This paper develops a higher-order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton's principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the system. This new model is validated with finite-element theory and Kirchhoff plate theory for a thin plate simulation and verified with comparison to experimental results for a 0.0191 m thick sectional plate.

References

References
1.
Graf
,
K. F.
,
1975
,
Wave Motion in Elastic Solids
,
Dover Publications
,
New York
.
2.
Junger
,
M. C.
, and
Feit
,
D.
,
1986
,
Sound, Structures, and Their Interaction
,
The MIT Press
,
Cambridge, MA
.
3.
Soedel
,
W.
,
2004
,
Vibrations of Shells and Plates
,
Marcel Dekker
,
New York
.
4.
Timoshenko
,
S. P.
,
1922
, “
On the Transverse Vibrations of Bars of Uniform Cross-Section
,”
Philos. Mag.
,
43
(
253
), pp.
125
131
.
5.
Bickford
,
W. B.
,
1982
, “
A Consistent Higher Order Beam Theory
,” 11th Southeastern Conference on Theoretical and Applied Mechanics, Huntsville, AL, Apr. 8–9, Vol.
11
, pp.
137
150
.
6.
Karama
,
M.
,
Afaq
,
K. S.
, and
Mistou
,
S.
,
2003
, “
Mechanical Behavior of Laminated Composite Beam by New Multi-Layered Laminated Composite Structures Model With Transverse Shear Stress Continuity
,”
Int. J. Solids Struct.
,
40
(
6
), pp.
1525
1546
.
7.
Mindlin
,
R. D.
,
1951
, “
Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic Elastic Plates
,”
ASME J. Appl. Mech.
,
18
(
1
), pp.
31
38
.
8.
Ambartsumyan
,
S. A.
,
1958
, “
On the Theory of Bending Plates
,”
Proc. Acad. Sci. USSR
,
5
, pp.
69
77
.
9.
Reddy
,
J. N.
, and
Phan
,
N. D.
,
1985
, “
Stability and Vibration of Isotropic, Orthotropic and Laminated Plates According to a Higher-Order Shear Deformation Theory
,”
J. Sound Vib.
,
98
(
2
), pp.
157
170
.
10.
Touratier
,
M.
,
1991
, “
An Efficient Standard Plate Theory
,”
Int. J. Eng. Sci.
,
29
(
8
), pp.
901
916
.
11.
Soldatos
,
K. P.
,
1992
, “
A Transverse Shear Deformation Theory for Homogeneous Monoclinic Plates
,”
Acta Mech.
,
94
(
3
), pp.
195
220
.
12.
Akavci
,
S. S.
,
2007
, “
Buckling and Free Vibration Analysis of Symmetric and Antisymmetric Laminated Composite Plates on an Elastic Foundation
,”
J. Reinf. Plast. Compos.
,
26
(
18
), pp.
1907
1919
.
13.
Mead
,
D. J.
, and
Pujara
,
K. K.
,
1971
, “
Space-Harmonic Analysis of Periodically Supported Beams: Response to Convected Random Loading
,”
J. Sound Vib.
,
14
(
4
), pp.
525
541
.
14.
Mace
,
B. R.
,
1980
, “
Periodically Stiffened Fluid-Loaded Plates—I: Response to Convected Harmonic Pressure and Free Wave Propagation
,”
J. Sound Vib.
,
73
(
4
), pp.
473
486
.
15.
Mace
,
B. R.
,
1980
, “
Periodically Stiffened Fluid-Loaded Plates—II: Response to Line and Point Forces
,”
J. Sound Vib.
,
73
(
4
), pp.
487
504
.
16.
Lin
,
G. G.
, and
Hayek
,
S. I.
,
1977
, “
Acoustic Radiation From Point Excited Rib-Reinforced Plate
,”
J. Acoust. Soc. Am.
,
62
(
1
), pp.
72
83
.
17.
Sylvia
,
J. E.
, and
Hull
,
A. J.
,
2013
, “
A Dynamic Model of a Reinforced Thin Plate With Ribs of Finite Width
,”
Int. J. Acoust. Vib.
,
18
(
2
), pp.
86
90
.
18.
Cray
,
B. A.
,
2015
, “
Experimental Verification of Acoustic Trace Wavelength Enhancement
,”
J. Acoust. Soc. Am.
,
138
(
6
), pp.
3765
3772
.
19.
Floquet
,
G.
,
1883
, “
On Linear Differential Equations With Periodic Coefficients
,”
Sci. Ann. Éc. Norm. Supér.
,
12
, pp.
47
88
.
20.
Bloch
,
F.
,
1928
, “
About the Quantum Mechanics of Electrons in Crystal Lattices
,”
Mag. Phys.
,
52
, pp.
555
600
.
21.
Hussein
,
M. I.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2014
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlooks
,”
ASME Appl. Mech. Rev.
,
66
(
4
), p.
040802
.
22.
Gei
,
M.
,
Movchan
,
A. B.
, and
Bigoni
,
D.
,
2009
, “
Band-Gap and Defect-Induced Annihilation in Prestressed Elastic Structures
,”
J. Appl. Phys.
,
105
(
6
), p.
063507
.
23.
Brun
,
M.
,
Giaccu
,
G. G.
,
Movchan
,
A. B.
, and
Movchan
,
N. V.
,
2012
, “
Asymptotics of Eigenfrequencies in the Dynamic Response of Elongated Structures
,”
Proc. R. Soc. A
,
468
(
2138
), pp.
378
394
.
24.
Brun
,
M.
,
Movchan
,
A. B.
, and
Jones
, I
. S.
,
2013
, “
Phononic Band Gap Systems in Structural Mechanics: Finite Slender Elastic Structures and Infinite Periodic Waveguides
,”
ASME J. Vib. Acoust.
,
135
(
4
), p.
041013
.
25.
Carta
,
G.
, and
Brun
,
M.
,
2015
, “
Bloch–Floquet Waves in Flexural Systems With Continuous and Discrete Elements
,”
Mech. Mater.
,
87
, pp.
11
26
.
26.
Carta
,
G.
,
Brun
,
M.
, and
Movchan
,
A. B.
,
2014
, “
Dynamic Response and Localization in Strongly Damaged Waveguides
,”
Proc. R. Soc. A
,
470
(
2167
), p.
20140136
.
27.
Carta
,
G.
,
Brun
,
M.
,
Movchan
,
A. B.
, and
Boiko
,
T.
,
2016
, “
Transmission and Localisation in Ordered and Randomly-Perturbed Structural Flexural Systems
,”
Int. J. Eng. Sci.
,
98
, pp.
126
152
.
28.
Reddy
,
J. N.
,
1984
,
Energy and Variational Methods in Applied Mechanics
,
Wiley
,
New York
.
29.
Cheng
,
Z.
, and
Shi
,
Z.
,
2013
, “
Influence of Parameter Mismatch on the Convergence of the Band Structures by Using the Fourier Expansion Method
,”
Compos. Struct.
,
106
, pp.
510
519
.
30.
Lamb
,
M.
, and
Rouillard
,
V.
,
2009
, “
Some Issues When Using Fourier Analysis for the Extraction of Model Parameters
,”
J. Phys.: Conf. Ser.
,
181
, p.
012007
.
You do not currently have access to this content.