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Solutions for a free-boundary problem modeling multilayer films with coherent and incoherent interfaces

Randy Llerena, Paolo Piovano

Abstract

In this paper we introduce a variational model for the study of multilayer films that allows for the treatment of both coherent and incoherent interfaces between layers. The model is designed in the framework of the theory of Stress Driven Rearrangement Instabilities, which are characterized by the competition between elastic and surface energy effects. The surface of each film layer is assumed to satisfy an ''exterior graph condition'', under which in particular bulk cracks are allowed to be of non-graph type. By applying the direct method of calculus of variations under a constraint on the number of connected components of the cracks not connected to the surface of the film layers the existence of energy minimizers is established in dimension 2. As a byproduct of the analysis the state of art on the variational modeling of single-layered films deposited on a fixed substrate is advanced by letting the substrate surface free, by addressing the presence of multiple layers of various materials, and by including the possibility of delamination between the various film layers.

Solutions for a free-boundary problem modeling multilayer films with coherent and incoherent interfaces

Abstract

In this paper we introduce a variational model for the study of multilayer films that allows for the treatment of both coherent and incoherent interfaces between layers. The model is designed in the framework of the theory of Stress Driven Rearrangement Instabilities, which are characterized by the competition between elastic and surface energy effects. The surface of each film layer is assumed to satisfy an ''exterior graph condition'', under which in particular bulk cracks are allowed to be of non-graph type. By applying the direct method of calculus of variations under a constraint on the number of connected components of the cracks not connected to the surface of the film layers the existence of energy minimizers is established in dimension 2. As a byproduct of the analysis the state of art on the variational modeling of single-layered films deposited on a fixed substrate is advanced by letting the substrate surface free, by addressing the presence of multiple layers of various materials, and by including the possibility of delamination between the various film layers.
Paper Structure (7 sections, 8 theorems, 137 equations, 2 figures)

This paper contains 7 sections, 8 theorems, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Mathematical setting and main results
  4. Multilayer model
  5. Main results
  6. Single-layer films with delamination
  7. Multilayered films

Key Result

Theorem 3.8

Fix $\alpha \in \mathbb{N}$, $\mathbf{m}= (m_0, \ldots,m_\alpha) \in \mathbb{N}^{\alpha+1}$, and $\sigma \in \{s,t\}$. Let $\{\mathbbm{v}_i\}_{i=0}^\alpha \subset [\mathcal{L}^2({\Omega})/2, \mathcal{L}^2({\Omega})]$ be such that $\mathbbm{v}_{i_1} \le \mathbbm{v}_{i_2}$ for every $0\le i_1< i_2 \le and the unconstrained minimum problem where $\mathcal{F}^{\alpha,\bm{\lambda}}_{ \sigma} : \mathca

Figures (2)

  • Figure 1: A multilayered film composite with 2 layers (on the substrate 0th layer $S_{h^0,K^0}$) associated to an admissible configuration $(S_{h^2,K^2}, S_{h^1,K^1}, S_{h^0,K^0},u)\in\mathcal{C}^2_\textbf{m}$ (see Definition \ref{['def:multilayers']}) is represented by indicating each $j$th layer with a gray color with decreasing value with respect to the increasing order of the index $j=0,1,2$. Furthermore, the $j$th layer is indicated with a thinner line with respect to the increasing order of the index $j=0,1,2$, and for the $0$th and 1st layer we distinguish between their coherent and incoherent portions by using a dashed or a continuous line, respectively.
  • Figure 2: A single-layer film (on the substrate 0th layer $S_{h^0,K^0}$) associated to an admissible configuration $(S_{h^1,K^1}, S_{h^0,K^0},u)\in\mathcal{C}^1_\textbf{m}$ (see Definition \ref{['def:multilayers']}) is represented by indicating each $j$th layer with a gray color with decreasing value with respect to the increasing order of the index $j=0,1$ and each $j$th layer with a thinner line with respect to the increasing order of the index $j=0,1$. Furthermore, in the $0$th layer we distinguish between its coherent and incoherent portions by using a dashed or a continuous line, respectively.

Theorems & Definitions (24)

  • Definition 3.1: Admissible multilayers and admissible configurations
  • Remark 3.2
  • Definition 3.3: $\tau_{\mathcal{B}^\alpha}$-Convergence
  • Definition 3.4: $\tau_{\mathcal{C}^{\alpha}}$-Convergence
  • Definition 3.5
  • Remark 3.6
  • Remark 3.7
  • Theorem 3.8: Existence of minimizers
  • Theorem 3.9
  • Theorem 3.10: Lower semicontinuity of $\mathcal{F}^\alpha_{\sigma}$
  • ...and 14 more