Mappings of Finite Distortion for Nonlinear Solid Mechanics

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.