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An existence result for accretive growth in elastic solids

Elisa Davoli, Katerina Nik, Ulisse Stefanelli, Giuseppe Tomassetti

Abstract

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality.

An existence result for accretive growth in elastic solids

Abstract

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality.
Paper Structure (7 sections, 4 theorems, 68 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Assumptions and notion of weak solution
  4. Main results
  5. The equilibrium problem for a given growth: Proof of Theorem \ref{['prop:equi']}
  6. The coupled equilibrium-growth problem: Proof of Theorem \ref{['thm:coupled']}
  7. Uniform Korn inequality

Key Result

Lemma 2.3

For all $\theta:{\mathbb R}^d \to [0,\infty)$ continuous the set $Q_S\coloneqq \cup_{t\in (0,S)} \Omega(t) \times \{t\}$ is measurable for every $S\in (0,T]$.

Figures (1)

  • Figure 1: A John domain in ${\mathbb R}^2$

Theorems & Definitions (7)

  • Definition 2.1: John domain
  • Definition 2.2: Viscosity solution
  • Lemma 2.3
  • proof
  • Proposition 2.4: Uniform Korn inequality
  • Theorem 2.5: Equilibrium, given the growth
  • Theorem 2.6: Coupled equilibrium and growth