13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- short-range deviations are those arising within a unit cell, referred to the center-of-mass of the unit cell, while the long-range deviations are the displacements of the centers-of-mass of unit cells over gauges much greater than the unit cell [13]. 3. Virtual specimen generation The art of formulating reconstruction algorithms or geometry generators for stochastic heterogeneous materials and related problems in statistical physics has a long history, mostly in the study of granular materials (e.g., [21, 22]. Textiles contrast with alloys in that they comprise long, continuous fiber bundles of essentially infinite aspect ratio, interlaced in complex but systematic patterns, a fact that calls for specialized algorithms in a virtual specimen generator, quite different in nature to those developed for statistically isotropic multi-phase materials (e.g., [23] and references therein). The simplest geometry generator for a textile directly exploits the linear continuity of tows: a Markov Chain formulation generates fluctuations in any tow cross-sectional characteristic by marching systematically along the tow’s length (Fig. 2) [24]. The key element of the Markov Chain is the Probability Transition Matrix (PTM), which determines the deviation and correlation length of any variable. The PTM is calibrated against measured statistical data, thus guaranteeing that the virtual specimens possess the same statistics as the real specimens that have been imaged. The Markov Chain formulation is very efficient and physically appropriate for the textile reconstruction problem, provided the dominant correlations are those along a tow, with correlations between tows relatively weak. It can be adapted to generate tows with 3D structure (Fig. 3) [25]. Figure 2. A μCT image yields statistics that are matched by generated virtual specimens. In this schematic, the generated tow structure at the left of the collage represents tows by 1D loci, suitable for use in the Binary Model [24, 26]. As it is described above, the Markov Chain formulation provides a purely empirical approach to virtual specimen generation: it simply matches statistical data from experiment. Models of the mechanics of fiber tow or preform deformation, which have been the basis for most other attempts to generate realistic geometric models of textiles, are not used. There is therefore no risk of errors flowing from uncertainty in the constitutive laws assumed to characterize such deformation, or in misrepresentation of the conditions during which preform fabrication or handling are carried out, including loading boundary conditions or the presence or absence of lubricating agents. The preform is analyzed as it is, in its final disposition.
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