{"title":"基于相场法的铁电材料柔电应变梯度有限元模型","authors":"Shuai Wang, Hengchang Su, Min Yi, Li-Hua Shao","doi":"10.1007/s10338-024-00485-5","DOIUrl":null,"url":null,"abstract":"<div><p>Flexoelectricity is a two-way coupling effect between the strain gradient and electric field that exists in all dielectrics, regardless of point group symmetry. However, the high-order derivatives of displacements involved in the strain gradient pose challenges in solving electromechanical coupling problems incorporating the flexoelectric effect. In this study, we formulate a phase-field model for ferroelectric materials considering the flexoelectric effect. A four-node quadrilateral element with 20 degrees of freedom is constructed without introducing high-order shape functions. The microstructure evolution of domains is described by an independent order parameter, namely the spontaneous polarization governed by the time-dependent Ginzburg–Landau theory. The model is developed based on a thermodynamic framework, in which a set of microforces is introduced to construct the constitutive relation and evolution equation. For the flexoelectric part of electric enthalpy, the strain gradient is determined by interpolating the mechanical strain at the node via the values of Gaussian integration points in the isoparametric space. The model is shown to be capable of reproducing the classic analytical solution of dielectric materials incorporating the flexoelectric contribution. The model is verified by duplicating some typical phenomena in flexoelectricity in cylindrical tubes and truncated pyramids. A comparison is made between the polarization distribution in dielectrics and ferroelectrics. The model can reproduce the solution to the boundary value problem of the cylindrical flexoelectric tube, and demonstrate domain twisting at domain walls in ferroelectrics considering the flexoelectric effect.</p></div>","PeriodicalId":50892,"journal":{"name":"Acta Mechanica Solida Sinica","volume":"37 4","pages":"570 - 579"},"PeriodicalIF":2.0000,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Strain Gradient Finite Element Formulation of Flexoelectricity in Ferroelectric Material Based on Phase-Field Method\",\"authors\":\"Shuai Wang, Hengchang Su, Min Yi, Li-Hua Shao\",\"doi\":\"10.1007/s10338-024-00485-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Flexoelectricity is a two-way coupling effect between the strain gradient and electric field that exists in all dielectrics, regardless of point group symmetry. However, the high-order derivatives of displacements involved in the strain gradient pose challenges in solving electromechanical coupling problems incorporating the flexoelectric effect. In this study, we formulate a phase-field model for ferroelectric materials considering the flexoelectric effect. A four-node quadrilateral element with 20 degrees of freedom is constructed without introducing high-order shape functions. The microstructure evolution of domains is described by an independent order parameter, namely the spontaneous polarization governed by the time-dependent Ginzburg–Landau theory. The model is developed based on a thermodynamic framework, in which a set of microforces is introduced to construct the constitutive relation and evolution equation. For the flexoelectric part of electric enthalpy, the strain gradient is determined by interpolating the mechanical strain at the node via the values of Gaussian integration points in the isoparametric space. The model is shown to be capable of reproducing the classic analytical solution of dielectric materials incorporating the flexoelectric contribution. The model is verified by duplicating some typical phenomena in flexoelectricity in cylindrical tubes and truncated pyramids. A comparison is made between the polarization distribution in dielectrics and ferroelectrics. The model can reproduce the solution to the boundary value problem of the cylindrical flexoelectric tube, and demonstrate domain twisting at domain walls in ferroelectrics considering the flexoelectric effect.</p></div>\",\"PeriodicalId\":50892,\"journal\":{\"name\":\"Acta Mechanica Solida Sinica\",\"volume\":\"37 4\",\"pages\":\"570 - 579\"},\"PeriodicalIF\":2.0000,\"publicationDate\":\"2024-04-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mechanica Solida Sinica\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10338-024-00485-5\",\"RegionNum\":3,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATERIALS SCIENCE, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mechanica Solida Sinica","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s10338-024-00485-5","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
Strain Gradient Finite Element Formulation of Flexoelectricity in Ferroelectric Material Based on Phase-Field Method
Flexoelectricity is a two-way coupling effect between the strain gradient and electric field that exists in all dielectrics, regardless of point group symmetry. However, the high-order derivatives of displacements involved in the strain gradient pose challenges in solving electromechanical coupling problems incorporating the flexoelectric effect. In this study, we formulate a phase-field model for ferroelectric materials considering the flexoelectric effect. A four-node quadrilateral element with 20 degrees of freedom is constructed without introducing high-order shape functions. The microstructure evolution of domains is described by an independent order parameter, namely the spontaneous polarization governed by the time-dependent Ginzburg–Landau theory. The model is developed based on a thermodynamic framework, in which a set of microforces is introduced to construct the constitutive relation and evolution equation. For the flexoelectric part of electric enthalpy, the strain gradient is determined by interpolating the mechanical strain at the node via the values of Gaussian integration points in the isoparametric space. The model is shown to be capable of reproducing the classic analytical solution of dielectric materials incorporating the flexoelectric contribution. The model is verified by duplicating some typical phenomena in flexoelectricity in cylindrical tubes and truncated pyramids. A comparison is made between the polarization distribution in dielectrics and ferroelectrics. The model can reproduce the solution to the boundary value problem of the cylindrical flexoelectric tube, and demonstrate domain twisting at domain walls in ferroelectrics considering the flexoelectric effect.
期刊介绍:
Acta Mechanica Solida Sinica aims to become the best journal of solid mechanics in China and a worldwide well-known one in the field of mechanics, by providing original, perspective and even breakthrough theories and methods for the research on solid mechanics.
The Journal is devoted to the publication of research papers in English in all fields of solid-state mechanics and its related disciplines in science, technology and engineering, with a balanced coverage on analytical, experimental, numerical and applied investigations. Articles, Short Communications, Discussions on previously published papers, and invitation-based Reviews are published bimonthly. The maximum length of an article is 30 pages, including equations, figures and tables