On the Stability of Objective Structures
The main focus of this thesis is the discussion of stability of an objective (atomic) structure consisting of single atoms which interact via a potential. We define atomistic stability using a second derivative test. More precisely, atomistic stability is equivalent to a vanishing first derivative o...
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| Format: | Online |
| Język: | angielski |
| Wydane: |
Logos Verlag Berlin
2021
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| Hasła przedmiotowe: | |
| Dostęp online: | ONIX_20211220_9783832553784_11 |
| Etykiety: |
Nie ma etykietki, Dołącz pierwszą etykiete!
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| _version_ | 1869518633161457664 |
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| author | Steinbach, Martin |
| author_browse | Steinbach, Martin |
| author_facet | Steinbach, Martin |
| author_sort | Steinbach, Martin |
| collection | Directory of Open Access Books |
| description | The main focus of this thesis is the discussion of stability of an objective (atomic) structure consisting of single atoms which interact via a potential. We define atomistic stability using a second derivative test. More precisely, atomistic stability is equivalent to a vanishing first derivative of the configurational energy (at the corresponding point) and the coerciveness of the second derivative of the configurational energy with respect to an appropriate semi-norm. Atomistic stability of a lattice is well understood, see, e. ,g., [40]. The aim of this thesis is to generalize the theory to objective structures. In particular, we first investigate discrete subgroups of the Euclidean group, then define an appropriate seminorm and the atomistic stability for a given objective structure, and finally provide an efficient algorithm to check its atomistic stability. The algorithm particularly checks the validity of the Cauchy-Born rule for objective structures. To illustrate our results, we prove numerically the stability of a carbon nanotube by applying the algorithm. |
| format | Online |
| id | doab-20.500.12854ir-75074 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Logos Verlag Berlin |
| publisherStr | Logos Verlag Berlin |
| record_format | ojs |
| spelling | doab-20.500.12854ir-750742024-04-04T19:18:37Z On the Stability of Objective Structures Steinbach, Martin Mathematical model Elasticity theory Stability theory Objective structure Discrete subgroup of the Euclidean group thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PH Physics::PHU Mathematical physics The main focus of this thesis is the discussion of stability of an objective (atomic) structure consisting of single atoms which interact via a potential. We define atomistic stability using a second derivative test. More precisely, atomistic stability is equivalent to a vanishing first derivative of the configurational energy (at the corresponding point) and the coerciveness of the second derivative of the configurational energy with respect to an appropriate semi-norm. Atomistic stability of a lattice is well understood, see, e. ,g., [40]. The aim of this thesis is to generalize the theory to objective structures. In particular, we first investigate discrete subgroups of the Euclidean group, then define an appropriate seminorm and the atomistic stability for a given objective structure, and finally provide an efficient algorithm to check its atomistic stability. The algorithm particularly checks the validity of the Cauchy-Born rule for objective structures. To illustrate our results, we prove numerically the stability of a carbon nanotube by applying the algorithm. 2021-12-20T15:23:16Z 2021-12-20T15:23:16Z 2021 book ONIX_20211220_9783832553784_11 9783832553784 https://directory.doabooks.org/handle/20.500.12854/75074 eng Augsburger Schriften zur Mathematik, Physik und Informatik image/jpeg Attribution-NonCommercial-NoDerivatives 4.0 International https://www.logos-verlag.de/ebooks/OA/978-3-8325-5378-4.pdf Logos Verlag Berlin Logos Verlag Berlin 10.30819/5378 10.30819/5378 04b263a1-7fba-4491-9eae-1c394ac42fc3 9783832553784 Logos Verlag Berlin 38 163 Berlin open access |
| spellingShingle | Mathematical model Elasticity theory Stability theory Objective structure Discrete subgroup of the Euclidean group thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PH Physics::PHU Mathematical physics Steinbach, Martin On the Stability of Objective Structures |
| title | On the Stability of Objective Structures |
| title_full | On the Stability of Objective Structures |
| title_fullStr | On the Stability of Objective Structures |
| title_full_unstemmed | On the Stability of Objective Structures |
| title_short | On the Stability of Objective Structures |
| title_sort | on the stability of objective structures |
| topic | Mathematical model Elasticity theory Stability theory Objective structure Discrete subgroup of the Euclidean group thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PH Physics::PHU Mathematical physics |
| topic_facet | Mathematical model Elasticity theory Stability theory Objective structure Discrete subgroup of the Euclidean group thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling thema EDItEUR::P Mathematics and Science::PH Physics::PHU Mathematical physics |
| url | ONIX_20211220_9783832553784_11 |
| work_keys_str_mv | AT steinbachmartin onthestabilityofobjectivestructures |