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

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

Geometric rigidity for incompatible fields and an application to strain-gradient plasticity

<p>In this paper we show that a strain-gradient plasticity model arises as the Gamma-limit of a nonlinear semi-discrete dislocation energy. We restrict our analysis to the case of plane elasticity, so that edge dislocations can be modelled as point singularities of the strain field.</p> <p>A key ingredient in the derivation is the extension of the rigidity estimate proved by Friesecke, James and Mueller to the case of fields with nonzero curl.</p&gt

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

Shape optimisation for nonlocal anisotropic energies

In this paper we consider shape optimisation problems for sets of prescribed mass, where the driving energy functional is nonlocal and anisotropic. More precisely, we deal with the case of attractive/repulsive interactions in two and three dimensions, where the attraction is quadratic and the repulsion is given by an anisotropic variant of the Coulomb potential. Under the sole assumption of strict positivity of the Fourier transform of the interaction potential, we show the existence of a threshold value for the mass above which the minimiser is an ellipsoid, and below which the minimiser does not exist. If, instead, the Fourier transform of the interaction potential is only nonnegative, we show the emergence of a dichotomy: either there exists a threshold value for the mass as in the case above, or the minimiser is an ellipsoid for any positive value of the mass

Homogenization of high-contrast Mumford-Shah energies

We prove a homogenization result for Mumford-Shah-type energies associated to a brittle composite material with weak inclusions distributed periodically at a scale ${\varepsilon}>0$. The matrix and the inclusions in the material have the same elastic moduli but very different toughness moduli, with the ratio of the toughness modulus in the matrix and in the inclusions being $1/\beta_{\varepsilon}$, with $\beta_{\varepsilon}>0$ small. We show that the high-contrast behaviour of the composite leads to the emergence of interesting effects in the limit: The volume and surface energy densities interact by $\Gamma$-convergence, and the limit volume energy is not a quadratic form in the critical scaling $\beta_{\varepsilon} = {\varepsilon}$, unlike the ${\varepsilon}$-energies, and unlike the extremal limit cases