102 research outputs found
Characterizing digital microstructures by the Minkowski‐based quadratic normal tensor
For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean field homogenization or as target value for generating synthetic microstructures
Characterization of Maximally Random Jammed Sphere Packings. III. Transport and Electromagnetic Properties via Correlation Functions
In the first two papers of this series, we characterized the structure of
maximally random jammed (MRJ) sphere packings across length scales by computing
a variety of different correlation functions, spectral functions, hole
probabilities, and local density fluctuations. From the remarkable structural
features of the MRJ packings, especially its disordered hyperuniformity,
exceptional physical properties can be expected. Here, we employ these
structural descriptors to estimate effective transport and electromagnetic
properties via rigorous bounds, exact expansions, and accurate analytical
approximation formulas. These property formulas include interfacial bounds as
well as universal scaling laws for the mean survival time and the fluid
permeability. We also estimate the principal relaxation time associated with
Brownian motion among perfectly absorbing traps. For the propagation of
electromagnetic waves in the long-wavelength limit, we show that a dispersion
of dielectric MRJ spheres within a matrix of another dielectric material forms,
to a very good approximation, a dissipationless disordered and isotropic
two-phase medium for any phase dielectric contrast ratio. We compare the
effective properties of the MRJ sphere packings to those of overlapping
spheres, equilibrium hard-sphere packings, and lattices of hard spheres.
Moreover, we generalize results to micro- and macroscopically anisotropic
packings of spheroids with tensorial effective properties. The analytic bounds
predict the qualitative trend in the physical properties associated with these
structures, which provides guidance to more time-consuming simulations and
experiments. They especially provide impetus for experiments to design
materials with unique bulk properties resulting from hyperuniformity, including
structural-color and color-sensing applications.Comment: 19 pages, 16 Figure
Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors
Gaussian random fields are among the most important models of amorphous
spatial structures and appear across length scales in a variety of physical,
biological, and geological applications, from composite materials to geospatial
data. Anisotropy in such systems can sensitively and comprehensively be
characterized by the so-called Minkowski tensors from integral geometry. Here,
we analytically calculate the expected Minkowski tensors of arbitrary rank for
the level sets of Gaussian random fields. The explicit expressions for
interfacial Minkowski tensors are confirmed in detailed simulations. We
demonstrate how the Minkowski tensors detect and characterize the anisotropy of
the level sets, and we clarify which shape information is contained in the
Minkowski tensors of different rank. Using an irreducible representation of the
Minkowski tensors in the Euclidean plane, we show that higher-rank tensors
indeed contain additional anisotropy information compared to a rank two tensor.
Surprisingly, we can nevertheless predict this information from the second-rank
tensor if we assume that the random field is Gaussian. This relation between
tensors of different rank is independent of the details of the model. It is,
therefore, useful for a null hypothesis test that detects non-Gaussianities in
anisotropic random fields
A computational multi-scale approach for brittle materials
Materials of industrial interest often show a complex microstructure which directly influences their macroscopic material behavior. For simulations on the component scale, multi-scale methods may exploit this microstructural information. This work is devoted to a multi-scale approach for brittle materials. Based on a homogenization result for free discontinuity problems, we present FFT-based methods to compute the effective crack energy of heterogeneous materials with complex microstructures
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