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On a family of numerical models for couple stress based flexoelectricity for continua and beams

Roman Poya, Antonio Gil Orcid Logo, Rogelio Ortigosa, Roberto Palma

Journal of the Mechanics and Physics of Solids, Volume: 125, Pages: 613 - 652

Swansea University Author: Antonio Gil Orcid Logo

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DOI (Published version): 10.1016/j.jmps.2019.01.013

Abstract

A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew-symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume...

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Published in: Journal of the Mechanics and Physics of Solids
ISSN: 0022-5096
Published: 2019
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URI: https://cronfa.swan.ac.uk/Record/cronfa48418
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first_indexed 2019-01-24T20:01:21Z
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In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material&#x2019;s centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the C0 continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation.</abstract><type>Journal Article</type><journal>Journal of the Mechanics and Physics of Solids</journal><volume>125</volume><paginationStart>613</paginationStart><paginationEnd>652</paginationEnd><publisher/><issnPrint>0022-5096</issnPrint><keywords>Flexoelectricity, Size-dependent piezoelectricity, Couple stress theory, Curvature-induced polarisation, Mixed high order finite elements</keywords><publishedDay>30</publishedDay><publishedMonth>4</publishedMonth><publishedYear>2019</publishedYear><publishedDate>2019-04-30</publishedDate><doi>10.1016/j.jmps.2019.01.013</doi><url/><notes/><college>COLLEGE NANME</college><department>Civil Engineering</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>CIVL</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2019-03-11T14:57:07.5073352</lastEdited><Created>2019-01-24T13:43:10.9238234</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering</level></path><authors><author><firstname>Roman</firstname><surname>Poya</surname><order>1</order></author><author><firstname>Antonio</firstname><surname>Gil</surname><orcid>0000-0001-7753-1414</orcid><order>2</order></author><author><firstname>Rogelio</firstname><surname>Ortigosa</surname><order>3</order></author><author><firstname>Roberto</firstname><surname>Palma</surname><order>4</order></author></authors><documents><document><filename>0048418-24012019134713.pdf</filename><originalFilename>poya2019.pdf</originalFilename><uploaded>2019-01-24T13:47:13.4070000</uploaded><type>Output</type><contentLength>11241665</contentLength><contentType>application/pdf</contentType><version>Accepted Manuscript</version><cronfaStatus>true</cronfaStatus><embargoDate>2020-01-24T00:00:00.0000000</embargoDate><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807>
spelling 2019-03-11T14:57:07.5073352 v2 48418 2019-01-24 On a family of numerical models for couple stress based flexoelectricity for continua and beams 1f5666865d1c6de9469f8b7d0d6d30e2 0000-0001-7753-1414 Antonio Gil Antonio Gil true false 2019-01-24 CIVL A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew-symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material’s centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the C0 continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation. Journal Article Journal of the Mechanics and Physics of Solids 125 613 652 0022-5096 Flexoelectricity, Size-dependent piezoelectricity, Couple stress theory, Curvature-induced polarisation, Mixed high order finite elements 30 4 2019 2019-04-30 10.1016/j.jmps.2019.01.013 COLLEGE NANME Civil Engineering COLLEGE CODE CIVL Swansea University 2019-03-11T14:57:07.5073352 2019-01-24T13:43:10.9238234 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering Roman Poya 1 Antonio Gil 0000-0001-7753-1414 2 Rogelio Ortigosa 3 Roberto Palma 4 0048418-24012019134713.pdf poya2019.pdf 2019-01-24T13:47:13.4070000 Output 11241665 application/pdf Accepted Manuscript true 2020-01-24T00:00:00.0000000 true eng
title On a family of numerical models for couple stress based flexoelectricity for continua and beams
spellingShingle On a family of numerical models for couple stress based flexoelectricity for continua and beams
Antonio Gil
title_short On a family of numerical models for couple stress based flexoelectricity for continua and beams
title_full On a family of numerical models for couple stress based flexoelectricity for continua and beams
title_fullStr On a family of numerical models for couple stress based flexoelectricity for continua and beams
title_full_unstemmed On a family of numerical models for couple stress based flexoelectricity for continua and beams
title_sort On a family of numerical models for couple stress based flexoelectricity for continua and beams
author_id_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2
author_id_fullname_str_mv 1f5666865d1c6de9469f8b7d0d6d30e2_***_Antonio Gil
author Antonio Gil
author2 Roman Poya
Antonio Gil
Rogelio Ortigosa
Roberto Palma
format Journal article
container_title Journal of the Mechanics and Physics of Solids
container_volume 125
container_start_page 613
publishDate 2019
institution Swansea University
issn 0022-5096
doi_str_mv 10.1016/j.jmps.2019.01.013
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Civil Engineering
document_store_str 1
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description A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew-symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material’s centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the C0 continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation.
published_date 2019-04-30T03:58:50Z
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