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Title: Behavior of Nonlinear Higher Harmonics in Plate and Rod Guided Waves
Authors: Ankit Srivastava
Ivan Bartoli
Salvatore Salamone
Francesco Lanza di Scalea
ABSTRACT
The study of nonlinear elastic wave propagation has been of considerable interest for the last four
decades. This, in part, is due to the fact that nonlinear parameters are, in general, more sensitive to
structural defects than linear parameters. Although guided waves combine the sensitivity of
nonlinear parameters with large inspection ranges there are very few studies of guided nonlinear
elastic waves due to the mathematical complexity of the problem.
This paper presents a theoretical study into nonlinear Lamb waves and cylindrical waves. It has
been known that the nonlinearity driven double harmonic in Lamb waves does not support
antisymmetric motion. However the proof of this has not been obvious. Moreover, little is known on
nonlinearity driven higher harmonics for either Lamb waves or cylindrical waves. These gaps are
here studied by the method of perturbation coupled with wavemode orthogonality and forced
response. For Lamb waves it is shown that antisymmetric motion is prohibited at all the higherorder even harmonics, whereas all the higher order odd harmonics allow both symmetric and
antisymmetric motions. Similarly, in the case of cylindrical waves, it is shown that the nature of the
primary generating mode severely restricts the families of modes that are generated at the higher
harmonics.
Finally, experimental results on measuring prestress levels in 7-wire PS tendons using nonlinear
ultrasonic guided waves in the 100 kHz – 2 MHz range are presented.
_____________
Ankit Srivastava, Graduate Student, NDE & SHM Laboratory, Department of Structural Engineering,
University of California, San Diego, La Jolla, CA 92093-0085
Ivan Bartoli, Ph.D., NDE & SHM Laboratory, Department of Structural Engineering, University of California,
San Diego, La Jolla, CA 92093-0085
Salvatore Salamone, Ph.D., NDE & SHM Laboratory, Department of Structural Engineering, University of
California, San Diego, La Jolla, CA 92093-0085
Francesco Lanza di Scalea (corresponding author), Professor, NDE & SHM Laboratory, Department of
Structural Engineering, University of California, San Diego, La Jolla, CA 92093-0085
INTRODUCTION
The study of nonlinear elastic wave propagation has been of considerable interest for the last four
decades. This, in part, is due to the fact that nonlinear parameters are, in general, more sensitive to
structural defects than linear parameters [1]. Guided waves combine the sensitivity of nonlinear
parameters with large inspection ranges [2]. Therefore, their application to nondestructive
evaluation and structural health monitoring has drawn considerable research interest [3] and [4].
There are very few studies of guided nonlinear elastic waves due to the mathematical complexity
of the problem. A recent investigation pertaining to the second harmonic generation in guided Lamb
waves was reported by Deng [5], [6], [7] and [8] but the physical manifestations were obscured due
to the complexity of their formulation. de Lima and Hamilton [9], [10] and subsequently Deng [8]
analyzed the problem of nonlinear guided waves in isotropic plates and rods by using normal mode
decomposition and forced response as suggested by Auld [11]. The authors used their formulation
to explain the generation of the double harmonic and the cumulative growth of a phase-matched
higher harmonic guided mode. However, their conclusions are limited to the double harmonics.
Srivastava and Lanza di Scalea [12] extended the analysis to higher harmonics to prove that
Rayleigh–Lamb antisymmetric motion is never allowed at even harmonics (2ω, 4ω,...) , whereas
both symmetric and antisymmetric Rayleigh–Lamb motions can exist at the odd harmonics (3ω,
5ω,...). They also concluded that, in the case of cylindrical waves, the nature of the primary
generating mode severely restricts the families of modes that are generated at the higher harmonics
[13].
Multi-wire steel strands are used in civil engineering as the tensioning components of pre-stressed
concrete structures and in cable systems of cable-stayed and suspension bridges. As documented in
several studies [14], [15], [16], [17], the presence of defects or the tendon breakage can induce
serious consequences for these structures. Recently, the attention has been focused on the behavior
of nonlinear ultrasonic guided waves for the monitoring of pre-stress levels in the strands. In fact, it
has been observed that the nonlinear features are more sensitive than the linear features to the prestress levels. The ultrasonic nonlinearities in the strands manifest themselves by the generation of
higher harmonics (2ω, 3ω,..) and/or sub-harmonics (0.5ω) when the excitation is at a primary
frequency of ω. Experimental studies will be presented showing that the nonlinearity increases with
decreasing inter-wire stresses (i.e. decreasing pre-stress level).
STATEMENT OF THE NONLINEAR PROBLEM
The equation of motion for nonlinear elasticity in a waveguide is given by (Fig. 1)
 2u
   2    u        u   f  0 2
t
(1) with stress free boundary conditions on the surface:
 S L  u   S  u   nr  0


(2)
z Ω Γ Figure 1. Schematic of a stress free waveguide.
where u is the particle displacement,  , and  are the Lame constants,  0 is the initial density of
the body, f is the body force, nr is the unit vector normal to the surface of the waveguide  , S L ,
and S are the linear and nonlinear parts of the second Piola-Kirchoff stress tensor, respectively.
Energy is written in Murnaghan potentials [18]:
E  2  3  4 ...
(3) where n corresponds to the set of terms in the energy expression which are of degree n in strain
multiples.
SOLUTION TO THE NONLINEAR PROBLEM
Following Auld [11] and de Lima and Hamilton [10] and using method of perturbation, the first
order nonlinear solution is written as a linear combination of the existing guided wavemodes at 2 :

 
 
v r , z , t  .5 Am ( z )v m (r )e i 2t


m 1
(4) 
where v m is the particle velocity of the mth mode at 2 , and Am is the higher order modal
amplitude given by:
*
Am ( z )  Am ( z )ei 2 z  Am (0)ei 2 n z
(5a) where, Am ( z )  i
Am
f
vol
n
 f nsurf

4 Pmn   2 
*
n
f
( z)  i
Pmn  .25

vol
n

 f nsurf
4 Pmn
;  n*  2  ;
*
n
 2    
vn*  Sm  vm  Sn* .nz d  
(5b) (5c)
(5d) 
f nsurf ( z )   vn*  S nr d 
(5e)

f nvol ( z )   vn* . f d 
(5f) 

 is the wavenumber of the primary generating mode,  n is the wavenumber of the wave that is

not orthogonal to the mth mode at the higher harmonic. S m is the stress tensor for the mth mode, nz
is the unit vector in the wave propagation direction. S and f are the nonlinear surface traction and
body force, respectively.
NONLINEAR HIGHER HARMONICS IN PLATES
Srivastava and Lanza di Scalea [12] proved that in the case of Rayleigh-Lamb dispersion relation,
the nonlinear solution (Eqs. 4, 5) is such that antisymmetric modes are nonexistent at even
harmonics (2ω, 4ω,...), whereas odd harmonics (3ω, 5ω,...) support both symmetric and
antisymmetric modes.
Two experiments were carried out to test the theoretical result. In the first experiment, one Pico
transducer (Physical Acoustics Corporation, 0.1–1 MHz, central frequency 0.543 MHz) was used to
generate Lamb waves in an aluminum plate of thickness 2.54 mm. The response was measured at a
distance of 25 cm by a Pinducer sensor (Valpey Fisher VP-1093). The plate was loaded quasistatiscally to a level large enough to induce measurable nonlinearity driven higher harmonics of the
primary Rayleigh–Lamb wave. Both the Pico and the Pinducer work by exciting and sensing out of
plane displacements, therefore, generate and receive predominantly antisymmetric motion. The
excitation was driven at a monochromatic frequency of 320 kHz.
Figure 2. Joint time–frequency analysis of the antisymmetric excitation and detection in the plate: (a) time history, (b)
wavelet scalogram of the signal in the DC-1 MHz range, (c) zoomed view of the wavelet scalogram. White lines are the
theoretical arrival times from the Rayleigh–Lamb formulation.
Fig. 2(b) shows the wavelet scalogram applied to the time signal depicted in Fig. 2(a). Fig 2(c)
shows a zoomed in view of the scalogram in the frequency range of 0.5–1 MHz. The white lines are
the theoretical arrival times of the pertinent modes according to Rayleigh–Lamb theory. It can be
seen that while a strong antisymmetric mode is present at the generation frequency (320 kHz), there
is no antisymmetric mode present at the double harmonic (640 kHz). The antisymmetric mode is,
instead, present at the triple harmonic (960 kHz), as predicted by the theoretical formulations.
In the second experiment, two macro-fiber composite (MFC) transducers (Smart Materials
Corporation, M2814P1) were used for both excitation and detection. MFCs work by generating and
detecting in-plane strains and hence they are preferentially sensitive to symmetric waves.
Figure 3. Joint time–frequency analysis of the symmetric excitation and detection in the plate: (a) time history, (b)
wavelet scalogram, (c) zoomed view of the wavelet scalogram. White lines are theoretical arrival times from Rayleigh–
Lamb formulation.
Fig. 3 shows the continuous wavelet scalogram of the received signal in the antisymmetric
experiment, along with the theoretical Rayleigh–Lamb curves of the pertinent modes. It can be seen
from Fig. 3(b) that while the primary harmonic (320 kHz) consists of both symmetric and
antisymmetric modes, the energy at the double harmonic (640 kHz) corresponds exclusively to the
fundamental symmetric mode (Fig. 3c). The energy at the triple harmonic (960 kHz), as expected,
consists of a combination of the fundamental antisymmetric, the fundamental symmetric, and the
first-order antisymmetric modes. Hence the experiments confirm that the antisymmetric modes can
only exist at odd harmonics, whereas symmetric modes can exist at both odd and even harmonics.
NONLINEAR HIGHER HARMONICS IN RODS
Srivastava and Lanza di Scalea [13] studied the problem of nonlinear higher harmonics in rods by
using wavemode orthogonality and perturbation techniques. The following conclusions, as
applicable to rod waves, were reached:
A primary generating mode in a rod with an angular order p will generate a rod mode with an
angular order l at the nth higher harmonic if and only if l  kp for some values of k where:
1. k spans all odd numbers from 1 to n when n is odd (odd harmonics).
2. k spans all even numbers from 0 to n when n is even (even harmonics).
Therefore, for odd harmonics:
1. A longitudinal primary generating mode will not produce any flexural modes. A
longitudinal primary generating mode can only produce longitudinal modes.
2. Only selected modes can be generated by flexural primary generating modes. For example,
a first order flexural mode ( p  1 ) at the triple harmonic ( n  3 ) can only generate the first
order ( l  1 ) and third order ( l  3 ) flexural modes. In general, a p th order flexural mode
can only generate at the nth harmonic flexural modes of orders equal to odd multiples of p ,
up to np .
For even harmonics:
1. Longitudinal modes can be generated irrespective of whether the primary generating mode
is longitudinal or flexural. Moreover, a longitudinal primary generating mode does not
produce flexural modes.
2. As in the case of odd harmonics, only selected modes can be generated by flexural primary
generating modes. For example, a first order flexural mode ( p  1 ) at the double harmonic
can only generate the longitudinal ( l  0 ) and second order ( l  2 ) flexural modes. In
general, a p th order flexural mode can only generate at the nth harmonic the longitudinal
mode and flexural modes of orders equal to even multiples of p , up to np .
PRE-STRESS LEVEL MONITORING IN SEVEN-WIRE STRAND
(a)
(b)
SEVEN-WIRE
STRAND
PZT5
D
182cm
PZT4
D
PZT3
D
PZT2
D
PZT1
PICO
CENT
PICO
PER
Figure 4. (a) The seven-wire strand on the SATEC machine. (b) Ultrasonic transducer lay-out and pictures.
Tests were performed at UCSD’s Powell Labs on the SATEC M600XWHVL, 600 kip capacity
testing machine (Figure 4a). A 0.6-in, 1.82m long, seven-wire strand (grade 270, U.T.S. 1.86 GPa270ksi) was subjected to load-unload cycles. The loading cycles were performed between zero
stress and a maximum stress of 70% U.T.S. and divided in 6 intervals. A number of ultrasonic
transducers were placed on the strand. The most successful configuration used ultrasonic generation
by transducer PZT1 and ultrasonic reception by transducers PZT2 through PZT5 (Figure 4b). A
special holder was designed to keep two additional ultra-mini broadband sensors of the Physical
Acoustic Corporation (PICO type) in place at the strand’s free end.
A 30-cycle ultrasonic toneburst, with center frequency controllable by software, was used to create
a narrowband excitation at the primary frequency of ω. The nonlinearity was quantified by the socalled “beta parameter”, β, defined as:

Amplitude of the 2 nd Harmonic at 2
Amplitude of the primary excitation at 
(6) Load = 0 %
0.05
Amplitude
Amplitude
0.04
0.03
0.02
0.01
0
0
500
1000
1500
Frequency [kHz]
2000
0
Load = 40 %
500
1000
1500
Frequency [kHz]
2000
2500
2000
2500
2000
2500
Load = 60 %
0.04
Amplitude
Amplitude
0
0.05
0.03
0.02
0.01
0.03
0.02
0.01
0
500
1000
1500
Frequency [kHz]
2000
0
2500
Load = 80 %
0.05
0
500
1000
1500
Frequency [kHz]
Load = 100 %
0.05
0.04
Amplitude
0.04
Amplitude
0.02
2500
0.04
0.03
0.02
0.01
0
0.03
0.01
0.05
0
Load = 20 %
0.05
0.04
0.03
0.02
0.01
0
500
1000
1500
Frequency [kHz]
2000
0
2500
0
500
1000
1500
Frequency [kHz]
nd
Non-linear Parameter referred to the 2 Harmonic versus Load Level
Loading Phase - Frequency Input = 550 kHz
0.014
0.013

0.012
0.011
0.01
0.009
0.008
0%
20%
40%
60%
Load Level
80%
100%
Figure 5. (a) FFT spectra of signals detected by PZT5 for a 30 cycle toneburst excitation centered at 550 kHz. (b)
Nonlinear parameter β from 2nd harmonic as a function of load applied to the strand.
Figure 5(a) shows the Fast-Fourier Transform (FFT) of the signal measured by PZT5 under an
excitation frequency of 550 kHz for different levels of load applied to the strand. It is apparent that
the FFT spectra reveal the generation of higher-harmonics, beyond the 550 kHz primary excitation,
indicating nonlinear behavior. Figure 5(b) shows that the nonlinear parameter β indeed decreases
with increasing load level from zero stress to 70% Ultimate Tensile Stress (U.T.S.). It can also be
seen that the trend of β vs. Load is reasonably linear, indicating the suitability for this parameter to
provide a direct indication of the level of pre-stress in the strand once the slope of the β vs. Load
line is known.
CONCLUSIONS
The nonlinear Rayleigh–Lamb guided wave problem was studied using the method of perturbation
coupled with wavemode orthogonality and forced response. It was shown that nonlinearity induced
even harmonics do not support antisymmetric motion whereas symmetric Rayleigh–Lamb waves
are allowed at all (odd or even) harmonics. For the case of rod waves it was concluded that the
condition of existence/nonexistence of nonlinearity-driven higher harmonics of longitudinal and
flexural waves in rods depends upon an angular order-based constraint. Finally, experimental results
were presented which indicated that the nonlinear ultrasonic parameter β, measuring the ratio
between higher order harmonics and the fundamental generated harmonic is a suitable feature for
monitoring prestress levels in free and embedded strands.
ACKNOWLEDGMENTS
This work was funded by Air Force Office of Scientific Research contract no. FA9550-07-1-0016
(Dr. Victor Giurgiutiu, Program Manager), Office of Naval Research contract no. N00014-08-10973 (Drs. Paul Hess and Liming Salvino, Program Managers), and by the California Department of
Transportation under contract # 59A0538 (Dr. C. Sikorsky, Program Manager).
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