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The quaternion beam model for hard-magnetic flexible cantilevers

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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.

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Authors and Affiliations

  1. State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wei Chen, Guozhen Wang, Yiqun Li & Zhouping Yin

  2. Flexible Electronics Research Center, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wei Chen, Guozhen Wang, Yiqun Li & Zhouping Yin

  3. Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Lin Wang

  4. Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan, 430074, China

    Lin Wang

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  1. Wei Chen
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  2. Guozhen Wang
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  3. Yiqun Li
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  4. Lin Wang
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  5. Zhouping Yin
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Correspondence to Zhouping Yin.

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Citation: CHEN, W., WANG, G. Z., LI, Y. Q., WANG, L., and YIN, Z. P. The quaternion beam model for hard-magnetic flexible cantilevers. Applied Mathematics and Mechanics (English Edition), 44(5), 787–808 (2023) https://doi.org/10.1007/s10483-023-2983-8

Project supported by the National Key Research and Development Program of China (No. 2018YFA0703200), the National Natural Science Foundation of China (Nos. 52205594 and 51820105008), the China National Postdoctoral Program for Innovative Talents (No. BX20220118), and the China Postdoctoral Science Foundation (No. 2021M701306)

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Chen, W., Wang, G., Li, Y. et al. The quaternion beam model for hard-magnetic flexible cantilevers. Appl. Math. Mech.-Engl. Ed. 44, 787–808 (2023). https://doi.org/10.1007/s10483-023-2983-8

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  • DOI: https://doi.org/10.1007/s10483-023-2983-8

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