2,519 research outputs found
Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density
Full text linkThe 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
Asymptotic derivation of models for materials with small length scales
Full text linkWe now give an overview of the content of this thesis, which consists of two parts.
In the first part we present some results concerning the derivation of asymptotic models for thin curved rods (see Chapters 2 and 3).
The second part is devoted to the study of homogenization problems for composite materials (see Chapters 4 and 5) and for porous media (see Chapter 6)
Γ-convergence of free-discontinuity problems
No full textWe study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our 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. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u. We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper
Line-tension model for plasticity as the Gamma-limit of a nonlinear dislocation energy
Full text linkIn 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
The nonlinear bending-torsion theory for curved rods as Gamma-limit of three-dimensional elasticity
Full text linkThe 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
A maximum-principle approach to the minimisation of a nonlocal dislocation energy
Full text linkIn this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies defined on probability measures in two dimensions. The purely nonlocal term in the energy is of convolution type, and is isotropic for the parameter alpha equal to 0 and anisotropic otherwise. The cases when alpha is equal to 0 and 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of I have been characterised by the same authors in an earlier paper, by exploiting some formal similarities with the Euler equation, and by means of complex-analysis techniques. We here propose a dierent approach, that we believe can be applied to more general energies
Damage as the Γ-limit of microfractures in linearized elasticity under the non-interpenetration constraint
Full text linkA 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 Equilibrium Measure for a Nonlocal Dislocation Energy
Full text linkIn this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well-known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels. © 2018 Wiley Periodicals, Inc
Stability of Ellipsoids as the Energy Minimizers of Perturbed Coulomb Energies
No full textIn this paper we characterize the minimizer for a class of nonlocal perturbations of the Coulomb energy. We show that the minimizer is the normalized characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, is smooth off the origin, and is sufficiently small. This result can be seen as the stability of ellipsoids as energy minimizers, since the minimizer of the Coulomb energy is the normalized characteristic function of a ball
Damage as Gamma-limit of microfractures in anti-plane linearized elasticity
Full text linkA homogenization result is given for a material having brittle inclusions arranged in a periodic structure.
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According to the relation between the softness parameter and the size of the microstructure, three different limit models are deduced via Gamma-convergence.
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In particular, damage is obtained as limit of periodically distributed
microfractures
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