Projektdetails
Abstract
1) Theoretical framework
The classical mathematical theory of hyperelasticity models the equilibrium configuration of a deformed body as a minimizer of an integral functional, subject to constraints ensuring noninterpenetration. This project combines this variational view with geometric analysis by augmenting the bulk stored energy by a curvature integral over the deformed boundary, resulting in a bulk-surface energy. Besides purely geometric applications, curvature-based functionals have been recently considered to describe crack formation inside an elastic material.
2) Research questions
The is a primarily theoretical project, devoted to establishing a rigorous mathematical existence theory for bulk-surface energies for simple and nonsimple materials within the framework of the calculus of variations. A particular goal is to find conditions that ensure that minimizers are globally invertible and to exploit this in the regularity discussion. The theory is then adapted to examine related mathematical questions, including an obstacle problem inspired by a hyperelastic cell model and gradient flows of bulksurface functionals. We introduce and examine a new concept of injectivity at the boundary and a notion of genus for curvature varifolds along the way.
3) Methods
The low regularity of deformations calls for a relaxation that models the deformed boundary as a varifold. We relate the boundary trace to this varifold in a precise way, based on the divergence theorem in the regular case and examine if the relation is closed in the appropriate weak topology, in order to prove existence of minimizers with the direct method. The essential tool in studying global injectivity is that curvature functionals (e.g., the Willmore energy, Canham-Helfrich energy, or Euler's elastic energy) can detect multiplicity through Li-Yau inequalities. Within a specific energy regime, injectivity at the boundary is established, yielding global invertibility by exploiting degree-type arguments à la Ball and Krömer.
4) Level of innovation
The project introduces a new variational theory relating hyperelastic bulk energies with curvature functionals at the deformed boundary and initiates the systematic study of boundary traces with geometric methods. The relation between the deformation and the varifold is also formulated for simple materials, allowing a transfer of injectivity and regularity from the boundary trace to the associated varifold and vice versa. These methods can be expected to be very useful for future variational problems involving bulk-interface interaction.
5) Primary researchers involved
Fabian Rupp (PI), Ulisse Stefanelli (mentor), Carlos Mora Corral and Matthias Röger (collaborators)
The classical mathematical theory of hyperelasticity models the equilibrium configuration of a deformed body as a minimizer of an integral functional, subject to constraints ensuring noninterpenetration. This project combines this variational view with geometric analysis by augmenting the bulk stored energy by a curvature integral over the deformed boundary, resulting in a bulk-surface energy. Besides purely geometric applications, curvature-based functionals have been recently considered to describe crack formation inside an elastic material.
2) Research questions
The is a primarily theoretical project, devoted to establishing a rigorous mathematical existence theory for bulk-surface energies for simple and nonsimple materials within the framework of the calculus of variations. A particular goal is to find conditions that ensure that minimizers are globally invertible and to exploit this in the regularity discussion. The theory is then adapted to examine related mathematical questions, including an obstacle problem inspired by a hyperelastic cell model and gradient flows of bulksurface functionals. We introduce and examine a new concept of injectivity at the boundary and a notion of genus for curvature varifolds along the way.
3) Methods
The low regularity of deformations calls for a relaxation that models the deformed boundary as a varifold. We relate the boundary trace to this varifold in a precise way, based on the divergence theorem in the regular case and examine if the relation is closed in the appropriate weak topology, in order to prove existence of minimizers with the direct method. The essential tool in studying global injectivity is that curvature functionals (e.g., the Willmore energy, Canham-Helfrich energy, or Euler's elastic energy) can detect multiplicity through Li-Yau inequalities. Within a specific energy regime, injectivity at the boundary is established, yielding global invertibility by exploiting degree-type arguments à la Ball and Krömer.
4) Level of innovation
The project introduces a new variational theory relating hyperelastic bulk energies with curvature functionals at the deformed boundary and initiates the systematic study of boundary traces with geometric methods. The relation between the deformation and the varifold is also formulated for simple materials, allowing a transfer of injectivity and regularity from the boundary trace to the associated varifold and vice versa. These methods can be expected to be very useful for future variational problems involving bulk-interface interaction.
5) Primary researchers involved
Fabian Rupp (PI), Ulisse Stefanelli (mentor), Carlos Mora Corral and Matthias Röger (collaborators)
| Kurztitel | Elast. Festkörper mit geometrischem Rand |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/04/24 → 31/03/27 |