1,721,099 research outputs found
Curvature theory of boundary phases: the two dimensional case
Full text linkWe describe the behaviour of minimum problems involving non-convex surface integrals in 2D,
singularly perturbed by a curvature term. We show that their limit is described by functionals which
take into account energies concentrated on vertices of polygons. Non-locality and non-compactness
effects are highlighted
Homogenization of quadratic convolution energies in periodically perforated domains
Full text linkWe prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The corresponding limit is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem from perforated domains belonging to a wide class containing compact periodic perforations
Perturbed minimizing movements of families of functionals
No full textWe consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a (Gamma-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters epsilon and tau, governing energy and time scales, respectively. We characterize the extreme cases when epsilon/tau and tau/epsilon converges to 0 sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of epsilon and tau. We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies
Homogenization of random convolution energies
Full text linkWe prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity, we prove that the Gamma-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, which can be extended to this 'asymptotically local' case. As a particular application, we derive a homogenization theorem on random perforated domains
Asymptotic behavior of the Dirichlet energy on Poisson point clouds
Full text linkWe prove that quadratic pair interactions for functions defined on planar
Poisson clouds and taking into account pairs of sites of distance up to a
certain (large-enough) threshold can be almost surely approximated by the
multiple of the Dirichlet energy by a deterministic constant. This is achieved
by scaling the Poisson cloud and the corresponding energies and computing a
compact discrete-to-continuum limit. In order to avoid the effect of
exceptional regions of the Poisson cloud, with an accumulation of sites or with
"disconnected sites", a suitable "coarse-grained" notion of convergence of
functions defined on scaled Poisson clouds must be given
Asymptotic behavior of the capacity in two-dimensional heterogeneous media
Full text linkWe describe the asymptotic behavior of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set Ω. This problem is governed by two small parameters: ", the size of the inclusion (which is not restrictive to assume to be a ball), and ı, the period of the inhomogeneity modeled by oscillating coefficients. We show that this capacity behaves as C jlog "j-1. The coefficient C is explicitly computed from the minimum of the oscillating coefficient and the determinant of the corresponding homogenized matrix, through a harmonic mean with a proportion depending on the asymptotic behavior of jlog ıj=jlog "j
Homogenization of cohesive fracture in masonry structures
Full text linkWe derive a homogenized mechanical model of a masonry-type structure constituted by a periodic assemblage of blocks with interposed mortar joints. The energy functionals in the model under investigation consist of (i) a linear elastic contribution within the blocks, (ii) a Barenblatt’s cohesive contribution at contact surfaces between blocks, and (iii) a suitable unilateral condition on the strain across contact surfaces, and are governed by a small parameter representing the typical ratio between the length of the blocks and the dimension of the structure. Using the terminology of Γ-convergence and within the functional setting supplied by the functions of bounded deformation, we analyze the asymptotic behavior of such energy functionals when the parameter tends to zero, and derive a simple homogenization formula for the limit energy. Furthermore, we highlight the main mathematical and mechanical properties of the homogenized energy, including its non-standard growth conditions under tension or compression. The key point in the limit process is the definition of macroscopic tensile and compressive stresses, which are determined by the unilateral conditions on contact surfaces and the geometry of the blocks
A Model for Craquelure: Brittle Layers on Elastic Substrates
No full textWe propose a model for a brittle layer of material on an elastic substrate, with in mind a layer of paint on canvas. The deformation of the underlying elastic material may trigger fracture on the brittle medium that are visible on its surface as a craquelure pattern. The geometry of this pattern may be influenced by the thickness of the brittle layer and its evolution history
Two geometric lemmas for
No full textWe prove two geometric lemmas for N−1-valued functions that allow to modify sequences of lattice spin functions on a small percentage of nodes during a discrete-to-continuum process so as to have a fixed average. This is used to simplify known formulas for the homogenization of spin systems
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