Three-dimensional elastic body and a Naghdi type shell interaction modelling

doctoral thesis

• Title (croatian): Modeliranje međudjelovanja trodimenzionalnog elastičnog tijela i ljuske Naghdijevog tipa
• Author: Matko Ljulj
• Mentor: Josip Tambača (mentor)
• Committee member: Eduard Marušić-Paloka (predsjednik povjerenstva)
• Committee member: Bojan Žugec (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-07, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

In this thesis we analyse an interaction problem between two elastic three-dimensional bodies of which one is much thinner than the other. Furthermore, the thin body is assumed to be more stiff with elastic coefficients related to the small thickness parameter. We consider this problem in context of both linear and nonlinear elasticity. We analyse asymptotic behaviour of solutions of these problems and obtain different limit models in various regimes as thickness tends to zero. Furthermore, we propose both linear and nonlinear models given by three-dimensional elasticity equations and two–dimensional elastic plate equations as a boundary condition instead of the three–dimensional body equations that includes the thin body and investigate the asymptotic behaviour of the proposed model as the thickness parameter goes to zero. We prove that proposed model has the same asymptotic behaviour as firstly observed three–dimensional model that includes the thin body in the same regimes. By using proposed models, one can avoid numerical challenges that implementing a scheme for a three–dimensional model including a thin domain can cause as in similar situations. The nonlinear model we propose is based on a nonlinear shell model we formulate and it is an additional contribution of the thesis. It is a nonlinear shell model of Naghdi type defined for shells which allow Lipschitz–continuous parametrizations of its middle surface, so it also models shells with middle surfaces having corners and folded plates and shells. Energy of the formulated model contains membrane, shear, drill and flexural terms, and (in appropriate regimes) it has the same asymptotic behaviour as models rigorously justified in the literature: membrane shell model, flexural shell model and constrained membrane plate model.

Abstract (croatian)

U ovom radu analiziramo problem međudjelovanja dva elastična trodimenzionalna tijela od kojih je jedno mnogo tanje od drugoga. Nadalje, za tanko tijelo pretpostavljamo da je kruće, s koeficijentima elastičnosti ovisnima o parametru male vrijednosti koji predstavlja debljinu tankog tijela. Promatramo taj problem koristeći i linearnu i nelinearnu elastičnost. Analiziramo asimptotsko ponašanje rješenja tih problema i dobivamo različite granične modele u raznim režimima kada vrijednost tog parametra teži k nuli. Nadalje, predlažemo i linearan i nelinearan model opisan jednadžbama trodimenzionalne elastičnosti i jednadžbama elastične ploče kao rubnim uvjetom umjesto jednadžbama trodimenzionalnog tijela koje uključuje tanko 3d tijelo, te istražujemo asimptotska svojstva predloženog modela kada parametar debljine teži k nuli. Dokazujemo da predloženi model ima jednako asimptotsko ponašanje kao trodimenzionalni model koji uključuje tanko 3d tijelo u istim režimima. Koristeći novopredložene modele moguće je izbjeći poteškoće kod numeričke implementacije trodimenzionalnih modela koji uključuju tanku domenu. Predloženi nelinearni model baziran je na nelinearnom modelu elastične ljuske koji smo zadali i koji je dodatni doprinos ovog rada. To je nelinearan model ljuske Naghdijeva tipa definiran za ljuske kojima je središnja ploha parametrizirana Lipschitzovom funkcijom, pa također modelira i ljuske kojima središnja ploha sadrži kutove te presavinute ploče i ljuske. Energija u zadanom modelu sadrži membranske i fleksijske efekte te efekte smicanja i uvrtanja, te u odgovarajućem režimu ima ista asimptotska svojstva kao modeli koji su strogo izvedeni u literaturi: membranski model ljuske, fleksijski model ljuske i uvjetni membranski model ploče.

• Keywords: interaction model; nonlinear elasticity; linearised elasticity; thin structure; justification; Γ–convergence; shells; Naghdi model; Cosserat model; membrane model; flexural model; Koiter model
• Keywords (croatian): model interakcije; nelinearna elastičnost; linearizirana elastičnost; tanka struktura; rigorozno opravdanje; Γ–konvergencija; model ljuske; Naghdijev model; Cosseratov model; membranski model; fleksijski model; Koiterov model
• Language: english
• URN:NBN: urn:nbn:hr:217:661602
• Promotion: 2020
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: viii, 192 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-07-13 12:45:41