Differential Geometry of Surfaces with Application to Shell Structures

authored by: Shahab Sahraee, Meisam Soleimani
Abstract: This work deals with infinitesimal deformations of an isotropic elastic shell in a differential geometry framework. It provides an introduction to differential geometry of embedded surfaces in the Euclidean three-dimensional space and serves as a tangible and practical “recipe” for those, particularly students, who are interested in invoking the old art of working with curvilinear coordinates. It seems that the fundamental concepts in this field have remained obscure and vague for many students and they do not dare employ this elegant tool. With regard to this, a shell structure composed of only a single element is taken into consideration represented geometrically exact in a curvilinear setting. The stress/strain are computed at the Gauss points using the framework of differential geometry and the results are validated against ones predicted by ANSYS as a commercial software. One should notice that, nowadays, this approach is not implemented in commercial software such as ANSYS due to its complications. Rather, such software prefers to use the so-called solid degenerated shell elements. Using such approach, one can circumvent dealing directly with objects such as Christoffel symbols which naturally emerge as soon as the derivative are taken in a curvilinear coordinate system.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Guilan University
Type: Contribution to book/anthology
Pages: 463-474
No. of pages: 12
Publication date: 13.03.2022
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Engineering(all), Computer Science(all)
Electronic version(s): https://doi.org/10.1007/978-3-030-87312-7_45 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@inbook{bdbef5b8bf574257ac7fd84d6ccd0414,
title = "Differential Geometry of Surfaces with Application to Shell Structures",
abstract = "This work deals with infinitesimal deformations of an isotropic elastic shell in a differential geometry framework. It provides an introduction to differential geometry of embedded surfaces in the Euclidean three-dimensional space and serves as a tangible and practical “recipe” for those, particularly students, who are interested in invoking the old art of working with curvilinear coordinates. It seems that the fundamental concepts in this field have remained obscure and vague for many students and they do not dare employ this elegant tool. With regard to this, a shell structure composed of only a single element is taken into consideration represented geometrically exact in a curvilinear setting. The stress/strain are computed at the Gauss points using the framework of differential geometry and the results are validated against ones predicted by ANSYS as a commercial software. One should notice that, nowadays, this approach is not implemented in commercial software such as ANSYS due to its complications. Rather, such software prefers to use the so-called solid degenerated shell elements. Using such approach, one can circumvent dealing directly with objects such as Christoffel symbols which naturally emerge as soon as the derivative are taken in a curvilinear coordinate system.",
author = "Shahab Sahraee and Meisam Soleimani",
year = "2022",
month = mar,
day = "13",
doi = "10.1007/978-3-030-87312-7_45",
language = "English",
isbn = "9783030873110",
pages = "463--474",
booktitle = "Current Trends and Open Problems in Computational Mechanics",
publisher = "Springer International Publishing AG",
address = "Switzerland",
}