From random microstructures to representative volume elements

Abstract

A unified treatment of random microstructures proposed in this contribution opens the way to efficient solutions of large-scale real world problems. The paper introduces a notion of statistically equivalent periodic unit cell (SEPUC) that replaces in a computational step the actual complex geometries on an arbitrary scale. A SEPUC is constructed such that its morphology conforms with images of real microstructures. Here, the appreciated two-point probability function and the lineal path function are employed to classify, from the statistical point of view, the geometrical arrangement of various material systems. Examples of statistically equivalent unit cells constructed for a unidirectional fibre tow, a plain weave textile composite and an irregular-coursed masonry wall are given. A specific result promoting the applicability of the SEPUC as a tool for the derivation of homogenized effective properties that are subsequently used in an independent macroscopic analysis is also presented.

Material characterization and modeling of carbon–carbon textile composites

“…A lucid presentation of individual steps enabling the substitution of real microstructures by their simplified artificial representatives -the SEPUCs -is available, e.g. in [2,3,5] and additional references given below. Herein, these steps are briefly reviewed concentrating on the specifics of multi-layer woven composites.…”

“…While the application of PUCs in problems of strictly periodic media has a rich history, their introduction in the field of random or imperfect microstructures is still very much on the frontier, despite the fact that the roots for incorporating basic features of random microstructures into the formulation of a PUC were planted already in the mid 1990s [1] and extended further in [2,3] to give rise to what we now call the concept of the Statistically Equivalent Periodic Unit Cell (SEPUC). In contrast with traditional approaches, where parameters of the unit cell model are directly measured from available material samples, the SEPUC approach is based on their statistical characterization.…”

Pragmatic multi-scale and multi-physics analysis of Charles Bridge in Prague

“…[26,27,28]. In this framework, each heterogeneity pattern is characterized by a statistically equivalent periodic unit cell [29], which is subject to a prescribed loading history parametrized by the macroscopic strain tensor E. The influence of the surrounding material is again accounted for by using periodic boundary conditions, now imposed on the boundary of a unit cell. The effective (homogenized) material behaviour is deduced from the relation between the overall strain and the corresponding average stress Σ in the unit cell, typically specified in the form of uniaxial stress-stress relations or failure envelopes, e.g.…”

“…4(b,c), were treated on the same footing employing the concept of a Statistically Equivalent Periodic Unit Cell, which approximates a non-periodic masonry texture with a Periodic Unit Cell sharing the same statistical response with the original sample, see [29] for further discussion.…”

Stochastic Modeling of Chaotic Masonry via Mesostructural Characterization

“…To the authors' best knowledge, the only masonry-related study available in this field was presented byŠejnoha et al [43] in the framework of stochastic re-formulation of Hashin-Shtrikman variational principles due to Willis [44]. Additionally, the notion of a stochastic representative element can be adopted, based either on matching spatial statistics as originally proposed by Povirk in [32] and subsequently applied to masonry structures in [43,48], or deduced from the convergence of apparent macroscopic properties. The latter concept was proposed by Huet [17] in the deterministic setting, extended by Sab [34] to random media and implemented for historical masonry structures by Cluni and Gusella [7,14].…”