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

Asymptotic behaviour of a pile-up of infinite walls of edge dislocations

We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x=0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number n of walls tends to infinity, and characterise this behaviour in terms of Gamma-convergence. There are five different cases, depending on the asymptotic behaviour of a single dimensionless parameter. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter

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