{"title":"外加磁场驱动下硬磁软梁的超弹性模型","authors":"Mingqi Lu, Saifei Zhang, Shujun Li","doi":"10.1016/j.apm.2025.116190","DOIUrl":null,"url":null,"abstract":"<div><div>Due to the hyperelastic behaviors of matrix materials, hard-magnetic soft beams driven by the external magnetic fields will exhibit large and complex deformation. To accurately and reasonably calculate the mechanical properties of this structure, a theoretical model that can simulate both the axial and bending deformation entirely based on the hyperelastic mechanics is needed for current applications. According to the finite deformation theory, a hard-magnetic soft beam model completely based on the hyperelastic material theory is proposed and its total energy equation of the hard-magnetic soft beam can be obtained. An iterative solution algorithm is also proposed to solve the deformation of the beam and an adaptive genetic algorithm is presented to enhance the solution efficiency. By comparing with the previous research results, the validity of the model is verified. On this basis, the structural responses of hard-magnetic soft beams driven by external magnetic fields are analyzed. Some new dimensionless quantities are defined to describe the structural properties and deformation. It is found that when the direction angle of the external magnetic field is not greater than 90°, the three dimensionless quantities: the dimensionless tangent angle of the cantilever end, the dimensionless secant angle of the cantilever end and the dimensionless distance between the two ends of the beam are all independent on the direction angle of the external magnetic field. From the study, it can also be concluded that the greater direction angle of the external magnetic field will result in the closer distance between the two ends of the beam and reduce the range extension reached by the cantilever end of the beam.</div></div>","PeriodicalId":50980,"journal":{"name":"Applied Mathematical Modelling","volume":"146 ","pages":"Article 116190"},"PeriodicalIF":4.4000,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hyperelastic model of hard-magnetic soft beams driven by the external magnetic fields\",\"authors\":\"Mingqi Lu, Saifei Zhang, Shujun Li\",\"doi\":\"10.1016/j.apm.2025.116190\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Due to the hyperelastic behaviors of matrix materials, hard-magnetic soft beams driven by the external magnetic fields will exhibit large and complex deformation. To accurately and reasonably calculate the mechanical properties of this structure, a theoretical model that can simulate both the axial and bending deformation entirely based on the hyperelastic mechanics is needed for current applications. According to the finite deformation theory, a hard-magnetic soft beam model completely based on the hyperelastic material theory is proposed and its total energy equation of the hard-magnetic soft beam can be obtained. An iterative solution algorithm is also proposed to solve the deformation of the beam and an adaptive genetic algorithm is presented to enhance the solution efficiency. By comparing with the previous research results, the validity of the model is verified. On this basis, the structural responses of hard-magnetic soft beams driven by external magnetic fields are analyzed. Some new dimensionless quantities are defined to describe the structural properties and deformation. It is found that when the direction angle of the external magnetic field is not greater than 90°, the three dimensionless quantities: the dimensionless tangent angle of the cantilever end, the dimensionless secant angle of the cantilever end and the dimensionless distance between the two ends of the beam are all independent on the direction angle of the external magnetic field. From the study, it can also be concluded that the greater direction angle of the external magnetic field will result in the closer distance between the two ends of the beam and reduce the range extension reached by the cantilever end of the beam.</div></div>\",\"PeriodicalId\":50980,\"journal\":{\"name\":\"Applied Mathematical Modelling\",\"volume\":\"146 \",\"pages\":\"Article 116190\"},\"PeriodicalIF\":4.4000,\"publicationDate\":\"2025-05-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematical Modelling\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0307904X25002653\",\"RegionNum\":2,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematical Modelling","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0307904X25002653","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
Hyperelastic model of hard-magnetic soft beams driven by the external magnetic fields
Due to the hyperelastic behaviors of matrix materials, hard-magnetic soft beams driven by the external magnetic fields will exhibit large and complex deformation. To accurately and reasonably calculate the mechanical properties of this structure, a theoretical model that can simulate both the axial and bending deformation entirely based on the hyperelastic mechanics is needed for current applications. According to the finite deformation theory, a hard-magnetic soft beam model completely based on the hyperelastic material theory is proposed and its total energy equation of the hard-magnetic soft beam can be obtained. An iterative solution algorithm is also proposed to solve the deformation of the beam and an adaptive genetic algorithm is presented to enhance the solution efficiency. By comparing with the previous research results, the validity of the model is verified. On this basis, the structural responses of hard-magnetic soft beams driven by external magnetic fields are analyzed. Some new dimensionless quantities are defined to describe the structural properties and deformation. It is found that when the direction angle of the external magnetic field is not greater than 90°, the three dimensionless quantities: the dimensionless tangent angle of the cantilever end, the dimensionless secant angle of the cantilever end and the dimensionless distance between the two ends of the beam are all independent on the direction angle of the external magnetic field. From the study, it can also be concluded that the greater direction angle of the external magnetic field will result in the closer distance between the two ends of the beam and reduce the range extension reached by the cantilever end of the beam.
期刊介绍:
Applied Mathematical Modelling focuses on research related to the mathematical modelling of engineering and environmental processes, manufacturing, and industrial systems. A significant emerging area of research activity involves multiphysics processes, and contributions in this area are particularly encouraged.
This influential publication covers a wide spectrum of subjects including heat transfer, fluid mechanics, CFD, and transport phenomena; solid mechanics and mechanics of metals; electromagnets and MHD; reliability modelling and system optimization; finite volume, finite element, and boundary element procedures; modelling of inventory, industrial, manufacturing and logistics systems for viable decision making; civil engineering systems and structures; mineral and energy resources; relevant software engineering issues associated with CAD and CAE; and materials and metallurgical engineering.
Applied Mathematical Modelling is primarily interested in papers developing increased insights into real-world problems through novel mathematical modelling, novel applications or a combination of these. Papers employing existing numerical techniques must demonstrate sufficient novelty in the solution of practical problems. Papers on fuzzy logic in decision-making or purely financial mathematics are normally not considered. Research on fractional differential equations, bifurcation, and numerical methods needs to include practical examples. Population dynamics must solve realistic scenarios. Papers in the area of logistics and business modelling should demonstrate meaningful managerial insight. Submissions with no real-world application will not be considered.