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Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. II

Kirill Cherednichenko, Alexander Kiselev, Igor Velčić, Josip Žubrinić

Abstract

We construct an order-sharp theory for a double-porosity model in the full linear elasticity setup. Crucially, we uncover time and frequency dispersive properties of highly oscillatory elastic composites.

Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. II

Abstract

We construct an order-sharp theory for a double-porosity model in the full linear elasticity setup. Crucially, we uncover time and frequency dispersive properties of highly oscillatory elastic composites.
Paper Structure (32 sections, 65 theorems, 355 equations, 4 figures)

This paper contains 32 sections, 65 theorems, 355 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Notation
  4. Operator of linear elasticity
  5. Main results
  6. Gelfand transform
  7. Operator theoretic approach: Ryzhov triples
  8. Abstract notion of a Ryzhov triple
  9. Operators associated with boundary value problems
  10. Differential operators of linear elasticity
  11. Lift operators
  12. Dirichlet-to-Neumann maps
  13. Transmission problem: reformulation in terms of Ryzhov triple
  14. Transmission problem: Ryzhov triple asymptotics
  15. Lift operators: asymptotic properties
  16. ...and 17 more sections

Key Result

Lemma 2.3

There tensor $\mathbb{A}_{\rm macro}$ is symmetric, in the sense that $[\mathbb{A}_{\rm macro}]_{ijkl} = [\mathbb{A}_{\rm macro}]_{jikl} =[\mathbb{A}_{\rm macro}]_{klij},$$i,j,k,l\in\{1,2,3\},$ and positive definite: there exists a constant $\eta>0$ such that $\mathbb{A}_{\rm macro}\boldsymbol \xi :

Figures (4)

  • Figure 1: The illustration of a highly oscillating composite material consisting of stiff matrix with soft inclusions
  • Figure 2: Examples of eigenfunctions $\varphi$ of $\mathcal{A}_{0,0}^{\rm soft}$ with $\langle \varphi \rangle\neq 0$ and the associated eigenvalues $\eta$. For each case, the first picture shows the deformation $x \mapsto x + \varphi(x)$ while the second shows the displacement vector field $x \mapsto \varphi(x)$.
  • Figure 3: Plots of the eigenvalues $\beta_1(z)$, $\beta_2(z)$ of the "truncation" $\mathcal{B}^n(z)$, see \ref{['Btruncation']}, for $n=11.$ The $x$-axis represents the frequency $z$, the blue curve is the function $\beta_1(z)$ and the red curve is $\beta_2(z)$. The regions of $z$ for which both $\beta_1(z)$ and $\beta_2(z)$ are negative yield no solutions to \ref{['drelation']}.
  • Figure 4: Solutions $(\chi,z)$ to the dispersion relation \ref{['drelation']}: $\chi$ is horizontal, $z$ is vertical. Left panel: side view in the direction of $\chi_2;$ right panel: 3D view. The number of dispersion surfaces at every $z$ is the number of non-negative eigenvalues of $\mathcal{B}(z)$. As $\varepsilon\to0,$ the gaps between the surfaces converge to the regions in which $\mathcal{B}(z)$ is negative-definite.

Theorems & Definitions (154)

  • Definition 2.2: Macroscopic operator
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • proof
  • Remark 2.8
  • ...and 144 more