Related papers: Variational problems in thin elastic structures
We consider a noninteracting disordered system designed to model particle diffusion, relaxation in glasses, and impurity bands of semiconductors. Disorder originates in the random spatial distribution of sites. We find strong numerical…
The piecewise linearity condition on the total energy with respect to the total magnetization of finite quantum systems is derived, using the infinite-separation-limit technique. This generalizes the well-known constancy condition, related…
A definition of K41 scaling law for suitable families of measures is given and investigated. First, a number of necessary conditions are proved. They imply the absence of scaling laws for 2D stochastic Navier-Stokes equations and for the…
An exact result concerning the energy transfers between non-linear waves of thin elastic plate is derived. Following Kolmogorov's original ideas in hydrodynamical turbulence, but applied to the F\"oppl-von K\'arm\'an equation for thin…
In the celebrated work of Friesecke, James and M\"uller '06 the authors derive a hierarchy of models for plates by carefully analyzing the $\Gamma$-convergence of the rescaled nonlinear elastic energy. The key ingredient of their proofs is…
The objective of this article is to compare different surface energies for multi-well singular perturbation problems associated with martensitic phase transformations involving higher order laminates. We deduce scaling laws in the singular…
We investigate Anderson localization on various 1D structures having flat bands. The main focus is on the scaling laws obeyed by the localization length at weak disorder in the vicinity of flat-band energies. A careful distinction is made…
We extend the theory of structured deformations to the setting of linearized elasticity by providing an integral representation for the underlying energy that features bulk and surface contributions. Our derivation is obtained both via a…
We study the scaling properties of forced folding of thin materials of different geometry. The scaling relations implying the topological crossovers from the folding of threedimensional plates to the folding of two-dimensional sheets, and…
This paper deals with the introduction of a decomposition of the deformations of curved thin beams, with section of order $\delta$, which takes into account the specific geometry of such beams. A deformation $v$ is split into an elementary…
The existing theory of incompatible elastic sheets uses the deviation of the surface metric from a reference metric to define the strain tensor [Efrati et al., J. Mech. Phys. Solids {\bf 57}, 762 (2009)]. For a class of simple axisymmetric…
In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimisation. For this purpose, we introduce a variational model for two-dimensional systems of edge dislocations, within the so-called core…
We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $\mathbb Z^d$ from the limiting Wulff shape in arbitrary dimensions. As the number $n$ of elements diverges, we prove that the symmetric difference to…
We consider a disk-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere…
Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is…
In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type…
We study 2d vortex sheets with unbounded support. First we show a version of the Biot- Savart law related to a class of objects including such vortex sheets. Next, we give a formula associating the kinetic energy of a very general class of…
We investigate thin plates where out-of-plane deformations arise due to membrane kinematic incompatibility of rotational type, specifically Volterra wedge disclinations, which are commonly observed in metal plates and graphene. We present…
We consider the elastic scattering of two open strings living on two D-branes separated by a distance $r$. We compute the high-energy behavior of the amplitude, to leading order in string coupling, as a function of the scattering angle…
We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness $h$ and around the mid-surface $S$ of arbitrary geometry, converge as $h\to 0$ to the critical points of the von K\'arm\'an functional…