arXiv subjects

Adriana Garroni

Publications and source records attributed to Adriana Garroni.

24 records · Page 2Linked to original sources

arXiv2015

Interfacial energies of systems of chiral molecules

We consider a simple model for the assembly of chiral molecules in two dimensions driven by maximization of the contact area. We derive a macroscopic model described by a parameter taking nine possible values corresponding to the possible minimal microscopic patterns and modulated phases of the chiral molecules. We describe the overall behaviour by means of an interaction energy of perimeter type between such phases. This energy is a crystalline perimeter energy, highlighting preferred directions for the interfaces between ensembles of molecules labelled by different values of the parameter.

math-ph↗

arXiv2014

Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity

In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.

math.AP↗

arXiv2010

Singular kernels, multiscale decomposition of microstructure, and dislocation models

We consider a model for dislocations in crystals introduced by Koslowski, Cuitiño and Ortiz, which includes elastic interactions via a singular kernel behaving as the $H^{1/2}$ norm of the slip. We obtain a sharp-interface limit of the model within the framework of $Γ$-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitté and Seppecher to which their rearrangement argument no longer applies. Instead we show that the microstructure must be approximately one-dimensional on most length scales and exploit this property to derive a sharp lower bound.

math.AP↗

arXiv2008

Gradient theory for plasticity via homogenization of discrete dislocations

In this paper, we deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal in exam, so that the mathematical formulation will involve a two dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so called core region. We show that the Gamma-limit as the core radius tends to zero and the number of dislocations tends to infinity of this energy (suitably rescaled), takes the form $$ E= \int_{\Om} (W(β^e) + \f (\Curl β^e)) dx, $$ where $β^e$ represents the elastic part of the macroscopic strain, and $\Curl β^e$ represents the geometrically necessary dislocation density. The plastic energy density $\f$ is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls.

math-ph↗

arXiv2005

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics

In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.

math.AP↗

arXiv1993

A Wiener estimate for relaxed Dirichlet problems in dimension $N\geq 2$

We prove a Wiener energy estimate for relaxed Dirichlet problems $Lu + μu =ν$ in $Ω$, with $L$ an uniformly elliptic operator with bounded coefficients, $μ$ a measure of ${\cal M}_0(Ω)$, $ν$ a Kato measure and $Ω$ a bounded open set of ${\bf R}^N$, $N \geq 2$. Choosing a particular $μ$, we obtain an energy estimate also for classical variational Dirichlet problems.

funct-an↗