Stochastic modelling of elasticity tensor fields.
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| Title: | Stochastic modelling of elasticity tensor fields. |
|---|---|
| Authors: | Shivanand, Sharana Kumar1 (AUTHOR) sharana.shivanand@kit.edu, Rosić, Bojana2 (AUTHOR), Matthies, Hermann G.3 (AUTHOR) |
| Source: | Mathematics & Mechanics of Solids. Jul2026, Vol. 31 Issue 7, p1500-1549. 50p. |
| Subjects: | Tensor fields, Lie algebras, Stochastic models, Rotational symmetry, Tensor algebra, Differentiable manifolds, Mechanical behavior of materials |
| Abstract: | We present a novel framework for the probabilistic modelling of random fourth-order material tensor fields, with a focus on tensors that are physically symmetric and positive definite (SPD), of which the elasticity tensor is a prime example. Given the critical role that spatial symmetries and invariances play in determining material behaviour, it is essential to incorporate these aspects into the probabilistic description and modelling of material properties. In particular, we focus on spatial point symmetries or invariances under rotations, a classical subject in elasticity. Following this, we formulate a stochastic modelling framework using a Lie algebra representation via a memory-less transformation that respects the requirements of positive definiteness and invariance. With this, it is shown how to generate a random ensemble of elasticity tensors that allows an independent control of strength, eigenstrain, and orientation. The procedure also accommodates the requirement to prescribe specific spatial symmetries and invariances for each member of the whole ensemble, while ensuring that the mean or expected value of the ensemble conforms to a potentially 'higher' class of spatial invariance. Furthermore, it is important to highlight that the set of SPD tensors forms a differentiable manifold, which geometrically corresponds to an open cone within the ambient space of symmetric tensors. Thus, we explore the mathematical structure of the underlying sample space of such tensors and introduce a new distance measure or metric, called the 'elasticity metric', between the tensors. Finally, we model and visualize a one-dimensional spatial field of orthotropic Kelvin matrices using interpolation based on the elasticity metric. [ABSTRACT FROM AUTHOR] |
| Copyright of Mathematics & Mechanics of Solids is the property of Sage Publications Inc. 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: 194777725 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Stochastic modelling of elasticity tensor fields. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Shivanand%2C+Sharana+Kumar%22">Shivanand, Sharana Kumar</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> sharana.shivanand@kit.edu</i><br /><searchLink fieldCode="AR" term="%22Rosić%2C+Bojana%22">Rosić, Bojana</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Matthies%2C+Hermann+G%2E%22">Matthies, Hermann G.</searchLink><relatesTo>3</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Mathematics+%26+Mechanics+of+Solids%22">Mathematics & Mechanics of Solids</searchLink>. Jul2026, Vol. 31 Issue 7, p1500-1549. 50p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Tensor+fields%22">Tensor fields</searchLink><br /><searchLink fieldCode="DE" term="%22Lie+algebras%22">Lie algebras</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+models%22">Stochastic models</searchLink><br /><searchLink fieldCode="DE" term="%22Rotational+symmetry%22">Rotational symmetry</searchLink><br /><searchLink fieldCode="DE" term="%22Tensor+algebra%22">Tensor algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Differentiable+manifolds%22">Differentiable manifolds</searchLink><br /><searchLink fieldCode="DE" term="%22Mechanical+behavior+of+materials%22">Mechanical behavior of materials</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We present a novel framework for the probabilistic modelling of random fourth-order material tensor fields, with a focus on tensors that are physically symmetric and positive definite (SPD), of which the elasticity tensor is a prime example. Given the critical role that spatial symmetries and invariances play in determining material behaviour, it is essential to incorporate these aspects into the probabilistic description and modelling of material properties. In particular, we focus on spatial point symmetries or invariances under rotations, a classical subject in elasticity. Following this, we formulate a stochastic modelling framework using a Lie algebra representation via a memory-less transformation that respects the requirements of positive definiteness and invariance. With this, it is shown how to generate a random ensemble of elasticity tensors that allows an independent control of strength, eigenstrain, and orientation. The procedure also accommodates the requirement to prescribe specific spatial symmetries and invariances for each member of the whole ensemble, while ensuring that the mean or expected value of the ensemble conforms to a potentially 'higher' class of spatial invariance. Furthermore, it is important to highlight that the set of SPD tensors forms a differentiable manifold, which geometrically corresponds to an open cone within the ambient space of symmetric tensors. Thus, we explore the mathematical structure of the underlying sample space of such tensors and introduce a new distance measure or metric, called the 'elasticity metric', between the tensors. Finally, we model and visualize a one-dimensional spatial field of orthotropic Kelvin matrices using interpolation based on the elasticity metric. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Mathematics & Mechanics of Solids is the property of Sage Publications Inc. 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: 10.1177/10812865251348030 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 50 StartPage: 1500 Subjects: – SubjectFull: Tensor fields Type: general – SubjectFull: Lie algebras Type: general – SubjectFull: Stochastic models Type: general – SubjectFull: Rotational symmetry Type: general – SubjectFull: Tensor algebra Type: general – SubjectFull: Differentiable manifolds Type: general – SubjectFull: Mechanical behavior of materials Type: general Titles: – TitleFull: Stochastic modelling of elasticity tensor fields. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Shivanand, Sharana Kumar – PersonEntity: Name: NameFull: Rosić, Bojana – PersonEntity: Name: NameFull: Matthies, Hermann G. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 07 Text: Jul2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 10812865 Numbering: – Type: volume Value: 31 – Type: issue Value: 7 Titles: – TitleFull: Mathematics & Mechanics of Solids Type: main |
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