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

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.

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