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

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.