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Stability under lamination and polycrystalline effective conductivity

Nathan Albin, Vincenzo Nesi, Mariapia Palombaro

TL;DR

This work proves stability under lamination for the inner bound ${\mathcal{L}}(S)$ of the set of attainable effective conductivities ${K^*(S)}$ in a three-dimensional polycrystal context. By leveraging invariant coordinates and constructing the curves $\Gamma_{\alpha}$ and $\Gamma_{\beta}$, the authors demonstrate that ${\mathcal{L}}(S)$ is closed under rank-one laminations, contributing to a more detailed understanding of the $G$-closure in 3D polycrystal conductivity. The analysis carefully analyzes rank-one connections along the $\gamma_{\alpha}$ and $\gamma_{\beta}$ paths, deriving explicit invariants and slope formulas that establish inward-tangent behavior and optimality of lamination trajectories. Together, these results provide a robust inner bound that advances the characterization of possible effective conductivities in complex polycrystalline composites.

Abstract

We prove the stability under lamination of a set of real, symmetric 3$\times$3 matrices that can be viewed as a subset of the effective conductivities of a polycrystal. Constructed in a companion paper, such set in combination with several previous constructions provides the best inner bound known so far on the $G$-closure of a three dimensional polycrystal.

Stability under lamination and polycrystalline effective conductivity

TL;DR

This work proves stability under lamination for the inner bound of the set of attainable effective conductivities in a three-dimensional polycrystal context. By leveraging invariant coordinates and constructing the curves and , the authors demonstrate that is closed under rank-one laminations, contributing to a more detailed understanding of the -closure in 3D polycrystal conductivity. The analysis carefully analyzes rank-one connections along the and paths, deriving explicit invariants and slope formulas that establish inward-tangent behavior and optimality of lamination trajectories. Together, these results provide a robust inner bound that advances the characterization of possible effective conductivities in complex polycrystalline composites.

Abstract

We prove the stability under lamination of a set of real, symmetric 33 matrices that can be viewed as a subset of the effective conductivities of a polycrystal. Constructed in a companion paper, such set in combination with several previous constructions provides the best inner bound known so far on the -closure of a three dimensional polycrystal.

Paper Structure

This paper contains 12 sections, 13 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary definitions and results
  3. The set ${\mathcal{L}}(S)$
  4. Stability under lamination of the set ${\mathcal{L}}(S)$
  5. Proving Steps 1-3
  6. Step 1: proving \ref{['tau0']}
  7. Step 2: proving \ref{['bohhh']}
  8. Step 3: proving \ref{['slopeEoptimal']}
  9. Computation of the invariants
  10. The $\gamma_{\beta}$-case and the proof of Lemma \ref{['davveroultimo']}
  11. The $\gamma_{\beta}$-case
  12. Proof of Lemma \ref{['davveroultimo']}

Key Result

Lemma 2.5

Let $F\in \mathcal{D}_d$, $G\in \mathcal{D}$. If $G\in \mathcal{T}^1(F)$, then $\lambda \in A(F,G)$. Conversely, if $A(F,G) \neq \emptyset$, then $G\in \mathcal{T}^1(F)$ and, in particular, for each $\lambda\in A(F,G)$, the vector $n=(n_1,n_2,n_3)$ that satisfies oldbp is determined, not uniquely In particular $F\in \mathcal{T}^{1}(F)$ since

Figures (1)

  • Figure 1: The left figure shows important fields in one sextant ($s_1\le s_2 \le s_3$) of the unit-trace plane. The outer quadrilateral connects the field $S$ to the isotropic field $\frac{1}{3}I$ and the two uni-axial points, $C_{\alpha}=(\frac{s_1+s_2}{2},\frac{s_1+s_2}{2},s_3)$ and $C_{\beta}=(s_1,\frac{s_2+s_3}{2},\frac{s_2+s_3}{2})$. The curves $\Gamma_{\alpha}$ and $\Gamma_{\beta}$ from Definition \ref{['def33']} are also shown, together with their intersections with the uniaxial lines (the dashed lines). The center figure shows the construction in Definition \ref{['def:Gamma-construction']}. The set ${\mathcal{L}}(S)$ is enclosed by the union of the reflected copies of $\Gamma_\alpha, \Gamma_\beta$. The rightmost figure shows the fields in the sextant once again, but in the plane of the matrix invariants, $(i_2,i_3)$, which are used in the proof of the stability. Abusing notation we denote points by the same letters in (a)-(c).

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • ...and 25 more