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Discontinuous finite element approximation of quasistatic crack growth in finite elasticity

Alessandro Giacomini, Marcello Ponsiglione

Abstract

We propose a time-space discretization of a general notion of quasistatic growth of brittle fractures in elastic bodies proposed in [13] by G. Dal Maso, G.A. Francfort, and R. Toader, which takes into account body forces and surface loads. We employ adaptive triangulations and prove convergence results for the total, elastic and surface energies. In the case in which the elastic energy is strictly convex, we prove also a convergence result for the deformations.

Discontinuous finite element approximation of quasistatic crack growth in finite elasticity

Abstract

We propose a time-space discretization of a general notion of quasistatic growth of brittle fractures in elastic bodies proposed in [13] by G. Dal Maso, G.A. Francfort, and R. Toader, which takes into account body forces and surface loads. We employ adaptive triangulations and prove convergence results for the total, elastic and surface energies. In the case in which the elastic energy is strictly convex, we prove also a convergence result for the deformations.

Paper Structure

This paper contains 8 sections, 13 theorems, 256 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The quasistatic crack growth of Dal Maso-Francfort-Toader
  4. The finite element space and the transfer of jump
  5. Preexisting cracks and their approximation
  6. The discontinuous finite element approximation
  7. The approximation result
  8. The strictly convex case

Key Result

Theorem 2.1

Let $A$ be an open and bounded subset of ${\mathbb R}^n$. Let $g(x,u): \, A\times {\mathbb R}^m \to [0,\infty]$ be a Borel function, lower semicontinuous in $u$ and satisfying the condition Let $(u_k)_{k \in {\mathbb N}}$ be a sequence in $GSBV^p(A;{\mathbb R}^m)$ such that Then there exists a subsequence $(u_{k_h})_{h \in {\mathbb N}}$ and a function $u \in GSBV^p (A;{\mathbb R}^m)$ such that

Theorems & Definitions (23)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • Proposition 5.1
  • proof
  • ...and 13 more