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

Walentyński Ryszard

Computer Assisted Methods in Engineering and Science

2019

vol. 26, no. 3-4

191--209

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.

Experimental and numerical investigation of the effect of the combined mechanism of circumferential expansion and folding on energy absorption parameters

Abolfathi M., Nia A. A., Attar A. A., Abbasi M.

Archives of Civil and Mechanical Engineering

2018

Vol. 18, no. 4

1464--1477

In this research, in order to increase energy absorption of thin-walled tubes, a combined deformation mechanism is proposed which involves the simultaneous combination of circumferential expansion and folding. Such a combined mechanism was not concerned in the literature. The study is carried out both experimentally and numerically. A special device was designed and made to conduct experimental tests on tubes. The samples were made of aluminum, and quasi-static loading was applied at two different speeds of 10 and 200 mm/min. Energy absorption parameters including specific energy absorption (SEA), crushing mean force, initial peak force, the deformation mode and crush force efficiency (CFE) were studied. Experimental results showed that combined mechanism (without lubrication) could increase absorbed energy up to 123% compared to the folding mechanism. If the lubricant is used, the increase will be up to 97%. The combined deformation mechanism (without lubrication) increases absorbed energy up to 94% compared to the circumstantial expansion. This value will be 107% with lubrication. In addition, the initial peak force in the combined mechanism decreases between 8% and 36% relative to the folding mechanism. The circumstantial expansion in the proposed mechanism is complete and the expansion stroke length is 100%, while this stroke was less in the previous researches due to design restrictions. Numerical simulations were conducted using LS-Dyna software and there is good agreement between the numerical results and experimental data.