This paper discusses the dynamic interaction between a monoatomic chain of solid particles and a thin-walled spherical pressure vessel. The objective is to find a relationship between the highly nonlinear solitary waves (HNSWs) propagating within the chain and the internal pressure of the vessel. The paper introduces first a general finite element model to predict the abovementioned interaction, and then a specific application to tennis balls. The scope is to demonstrate a new nondestructive testing (NDT) method to infer the internal pressure of the balls. The overarching idea is that a mechanically induced solitary pulse propagating within the chain interacts with the thin-walled ball to be probed. At the chain–ball interface, the acoustic pulse is partially reflected back to the chain and partially deforms the rubber giving rise to secondary pulses. The research hypothesis is that one or more features of the reflected waves are monotonically dependent on the internal pressure. Both numerical and experimental results demonstrate a monotonic relationship between the time of flight (TOF) of the solitary waves and the internal pressure of the tennis balls. In addition, the pressure inferred nondestructively with the HNSWs matches very well the pressure measured destructively with an ad hoc pressure gauge needle. In the future, the results presented in this study could be used to develop a portable device to infer anytime anywhere the internal pressure of deformable systems (including biological systems) for which conventional pressure gages cannot be used noninvasively.

References

References
1.
Hussein
,
M. I.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2014
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook
,”
ASME Appl. Mech. Rev.
,
66
(
4
), p.
040802
.
2.
Cummer
,
S. A.
,
Christensen
,
J.
, and
Alù
,
A.
,
2016
, “
Controlling Sound With Acoustic Metamaterials
,”
Nat. Rev. Mater.
,
1
, p.
16001
.
3.
Brûlé
,
S.
,
Javelaud
,
E.
,
Enoch
,
S.
, and
Guenneau
,
S.
,
2014
, “
Experiments on Seismic Metamaterials: Molding Surface Waves
,”
Phys. Rev. Lett.
,
112
(
13
), p.
133901
.
4.
Bell
,
J. S.
,
Summers
,
I.
,
Murray
,
A. R.
,
Hendry
,
E.
,
Sambles
,
J. R.
, and
Hibbins
,
A. P.
,
2012
, “
Low Acoustic Transmittance Through a Holey Structure
,”
Phys. Rev. B
,
85
(
21
), p.
214305
.
5.
Zigoneanu
,
L.
,
Popa
,
B.-I.
, and
Cummer
,
S. A.
,
2014
, “
Three-Dimensional Broadband Omnidirectional Acoustic Ground Cloak
,”
Nat. Mater.
,
13
(
4
), pp.
352
355
.
6.
Guenneau
,
S. R.
,
Enoch
,
S.
, and
Brule
,
S.
,
2014
, “
Seismic Metamaterials: Shielding and Focusing Surface Elastic Waves in Structured Soils
,”
J. Acoust. Soc. Am.
,
136
(
4
), pp.
2077
2077
.
7.
Deymier
,
P. A.
,
2013
,
Acoustic Metamaterials and Phononic Crystals
,
Springer Science & Business Media
,
Springer, Berlin
.
8.
Aravantinos-Zafiris
,
N.
, and
Sigalas
,
M.
,
2015
, “
Large Scale Phononic Metamaterials for Seismic Isolation
,”
J. Appl. Phys.
,
118
(
6
), p.
064901
.
9.
Fraternali
,
F.
,
Carpentieri
,
G.
,
Amendola
,
A.
,
Skelton
,
R. E.
, and
Nesterenko
,
V. F.
,
2014
, “
Multiscale Tunability of Solitary Wave Dynamics in Tensegrity Metamaterials
,”
Appl. Phys. Lett.
,
105
(
20
), p.
201903
.
10.
Casadei
,
F.
,
Delpero
,
T.
,
Bergamini
,
A.
,
Ermanni
,
P.
, and
Ruzzene
,
M.
,
2012
, “
Piezoelectric Resonator Arrays for Tunable Acoustic Waveguides and Metamaterials
,”
J. Appl. Phys.
,
112
(
6
), p.
064902
.
11.
Hussein
,
M. I.
, and
Frazier
,
M. J.
,
2013
, “
Metadamping: An Emergent Phenomenon in Dissipative Metamaterials
,”
J. Sound Vib.
,
332
(
20
), pp.
4767
4774
.
12.
Daraio
,
C.
, and
Rizzo
,
P.
,
2012
, “Method and Apparatus for Nondestructive Evaluation and Monitoring of Materials and Structures,” University of Pittsburgh/California Institute of Technology, Pittsburgh, PA/Pasadena, CA, U.S. Patent No.
8,327,709
.https://www.google.com/patents/US8327709
13.
Bagheri
,
A.
,
Rizzo
,
P.
, and
Al-Nazer
,
L.
,
2016
, “
A Numerical Study on the Optimization of a Granular Medium to Infer the Axial Stress in Slender Structures
,”
Mech. Adv. Mater. Struct.
,
23
(
10
), pp.
1131
1143
.
14.
Bagheri
,
A.
,
Ribolla
,
E. L. M.
,
Rizzo
,
P.
,
Al-Nazer
,
L.
, and
Giambanco
,
G.
,
2015
, “
On the Use of L-Shaped Granular Chains for the Assessment of Thermal Stress in Slender Structures
,”
Exp. Mech.
,
55
(
3
), pp.
543
558
.
15.
Bagheri
,
A.
,
Rizzo
,
P.
, and
Al-Nazer
,
L.
,
2014
, “
Determination of the Neutral Temperature of Slender Beams by Using Nonlinear Solitary Waves
,”
J. Eng. Mech.
,
141
(
6
), p.
04014163
.
16.
Rizzo
,
P.
,
Ni
,
X.
,
Nassiri
,
S.
, and
Vandenbossche
,
J.
,
2014
, “
A Solitary Wave-Based Sensor to Monitor the Setting of Fresh Concrete
,”
Sensors
,
14
(
7
), pp.
12568
12584
.
17.
Li
,
K.
,
Rizzo
,
P.
, and
Ni
,
X.
,
2014
, “
Alternative Designs of Acoustic Lenses Based on Nonlinear Solitary Waves
,”
ASME J. Appl. Mech.
,
81
(
7
), p.
071011
.
18.
Cai
,
L.
,
Rizzo
,
P.
, and
Al-Nazer
,
L.
,
2013
, “
On the Coupling Mechanism Between Nonlinear Solitary Waves and Slender Beams
,”
Int. J. Solids Struct.
,
50
(
25
), pp.
4173
4183
.
19.
Cai
,
L.
,
Yang
,
J.
,
Rizzo
,
P.
,
Ni
,
X.
, and
Daraio
,
C.
,
2013
, “
Propagation of Highly Nonlinear Solitary Waves in a Curved Granular Chain
,”
Granular Matter
,
15
(
3
), pp.
357
366
.
20.
Rozina
,
S.
,
Andrei
,
D. N.
,
Nicoleta
,
I.
,
Catalin-Andrei
,
T.
,
Frantisek
,
N.
,
Stanislava
,
F.
,
Petrica
,
V.
, and
Adriana
,
S.
, “
Nondestructive Testing of Advanced Materials Using Sensors With Metamaterials
,”
IOP Conf. Ser.: Mater. Sci. Eng.
, p.
012060
.
21.
Madeo
,
A.
,
Placidi
,
L.
, and
Rosi
,
G.
,
2014
, “
Towards the Design of Metamaterials With Enhanced Damage Sensitivity: Second Gradient Porous Materials
,”
Res. Nondestr. Eval.
,
25
(
2
), pp.
99
124
.
22.
Savin
,
A.
,
Bruma
,
A.
,
Steigmann
,
R.
,
Iftimie
,
N.
, and
Faktorova
,
D.
,
2015
, “
Enhancement of Spatial Resolution Using a Metamaterial Sensor in Nondestructive Evaluation
,”
Appl. Sci.
,
5
(
4
), pp.
1412
1430
.
23.
Li
,
K.
, and
Rizzo
,
P.
,
2015
, “
Energy Harvesting Using Arrays of Granular Chains and Solid Rods
,”
J. Appl. Phys.
,
117
(
21
), p.
215101
.
24.
Li
,
K.
, and
Rizzo
,
P.
,
2015
, “
Energy Harvesting Using an Array of Granules
,”
ASME J. Vib. Acoust.
,
137
(
4
), p.
041002
.
25.
Chen
,
Z.
,
Guo
,
B.
,
Yang
,
Y.
, and
Cheng
,
C.
,
2014
, “
Metamaterials-Based Enhanced Energy Harvesting: A Review
,”
Phys. B: Condens. Matter
,
438
, pp.
1
8
.
26.
Carrara
,
M.
,
Cacan
,
M.
,
Toussaint
,
J.
,
Leamy
,
M.
,
Ruzzene
,
M.
, and
Erturk
,
A.
,
2013
, “
Metamaterial-Inspired Structures and Concepts for Elastoacoustic Wave Energy Harvesting
,”
Smart Mater. Struct.
,
22
(
6
), p.
065004
.
27.
Hu
,
G.
,
Tang
,
L.
,
Banerjee
,
A.
, and
Das
,
R.
,
2017
, “
Metastructure With Piezoelectric Element for Simultaneous Vibration Suppression and Energy Harvesting
,”
ASME J. Vib. Acoust.
,
139
(
1
), p.
011012
.
28.
Bigoni
,
D.
,
Guenneau
,
S.
,
Movchan
,
A. B.
, and
Brun
,
M.
,
2013
, “
Elastic Metamaterials With Inertial Locally Resonant Structures: Application to Lensing and Localization
,”
Phys. Rev. B
,
87
(
17
), p.
174303
.
29.
Rosenblatt
,
G.
, and
Orenstein
,
M.
,
2015
, “
Perfect Lensing by a Single Interface: Defying Loss and Bandwidth Limitations of Metamaterials
,”
Phys. Rev. Lett.
,
115
(
19
), p.
195504
.
30.
Bénédicto
,
J.
,
Centeno
,
E.
,
Pollès
,
R.
, and
Moreau
,
A.
,
2013
, “
Ultimate Resolution of Indefinite Metamaterial Flat Lenses
,”
Phys. Rev. B
,
88
(
24
), p.
245138
.
31.
Kaina
,
N.
,
Lemoult
,
F.
,
Fink
,
M.
, and
Lerosey
,
G.
,
2015
, “
Negative Refractive Index and Acoustic Superlens From Multiple Scattering in Single Negative Metamaterials
,”
Nature
,
525
(
7567
), p.
77
.
32.
Nesterenko
,
V.
,
1983
, “
Propagation of Nonlinear Compression Pulses in Granular Media
,”
J. Appl. Mech. Tech. Phys.
,
24
(
5
), pp.
733
743
.
33.
Coste
,
C.
,
Falcon
,
E.
, and
Fauve
,
S.
,
1997
, “
Solitary Waves in a Chain of Beads Under Hertz Contact
,”
Phys. Rev. E
,
56
(
5
), p.
6104
.
34.
Daraio
,
C.
,
Nesterenko
,
V.
,
Herbold
,
E.
, and
Jin
,
S.
,
2006
, “
Tunability of Solitary Wave Properties in One-Dimensional Strongly Nonlinear Phononic Crystals
,”
Phys. Rev. E
,
73
(
2
), p.
026610
.
35.
Khatri
,
D.
,
Ngo
,
D.
, and
Daraio
,
C.
,
2012
, “
Highly Nonlinear Solitary Waves in Chains of Cylindrical Particles
,”
Granular Matter
,
14
(1), pp.
63
69
.
36.
Ngo
,
D.
,
Griffiths
,
S.
,
Khatri
,
D.
, and
Daraio
,
C.
,
2013
, “
Highly Nonlinear Solitary Waves in Chains of Hollow Spherical Particles
,”
Granular Matter
,
15
(
2
), pp.
149
155
.
37.
Ngo
,
D.
,
Khatri
,
D.
, and
Daraio
,
C.
,
2011
, “
Highly Nonlinear Solitary Waves in Chains of Ellipsoidal Particles
,”
Phys. Rev. E
,
84
(
2
), p.
026610
.
38.
Nesterenko
,
V.
,
2013
,
Dynamics of Heterogeneous Materials
,
Springer Science & Business Media
,
Berlin
.
39.
Schiffer
,
A.
,
Alkhaja
,
A.
,
Yang
,
J.
,
Esfahani
,
E.
, and
Kim
,
T.-Y.
,
2017
, “
Interaction of Highly Nonlinear Solitary Waves With Elastic Solids Containing a Spherical Void
,”
Int. J. Solids Struct.
,
118–119
, pp.
204
212
.
40.
Yang
,
J.
,
Silvestro
,
C.
,
Khatri
,
D.
,
De Nardo
,
L.
, and
Daraio
,
C.
,
2011
, “
Interaction of Highly Nonlinear Solitary Waves With Linear Elastic Media
,”
Phys. Rev. E
,
83
(
4
), p.
046606
.
41.
Job
,
S.
,
Melo
,
F.
,
Sokolow
,
A.
, and
Sen
,
S.
,
2005
, “
How Hertzian Solitary Waves Interact With Boundaries in a 1D Granular Medium
,”
Phys. Rev. Lett.
,
94
(
17
), p.
178002
.
42.
Job
,
S.
,
Melo
,
F.
,
Sokolow
,
A.
, and
Sen
,
S.
,
2007
, “
Solitary Wave Trains in Granular Chains: Experiments, Theory and Simulations
,”
Granular Matter
,
10
(
1
), pp.
13
20
.
43.
Manciu
,
F. S.
, and
Sen
,
S.
,
2002
, “
Secondary Solitary Wave Formation in Systems With Generalized Hertz Interactions
,”
Phys. Rev. E
,
66
(
1
), p.
016616
.
44.
Falcon
,
E.
,
Laroche
,
C.
,
Fauve
,
S.
, and
Coste
,
C.
,
1998
, “
Collision of a 1-D Column of Beads With a Wall
,”
Eur. Phys. J. B-Condens. Matter Complex Syst.
,
5
(
1
), pp.
111
131
.
45.
Ni
,
X.
,
Rizzo
,
P.
,
Yang
,
J.
,
Katri
,
D.
, and
Daraio
,
C.
,
2012
, “
Monitoring the Hydration of Cement Using Highly Nonlinear Solitary Waves
,”
NDT&E Int.
,
52
, pp.
76
85
.
46.
Ni
,
X.
, and
Rizzo
,
P.
,
2012
, “
Use of Highly Nonlinear Solitary Waves in Nondestructive Testing
,”
Mater. Eval.
,
70
(
5
), pp.
561
569
.
47.
Nasrollahi
,
A.
,
Deng
,
W.
,
Rizzo
,
P.
,
Vuotto
,
A.
, and
Vandenbossche
,
J. M.
,
2017
, “
Nondestructive Testing of Concrete Using Highly Nonlinear Solitary Waves
,”
Nondestr. Test. Eval.
,
32
(
4
), pp.
381
399
.
48.
Rizzo
,
P.
,
Nasrollahi
,
A.
,
Deng
,
W.
, and
Vandenbossche
,
J. M.
,
2016
, “
Detecting the Presence of High Water-to-Cement Ratio in Concrete Surfaces Using Highly Nonlinear Solitary Waves
,”
Appl. Sci.
,
6
(
4
), pp.
104
120
.
49.
Deng
,
W.
,
Nasrollahi
,
A.
,
Rizzo
,
P.
, and
Li
,
K.
,
2016
, “
On the Reliability of a Solitary Wave Based Transducer to Determine the Characteristics of Some Materials
,”
Sensors
,
16
(
1
), p.
5
.
50.
Vergara
,
L.
,
2005
, “
Scattering of Solitary Waves From Interfaces in Granular Media
,”
Phys. Rev. Lett.
,
95
(
10
), p.
108002
.
51.
Bagheri
,
A.
, and
Rizzo
,
P.
,
2017
, “
Assessing the Pressure of Tennis Balls Using Nonlinear Solitary Waves: A Numerical Study
,”
Sports Eng.
,
20
(
1
), pp.
53
62
.
52.
Belytschko
,
T.
,
Liu
,
W. K.
,
Moran
,
B.
, and
Elkhodary
,
K.
,
2013
,
Nonlinear Finite Elements for Continua and Structures
,
Wiley
,
Hoboken, NJ
.
53.
Frey
,
P. J.
, and
George
,
P. L.
,
2000
,
Mesh Generation: Application to Finite Elements
, Wiley, Hoboken, NJ.
54.
Goodwill
,
S.
,
Kirk
,
R.
, and
Haake
,
S.
,
2005
, “
Experimental and Finite Element Analysis of a Tennis Ball Impact on a Rigid Surface
,”
Sports Eng.
,
8
(
3
), pp.
145
158
.
55.
Herbold
,
E. B.
,
2008
, “
Optimization of the Dynamic Behavior of Strongly Nonlinear Heterogeneous Materials
,”
Ph.D. thesis
, University of California, San Diego, CA.https://cloudfront.escholarship.org/dist/prd/content/qt03f6q1m0/qt03f6q1m0.pdf
56.
Gamma, 2017, “
GAMMA
,” GAMMA Sports, Pittsburgh, PA, accessed Dec. 1, 2017, https://gammasports.com/tennis/kids-and-training/
57.
Andreaus
,
U.
,
Chiaia
,
B.
, and
Placidi
,
L.
,
2013
, “
Soft-Impact Dynamics of Deformable Bodies
,”
Continuum Mech. Thermodyn.
,
25
(
2
), pp.
375
398
.
You do not currently have access to this content.