The quaternion beam model for hard-magnetic flexible cantilevers

doi:10.1007/s10483-023-2983-8

Abstract

Abstract: The recently developed hard-magnetic soft (HMS) materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields, e.g., soft robotics, flexible electronics, and biomedicine. Theoretical investigations on large deformations of HMS structures are significant foundations of their applications. This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams. A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact (GE) beam model. The singularity-free governing equations for the three-dimensional (3D) large deformations of an HMS beam are first derived, and then solved with the Galerkin discretization method and the trust-region-dogleg iterative algorithm. The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones. The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples. It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field. The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.

Key words: quaternion beam model, singularity-free formulation, hard-magnetic soft (HMS) beam, geometrically exact (GE) equation, three-dimensional (3D) large deformation

2010 MSC Number:

Wei CHEN, Guozhen WANG, Yiqun LI, Lin WANG, Zhouping YIN. The quaternion beam model for hard-magnetic flexible cantilevers. Applied Mathematics and Mechanics (English Edition), 2023, 44(5): 787-808.

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References

[1] SUN, W., MA, W., ZHANG, F., HONG, W., and LI, B. Snap-through path in a bistable dielectric elastomer actuator. Applied Mathematics and Mechanics] (English Edition), 43(8), 1159-1170(2022) https://doi.org/10.1007/s10483-022-2888-6
[2] YIN, Y., ZHAO, D., LIU, J., and XU, Z. Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction. Applied Mathematics and Mechanics (English Edition), 43(6), 793-812(2022) https://doi.org/10.1007/s10483-022-2853-9
[3] GUO, Z., NI, Q., CHEN, W., DAI, H., and WANG, L. Dynamic analysis and regulation of the flexible pipe conveying fluid with a hard-magnetic soft segment. Applied Mathematics and Mechanics (English Edition), 43(9), 1415-1430(2022) https://doi.org/10.1007/s10483-022-2901-9
[4] LI, C., LAU, G. C., YUAN, H., AGGARWAL, A., DOMINGUEZ, V. L., LIU, S., SAI, H., PALMER, L. C., SATHER, N. A., PEARSON, T. J., FREEDMAN, D. E., AMIRI, P. K., CRUZ, M. O. D. L., and STUPP, S. I. Fast and programmable locomotion of hydrogel-metal hybrids under light and magnetic fields. Science Robotics, 5(49), eabb9822(2020)
[5] CIANCHETTI, M., LASCHI, C., MENCIASSI, A., and DARIO, P. Biomedical applications of soft robotics. Nature Reviews Materials, 3(6), 143-153(2018)
[6] BAKHTIYARI, A., BAGHANI, M., and SOHRABPOUR, S. An investigation on multilayer shape memory polymers under finite bending through nonlinear thermo-visco-hyperelasticity. Applied Mathematics and Mechanics (English Edition), 44(1), 73-88(2023) https://doi.org/10.1007/s10483-023-2952-6
[7] LI, Y., CHEN, Y., REN, T., LI, Y., and CHOI, S. H. Precharged pneumatic soft actuators and their applications to untethered soft robots. Soft Robotics, 5(5), 567-575(2018)
[8] LI, T., LI, G., LIANG, Y., CHENG, T., DAI, J., YANG, X., LIU, B., ZENG, Z., HUANG, Z., LUO, Y., XIE, T., and YANG, W. Fast-moving soft electronic fish. Science Advances, 3(4), e1602045(2017)
[9] LI, G., CHEN, X., ZHOU, F., LIANG, Y., XIAO, Y., CAO, X., ZHANG, Z., ZHANG, M., WU, B., YIN, S., XU, Y., FAN, H., CHEN, Z., SONG, W., YANG, W., PAN, B., HOU, J., ZOU, W., HE, S., YANG, X., MAO, G., JIA, Z., ZHOU, H., LI, T., QU, S., XU, Z., HUANG, Z., LUO, Y., XIE, T., GU, J., ZHU, S., and YANG, W. Self-powered soft robot in the Mariana Trench. nature, 591(7848), 66-71(2021)
[10] CHEN, J., YANG, S., LI, Y., HUANG, Y., and YIN, Z. Active curved surface deforming of flexible conformal electronics by multi-fingered actuator. Robotics and Computer-Integrated Manufacturing, 64, 101942(2020)
[11] ZHOU, C., YANG, Y., WANG, J., WU, Q., GU, Z., ZHOU, Y., LIU, X., YANG, Y., TANG, H., LING, Q., WANG, L., and ZANG, J. Ferromagnetic soft catheter robots for minimally invasive bioprinting. Nature Communications, 12(1), 1-12(2021)
[12] ILAMI, M., BAGHERI, H., AHMED, R., SKOWRONEK, E. O., and MARVI, H. Materials, actuators, and sensors for soft bioinspired robots. Advanced Materials, 33(19), 2003139(2021)
[13] RIGBI, Z. and JILKEN, L. The response of an elastomer filled with soft ferrite to mechanical and magnetic influences. Journal of Magnetism and Magnetic Materials, 37(3), 267-276(1983)
[14] BORCEA, L. and BRUNO, O. On the magneto-elastic properties of elastomer-ferromagnet composites. Journal of the Mechanics and Physics of Solids, 49(12), 2877-2919(2001)
[15] HU, W., LUM, G. Z., MASTRANGELI, M., and SITTI, M. Small-scale soft-bodied robot with multimodal locomotion. nature, 554(7690), 81-85(2018)
[16] KIM, Y., YUK, H., ZHAO, R., CHESTER, S. A., and ZHAO, X. Printing ferromagnetic domains for untethered fast-transforming soft materials. nature, 558(7709), 274-279(2018)
[17] KUANG, X., WU, S., ZE, Q., YUE, L., JIN, Y., MONTGOMERY, S. M., YANG, F., QI, H. J., and ZHAO, R. Magnetic dynamic polymers for modular assembling and reconfigurable morphing architectures. Advanced Materials, 33(30), 2102113(2021)
[18] WU, S., ZE, Q., DAI, J., UDIPI, N., PAULINO, G. H., and ZHAO, R. Stretchable origami robotic arm with omnidirectional bending and twisting. Proceedings of the National Academy of Sciences, 118(36), e2110023118(2021)
[19] ALAPAN, Y., KARACAKOL, A. C., GUZELHAN, S. N., ISIK, I., and SITTI, M. Reprogrammable shape morphing of magnetic soft machines. Science Advances, 6(38), eabc6414(2020)
[20] WANG, X., MAO, G., GE, J., DRACK, M., CAÑÓN BERMÚDEZ, G. S., WIRTHL, D., ILLING, R., KOSUB, T., BISCHOFF, L., WANG, C., FASSBENDER, J., KALTENBRUNNER, M., and MAKAROV, D. Untethered and ultrafast soft-bodied robots. Communications Materials, 1(1), 1-10(2020)
[21] ZHAO, R., KIM, Y., CHESTER, S. A., SHARMA, P., and ZHAO, X. Mechanics of hard-magnetic soft materials. Journal of the Mechanics and Physics of Solids, 124, 244-263(2019)
[22] YAN, D., ABBASI, A., and REIS, P. M. A comprehensive framework for hard-magnetic beams: reduced-order theory, 3D simulations, and experiments. International Journal of Solids and Structures, 257, 111319(2022)
[23] LUCARINI, S., HOSSAIN, M., and GARCIA-GONZALEZ, D. Recent advances in hard-magnetic soft composites: synthesis, characterisation, computational modelling, and applications. Composite Structures, 279, 114800(2022)
[24] CHEN, W. and WANG, L. Theoretical modeling and exact solution for extreme bending deformation of hard-magnetic soft beams. Journal of Applied Mechanics--Transactions of the ASME, 87(4), 041002(2020)
[25] WANG, L., KIM, Y., GUO, C. F., and ZHAO, X. Hard-magnetic elastica. Journal of the Mechanics and Physics of Solids, 142, 104045(2020)
[26] CHEN, W., YAN, Z., and WANG, L. Complex transformations of hard-magnetic soft beams by designing residual magnetic flux density. Soft Matter, 16(27), 6379-6388(2020)
[27] CHEN, W., YAN, Z., and WANG, L. On mechanics of functionally graded hard-magnetic soft beams. International Journal of Engineering Science, 157, 103391(2020)
[28] WANG, L., ZHENG, D., HARKER, P., PATEL, A. B., GUO, C. F., and ZHAO, X. Evolutionary design of magnetic soft continuum robots. Proceedings of the National Academy of Sciences, 118(21), e2021922118(2021)
[29] CHEN, W., WANG, L., and PENG, Z. R. A magnetic control method for large-deformation vibration of cantilevered pipe conveying fluid. Nonlinear Dynamics, 105(2), 1459-1481(2021)
[30] CHEN, W., WANG, L., YAN, Z., and LUO, B. Three-dimensional large-deformation model of hard-magnetic soft beams. Composite Structures, 266, 113822(2021)
[31] SANO, T. G., PEZZULLA, M., and REIS, P. M. A Kirchhoff-like theory for hard magnetic rods under geometrically nonlinear deformation in three dimensions. Journal of the Mechanics and Physics of Solids, 160, 104739(2022)
[32] GOLDSTEIN, H., POOLE, C. P., and SAFKO, J. L. Classical Mechanics, 3rd ed., Addison-Wesley, New York (2001)
[33] HEARD, W. B. Rigid Body Mechanics: Mathematics, Physics and Applications, John Wiley & Sons, Weinheim (2006)
[34] CHELNOKOV, Y. N. Quaternion methods and models of regular celestial mechanics and astrodynamics. Applied Mathematics and Mechanics (English Edition), 43(1), 21-80(2022) https://doi.org/10.1007/s10483-021-2797-9
[35] COTTANCEAU, E., THOMAS, O., VÉRON, P., ALOCHET, M., and DELIGNY, R. A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables. Finite Elements in Analysis and Design, 139, 14-34(2018)
[36] GHOSH, S. and ROY, D. Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam. Computer Methods in Applied Mechanics and Engineering, 198(3-4), 555-571(2008)
[37] MALIKAN, M. and EREMEYEV, V. A. Flexomagneticity in buckled shear deformable hard-magnetic soft structures. Continuum Mechanics and Thermodynamics, 34(1), 1-16(2022)
[38] CHEN, W. and WANG, L. Large bending deformation of a cantilevered soft beam under external load: the applicability of inextensibility assumption of the centerline. Current Mechanics and Advanced Materials, 1(1), 24-38(2021)
[39] NAYFEH, A. H. and PAI, P. F. Linear and Nonlinear Structural Mechanics, 1st ed., John Wiley & Sons, Weinheim, 195-199, 226-234(2004)
[40] PAI, P. F. and PALAZOTTO, A. Large-deformation analysis of flexible beams. International Journal of Solids and Structures, 33(9), 1335-1353(1996)
[41] WANG, X. C. Finite Element Method (in Chinese), 1st ed., Tsinghua University Press, Beijing, 272-277(2003)
[42] DING, G. T. Two ways to introduce undertermined multipliers to constrained mechanical systems (in Chinese). Mechanics in Engineering, 38(1), 83-86(2016)
[43] CHEN, W., DAI, H. L., JIA, Q. Q., and WANG, L. Geometrically exact equation of motion for large-amplitude oscillation of cantilevered pipe conveying fluid. Nonlinear Dynamics, 98(3), 2097-2114(2019)
[44] CHEN, W., DAI, H. L., and WANG, L. Three-dimensional dynamical model for cantilevered pipes conveying fluid under large deformation. Journal of Fluids and Structures, 105, 103329(2021)
[45] CHEN, W., ZHOU, K., WANG, L., and YIN, Z. Geometrically exact model and dynamics of cantilevered curved pipe conveying fluid. Journal of Sound and Vibration, 534, 117074(2022)
[46] CHEN, W., WANG, L., and YAN, Z. On the dynamics of curved magnetoactive soft beams. International Journal of Engineering Science, 183, 103792(2023)
[47] NI, Q., WANG, Y. K., TANG, M., LUO, Y. Y., YAN, H., and WANG, L. Nonlinear impacting oscillations of a fluid-conveying pipe subjected to distributed motion constraints. Nonlinear Dynamics, 81(1), 893-906(2015)
[48] DAI, H. L., HE, Y. X., ZHOU, K., PENG, Z. R., WANG, L., and HAGEDORN, P. Utilization of nonlinear vibrations of soft pipe conveying fluid for driving underwater bio-inspired robot. Applied Mathematics and Mechanics (English Edition), 43(7), 1109-1124(2022) https://doi.org/10.1007/s10483-022-2866-7
[49] HAO, M. Y., DING, H., MAO, X. Y., and CHEN, L. Q. Stability and nonlinear response analysis of parametric vibration for elastically constrained pipes conveying pulsating fluid. Acta Mechanica Solida Sinica (2022) https://doi.org/10.1007/s10338-022-00370-z
[50] YUAN, J. R. and DING, H. Dynamic model of curved pipe conveying fluid based on the absolute nodal coordinate formulation. International Journal of Mechanical Sciences, 232, 107625(2022)
[51] WEI, S., YAN, X., FAN, X., MAO, X. Y., DING, H., and CHEN, L. Q. Vibration of fluid-conveying pipe with nonlinear supports at both ends. Applied Mathematics and Mechanics (English Edition), 43(6), 845-862(2022) https://doi.org/10.1007/s10483-022-2857-6