In this paper, a spectral finite element method (SFEM) based on the alternating frequency–time (AFT) framework is extended to study impact wave propagation in a rod structure with a general material nonlinearity. The novelty of combining AFT and SFEM successfully solves the computational issue of existing nonlinear versions of SFEM and creates a high-fidelity method to study impact response behavior. The validity and efficiency of the method are studied through comparison with the prediction of a qualitative analytical study and a time-domain finite element method (FEM). A new analytical approach is also proposed to derive an analytical formula for the wavenumber. By using the wavenumber equation and with the help of time–frequency analysis techniques, the physical meaning of the nonlinear behavior is studied. Through this combined effort with both analytical and numerical components, distortion of the wave shape and dispersive behavior have been identified in the nonlinear response. The advantages of AFT-FEM are (1) high-fidelity results can be obtained with fewer elements for high-frequency impact shock response conditions; (2) dispersion or dissipation is not erroneously introduced into the response as can occur with time-domain FEM; (3) the high-fidelity properties of SFEM enable it to provide a better interpretation of nonlinear behavior in the response; and (4) the AFT framework makes it more computationally efficient when compared to existing nonlinear versions of SFEM which often involve convolution operations.

References

References
1.
Li
,
Q.
,
Chan
,
C.
,
Ho
,
K.
, and
Soukoulis
,
C.
,
1996
, “
Wave Propagation in Nonlinear Photonic Band-Gap Materials
,”
Phys. Rev. B
,
53
(
23
), pp.
15577
15585
.10.1103/PhysRevB.53.15577
2.
Adams
,
D.
,
2007
,
Health Monitoring of Structural Materials and Components: Methods With Applications
,
Wiley
,
Chichester, UK
.10.1002/9780470511589
3.
Meo
,
M.
, and
Zumpano
,
G.
,
2005
, “
Nonlinear Elastic Wave Spectroscopy Identification of Impact Damage on a Sandwich Plate
,”
Compos. Struct.
,
71
(
3–4
), pp.
469
474
.10.1016/j.compstruct.2005.09.027
4.
DeSalvo
,
R.
,
2007
, “
Passive, Nonlinear, Mechanical Structures for Seismic Attenuation
,”
ASME J. Comput. Nonlinear Dyn.
,
2
(
4
), pp.
290
298
.10.1115/1.2754305
5.
Ramabathiran
,
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
6.
Nayfeh
,
A.
, and
Mook
,
D.
,
1995
,
Nonlinear Oscillations
,
Wiley
,
New York
.10.1002/9783527617586
7.
Abdusalam
,
H. A.
,
2005
, “
On an Improved Complex Tanh-Function Method
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
6
(
2
), pp.
99
106
.10.1515/IJNSNS.2005.6.2.99
8.
Malfliet
,
W.
, and
Hereman
,
W.
,
1996
, “
The Tanh Method: 1. Exact Solutions of Nonlinear Evolution and Wave Equations
,”
Phys. Scr.
,
54
(
6
), pp.
563
568
.10.1088/0031-8949/54/6/003
9.
Abdou
,
M.
,
2009
, “
Exact Travelling Wave Solutions in a Nonlinear Elastic Rod Equation
,”
Int. J. Nonlinear Sci.
,
7
(
2
), pp.
167
173
.
10.
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
11.
He
,
J.-H.
, and
Wu
,
X.-H.
,
2006
, “
Exp-Function Method for Nonlinear Wave Equations
,”
Chaos, Solitons Fractals
,
30
(
3
), pp.
700
708
.10.1016/j.chaos.2006.03.020
12.
Saraç
,
Y.
,
2012
, “
Symbolic and Numeric Computation of Optimal Initial Velocity in a Wave Equation
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
1
), p.
011018
.10.1115/1.4006786
13.
Kudryashov
,
N.
,
2009
, “
Seven Common Errors in Finding Exact Solutions of Nonlinear Differential Equations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
9–10
), pp.
3507
3529
.10.1016/j.cnsns.2009.01.023
14.
Ham
,
S.
, and
Bathe
,
K.-J.
,
2012
, “
A Finite Element Method Enriched for Wave Propagation Problems
,”
Comput. Struct.
,
94–95
, pp.
1
12
.10.1016/j.compstruc.2012.01.001
15.
Idesman
,
A.
,
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
16.
Mahapatra
,
D.
, and
Gopalakrishnan
,
S.
,
2003
, “
A Nonlinear Spectral Finite Element Model for Analysis of Wave Propagation in Solid With Internal Friction and Dissipation
,”
International Conference on Computational Science and Its Applications 2003
, Montreal, QC, pp.
745
754
.
17.
Doyle
,
J.
,
1997
,
Wave Propagation in Structures
,
Springer
,
Berlin, Germany
.10.1007/978-1-4612-1832-6
18.
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
19.
Oppenheim
,
A.
, and
Schafer
,
R.
,
2009
,
Discrete-Time Signal Processing
,
Prentice Hall
,
Upper Saddle River
,
NJ
.
20.
Lee
,
U.
,
2009
,
Spectral Element Method in Structural Dynamics
,
Wiley
,
Hoboken, NJ
.10.1002/9780470823767
21.
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
22.
Lee
,
U.
, and
Jang
,
I.
,
2011
, “
Nonlinear Spectral Element Model for the Blood Flow in Human Arteries
,”
AIP Conf. Proc.
,
1371
, pp.
136
145
.10.1063/1.3596636
23.
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
24.
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
25.
Reddy
,
J.
,
2004
,
An Introduction to Nonlinear Finite Element Analysis
,
Oxford University Press
,
Oxford, UK
.10.1093/acprof:oso/9780198525295.001.0001
You do not currently have access to this content.