Abstract
We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional low-contrast periodic environment, by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuum analysis. As in a recent paper by Braides and Scilla dealing with high-contrast periodic media, we give an example showing that in general the effective motion does not depend only on the $\Gamma$-limit, but also on geometrical features that are not detected in the static description. We show that there exists a critical value $\widetilde{\delta}$ of the contrast parameter $\delta$ above which the discrete motion is constrained and coincides with the high-contrast case. If $\delta<\widetilde{\delta}$ we have a new pinning threshold and a new effective velocity both depending on $\delta$. We also consider the case of non-uniform inclusions distributed into periodic uniform layers.
Highlights
In this paper we study a problem of homogenization for a discrete crystalline flow
We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional low-contrast periodic environment, by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuum analysis
As in a recent paper by Braides and Scilla dealing with high-contrast periodic media, we give an example showing that in general the effective motion does not depend only on the Γ-limit, and on geometrical features that are not detected in the static description
Summary
In this paper we study a problem of homogenization for a discrete crystalline flow. The analysis will be carried over by using the minimizing-movement scheme of Almgren, Taylor and Wang [3]. In recent papers by Braides, Gelli and Novaga [9] and Braides and Scilla [11], the Almgren-Taylor-Wang approach has been used coupled to a homogenization procedure In this case the perimeters and the distances depend on a small parameter ε (interpreted as a space scale), and after introducing a time scale τ , the time-discrete motions are the Ekτ,ε defined iteratively by. Where the effective velocity function f , obtained as solution of a one-dimensional homogenization problem, is a locally constant function on compact subsets of (0, +∞) which depends on α, the period and size of the inclusions but not on γ (neither on the value β) This function has been computed, by means of algebraic formulas, in the simpler cases Nβ = 1 and Nβ = 2, with varying Nα.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
