ZAMM · Z. Angew. Math. Mech. 93, No. 5, 346 – 366 (2013) / DOI 10.1002/zamm.201100190 Characterization and statistical modeling of irregular porosity in carbon/carbon composites based on X-ray microtomography data Borys Drach, Andrew Drach, and Igor Tsukrov ∗ Mechanical Engineering Department, University of New Hampshire, Durham, NH (USA) University of New Hampshire, Durham, NH 03824, USA Received 27 December 2011, revised 9 March 2012, accepted 13 March 2012 Published online 29 May 2012 Key words Carbon-carbon composites, microcomputed tomography, 3D pores of irregular shapes, statistical analysis, effective elastic properties, design of experiments. Statistical analysis procedure is proposed to characterize volume, shape and orientation distribution of pores in chemical vapor infiltrated carbon/carbon composites. The microstructure data is provided by X-ray microtomography. To charac- terize orientation distribution of pores, probability distribution functions of pore volume, orientation angles and principal moments of inertia are constructed. Based on this information, a statistically significant range of pore geometry parameters is determined for evaluation of their contribution to the effective elastic properties of the material. Using the design of ex- periment approach, a subset of 53 pores is selected for the finite element simulations. The results are analyzed to construct a 3-factor stochastic model of a pore contribution to the overall elastic response based on its geometric parameters. It is determined that the non-dimensionalized surface to volume ratio factor plays an important role for pores in this type of ma- terial. A 4-factor model incorporating this ratio is proposed. The model is validated by direct finite element simulations for a set of 150 randomly selected pores not included in the initial subset. The accuracy of the proposed approach is compared with the traditionally used approximation of pores by equivalent ellipsoids. c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction In this paper we propose an approach to characterization and statistical modeling of contribution of irregularly shaped pores to the overall elastic properties of porous materials. We illustrate our approach by considering a sample of the carbon/carbon composite (C/C) manufactured by chemical vapor infiltration (CVI) of carbon fiber preform. The CVI process includes deposition of carbon particles from the carbon containing gas, such as methane or ethanol, on the carbon fibers placed in a reactor with elevated temperature and pressure (see Benzinger and H¨ uttinger [2], Li et al. [20] This manufacturing process results in a porous material (typical porosities are in the 2–15% range) with pores having highly irregular shapes, as illustrated in Fig. 1 (Note that other C/C manufacturing methods also produce irregularly shaped pores, see Tomkova et al. [38]). The shapes of the pores are influenced by the local arrangement of fibers, the infiltration parameters and carbon deposition rates, and, possibly, by the consecutive thermal treatment of the composite. Characterization and modeling of C/C material constituents including carbon fiber bundles (Hashin [15], Tsukrov and Drach [39]), pyrolytic carbon (Reznik and H¨ uttinger [32], B¨ ohlke et al. [5], Gross et al. [14]) and pores approximated by ellipsoids (Piat et al. [29, 30]) are discussed elsewhere. The objective of this paper is to characterize distribution of irregular pores in the material and develop a statistical micromechanical model to predict the overall stiffness based on the proper choice of morphological parameters reflecting pore shapes and orientations. Note that the spatial distribution effects, e.g. clustering of pores, are not included in the scope of the paper as they were not observed in the considered specimen; the spatial distribution of the voids is assumed to be uniform. Our studies are based on the X-ray microtomography (μCT) information obtained as described in Gebert et al. [13]). Microtomography is routinely utilized in micromechanical modeling to characterize microstructure of heterogeneous mate- rials. The obtained data is usually either directly used to develop a finite element mesh reproducing the scanned microstruc- ture as in Pahr and Zysset [27] or processed to determine a typical ellipsoidal inhomogeneity and utilize it in a certain homogenization procedure as in Borbely et al. [4]. To the best of the authors’ knowledge, no statistical models directly incorporating the μCT-determined irregular heterogeneity shapes in the predictions of elastic properties have been reported in the literature. One of the techniques developed for irregularly shaped pores involves combination of analytical micromechanical mod- eling with numerical approaches to evaluate contributions of individual pores to the effective elastic properties, see Tsukrov and Novak [40], Sevostianov et al. [36], Drach et al. [7]. In particular, in Drach et al. [7] the finite element analysis (FEA) ∗ Corresponding author E-mail: igor.tsukrov@unh.edu, phone: +1 (603) 862-2086, fax: +1 (603) 862-1865 c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim