Minimising movements for the motion of discrete screw dislocations along glide directions

In [3] a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in [8] and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on $\Gamma$-convergence techniques

Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems

We study, through a !-convergence procedure, the discrete to continuum limit of Ising-type energies of the form F"(u) = −Pi,j c" i,juiuj , where u is a spin variable defined on a portion of a cubic lattice "Zd 8 ⌦, ⌦ being a regular bounded open set, and valued in {−1, 1}. If the constants c" i,j are nonnegative and satisfy suitable coercivity and decay assumptions, we show that all possible !-limits of surface scalings of the functionals F" are finite on BV (⌦; {±1}) and of the formR Su'(x, ⌫u) dH11 d−1. If such decay assumptions are violated, we show that we may approximate nonlocal functionals of the form R Su '(⌫u) dHd−1+ K(x, y)g(u(x), u(y)) dxdy. We focus on the approximation of two relevant examples: fractional perimeters and Ohta–Kawasaki-type energies. Eventually, we provide a general criterion for a ferromagnetic behavior of the energies F" even when the constants c" i,j change sign. If such a criterion is satisfied, the ground states of F" are still the uniform states 1 and −1 and the continuum limit of the scaled energies is an integral surface energy of the form above

Convergence analysis of the B\"ol-Reese discrete model for rubber

In a recent work, B\"ol and Reese have introduced a discrete model for polymer networks by means of a finite element modeling. They have also provided a comparison with real experiments. A key parameter of their model is the size h of the finite element mesh, that is meant to be small in practice. The aim of the present work is to study the asymptotic behaviour (and the convergence of the finite element method) when the meshsize goes to zero. In particular, we address the properties satisfied by the model at the limit, depending on the properties of the mesh

Convergence analysis of the Böl-Reese discrete model for rubber

International audienceIn a recent work, Böl and Reese have introduced a discrete model for polymer networks by means of a finite element modeling. They have also provided a comparison with real experiments. A key parameter of their model is the size h of the finite element mesh, that is meant to be small in practice. The aim of the present work is to study the asymptotic behaviour (and the convergence of the finite element method) when the meshsize goes to zero. In particular, we address the properties satisfied by the model at the limit, depending on the properties of the mesh

Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg-Landau framework. © 2014 Springer-Verlag Berlin Heidelberg

Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg-Landau framework. © 2014 Springer-Verlag Berlin Heidelberg