Abstract

Hard-magnetic soft materials (HMSMs) manufactured by embedding hard-magnetic particles in soft materials belong to a new type of soft active materials. The abilities of fast and complicated transformations of hard-magnetic soft structures provide a promising technology for soft robotics, flexible electronics, and biomedical devices. It is significant to investigate the mechanical behaviors of hard-magnetic soft structures for their better applications. In this work, a hard-magnetic soft beam under an external magnetic field is theoretically modeled and the exact solutions for its mechanical responses are presented. First, the governing equations and boundary conditions are derived based on the principle of minimum potential energy. To solve the derived governing equations analytically, a new polynomial fitting model for hyperelastic materials is proposed for the hard-magnetic soft beam. Then, the exact solutions of a cantilevered hard-magnetic soft beam actuated by a uniform magnetic field in any direction are obtained. The newly derived exact solutions are further verified by comparing current results with those from recent simulations and experiments. For large bending angles up to 90 deg and extreme bending angle up to 180 deg, quite consistent agreement among exact solutions, numerical simulations, and experimental observations can be achieved. Finally, using our theoretical model, the deformation of the hard-magnetic soft beam actuated by magnetic fields in an arbitrary direction with non-zero magnetic declination is explored. When the magnetic actuation is increased from a small level gradually, the hard-magnetic soft beam deflects and it would undergo small, large, and extreme bending deformations in sequence. It is very interesting that, when the magnetic actuation is sufficiently large, the hard-magnetic soft beam is stretched and its centerline tends to align with the external magnetic field direction, implying that the hard-magnetic soft beam undergoes a uniaxial tension. The theoretical modeling and exact solutions for hard-magnetic soft beams are expected to be useful in the analysis and design of soft materials and structures.

References

References
1.
Park
,
S. J.
,
Gazzola
,
M.
,
Park
,
K. S.
,
Park
,
S.
,
Di Santo
,
V.
,
Blevins
,
E. L.
,
Lind
,
J. U.
,
Campbell
,
P. H.
,
Dauth
,
S.
,
Capulli
,
A. K.
,
Pasqualini
,
F. S.
,
Ahn
,
S.
,
Cho
,
A.
,
Yuan
,
H.
,
Maoz
,
B. M.
,
Vijaykumar
,
R.
,
Choi
,
J.
,
Deisseroth
,
K.
,
Lauder
,
G. V.
,
Mahadevan
,
L.
, and
Parker
,
K. K.
,
2016
, “
Phototactic Guidance of a Tissue-Engineered Soft-Robotic Ray
,”
Science
,
353
(
6295
), pp.
158
162
. 10.1126/science.aaf4292
2.
Rus
,
D.
, and
Tolley
,
M. T.
,
2015
, “
Design, Fabrication and Control of Soft Robots
,”
Nature
,
521
(
7553
), pp.
467
475
. 10.1038/nature14543
3.
Wehner
,
M.
,
Truby
,
R. L.
,
Fitzgerald
,
D. J.
,
Mosadegh
,
B.
,
Whitesides
,
G. M.
,
Lewis
,
J. A.
, and
Wood
,
R. J.
,
2016
, “
An Integrated Design and Fabrication Strategy for Entirely Soft, Autonomous Robots
,”
Nature
,
536
(
7617
), pp.
451
455
. 10.1038/nature19100
4.
Chen
,
Z.
,
Liang
,
X.
,
Wu
,
T.
,
Yin
,
T.
,
Xiang
,
Y.
, and
Qu
,
S.
,
2018
, “
Pneumatically Actuated Soft Robotic Arm for Adaptable Grasping
,”
Acta Mech. Solida Sin.
,
31
(
5
), pp.
608
622
. 10.1007/s10338-018-0052-4
5.
Ma
,
M.
,
Guo
,
L.
,
Anderson
,
D. G.
, and
Langer
,
R.
,
2013
, “
Bio-Inspired Polymer Composite Actuator and Generator Driven by Water Gradients
,”
Science
,
339
(
6116
), pp.
186
189
. 10.1126/science.1230262
6.
Zarek
,
M.
,
Layani
,
M.
,
Cooperstein
,
I.
,
Sachyani
,
E.
,
Cohn
,
D.
, and
Magdassi
,
S.
,
2016
, “
3D Printing of Shape Memory Polymers for Flexible Electronic Devices
,”
Adv. Mater.
,
28
(
22
), pp.
4449
4454
. 10.1002/adma.201503132
7.
Suh
,
J. K.
, and
DiSilvestro
,
M. R.
,
1999
, “
Biphasic Poroviscoelastic Behavior of Hydrated Biological Soft Tissue
,”
ASME J. Appl. Mech.
,
66
(
2
), pp.
528
535
. 10.1115/1.2791079
8.
Spasic
,
D. T.
,
Kovincic
,
N. I.
, and
Dankuc
,
D. V.
,
2016
, “
A New Material Identification Pattern for the Fractional Kelvin–Zener Model Describing Biomaterials and Human Tissues
,”
Commun. Nonlinear Sci. Numer. Simul.
,
37
, pp.
193
199
. 10.1016/j.cnsns.2016.01.004
9.
Stayton
,
P. S.
,
Shimoboji
,
T.
,
Long
,
C.
,
Chilkoti
,
A.
,
Ghen
,
G.
,
Harris
,
J. M.
, and
Hoffman
,
A. S.
,
1995
, “
Control of Protein–Ligand Recognition Using a Stimuli-Responsive Polymer
,”
Nature
,
378
(
6556
), pp.
472
474
. 10.1038/378472a0
10.
Zhao
,
X.
,
Kim
,
J.
,
Cezar
,
C. A.
,
Huebsch
,
N.
,
Lee
,
K.
,
Bouhadir
,
K.
, and
Mooney
,
D. J.
,
2011
, “
Active Scaffolds for On-Demand Drug and Cell Delivery
,”
Proc. Natl. Acad. Sci. USA
,
108
(
1
), pp.
67
72
. 10.1073/pnas.1007862108
11.
Jochum
,
F. D.
, and
Theato
,
P.
,
2013
, “
Temperature- and Light-Responsive Smart Polymer Materials
,”
Chem. Soc. Rev.
,
42
(
17
), pp.
7468
7483
. 10.1039/C2CS35191A
12.
Sodhi
,
J. S.
,
Cruz
,
P. R.
, and
Rao
,
I. J.
,
2015
, “
Inhomogeneous Deformations of Light Activated Shape Memory Polymers
,”
Int. J. Eng. Sci.
,
89
, pp.
1
17
. 10.1016/j.ijengsci.2014.11.010
13.
Ribeiro de Almeida
,
R. R.
,
Evangelista
,
L. R.
,
Lenzi
,
E. K.
,
Zola
,
R. S.
, and
Jákli
,
A.
,
2019
, “
Electrical Transport Properties and Fractional Dynamics of Twist-Bend Nematic Liquid Crystal Phase
,”
Commun. Nonlinear Sci. Numer. Simul.
,
70
, pp.
248
256
. 10.1016/j.cnsns.2018.10.021
14.
Silveyra
,
J. M.
,
Ferrara
,
E.
,
Huber
,
D. L.
, and
Monson
,
T. C.
,
2018
, “
Soft Magnetic Materials for a Sustainable and Electrified World
,”
Science
,
362
(
6413
), p.
eaao0195
. 10.1126/science.aao0195
15.
Zhang
,
Q.
,
Liu
,
L.
,
Pan
,
C.
,
Li
,
D.
, and
Gai
,
G.
,
2018
, “
Thermally Sensitive, Adhesive, Injectable, Multiwalled Carbon Nanotube Covalently Reinforced Polymer Conductors With Self-Healing Capabilities
,”
J. Mater. Chem. C
,
6
(
7
), pp.
1746
1752
. 10.1039/C7TC05432G
16.
Zhuo
,
H.
,
Mei
,
Z.
,
Chen
,
H.
, and
Chen
,
S.
,
2018
, “
Chemically-Crosslinked Zwitterionic Polyurethanes With Excellent Thermally-Induced Multi-Shape Memory Effect and Moisture-Induced Shape Memory Effect
,”
Polymer
,
148
, pp.
119
126
. 10.1016/j.polymer.2018.06.037
17.
Sommer
,
M. R.
,
Alison
,
L.
,
Minas
,
C.
,
Tervoort
,
E.
,
Rühs
,
P. A.
, and
Studart
,
A. R.
,
2017
, “
3D Printing of Concentrated Emulsions Into Multiphase Biocompatible Soft Materials
,”
Soft Matter
,
13
(
9
), pp.
1794
1803
. 10.1039/C6SM02682F
18.
Hong
,
S.
,
Sycks
,
D.
,
Chan
,
H. F.
,
Lin
,
S.
,
Lopez
,
G. P.
,
Guilak
,
F.
,
Leong
,
K. W.
, and
Zhao
,
X.
,
2015
, “
3D Printing of Highly Stretchable and Tough Hydrogels Into Complex, Cellularized Structures
,”
Adv. Mater.
,
27
(
27
), pp.
4035
4040
. 10.1002/adma.201501099
19.
Ding
,
Z.
,
Yuan
,
C.
,
Peng
,
X.
,
Wang
,
T.
,
Qi
,
H. J.
, and
Dunn
,
M. L.
,
2017
, “
Direct 4D Printing via Active Composite Materials
,”
Sci. Adv.
,
3
(
4
), p.
e1602890
. 10.1126/sciadv.1602890
20.
Moon
,
S.
,
Cui
,
F.
, and
Rao
,
I. J.
,
2019
, “
A Thermodynamic Framework for the Modeling of Crystallizable Triple Shape Memory Polymers
,”
Int. J. Eng. Sci.
,
134
, pp.
1
30
. 10.1016/j.ijengsci.2018.10.003
21.
Li
,
Y.
,
Liu
,
R.
, and
Liu
,
Z.
,
2018
, “
The Dynamic Behaviors of a Shape Memory Polymer Membrane
,”
Acta Mech. Solida Sin.
,
31
(
5
), pp.
635
651
. 10.1007/s10338-018-0042-6
22.
Qiu
,
B.
,
Kan
,
Q.
,
Zhao
,
T.
,
Xie
,
X.
, and
Kang
,
G.
,
2018
, “
Investigation on the Anisotropic Transformation Surfaces of Super-Elastic Niti Shape Memory Alloys Under Multiaxial Cyclic Loading Conditions
,”
Acta Mech. Solida Sin.
,
31
(
6
), pp.
744
757
. 10.1007/s10338-018-0034-6
23.
Hasan
,
M. M.
, and
Baxevanis
,
T.
,
2019
, “
Actuation Fatigue Life Prediction of Notched Shape Memory Alloy Members
,”
ASME J. Appl. Mech.
,
86
(
6
), p.
064501
. 10.1115/1.4042994
24.
Yu
,
C.
,
Kang
,
G.
, and
Fang
,
D.
,
2018
, “
A Thermo-Magneto-Mechanically Coupled Constitutive Model of Magnetic Shape Memory Alloys
,”
Acta Mech. Solida Sin.
,
31
(
5
), pp.
535
556
. 10.1007/s10338-018-0046-2
25.
Sirrine
,
J. M.
,
Meenakshisundaram
,
V.
,
Moon
,
N. G.
,
Scott
,
P. J.
,
Mondschein
,
R. J.
,
Weiseman
,
T. F.
,
Williams
,
C. B.
, and
Long
,
T. E.
,
2018
, “
Functional Siloxanes With Photo-Activated, Simultaneous Chain Extension and Crosslinking for Lithography-Based 3D Printing
,”
Polymer
,
152
, pp.
25
34
. 10.1016/j.polymer.2018.02.056
26.
Li
,
K.
, and
Cai
,
S.
,
2016
, “
Modeling of Light-Driven Bending Vibration of a Liquid Crystal Elastomer Beam
,”
ASME J. Appl. Mech.
,
83
(
3
), p.
031009
. 10.1115/1.4032073
27.
DeSimone
,
A.
,
Gidoni
,
P.
, and
Noselli
,
G.
,
2015
, “
Liquid Crystal Elastomer Strips as Soft Crawlers
,”
J. Mech. Phys. Solids
,
84
, pp.
254
272
. 10.1016/j.jmps.2015.07.017
28.
White
,
T. J.
, and
Broer
,
D. J.
,
2015
, “
Programmable and Adaptive Mechanics With Liquid Crystal Polymer Networks and Elastomers
,”
Nat. Mater.
,
14
(
11
), pp.
1087
1098
. 10.1038/nmat4433
29.
Carrico
,
J. D.
,
Traeden
,
N. W.
,
Aureli
,
M.
, and
Leang
,
K. K.
,
2015
, “
Fused Filament 3D Printing of Ionic Polymer-Metal Composites (IPMCs)
,”
Smart Mater. Struct.
,
24
(
12
), p.
125021
. 10.1088/0964-1726/24/12/125021
30.
Porfiri
,
M.
,
Sharghi
,
H.
, and
Zhang
,
P.
,
2018
, “
Modeling Back-Relaxation in Ionic Polymer Metal Composites: The Role of Steric Effects and Composite Layers
,”
J. Appl. Phys.
,
123
(
1
), p.
014901
. 10.1063/1.5004573
31.
Volpini
,
V.
,
Bardella
,
L.
,
Rodella
,
A.
,
Cha
,
Y.
, and
Porfiri
,
M.
,
2017
, “
Modelling Compression Sensing in Ionic Polymer Metal Composites
,”
Smart Mater. Struct.
,
26
(
3
), p.
035030
. 10.1088/1361-665X/26/3/035030
32.
Liu
,
F.
, and
Zhou
,
J.
,
2018
, “
Shooting and Arc-Length Continuation Method for Periodic Solution and Bifurcation of Nonlinear Oscillation of Viscoelastic Dielectric Elastomers
,”
ASME J. Appl. Mech.
,
85
(
1
), p.
011005
. 10.1115/1.4038327
33.
Jiang
,
Y.
, and
Liu
,
Y.
,
2019
, “
Effect of Dielectric Imperfections on the Electroactive Deformations of Polar Dielectric Elastomers
,”
ASME J. Appl. Mech.
,
86
(
8
), p.
081007
. 10.1115/1.4043720
34.
Zhang
,
M.
,
Xie
,
Y.
,
Yao
,
T.
,
Cao
,
X.
,
Zhang
,
Z.
,
Li
,
G.
,
Ma
,
Z.
,
Mao
,
J.
,
Yang
,
T.
,
Luo
,
Y.
, and
Li
,
T.
,
2018
, “
Scar-Like Self-Reinforced and Failure-Tolerant Dielectric Elastomer Actuator With AgNWs Electrode
,”
ASME J. Appl. Mech.
,
85
(
3
), p.
031006
. 10.1115/1.4038809
35.
Jayaneththi
,
V. R.
,
Aw
,
K. C.
, and
McDaid
,
A. J.
,
2019
, “
Coupled Magneto-Mechanical Modeling of Non-Linear Ferromagnetic Diaphragm Systems
,”
Int. J. Mech. Sci.
,
155
, pp.
360
369
. 10.1016/j.ijmecsci.2019.03.003
36.
Kashima
,
S.
,
Miyasaka
,
F.
, and
Hirata
,
K.
,
2012
, “
Novel Soft Actuator Using Magnetorheological Elastomer
,”
IEEE Trans. Magn.
,
48
(
4
), pp.
1649
1652
. 10.1109/TMAG.2011.2173669
37.
Makarova
,
L. A.
,
Nadzharyan
,
T. A.
,
Alekhina
,
Y. A.
,
Stepanov
,
G. V.
,
Kazimirova
,
E. G.
,
Perov
,
N. S.
, and
Kramarenko
,
E. Y.
,
2017
, “
Magnetoactive Elastomer as an Element of a Magnetic Retina Fixator
,”
Smart Mater. Struct.
,
26
(
9
), p.
95054
. 10.1088/1361-665X/aa82e9
38.
Rudykh
,
S.
, and
Bertoldi
,
K.
,
2013
, “
Stability of Anisotropic Magnetorheological Elastomers in Finite Deformations: A Micromechanical Approach
,”
J. Mech. Phys. Solids
,
61
(
4
), pp.
949
967
. 10.1016/j.jmps.2012.12.008
39.
Coquelle
,
E.
, and
Bossis
,
G.
,
2006
, “
Mullins Effect in Elastomers Filled With Particles Aligned by a Magnetic Field
,”
Int. J. Solids Struct.
,
43
(
25–26
), pp.
7659
7672
. 10.1016/j.ijsolstr.2006.03.020
40.
Schubert
,
G.
, and
Harrison
,
P.
,
2016
, “
Magnetic Induction Measurements and Identification of the Permeability of Magneto-Rheological Elastomers Using Finite Element Simulations
,”
J. Magn. Magn. Mater.
,
404
, pp.
205
214
. 10.1016/j.jmmm.2015.12.003
41.
Ying
,
Z. G.
,
Ni
,
Y. Q.
, and
Sajjadi
,
M.
,
2013
, “
Nonlinear Dynamic Characteristics of Magneto-Rheological Visco-Elastomers
,”
Sci. China: Technol. Sci.
,
56
(
4
), pp.
878
883
. 10.1007/s11431-013-5168-7
42.
Cantera
,
M. A.
,
Behrooz
,
M.
,
Gibson
,
R. F.
, and
Gordaninejad
,
F.
,
2017
, “
Modeling of Magneto-Mechanical Response of Magnetorheological Elastomers (MRE) and MRE-Based Systems: A Review
,”
Smart Mater. Struct.
,
26
(
2
), p.
023001
. 10.1088/1361-665X/aa549c
43.
Lum
,
G. Z.
,
Ye
,
Z.
,
Dong
,
X.
,
Marvi
,
H.
,
Erin
,
O.
,
Hu
,
W.
, and
Sitti
,
M.
,
2016
, “
Shape-Programmable Magnetic Soft Matter
,”
Proc. Natl. Acad. Sci. USA
,
113
(
41
), pp.
E6007
E6015
. 10.1073/pnas.1608193113
44.
Kim
,
Y.
,
Yuk
,
H.
,
Zhao
,
R.
,
Chester
,
S. A.
, and
Zhao
,
X.
,
2018
, “
Printing Ferromagnetic Domains for Untethered Fast-Transforming Soft Materials
,”
Nature
,
558
(
7709
), pp.
274
279
. 10.1038/s41586-018-0185-0
45.
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
. 10.1016/j.jmps.2018.10.008
46.
Kim
,
Y.
,
Parada
,
G. A.
,
Liu
,
S.
, and
Zhao
,
X.
,
2019
, “
Ferromagnetic Soft Continuum Robots
,”
Sci. Rob.
,
4
(
33
), p.
eaax7329
. 10.1126/scirobotics.aax7329
47.
Garcia-Gonzalez
,
D.
,
2019
, “
Magneto-Visco-Hyperelasticity for Hard-Magnetic Soft Materials: Theory and Numerical Applications
,”
Smart Mater. Struct.
,
28
(
8
), p.
085020
. 10.1088/1361-665X/ab2b05
48.
Furusawa
,
M.
,
Maeda
,
K.
,
Azukizawa
,
S.
,
Shinoda
,
H.
, and
Tsumori
,
F.
,
2019
, “
Bio-Mimic Motion of Elastic Material Dispersed With Hard-Magnetic Particles
,”
J. Photopolym. Sci. Technol.
,
32
(
2
), pp.
309
313
. 10.2494/photopolymer.32.309
49.
Bertotti
,
G.
,
1998
,
Hysteresis in Magnetism: for Physicists, Materials Scientists, and Engineers
,
Academic Press
,
Cham
.
50.
Wang
,
H.
,
Ning
,
X.
,
Li
,
H.
,
Luan
,
H.
,
Xue
,
Y.
,
Yu
,
X.
,
Fan
,
Z.
,
Li
,
L.
,
Rogers
,
J. A.
,
Zhang
,
Y.
, and
Huang
,
Y.
,
2018
, “
Vibration of Mechanically-Assembled 3D Microstructures Formed by Compressive Buckling
,”
J. Mech. Phys. Solids
,
112
, pp.
187
208
. 10.1016/j.jmps.2017.12.002
51.
Marckmann
,
G.
, and
Verron
,
E.
,
2006
, “
Comparison of Hyperelastic Models for Rubber-Like Materials
,”
Rubber Chem. Technol.
,
79
(
5
), pp.
835
858
. 10.5254/1.3547969
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