The quaternion beam model for hard-magnetic flexible cantilevers

doi:10.1007/s10483-023-2983-8

Applied Mathematics and Mechanics (English Edition) ›› 2023, Vol. 44 ›› Issue (5): 787-808.doi: https://doi.org/10.1007/s10483-023-2983-8

The quaternion beam model for hard-magnetic flexible cantilevers

Wei CHEN^1,2, Guozhen WANG^1,2, Yiqun LI^1,2, Lin WANG^3,4, Zhouping YIN^1,2

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;
2. Flexible Electronics Research Center, Huazhong University of Science and Technology, Wuhan 430074, China;
3. Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
4. Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, China

收稿日期:2022-11-20 修回日期:2023-02-24 发布日期:2023-04-24
通讯作者: Zhouping YIN, E-mail: yinzhp@mail.hust.edu.cn
基金资助:
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)

The quaternion beam model for hard-magnetic flexible cantilevers

Wei CHEN^1,2, Guozhen WANG^1,2, Yiqun LI^1,2, Lin WANG^3,4, Zhouping YIN^1,2

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;
2. Flexible Electronics Research Center, Huazhong University of Science and Technology, Wuhan 430074, China;
3. Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
4. Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2022-11-20 Revised:2023-02-24 Published:2023-04-24
Contact: Zhouping YIN, E-mail: yinzhp@mail.hust.edu.cn
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)

摘要/Abstract

摘要： 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.

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

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

中图分类号:

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

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The quaternion beam model for hard-magnetic flexible cantilevers

The quaternion beam model for hard-magnetic flexible cantilevers

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