Reflection on 'finite rotation problem' in plate and shell theories and in finite element formulation - Back to basics

Li, Shuguang

Abstract

This paper results from a reflection on the problem of finite rotations in plates and shells and its presentation in finite element formulation, a well-attended subject with a wide diversity of treatments, often extremely complicated. For instance, the very topic attracted 3 out of total 15 chapters over 120 out of total 494 pages in a monograph on nonlinear FEA [5]. In another instance, a 71 page journal paper was published [2] specifically on this topic alone. Review papers on this matter can be found in various sources including Appl Mech Rev, (e.g. Ibrahimbegovic, 1997 [8]). No attempt is to be made on reviewing these sophisticated approaches but the present paper will bring answers to questions, if they have ever been asked, such as whether the sophistication in existing approaches is really necessary and whether there is a much simpler and conceptually more direct and accurate approach. A proper re-examination of the existing approaches would reveal a fundamental inconsistency: rigid body rotations have been assumed without reservation to describe the deformations of deformable plates and shells. After re-establishing the consistency based on the very basics of conventional plate and shell theories as a simple reflection, one can conclude surprisingly that the whole issue on ‘finite rotations’ results from a logic fallacy of faulty generalisation. The so-called ‘rotations’ should be displacement gradients instead. They can be considered as ‘rotations’ as conventionally perceived hitherto under the condition of small deformation. Somehow, the concept of ‘rotations’ got generalised regardless the magnitude of the deformation. Instead of calling the problem as ‘finite displacement gradients’ as it should be called, a falsely generalised term of ‘finite rotations’ have been used. Since they were called ‘rotations’, the definition of those of rigid body kinematics has been taken for granted. Finite displacement gradients should not present any additional problem apart from introducing nonlinear nature into the problem, which can be addressed as a geometrically nonlinear problem in a conventional manner. However, along the line of a falsely generalised concept of finite rigid body rotations, complications have been the norm. The complicated accounts on the ‘finite rotation’ problem in the literature, which might have enhanced the understanding of rigid body kinematics, are entirely unnecessary as far as the deformable plates and shells are concerned, involving infinitesimal or finite deformations.

• Journal Article Type: Article
• Acceptance Date: Jul 14, 2014
• Online Publication Date: Jul 23, 2014
• Publication Date: 2015-02
• Deposit Date: Jul 11, 2018
• Journal: International Journal of Mechanical Sciences
• Print ISSN: 0020-7403
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 91
• Pages: 12-17
• DOI: https://doi.org/10.1016/j.ijmecsci.2014.07.004
• Public URL: http://www.scopus.com/inward/record.url?eid=2-s2.0-84923864394&partnerID=40&md5=1c17c47b8c20a31e7f38e7b98bf18855
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0020740314002471?via%3Dihub