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Asymptotic analysis of a clamped thin multidomain allowing for fractures and discontinuities

G. Carvalho, J. Matias, E. Zappale

Abstract

We consider a thin multidomain of $\mathbb R^3,$ consisting of a vertical rod upon a horizontal disk. The equilibrium configurations of the thin hyperelastic multidomain, allowing for fracture and damage, are described by means of a bulk energy density of the kind $W(\nabla U)$, where $W$ is a Borel function with linear growth and $\nabla U$ denotes the gradient of the displacement, i.e. a vector valued function $U:Ω\to \mathbb R^3$. By assuming that the two volumes tend to zero, under suitable boundary conditions and loads, and suitable assumptions of the rate of convergence of the two volumes, we prove that the limit model is well posed in the union of the limit domains, with dimensions, respectively, $1$ and $2$.

Asymptotic analysis of a clamped thin multidomain allowing for fractures and discontinuities

Abstract

We consider a thin multidomain of consisting of a vertical rod upon a horizontal disk. The equilibrium configurations of the thin hyperelastic multidomain, allowing for fracture and damage, are described by means of a bulk energy density of the kind , where is a Borel function with linear growth and denotes the gradient of the displacement, i.e. a vector valued function . By assuming that the two volumes tend to zero, under suitable boundary conditions and loads, and suitable assumptions of the rate of convergence of the two volumes, we prove that the limit model is well posed in the union of the limit domains, with dimensions, respectively, and .
Paper Structure (13 sections, 28 theorems, 172 equations, 1 figure)

This paper contains 13 sections, 28 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The original problem and the rescalings
  3. Preliminary results
  4. Convexity notions
  5. Main results
  6. Finite rate
  7. Infinite rate: $\ell=+\infty$.
  8. Case $\lim_{n\to \infty}\frac{h_n}{r_n}= 0$
  9. Case $\frac{h_n}{r_n} \to \infty$
  10. Case $l = 0$
  11. The super-linear case
  12. Case $\ell = +\infty$
  13. Case $\ell = 0$

Key Result

Theorem 1.3

Let $W\;:\; {\mathbb R}^{3{\times}3}\to {\mathbb R}$ be a Borel function satisfying coerci, and $F_n^\ell$ be as in Definition def:energies depending on the superscript $\ell$ defined in ell. Denote by $\hat{W}:{\mathbb R}^3\to \mathbb R$ the function given by and let $W_0:\mathbb R^{3\times 2}\to \mathbb R$ be the function defined Assume that the recession functions of $\hat{W}$ and $W_0$, $\ha

Figures (1)

  • Figure 1: Multidomain $\Omega_n$

Theorems & Definitions (63)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Definition 2.6
  • ...and 53 more