This work focuses on elastic wave propagation in three-dimensional (3D) low-density lattices and explores their wave directionality and energy flow characteristics. In particular, we examine the dynamic response of Kelvin foam, a simple-and framed-cubic lattice, as well as the octet lattice, spanning this way a range of average nodal connectivities and both stretching-and bending-dominated behavior. Bloch wave analysis on unit periodic cells is employed and frequency diagrams are constructed. Our results show that in the low relative-density regime analyzed here, only the framed-cubic lattice displays a complete bandgap in its frequency diagram. New representations of iso-frequency contours and group-velocity plots are introduced to further analyze dispersive behavior, wave directionality, and the presence of partial bandgaps in each lattice. Significant wave beaming is observed for the simple-cubic and octet lattices in the low frequency regime, while Kelvin foam exhibits a nearly isotropic behavior in low frequencies for the first propagating mode. Results of Bloch wave analysis are verified by explicit numerical simulations on finite size domains under a harmonic perturbation.

References

References
1.
Mead
,
D. M.
,
1996
, “
Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964–1995
,”
J. Sound Vib.
,
190
(
3
), pp.
495
524
.
2.
Langley
,
R. S.
,
1994
, “
On the Modal Density and Energy Flow Characteristics of Periodic Structures
,”
J. Sound Vib.
,
172
(
4
), pp.
491
511
.
3.
Brillouin
,
L.
,
1953
,
Wave Propagation in Periodic Structures
,
2nd ed.
,
Dover Publications
,
Mineola, NY
.
4.
Hussein
,
M. I.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2014
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook
,”
ASME Appl. Mech. Rev.
,
66
(
4
), p.
040802
.
5.
Sigalas
,
M.
, and
Economou
,
E. N.
,
1993
, “
Band Structure of Elastic Waves in Two Dimensional Systems
,”
Solid State Commun.
,
86
(
3
), pp.
141
143
.
6.
Phani
,
A. S.
,
Woodhouse
,
J.
, and
Fleck
,
N. A.
,
2006
, “
Wave Propagation in Two-Dimensional Periodic Lattices
,”
J. Acoust. Soc. Am.
,
119
(
4
), pp.
1995
2005
.
7.
Spadoni
,
A.
,
Ruzzene
,
M.
,
Gonella
,
S.
, and
Scarpa
,
F.
,
2009
, “
Phononic Properties of Hexagonal Chiral Lattices
,”
Wave Motion
,
46
(
7
), pp.
435
450
.
8.
Gonella
,
S.
, and
Ruzzene
,
M.
,
2008
, “
Analysis of in-Plane Wave Propagation in Hexagonal and Re-Entrant Lattices
,”
J. Sound Vib.
,
312
(
1
), pp.
125
139
.
9.
Baravelli
,
E.
, and
Ruzzene
,
M.
,
2013
, “
Internally Resonating Lattices for Bandgap Generation and Low-Frequency Vibration Control
,”
J. Sound Vib.
,
332
(
25
), pp.
6562
6579
.
10.
Martinsson
,
P. G.
, and
Movchan
,
A. B.
,
2003
, “
Vibrations of Lattice Structures and Phononic Band Gaps
,”
Q. J. Mech. Appl. Math.
,
56
(
1
), pp.
45
64
.
11.
Langley
,
R. S.
,
Bardell
,
N. S.
, and
Ruivo
,
H. M.
,
1997
, “
The Response of Two-Dimensional Periodic Structures to Harmonic Point Loading: A Theoretical and Experimental Study of a Beam Grillage
,”
J. Sound Vib.
,
207
(
4
), pp.
521
535
.
12.
Zelhofer
,
A. J.
, and
Kochmann
,
D. M.
,
2017
, “
On Acoustic Wave Beaming in Two-Dimensional Structural Lattices
,”
Int. J. Solids Struct.
,
115–116
, pp.
248
269
.
13.
Wang
,
Y. F.
,
Wang
,
Y. S.
, and
Zhang
,
C.
,
2014
, “
Bandgaps and Directional Properties of Two-Dimensional Square Beam-like Zigzag Lattices
,”
AIP Adv.
,
4
(
12
), p.
124403
.
14.
Trainiti
,
G.
,
Rimoli
,
J. J.
, and
Ruzzene
,
M.
,
2016
, “
Wave Propagation in Undulated Structural Lattices
,”
Int. J. Solids Struct.
,
97–98
, pp.
431
444
.
15.
Ruzzene
,
M.
,
Scarpa
,
F.
, and
Soranna
,
F.
,
2003
, “
Wave Beaming Effects in Two-Dimensional Cellular Structures
,”
Smart Mater. Struct.
,
12
(
3
), p.
363
.
16.
Casadei
,
F.
, and
Rimoli
,
J. J.
,
2013
, “
Anisotropy-Induced Broadband Stress Wave Steering in Periodic Lattices
,”
Int. J. Solids Struct.
,
50
(
9
), pp.
1402
1414
.
17.
Messner
,
M. C.
,
Barham
,
M. I.
,
Kumar
,
M.
, and
Barton
,
N. R.
,
2015
, “
Wave Propagation in Equivalent Continuums Representing Truss Lattice Materials
,”
Int. J. Solids Struct.
,
73–74
, pp.
55
66
.
18.
Delpero
,
T.
,
Schoenwald
,
S.
,
Zemp
,
A.
, and
Bergamini
,
A.
,
2016
, “
Structural Engineering of Three-Dimensional Phononic Crystals
,”
J. Sound Vib.
,
363
, pp.
156
165
.
19.
D'Alessandro
,
L.
,
Belloni
,
E.
,
Ardito
,
R.
,
Corigliano
,
A.
, and
Braghin
,
F.
,
2016
, “
Modeling and Experimental Verification of an Ultra-Wide Bandgap in 3D Phononic Crystal
,”
Appl. Phys. Lett.
,
109
(
22
), p.
221907
.
20.
Lucklum
,
F.
, and
Vellekoop
,
M. J.
,
2016
, “
Realization of Complex 3-D Phononic Crystals With Wide Complete Acoustic Band Gaps
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
,
63
(
5
), pp.
796
797
.
21.
Matlack
,
K. H.
,
Bauhofer
,
A.
,
Krödel
,
S.
,
Palermo
,
A.
, and
Daraio
,
C.
,
2016
, “
Composite 3D-Printed Metastructures for Low-Frequency and Broadband Vibration Absorption
,”
Proc. Natl. Acad. Sci. USA
,
113
(
30
), pp. 8386–8390.
22.
Thomson
,
W.
,
1887
, “
On the Division of Space With Minimum Partitional Area
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
,
24
(
151
), pp.
503
514
.
23.
Weaire
,
D.
, and
Phelan
,
R.
,
1994
, “
A Counter-Example to Kelvin's Conjecture on Minimal Surfaces
,”
Philos. Mag. Lett.
,
69
(
2
), pp.
107
110
.
24.
Gong
,
L.
,
Kyriakides
,
S.
, and
Jang
,
W. Y.
,
2005
, “
Compressive Response of Open-Cell Foams. Part I: Morphology and Elastic Properties
,”
Int. J. Solids Struct.
,
42
(
5
), pp.
1355
1379
.
25.
Gong
,
L.
, and
Kyriakides
,
S.
,
2005
, “
Compressive Response of Open Cell Foams—Part II: Initiation and Evolution of Crushing
,”
Int. J. Solids Struct.
,
42
(
5
), pp.
1381
1399
.
26.
Jang
,
W. Y.
, and
Kyriakides
,
S.
,
2009
, “
On the Crushing of Aluminum Open-Cell Foams: Part II Analysis
,”
Int. J. Solids Struct.
,
46
(
3
), pp.
635
650
.
27.
Kittel
,
C.
,
2005
,
Introduction to Solid State Physics
,
8th ed.
,
Wiley
,
Hoboken, NJ
.
28.
Åberg
,
M.
, and
Gudmundson
,
P.
,
1997
, “
The Usage of Standard Finite Element Codes for Computation of Dispersion Relations in Materials With Periodic Microstructure
,”
J. Acoust. Soc. Am.
,
102
(
4
), pp.
2007
2013
.
29.
Wang
,
P.
,
Casadei
,
F.
,
Kang
,
S. H.
, and
Bertoldi
,
K.
,
2015
, “
Locally Resonant Band Gaps in Periodic Beam Lattices by Tuning Connectivity
,”
Phys. Rev. B
,
91
(
2
), p.
020103(R)
.
You do not currently have access to this content.