The event of rigid solids impacting a viscoelastic body is encountered in many engineering disciplines. However, for this problem no analytical solution is available, making the proper design of, e.g., protective bodies a difficult task. As a remedy, generally valid solutions in form of nondimensional response curves are presented in this paper, serving as reference for validation purposes or as design charts for the performance-oriented development of impact absorbers. Hereby, the problem of a frictionless rigid sphere impinging a viscoelastic half-space is investigated by finite-element analyses. The general applicability of the results is assured by a transformation to dimensionless problem parameters. For this purpose, the analytical solution by Hertz (1881, “Über die Berührung fester elastischer Körper,” J. die Reine Angew. Math., 1882(92), pp. 156–171) for the purely elastic impact is taken into account. The chosen nondimensional format leads to a reduced number of system parameters, allowing for a compact representation by so-called master curves. From these, optimal material characteristics are found for three different design objectives.

References

1.
Argatov
,
I. I.
,
2013
, “
Mathematical Modeling of Linear Viscoelastic Impact: Application to Drop Impact Testing of Articular Cartilage
,”
Tribol. Int.
,
63
, pp.
213
225
.
2.
Hertz
,
H.
,
1881
, “
Über die Berührung fester elastischer Körper
,”
J. die Reine Angew. Math.
,
1882
(
92
), pp.
156
171
.
3.
Pao
,
Y.-H.
,
1955
, “
Extension of the Hertz Theory of Impact to the Viscoelastic Case
,”
J. Appl. Phys.
,
26
(
9
), pp.
1083
1088
.
4.
Radok
,
J.
,
1957
, “
Visco-Elastic Stress Analysis
,”
Q. Appl. Math.
,
15
(2), pp.
198
202
.
5.
Lee
,
E. H.
, and
Radok
,
J. R. M.
,
1960
, “
The Contact Problem for Viscoelastic Bodies
,”
ASME J. Appl. Mech.
,
27
(
3
), pp.
438
444
.
6.
Hunter
,
S. C.
,
1960
, “
The Hertz Problem for a Rigid Spherical Indenter and a Viscoelastic Half-Space
,”
J. Mech. Phys. Solids
,
8
(
4
), pp.
219
234
.
7.
Graham
,
G. A. C.
,
1965
, “
The Contact Problem in the Linear Theory of Viscoelasticity
,”
Int. J. Eng. Sci.
,
3
(
1
), pp.
27
46
.
8.
Yang
,
W. H.
,
1966
, “
The Contact Problem for Viscoelastic Bodies
,”
ASME J. Appl. Mech.
,
33
(
2
), pp.
395
401
.
9.
Ting
,
T. C. T.
,
1966
, “
The Contact Stresses Between a Rigid Indenter and a Viscoelastic Half-Space
,”
ASME J. Appl. Mech.
,
33
(
4
), pp.
845
854
.
10.
Graham
,
G. A. C.
,
1967
, “
The Contact Problem in the Linear Theory of Viscoelasticity When the Time Dependent Contact Area Has Any Number of Maxima and Minima
,”
Int. J. Eng. Sci.
,
5
(
6
), pp.
495
514
.
11.
Ting
,
T. C. T.
,
1968
, “
Contact Problems in Linear Theory of Viscoelasticity
,”
ASME J. Appl. Mech.
,
35
(
2
), pp.
248
254
.
12.
Hunter
,
S. C.
,
1967
, “
The Solution of Boundary Value Problems in Linear Viscoelasticity
,”
Mechanics and Chemistry of Solid Propellants: Proceedings of the Fourth Symposium on Naval Structural Mechanics
,
A. C.
Eringen
,
H.
Liebowitz
,
S. L.
Koh
, and
J. M.
Crowley
, eds.,
Pergamon Press
,
New York
, pp.
257
295
.
13.
Ogibalov
,
P.
,
Koltunov
,
M.
, and
Soldatov
,
M.
,
1969
, “
Formulation and Solution of Boundary Value Problems of Polymer Mechanics
,”
Mech. Compos. Mater.
,
5
(
1
), pp.
43
51
.
14.
Kalker
,
J.
,
1977
, “
A Survey of the Mechanics of Contact Between Solid Bodies
,”
ZAMM
,
57
(
5
), pp.
T3
T17
.
15.
Calvit
,
H. H.
,
1967
, “
Numerical Solution of the Problem of Impact of a Rigid Sphere Onto a Linear Viscoelastic Half-Space and Comparison With Experiment
,”
Int. J. Solids Struct.
,
3
(
6
), pp.
951
960
.
16.
Aboudi
,
J.
,
1979
, “
The Dynamic Indentation and Impact of a Viscoelastic Half-Space by an Axisymmetric Rigid Body
,”
Comput. Methods Appl. Mech. Eng.
,
20
(
2
), pp.
135
150
.
17.
Sabin
,
G. C. W.
,
1987
, “
The Impact of a Rigid Axisymmetric Indentor on a Viscoelastic Half-Space
,”
Int. J. Eng. Sci.
,
25
(
2
), pp.
235
251
.
18.
Aksel
,
N.
,
1986
, “
On the Impact of a Rigid Sphere on a Viscoelastic Half-Space
,”
Arch. Appl. Mech.
,
56
(1), pp.
38
54
.
19.
Chen
,
C. P.
, and
Lakes
,
R. S.
,
1990
, “
Design of Viscoelastic Impact Absorbers: Optimal Material Properties
,”
Int. J. Solids Struct.
,
26
(
12
), pp.
1313
1328
.
20.
Argatov
,
I.
,
2012
, “
An Analytical Solution of the Rebound Indentation Problem for an Isotropic Linear Viscoelastic Layer Loaded With a Spherical Punch
,”
Acta Mech.
,
223
(
7
), pp.
1441
1453
.
21.
Findley
,
W. N.
,
Lai
,
J. S.
, and
Onaran
,
K.
,
1976
,
Creep and Relaxation of Nonlinear Viscoelastic Materials—With an Introduction to Linear Viscoelasticity
,
North Holland Publishing Company
,
Amsterdam
.
22.
Christensen
,
R.
,
1982
,
Theory of Viscoelasticity: An Introduction
,
Academic Press
,
New York
.
23.
Lakes
,
R. S.
,
1998
,
Viscoelastic Solids
,
CRC Press
,
Boca Raton, FL
.
24.
Deresiewicz
,
H.
,
1968
, “
A Note on Hertz's Theory of Impact
,”
Acta Mech.
,
6
(
1
), pp.
110
112
.
25.
Johnson
,
K. L.
,
1985
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge, UK
.
26.
ansys
12.1.
,
2009
,
Theory Reference for the Mechanical APDL and Mechanical Applications
,
ANSYS
,
Canonsburg, PA
.
27.
Cheng
,
Y.-T.
, and
Cheng
,
C.-M.
,
2004
, “
Scaling, Dimensional Analysis, and Indentation Measurements
,”
Mater. Sci. Eng.: R: Rep.
,
44
(4–5), pp.
91
149
.
28.
Herrenbrück
,
M.
,
2013
, “
Finite-Element Based Determination of Response Spectra of Viscoelastic Materials Subjected to Low-Velocity Impact Loading
,” Ph.D. thesis, Technische Universität München, Munich, Germany.
29.
Ashby
,
M. F.
,
2005
,
Materials Selection in Mechanical Design
,
Butterworth-Heinemann
,
Boston, MA
.
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