Large deformations description of the continuum, shells and thin-wall structures and their visualisation with Mathematica

Proper description of large deformation of the continuum or the shell requires dealing with curved spaces and application of tensor analysis and distinguishing of covariant and contravariant basis. Thanks to symbolic computations and visualization capabilities of the Mathematica system, this task can be carried out straightfor- ward. It has been already discussed in [1] and [2]. This paper is a further extension of them. First it will be shown that the deformation is indeed changing of curvature of the considered space. Next, there will be shown how Cartesian basis of the flat undeformed space splits to the covariant and contravariant ones and that this basis changes in the space. It makes possible to explain why we have to introduce covariant derivatives and Christoffel symbols, for example. It is important in 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 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 maniputation of the plots provided within the Mathematica system. This paper can be also usefull inspiration both in teaching and learning of the Continuum Mechanics, the Theory of Shells and Thin-Wall Structures. The work has been presented at the conference Polish Congress of Mechanics, Computer Methods in Mechanics PCM-CMM-2019 in Krakow.