On the variational limits of lattice energies on prestrained elastic bodies

We study the asymptotic behaviour of the discrete elastic energies in presence of the prestrain metric $G$, assigned on the continuum reference configuration $\Omega$. When the mesh size of the discrete lattice in $\Omega$ goes to zero, we obtain the variational bounds on the limiting (in the sense of $\Gamma$-limit) energy. In case of the nearest-neighbour and next-to-nearest-neibghour interactions, we derive a precise asymptotic formula, and compare it with the non-Euclidean model energy relative to $G$

Random parking, Euclidean functionals, and rubber elasticity

We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\mathbb{R}^d$ and of point sets that are (almost) subadditive in their first variable. Denoting by $\xi$ the random parking measure in $\mathbb{R}^d$, and by $\xi^R$ the random parking measure in the cube $Q_R=(-R,R)^d$, we show, under some natural assumptions on $S$, that there exists a constant $\bar{S}\in \mathbb{R}$ such that % $$\lim_{R\to +\infty} \frac{S(Q_R,\xi)}{|Q_R|}\,=\,\lim_{R\to +\infty}\frac{S(Q_R,\xi^R)}{|Q_R|}\,=\,\bar{S}$$ % almost surely. If $\zeta \mapsto S(Q_R,\zeta)$ is the counting measure of $\zeta$ in $Q_R$, then we retrieve the result by the second author on the existence of the jamming limit. The present work generalizes this result to a wide class of (almost) subadditive functions. In particular, classical Euclidean optimization problems as well as the discrete model for rubber previously studied by Alicandro, Cicalese, and the first author enter this class of functions. In the case of rubber elasticity, this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit, as well as a generalization to stochastic networks generated on bounded sets.Comment: 28 page

On the effect of interactions beyond nearest neighbours on non-convex lattice systems

We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation

Domain formation in magnetic polymer composites: an approach via stochastic homogenization

We study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a stochastic lattice scaled by a small parameter $\varepsilon$ and the magnets as classical $\pm 1$ spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of $\Gamma$-convergence that, up to subsequences, the (continuum) $\Gamma$-limit of these energies is finite on the set of Caccioppoli partitions representing the magnetic Weiss domains where it has a local integral structure. Assuming stationarity of the stochastic lattice, we can make use of ergodic theory to further show that the $\Gamma$-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic.Comment: 31 page

Derivation of linear elasticity for a general class of atomistic energies

The purpose of this paper is the derivation, in the framework of Gamma-convergence, of linear elastic continuum theories from a general class of atomistic models, in the regime of small deformations. Existing results are available only in the special case of one-well potentials accounting for very short interactions. We consider here the general case of multi-well potentials accounting for interactions of finite but arbitrarily long range. The extension to this setting requires a novel idea for the proof of the Gamma-convergence which is interesting in its own right and potentially relevant in other applications

Automatic shoreline detection from eight-band VHR satellite imagery

Coastal erosion, which is naturally present in many areas of the world, can be significantly increased by factors such as the reduced transport of sediments as a result of hydraulic works carried out to minimize flooding. Erosion has a significant impact on both marine ecosystems and human activities; for this reason, several international projects have been developed to study monitoring techniques and propose operational methodologies. The increasing number of available high-resolution satellite platforms (i.e., Copernicus Sentinel) and algorithms to treat them allows the study of original approaches for the monitoring of the land in general and for the study of the coastline in particular. The present project aims to define a methodology for identifying the instantaneous shoreline, through images acquired from the WorldView 2 satellite, on eight spectral bands, with a geometric resolution of 0.5mfor the panchromatic image and 1.8mfor the multispectral one. A pixel-based classification methodology is used to identify the various types of land cover and to make combinations between the eight available bands. The experiments were carried out on a coastal area with contrasting morphologies. The eight bands in which the images are taken produce good results both in the classification process and in the combination of the bands, through the algorithms of normalized difference vegetation index (NDVI), normalized difference water index (NDWI), spectral angle mapper (SAM), and matched filtering (MF), with regard to the identification of the various soil coverings and, in particular, the separation line between dry and wet sand. In addition, the real applicability of an algorithm that extracts bathymetry in shallow water using the "coastal blue" band was tested. These data refer to the instantaneous shoreline and could be corrected in the future with morphological and tidal data of the coastal areas under study

Coarse-Graining of a Discrete Model for Edge Dislocations in the Regular Triangular Lattice

We consider a discrete model of planar elasticity where the particles, in the reference configuration, sit on a regular triangular lattice and interact through nearest-neighbor pairwise potentials, with bonds modeled as linearized elastic springs. Within this framework, we introduce plastic slip fields, whose discrete circulation around each tri-angle detects the possible presence of an edge dislocation. We provide a gamma-convergence analysis, as the lattice spacing tends to zero, of the elastic energy induced by edge dislocations in the energy regime corresponding to a finite number of geometrically necessary dislocations

ALBA FUCENS ARCHAEOLOGICAL SITE: MULTISCALE AND MULTIDISCIPLINARY APPROACH FOR RISK ASSESSMENT AND CONSERVATION

Abstract. The Latin Colony (303 BC) of Alba Fucens (L'Aquila, Italy) is the largest archaeological area of the whole Apennines. Due to its extension, location and environmental context, the conservation of the site is particularly complex.For these reasons, in the paper a multiscale and multidisciplinary geoarchaeological study (remote sensing and UAV photogrammetry) of the site, to extract and measure morphostructural information to be associated to the environmental context, risk assessment and conservation, is reported.The study area is located on a higher geostructure with a subangular shape, which suggests a tectonic origin, with respect to the surrounding plain and bounded to the East by a large fan that takes place towards the Piana del Fucino.First, the geo-structural analysis, using the Landsat-8 and GeoEye multispectral sensors, was performed. The GeoEye satellite image allowed carrying out the morphological analysis of the archaeological area, its physical characteristics, the drainage pattern and the land use. Subsequently, after image processing of satellite data, a UAV survey was carried out in some relevant zones. Considering the UAV photogrammetry accuracy information, it was possible to extract data as map producing with several advantages (economic and time saving, minimum field work). With a multiscale and metric approach, the geomatics techniques allowed to deeply investigate some areas, creating detailed 3D models for evaluate risks and the decay. Finally, a general discussion about risk mitigation and conservation is reported.</p