Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures
The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from...
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| Format: | Article |
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MDPI AG
2022-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/10/1670 |
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| author | Klaus Gürlebeck Dmitrii Legatiuk Kemmar Webber |
| author_facet | Klaus Gürlebeck Dmitrii Legatiuk Kemmar Webber |
| author_sort | Klaus Gürlebeck |
| collection | DOAJ |
| description | The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions. |
| first_indexed | 2024-03-10T03:29:05Z |
| format | Article |
| id | doaj.art-9128c21db9fb4d31a74bc57fe6ea12b0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:29:05Z |
| publishDate | 2022-05-01 |
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| spelling | doaj.art-9128c21db9fb4d31a74bc57fe6ea12b02023-11-23T12:00:43ZengMDPI AGMathematics2227-73902022-05-011010167010.3390/math10101670Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry StructuresKlaus Gürlebeck0Dmitrii Legatiuk1Kemmar Webber2Chair of Applied Mathematics, Bauhaus-Universität Weimar, 99423 Weimar, GermanyChair of Mathematics, Universität Erfurt, 99089 Erfurt, GermanyChair of Advanced Structures, Bauhaus-Universität Weimar, 99423 Weimar, GermanyThe solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions.https://www.mdpi.com/2227-7390/10/10/1670quaternionic analysismathematical modellingoperator calculusmodel comparisonmasonry structureselasticity theory |
| spellingShingle | Klaus Gürlebeck Dmitrii Legatiuk Kemmar Webber Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures Mathematics quaternionic analysis mathematical modelling operator calculus model comparison masonry structures elasticity theory |
| title | Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures |
| title_full | Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures |
| title_fullStr | Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures |
| title_full_unstemmed | Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures |
| title_short | Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures |
| title_sort | operator calculus approach to comparison of elasticity models for modelling of masonry structures |
| topic | quaternionic analysis mathematical modelling operator calculus model comparison masonry structures elasticity theory |
| url | https://www.mdpi.com/2227-7390/10/10/1670 |
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