Differential inclusions and polycrystals

TL;DR

This work analyzes the differential inclusion $\nabla u\in K(S)$ for a diagonal $S$ with distinct eigenvalues, where $K(S)=\{\lambda R^tSR: \lambda\in\mathbb{R}, R\in SO(3)\}$, motivated by polycrystal conductivity bounds. It develops an infinite-rank-lamination framework using seed sets ${\mathcal T}^{2}(S)$ to build trajectories in the unit-trace set $K^*(S)$ and introduces a new inner bound ${\mathcal L}(S)\subset K^*(S)$. The key advance is that ${\mathcal L}(S)$ strictly enlarges prior bounds (notably NM's ${\mathcal L}_{MN}(S)$) for non-uniaxial $S$ by constructing explicit rank-one connections and laminations from $S$ to seed uniaxial points $U_\alpha,U_\beta$, along with a straight-line attainability property for interior points. These results yield sharper inner bounds on the $G$-closure of polycrystal conductivities and provide a rapid geometrical scheme for designing optimal microgeometries, with stability under lamination to be addressed in future work.

Abstract

We study the differential inclusion $Du\in K$, where $K$ is an unbounded and rotationally invariant subset of the real symmetric $3\times 3$ matrices. We exhibit a subset of all possible average fields. The corresponding microgeometries are laminates of infinite rank. The problem originated in the search for the effective conductivity of polycrystalline composites. In the latter context, our result is an improvement of the previously known bounds established by Nesi $\&$ Milton, hence proving the optimality of a new full-measure class of microgeometries.

Figures (4)

• Figure 1: The projection of the unit-trace plane onto $\mathbb{R}^2$. Every $3\times 3$ real symmetric matrix, $S$, is visualized as a point in $\mathbb{R}^3$ corresponding to its eigenvalues, $(s_1,s_2,s_3)$ (up to permutation). The shaded triangle is the portion of the unit-trace plane formed by the convex hull of the canonical basis vectors. Figure \ref{['fig:2d-projection-a']} shows the plane and a matrix, $S$, on the plane viewed from the $(+,+,+)$-octant. Figure \ref{['fig:2d-projection-b']} shows the same thing but viewed from the $(-,-,+)$-octant. Figure \ref{['fig:2d-projection-c']} shows a third view of the same configuration giving rise to the 2D projection. Here, the viewer sees the plane orthogonally from behind the origin. In this figure, $0<s_1<s_2<s_3$.
• Figure 2: The dashed curves in both figures show the set of $\mathcal{T}^{2}(S)$ fields defined in \ref{['condnec1']} and \ref{['condnec2']}. The solid curves in the right figure show rank-one curves that are formed by laminating $S$ with $\mathcal{T}^{2}(S)$ fields, $T_\alpha$ satisfying \ref{['condnec1']} and $T_\beta$ satisfying \ref{['condnec2']}.
• Figure 3: 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, $U_\alpha$ and $U_\beta$ respectively, with the uniaxial 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 right figure compares the curves from Figure \ref{['fig:sextant-annotated']} to the set of $\mathcal{T}^{2}(S)$ fields shown in Figure \ref{['fig:T2-locations']}.
• Figure 4: The left two figures show comparisons between Figure \ref{['fig:hexagon-annotated']} with the constructions in NM. The shaded gray region is formed by the procedure described in NM. The blue and orange regions are additional fields found by the constructions in the present paper. They are new. The right figure shows the comparison between $\Gamma$ (solid lines), $\mathcal{T}^2(S)$ (dashed lines) and the boundary of $\mathcal{L}_{MN}(S)$ (dotted lines).

Theorems & Definitions (32)

• Theorem 2.1
• Lemma 2.2
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Lemma 3.4
• Remark 3.5
• Definition 4.1
• Remark 4.2
• Remark 4.3