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Systematic derivation of one‐dimensional strain‐gradient models for non‐linear elastic structures

We present a general method for deriving one‐dimensional models for non‐linear structures, that capture the strain energy associated not only with the macroscopic strain as in traditional structural models, but also with the strain gradient. The method is based on a two‐scale expansion, and is asymptotically exact. It is applied to various types of non‐linear structures featuring localization, such as a soft elastic beam that necks in the presence of surface tension, or a tape spring that snaps. In contrast to traditional models, the one‐dimensional models obtained in this way can account for significant and non‐uniform deformations of the cross‐sections, and can therefore capture localization phenomena accurately.

The energy of a Möbius band

The talk is based on ongoing joint work with L. Freddi, P. Hornung, and M.G. Mora.

Quasi‐rods

Total Curvature of Curves in the $C^1$‐Closure of Knot Classes

Let $(\gamma_{k})_{k \in \mathbb{N}}$ be a sequence of simple closed curves in the same knot class, i.e. $[\gamma_k]=[\gamma_{k+1}]$ for all $k \in \mathbb{N}$, such that $\gamma_k \longrightarrow \gamma$ in $C^1$. Notice that the limit $\gamma$ might contain self-intersections. Then $$ TC(\gamma) \geq 2 \pi \mu( [\gamma_k]) $$ holds for $\gamma \in W^{2,2}$ with only finitely many, isolated self-intersections.

[Milnor 1950] Milnor, J. W.: On the total curvature of knots. In: The Annals of Mathematics, 2nd Ser., Vol. 52, No. 2. (1950), S. 248-257. doi.org/10.2307/1969467.

Elastic Curves Confined to Spheres

Narrow Elastic Ribbons

This is joint work with L. Freddi, M.G. Mora and R. Paroni.

A bending‐torsion model for elastic rods

This is joint work with Sören Bartels.

Boundary value problems for the one‐dimensional Helfrich functional

This is joint work with Anna Dall'Acqua (Ulm).

A Resolution of the Poisson Problem for Elastic Plates

This is a joint work with F. Da Lio & T. Rivière.

Long Time Existence of Solutions to an Elastic Flow of Networks

This is joint work with Harald Garcke and Alessandra Pluda.

Asymptotic estimates for the Willmore flow with small Energy

The gradient flow of $p$‐elastic energies

This is joint work with Christoher Hopper und Nicole Vorderobermeier.

Some examples of Riemannian metrics of curves

Handout

Effective bending–torsion theory for rods with micro‐heterogeneous prestrain

This joint work with R. Bauer (TU Dresden) and M. Schäffner (U Leipzig).