Open AccessNotes on the possibilities and limitations of using topological tools in mechanical modeling of media
Author(s) -
Géza Lámer
Publication year - 2019
Publication title -
iop conference series. materials science and engineering
Language(s) - English
Resource type - Journals
eISSN - 1757-899X
pISSN - 1757-8981
DOI - 10.1088/1757-899x/568/1/012057
Subject(s) - basis (linear algebra) , grid , position (finance) , topology (electrical circuits) , sort , topological space , computer science , set (abstract data type) , mathematics , pure mathematics , geometry , finance , combinatorics , economics , information retrieval , programming language
In the mechanical modeling of the media, the atomic-molecular structure of the material is usually neglected, furthermore, with regard to the characteristic distance of the atomic-molecular structure, the media is considered to be a continuous material distribution. Referring to the continuous distribution, we use the tools of differential and integral calculus for the mechanical-mathematical description of the media. The use of these mathematical tools has conditions and limitations: there must be a sort of topological order in the examined space. This order records the relative position of the points. The fixed position relative to each other makes it possible to describe only such mappings that retain the fixed position relative to one another. Thus, it is possible to distinguish between strain and rearrangement, and at the same time to define the boundaries of the possibilities offered by the applied mathematical tools: the continuum many points set provides the basis of the topological description for the classical continuum, a set of finite points in the grid network with its internal structural features, provides the basis of the topological description for the grid (that is generalized) continuum.