Abstract

The application of guided waves to investigate commonly used plate shapes in the aerospace, mechanical, and civil industries is plates with bend shapes. This article investigates the interaction of fundamental Lamb waves with notches in bent plates, commonly found in aerospace, mechanical, and civil engineering applications. These areas are particularly susceptible to failure due to defects such as cracks and notches, which often manifest as semicircular corrosion patches or 90-deg notches. The presence of notches affects stress distribution, necessitating thorough analysis to prevent accidents. Accordingly, this article focuses on the interaction of fundamental Lamb waves through two types of notches that could be present inside a bent metal plate section. To explore this, a hybrid numerical framework is employed which combines semianalytical finite elements (SAFEs) with the finite element method (FEM). A bent plate section with various notch types is simulated using FEM, while SAFEs facilitate the definition of wave propagation through healthy regions of the plate. The study analyzes the scattering behavior of Lamb waves for different notch configurations and examines both fundamental modes over a specified frequency range. With a change in the interrogation signal parameters, there is a noticeable difference in the sensitivity of scattered waves with different notch types. Formulating a strategy for identifying and locating a notch inside a bent plate may need careful consideration of the important conclusions drawn. Understanding these interactions, the aim of the article is to enhance the integrity assessment of structural components subject to such defects.

References

1.
Hu
,
C.
,
Yang
,
B.
,
Xuan
,
F.-Z.
,
Yan
,
J.
, and
Xiang
,
Y.
,
2020
, “
Damage Orientation and Depth Effect on the Guided Wave Propagation Behavior in 30crmo Steel Curved Plates
,”
Sensors
,
20
(
3
), p.
849
.
2.
Yuan
,
Q.
,
Kato
,
B.
,
Fan
,
K.
, and
Wang
,
Y.
,
2023
, “
Phased Array Guided Wave Propagation in Curved Plates
,”
Mech. Syst. Signal Process.
,
185
, pp.
1
22
.
3.
Cao
,
X.
,
Zeng
,
L.
, and
Lin
,
J.
,
2022
, “
Generalized Scattering Matrix Method for Lamb Wave Scattering Analysis at Cascaded Notches
,”
J. Vib. Control
,
28
(
23–24
), pp.
3638
3651
.
4.
Gangwar
,
A. S.
,
Agrawal
,
Y.
, and
Joglekar
,
D.
,
2021
, “
Nonlinear Interactions of Lamb Waves With a Delamination in Composite Laminates
,”
ASME J. Nondestr. Eval. Diagn. Progn. Eng. Syst.
,
4
(
3
), p.
031008
.
5.
Gangwar
,
A. S.
, and
Joglekar
,
D. M.
,
2024
, “
Probabilistic Bayesian Approach for Delamination Localization in GFRP Composites Using Nonlinear Guided Waves
,”
ASME J. Appl. Mech.
,
91
(
3
), p.
031001
.
6.
Saito
,
O.
,
Sen
,
E.
, and
Okabe
,
Y.
,
2022
, “
2D Slowness Visualization of Ultrasonic Wave Propagation for Delamination Detection in CFRP Laminates
,”
NDT E Int.
,
131
, pp.
1
8
.
7.
Lowe
,
M.
,
Cawley
,
P.
,
Kao
,
J.
, and
Diligent
,
O.
,
2000
, “
Prediction and Measurement of the Reflection of the Fundamental Anti-Symmetric Lamb Wave From Cracks and Notches
,”
AIP Conf. Proc.
,
509
(
1
), pp.
193
200
.
8.
Lu
,
Y.
,
Ye
,
L.
,
Wang
,
D.
,
Wang
,
X.
, and
Su
,
Z.
,
2010
, “
Conjunctive and Compromised Data Fusion Schemes for Identification of Multiple Notches in an Aluminum Plate Using Lamb Wave Signals
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
57
(
9
), pp.
2005
2016
.
9.
Gravenkamp
,
H.
,
2018
, “
Efficient Simulation of Elastic Guided Waves Interacting With Notches, Adhesive Joints, Delaminations and Inclined Edges in Plate Structures
,”
Ultrasonics
,
82
, pp.
101
113
.
10.
Park
,
C. Y.
,
Palazotto
,
A. N.
,
Hale
,
C. S.
, and
Jung
,
H. K.
,
2018
, “
Internal Longitudinal Damage Detection in a Steel Beam Using Lamb Waves: Simulation and Test Study
,”
J. Intell. Mater. Syst. Struct.
,
29
(
3
), pp.
411
422
.
11.
Fromme
,
P.
, and
Rouge
,
C.
,
2011
, “
Directivity of Guided Ultrasonic Wave Scattering at Notches and Cracks
,”
J. Phys. Conf. Ser.
,
269
, pp.
1
11
.
12.
Fromme
,
P.
,
Wilcox
,
P.
,
Lowe
,
M.
, and
Cawley
,
P.
,
2004
, “
On the Scattering and Mode Conversion of the A0 Lamb Wave Mode at Circular Defects in Plates
,”
Am. Inst. Phys., Conf. Proc.
,
700
(
1
), pp.
142
149
.
13.
Xu
,
K.
,
Ta
,
D.
,
Su
,
Z.
, and
Wang
,
W.
,
2014
, “
Transmission Analysis of Ultrasonic Lamb Mode Conversion in a Plate With Partial-Thickness Notch
,”
Ultrasonics
,
54
(
1
), pp.
395
401
.
14.
Glushkov
,
E.
,
Glushkova
,
N.
,
Eremin
,
A.
, and
Giurgiutiu
,
V.
,
2015
, “
Low-Cost Simulation of Guided Wave Propagation in Notched Plate-Like Structures
,”
J. Sound Vib.
,
352
, pp.
80
91
.
15.
Soleimanpour
,
R.
, and
Ng
,
C.-T.
,
2016
, “
Scattering of the Fundamental Anti-Symmetric Lamb Wave at Through-Thickness Notches in Isotropic Plates
,”
J. Civil Struct. Health Monit.
,
6
(
3
), pp.
447
459
.
16.
Lee
,
B.
, and
Staszewski
,
W.
,
2007
, “
Lamb Wave Propagation Modelling for Damage Detection: II. Damage Monitoring Strategy
,”
Smart Mater. Struct.
,
16
(
2
), pp.
260
274
.
17.
Rattanawangcharoen
,
N.
,
Shah
,
A.
, and
Datta
,
S.
,
1994
, “
Reflection of Waves at the Free Edge of a Laminated Circular Cylinder
,”
ASME J. Appl. Mech.
,
61
(
2
), pp.
323
329
.
18.
Alleyne
,
D. N.
, and
Cawley
,
P.
,
1992
, “
The Interaction of Lamb Waves With Defects
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
,
39
(
3
), pp.
381
397
.
19.
Diligent
,
O.
,
Grahn
,
T.
,
Boström
,
A.
,
Cawley
,
P.
, and
Lowe
,
M. J.
,
2002
, “
The Low-Frequency Reflection and Scattering of the S0 Lamb Mode From a Circular Through-Thickness Hole in a Plate: Finite Element, Analytical and Experimental Studies
,”
J. Acoust. Soc. Am.
,
112
(
6
), pp.
2589
2601
.
20.
Ghosh
,
T.
,
Kundu
,
T.
, and
Karpur
,
P.
,
1998
, “
Efficient Use of Lamb Modes for Detecting Defects in Large Plates
,”
Ultrasonics
,
36
(
7
), pp.
791
801
.
21.
Benmeddour
,
F.
,
Grondel
,
S.
,
Assaad
,
J.
, and
Moulin
,
E.
,
2008
, “
Study of the Fundamental Lamb Modes Interaction With Symmetrical Notches
,”
NDT E Int.
,
41
(
1
), pp.
1
9
.
22.
Lowe
,
M.
, and
Diligent
,
O.
,
2002
, “
Low-Frequency Reflection Characteristics of the S0 Lamb Wave From a Rectangular Notch in a Plate
,”
J. Acoust. Soc. Am.
,
111
(
1
), pp.
64
74
.
23.
Demma
,
A.
,
Cawley
,
P.
,
Lowe
,
M.
, and
Roosenbrand
,
A.
,
2003
, “
The Reflection of the Fundamental Torsional Mode From Cracks and Notches in Pipes
,”
J. Acoust. Soc. Am.
,
114
(
2
), pp.
611
625
.
24.
Lowe
,
M.
,
Challis
,
R.
, and
Chan
,
C.
,
2000
, “
The Transmission of Lamb Waves Across Adhesively Bonded Lap Joints
,”
J. Acoust. Soc. Am.
,
107
(
3
), pp.
1333
1345
.
25.
Gan
,
C.
,
Wei
,
Y.
, and
Yang
,
S.
,
2016
, “
Longitudinal Wave Propagation in a Multi-Step Rod With Variable Cross-Section
,”
J. Vib. Control
,
22
(
3
), pp.
837
852
.
26.
Agrawal
,
Y.
,
Gangwar
,
A. S.
, and
Joglekar
,
D.
,
2022
, “
Localization of a Breathing Delamination Using Nonlinear Lamb Wave Mixing
,”
ASME J. Nondestruct. Eval. Diagn. Progn. Eng. Syst.
,
5
(
3
), p.
031005
.
27.
Akbaş
,
S. D.
,
2016
, “
Wave Propagation in Edge Cracked Functionally Graded Beams Under Impact Force
,”
J. Vib. Control
,
22
(
10
), pp.
2443
2457
.
28.
Karunasena
,
W.
,
2004
, “
Numerical Modelling of Obliquely Incident Guided Wave Scattering by a Crack in a Laminated Composite Plate
,”
Structural Integrity and Fracture: Proceedings of the International Conference, SIF 2004
,
Brisbane, Australia
,
Sept. 26–29
, Australian Fracture Group Inc., pp.
181
187
.
29.
Mori
,
N.
,
Biwa
,
S.
, and
Hayashi
,
T.
,
2013
, “
Reflection and Transmission of Lamb Waves at an Imperfect Joint of Plates
,”
J. Appl. Phys.
,
113
(
7
), p.
074901
.
30.
Ramadas
,
C.
,
Hood
,
A.
,
Khan
,
I.
,
Balasubramaniam
,
K.
, and
Joshi
,
M.
,
2013
, “
Transmission and Reflection of the Fundamental Lamb Modes in a Metallic Plate With a Semi-Infinite Horizontal Crack
,”
Ultrasonics
,
53
(
3
), pp.
773
781
.
31.
Gunawan
,
A.
, and
Hirose
,
S.
,
2005
, “
Scattering Analysis of Lamb Waves in a Perpendicularly Bent Plate
,”
J. Appl. Mech.
,
8
(
1
), pp.
959
966
.
32.
Gridin
,
D.
, and
Craster
,
R. V.
,
2004
, “
Lamb Quasi–Modes in Curved Plates
,”
Proc. R. Soc. London, Ser. A
,
460
(
2046
), pp.
1831
1847
.
33.
Demma
,
A.
,
Cawley
,
P.
,
Lowe
,
M.
, and
Pavlakovic
,
B.
,
2005
, “
The Effect of Bends on the Propagation of Guided Waves in Pipes
,”
ASME J. Pressure Vessel Technol.
,
127
(
3
), pp.
328
335
.
34.
Furuhashi
,
S.
,
Sorimachi
,
K.
, and
Sugiura
,
T.
,
2010
, “
Change in Mode Configurations and Propagation Velocity of Guided Waves Through an Elbow Section of a Pipe
,”
2010 IEEE International Ultrasonics Symposium
,
San Diego, CA
,
Oct. 11–14
, pp.
2211
2214
.
35.
Nazeer
,
N.
,
Ratassepp
,
M.
, and
Fan
,
Z.
,
2017
, “
Damage Detection in Bent Plates Using Shear Horizontal Guided Waves
,”
Ultrasonics
,
75
, pp.
155
163
.
36.
Towfighi
,
S.
,
Kundu
,
T.
, and
Ehsani
,
M.
,
2002
, “
Elastic Wave Propagation in Circumferential Direction in Anisotropic Cylindrical Curved Plates
,”
ASME J. Appl. Mech.
,
69
(
3
), pp.
283
291
.
37.
Harris
,
J. G.
,
2002
, “
Rayleigh Wave Propagation in Curved Waveguides
,”
Wave Motion
,
36
(
4
), pp.
425
441
.
38.
Lee
,
J.-W.
,
Na
,
W.-B.
, and
Kim
,
J.-T.
,
2010
, “
Curvature Effect on Guided Wave Propagation in Curved Steel Plates and Hollow Steel Cylinders
,”
Struct. Eng. Mech.
,
34
(
3
), pp.
395
398
.
39.
Tembhare
,
G. U.
,
Mitra
,
A. K.
, and
Joglekar
,
D.
,
2023
, “
Interaction of Lamb Waves With a Notch in a Bent Plate
,”
Int. J. Appl. Mech.
,
15
(
9
), p.
2350073
.
40.
Bahrami
,
A.
,
Ilkhani
,
M. R.
, and
Bahrami
,
M. N.
,
2015
, “
Wave Propagation Technique for Free Vibration Analysis of Annular Circular and Sectorial Membranes
,”
J. Vib. Control
,
21
(
9
), pp.
1866
1872
.
41.
Mitra
,
M.
, and
Gopalakrishnan
,
S.
,
2016
, “
Guided Wave Based Structural Health Monitoring: A Review
,”
Smart Mater. Struct.
,
25
(
5
), p.
053001
.
42.
Gavrić
,
L.
,
1994
, “
Finite Element Computation of Dispersion Properties of Thin-Walled Waveguides
,”
J. Sound Vib.
,
173
(
1
), pp.
113
124
.
43.
Barbieri
,
E.
,
Cammarano
,
A.
,
De Rosa
,
S.
, and
Franco
,
F.
,
2009
, “
Waveguides of a Composite Plate by Using the Spectral Finite Element Approach
,”
J. Vib. Control
,
15
(
3
), pp.
347
367
.
44.
Patra
,
A. K.
,
Sharma
,
A. K.
,
Joglekar
,
D.
, and
Joglekar
,
M.
,
2023
, “
A Semi-Analytical Finite Element Framework for Lamb Waves in Soft Compressible Plates Considering Strain Stiffening Effect
,”
Int. J. Appl. Mech.
,
15
(
01
), p.
2250102
.
45.
Rajagopal
,
P.
,
Drozdz
,
M.
,
Skelton
,
E. A.
,
Lowe
,
M. J.
, and
Craster
,
R. V.
,
2012
, “
On the use of Absorbing Layers to Simulate the Propagation of Elastic Waves in Unbounded Isotropic Media Using Commercially Available Finite Element Packages
,”
NDT E Int.
,
51
, pp.
30
40
.
46.
Drewry
,
M. A.
, and
Wilcox
,
P. D.
,
2014
, “
One-Dimensional Time-Domain Finite-Element Modelling of Nonlinear Wave Propagation for Non-Destructive Evaluation
,”
NDT E Int.
,
61
, pp.
45
52
.
47.
Demma
,
A.
,
Cawley
,
P.
, and
Lowe
,
M.
,
2003
, “
Scattering of the Fundamental Shear Horizontal Mode From Steps and Notches in Plates
,”
J. Acoust. Soc. Am.
,
113
(
4
), pp.
1880
1891
.
48.
Gresil
,
M.
, and
Giurgiutiu
,
V.
,
2013
, “
Time-Domain Hybrid Global–Local Concept for Guided-Wave Propagation With Piezoelectric Wafer Active Sensor
,”
J. Intell. Mater. Syst. Struct.
,
24
(
15
), pp.
1897
1911
.
49.
Cho
,
Y.
, and
Rose
,
J. L.
,
1996
, “
A Boundary Element Solution for a Mode Conversion Study on the Edge Reflection of Lamb Waves
,”
J. Acoust. Soc. Am.
,
99
(
4
), pp.
2097
2109
.
50.
Galán
,
J. M.
, and
Abascal
,
R.
,
2002
, “
Numerical Simulation of Lamb Wave Scattering in Semi Infinite Plates
,”
Int. J. Numer. Methods Eng.
,
53
(
5
), pp.
1145
1173
.
51.
Hayashi
,
T.
,
Song
,
W.-J.
, and
Rose
,
J. L.
,
2003
, “
Guided Wave Dispersion Curves for a Bar With an Arbitrary Cross-Section, a Rod and Rail Example
,”
Ultrasonics
,
41
(
3
), pp.
175
183
.
52.
Yan
,
F.
,
Mu
,
J.
, and
Rose
,
J. L.
,
2008
, “
Guided Wave Propagation in Curved Plate-Like Structures
,”
AIP Conf. Proc.
,
975
(
1
), pp.
92
98
.
53.
Loveday
,
P. W.
,
Long
,
C. S.
, and
Ramatlo
,
D. A.
,
2020
, “
Ultrasonic Guided Wave Monitoring of an Operational Rail Track
,”
Struct. Health. Monit.
,
19
(
6
), pp.
1666
1684
.
54.
Zhao
,
X. “G.”
, and
Rose
,
J. L.
,
2003
, “
Boundary Element Modeling for Defect Characterization Potential in a Wave Guide
,”
Int. J. Solids Struct.
,
40
(
11
), pp.
2645
2658
.
55.
Cho
,
Y.
, and
Rose
,
J. L.
,
2000
, “
An Elastodynamic Hybrid Boundary Element Study for Elastic Guided Wave Interactions With a Surface Breaking Defect
,”
Int. J. Solids Struct.
,
37
(
30
), pp.
4103
4124
.
56.
Liu
,
Z.
,
2004
, “
Reflection and Transmission of Lamb Waves at Discontinuity in Plate
,”
16th World Conference on NDT
,
Montreal, Canada
,
Aug. 31–Sept. 3
, pp.
1
8
.
57.
Terrien
,
N.
,
Osmont
,
D.
,
Royer
,
D.
,
Lepoutre
,
F.
, and
Déom
,
A.
,
2007
, “
A Combined Finite Element and Modal Decomposition Method to Study the Interaction of Lamb Modes With Micro-Defects
,”
Ultrasonics
,
46
(
1
), pp.
74
88
.
58.
Karunasena
,
W.
,
2008
, “
Elastodynamic Reciprocity Relations for Wave Scattering by Flaws in Fiber Reinforced Composite Plates
,”
J. Mech. Mater. Struct.
,
3
(
10
), pp.
1831
1846
.
59.
Tian
,
J.
,
Gabbert
,
U.
,
Berger
,
H.
, and
Su
,
X.
,
2004
, “
Lamb Wave Interaction With Delaminations in CFRP Laminates
,”
Comput. Mater. Continua
,
1
(
4
), pp.
327
336
.
60.
Liu
,
G.
,
2002
, “
A Combined Finite Element/Strip Element Method for Analyzing Elastic Wave Scattering by Cracks and Inclusions in Laminates
,”
Comput. Mech.
,
28
(
1
), pp.
76
82
.
61.
Renno
,
J. M.
, and
Mace
,
B. R.
,
2013
, “
Calculation of Reflection and Transmission Coefficients of Joints Using a Hybrid Finite Element/Wave and Finite Element Approach
,”
J. Sound Vib.
,
332
(
9
), pp.
2149
2164
.
62.
Aour
,
B.
,
Rahmani
,
O.
, and
Nait-Abdelaziz
,
M.
,
2007
, “
A Coupled FEM/BEM Approach and Its Accuracy for Solving Crack Problems in Fracture Mechanics
,”
Int. J. Solids Struct.
,
44
(
7–8
), pp.
2523
2539
.
63.
Bratton
,
R.
,
Datta
,
S.
, and
Shah
,
A.
,
1991
, “Scattering of Lamb Waves in a Composite Plate,”
Review of Progress in Quantitative Nondestructive Evaluation
,
D. O.
Thompson
, and
D. E.
Chimenti
, eds.,
Springer
,
Netherlands
, pp.
1507
1514
.
64.
Castaings
,
M.
,
Le Clezio
,
E.
, and
Hosten
,
B.
,
2002
, “
Modal Decomposition Method for Modeling the Interaction of Lamb Waves With Cracks
,”
J. Acoust. Soc. Am.
,
112
(
6
), pp.
2567
2582
.
65.
Flores-López
,
M. A.
, and
Douglas Gregory
,
R.
,
2006
, “
Scattering of Rayleigh-Lamb Waves by a Surface Breaking Crack in an Elastic Plate
,”
J. Acoust. Soc. Am.
,
119
(
4
), pp.
2041
2049
.
66.
Ahmad
,
Z.
,
Vivar-Perez
,
J. M.
, and
Gabbert
,
U.
,
2013
, “
Semi-Analytical Finite Element Method for Modeling of Lamb Wave Propagation
,”
CEAS Aeronaut. J.
,
4
(
1
), pp.
21
33
.
67.
Willberg
,
C.
,
Vivar-Perez
,
J. M.
, and
Gabbert
,
U.
,
2009
, “
Lamb Wave Interaction With Defects in Homogeneous Plates
,”
International Conference on Structural Engineering Dynamics
,
Rhodes, Greece
,
June 22–24
, pp.
1
5
.
68.
Joglekar
,
D.
,
2022
, “
Scattering of the Fundamental Lamb Modes in Bent Metallic Plates
,”
ASME J. Appl. Mech.
,
89
(
12
), p.
121004
.
69.
Ahmad
,
Z. A. B.
,
2011
, “
Numerical Simulations of Lamb Waves in Plates Using a Semi-Analytical Finite Element Method
,”
Ph.D. thesis
,
Otto von Guericke Universitat
,
Magdeburg
.
70.
Bartoli
,
I.
,
Marzani
,
A.
,
Di Scalea
,
F. L.
, and
Viola
,
E.
,
2006
, “
Modeling Wave Propagation in Damped Waveguides of Arbitrary Cross-Section
,”
J. Sound Vib.
,
295
(
3–5
), pp.
685
707
.
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