In this paper, transient wave propagation in nonlinear one-dimensional (1D) waveguides is studied. A complete nonlinear (CN) 1D model accounting for both axial and transverse displacements is developed and geometric and material nonlinearities are separately modeled. The alternating frequency-time finite element method (AFT-FEM) is implemented for this complete 1D model. Numerical simulations are conducted and the response behaviors for axial and transverse motions are analyzed. Comparison of the responses for the geometrically nonlinear (GN) model with a corresponding linear model supports predictions made from the previous analytical studies that the geometric nonlinearity has limited influence on the response of transient transverse waves in the intermediate strain regime. On the contrary, strong nonlinear behavior appears in the response for the materially nonlinear (MN) models. Depending on the local nonlinear property of the material in the intermediate strain regime, the amplitude of the response can be significantly influenced and additional dispersion can be introduced into the response. An exploration of the interaction between the geometric nonlinearity and the material nonlinearity for a rod model in a large strain regime is also conducted and the responses are analyzed by using time-frequency analysis. The competing effect of the geometric nonlinearity and the material nonlinearity can result in a pseudolinear response in a strong nonlinear system for a given range of impact loading.

References

References
1.
Humphrey
,
V.
,
2000
, “
Nonlinear Propagation in Ultrasonic Fields: Measurements, Modelling and Harmonic Imaging
,”
Ultrasonics
,
38
(
1
), pp.
267
272
.10.1016/S0041-624X(99)00122-5
2.
Ward
,
B.
,
Baker
,
A.
, and
Humphrey
,
V.
,
1997
, “
Nonlinear Propagation Applied to the Improvement of Resolution In Diagnostic Medical Ultrasound
,”
J. Acoust. Soc. Am.
,
101
, pp.
143
154
.10.1121/1.417977
3.
Meo
,
M.
, and
Zumpano
,
G.
,
2005
, “
Nonlinear Elastic Wave Spectroscopy Identification of Impact Damage on a Sandwich Plate
,”
Compos. Struct.
,
71
(
3
), pp.
469
474
.10.1016/j.compstruct.2005.09.027
4.
Li
,
Q.
,
Chan
,
C.
,
Ho
,
K.
, and
Soukoulis
,
C.
,
1996
, “
Wave Propagation in Nonlinear Photonic Band-Gap Materials
,”
Phys. Rev. B
,
53
(
23
), p. 15577.10.1103/PhysRevB.53.15577
5.
Rosakis
,
A.
,
Samudrala
,
O.
, and
Coker
,
D.
,
1999
, “
Cracks Faster Than the Shear Wave Speed
,”
Science
,
284
(
5418
), pp.
1337
1340
.10.1126/science.284.5418.1337
6.
Sen
,
M.
, and
Stoffa
,
P.
,
1991
, “
Nonlinear One-Dimensional Seismic Waveform Inversion Using Simulated Annealing
,”
Geophysics
,
56
(
10
), pp.
1624
1638
.10.1190/1.1442973
7.
Porubov
,
A.
, and
Pastrone
,
F.
,
2004
, “
Non-Linear Bell-Shaped and Kink-Shaped Strain Waves in Microstructured Solids
,”
Int. J. Non-Linear Mech.
,
39
(
8
), pp.
1289
1299
.10.1016/j.ijnonlinmec.2003.09.002
8.
Samsonov
,
A.
,
Dreiden
,
G.
,
Porubov
,
A.
, and
Semenova
,
I.
,
1998
, “
Longitudinal-Strain Soliton Focusing in a Narrowing Nonlinearly Elastic Rod
,”
Phys. Rev. B
,
57
(
10
), p.
5778
.10.1103/PhysRevB.57.5778
9.
Destrade
,
M.
, and
Saccomandi
,
G.
,
2008
, “
Nonlinear Transverse Waves in Deformed Dispersive Solids
,”
Wave Motion
,
45
(
3
), pp.
325
336
.10.1016/j.wavemoti.2007.07.002
10.
Abedinnasab
,
M. H.
, and
Hussein
,
M. I.
,
2012
, “
Wave Dispersion Under Finite Deformation
,”
Wave Motion
,
50
(
3
), pp.
374
388
.10.1016/j.wavemoti.2012.10.008
11.
Liu
,
Y.
,
Ghaderi
,
P.
, and
Dick
,
A.
,
2012
, “
High Fidelity Methods for Modeling Nonlinear Wave Propagation in One-Dimensional Waveguides
,”
ASME
Paper No. IMECE2012-88162.10.1115/IMECE2012-88162
12.
Porubov
,
A.
, and
Maugin
,
G.
,
2005
, “
Longitudinal Strain Solitary Waves in Presence of Cubic Non-Linearity
,”
Int. J. Non-Linear Mech.
,
40
(
7
), pp.
1041
1048
.10.1016/j.ijnonlinmec.2005.03.001
13.
Xu
,
H.
,
Day
,
S. M.
, and
Minster
,
J.-B. H.
,
2000
, “
Hysteresis and Two-Dimensional Nonlinear Wave Propagation in Berea Sandstone
,”
J. Geophys. Res.
,
105
(
B3
), pp.
6163
6175
.10.1029/1999JB900363
14.
Catheline
,
S.
,
Gennisson
,
J.-L.
,
Tanter
,
M.
, and
Fink
,
M.
,
2003
, “
Observation of Shock Transverse Waves in Elastic Media
,”
Phys. Rev. Lett.
,
91
(
16
), p.
164301
.10.1103/PhysRevLett.91.164301
15.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
2008
,
Nonlinear Oscillations
,
Wiley-VCH
,
Weinheim, Germany
.
16.
Liu
,
Y.
, and
Dick
,
A. J.
,
2014
, “
On the Role of Boundary Conditions in the Nonlinear Dynamic Response of Simple Structures
,”
Proceedings of the SEM IMAC XXXII A Conference and Exposition on Structural Dynamics
, pp.
135
143
.10.1007/978-3-319-04774-4_13
17.
Idesman
,
A. V.
,
2007
, “
A New High-Order Accurate Continuous Galerkin Method for Linear Elastodynamics Problems
,”
Comput. Mech.
,
40
(
2
), pp.
261
279
.10.1007/s00466-006-0096-z
18.
Ham
,
S.
, and
Bathe
,
K.-J.
,
2012
, “
A Finite Element Method Enriched for Wave Propagation Problems
,”
Comput. Struct.
,
94
, pp.
1
12
.10.1016/j.compstruc.2012.01.001
19.
Doyle
,
J.
,
1997
,
Wave Propagation in Structures
,
Springer
,
New York
.
20.
Kudela
,
P.
,
Krawczuk
,
M.
, and
Ostachowicz
,
W.
,
2007
, “
Wave Propagation Modelling in 1D Structures Using Spectral Finite Elements
,”
J. Sound Vib.
,
300
(
1
), pp.
88
100
.10.1016/j.jsv.2006.07.031
21.
Ramabathiran
,
A. A.
, and
Gopalakrishnan
,
S.
,
2012
, “
Time and Frequency Domain Finite Element Models for Axial Wave Analysis in Hyperelastic Rods
,”
Mech. Adv. Mater. Struct.
,
19
(
1–3
), pp.
79
99
.10.1080/15376494.2011.572239
22.
Lee
,
U.
, and
Jang
,
I.
,
2011
, “
Nonlinear Spectral Element Model for the Blood Flow in Human Arteries
,”
AIP Conf. Proc.
,
1371
, p.
136
.
23.
Cameron
,
T.
, and
Griffin
,
J.
,
1989
, “
An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems
,”
ASME J. Appl. Mech.
,
56
(
1
), pp.
149
154
.10.1115/1.3176036
24.
Liu
,
Y.
, and
Dick
,
A. J.
,
2013
, “
Transient Wave Propagation in a Materially Nonlinear Beam
,”
ASME
Paper No. IMECE2013-64975.10.1115/IMECE2013-64975
25.
Lee
,
U.
,
2009
, Spectral Element Method in Structural Dynamics,
Wiley
,
Chichester, UK
.10.1002/9780470823767
26.
Narisetti
,
R. K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2010
, “
A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures
,”
ASME J. Vib. Acoust.
,
132
(
3
), p.
031001
.10.1115/1.4000775
27.
Narisetti
,
R. K.
,
Ruzzene
,
M.
, and
Leamy
,
M. J.
,
2011
, “
A Perturbation Approach for Analyzing Dispersion and Group Velocities in Two-Dimensional Nonlinear Periodic Lattices
,”
ASME J. Vib. Acoust.
,
133
(
6
), p.
061020
.10.1115/1.4004661
28.
Reddy
,
J.
,
2004
,
Nonlinear Finite Element Analysis
,
Oxford University Press
,
New York
.
29.
Schneider
,
S.
,
Schneider
,
S.
,
Silva
,
H.
, and
Moura Neto
,
C.
,
2005
, “
Study of the Non-Linear Stress-Strain Behavior in Ti-Nb-Zr Alloys
,”
Mater. Res.
,
8
(
4
), pp.
435
438
.10.1590/S1516-14392005000400013
30.
aluMATTER
,
2014
, “
Aluminium in Structural Applications
,” Technical Report, April. Available at: http://aluminium.matter.org.uk
31.
Li
,
J.
, and
Zhang
,
Y.
,
2008
, “
Exact Travelling Wave Solutions in a Nonlinear Elastic Rod Equation
,”
Appl. Math. Comput.
,
202
(
2
), pp.
504
510
.10.1016/j.amc.2008.02.027
32.
Shanyuan
,
Z.
, and
Wei
,
Z.
,
1987
, “
The Strain Solitary Waves in a Nonlinear Elastic Rod
,”
Acta Mech. Sin.
,
3
(
1
), pp.
62
72
.10.1007/BF02486784
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