PSI - Issue 13

Alexey Smolin et al. / Procedia Structural Integrity 13 (2018) 680–685

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Alexey Smolin et al. / Structural Integrity Procedia 00 (2018) 000 – 000

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where η is the scale parameter, and β is the shape parameter. To estimate the values of η and β in the case of a small random sample size, it is recommended to use the median rank regression, which is reduced to the transformation of Eq. (1) to a linear equation, and to the linear approximation of this equation by means of simple least-square regression. In our work, we used a special package designed for the statistical programming language R , providing basic functionality needed to perform Weibull analysis available at http://r-forge.r-project.org/projects/abernethy/. 3. Modeling results Figure 2 s hows normalized Young’s modulus and compression strength of the composite versus volume fraction of inclusions for all range used in the computations (here E 0 and σ 0 are the values of Young’s modulus and strength limit of the porous ceramics, determined at first scale of the model). As one can see, the scatter for strength value (Fig. 2, b) is much larger than for elastic modulus (Fig. 2, a). Nevertheless, the modeling results in Fig. 2 can be well approximated by the power function (Smolin, 2014)   m C C С max 0 0     (2) where C max , C 0 , and m are adjustable parameters. The fitting curves are shown in Fig. 2 as cyan lines for the specimens with inclusion s oriented in one direction (these points are marked as “α y =45°”) and as magenta lines for the specimens of the second type of the composite structure (these points are marked as “rand”). As one can see, random orientation of cylindrical inclusions along the compression direction has no influence on the elastic modulus along this direction (Fig. 1, a), but increase the strength value up to 10 % (Fig. 1, b).

a b Fig. 2. (a) Normalized Young’s modulus versus volume fraction of inclusions and (b) normalized compression strength versus volume fraction of inclusions for two group of the composite specimens with unidirectional cylindrical inclusions (α y =45°) and randomly oriented inclusions (rand). Figure 3 depicts the resulting Poisson’s ratios along two lateral directions (axes X and Y ) versus volume fraction of soft inclusions for two types of the composite structure. First, this figures clearly show, that scatters of the Poisson’s ratio values is rather large for the model specimens and its dependence on volume fraction can be well approximated by linear function. For unidirectional inclusions, the best fit gives an approximately constant value of Poison’s ratios along both directions. T he average value of Poisson’s ratio ν x along axis X (Fig. 3, a) is just a little bit smaller than Poisson’s ratio ν y along axis Y (Fig. 3, b). For the specimens with randomly oriented inclusions, the values of Poisson’s ratio along both axes are approximately the same taking into account a large scatter of the values. It is interesting to note that for this type of the composite structure the magn itude of Poisson’s ratio decreases with increasing the volume fraction of soft inclusions (magenta lines in Fig. 3). But we have to note, that this may be just a consequence of a smaller range of the modeled fraction of this type of inclusions. Indeed, if we try