Dynamics of Hard-Magnetic Soft Materials

Dynamic modelling and analysis of viscoelastic Hard-Magnetic
Soft Actuators
Presentation By:
Shivendra Nandan
(M21ME009)
Department of Mechanical Engineering
Indian Institute of Technology Jodhpur
Under the guidance of:
Dr. Atul Kumar Sharma
Assistant professor
Department of Mechanical Engineering
Indian Institute of Technology Jodhpur
A Presentation on

INTRODUCTION TO HARD-MAGNETIC SOFT ACTUATORS
• High Saturation Magnetization (Ms)
• Low Coercivity (Hc)
• Low Remanence (Mr) With Narrow
Hysteresis Curves
• Mostly use carbonyl iron or iron oxide
particles embedded in soft polymer matrices
Soft magnetic soft materials
• Large hysteresis
• High coercivity
• High remanence With wide Hysteresis Curves
• Mostly uses samarium–cobalt (SmCo5 or
Sm2Co17) or neodymium–iron–boron (NdFeB)
Hard magnetic soft materials
Applications
• Soft Robotics
• Biomedical
• Medical
Devices
Cited from: https://pubs.acs.org/doi/10.1021/acs.chemrev.1c00481 Cited from: https://www.science.org/doi/abs/10.1126/scirobotics.aax7329

THESIS HIGHLIGHTS
Hard magnetic
soft actuator
1. Without
prestress
2. With prestress
Finite
Extensibility
Polymer Chain
entanglements
Viscoelasticity
Crosslinks
Research Contribution
Research Gap & Motivation
• To develop an analytical framework for modeling the dynamic behavior of HMSAs incorporating the effect of
polymer chain entanglements, cross-links, and finite extensibility.
• To incorporate the visco-elastic and prestres effects in the developed dynamic model of HMS actuator.
• To perform a parametric study to investigate the effect of the inherent material properties on the nonlinear
oscillations of a hard-magnetic soft actuator under constant and periodic magnetic fluxes.
Research Objective
From detailed literature survey:
• Wealth of literature expounds on the modeling of the HMSA in the quasi-static
mode of loading.
• The inherent properties of the soft material significantly affects the time dependent
response. It dissipates the energy during actuation.
• The dynamic motion of actuator is significantly affected by the viscosity, polymer
chain entanglements and cross-links, prestress, anisotropy and temperature.
• It becomes necessary to explore the effect of inherent material properties on the
nonlinear dynamic behavior of the hard-magnetic soft actuator for the design of
efficient and intelligent devices.

ALGORITHM
Parametric Study
Analytical Framework Modeling
Problem definition Dynamic Modeling
• Define the co-ordinate system
• Establish relationship from reference and current
configuration for any point on the object
• Using continuum mechanics, define stretch
parameter, deformation gradient, etc.
• Define direction of mechanical loading (prestress)
and magnetic loading (alongside / opposite relation)
• Find strain energy density function of magnetic flux
• Define the characteristics of “Material model”
• Define the work potential
• Finally, find the total strain energy density function
• Now, the expression will be used in dynamic
modeling
• Based on principle of least
action, define expression of
Euler-Lagrange’s equation
• Use the total energy density
function to obtain kinetic and
potential energy functions
• Find ODE in the form of
dimensional parameters
• Define dimensionless parameters
and convert equation in non-
dimensional form
• Define the boundary condition of
the HMS actuator
• Further, use in parametric study
• Define material parameters
• Define the applied external
loading characteristics (prestress,
periodic/constant)
• Use ODE45 solver in MATLAB
to solve the nonlinear governing
equations
• Obtain the results for analysis:
time history graph, phase paths
and Poincaré maps.
• Investigate the dynamic
properties, such as the dynamic
response, dynamic stability,
periodicity, and resonance.

Methodology: Kinematics
1 1 1
x X


2 2 2
x X


3 3 3
x X


Based on concept of continuum mechanics
Under Compressible
conditions
In virtue of the volumetric
incompressibility
1 2 3 1
   
1 2
  
 
2
3
2
1
1
 
 

 
External Stimuli Expression Direction
Applied Magnetic Flux Vector Alongside Positive and Negative X3-
direction
Residual Magnetic Flux Density Parallel to Positive X3-direction
Mechanical Pre-stress Lateral dimensions of the cuboid
0
0
applied
applied
B
B
 
 
  
 
 
 
0 0
r r
B B

P
A
 
1
2
2
3
0 0 0 0
0 0 0 0
0 0 0 0
F
 
 
 
   
   
 
   
   
   
Decomposition of Deformation Gradient
Implementation of Zener Rheological Model
Deformation
Gradient
F = Fe Fv
Elastic
component
Viscous
component
1 1
2 2
3 3
0 0 0 0
0 0 0 0
0 0 0 0
e v
e v
e v
F
 
 
 
   
   
    
   
   
1
ei i i

  

Viscous principal stretch
associated with the viscous
damper
Principal stretch with the
nonlinear spring of the
Maxwell element
Incompressible stretch
Parameter

Strain Energy Density Function: Material Models
Viscoelastic Analysis Polymer chain Analysis
The neo-Hookean material model: [9] Physics based nonaffine material model: [8]
     
2 2 2 2 1
max max
1
ln 3
6
Polyme c c e
r i i i i
        
   
 
 
 
 
2 2 2 2 2
max max 1 2 3
2 2 2
1 2 3 1 1 1
1 2 3 1 2 3
ln 3
1
6
Polymer
c
c
e
     
   
      
  
 

 

 
  
   
  
 
 

μc : The network modulus which depends on the density of crosslinks,
λmax : The limited extensibility of the polymer chains,
μe : the entanglement modulus that modifies with the levels of
entanglements in the polymer network
 
1
2
3
2
2
2
3
2
neo Hookean

  

    
strain equilibrium viscous





   
2
1 2 3
2 2 2 2 2 2 2 2
1 1 2 3
2 3
2
3 3
2
v v v
strain
 
        
  
 
      
 




 
2
2
3
neo Hookean i


  

For incompressible
material:
For incompressible
material:


: The effective shear modulus of the polymer,
: The non-dimensional proportionality constant.

The proposed material model:
The strain energy
density of the visco-
elastomer
2 2 2 2 2 2 2 2 2
2
1 2 3 2 3
1 1 3
2
3
2
3 v v
tr i v
s a n
 
        
  
 
      
 
   
  

 


strain equilibrium viscous





Equilibrium part
of strain energy
Viscous parts of
strain energy
The work potential of
the HMS actuator
Pre-stress part
of external
stimuli
Magnetic parts of
external stimuli
Workdone (under Biaxial prestress):
external t
pres ress
WP
WP 
   
 
1 2 2
1 1 1
prestress
WP     
 
 
 
 
Upon invoking the condition of equal biaxial
prestress and equi-lateral stretch:
1 2
( )
  
 
1 2
( )
  
 
 
 
2 1
prestress
WP  
 
 

 
The viscous stretch also follows the
incompressibility constraint: 1 2
v v v
  
  1
2
1
3 1 2 )
( ( )
v v v v
   
 
 
Magnetic flux density:
1 2 0
appl
r
magneti
i d
c
e
B B
  
 
 

 

0
r a
ma
i
gnetic
ppl ed
FB B

 
  



Total
Energy
Density
Function:
 
 
2 2 2
0
4 4 4
2
2 2 2 1
2 2
3 3 d
r
strain exter
appli
a
e
v v
n l
B B
WP
 
       
 
 

 
 
      
 
 
   
       
 




 
 
 
(Viscoelastic analysis)
Methodology
       
 
2 4 2 2 2 4 2 1 2
max max
0
2
1
2 ln 3 2 2 2 2 1
6
r
strain externa
i
l c c
appl ed
e
B B
WP               
 
   
 
 
           
 
 
 


  



     
2 2 2 2 2 2 2 2 1 1 1
1 2 3 max max 1 2 3 1 2 3 1 2 3
1
ln 3
6
str i c c e
a n                 
  
 
         
 
 
 
  2
0
applied
r
magnetic
B B
 
 
 

 

(Polymer chain
analysis)
(Viscoelastic
analysis)
(Polymer chain
analysis)

Methodology: Dynamic Modeling
Dynamic Governing Equation of Motion
     
 
4
2 2 2
0
2 4 4
2
8 2 2 2 1
2
3 3
2
r applied
v v
B B
U HL
 
       
 



 
    
 
    



 

2
2 2 2
6
2
1
8
3 3
2
T HL L H

 

 




 
 

 



&
&
L : Lagrangian (L = T - U)
T : Associated total kinetic
energy
U : Associated total potential
energy
: Time derivative of principal
lateral stretch
D : Rayleigh dissipation function
2
2
8
2
v
D HL




 
 
d
0
dt
L L D
  
  
 
  
 
  
 
Euler–Lagrange’s (E–L) equation of motion:
Total Potential Energy Total Kinetic Energy
       
 
2 2 4 2 2 2 4 2 1
x 2
2
ma max
0
1
8 2 ln 3 2 2 2 2 1
6
r
c c e
applied
B B
U HL               
 
   
 
 
        
 
 
 


 



Defining dimensionless
parameters:
2
t
L




2
2
H
c
L

Geometric
constant:
Dimensionless time:
Applied
dimensionless
magnetic Flux
density:
0
r applied
B B
b


Dimensionless
mechanical pre-
stress:
S




(Viscoelastic
analysis)
(Polymer chain
analysis)

Equation governing the dynamic
motion:
Three initial conditions (At rest condition):
0
0
d
t
dt



0
1
t
 

0
1
v t
 

Equation in the non-dimensional
form:
 
2 3
2
4 3
d 2
0
d
V
v v
L
  
   
  
 
 

   
 
2
5 5
2
2 2 3
2
4
6 7
d
0
2 d 6
1
d
3
d
v v
c c
S b
 
      
   
 
 
   
   
   


 
   

Methodology: Dynamic Modeling
 
 
   
2
2
2 2 5
2
2 3
max
6 7 3
2 2
ma
¨
0
x
4
2
1
3 0
3
6
2
2
1
3
applied
c e
r
B B
H
L H

 
          
   
  
 


 
 
 
 
 
    
 
   
      
 
 
 
    

 
   
 
 
2
¨
2
2
5 5 4
2
6
3
7
2
0
2
3
6
0
v v
ap
r plied
H
L
B B
H

      
 

 
  

 

 
 
 
 
 
    


   
   


   
   
 
 
  
 
 
   
 
   
 
 
 
Equation governing the dynamics of the
motion:
 
3 2 3
4
2
0
v v
v

    

 

 
(Viscoelastic
analysis)
(Polymer chain
analysis)
(Viscoelastic
analysis)
 
 
     
2
5 2 3 3
max
6 7 2 2
ma
2
x
2
2
2 4
2
1
3 0
d 2 d 6
1
3
1
d d 2
3
e
c
c c
S b
 
 
     
    
  
  


 
 
 
 
    
 
 
 

   
      
 
 

 
   

Analysis : Defining Material parameters
3 3
min
3
max
2
a  



Thickness Stretch Amplitude Nondimensional Excitation Angular
Frequency:
*
*
2



*
 : Nondimensional oscillation period of actuator for a specified time
Defining the parameters: For Constant and Periodic Magnetic Flux Density
Viscous Effect
Polymer Chain Effect
 
si .
n
a t
b b 

Case-1: 2
0
b 
Case-2:
2
0
b 
(Applied magnetic flux density
along residual flux density)
(Applied magnetic flux density
opposite to residual flux density)
Without prestress:
S = 0
With prestress:
S = 1
Magnetic flux density:
b = 0.4
Material parameter:
c = 1.0
5
10
50






Viscosity
Parameter:
𝛽 = 2
Proportionality
constant of the
material:
Effect of
entanglements:
0
0.5
1
e
c
e
c
e
c









0
0.5
1
c
e
c
e
c
e









max
max
max
2
5
10






Effect of
crosslinks:
Effect of
extensibility:
Pa-s

Analysis and Results: Constant Magnetic Flux Density
(Viscoelastic
analysis)
(Polymer chain
analysis)
Magnetic Flux Density alongside the residual
Magnetic Flux Density opposite to the residual
Time–history responses
• For each considered value of η in both of cases, i.e., S = 0, and 1, the stretch λ3 decreases
spontaneously and further evolves with time for achieving the equilibrium position.
• With the decrease in the value of viscosity parameter, the vibration attenuates more
quickly and the actuator takes less time for attaining the position of equilibrium.
• The actuator with prestress condition attains a higher level of deformation at the
equilibrium state compared to the case in which actuator is not prestressed for any value of
viscosity parameter.
• The actuator with prestress (S = 1) undergoes compression: large compressive deformation
because of mechanical loading over expansion due to magnetic loading.
Inferences
𝑏2 > 0
• In polymer chain analysis, the actuator undergoes oscillations from initial
configuration to final configuration and increase in crosslinks/entanglements
reduces oscillation stretch.
(Viscoelastic
analysis)

Analysis and Results: Constant Magnetic Flux Density
(Viscoelastic
analysis)
(Polymer chain
analysis)
Magnetic Flux Density alongside the residual
Magnetic Flux Density opposite to the residual
Phase-plane plot
• The phase–plane portraits depict that the planar HMSA oscillates about its mean position
prior to attaining its state of stabilized equilibrium for both with prestress and without
prestress conditions.
• The number of oscillation cycles is directly proportional to the viscosity parameter of the
system, i.e., the lower degree of η develops lesser oscillations.
• The large viscosity parameter leads to the higher number of oscillations before achieving
the equilibrium state.
• Similar behavior of vibrations in both mode of actuation except for the change in
amplitude of vibrations.
Inferences
• In polymer chain analysis, the phase-plane plot forms a close loop indicating the
periodic motion.

Analysis and Results: Periodic Magnetic Flux Density
(Viscoelastic
• The actuator without prestress exhibits small amplitude of oscillations and the resonant
actuation frequency as compared to prestressed actuator.
• For the actuator without prestress, the dimensionless resonant frequency is obtained to
be ω0 = 2.14, while that for with prestress case is ω0 = 2.20.
• The actuator without prestress (S = 0) attains stability more quickly than the
prestressed actuator (S = 1).
• The actuator exhibits a stable periodic oscillation for both states of prestress and
deviates from its normal response by forming a closed spiral, indicating the
occurrence of resonance in the planar actuator.

---- Sub-harmonic
---- Harmonic
---- Super-harmonic
Same Response (Poincare,
phase-plane, time history)
as without prestress
condition
The variation of stretch amplitude of a
prestressed hyperelastic actuator with
the nondimensional frequency of
excitation (ω*).
The angular frequency of the
excitation magnetic flux density is
approximately twice or half of the
resonance frequency, it exhibits peaks
in oscillation amplitude and represents
the subharmonic and superharmonic
resonances.
The vibration remains stable. Further,
the Poincaré maps form a closed loop
rather than isolated points, which
shows that the motion is quasi-
periodic.
(Viscoelastic

(Polymer chain
analysis)
Effect of Crosslinks

(Polymer chain
analysis)
Effect of Entanglements

(Polymer chain
analysis)
Effect of Finite extensibility

Conclusions
In this presentation, a parametric study exploring the effect of varying levels of viscosity parameter and mechanical prestress on the nonlinear
dynamic responses of the actuator subjected to the constant and the periodic magnetic fields was conducted. The key inferences drawn from the
analysis are summarized below:
• The inherent viscosity of the hard-magnetic soft materials appreciably affects the time taken by the actuator to acquire the stable equilibrium
configuration.
• The increase in the level of viscosity parameter increases the stiffness of the actuator material and thereby the actuator's deformation level is
depleted to some extent.
• An increase in the viscoelasticity of the actuator increases the resonant frequencies and results in a depletion of the vibration amplitude.
• In the absence of mechanical pre-stress, the actuator attains the stable periodic oscillation more rapidly in comparison to the actuator with
mechanical prestress subjected to the resonant periodic magnetic flux density.
• The planar actuator forms a closed spiral and departs from its regular response when it is excited with the resonant frequencies.
• A strong crosslinks of polymer chains enhances the resonant frequency of HMS actuator, but weakens the peak value of resonance. Next, the
special beating also emerges in the dynamic response, and can be eliminated if the entanglements increase.
• The phase paths of HMS actuator with different entanglements do not go to infinity, indicative of a stable vibration. As the entanglements
strengthen, the periodicity of HMS actuator transfers from aperiodic vibration to the quasi-periodic vibration.
The underlying analytical dynamic model along with the inferences reported in the current analysis can be useful in the design and development of
the futuristic remotely driven soft actuators executing time-dependent motion.

Recommendations for future work
Future work includes:
• The development of dynamic modeling framework for hard-magnetic soft actuators to perform a parametric study and analyze the effect of
inclined magnetic flux density in planar actuator, HMS beam, HMS resonator and HMS energy structures.
• Analysis of in-homogeneously deformed HMS planar actuators, HMS beam, HMS resonator and HMS energy structures.
• Experimental validation of the developed dynamic framework and parametric study considering the effects of crosslinks, finite extensibility,
entanglements, viscoelasticity, material anisotropy and temperature on the nonlinear oscillations of a hard-magnetic soft actuator.
• Experimental validation of the proposed dynamic framework and parametric study considering the effects of inclined magnetic flux density in
planar actuator, HMS beam, HMS resonator and HMS energy structures.

Publications
1. Shivendra Nandan, Divyansh Sharma, and Atul Kumar Sharma. ”Viscoelastic effects on the nonlinear oscillations of hard-magnetic
soft actuators.” ASME, Journal of Applied Mechanics 90.6 (2023): 061001.
2. Shivendra Nandan, Divyansh Sharma, and Atul Kumar Sharma. ”Effects of the polymer chain entanglements, crosslinks and finite
extensibility on the nonlinear oscillations of hard-magnetic soft actuators”, Submitted to an international journal.
Publications from thesis:
Other publications in the area of thesis:
1. Shivendra Nandan, Divyansh Sharma, and Atul Kumar Sharma. ”Thermal effects on the nonlinear oscillations of hard-magnetic soft
actuators”, In preparation.
2. Shivendra Nandan, Divyansh Sharma, and Atul Kumar Sharma. ”Effects of material transverse anisotropy on the nonlinear oscillations
of hard-magnetic soft actuators”, In preparation.
3. Shivendra Nandan and Atul Kumar Sharma, Manish M. Joglekar. “A state of art review on Pull-in instability in dielectric elastomer
actuators”, In preparation.
4. Aman Khurana, Shivendra Nandan and Atul Kumar Sharma, Manish M. Joglekar. ”A Comprehensive Review on Nonlinear Dynamic
Modelling of DE-Based Minimum Energy Structures”, In preparation.

References and Bibilography
References: 1. Ruike Zhao, Yoonho Kim, Shawn A Chester, Pradeep Sharma, and Xuanhe Zhao. Mechanics of hard-magnetic soft
materials. Journal of the Mechanics and Physics of Solids, 124:244–263, 2019.
2. Aman Khurana, Ajay Kumar, Santosh Kumar Raut, Atul Kumar Sharma, and Manish M Joglekar. Effect of
viscoelasticity on the nonlinear dynamic behavior of dielectric elastomer minimum energy structures. International
Journal of Solids and Structures, 208:141–153, 2021.
3. Daniel Garcia-Gonzalez. Magneto-visco-hyperelasticity for hard-magnetic soft materials: theory and numerical
applications. Smart Materials and Structures, 28(8):085020, 2019.
4. Zeinab Alameh, Shengyou Yang, Qian Deng, and Pradeep Sharma. Emergent magnetoelectricity in soft materials,
instability, and wireless energy harvesting. Soft Matter, 14(28):5856–5868, 2018.
5. Deepak Kumar and Somnath Sarangi. Electro-magnetostriction under large deformation: modeling with
experimental validation. Mechanics of Materials, 128:1–10, 2019.
6. Khurana, A., Kumar, D., Sharma, A. K., & Joglekar, M. M. (2021). Nonlinear oscillations of particle-reinforced
electro-magneto-viscoelastomer actuators. Journal of Applied Mechanics, 88(12).
7. Davidson, Jacob D., and Nakhiah C. Goulbourne. "A nonaffine network model for elastomers undergoing finite
deformations." Journal of the Mechanics and Physics of Solids 61.8 (2013): 1784-1797.
8. DeBotton, G., Hariton, I., & Socolsky, E. A. (2006). Neo-Hookean fiber-reinforced composites in finite
elasticity. Journal of the Mechanics and Physics of Solids, 54(3), 533-559.

THANK YOU
S h i v e n d ra N a n d a n + 9 1 8 8 0 8 8 3 6 4 6 8 n a n d a n . 1 @ i i t j . a c . i n

Dynamic Properties Investigation: Parametric Study
Nonlinear Oscillations Identification Techniques:
Periodic motion Quasi-periodic
motion
Chaotic motion
Time
History
Graph:
Phase-
Plane
Portraits:
Poincaré
maps:
The trajectories of the
system versus dimensionless
time are plotted.
The phase-plane diagram
is the velocity versus
displacement plot.
The Poincaré map is attained by choosing the
period of external magnetic flux as time step
(sampling at every periodic of external force).
The time history: It includes regular
and predictable trajectories;
The phase-plane portrait: It
indicates a closed loop;
The Poincaré maps: It consists of a
single point;
The time history: It includes
irregular trajectories;
The phase-plane portraits: a
torus shape appears;
The Poincaré maps: a closed
curve of points is seen;
The time history: It includes irregular
and unpredictable trajectories;
The phase-plane portraits: irregular
and complicated shapes arises;
The Poincaré maps: It consists of a
large number of points;

OUTLINE OF THE PRESENTATION
 Graphical Abstract
 Introduction: What are hard magnetic soft materials?
 Research Gap and Motivation: Why Hard magnetic soft material actuators?
 Objectives of the proposed research
 Methodology and Mathematical formulation
 Analysis and Results
 Conclusion and Future works
 References and Publications

𝑏2 =
𝐵𝑟𝐵𝑎𝑝𝑝𝑙𝑖𝑒𝑑
𝜇𝜇0
𝑏2 > 0
𝜆3 > 1
𝑏2 < 0
𝜆3 < 1
Contraction in Thickness
Expansion in Thickness
S = 0
S = 1
GRAPHICAL ABSTRACT

Appendix: Field of Use: Possible Applications

Static Analysis:
• Zhao et al. presented an asymmetric Cauchy stress-based continuum formulation to establish a direct
relationship between the expression of an external magnetic induction and stress tensor [1].
• Further, Hossain et al. [2] improved the proposed formulation of Zhao et al. [1] so that it can account for
viscoelastic effects and determine the time-dependent dissipative response of soft beams.
• Mukherjee et al. presented a dissipative model for HMSMs that explored the impact of ferromagnetic hysteresis
and particle volume fraction [3].
• Zhang et al. [4] demonstrated a theoretical model based on micromechanics to investigate the
influence of the hard-magnetic particles interacting with the soft membrane and its response on the actuation
performance of HMSMs.
• Kadapa et al. [5] presented a unified FEA model for the investigation of hard and soft particle-based magneto-
active membranes considering the visco-elastic effects to simulate a magnetically driven robotic gripper.
Appendix: Literature Review: (Hard-Magnetic Coupling Theories)

Appendix: Literature Review: (Hard-Magnetic Coupling Theories)
Dynamic Analysis:
• Khurana et al. investigated the viscoelasticity of filler material-based polymers to find the suitable volume
fraction. Further, they studied the nonlinear oscillations of actuators under dynamic conditions [6].
• Recently, the Zener model has been employed to analyze the performance of actuators and study the effects
of viscoelastic behavior on the design of such actuators [6,7].
• Khurana et al. [7] investigated the electro-magneto-viscoelastic actuators to examine the non-linear
dynamic response and nonlinear oscillations under DC and AC dynamic modes of actuation.

Appendix: Research Gap & Motivation
1) Based on the detailed literature survey, it is observed that a wealth of literature expounds on the modeling
of the hard-magnetic soft actuators in the quasi-static mode of loading, and also very few researchers has examined the
impact of viscoelasticity on the nonlinear dynamic behavior of the actuator across all of the research domain.
2) Viscosity is inherent property of the soft material and significantly affects the time dependent behavior of the soft
actuators. It dissipates the energy during actuation and can alter the nonlinear response of the actuator. Hard-magnetic
soft actuator undergo dynamic motion which is significantly affected by the viscoelastic
behavior of the material, polymer chain entanglements and cross-links, prestress, and temperature.
3) It becomes necessary to explore the effect of viscoelasticity on the nonlinear dynamic behavior of the viscoelastic
actuator for the design of efficient and intelligent devices because there is a scarcity of research on the dynamic
analysis of magneto-viscoelastic actuators considering the viscous effects.

Appendix: Research Objectives
• To develop an analytical framework for modeling the dynamic behavior of HMSMs incorporating
the viscoelastic effects.
• To develop an analytical framework for modeling the dynamic behavior of HMSMs incorporating the effect of
polymer chain entanglements and cross-links.
• To perform a parametric study to investigate the effect of various parameters such as viscoelasticity, equi-biaxial
prestressing, polymer chain entanglements, material anisotropy and cross-links on the nonlinear oscillations of a
hard-magnetic soft actuator under constant and periodic magnetic fluxes.

Dynamics of Hard-Magnetic Soft Materials

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