Graphical Abstract Figure
Graphical Abstract Figure
Close modal

Abstract

Hard-magnetic soft materials, which exhibit finite deformation under magnetic loading, have emerged as a promising class of soft active materials for the development of phononic structures with tunable elastic wave band gap characteristics. In this paper, we present a gradient-based topology optimization framework for designing the hard-magnetic soft materials-based two-phase phononic structures with wide and magnetically tunable anti-plane shear wave band gaps. The incompressible Gent hyperelastic material model, along with the ideal hard-magnetic soft material model, is used to characterize the constitutive behavior of the hard-magnetic soft phononic structure phases. To extract the dispersion curves, an in-house finite element model in conjunction with Bloch’s theorem is employed. The method of moving asymptotes is used to iteratively update the design variables and obtain the optimal distribution of the hard-magnetic soft phases within the phononic structure unit cell. Analytical sensitivity analysis is performed to evaluate the gradient of the band gap maximization function with respect to each one of the design variables. Numerical results show that the optimized phononic structures exhibit a wide band gap width in comparison to a standard hard-magnetic soft phononic structure with a central circular inclusion, demonstrating the effectiveness of the proposed numerical framework. The numerical framework presented in this study, along with the derived conclusions, can serve as a valuable guide for the design and development of futuristic tunable wave manipulators.

References

1.
Sitti
,
M.
, and
Wiersma
,
D. S.
,
2020
, “
Pros and Cons: Magnetic Versus Optical Microrobots
,”
Adv. Mater.
,
32
(
20
), p.
1906766
.
2.
Wu
,
S.
,
Hu
,
W.
,
Ze
,
Q.
,
Sitti
,
M.
, and
Zhao
,
R.
,
2020
, “
Multifunctional Magnetic Soft Composites: A Review
,”
Multifunct. Mater.
,
3
(
4
), p.
042003
.
3.
Wang
,
L.
,
Kim
,
Y.
,
Guo
,
C. F.
, and
Zhao
,
X.
,
2020
, “
Hard-Magnetic Elastica
,”
J. Mech. Phys. Solids
,
142
, p.
104045
.
4.
Zhao
,
R.
,
Kim
,
Y.
,
Chester
,
S. A.
,
Sharma
,
P.
, and
Zhao
,
X.
,
2019
, “
Mechanics of Hard-Magnetic Soft Materials
,”
J. Mech. Phys. Solids
,
124
, pp.
244
263
.
5.
Rahmati
,
A. H.
,
Jia
,
R.
,
Tan
,
K.
,
Zhao
,
X.
,
Deng
,
Q.
,
Liu
,
L.
, and
Sharma
,
P.
,
2023
, “
Theory of Hard Magnetic Soft Materials to Create Magnetoelectricity
,”
J. Mech. Phys. Solids
,
171
, p.
105136
.
6.
Yan
,
D.
,
Aymon
,
B. F.
, and
Reis
,
P. M.
,
2023
, “
A Reduced-Order, Rotation-Based Model for Thin Hard-Magnetic Plates
,”
J. Mech. Phys. Solids
,
170
, p.
105095
.
7.
Rahmati
,
A. H.
,
Jia
,
R.
,
Tan
,
K.
,
Liu
,
L.
,
Zhao
,
X.
,
Deng
,
Q.
, and
Sharma
,
P.
,
2023
, “
Giant Magnetoelectricity in Soft Materials Using Hard Magnetic Soft Materials
,”
Mater. Today Phys.
,
31
, p.
100969
.
8.
Yang
,
Y.
,
Li
,
M.
, and
Xu
,
F.
,
2024
, “
A Solid-Shell Model of Hard-Magnetic Soft Materials
,”
Int. J. Mech. Sci.
,
271
, p.
109129
.
9.
Hines
,
L.
,
Petersen
,
K.
,
Lum
,
G. Z.
, and
Sitti
,
M.
,
2017
, “
Soft Actuators for Small-Scale Robotics
,”
Adv. Mater.
,
29
(
13
), p.
1603483
.
10.
Erb
,
R. M.
,
Martin
,
J. J.
,
Soheilian
,
R.
,
Pan
,
C.
, and
Barber
,
J. R.
,
2016
, “
Actuating Soft Matter With Magnetic Torque
,”
Adv. Funct. Mater.
,
26
(
22
), pp.
3859
3880
.
11.
Boyraz
,
P.
,
Runge
,
G.
, and
Raatz
,
A.
,
2018
, “
An Overview of Novel Actuators for Soft Robotics
,”
Actuators
,
7
, p.
48
.
12.
Nandan
,
S.
,
Sharma
,
D.
, and
Sharma
,
A. K.
,
2023
, “
Viscoelastic Effects on the Nonlinear Oscillations of Hard-Magnetic Soft Actuators
,”
ASME J. Appl. Mech.
,
90
(
6
), p.
061001
.
13.
Nagal
,
N.
,
Srivastava
,
S.
,
Pandey
,
C.
,
Gupta
,
A.
, and
Sharma
,
A. K.
,
2022
, “
Alleviation of Residual Vibrations in Hard-Magnetic Soft Actuators Using a Command-Shaping Scheme
,”
Polymers
,
14
(
15
), p.
3037
.
14.
Nandan
,
S.
,
Sharma
,
D.
, and
Sharma
,
A. K.
,
2023
, “
Dynamic Modeling of Hard-Magnetic Soft Actuators: Unraveling the Role of Polymer Chain Entanglements, Crosslinks, and Finite Extensibility
,”
J. Magn. Magn. Mater.
,
587
, p.
171237
.
15.
Khurana
,
A.
,
Kumar
,
D.
,
Sharma
,
A. K.
, and
Joglekar
,
M. M.
,
2021
, “
Nonlinear Oscillations of Particle-Reinforced Electro-Magneto-Viscoelastomer Actuators
,”
ASME J. Appl. Mech.
,
88
(
12
), p.
121002
.
16.
Sharma
,
D.
, and
Sharma
,
A. K.
,
2024
, “
Dynamic Modeling and Analysis of Viscoelastic Hard-Magnetic Soft Actuators With Thermal Effects
,”
Int. J. Non-Linear Mech.
,
165
, p.
104801
.
17.
Kim
,
J. G.
,
Park
,
J. E.
,
Won
,
S.
,
Jeon
,
J.
, and
Wie
,
J. J.
,
2019
, “
Contactless Manipulation of Soft Robots
,”
Materials
,
12
(
19
), p.
3065
.
18.
Wang
,
X.
,
Mao
,
G.
,
Ge
,
J.
,
Drack
,
M.
,
Cañón Bermúdez
,
G. S.
,
Wirthl
,
D.
,
Illing
,
R.
,
Kosub
,
T.
,
Bischoff
,
L.
,
Wang
,
C.
, and
Fassbender
,
J.
,
2020
, “
Untethered and Ultrafast Soft-Bodied Robots
,”
Commun. Mater.
,
1
(
1
), p.
67
.
19.
Tian
,
J.
,
Li
,
M.
,
Han
,
Z.
,
Chen
,
Y.
,
Gu
,
X. D.
,
Ge
,
Q.
, and
Chen
,
S.
,
2022
, “
Conformal Topology Optimization of Multi-material Ferromagnetic Soft Active Structures Using an Extended Level Set Method
,”
Comput. Methods Appl. Mech. Eng.
,
389
, p.
114394
.
20.
Wu
,
Z.
,
Chen
,
Y.
,
Mukasa
,
D.
,
Pak
,
O. S.
, and
Gao
,
W.
,
2020
, “
Medical Micro/Nanorobots in Complex Media
,”
Chem. Soc. Rev.
,
49
(
22
), pp.
8088
8112
.
21.
Wang
,
Y.-F.
,
Wang
,
Y.-Z.
,
Wu
,
B.
,
Chen
,
W.
, and
Wang
,
Y.-S.
,
2020
, “
Tunable and Active Phononic Crystals and Metamaterials
,”
ASME Appl. Mech. Rev.
,
72
(
4
), p.
040801
.
22.
Graczykowski
,
B.
,
2021
, “
Progress and Perspectives on Phononic Crystals
,”
J. Appl.
,
129
(
16
), p.
160901
.
23.
Oudich
,
M.
,
Gerard
,
N. J.
,
Deng
,
Y.
, and
Jing
,
Y.
,
2023
, “
Tailoring Structure-Borne Sound Through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review
,”
Adv. Funct. Mater.
,
33
(
2
), p.
2206309
.
24.
Zhang
,
Q.
,
Cherkasov
,
A. V.
,
Xie
,
C.
,
Arora
,
N.
, and
Rudykh
,
S.
,
2023
, “
Nonlinear Elastic Vector Solitons in Hard-Magnetic Soft Mechanical Metamaterials
,”
Int. J. Solids Struct.
,
280
, p.
112396
.
25.
Patra
,
A. K.
,
Sharma
,
A. K.
,
Joglekar
,
D. M.
, and
Joglekar
,
M. M.
,
2024
, “
Propagation of the Fundamental Lamb Modes in Strain Stiffened Hard-Magnetic Soft Plates
,”
ASME J. Appl. Mech.
,
91
(
6
), p.
061007
.
26.
Lucarini
,
S.
,
Hossain
,
M.
, and
Garcia-Gonzalez
,
D.
,
2022
, “
Recent Advances in Hard-Magnetic Soft Composites: Synthesis, Characterisation, Computational Modelling, and Applications
,”
Comp. Struct.
,
279
, p.
114800
.
27.
Alam
,
Z.
, and
Sharma
,
A. K.
,
2022
, “
Functionally Graded Soft Dielectric Elastomer Phononic Crystals: Finite Deformation, Electro-Elastic Longitudinal Waves, and Band Gaps Tunability Via Electro-Mechanical Loading
,”
Int. J. Appl. Mech.
,
14
(
06
), p.
2250050
.
28.
Bortot
,
E.
, and
Shmuel
,
G.
,
2017
, “
Tuning Sound With Soft Dielectrics
,”
Smart Mater. Struct.
,
26
(
4
), p.
045028
.
29.
Kushwaha
,
M. S.
,
Halevi
,
P.
,
Martinez
,
G.
,
Dobrzynski
,
L.
, and
Djafari-Rouhani
,
B.
,
1994
, “
Theory of Acoustic Band Structure of Periodic Elastic Composites
,”
Phys. Rev. B
,
49
(
4
), p.
2313
.
30.
Khelif
,
A.
,
Choujaa
,
A.
,
Benchabane
,
S.
,
Djafari-Rouhani
,
B.
, and
Laude
,
V.
,
2004
, “
Guiding and Bending of Acoustic Waves in Highly Confined Phononic Crystal Waveguides
,”
Appl. Phys. Lett.
,
84
(
22
), pp.
4400
4402
.
31.
Montgomery
,
S. M.
,
Wu
,
S.
,
Kuang
,
X.
,
Armstrong
,
C. D.
,
Zemelka
,
C.
,
Ze
,
Q.
,
Zhang
,
R.
,
Zhao
,
R.
, and
Qi
,
H. J.
,
2021
, “
Magneto-Mechanical Metamaterials With Widely Tunable Mechanical Properties and Acoustic Bandgaps
,”
Adv. Funct. Mater.
,
31
(
3
), p.
2005319
.
32.
Pennec
,
Y.
,
Djafari-Rouhani
,
B.
,
Vasseur
,
J. O.
,
Khelif
,
A.
, and
Deymier
,
P. A.
,
2004
, “
Tunable Filtering and Demultiplexing in Phononic Crystals With Hollow Cylinders
,”
Phys. Rev. E
,
69
(
4
), p.
046608
.
33.
Zhang
,
P.
, and
To
,
A. C.
,
2013
, “
Broadband Wave Filtering of Bioinspired Hierarchical Phononic Crystal
,”
Appl. Phys. Lett.
,
102
(
12
), p.
121910
.
34.
Chen
,
Z.-G.
,
Zhao
,
J.
,
Mei
,
J.
, and
Wu
,
Y.
,
2017
, “
Acoustic Frequency Filter Based on Anisotropic Topological Phononic Crystals
,”
Sci. Rep.
,
7
(
1
), p.
15005
.
35.
Yu
,
X.
,
Lu
,
Z.
,
Cui
,
F.
,
Cheng
,
L.
, and
Cui
,
Y.
,
2017
, “
Tunable Acoustic Metamaterial With an Array of Resonators Actuated by Dielectric Elastomer
,”
Extr. Mech. Lett.
,
12
, pp.
37
40
.
36.
Elnady
,
T.
,
Elsabbagh
,
A.
,
Akl
,
W.
,
Mohamady
,
O.
,
Garcia-Chocano
,
V. M.
,
Torrent
,
D.
,
Cervera
,
F.
, and
Sánchez-Dehesa
,
J.
,
2009
, “
Quenching of Acoustic Bandgaps by Flow Noise
,”
Appl. Phys. Lett.
,
94
(
13
), p.
134104
.
37.
Badreddine Assouar
,
M.
,
Senesi
,
M.
,
Oudich
,
M.
,
Ruzzene
,
M.
, and
Hou
,
Z.
,
2012
, “
BroadBand Plate-Type Acoustic Metamaterial for Low-Frequency Sound Attenuation
,”
Appl. Phys. Lett.
,
101
(
17
), p.
173505
.
38.
Zheng
,
L.-Y.
,
Wu
,
Y.
,
Ni
,
X.
,
Chen
,
Z.-G.
,
Lu
,
M.-H.
, and
Chen
,
Y.-F.
,
2014
, “
Acoustic Cloaking by a Near-Zero-Index Phononic Crystal
,”
Appl. Phys. Lett.
,
104
(
16
), p.
161904
.
39.
Zhang
,
Q.
,
Hu
,
G.
, and
Rudykh
,
S.
,
2024
, “
Magnetoactive Asymmetric Mechanical Metamaterial for Tunable Elastic Cloaking
,”
Int. J. Solids Struct.
,
289
, p.
112648
.
40.
Zhang
,
Q.
, and
Rudykh
,
S.
,
2022
, “
Magneto-Deformation and Transverse Elastic Waves in Hard-Magnetic Soft Laminates
,”
Mech. Mater.
,
169
, p.
104325
.
41.
Padmanabhan
,
S.
,
Alam
,
Z.
, and
Sharma
,
A. K.
,
2024
, “
Tunable Anti-Plane Wave Bandgaps in 2D Periodic Hard-Magnetic Soft Composites
,”
Int. J. Mech. Sci.
,
261
, p.
108686
.
42.
Alam
,
Z.
,
Padmanabhan
,
S.
, and
Sharma
,
A. K.
,
2023
, “
Magnetically Tunable Longitudinal Wave Band Gaps in Hard-Magnetic Soft Laminates
,”
Int. J. Mech. Sci.
,
249
, p.
108262
.
43.
Li
,
B.
,
Yan
,
W.
, and
Gao
,
Y.
,
2022
, “
Tunability of Band Gaps of Programmable Hard-Magnetic Soft Material Phononic Crystals
,”
Acta Mech. Solida Sinica
,
35
(
5
), pp.
719
732
.
44.
Li
,
B.
, and
Gao
,
Y.
,
2023
, “
Magnetic-Controlled Programmable Soft Lattice Phononic Crystals With Sinusoidally-Shaped-Like Ligaments for Band Gap Control
,”
J. Magn. Magn. Mater.
,
580
, p.
170945
.
45.
Sim
,
J.
, and
Zhao
,
R. R.
,
2024
, “
Magneto-Mechanical Metamaterials: A Perspective
,”
ASME J. Appl. Mech.
,
91
(
3
), p.
031004
.
46.
Sigmund
,
O.
, and
Søndergaard Jensen
,
J.
,
2003
, “
Systematic Design of Phononic Band-Gap Materials and Structures by Topology Optimization
,”
Philos. Trans. R. Soc. Lond., Ser. A.
,
361
(
1806
), pp.
1001
1019
.
47.
Wu
,
Q.
,
He
,
J.
,
Chen
,
W.
,
Li
,
Q.
, and
Liu
,
S.
,
2023
, “
Topology Optimization of Phononic Crystal With Prescribed Band Gaps
,”
Comput. Methods Appl. Mech. Eng.
,
412
, p.
116071
.
48.
Liu
,
W.
,
Yoon
,
G. H.
,
Yi
,
B.
,
Choi
,
H.
, and
Yang
,
Y.
,
2020
, “
Controlling Wave Propagation in One-Dimensional Structures Through Topology Optimization
,”
Comput. Struct.
,
241
, p.
106368
.
49.
Yi
,
G.
,
Shin
,
Y. C.
,
Yoon
,
H.
,
Jo
,
S.-H.
, and
Youn
,
B. D.
,
2019
, “
Topology Optimization for Phononic Band Gap Maximization Considering a Target Driving Frequency
,”
JMST Adv.
,
1
, pp.
153
159
.
50.
Halkjær
,
S.
,
Sigmund
,
O.
, and
Jensen
,
J. S.
,
2006
, “
Maximizing Band Gaps in Plate Structures
,”
Struct. Multidiscipl. Optim.
,
32
, pp.
263
275
.
51.
Gazonas
,
G. A.
,
Weile
,
D. S.
,
Wildman
,
R.
, and
Mohan
,
A.
,
2006
, “
Genetic Algorithm Optimization of Phononic Bandgap Structures
,”
Int. J. Solids Struct.
,
43
(
18–19
), pp.
5851
5866
.
52.
Bilal
,
O. R.
, and
Hussein
,
M. I.
,
2011
, “
Ultrawide Phononic Band Gap for Combined In-Plane and Out-of-Plane Waves
,”
Phys. Rev. E
,
84
(
6
), p.
065701
.
53.
Vatanabe
,
S. L.
,
Paulino
,
G. H.
, and
Silva
,
E. C.
,
2014
, “
Maximizing Phononic Band Gaps in Piezocomposite Materials by Means of Topology Optimization
,”
J. Acoust. Soc. Am.
,
136
(
2
), pp.
494
501
.
54.
Hedayatrasa
,
S.
,
Abhary
,
K.
, and
Uddin
,
M.
,
2015
, “
Numerical Study and Topology Optimization of 1D Periodic Bimaterial Phononic Crystal Plates for Bandgaps of Low Order Lamb Waves
,”
Ultrasonics
,
57
, pp.
104
124
.
55.
Quinteros
,
L.
,
Meruane
,
V.
, and
Cardoso
,
E. L.
,
2021
, “
Phononic Band Gap Optimization in Truss-Like Cellular Structures Using Smooth P-Norm Approximations
,”
Struct. Multidiscipl. Optim.
,
64
(
1
), pp.
113
124
.
56.
Li
,
W.
,
Meng
,
F.
,
Chen
,
Y.
,
Li
,
Y. F.
, and
Huang
,
X.
,
2019
, “
Topology Optimization of Photonic and Phononic Crystals and Metamaterials: A Review
,”
Adv. Theory Simul.
,
2
(
7
), p.
1900017
.
57.
Bortot
,
E.
,
Amir
,
O.
, and
Shmuel
,
G.
,
2018
, “
Topology Optimization of Dielectric Elastomers for Wide Tunable Band Gaps
,”
Int. J. Solids Struct.
,
143
, pp.
262
273
.
58.
Sharma
,
A. K.
,
Kosta
,
M.
,
Shmuel
,
G.
, and
Amir
,
O.
,
2022
, “
Gradient-Based Topology Optimization of Soft Dielectrics as Tunable Phononic Crystals
,”
Comp. Struct.
,
280
, p.
114846
.
59.
Dalklint
,
A.
,
Wallin
,
M.
,
Bertoldi
,
K.
, and
Tortorelli
,
D.
,
2022
, “
Tunable Phononic Bandgap Materials Designed Via Topology Optimization
,”
J. Mech. Phys. Solids
,
163
, p.
104849
.
60.
Sharma
,
A. K.
,
Joglekar
,
M. M.
,
Joglekar
,
D. M.
, and
Alam
,
Z.
,
2022
, “
Topology Optimization of Soft Compressible Phononic Laminates for Widening the Mechanically Tunable Band Gaps
,”
Comp. Struct.
,
289
, p.
115389
.
61.
Dorfmann
,
L.
, and
Ogden
,
R. W.
,
2014
,
Nonlinear Theory of Electroelastic and Magnetoelastic Interactions
, Vol.
1
,
Springer
,
New York
.
62.
Gent
,
A. N.
,
1996
, “
A New Constitutive Relation for Rubber
,”
Rubber. Chem. Technol.
,
69
(
1
), pp.
59
61
.
63.
Kittel
,
C.
,
2005
, “Introduction to Solid State Physics,”
John Wiley & Sons Inc
,
Hoboken, NJ
.
64.
Amir
,
O.
,
Bendsøe
,
M. P.
, and
Sigmund
,
O.
,
2009
, “
Approximate Reanalysis in Topology Optimization
,”
Int. J. Numer. Methods Eng.
,
78
(
12
), pp.
1474
1491
.
65.
Svanberg
,
K.
,
1987
, “
The Method of Moving Asymptotes–A New Method for Structural Optimization
,”
Int. J. Numer. Methods Eng.
,
24
(
2
), pp.
359
373
.
66.
Sigmund
,
O.
,
2001
, “
A 99 Line Topology Optimization Code Written in Matlab
,”
Struct. Multidiscipl. Optim.
,
21
(
2
), pp.
120
127
.
67.
Li
,
X.
,
Yu
,
W.
,
Liu
,
J.
,
Zhu
,
X.
,
Wang
,
H.
,
Sun
,
X.
,
Liu
,
J.
, and
Yuan
,
H.
,
2023
, “
A Mechanics Model of Hard-magnetic Soft Rod With Deformable Cross-Section Under Three-Dimensional Large Deformation
,”
Int. J. Solids Struct.
,
279
, p.
112344
.
68.
Long
,
K.
,
Wang
,
X.
, and
Gu
,
X.
,
2018
, “
Local Optimum in Multi-material Topology Optimization and Solution by Reciprocal Variables
,”
Struct. Multidiscipl. Optim.
,
57
(
3
), pp.
1283
1295
.
69.
Zhang
,
S.
, and
Norato
,
J. A.
,
2018
, “
Finding Better Local Optima in Topology Optimization Via Tunneling
,” International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol.
51760
,
American Society of Mechanical Engineers
, p.
V02BT03A014
.
70.
Meng
,
F.
,
Huang
,
X.
, and
Jia
,
B.
,
2015
, “
Bi-Directional Evolutionary Optimization for Photonic Band Gap Structures
,”
J. Comput. Phys.
,
302
, pp.
393
404
.
71.
Li
,
Y. F.
,
Huang
,
X.
,
Meng
,
F.
, and
Zhou
,
S.
,
2016
, “
Eolutionary Topological Design for Phononic Band Gap Crystals
,”
Struct. Multidiscipl. Optim.
,
54
(
3
), pp.
595
617
.
You do not currently have access to this content.