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Annali di Matematica Pura ed Applicata
No full textIn this paper we propose a variational model for the irreversible quasi static growth in brittle fractures for a linearly elastic homogeneous isotropic plate, subject to a time dependent vertical displacement on a part of its lateral surface. The model is based on the Griffith’s criterion for crack growth and is inspired to the model proposed in [11] by Francfort and Marigo in the case of 3-D elasticity. We give a precise mathematical formulation of the model and in this framework we prove an existence resul
Γ -Convergence of the Heitmann–Radin Sticky Disc Energy to the Crystalline Perimeter
Full text linkWe consider low-energy configurations for the Heitmann–Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann–Radin potential by subtracting the minimal energy per particle, i.e. the so-called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function. Whenever the limit configuration is a single crystal, i.e. it has constant orientation, we show that the Γ-limit is the anisotropic perimeter, corresponding to the Finsler metric determined by the orientation of the single crystal
The core-radius approach to supercritical fractional perimeters, curvatures and geometric flows
Get PDFWe consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0<1, to the case s≥1. We show that, as the core-radius vanishes, such core-radius regularized s-fractional perimeters, suitably scaled, Γ-converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized s-fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Furthermore, we show the same asymptotic behavior as the core-radius vanishes and s→s̄≥1 simultaneously. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics
Quantitative stability in the isodiametric inequality via the isoperimetric inequality
No full textThe isodiametric inequality is derived from the isoperimetric inequality through a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter. This principle also brings quantitative improvements to the isodiametric inequality, shown to be sharp by explicit nearly optimal sets
Long time behavior of discrete volume preserving mean curvature flows
Full text linkIn this paper we analyze the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter
Uniform distribution of dislocations in Peierls–Nabarro models for semi-coherent interfaces
Full text linkIn this paper we introduce Peierls–Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy Γ -converges to a limit functional comprised of two contributions: one is given by a constant c∞> 0 gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that c∞ is reached when dislocations are evenly-spaced on the one dimensional circle
A Variational Model for Dislocations at Semi-coherent Interfaces
No full textWe propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far-field strain vanishes as the interface size increases
From 1-homogeneous supremal functionals to difference quotients: relaxation and -convergence.
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