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Large deformations in terms of stretch and rotation and local solution to the non-stationary problem

Abramo Agosti, Michel Fremond

Abstract

In this paper we consider and generalize a model, recently proposed and analytically investigated in its quasi-stationary approximation by the authors, for visco-elasticity with large deformations and conditional compatibility, where the independent variables are the stretch and the rotation tensors. The model takes the form of a system of integro-differential coupled equations. Here, its derivation is generalized to consider mixed boundary conditions, which may represent a wider range of physical applications then the case with Dirichlet boundary conditions considered in our previous contribution. This also introduces nontrivial technical difficulties in the theoretical framework, related to the definition and the regularity of the solutions of elliptic operators with mixed boundary conditions. As a novel contribution, we develop the analysis of the fully non-stationary version of the system where we consider inertia. In this context, we prove the existence of a local in time weak solution in three space dimensions, employing techniques from PDEs and convex analysis.

Large deformations in terms of stretch and rotation and local solution to the non-stationary problem

Abstract

In this paper we consider and generalize a model, recently proposed and analytically investigated in its quasi-stationary approximation by the authors, for visco-elasticity with large deformations and conditional compatibility, where the independent variables are the stretch and the rotation tensors. The model takes the form of a system of integro-differential coupled equations. Here, its derivation is generalized to consider mixed boundary conditions, which may represent a wider range of physical applications then the case with Dirichlet boundary conditions considered in our previous contribution. This also introduces nontrivial technical difficulties in the theoretical framework, related to the definition and the regularity of the solutions of elliptic operators with mixed boundary conditions. As a novel contribution, we develop the analysis of the fully non-stationary version of the system where we consider inertia. In this context, we prove the existence of a local in time weak solution in three space dimensions, employing techniques from PDEs and convex analysis.
Paper Structure (16 sections, 4 theorems, 175 equations)

This paper contains 16 sections, 4 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Geometrical and functional setting
  4. Green functions with mixed boundary conditions
  5. Helmholtz--Hodge decomposition for vector fields with mixed boundary conditions.
  6. Functional inequalities
  7. Model derivation and main result
  8. Proof of the main result
  9. Regularization
  10. Faedo--Galerkin approximation
  11. A priori estimates
  12. First a priori estimate
  13. Second a priori estimate
  14. Higher order estimates for the displacement and the defect variables
  15. Passing to the limit
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\mathcal{D}_a\subset \mathbb{R}^3$ be an open bounded and simply connected domain with Lipschitz boundary $\Gamma_a:=\partial \mathcal{D}_a$. Let us assume that $\Gamma_a=\Gamma_D\cup \Gamma_N$, where $\Gamma_D,\Gamma_N$ are connected and Lipschitz continuous subsets of $\Gamma_a$ with positive where $c\in \mathbb{R}$ is a constant. For any $\vec{\xi}\in L^2(\mathcal{D}_a,\mathbb{R}^3)$, ther

Theorems & Definitions (13)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 3 more