Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models
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| Title: | Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models |
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| Authors: | Polyzos, D.1 demosthenes.polyzos@gmail.com, Fotiadis, D.I.2 |
| Source: | International Journal of Solids & Structures. Feb2012, Vol. 49 Issue 3/4, p470-480. 11p. |
| Subjects: | Mindlin theory, Physiologic strain, Lattice theory, Mathematical continuum, Models & modelmaking, Asymptotic homogenization |
| Abstract: | Abstract: Mindlin, in his celebrated papers of Arch. Rat. Mech. Anal. 16, 51–78, 1964 and Int. J. Solids Struct. 1, 417–438, 1965, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro-structural effects. Since then, many works dealing with strain gradient elastic theories, derived either from lattice models or homogenization approaches, have appeared in the literature. Although elegant, none of them reproduces entirely the equation of motion as well as the classical and non-classical boundary conditions appearing in Mindlin theory, in terms of the considered lattice or continuum unit cell. Furthermore, no lattice or continuum models that confirm the second gradient elastic theory of Mindlin have been reported in the literature. The present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin. The first is based on the standard continualization of the equation of motion taken for a sequence of mass-spring lattices, while the second one exploits average processes valid in continuum mechanics. Furthermore, Mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro-structural parameter without however providing explicit expressions that correlate micro-structure with macro-structure. This is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell. [Copyright &y& Elsevier] |
| Copyright of International Journal of Solids & Structures is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 70043223 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Polyzos%2C+D%2E%22">Polyzos, D.</searchLink><relatesTo>1</relatesTo><i> demosthenes.polyzos@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Fotiadis%2C+D%2EI%2E%22">Fotiadis, D.I.</searchLink><relatesTo>2</relatesTo> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Solids+%26+Structures%22">International Journal of Solids & Structures</searchLink>. Feb2012, Vol. 49 Issue 3/4, p470-480. 11p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Mindlin+theory%22">Mindlin theory</searchLink><br /><searchLink fieldCode="DE" term="%22Physiologic+strain%22">Physiologic strain</searchLink><br /><searchLink fieldCode="DE" term="%22Lattice+theory%22">Lattice theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+continuum%22">Mathematical continuum</searchLink><br /><searchLink fieldCode="DE" term="%22Models+%26+modelmaking%22">Models & modelmaking</searchLink><br /><searchLink fieldCode="DE" term="%22Asymptotic+homogenization%22">Asymptotic homogenization</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Abstract: Mindlin, in his celebrated papers of Arch. Rat. Mech. Anal. 16, 51–78, 1964 and Int. J. Solids Struct. 1, 417–438, 1965, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro-structural effects. Since then, many works dealing with strain gradient elastic theories, derived either from lattice models or homogenization approaches, have appeared in the literature. Although elegant, none of them reproduces entirely the equation of motion as well as the classical and non-classical boundary conditions appearing in Mindlin theory, in terms of the considered lattice or continuum unit cell. Furthermore, no lattice or continuum models that confirm the second gradient elastic theory of Mindlin have been reported in the literature. The present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin. The first is based on the standard continualization of the equation of motion taken for a sequence of mass-spring lattices, while the second one exploits average processes valid in continuum mechanics. Furthermore, Mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro-structural parameter without however providing explicit expressions that correlate micro-structure with macro-structure. This is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell. [Copyright &y& Elsevier] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of International Journal of Solids & Structures is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.ijsolstr.2011.10.021 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 11 StartPage: 470 Subjects: – SubjectFull: Mindlin theory Type: general – SubjectFull: Physiologic strain Type: general – SubjectFull: Lattice theory Type: general – SubjectFull: Mathematical continuum Type: general – SubjectFull: Models & modelmaking Type: general – SubjectFull: Asymptotic homogenization Type: general Titles: – TitleFull: Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Polyzos, D. – PersonEntity: Name: NameFull: Fotiadis, D.I. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 02 Text: Feb2012 Type: published Y: 2012 Identifiers: – Type: issn-print Value: 00207683 Numbering: – Type: volume Value: 49 – Type: issue Value: 3/4 Titles: – TitleFull: International Journal of Solids & Structures Type: main |
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