Damage as the Γ-limit of microfractures in linearized elasticity under the non-interpenetration constraint

A homogenization result is given for a material with brittle periodic inclusions, under the requirement that the interpenetration of matter is forbidden. According to the ratio between the softness of the inclusions and the size of the microstucture, three different limit models are deduced via Gamma-convergence. In particular it is shown that in the limit the non-interpenetration constraint breaks the symmetry between states where the material is in extension and in compression

The nonlinear bending-torsion theory for curved rods as Gamma-limit of three-dimensional elasticity

The problem of the rigorous derivation of one-dimensional models for nonlinearly elastic curved beams is studied in a variational setting. Considering different scalings of the three-dimensional energy and passing to the limit as the diameter of the beam goes to zero, a nonlinear model for strings and a bending-torsion theory for rods are deduced

Line-tension model for plasticity as the Gamma-limit of a nonlinear dislocation energy

In this paper we rigorously derive a line-tension model for plasticity as the Gamma-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius. The Gamma-limit we obtain as the length of the Burgers vector tends to zero has the same form as the Gamma-limit obtained by starting from a linear, semi-discrete dislocation energy. The nonlinearity, however, creates several mathematical difficulties, which we tackled by proving suitable versions of the Rigidity Estimate in non-simply-connected domains and by performing a rigorous two-scale linearisation of the energy around an equilibrium configuration

Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density

The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness of the plate goes to zero. More precisely, it is shown that critical points of the nonlinear elastic functional converge to critical points of the Γ-limit. This is proved under the physical assumption that the energy density blows up as the determinant of the deformation gradient becomes infinitesimally small

Damage as Gamma-limit of microfractures in anti-plane linearized elasticity

A homogenization result is given for a material having brittle inclusions arranged in a periodic structure. <br/> According to the relation between the softness parameter and the size of the microstructure, three different limit models are deduced via Gamma-convergence. <br/> In particular, damage is obtained as limit of periodically distributed microfractures

Preliminary orbital elements of six visual binary stars

International audiencePreliminary new orbital elements were computed for the visual binary stars A 1 - ADS 1345, A 2629 - ADS 3610, BU 560 - ADS 4371, STF 3115 - ADS 4376, STF 1426 AB - ADS 7730 and STF 2437 - ADS 11956. Using Straizys and Kuriliene's data, we derived new formulae for computing dynamical parallaxes for luminosity classes IV and V. The values found for those systems are in agreement with the {Hipparcos} parallaxes and the corresponding systemic masses are consistent with the spectral types

A global method for deterministic and stochastic homogenisation in BV

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation

The Ellipse Law: Kirchhoff Meets Dislocations

In this paper we consider a nonlocal energy I \u3b1 whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter \u3b1 08 R. The case \u3b1 = 0 corresponds to purely logarithmic interactions, minimised by the circle law; \u3b1 = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for \u3b1 08 (0 , 1) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes 1-\u3b1 and 1+\u3b1. This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff\u2019s celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses