Symmetric unisolvent equations for linear elasticity purely in stresses.
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| Title: | Symmetric unisolvent equations for linear elasticity purely in stresses. |
|---|---|
| Authors: | Sky, Adam1 (AUTHOR) adam.sky@uni.lu, Zilian, Andreas1 (AUTHOR) andreas.zilian@uni.lu |
| Source: | International Journal of Solids & Structures. Jun2024, Vol. 295, pN.PAG-N.PAG. 1p. |
| Subjects: | Linear equations, Differential equations, Boundary value problems, Differential operators, Elasticity, Functional analysis |
| Abstract: | In this work we introduce novel stress-only formulations of linear elasticity with special attention to their approximate solution using weighted residual methods. We present four sets of boundary value problems for a pure stress formulation of three-dimensional solids, and in two dimensions for plane stress and plane strain. The associated governing equations are derived by modifications and combinations of the Beltrami–Michell equations and the Navier–Cauchy equations. The corresponding variational forms of dimension d ∈ { 2 , 3 } allow to approximate the stress tensor directly, without any presupposed potential stress functions, and are shown to be well-posed in H 1 ⊗ Sym (d) in the framework of functional analysis via the Lax–Milgram theorem, making their finite element implementation using C 0 -continuous elements straightforward. Further, in the finite element setting we provide a treatment for constant and piece-wise constant body forces via distributions. The operators and differential identities in this work are provided in modern tensor notation and rely on exact sequences, making the resulting equations and differential relations directly comprehensible. Finally, numerical benchmarks for convergence as well as spectral analysis are used to test the limits and identify viable use-cases of the equations. • New well-posed symmetric variational forms for linear elasticity purely in stresses. • Novel boundary value problems for plane stress and plane strain purely in stresses. • Well-posedness proofs and stabilisation of the planar variational forms. • Treatment of constant and piece-wise constant body forces via distributions. • Numerical convergence benchmarks and spectral analysis of the discretised forms. [ABSTRACT FROM AUTHOR] |
| 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: 176720166 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Symmetric unisolvent equations for linear elasticity purely in stresses. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Sky%2C+Adam%22">Sky, Adam</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> adam.sky@uni.lu</i><br /><searchLink fieldCode="AR" term="%22Zilian%2C+Andreas%22">Zilian, Andreas</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> andreas.zilian@uni.lu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Solids+%26+Structures%22">International Journal of Solids & Structures</searchLink>. Jun2024, Vol. 295, pN.PAG-N.PAG. 1p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Linear+equations%22">Linear equations</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+equations%22">Differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+operators%22">Differential operators</searchLink><br /><searchLink fieldCode="DE" term="%22Elasticity%22">Elasticity</searchLink><br /><searchLink fieldCode="DE" term="%22Functional+analysis%22">Functional analysis</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this work we introduce novel stress-only formulations of linear elasticity with special attention to their approximate solution using weighted residual methods. We present four sets of boundary value problems for a pure stress formulation of three-dimensional solids, and in two dimensions for plane stress and plane strain. The associated governing equations are derived by modifications and combinations of the Beltrami–Michell equations and the Navier–Cauchy equations. The corresponding variational forms of dimension d ∈ { 2 , 3 } allow to approximate the stress tensor directly, without any presupposed potential stress functions, and are shown to be well-posed in H 1 ⊗ Sym (d) in the framework of functional analysis via the Lax–Milgram theorem, making their finite element implementation using C 0 -continuous elements straightforward. Further, in the finite element setting we provide a treatment for constant and piece-wise constant body forces via distributions. The operators and differential identities in this work are provided in modern tensor notation and rely on exact sequences, making the resulting equations and differential relations directly comprehensible. Finally, numerical benchmarks for convergence as well as spectral analysis are used to test the limits and identify viable use-cases of the equations. • New well-posed symmetric variational forms for linear elasticity purely in stresses. • Novel boundary value problems for plane stress and plane strain purely in stresses. • Well-posedness proofs and stabilisation of the planar variational forms. • Treatment of constant and piece-wise constant body forces via distributions. • Numerical convergence benchmarks and spectral analysis of the discretised forms. [ABSTRACT FROM AUTHOR] – 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.2024.112808 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 1 StartPage: N.PAG Subjects: – SubjectFull: Linear equations Type: general – SubjectFull: Differential equations Type: general – SubjectFull: Boundary value problems Type: general – SubjectFull: Differential operators Type: general – SubjectFull: Elasticity Type: general – SubjectFull: Functional analysis Type: general Titles: – TitleFull: Symmetric unisolvent equations for linear elasticity purely in stresses. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Sky, Adam – PersonEntity: Name: NameFull: Zilian, Andreas IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2024 Type: published Y: 2024 Identifiers: – Type: issn-print Value: 00207683 Numbering: – Type: volume Value: 295 Titles: – TitleFull: International Journal of Solids & Structures Type: main |
| ResultId | 1 |
