1,720,990 research outputs found
Free-discontinuity problems generated by singular perturbation: the N-dimensional case
No full textWe provide an approximation of some free discontinuity problems by local functionals with a singular perturbation of higher order. More precisely, we study the limiting behaviour of energies of the form F epsilon(u) = 1/epsilon integral(Omega) f(epsilon\del u\(2)) dx + epsilon(2) integral(Omega)\Hu\(2)dx,
where Hu denotes the Hessian matrix of u
Derivation of Linear Elasticity for a General Class of Atomistic Energies
No full textThe 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 multiwell 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
On the effect of interactions beyond nearest neighbours on non-convex lattice systems
No full textWe 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
Derivation of a Rod theory from lattice systems with interactions beyond nearest neighbours
No full textWe study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-tonearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires
Finite difference approximation of energies in Fracture Mechanics
No full textWe provide a variational approximation by finite-difference energies of
functionals of the type
defined for u E SBD(Q), which are related to variational models in ’ fracture
mechanics for linearly-elastic materials. We perform this approximation in di-
mension 2 via both discrete and continuous functionals. In the discrete scheme we
treat also boundary value problems and we give an extension of the approximation
result to dimension 3
Free-discontinuity problems generated by singular perturbation
No full textWe show that some free discontinuity problems can be obtained as a limit of nonconvex local functionals with a singular perturbation of higher order
Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions
Full text linkWe propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence
Correction to: Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals (Archive for Rational Mechanics and Analysis, (2018), 230, 1, (1-45), 10.1007/s00205-018-1240-6)
No full textIn the original version there was a mistake in the displacement of the tables.The original version of this article unfortunately contained mistakes
Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals
No full textLinear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours
Convergence analysis of the B"ol-Reese discrete model for rubber
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