Related papers: Dimension reduction for thin films prestrained by …
We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the…
We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective…
Evaluation of residual elastic strain within the bulk of engineering components or natural objects is a challenging task, since in general it requires mapping a six-component tensor quantity in three dimensions. A further challenge concerns…
We study the linearized Fopl - von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle $2 \alpha$. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness…
Starting from the three-dimensional Cosserat elasticity, we derive a two-dimensional model for isotropic elastic shells. For the dimensional reduction, we employ a derivation method similar to that used in classical shell theory, as…
We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flat substrate inclined at a non-zero angle to the horizontal. The dynamics of the thin film is influenced by an electric field which is set…
We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of…
The asymptotic behaviour of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness $h$ of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values…
We consider a class of models for nonlinearly elastic surfaces in this work. We have in mind thin, highly deformable structures modeled directly as two-dimensional nonlinearly elastic continua, accounting for finite membrane and bending…
This paper is concerned with the study of linear geometric rigidity of shallow thin domains under zero Dirichlet boundary conditions on the displacement field on the thin edge of the domain. A shallow thin domain is a thin domain that has…
It is well known that an elastic sheet loaded in tension will wrinkle and that the length scale of the wrinkles tends to zero with vanishing thickness of the sheet [Cerda and Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)]. We give the first…
We study isometric immersions of a Riemannian surface $(\Omega,\frak{g})$, where $\Omega \subset \mathbb{R}^2$, into $\mathbb{R}^3$. We consider their bending energy, i.e., the square of the $L^2$-norm of their second fundamental form,…
This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy…
We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, we often use a thin film approximation, which…
We consider a class of kinetic models for polymeric fluids motivated by the Peterlin dumbbell theories for dilute polymer solutions with a nonlinear spring law for an infinitely extensible spring. The polymer molecules are suspended in an…
Unsupervised machine learning lacks ground truth by definition. This poses a major difficulty when designing metrics to evaluate the performance of such algorithms. In sharp contrast with supervised learning, for which plenty of quality…
Paints and coatings often feature interfacial defects due to disturbances during the deposition process which, if they persist until solidification, worsen the product quality. In this article, we investigate the stability of a thin liquid…
Motivated by experiments and formal asymptotic expansions in the physics literature, Maor and Shachar (J. Elasticity 134 (2019), 149-173) studied the behaviour of a model elastic energy of maps between manifolds with incompatible metrics.…
Linearized elasticity models are derived, via Gamma-convergence, from suitably rescaled nonlinear energies when the corresponding energy densities have a multiwell structure and satisfy a weak coercivity condition, in the sense that the…
On a two-dimensional Riemannian manifold without boundary we consider the variational limit of a family of functionals given by the sum of two terms: a Ginzburg-Landau and a perimeter term. Our scaling allows low-energy states to be…