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Evolution of elastic thin films with curvature regularization via minimizing movements
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Microscopical justification of the Winterbottom problem for well-separated lattices
No full textWe consider the discrete atomistic setting introduced in Piovano and Velcic (2022) to microscopically justify the continuum model related to the Winterbottom problem, i.e., the problem of determining the equilibrium shape of crystalline film drops resting on a substrate, and relax the rigidity assumption considered in Piovano and Velcic (2022) to characterize the wetting and dewetting regimes and to perform the discrete to continuum passage. In particular, all results of Piovano and Velcic (2022) are extended to the setting where the distance between the reference lattices for the film and the substrate is not smaller than the optimal bond length between a film and a substrate atom. Such optimal film-substrate bonding distance is prescribed together with the optimal film-film distance by means of two-body atomistic interaction potentials of Heitmann-Radin type, which are both taken into account in the discrete energy, and in terms of which the wetting-regime threshold and the effective expression for the wetting parameter in the continuum energy are determined
Microscopic validation of a variational model of epitaxially strained crystalline films
Full text linkA discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates is performed by Γ-convergence. The discrete model introduced here is characterized by an energy with two contributions, the surface and the elastic-bulk energy, and it is formally justified starting from atomistic interactions. The surface energy counts missing bonds at the film and substrate boundaries, while the elastic energy models the fact that for film atoms there is a preferred interatomic distance different from the preferred interatomic distance for substrate atoms. In the regime of small mismatches between the film and the substrate optimal lattices, a discrete rigidity estimate is established by regrouping the elastic energy in triangular-cell energies and by locally applying rigidity estimates from the literature. This is crucial to establish precompactness for sequences with equibounded energy and to prove that the limiting deformation is one single rigid motion. By properly matching the convergence scaling of the different terms in the discrete energy, both surface and elastic contributions appear also in the resulting continuum limit in agreement (and in a form consistent) with literature models. Thus, the analysis performed here is a microscopical justification of such models
A Unified Model for Stress-Driven Rearrangement Instabilities
Full text linkA variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin films to crystalline cavities, and from capillarity problems to fracture models. The existence of minimizing configurations is established by adopting the direct method of the Calculus of Variations. The compactness of energy-equibounded sequences and energy lower semicontinuity are shown with respect to a proper selected topology in a class of admissible configurations that extends the classes previously considered in the literature. In particular, graph-like constraints previously considered for the setting of thin films and crystalline cavities are substituted by the more general assumption that the free crystalline interface is the boundary, consisting of an at most fixed finite number m of connected components, of sets of finite perimeter. Finally, it is shown that, as m tend to infinite, the energy of minimal admissible configurations tends to the minimum energy in the general class of configurations without the bound on the number of connected components for the free interface
Derivation of a heteroepitaxial thin-film model
Full text linkA variational model for epitaxially-strained thin films on rigid substrates is derived both by Γ-convergence from a transition-layer setting, and by relaxation from a sharp-interface description available in the literature for regular configurations. The model is characterized by a configurational energy that accounts for both the competing mechanisms responsible for the film shape. On the one hand, the lattice mismatch between the film and the substrate generate large stresses, and corrugations may be present because film atoms move to release the elastic energy. On the other hand, flatter profiles may be preferable to minimize the surface energy. Some first regularity results are presented for energetically-optimal film profiles
Carbon-Nanotube Geometries as Optimal Configurations
Full text linkThe fine geometry of carbon nanotubes is investigated from the viewpoint of Molecular Mechanics. Actual nanotube configurations are characterized as being locally minimizing a given configurational energy, including both two- and three-body contributions. By focusing on so-called zigzag and armchair topologies, we prove that the configurational energy is strictly minimized within specific, one-parameter families of periodic configurations. Such
optimal configurations are checked to be stable with respect to a large class of small nonperiodic perturbations and do not coincide with classical rolled-up nor polyhedral geometries
Sharp N 3 / 4 Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice
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