Guided waves are used for the non-destructive evaluation in automotive and aerospace industries. There is a trend leaning away from isotropic materials to the manufacturing based on composites. However, the elastic wave dynamics in such materials is considerably more complicated. Much effort has been committed to the calculation of guided waves' dispersion curves in composites. Lots of methods and tools are available, but it becomes difficult when there are more than one hundred layers. In this paper the calculation of dispersion diagrams and mode shapes using the stiffness matrix method is demonstrated. Boundary conditions are implemented into the stiffness matrix method that allow for the separate tracing of the various mode families. Shear horizontal modes are modeled with the transfer matrix method without facing any numerical instability. It is elucidated just how the occurrence of the mode families depends on the system's symmetry and wave propagation direction. As a result, the robustness and reliability of guided wave modeling by using the stiffness method is improved, and more information about the modes is yielded. This is demonstrated on exemplary layups of the fiber reinforced polymer T800/913, with up to 400 layers. Referencing is made against results from DISPERSE® (Imperial College London, London, UK) for selected cases.

1.
I.
Viktorov
,
Rayleigh and Lamb Waves: Physical Theory and Applications
(
Plenum
,
New York
,
1967
), pp.
123
144
.
2.
M.
Luukkala
,
P.
Heikkila
, and
J.
Surakka
, “
Plate wave resonance—A contactless test method
,”
Ultrasonics
9
(
4
),
201
208
(
1971
).
3.
M.
Luukkala
and
P.
Meriläinen
, “
Metal plate testing using airborne ultrasound
,”
Ultrasonics
11
(
5
),
218
221
(
1973
).
4.
P. D.
Wilcox
,
M. J. S.
Lowe
, and
P.
Cawley
, “
Mode and transducer selection for long range Lamb wave inspection
,”
J. Intell. Mater. Syst. Struct.
12
,
553
565
(
2001
).
5.
M. J. S.
Lowe
,
D. N.
Alleyne
, and
P.
Cawley
, “
Defect detection in pipes using guided waves
,”
Ultrasonics
36
,
147
154
(
1998
).
6.
M. J. S.
Lowe
and
P.
Cawley
, “
The applicability of plate wave techniques for the inspection of adhesive and diffusion bonded joints
,”
J. Nondestr. Eval.
13
(
4
),
185
200
(
1994
).
7.
T.
Kundu
,
A.
Maji
,
T.
Gosh
, and
K.
Maslov
, “
Detection of kissing bonds by Lamb waves
,”
Ultrasonics
35
,
573
580
(
1998
).
8.
K.
Maslov
and
T.
Kundu
, “
Selection of Lamb modes for detecting internal defects in composite laminates
,”
Ultrasonics
35
,
141
150
(
1997
).
9.
S. S.
Kessler
,
S. M.
Spearing
, and
C.
Soutis
, “
Damage detection in composite materials using Lamb wave methods
,”
Smart Mater. Struct.
11
,
269
278
(
2002
).
10.
N.
Toyama
,
J.
Noda
, and
T.
Okabe
, “
Quantitative damage detection in cross-ply laminates using Lamb wave method
,”
Compos. Sci. Technol.
63
,
1473
1479
(
2003
).
11.
Z.
Su
,
L.
Ye
, and
Y.
Lu
, “
Guided Lamb waves for identification of damage in composite structures: A review
,”
J. Sound Vib.
295
,
753
780
(
2006
).
12.
K.
Diamanti
and
C.
Soutis
, “
Structural health monitoring techniques for aircraft composite structures
,”
Prog. Aeronaut. Sci.
46
,
342
352
(
2010
).
13.
A. S.
Purekar
and
D. J.
Pines
, “
Damage detection in thin composite laminates using piezoelectric phased sensor arrays and guided Lamb wave interrogation
,”
J. Intell. Mater. Syst. Struct.
21
,
995
1010
(
2010
).
14.
C.
Ramadas
,
K.
Balasubramaniam
,
M.
Joshi
, and
C. V.
Krishnamurthy
, “
Sizing of interface delamination in a composite T-joint using time-of-flight of Lamb waves
,”
J. Intell. Mater. Syst. Struct.
22
,
757
768
(
2011
).
15.
H.
Cunfu
,
L.
Hongye
,
L.
Zenghua
, and
W.
Bin
, “
The propagation of coupled Lamb waves in multilayered arbitrary anisotropic composite laminates
,”
J. Sound Vib.
332
,
7243
7256
(
2013
).
16.
M.
Castaings
and
P.
Cawley
, “
The generation, propagation, and detection of Lamb waves in plates using air-coupled ultrasonic transducers
,”
J. Acoust. Soc. Am.
100
(
5
),
3070
3077
(
1996
).
17.
M.
Castaings
,
P.
Cawley
,
R.
Farlow
, and
G.
Hayward
, “
Single sided inspection of composite materials using air coupled ultrasound
,”
J. Nondestr. Eval.
17
(
1
),
37
45
(
1998
).
18.
M.
Castaings
and
B.
Hosten
, “
Lamb and SH waves generated and detected by air-coupled ultrasonic transducers in composite material plates
,”
NDT&E Int.
34
,
249
258
(
2001
).
19.
M.
Castaings
and
B.
Hosten
, “
Ultrasonic guided waves for health monitoring of high-pressure composite tanks
,”
NDT&E Int.
41
,
648
655
(
2008
).
20.
I.
Solodov
,
R.
Stoessel
, and
G.
Busse
, “
Material characterization and NDE using focused slanted transmission mode of air-coupled ultrasound
,”
Res. Nondestr. Eval.
15
(
2
),
65
85
(
2004
).
21.
I.
Solodov
,
K.
Pfleiderer
,
H.
Gerhard
, and
G.
Busse
, “
Nonlinear acoustic approach to material characterisation of polymers and composites in tensile tests
,”
Ultrasonics
42
,
1011
1015
(
2004
).
22.
I.
Solodov
,
K.
Pfleiderer
,
H.
Gerhard
,
S.
Predak
, and
G.
Busse
, “
New opportunities for NDE with air-coupled ultrasound
,”
NDT&E Int.
39
(
3
),
176
183
(
2006
).
23.
M.
Rheinfurth
,
F.
Schmidt
,
D.
Döring
,
I.
Solodov
,
G.
Busse
, and
P.
Horst
, “
Air-coupled guided waves combined with thermography for monitoring fatigue in biaxially loaded composite tubes
,”
Compos. Sci. Technol.
71
(
5
),
600
608
(
2011
).
24.
M.
Rheinfurth
,
N.
Kosmann
,
D.
Sauer
,
G.
Busse
, and
K.
Schulte
, “
Lamb waves for non-contact fatigue state evaluation of composites under various mechanical loading conditions
,”
Composites: Part A
43
,
1203
1211
(
2012
).
25.
S. D.
Holland
and
D. E.
Chimenti
, “
High contrast air-coupled acoustic imaging with zero group velocity Lamb modes
,”
Ultrasonics
42
,
957
960
(
2004
).
26.
R.
Raisutis
,
R.
Kazys
,
E.
Zukauskas
, and
L.
Mazeika
, “
Ultrasonic air-coupled testing of square-shape CFRP composite rods by means of guided waves
,”
NDT&E Int.
44
,
645
654
(
2011
).
27.
L.
Zenghua
,
Y.
Hongtao
,
H.
Cunfu
, and
W.
Bin
, “
Delamination damage detection of laminated composite beams using air-coupled ultrasonic transducers
,”
Sci. China: Phys., Mech. Astron.
56
(
7
),
1269
1279
(
2013
).
28.
Dispersion Calculator Rev. 1.0, available at https://www.dlr.de/bt/en/desktopdefault.aspx/tabid-2478/11208_read-53373/ (Last viewed 4 December 2018).
29.
S. I.
Rokhlin
and
L.
Wang
, “
Stable recursive algorithm for elastic wave propagation in layered anisotropic media: Stiffness matrix method
,”
J. Acoust. Soc. Am.
112
(
3
),
822
834
(
2002
).
30.
L.
Wang
and
S. I.
Rokhlin
, “
Stable reformulation of transfer matrix method for wave propagation in layered anisotropic media
,”
Ultrasonics
39
,
413
424
(
2001
).
31.
W. T.
Thomson
, “
Transmission of elastic waves through as stratified solid medium
,”
J. Appl. Phys.
21
,
89
93
(
1950
).
32.
N. A.
Haskell
, “
The dispersion of surface waves on multilayered media
,”
Bull. Seism. Soc. Am.
43
(
1
),
17
34
(
1953
).
33.
A. H.
Nayfeh
,
Wave Propagation in Layered Anisotropic Media with Applications to Composites
(
North-Holland
,
Amsterdam
,
1995
), pp.
1
152
.
34.
L.
Knopoff
, “
A matrix method for elastic wave problems
,”
Bull. Seism. Soc. Am.
54
(
1
),
431
438
(
1964
).
35.
H.
Schmidt
and
G.
Tango
, “
Efficient global matrix approach to the computation of synthetic seismograms
,”
Geophys. J. R. Astr. Soc.
84
,
331
359
(
1986
).
36.
A. K.
Mal
, “
Wave propagation in layered composite laminates under periodic surface loads
,”
Wave Motion
10
(
3
),
257
266
(
1988
).
37.
E.
Kausel
and
J. M.
Roesset
, “
Stiffness matrices for layered soils
,”
Bull. Seis. Soc. Am.
71
(
6
),
1743
1761
(
1981
).
38.
V. G. A.
Kamal
and
V.
Giurgiutiu
, “
Stiffness transfer matrix method (STMM) for stable dispersion curves solution in anisotropic composites
,”
Proc. SPIE
9064
(
2014
).
39.
M.
Barski
and
P.
Pajak
, “
Determination of dispersion curves for composite materials with the use of stiffness matrix method
,”
Acta Mech. Autom.
11
(
2
),
121
128
(
2017
).
40.
L.
Gavric
, “
Computation and propagative waves in free rail using a finite element technique
,”
J. Sound Vib.
185
(
3
),
531
543
(
1995
).
41.
F. H.
Quintanilla
,
M. J. S.
Lowe
, and
R. V.
Craster
, “
Modeling guided elastic waves in generally anisotropic media using a spectral collocation method
,”
J. Acoust. Soc. Am.
137
(
3
),
1180
1194
(
2015
).
42.
F. H.
Quintanilla
,
M. J. S.
Lowe
, and
R. V.
Craster
, “
The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides
,”
J. Acoust. Soc. Am.
141
(
1
),
406
418
(
2017
).
43.
M.
Lowe
and
B.
Pavlakovic
,
Disperse User's Manual
(
Imperial College London
,
London
,
2013
), pp.
1
207
.
44.
M. J. S.
Lowe
, “
Matrix techniques for modeling ultrasonic waves in multilayered media
,”
IEEE Trans. Ultrason., Ferroelect., Freq. Control
42
(
4
),
525
542
(
1995
).
45.
B. A.
Auld
,
Acoustic Fields and Waves in Solids
, 2nd ed. (
Krieger
,
Malabar, FL
,
1990
), p.
165
.
46.
M.
Sause
and
M.
Hamstad
,
Acoustic Emission Analysis
(
Academic
,
Oxford
,
2018
), p.
300
.
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