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On the Stability of Objective Structures
Martin Steinbach
2021
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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.
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Keywords
- applied mathematics
- Discrete subgroup of the Euclidean group
- Elasticity theory
- Mathematical model
- mathematical modelling
- Mathematical physics
- Mathematics
- Mathematics & science
- Objective structure
- Physics
- Stability Theory
- 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
Links
DOI: 10.30819/5378Editions