Description of large deformations of continuum and shellsand their visualisation with Mathematica

• Autorzy: Walentyński Ryszard

Identyfikatory

DOI

10.24423/cames.272

• Języki publikacji: EN

Abstrakty

EN

A proper description of large deformation of continuum or shell requires dealing with curved spacesand application of tensor analysis and distinguishing of covariant and contravariant bases. Thanks tosymbolic computations and visualization capabilities of theMathematicasystem, this task can be carriedout in a straightforward manner. This has been already discussed in [9] and [10]. This paper is a furtherextension of these researches. First, it will be shown that the deformation is indeed changing a curvatureof the considered space. Next, there will be shown how the Cartesian basis of the undeformed flat spacesplits into the covariant and contravariant ones and this basis changes in the space. This makes it possibleto explain why we have to introduce covariant derivatives and Christoffel symbols, for example. This isimportant in the case of the optical analysis of large deformations of thin-wall structures. Moreover, itis possible to easily explain that strain tensor is defined with a change of metric tensor. It also helps to showthe idea of material (Lagrangian) and spatial (Eulerian) description of the deformation and the motion,and avoid misunderstandings in this matter. Everything is visualised with 3D graphical capabilities andinteractive manipulation of the plots provided within theMathematicasystem. This paper can also bea useful inspiration both in teaching and learning of continuum mechanics, the theory of shells and thin-wall structures. This work has been presented at the conference “4th Polish Congress of Mechanics, 23rd International Conference on Computer Methods in Mechanics” PCM-CMM-2019 in Kraków.

Słowa kluczowe

EN

continuum mechanics; theory of shells; tensor analysis; thin wall structure

PL

mechanika kontinuum; teoria powłok; analiza tensorowa; struktura cienkościenna

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Methods in Engineering and Science
• Rocznik: 2019
• Tom: vol. 26, no. 3-4
• Strony: 191--209
• Opis fizyczny: Bibliogr. 11 poz., rys.

Twórcy

autor

Walentyński Ryszard

• Silesian University of Technology Faculty of Civil Engineering Department of Mechanics and Bridges Akademicka 5, 44-100 Gliwice, Poland

Bibliografia

• [1] S. Bielak. Theory of Shells. Part II, Theory and Applications [in Polish]. Civil Engineering. Silesian University of Technology, 2nd ed., 1993.
• [2] S.P. Kiselev, E.V. Vorozhtsov, V.M. Fomin. Foundations of Fluid Mechanics with Applications. Problem Solving Using Mathematica . Modelling and Simulation in Science, Engineering and Technology. Birkhäuser Basel, 1st ed., 1999, doi: 10.1007/978-1-4612-1572-1.
• 3] J. Kuczmarski. Cells to TeX, Jan 2019. https://github.com/jkuczm/MathematicaCellsToTeX.
• [4] J. Kuczmarski. Mathematica cells in TeX, January 2017. https://github.com/jkuczm/mmacells.
• [5] S. McManus, M. Cook. Raspberry Pi for Dummies . John Wiley & Sons, 3rd ed., 2017.
• [6] L. Parker, S.M. Christensen. MathTensor: A System for Doing Tensor analysis by Computer . Addison- Wesley, 1994.
• [7] UpSkill Learning. Raspberry Pi 3: Get Started with Raspberry Pi 3 a Simple Guide to Understanding and Programming Raspberry Pi 3 (Raspberry Pi 3 User Guide, Python Programming, Mathematica Programming) . CreateSpace Independent Publishing Platform, 2016.
• [8] R. Walentyński. Application of computer algebra in symbolic computation and boundary-value problems of the theory of shells. Zeszyty Naukowe. Budownictwo , 100 : 13–198, Silesian University of Technology, 2003.
• [9] R. Walentyński. Description and visualization of large deformations of continuum with Mathematica . In: Proceedings of 12th International Mathematica Symposium (IMS 2015) , Prague, January 12–14, 2015.
• [10] R. Walentyński. Lecturing continuum mechanics with Mathematica . In: Computer Algebra Systems in Teaching and Research, Mathematical Modelling in Physics, Civil Engineering, Economics and Finance , L. Gadomski et al. [Eds], pp. 184–193. Collegium Mazovia, Siedlce, 2011.
• [11] S. Wolfram. An Elementary Introduction to the Wolfram Language. Wolfram Media, Inc., 2015. Available online.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).