A sequential addition and migration method for generating microstructures of short fibers with prescribed length distribution

Abstract

We describe an algorithm for generating fiber-filled volume elements for use in computational homogenization schemes. The algorithm permits to prescribe both a length distribution and a fiber-orientation tensor of second order, and composites with industrial filler fraction can be generated. Typically, for short-fiber composites, data on the fiber-length distribution and on the volume-weighted fiber-orientation tensor of second order is available. We consider a model where the fiber orientation and the fiber length distributions are independent, i.e., uncoupled. We discuss the use of closure approximations for this case and report on identifying the describing parameters of the frequently used Weibull distribution for modeling the fiber-length distribution. We discuss how to integrate these procedures in the Sequential Addition and Migration algorithm, developed for fibers of equal length, and work out algorithmic modifications accounting for possibly rather long fibers. We investigate the capabilities of the introduced methodology for industrial short-fiber composites, demonstrating the rather low dispersion of the effective elastic moduli for the generated unit cells.

On the phase space of fourth-order fiber-orientation tensors

“…For a start, it permits assessing the realizability of a given fourth-order tensor as a fiber-orientation tensor in a simple and straightforward way. Consider, for instance, fiber-microstructure generators [18,19,73], which serve as the basis for full-field simulations and where the fourth-order fiber-orientation tensor serves as the input. Then, if the prescribed tensor does not belong to the set Cand(d), it cannot be realized by a fiber microstructure at all, independent of the size of the microstructure and the number of considered fibers.…”

“…Fiber-orientation tensors [1] date back to the far-reaching works [2,3] and describe the relevant features of the fiber-orientation distribution of discontinuous fiber-reinforced composites. Within the virtual development and design process of such composites [1,[4][5][6], fiber-orientation tensors appear in material modeling [7][8][9][10][11][12], microstructure generation [13][14][15][16][17][18][19], mold filling or flow simulations [20][21][22][23][24][25] and the experimental computer tomography analysis [26][27][28]. This wide field of application motivates a detailed understanding of the mathematical properties of fiber-orientation tensors.…”

“…A scientific topic which is intimately connected to the question on the phase space of fourth-order fiberorientation tensors, is fiber-orientation closure approximations. Such closure approximations [22,37,[56][57][58][59][60][61][62] are tensor-valued functions which postulate a functional relationship between a given second-order fiber-orientation tensor and an unknown fourth-order fiber-orientation tensor [19]. Identifying the phase space of fourth-order fiber-orientation tensors is essential to solve a fundamental problem of closure approximations -which fourth-order fiber-orientation tensors can be realized?…”

“…In particular, the fiber-orientation realization problem (2.18) may be studied in terms of the sets (2 19…”

An algorithm for generating microstructures of fiber‐reinforced composites with long fibers

“…For fiber segments whose length is smaller than the half of the minimum of the $$$ {Q}_i $$$ $$$ \left(i=1,2,3\right) $$$, it is therefore appropriate to replace the distance between the centers by the version modified with $$$ \mathcal{M} $$$ component‐wise. For longer fiber segments, a decomposition approach may be used, described in Mehta and Schneider 67(§3.3) …”