1,720,993 research outputs found
Homogenization of Energies Defined on 1-Rectifiable Currents
No full textWe study the homogenization of a class of energies concentrated on lines. In dimension 2 (i.e., in codimension 1) the problem reduces to the homogenization of partition energies studied by L. Ambrosio and A. Braides [Functionals defined on partitions in sets of finite perimeter. II: Semicontinuity, relaxation and homogenization, J. Math. Pures Appl. 69 (1990) 307–333.] There, the key tool is the representation of partitions in terms of BV functions with values in a discrete set. In our general case the key ingredient is the representation of closed loops with discrete multiplicity either as divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In the 3 dimensional case the main motivation for the analysis of this class of energies is the study of line defects in crystals, the so called dislocations
Sharp rigidity estimates for incompatible fields as a consequence of the Bourgain Brezis div-curl result
No full textIn this note we show that a sharp rigidity estimate and a sharp Korn s inequality for matrixvalued fields whose incompatibility is a bounded measure can be obtained as a consequence of a Hodge decomposition with critical integrability due to Bourgain and Brezis
Homogenization of line tension energies
No full textWe prove an homogenization result, in terms of Gamma-convergence, for energies concentrated on rectifiable lines in R-3 without boundary. The main application of our result is in the context of dislocation lines in dimension 3. The result presented here shows that the line tension energy of unions of single line defects converge to the energy associated to macroscopic densities of dislocations carrying plastic deformation. As a byproduct of our construction for the upper bound for the Gamma-Limit, we obtain an alternative proof of the density of rectifiable 1-currents without boundary in the space of divergence free fields
Derivation of a line-tension model for dislocations from a nonlinear three-dimensional energy: The case of quadratic growth
Full text linkIn this paper we derive a line tension model for dislocations in 3D starting from a geometrically nonlinear elastic energy with quadratic growth. In the asymptotic analysis, as the amplitude of the Burgers vectors (proportional to the lattice spacing) tends to zero, we show that the elastic energy linearizes and the line tension energy density, up to an overall constant rotation, is identi_ed by the linearized cell problem formula given in [S. Conti, A. Garroni, and M. Ortiz, Arch. Ration. Mech. Anal., 218 (2015), pp. 699{755]
From 1-homogeneous supremal functionals to difference quotients: relaxation and -convergence.
No full textGradient bounds for minimizers of variational problems related to cohesive zone models in fracture mechanics
No full textIn 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
The capacity method for asymptotic Dirichlet problems
No full textWe prove that the asymptotic behaviour of the solutions of Dirichlet problems for second-order, linear, not necessarily symmetric elliptic equations in perforated domains of the form Omega(h) = OmegaE-h is uniquely determined by the asymptotic behaviour, as h --> infinity, of suitable capacities of the sets B boolean AND E-h, where B runs in a conveniently large class of subsets of Omega
A multi-phase transition model for dislocations with interfacial microstructure
No full textWe study, by means Gamma-convergence, the asymptotic behavior of a variational model for dislocations moving on a slip plane. The variational functional is a two-dimensional multi-phase transition-type energy given by a nonlocal term and a nonlinear potential which penalizes noninteger values for the components of the phase. In the limit we obtain an anisotropic sharp interfaces model. The relevant feature of this problem is that optimal sequences in general are not given by a one-dimensional profile, but they can create microstructure
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