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In recent years, there has been increasing interest in hard-magnetic soft materials (HMSMs) due to their ability to retain high residual magnetization and undergo large deformations under external magnetic loading. The performance of these materials in the dynamic mode of actuation is significantly influenced by internal properties, such as entanglements, crosslinks, and the finite extensibility of polymer chains. This article presents a theoretical framework for modeling the dynamic behavior of a hard-magnetic soft material-based planar actuator. A physics-based nonaffine material model is utilized to consider the inherent properties of polymer chain networks. The governing equation for dynamic motion is derived using Euler–Lagrange’s equation of motion for conservative systems. The devised dynamic model is utilized to examine the dynamic response, stability, periodicity, and resonance properties of a planar hard-magnetic soft actuator for different values of polymer chain entanglements, crosslinks, and finite extensibility parameters. The Poincaré maps and phase-plane plots are presented to analyze the stability and periodicity of the nonlinear vibrations of the actuator. The results reveal that transitions between aperiodic and quasi-periodic oscillations occur when the density of polymer chain entanglements and cross-linking changes. The findings from the present investigation can serve as an initial step towards the design and manufacturing of remotely controlled actuators for various futuristic applications.

- 1. 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
- 2. 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
- 3. 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.
- 4. 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.
- 5. 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
- 6. 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.
- 7. 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)
- 8. 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)
- 9. 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
- 10. 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
- 11. 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)
- 12. 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.
- 13. 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.
- 14. ---- Sub-harmonic ---- Harmonic ---- Super-harmonic Analysis and Results: Periodic Magnetic Flux Density 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
- 15. Analysis and Results: Periodic Magnetic Flux Density (Polymer chain analysis) Effect of Crosslinks
- 16. Analysis and Results: Periodic Magnetic Flux Density (Polymer chain analysis) Effect of Entanglements
- 17. Analysis and Results: Periodic Magnetic Flux Density (Polymer chain analysis) Effect of Finite extensibility
- 18. 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.
- 19. 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.
- 20. 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.
- 21. 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.
- 22. 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
- 23. 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;
- 24. 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
- 25. 𝑏2 = 𝐵𝑟𝐵𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝜇𝜇0 𝑏2 > 0 𝜆3 > 1 𝑏2 < 0 𝜆3 < 1 Contraction in Thickness Expansion in Thickness S = 0 S = 1 GRAPHICAL ABSTRACT
- 26. Appendix: Field of Use: Possible Applications
- 27. Appendix: Field of Use: Possible Applications
- 28. Appendix: Field of Use: Possible Applications
- 29. 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)
- 30. 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.
- 31. 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.
- 32. 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.