Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global existence and uniqueness of weak solutions for a Willis-type model of elastodynamics

Thomas Blesgen, Patrizio Neff

TL;DR

This work establishes global existence and uniqueness of weak solutions for a Willis-type elastodynamics system in both the whole space and bounded domains by recasting the model as a linear symmetric hyperbolic first-order system. A novel symmetry requirement, $S_{ijk}=S_{jki}$ in addition to $S_{ijk}=S_{jik}$, makes the Willis coupling tensor totally symmetric, enabling a rigorous hyperbolic formulation and application of standard well-posedness theory. Under these structural assumptions and suitable regularity of data, the authors obtain weak-solution results, with classical solutions for regular data, and detailed tangential-regularity results in bounded domains via Shizuta95/Rauch85-type theory. The analysis provides a mathematically rigorous foundation for homogenized Willis closures in elastodynamics of metamaterials, while highlighting the need to verify the new symmetry condition experimentally and physically interpret its implications.

Abstract

The existence and uniqueness of weak solutions is shown for a system related to the Willis model of elastodynamics. Both the whole space case and the case of a bounded smooth domain are studied. To this end the equations are reformulated as a linear symmetric hyperbolic system of first order and the existing theory for such systems is applied. If the initial and boundary data is regular enough, classical solutions are obtained. The possibility to transform the problem to a linear symmetric hyperbolic system hinges on a new symmetry condition on the Willis coupling tensor S, not yet considered in the literature. This condition demands that S is a totally symmetric third-order tensor.

Global existence and uniqueness of weak solutions for a Willis-type model of elastodynamics

TL;DR

This work establishes global existence and uniqueness of weak solutions for a Willis-type elastodynamics system in both the whole space and bounded domains by recasting the model as a linear symmetric hyperbolic first-order system. A novel symmetry requirement, in addition to , makes the Willis coupling tensor totally symmetric, enabling a rigorous hyperbolic formulation and application of standard well-posedness theory. Under these structural assumptions and suitable regularity of data, the authors obtain weak-solution results, with classical solutions for regular data, and detailed tangential-regularity results in bounded domains via Shizuta95/Rauch85-type theory. The analysis provides a mathematically rigorous foundation for homogenized Willis closures in elastodynamics of metamaterials, while highlighting the need to verify the new symmetry condition experimentally and physically interpret its implications.

Abstract

The existence and uniqueness of weak solutions is shown for a system related to the Willis model of elastodynamics. Both the whole space case and the case of a bounded smooth domain are studied. To this end the equations are reformulated as a linear symmetric hyperbolic system of first order and the existing theory for such systems is applied. If the initial and boundary data is regular enough, classical solutions are obtained. The possibility to transform the problem to a linear symmetric hyperbolic system hinges on a new symmetry condition on the Willis coupling tensor S, not yet considered in the literature. This condition demands that S is a totally symmetric third-order tensor.

Paper Structure

This paper contains 9 sections, 6 theorems, 110 equations.

Table of Contents

  1. The Willis model in elastomechanics
  2. Background for the Willis approach
  3. Invariance considerations
  4. Prerequisites and assumptions
  5. Reformulation of the problem
  6. Existence and uniqueness of weak solutions
  7. Existence theory for $\Omega=\mathbb{R}^n$
  8. Existence theory in a bounded domain
  9. Concluding remarks

Key Result

Lemma 1

Assume that the Willis coupling tensor satisfies the further symmetry relation Then the equations (mod)--(mod3) constitute a linear symmetric hyperbolic system of first order, i.e. they can be written as the mixed initial boundary value problem where $L$ is given by (linsym), $M(x)\in\mathbb{R}^{m\times m}$ for $x\in\Gamma$, and $w=w(x,t)\in\mathbb{R}^m$, $v_0\in\mathbb{R}^m$ are suitable vector

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 1: Existence and uniqueness for $\Omega=\mathbb{R}^n$
  • Corollary 1
  • Remark 3
  • Corollary 2: Classical Dirichlet boundary conditions
  • Definition 1
  • Definition 2
  • Definition 3
  • ...and 4 more