# Description of large deformations of continuum and shells and their visualisation with Mathematica

• Ryszard Walentyński Silesian University of Technology

### Abstract

A proper description of large deformation of continuum or shell requires dealing with curved spaces and application of tensor analysis and distinguishing of covariant and contravariant bases. Thanks to symbolic computations and visualization capabilities of the Mathematica system, this task can be carried out in a straightforward manner. This has been already discussed in [9] and [10]. This paper is a further extension of these researches. First, it will be shown that the deformation is indeed changing a curvature of the considered space. Next, there will be shown how the Cartesian basis of the undeformed flat space splits into the covariant and contravariant ones and this basis changes in the space. This makes it possible to explain why we have to introduce covariant derivatives and Christoffel symbols, for example. This is important in the case of the optical analysis of large deformations of thin-wall structures. Moreover, it is possible to easily explain that strain tensor is defined with a change of metric tensor. It also helps to show the 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 and interactive manipulation of the plots provided within the Mathematica system. This paper can also be a 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.

### Keywords

continuum mechanics, theory of shells, Mathematica, tensor analysis, thin-wall structures,

### References

[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. Birkhauser Basel, 1st ed., 1999, https://doi.org/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.
Published
Dec 31, 2019
How to Cite
WALENTYŃSKI, Ryszard. Description of large deformations of continuum and shells and their visualisation with Mathematica. Computer Assisted Methods in Engineering and Science, [S.l.], v. 26, n. 3–4, p. 191–209, dec. 2019. ISSN 2956-5839. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/272>. Date accessed: 14 sep. 2024. doi: http://dx.doi.org/10.24423/cames.272.
Citation Formats
Issue
Section
Articles