Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Phase separation in multiply periodic materials with fine microstructures

Riccardo Cristoferi, Luca Pignatelli

Abstract

We study a Cahn-Hilliard model for phase separation in composite materials with multiple periodic microstructures. These are modeled by considering a highly oscillating potential. The focus of this paper is in the case where the scales of the microstructures are smaller than that of phase separation. We provide a compactness result and prove that the Γ-limit of the energy is a multiple of the perimeter. In particular, using the recently introduced unfolding operator for multiple scales, we show that the taking the limit of all of the scales together is equivalent to taking one limit at the time, starting from the smaller scale and keeping the larger fixed.

Phase separation in multiply periodic materials with fine microstructures

Abstract

We study a Cahn-Hilliard model for phase separation in composite materials with multiple periodic microstructures. These are modeled by considering a highly oscillating potential. The focus of this paper is in the case where the scales of the microstructures are smaller than that of phase separation. We provide a compactness result and prove that the Γ-limit of the energy is a multiple of the perimeter. In particular, using the recently introduced unfolding operator for multiple scales, we show that the taking the limit of all of the scales together is equivalent to taking one limit at the time, starting from the smaller scale and keeping the larger fixed.

Paper Structure

This paper contains 15 sections, 23 theorems, 328 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Outline of the strategy
  4. Preliminaries
  5. Unfolding operator
  6. $\Gamma$-convergence
  7. Sets of finite perimeter
  8. Technical results
  9. Estimates for sequences with uniformly bounded energies
  10. Definition and properties of the auxiliary cell problem
  11. Limit of the auxiliary surface tension.
  12. Compactness
  13. Liminf inequality
  14. Limsup inequality
  15. Mass-constrained case

Key Result

Theorem 8

(Compactness) Let $(\eta_n)_n$, $(\delta_n)_n$, $(\varepsilon_n)_n$ be infinitesimal sequences such that $\eta_n \ll \delta_n \ll \varepsilon_n$, that is Assume that itm:1_H1, itm:1_H2, itm:1_H5, and itm:1_H5 hold. Let $(u_n)_n \subset L^1 (\Omega; \mathbb R^M)$ be a sequence of functions such that Then, there exists a subsequence $(u_{n_k})_k \subset W^{1,2}(\Omega; \mathbb R^M)$ and a function

Figures (7)

  • Figure 1: Left: A composite material with a periodic microstructure. Right: A microstructure with materials inclusions. Different colors correspond to different materials.
  • Figure 2: Left: A composite material with a two nested periodic microstructures. Center: A microstructure with materials inclusions and a nested microstructure. Right: A microstructure with periodic inclusions. Different colors correspond to different materials.
  • Figure 3: The regime considered in this paper: $\eta \ll \delta \ll \varepsilon$
  • Figure 4: Left: Integer and fractional decomposition in $\mathbb R^N$. Right: Integer and fractional decomposition in $\Omega$.
  • Figure 5: Subdivision of the interface layer
  • ...and 2 more figures

Theorems & Definitions (59)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Theorem 8
  • Definition 9
  • Remark 10
  • ...and 49 more