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Mappings of Finite Distortion for Nonlinear Solid Mechanics

Mappings of Finite Distortion for Nonlinear Solid Mechanics

Anastasia Molchanova (ORCID: 0000-0003-4121-2192)
  • Grant DOI 10.55776/M2670
  • Funding program Lise Meitner
  • Status ended
  • Start June 1, 2019
  • End April 30, 2021
  • Funding amount € 159,340
  • Project website
  • E-mail

Disciplines

Mechanical Engineering (45%); Mathematics (55%)

Keywords

    Quasiconformal Analysis, Mapping Of Finite Distortion, Nonlinear Elasticity

Abstract Final report

The project Mappings of finite distortion for Nonlinear Solid Mechanics focuses on the mathematical analysis of the deformation of solids under prescribed forces, in the context of mappings of finite distortion. These are a special class of mappings which provide a favorable alternative to those normally used for describing elastic deformations, as they directly encode much of the underlying physics. As the project promotes an unconventional viewpoint on elasticity problems, it will foster the development of continuum mechanics in a novel way, as well as ensure a deeper understanding of geometrical and analytical properties of mappings with finite distortion. The goals of the project include identifying classes of nonlinear elastic materials explicitly calling for finite-distortion formulations. The proposed research targets a family of novel problems appearing at the interface of geometric analysis and mechanics. It will therefore be necessary to borrow and combine techniques from these fields. To be more specific, modern and classical techniques, as well as established results from quasiconformal analysis, from the calculus of variations, and from the theory of partial differential equations will be used. In particular, analytical and geometrical properties of mappings will be investigated mainly by means of the theory of mappings of finite distortion. Initially, the project will be guided by a novel approach, developed recently by the applicant and Prof. Vodopyanov. The method is based on the theory of mappings inducing the boundedness of the composition operator. This approach provides a novel operator-theory perspective in geometrical and analytical issues. The primary focus will be on analytical and geometrical problems associated with the modelling of various materials involving a mixed Eulerian and Lagrangian formulation, such as magnetoelastic materials and nematic elastomers, and much attention will be paid to the physical interpretation of mathematical properties related to the structures considered.

The project "Mappings with finite distortion for Nonlinear Solid Mechanics" focused on the mathematical analysis of the deformation of solids under prescribed forces, in the context of mappings of finite distortion. These are a special class of mappings which provide a favourable alternative to those normally used for describing elastic deformations, as they directly encode much of the underlying physics. An unconventional viewpoint on elasticity problems, promoted in the project, yielded to the development of continuum mechanics in a novel way, as well as a deeper understanding of geometrical and analytical properties of mappings with finite distortion. The main outcomes of the project are a study of the injectivity proper ties of limits of Sobolev homeomorphisms, a mathematical model of charged deformable materials and the proof of the existence of their equilibria, an analysis of the regularity of the inverse of a bilipschitz mapping belonging to a given Banach function space, the pointwise characterisation of Sobolev spaces defined on Banach lattices and an extended variational theory of evolution equations by means of modular spaces.

Research institution(s)
  • Technische Universität Wien - 100%
International project participants
  • Sergey Vodopyanov, Siberian Branch of the Russion Academy of Sciences - Russia

Research Output

  • 66 Citations
  • 14 Publications
  • 9 Scientific Awards
Publications
  • 2022
    Title Equilibria of Charged Hyperelastic Solids
    DOI 10.1137/21m1413286
    Type Journal Article
    Author Davoli E
    Journal SIAM Journal on Mathematical Analysis
    Pages 1470-1487
    Link Publication
  • 2021
    Title Equilibria of charged hyperelastic solids
    DOI 10.48550/arxiv.2104.08079
    Type Preprint
    Author Davoli E
  • 2021
    Title Equilibria of charged hyperelastic solids
    Type Other
    Author Davoli E
    Link Publication
  • 2020
    Title An extended variational theory for nonlinear evolution equations via modular spaces
    Type Other
    Author Menovschikov A
    Link Publication
  • 2020
    Title An extended variational theory for nonlinear evolution equations via modular spaces
    DOI 10.48550/arxiv.2012.05518
    Type Preprint
    Author Menovschikov A
  • 2021
    Title Regularity of the inverse mapping in Banach function spaces
    DOI 10.1002/mana.201900374
    Type Journal Article
    Author Molchanova A
    Journal Mathematische Nachrichten
    Pages 2382-2395
    Link Publication
  • 2019
    Title Injectivity almost everywhere for weak limits of Sobolev homeomorphisms
    DOI 10.48550/arxiv.1912.05413
    Type Preprint
    Author Bouchala O
  • 2020
    Title On grand Sobolev spaces and pointwise description of Banach function spaces
    DOI 10.48550/arxiv.2004.12712
    Type Preprint
    Author Jain P
  • 2020
    Title The Routledge Companion to Digital Humanities and Art History
    DOI 10.4324/9780429505188
    Type Book
    Publisher Taylor & Francis
    Link Publication
  • 2020
    Title Injectivity almost everywhere for weak limits of Sobolev homeomorphisms
    DOI 10.1016/j.jfa.2020.108658
    Type Journal Article
    Author Bouchala O
    Journal Journal of Functional Analysis
    Pages 108658
    Link Publication
  • 2021
    Title An Extended Variational Theory for Nonlinear Evolution Equations via Modular Spaces
    DOI 10.1137/20m1385251
    Type Journal Article
    Author Menovschikov A
    Journal SIAM Journal on Mathematical Analysis
    Pages 4865-4907
    Link Publication
  • 2021
    Title On grand Sobolev spaces and pointwise description of Banach function spaces
    DOI 10.1016/j.na.2020.112100
    Type Journal Article
    Author Jain P
    Journal Nonlinear Analysis
    Pages 112100
    Link Publication
  • 2019
    Title Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity
    DOI 10.1007/s00526-019-1671-4
    Type Journal Article
    Author Molchanova A
    Journal Calculus of Variations and Partial Differential Equations
    Pages 17
  • 2019
    Title Regularity of the inverse mapping in Banach function spaces
    Type Other
    Author Molchanova A
    Link Publication
Scientific Awards
  • 2021
    Title Charged elastic materials: a variational view point
    Type Personally asked as a key note speaker to a conference
    Level of Recognition Continental/International
  • 2021
    Title Invertibility properties of limits of Sobolev homeomorphisms
    Type Personally asked as a key note speaker to a conference
    Level of Recognition Continental/International
  • 2021
    Title On the non-interpenetration condition for limits of Sobolev homeomorphisms
    Type Personally asked as a key note speaker to a conference
    Level of Recognition Continental/International
  • 2021
    Title Well-posedness of variational formulation of "modular" PDEs
    Type Personally asked as a key note speaker to a conference
    Level of Recognition National (any country)
  • 2020
    Title An extended variational approach for nonlinear PDE via modular spaces
    Type Personally asked as a key note speaker to a conference
    Level of Recognition Continental/International
  • 2020
    Title Failure of injectivity for limits of Sobolev homeomorphisms,
    Type Personally asked as a key note speaker to a conference
    Level of Recognition Continental/International
  • 2020
    Title Invertibility properties of Sobolev mappings
    Type Personally asked as a key note speaker to a conference
    Level of Recognition National (any country)
  • 2020
    Title Limits of Sobolev homeomorphisms: global injectivity
    Type Personally asked as a key note speaker to a conference
    Level of Recognition Continental/International
  • 2020
    Title Pointwice description of Banach spaces with lattice property
    Type Personally asked as a key note speaker to a conference
    Level of Recognition Continental/International

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