On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution
Among various magneto-elastic phenomena, flexomagnetic (FM) coupling can be defined as a dependence between strain gradient and magnetic polarization and, contrariwise, elastic strain and magnetic field gradient. This feature is a higher-order one than piezomagnetic, which is the magnetic response t...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-09-01
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| Series: | Nanomaterials |
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| Online Access: | https://www.mdpi.com/2079-4991/10/9/1762 |
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| author | Mohammad Malikan Victor A. Eremeyev |
| author_facet | Mohammad Malikan Victor A. Eremeyev |
| author_sort | Mohammad Malikan |
| collection | DOAJ |
| description | Among various magneto-elastic phenomena, flexomagnetic (FM) coupling can be defined as a dependence between strain gradient and magnetic polarization and, contrariwise, elastic strain and magnetic field gradient. This feature is a higher-order one than piezomagnetic, which is the magnetic response to strain. At the nanoscale, where large strain gradients are expected, the FM effect is significant and could be even dominant. In this article, we develop a model of a simultaneously coupled piezomagnetic–flexomagnetic nanosized Euler–Bernoulli beam and solve the corresponding problems. In order to evaluate the FM on the nanoscale, the well-known nonlocal model of strain gradient (NSGT) is implemented, by which the nanosize beam can be transferred into a continuum framework. To access the equations of nonlinear bending, we use the variational formulation. Converting the nonlinear system of differential equations into algebraic ones makes the solution simpler. This is performed by the Galerkin weighted residual method (GWRM) for three conditions of ends, that is to say clamp, free, and pinned (simply supported). Then, the system of nonlinear algebraic equations is solved on the basis of the Newton–Raphson iteration technique (NRT) which brings about numerical values of nonlinear deflections. We discovered that the FM effect causes the reduction in deflections in the piezo-flexomagnetic nanobeam. |
| first_indexed | 2024-03-10T16:31:24Z |
| format | Article |
| id | doaj.art-0dbb07dbba3a4b108a990dd60528843b |
| institution | Directory Open Access Journal |
| issn | 2079-4991 |
| language | English |
| last_indexed | 2024-03-10T16:31:24Z |
| publishDate | 2020-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Nanomaterials |
| spelling | doaj.art-0dbb07dbba3a4b108a990dd60528843b2023-11-20T12:46:23ZengMDPI AGNanomaterials2079-49912020-09-01109176210.3390/nano10091762On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical SolutionMohammad Malikan0Victor A. Eremeyev1Department of Mechanics of Materials and Structures, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, PolandDepartment of Mechanics of Materials and Structures, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, PolandAmong various magneto-elastic phenomena, flexomagnetic (FM) coupling can be defined as a dependence between strain gradient and magnetic polarization and, contrariwise, elastic strain and magnetic field gradient. This feature is a higher-order one than piezomagnetic, which is the magnetic response to strain. At the nanoscale, where large strain gradients are expected, the FM effect is significant and could be even dominant. In this article, we develop a model of a simultaneously coupled piezomagnetic–flexomagnetic nanosized Euler–Bernoulli beam and solve the corresponding problems. In order to evaluate the FM on the nanoscale, the well-known nonlocal model of strain gradient (NSGT) is implemented, by which the nanosize beam can be transferred into a continuum framework. To access the equations of nonlinear bending, we use the variational formulation. Converting the nonlinear system of differential equations into algebraic ones makes the solution simpler. This is performed by the Galerkin weighted residual method (GWRM) for three conditions of ends, that is to say clamp, free, and pinned (simply supported). Then, the system of nonlinear algebraic equations is solved on the basis of the Newton–Raphson iteration technique (NRT) which brings about numerical values of nonlinear deflections. We discovered that the FM effect causes the reduction in deflections in the piezo-flexomagnetic nanobeam.https://www.mdpi.com/2079-4991/10/9/1762flexomagneticnanobeamlarge deflectionNSGTGalerkin methodNewton–Raphson method |
| spellingShingle | Mohammad Malikan Victor A. Eremeyev On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution Nanomaterials flexomagnetic nanobeam large deflection NSGT Galerkin method Newton–Raphson method |
| title | On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution |
| title_full | On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution |
| title_fullStr | On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution |
| title_full_unstemmed | On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution |
| title_short | On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution |
| title_sort | on nonlinear bending study of a piezo flexomagnetic nanobeam based on an analytical numerical solution |
| topic | flexomagnetic nanobeam large deflection NSGT Galerkin method Newton–Raphson method |
| url | https://www.mdpi.com/2079-4991/10/9/1762 |
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