Spectral analysis of thin heterogeneous elastic structures

doctoral thesis

• Title (croatian): Spektralna analiza tankih heterogenih elastičnih struktura
• Author: Josip Žubrinić
• Mentor: Igor Velčić (mentor)
• Committee member: Kirill Cherednichenko https://orcid.org/0000-0002-0998-7820 (predsjednik povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Igor Velčić (član povjerenstva)
• Committee member: Josipa-Pina Milišić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-05-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

This thesis consists of two parts. In the first part of the thesis, we analyse the behavior of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the resolvent equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, first we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes. In the second part of the thesis, we provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing the thin infinite elastic rod with material coefficients which periodically highly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the thickness of the rod and the period of material oscillations. These two parameters are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under some additional assumptions, as well as in the general case where these two sorts of displacements intertwine inseparably.

Abstract (croatian)

Ovaj rad sastoji se od dva dijela. U prvom dijelu rada analiziramo ponašanje tankih kompozitnih ploča čija svojstva materijala periodično variraju u ravnini i posjeduju visok stupanj kontrasta između pojedinih komponenti. Polazeći od rezolventnih jednadžbi trodimenzionalne linearne elastičnosti koje opisuju meke inkluzije ugrađene u relativno krutu matricu tanke ploče, izvodimo odgovarajuće asimptotski ekvivalentne jednadžbe dvodimenzionalne ploče. Naš pristup temelji se na nedavnim rezultatima o dekompoziciji deformacija s ograničenim simetriziranim gradijentima. Koristeći pristup teorije operatora, najprije izračunavamo limes rezolventu te analiziramo pridruženi limes spektar i efektivne evolucijske jednadžbe. Naše rezultate dobivamo pod različitim asimptotičkim odnosima između veličine mekih inkluzija (perioda oscilacija) i debljine ploče, kao i pod različitim kombinacijama skaliranja između kontrasta, spektra i vremena. Također pokazujemo značajne kvalitativne razlike između asimptotskih modela dobivenih u različitim režimima. U drugom dijelu rada izvodimo asimptotiku rezolventi kao i razne ocjene u operatorskim normama za sustav linearnih parcijalnih diferencijalnih jednadžbi koje opisuju tanki beskonačni elastični štap s materijalnim koeficijentima koji periodično jako osciliraju duž stapa. Rezolventnu asimptotiku izvodimo simultano s obzirom na debljinu štapa i period oscilacija materijala. Uzimamo da su ova dva parametra istog reda. Analizu provodimo zasebno na dva invarijantna podprostora koji se odnose na pomake duž prostiranja stapa i pomake okomite na prostiranje štapa, pri čemu pretpostavljamo neke dodatne pretpostavke. Također provodimo analizu i u općem slučaju kada se ove dvije vrste pomaka neraskidivo isprepliću.

• Keywords: Homogenisation; Dimension reduction; Two-scale convergence; Highcontrast; Resolvent asymptotics; Elastic heterogeneous rods and plates
• Keywords (croatian): Homogenizacija; Redukcija dimenzije; ·Dvoskalna konvergencija; Visoki kontrast; Rezolventna asimptotika; Elastični heterogeni štapovi i ploče
• Language: english
• URN:NBN: urn:nbn:hr:217:330220
• Promotion: 2022
• 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: v, 225 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-06-14 09:33:16