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A free boundary approach to the quasistatic evolution of debonding models

Eleonora Maggiorelli, Filippo Riva, Edoardo Giovanni Tolotti

TL;DR

The work addresses quasistatic debonding of an adhesive membrane from a substrate by recasting the problem as a free boundary issue of Alt–Caffarelli type and solving via Minimizing Movements in function spaces. The authors define a time-dependent driving energy 𝓔(t,A) as the minimal Dirichlet energy under boundary data w(t) on Γ and a dissipation distance that enforces irreversibility, then formulate shape energetic solutions (SES) and, through a displacement field u(t)=𝔥_{A(t),w(t)}, displacement energetic solutions (DES) linked to a one-phase Bernoulli free boundary problem. Existence of DES (and thus SES) is established via a time-discrete scheme, a Helly-type compactness argument for the evolution of the debonded region, and a careful passage to the limit; a safe-assumption guarantees a full energy balance, while a general conditional result is obtained by approximating the initial state. The approach simplifies previous treatments by working with open sets and a displacement reformulation, enabling a direct connection to classical free boundary theory and providing explicit 1D behavior for the front. Overall, the paper contributes a robust variational framework for rate-independent debonding with energy balance and a practical pathway to handle irreversibility in a free boundary context.

Abstract

The mechanical process of progressively debonding an adhesive membrane from a substrate is described as a quasistatic variational evolution of sets and herein investigated. Existence of energetic solutions, based on global minimisers of a suitable functional together with an energy balance, is obtained within the natural class of open sets, improving and simplifying previous results known in literature. The proposed approach relies on an equivalent reformulation of the model in terms of the celebrated one-phase Bernoulli free boundary problem. This point of view allows performing the Minimizing Movements scheme in spaces of functions instead of the more complicated framework of sets. Nevertheless, in order to encompass irreversibility of the phenomenon, it remains crucial to keep track of the debonded region at each discrete time-step, thus actually resulting in a coupled algorithm.

A free boundary approach to the quasistatic evolution of debonding models

TL;DR

The work addresses quasistatic debonding of an adhesive membrane from a substrate by recasting the problem as a free boundary issue of Alt–Caffarelli type and solving via Minimizing Movements in function spaces. The authors define a time-dependent driving energy 𝓔(t,A) as the minimal Dirichlet energy under boundary data w(t) on Γ and a dissipation distance that enforces irreversibility, then formulate shape energetic solutions (SES) and, through a displacement field u(t)=𝔥_{A(t),w(t)}, displacement energetic solutions (DES) linked to a one-phase Bernoulli free boundary problem. Existence of DES (and thus SES) is established via a time-discrete scheme, a Helly-type compactness argument for the evolution of the debonded region, and a careful passage to the limit; a safe-assumption guarantees a full energy balance, while a general conditional result is obtained by approximating the initial state. The approach simplifies previous treatments by working with open sets and a displacement reformulation, enabling a direct connection to classical free boundary theory and providing explicit 1D behavior for the front. Overall, the paper contributes a robust variational framework for rate-independent debonding with energy balance and a practical pathway to handle irreversibility in a free boundary context.

Abstract

The mechanical process of progressively debonding an adhesive membrane from a substrate is described as a quasistatic variational evolution of sets and herein investigated. Existence of energetic solutions, based on global minimisers of a suitable functional together with an energy balance, is obtained within the natural class of open sets, improving and simplifying previous results known in literature. The proposed approach relies on an equivalent reformulation of the model in terms of the celebrated one-phase Bernoulli free boundary problem. This point of view allows performing the Minimizing Movements scheme in spaces of functions instead of the more complicated framework of sets. Nevertheless, in order to encompass irreversibility of the phenomenon, it remains crucial to keep track of the debonded region at each discrete time-step, thus actually resulting in a coupled algorithm.

Paper Structure

This paper contains 8 sections, 19 theorems, 142 equations, 1 figure.

Table of Contents

  1. Notations and preliminaries
  2. minimisers of one-phase free boundary functionals
  3. Setting and main results
  4. One-dimensional framework
  5. Properties of displacement energetic solutions
  6. Construction of displacement energetic solutions
  7. Proof of \ref{['thm:general']}
  8. Proof of \ref{['thm:conditional']}

Key Result

Lemma 1.1

Let $\eta\in H^1(\Omega)$ and $A \in \mathcal{M}_{\Gamma,\eta}$. There exists a unique minimiser, denoted by $\mathfrak{h}_{A,\eta}$, of the Dirichlet functional over the set $H^1_{\Gamma,\eta}(\Omega,A)$. The minimiser $\mathfrak{h}_{A, \eta}$ is equivalently characterized by the Euler-Lagrange equation If in addition $A$ is open, i.e. $A\in\mathcal{O}_{\Gamma,\eta}$, then $\mathfrak{h}_{A,\eta

Figures (1)

  • Figure 1: The graphs of $w(t)$ and $w^\ast(t)$.

Theorems & Definitions (45)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 35 more