Asymptotic models for curved rods derived from nonlinear elasticity by Gamma-convergence

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elasticity. Finally we also address the case of thin elastic rings.Comment: 25 page

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

Asymptotic derivation of models for materials with small length scales

We 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)

Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy $E_\gamma$ modelling the interactions$-$at a typical length-scale of $1/\gamma$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_\gamma$ in powers of $1/\gamma$ by $\Gamma$-convergence, in the limit $\gamma\to\infty$. While the zero-order term in the expansion, the $\Gamma$-limit of $E_\gamma$, captures the `bulk' profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a `boundary-layer' energy that captures the profile of the density in the proximity of the lock. This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.Comment: 25 page

Paleomagnetic investigations on the Pleistocene lacustrine sequence of Piànico-Sèllere (northern Italy)

The Piànico-Sèllere is a lacustrine succession from northern Italy that records a sequence of climatic transitions across two Pleistocene glacial stages. The intervening interglacial stage is represented by well-preserved varves with calcitic (summer) and clastic (winter) laminae. There is a tight coupling between climate-driven lithologic changes and magnetic susceptibility variations, and stable paleomagnetic components were retrieved from all investigated lithologies including the largely diamagnetic calcite varves. These components were used to delineate a sequence of magnetic polarity reversals that was interpreted as a record of excursions of the Earth’s magnetic field. Comparison of the magnetostratigraphic results with previously published data allows discussion of two possible models which have generated controversy regarding the age of the Piànico Formation. The data indicates that the Piànico Formation magnetostratigraphy correlates to geomagnetic field excursions across the Brunhes/Matuyama transition, and consequently the Piànico interglacial correlates to marine isotope stage 19. This correlation option is substantially consistent with K-Ar radiometric age estimates recently obtained from a tepha layer interbedded in the Piànico Formation. The alternative option, considering the Piànico interglacial correlative to marine isotope stage 11 within the Brunhes Chron as supported by tephrochronological dating reported in the literature, is not supported by the magnetostratigraphic results