b) plastic contact of the sphere with penetration depth hp and contact radius a, and (c) elastic deformation around the contact region caused by an equivalent flat punch of radius a

Close modal

Abstract

Elastoplastic deformation during particle impact occurs widely in many engineering applications. The material properties characterizing both the elastic and plastic behavior play an important role in particle impact. A non-linear contact stiffness-based model representing the elastic and plastic deformation of the material is used to obtain the coefficient of restitution during the impact of a sphere on a deformable substrate. The model consists of the Maxwell combination of perfectly plastic component and a non-linear elastic component. The proposed model is used to estimate the plastic energy dissipation during the impact. An analytical solution is obtained for residual contact radius and coefficient of restitution expressed in terms of experimentally determinable parameters. Our approach yields a single dimensionless parameter referred to as the “indentation parameter,” Λ, and it is shown that the impact response and coefficient of restitution for various impact situations can be determined based on this indentation parameter. The proposed model accurately predicts the residual contact radius and coefficient of restitution, validated through experimental results of low-velocity impacts (1–4 m/s) over a flat sample of aluminum alloy (Al6061) impacted by steel and zirconia balls. The present model is further compared with other existing theoretical contact models for the elastoplastic impact and the extension of the present model for other dissipative systems is also discussed.

References

1.
Melosh
,
H. J.
,
1989
,
Impact Cratering: A Geologic Process
,
Oxford University Press
,
New York
.
2.
Suárez-Cortés
,
A.
,
Flandes
,
l. D.
, and
Durand-Manterola
,
H. J.
,
2021
, “
Planetary Impact Craters Study Through Low-Speed Laboratory Experiments
,”
Int. J. Impact Eng.
,
156
, p.
103954
.
3.
Friend
,
R. D.
, and
Kinra
,
V. K.
,
2000
, “
Particle Impact Damping
,”
J. Sound Vib.
,
233
(
1
), pp.
93
118
.
4.
Carré
,
M. J.
,
James
,
D. M.
, and
Haake
,
S. J.
,
2004
, “
Impact of a Non-Homogeneous Sphere on a Rigid Surface
,”
Proc. Inst. Mech. Eng., Part C
,
218
(
3
), pp.
273
281
.
5.
Cross
,
R.
,
2014
, “
Impact of Sports Balls With Striking Implements
,”
Sports Eng.
,
17
(
1
), pp.
3
22
.
6.
Goldman
,
D. I.
, and
Umbanhowar
,
P.
,
2008
, “
Scaling and Dynamics of Sphere and Disk Impact Into Granular Media
,”
Phys. Rev. E
,
77
(
2
), p.
021308
.
7.
Kondic
,
L.
,
Fang
,
X.
,
Losert
,
W.
,
O’Hern
,
C. S.
, and
Behringer
,
R. P.
,
2012
, “
Microstructure Evolution During Impact on Granular Matter
,”
Phys. Rev. E
,
85
(
1
), p.
011305
.
8.
Wojtkowski
,
M.
,
Pecen
,
J.
,
Horabik
,
J.
, and
Molenda
,
M.
,
2010
, “
Rapeseed Impact Against a Flat Surface: Physical Testing and DEM Simulation With Two Contact Models
,”
Powder Technol.
,
198
(
1
), pp.
61
68
.
9.
Newton
,
I.
,
1729
,
The Mathematical Principles of Natural Philosophy
,
University of California Press
,
New York
.
10.
Poisson
,
S. D.
,
1842
,
A Treatise of Mechanics
,
Longman
,
London
.
11.
Stronge
,
W. J.
,
1990
, “
Rigid Body Collisions With Friction
,”
Proc. R. Soc. Lond. A
,
431
(
1881
), pp.
169
181
.
12.
Hertz
,
H.
,
1882
, “
Über Die Berührung Fester Elastischer Körper. (On the Contact of Elastic Solids)
,”
J. reine und angewandte Mathematik
,
92
, pp.
156
171
. (For English Translation, see Misc. Papers by H. Hertz, Jones and Schott, Macmillan, London, UK, 146–162, 1896).
13.
Love
,
A. E. H.
,
1892
,
A Treatise on the Mathematical Theory of Elasticity
,
Cambridge University Press
,
Cambridge
.
14.
Johnson
,
K. L.
,
1987
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge
.
15.
Raman
,
C. V.
,
1920
, “
On Some Applications of Hertz’s Theory of Impact
,”
Phys. Rev.
,
15
(
4
), pp.
277
284
.
16.
Tillett
,
J. P. A.
,
1954
, “
A Study of the Impact of Spheres on Plates
,”
Proc. Phys. Soc. B
,
67
(
9
), pp.
677
688
.
17.
Hunter
,
S. C.
,
1957
, “
Energy Absorbed by Elastic Waves During Impact
,”
J. Mech. Phys. Solids
,
5
(
3
), pp.
162
171
.
18.
Reed
,
J. R.
,
1985
, “
Energy Losses Due to Elastic Wave Propagation During an Elastic Impact
,”
J. Phys. D: Appl. Phys.
,
18
(
12
), pp.
2329
2337
.
19.
Hutchings
,
I.
,
1979
, “
Energy Absorbed by Elastic Waves During Plastic Impact
,”
J. Phys. D: Appl. Phys.
,
12
(
11
), pp.
1819
1824
.
20.
Stronge
,
W. J.
,
2018
,
Impact Mechanics
,
Cambridge University Press
,
Cambridge
.
21.
Tabor
,
D.
,
1951
,
The Hardness of Metals
,
Oxford University Press
,
Oxford
.
22.
Pethicai
,
J. B.
,
Hutchings
,
R.
, and
Oliver
,
W. C.
,
1983
, “
Hardness Measurement at Penetration Depths As Small as 20 Nm
,”
Philos. Mag. A
,
48
(
4
), pp.
593
606
.
23.
Doerner
,
M.
, and
Nix
,
W.
,
1986
, “
A Method for Interpreting the Data From Depth-Sensing Indentation Instruments
,”
J. Mater. Res.
,
1
(
4
), pp.
601
609
.
24.
Oliver
,
W.
, and
Pharr
,
G.
,
1992
, “
An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments
,”
J. Mater. Res.
,
7
(
6
), pp.
1564
1583
.
25.
Field
,
J.
, and
Swain
,
M.
,
1995
, “
Determining the Mechanical Properties of Small Volumes of Material From Submicrometer Spherical Indentations
,”
J. Mater. Res.
,
10
(
1
), pp.
101
112
.
26.
Bobji
,
M. S.
, and
Biswas
,
S. K.
,
1998
, “
Estimation of Hardness by Nanoindentation of Rough Surfaces
,”
J. Mater. Res.
,
13
(
11
), pp.
3227
3233
.
27.
Wolf
,
B.
,
2000
, “
Inference of Mechanical Properties From Instrumented Depth Sensing Indentation at Tiny Loads and Indentation Depths
,”
Cryst. Res. Technol
,
35
(
4
), pp.
377
399
.
28.
Yigit
,
A.
, and
Christoforou
,
A.
,
1994
, “
On The Impact of a Spherical Indenter and an Elastic-Plastic Transversely Isotropic Half-space
,”
Compos. Eng.
,
4
(
11
), pp.
1143
1152
.
29.
Thornton
,
C.
,
1997
, “
Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres
,”
ASME J. Appl. Mech.
,
64
(
2
), pp.
383
386
.
30.
Li
,
L. Y.
,
Wu
,
C.
, and
Thornton
,
C.
,
2001
, “
A Theoretical Model for the Contact of Elastoplastic Bodies
,”
Proc. Inst. Mech. Eng. C
,
216
(
4
), pp.
421
431
.
31.
Brake
,
M. R.
,
2012
, “
An Analytical Elastic-perfectly Plastic Contact Model
,”
Int. J. Solids Struct.
,
49
(
22
), pp.
3129
3141
.
32.
Brake
,
M. R.
,
2015
, “
An Analytical Elastic Plastic Contact Model With Strain Hardening and Frictional Effects for Normal and Oblique Impacts
,”
Int. J. Solids Struct.
,
62
, pp.
104
123
.
33.
Ma
,
D.
, and
Liu
,
C.
,
2015
, “
Contact Law and Coefficient of Restitution in Elastoplastic Spheres
,”
ASME J. Appl. Mech.
,
82
(
12
), p.
121006
.
34.
Wang
,
H.
,
Yin
,
X.
,
Hao
,
H.
,
Chen
,
W.
, and
Yu
,
B.
,
2020
, “
The Correlation of Theoretical Contact Models for Normal Elastic-Plastic Impacts
,”
Int. J. Solids Struct.
,
182–183
, pp.
15
33
.
35.
Wu
,
C.
,
Li
,
L. Y.
, and
Thornton
,
C.
,
2003
, “
Rebound Behaviour of Spheres for Plastic Impacts
,”
Int. J. Impact Eng.
,
28
(
9
), pp.
929
946
.
36.
Gharib
,
M.
, and
Hürmüzlü
,
Y.
,
2012
, “
A New Contact Force Model for Low Coefficient of Restitution Impact
,”
ASME J. Appl. Mech.
,
79
(
6
), p.
064506
.
37.
Ahmad
,
M.
,
Ismail
,
K. A.
, and
Mat
,
F.
,
2016
, “
Impact Models and Coefficient of Restitution: A Review
,”
ARPN J. Eng. Appl. Sci.
,
11
(
10
), pp.
6549
6555
.
38.
Hunt
,
K. H.
, and
Crossley
,
F.
,
1975
, “
Coefficient of Restitution Interpreted as Damping in Vibroimpact
,”
ASME J. Appl. Mech.
,
42
(
2
), pp.
440
445
.
39.
Findley
,
W. N.
,
Lai
,
J. S.
, and
Onaran
,
K.
,
1989
,
Creep and Relaxation of Nonlinear Viscoelastic Materials: With an Introduction to Linear Viscoelasticity
,
Dover Publications
,
New York
.
40.
Shimizu
,
S.
,
Yanagimoto
,
T.
, and
Sakai
,
M.
,
1999
, “
Pyramidal Indentation Load-Depth Curve of Viscoelastic Materials
,”
J. Mater. Res.
,
14
(
10
), pp.
4075
4086
.
41.
Loubet
,
J.
,
Georges
,
J. M.
, and
Meille
,
G.
,
1986
, “Vickers Indentation Curves of Elastoplastic Materials,”
Microindentation Techniques in Materials Science and Engineering
,
P. J.
Blau
and
B. R.
Lawn
, eds.,
ASTM Special Technical Publication
,
Philadelphia
, PA, pp.
72
89
.
42.
Sakai
,
M.
,
1993
, “
Energy Principle of the Indentation-Induced Inelastic Surface Deformation and Hardness of Brittle Materials
,”
Acta Metall. Mater.
,
41
(
6
), pp.
1751
1758
.
43.
Sakai
,
M.
,
1999
, “
The Meyer Hardness: A Measure for Plasticity?
J. Mater. Res.
,
14
(
9
), pp.
3630
3639
.
44.
Oyen
,
M. L.
, and
Cook
,
R. F.
,
2003
, “
Load-Displacement Behavior During Sharp Indentation of Viscous–Elastic-Plastic Materials
,”
J. Mater. Res.
,
18
(
1
), pp.
139
150
.
45.
Bobji
,
M. S.
, and
Biswas
,
S.
,
1996
, “
Determination and Study of the Strength of the Blister Field Generated by Conical Indentation
,”
Philos. Mag. A
,
73
(
2
), pp.
399
413
.
46.
Sneddon
,
I. N.
,
1965
, “
The Relation Between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile
,”
Int. J. Eng. Sci.
,
3
(
1
), pp.
47
57
.
47.
Stilwell
,
N. A.
, and
Tabor
,
D.
,
1961
, “
Elastic Recovery of Conical Indentations
,”
Proc. Phys. Soc.
,
78
(
2
), pp.
169
179
.
48.
Atherton
,
T.
, and
Kerbyson
,
D.
,
1999
, “
Size Invariant Circle Detection
,”
Image Vis. Comput.
,
17
(
11
), pp.
795
803
.
49.
Wang
,
Y.
,
Xin
,
L.
,
Liu
,
H.
,
Su
,
B.
,
Jin
,
T.
,
Zhang
,
S.
,
Chen
,
J.
, and
Li
,
Z.
,
2024
, “
Modeling of Strain Hardening Behaviors of 6061 Aluminum Alloy Considering Strain Rate and Temperature Effects
,”
J. Mater. Res. Technol.
,
30
(
4
), pp.
4973
4985
.
50.
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
.
51.
Minamoto
,
H.
, and
Kawamura
,
S.
,
2011
, “
Moderately High Speed Impact of Two Identical Spheres
,”
Int. J. Impact Eng.
,
38
(
2–3
), pp.
123
129
.
52.
Wong
,
C.
,
Daniel
,
M.
, and
Rongong
,
J.
,
2009
, “
Energy Dissipation Prediction of Particle Dampers
,”
J. Sound Vib.
,
319
(
1-2
), pp.
91
118
.
53.
Kharaz
,
A. H.
, and
Gorham
,
D.
,
2000
, “
A Study of the Restitution Coefficient in Elastic-Plastic Impact
,”
Philos. Mag. Lett.
,
80
(
8
), pp.
549
559
.
54.
Jayadeep
,
U. B.
,
Bobji
,
M. S.
, and
Jog
,
C. S.
,
2013
, “
Energy Loss in the Impact of Elastic Spheres on a Rigid Half-Space in Presence of Adhesion
,”
Tribol. Lett.
,
53
(
1
), pp.
79
89
.
55.
Johnson
,
K. L.
,
Kendall
,
K.
, and
Roberts
,
A. D.
,
1971
, “
Surface Energy and the Contact of Elastic Solids
,”
Proc. R. Soc. Lond. A
,
324
(
1558
), pp.
301
313
.
56.
Derjaguin
,
B.
,
Muller
,
V.
, and
Toporov
,
Y.
,
1975
, “
Effect of Contact Deformations on the Adhesion of Particles
,”
J. Colloid. Interface Sci.
,
53
(
2
), pp.
314
326
.
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