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Statistical three-dimensional reconstruction of co-continuous ceramic composites
Finite Elements in Analysis and Design 114 (2016) 85–91
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel
Review
Statistical three-dimensional reconstruction of co-continuous ceramic composites Qingxiang Wang, Hongmei Zhang n, Hongnian Cai, Qunbo Fan, Xu Zhang National Key Laboratory of Science and Technology on Materials under Shock and Impact, School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 April 2015 Received in revised form 14 December 2015 Accepted 17 December 2015 Available online 19 March 2016
In this study, a 3D solid model of co-continuous ceramic composites was reconstructed using a single 2D microstructure section image. First, the volume fractions and distribution functions of both metallic alloy and ceramic phases were statistically obtained by analyzing a single 2D microstructure SEM image after binarization. The distribution functions of the two phases were then revised to generate a Gaussian random field. A 3D solid model of co-continuous ceramic composites was obtained through 3D fast Fourier transform. A Micro-CT model was proposed for comparison. Volume fraction and distribution function differences between the reconstructed model and the Micro-CT model were acceptable within the tolerance range. Results confirmed that the reproduced solid model retained the original geometrical characteristics of the real 2D microstructure SEM image as well as the micro-CT model. & 2016 Elsevier B.V. All rights reserved.
Keywords: Co-continuous ceramic composites Mathematical statistics Microstructure image 3D reconstruction
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric characteristics of the statistics of co-continuous ceramic composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Digitalization of metallic alloy and ceramic phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Distribution statistics of metallic alloy and ceramic phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Revision of covariance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Establishment of a co-continuous ceramic composite model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Binarization of 2D microstructure SEM image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Calculation and revision of covariance function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Calculation of spectral function and construction of complex conjugate matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Non-linear filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Visualization of 3D binary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Model verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Three-dimensional co-continuous ceramic composites are ceramic matrix composites consisting of metallic alloy and ceramic phases. Both phases exhibit 3D interpenetrating characteristics.
n
Corresponding author. Tel.: þ86 10 68913951 605; fax.: þ86 10 6891 3951 608. E-mail address: [email protected] (H. Zhang).
http://dx.doi.org/10.1016/j.finel.2015.12.010 0168-874X/& 2016 Elsevier B.V. All rights reserved.
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The 3D co-continuous ceramic composites were discovered by Breslin et al. at Ohio University [1]. For different service conditions, the ingredient and composition of the metallic alloy and ceramic phases vary in different kinds of 3D co-continuous ceramic composites. Prior to the application of the 3D co-continuous ceramic composites, experiments should be repeated to adjust the ingredient and composition of the metallic alloy and ceramic phases; the process is usually long and costly. Numerical simulation, an important method in studying the properties of co-continuous
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ceramic composites, can effectively shorten the test cycle, thereby reducing the costs. The 3D solid model construction is the first as well as an important step of the numerical simulation for the accuracy in simulation results. The model construction of cocontinuous ceramic composites is one of the difficult problems because of the complex spatial geometric topology of the composites. Metal–ceramic composites usually include particle-reinforced composites, fiber-reinforced composites, and co-continuous ceramic composites. For particle-reinforced composites, researchers often simplify the solid model into a 3D model of uniform particle distribution with periodic arrangement or completely random distribution. For example, Guild [2] has adopted a cylindrical unit cell model to predict the properties of polyethylene hydroxyapatite composites. Natarajan et al. [3] have used a simulated annealing method to reconstruct the 3D model of particlereinforced composites with random distribution of four-phase particles; the proposed method demonstrated good convergence and repeatability. Chawla et al. [4–6] have established a particlereinforced composite model to predict the Young's modulus and stress–strain behavior by using a serial sectioning method. For fiber-reinforced composites, researchers usually use 2D axisymmetric unit cell models for numerical simulation [7–9] of the mechanical behavior under loads. For 3D co-continuous ceramic composites, a 2D model hardly describes the spatial structure characteristics because the ceramic and metallic alloy phases are continuously distributed in the 3D space and interpenetrated with each other to form a complex interwoven net structure. The microstructure-based 3D co-continuous ceramic composite solid model is more difficult to reconstruct than that with particlereinforced composites. Researchers often construct a 3D solid model based on the real microstructure of 3D co-continuous ceramic composites and other materials which have similar microstructure with the composites. Węglewski et al. [10] and Altaf et al. [11] have built a 3D composite solid model via CT scanning method. Joshi [12] and Quiblier [13] proposed and modified Gaussian method firstly to research porous media reconstruction problem to get a 3D porous media model. Hazlett et al. [14] applied simulated annealing method to reconstruct 3D model of reservoir rock, and flow properties were computed from an accurate depiction of the porosity network in three dimensions. Wang et al. [15] had proposed a quartet structure generation set method based on the stochastic cluster growth theory for generating more realistic microstructures of porous media, and predicted the effective thermal conductivities of porous media with multiphase structure and stochastic complex geometries. According to the above researches, these physical experimental methods subject to be limited by the experimental condition, resolution and accuracy. For serial sectioning method, a time-consuming grind and polish process may change microstructure of the material. Because these automatic serial sectioning instruments [16] are merely a few around the globe, most of the studies are accomplished by manual work. For the CT tomography method such as micro-CT, the scanning result depends on the absorptivity of the sample. If phases of the sample have a close absorptivity, the contrast of the phases is not obvious under X-ray such as the SiC/Al composites. So the physical method is confined in modeling process of co-continuous ceramic composites. Among the numerical methods, the Gaussian statistical method is the most appropriate for the modeling process of co-continuous ceramic composites which can save much time and improve the accuracy. The present study proposed a model reconstruction method, namely, statistical method [17,18] of 3D co-continuous ceramic composites based on a single 2D microstructure SEM image. Distribution functions and covariance functions of metallic alloy and ceramic phases were statistically calculated. A random function
was used to construct the Fourier transform coefficient matrix. After the nonlinear filter process, a 3D solid model was generated in line with Gaussian field distribution. Statistical method can avoid repeated capturing of 2D images and can simply and quickly reconstruct a 3D co-continuous ceramic composite solid model with the same characteristics as those of the real structure of cocontinuous ceramic composites under low-cost condition. The reconstructed process is a part of pre-processing in numerical simulation. The reconstructed model can be imported into the finite element software for a further study of its properties.
2. Geometric characteristics of the statistics of co-continuous ceramic composites Co-continuous ceramic composites have a complex spatial topological geometry microstructure. The metallic and ceramic phases are both constituent and interpenetrating. The special structure is obtained by its forming process. The process is indicated as following: firstly, the ceramic particles are accumulated by a stacking manner. Then the ceramic phase is prepared by sintering to form a skeleton. Interpenetrating ceramic phase is formed as well as the interpenetrating pores. Secondly, melt metal is infiltrated in the pores by vacuum pressure infiltration. As a result, both ceramic and metallic phases are continuous in the composites. For the complex continuous character, the 3D reconstruction of the composites is a bottleneck up to now. In order to reconstruct 3D model of co-continuous ceramic composites in this paper, the first step is to calculate geometric characteristic statistics. SEM image is the basement of the algorithm which is shown in Fig. 1. In this paper, the metallic and ceramic phases of the co-continuous ceramic composites are Al and SiC respectively. 2.1. Digitalization of metallic alloy and ceramic phases Metallic alloy and ceramic phases are interpenetrating in the 3D co-continuous ceramic composites exhibiting complex geometric topology. In the reconstruction process of 3D geometric model, the spatial distribution of the two phases can be represented as: 1 P A metallicalloyphase Z ðP Þ ¼ ð1Þ P A ceramicphase 0 where P represents an arbitrary point P(x, y, z) in space, and Z(P)
Fig. 1. SEM image of co-continuous ceramic composites.
Q. Wang et al. / Finite Elements in Analysis and Design 114 (2016) 85–91
represents the material property for the point P(x, y, z) of the 3D co-continuous ceramic composites. Eq. (1) shows the spatial distribution of the two phases of any 3D co-continuous ceramic composites. By traversing all the points of co-continuous ceramic composites, the volume fractions of the metallic alloy and ceramic phases can be calculated using a statistical method. Volume fraction of the metallic alloy phase can be obtained as Eq. (2) by calculating the average of Eq. (1). The volume fraction can be calculated by summing up the number of metallic alloy phase points and dividing by the total number of points. Accordingly, the volume fraction of the ceramic phase can be obtained by Eq. (3).
εmetal ¼ Z ðP Þ
ð2Þ
εceramic ¼ 1 εmetal
ð3Þ
2.2. Distribution statistics of metallic alloy and ceramic phases The covariance function is used to characterize the distribution of the metallic alloy and ceramic phases by analyzing the 2D SEM image of 3D co-continuous ceramic composites. The covariance function represented as Rz(u) can be calculated by the following Eq. (4): Rz ðuÞ ¼
½Z ðP Þ ε∙½Z ðP þuÞ ε ðε ε2 Þ
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the covariance function Rz(u) should be calculated to the revised function which is denoted as Ry(u) though mathematics. The mathematical processes [18] are explained in Eqs. (5)–(9): Rz ðuÞ ¼
1 X
ðC m Þ2 ∙Rm y ðuÞ
ð5Þ
m¼0
1 C m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2π ∙m!
cðyÞ ¼
Z
þ1 1
8 ε1 ffi > < pffiffiffiffiffiffiffiffiffiffiffi εð1 εÞ ε ffi > : pffiffiffiffiffiffiffiffiffiffiffi εð1 εÞ
y2 H m ðyÞdy cðyÞexp 2
pðyÞ r εmetal pðyÞ 4 εmetal
2 m y d y2 H m ðyÞ ¼ ð 1Þm exp m exp 2 dy 2 1 pðyÞ ¼ pffiffiffiffiffiffi 2π
Z
y2 dy exp 2 1 y
ð6Þ
ð7Þ
ð8Þ
ð9Þ
where the covariance function Rz(u) and volume fraction of metallic alloy phase εmetal can be obtained directly from Eqs. (2) and (4). Rz(u) is an m-order polynomial function based on Ry(u). The revised covariance function can be obtained by using Eqs. (5)–(9).
ð4Þ
where u represents a distance, and (P þu) represents the point with a distance u from the point P. Covariance function is the distribution relationship between the local phase particle and the one with a distance u from the local phase particle. This function contains the distribution characteristics of the metallic alloy and ceramic phases. The positive or negative result of the function indicates the correlation of the two particles in different spatial positions. If the value of the function is positive, a positive correlation is indicated, and the two particles belong to the same phase, which is either the metallic alloy or the ceramic phase. Conversely, if the value of the covariance function is negative, a negative correlation is indicated, and the two particles belong to different phases: one is metallic alloy phase, and the other is ceramic phase. This reconstruction method is based on the isotropic and homogeneous distribution of the metallic alloy and ceramic phases of the 3D co-continuous ceramic composites.
3. Establishment of a co-continuous ceramic composite model In this study, a 3D co-continuous ceramic composite solid geometry model was established by a statistical method. The flow chart of the statistical method is illustrated in Fig. 2. 3.1. Binarization of 2D microstructure SEM image The 2D SEM image of the 3D co-continuous ceramic composites is shown in Fig. 1. To facilitate computer recognition and programming, the 2D SEM image was binarized firstly. The result of binarization is shown in Fig. 3. The white area is the metallic alloy phase with volume fraction εmetal of 12.46%, and the black area is the ceramic phase with volume fraction εceramic of 87.54%. After binarization, the 2D gray image with 256 brightness levels was transformed into a binary image of two brightness levels consisting of only 0 and 1.
2.3. Revision of covariance function 3.2. Calculation and revision of covariance function To obtain a nearly similar 3D co-continuous ceramic composite solid model of the real microstructure, Gaussian field is a necessary calculating process. Hence, before the covariance function Rz(u) is applied to the 3D space, the distribution characteristics must meet some specific mathematical requirements. Furthermore,
The covariance functions of the x-axis and y-axis directions are compared in Fig. 4. Notably, the covariance functions of the x-axis and y-axis are the same because the 3D co-continuous ceramic composites are isotropic.
Fig. 2. Flow chart of the reconstruction process. (FFT, fast Fourier transform; iFFT, inverse fast Fourier transform).
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The revision of the covariance function is presented in Table 1. Different values of m were calculated. An m¼ 17 indicates that the result met the accuracy requirements substantially. When m 417, the result suggests that the calculating time was very long, and the result slightly changed. To save computing time and ensure the accuracy of the calculation results, m ¼17 was selected for this study. The revised covariance function after revision is shown in Fig. 5.
non-linear filter of Fourier coefficient matrix A. Non-linear filter process can be described by Eq. (10). After the non-linear filter transformation, 3D binary matrix, which showed hexahedral-like 3D matrix, was consisted of 0 and 1 in geometric space (Fig. 6). ( 1 PðyÞ r εmetal Z¼ ð10Þ 0 PðyÞ 4 εmetal
3.3. Calculation of spectral function and construction of complex conjugate matrix
3.5. Visualization of 3D binary matrix
The spectral density S(u) can be calculated from the covariance function Ry(u) using fast Fourier transform (FFT). The amplitude spectrum S0 (u) of covariance function Ry(u) can then be obtained by the spectral density S(u). Furthermore, amplitude spectrum S0 (u) can be transformed into complex Fourier coefficient matrix A by creating a random angle α. After iFFT, Fourier coefficient matrix Y can be finally obtained. To ensure that Y is a real matrix, the matrix A should meet the requirements of Hermitian matrix [17]. The structure of the algorithm is presented in Table 2. 3.4. Non-linear filter The 3D binary matrix Z, which exhibits the same characteristic functions as those of 2D real SEM image, can be obtained after a
The 3D binary matrix Z is a numerical matrix consisting of 0 and 1. Therefore, the 3D solid model is not a solid model. Binary matrix Z can be transformed into a visual 3D model via Eq. (1) by assigning the material properties to 0 and 1 of the binary matrix. Finally, the 3D solid model of co-continuous ceramic composites was obtained as Fig. 7a, and the solid model of the metallic alloy and ceramic phases is shown in Fig. 7b and c. The volume fractions of the metallic alloy and ceramic phases of the 3D solid model were verified. As a result, the volume fraction of the metallic alloy phase was 13.24%, and the ceramic phase was 86.76%. As comparative figures in the real 3D co-continuous ceramic composites, the volume fraction of the metallic alloy and ceramic phases were 17.07% and 82.93%, respectively. Compared with the reconstructed 3D model and real co-continuous ceramic composites, the volume fraction of the metallic alloy phase error was 4.4%, which is within the acceptable error range.
Table 1 Revised algorithm. Algorithm revision 1 2 3 4 5 6 7 8 9 10 11 12 13 Fig. 3. Binary image of 3D co-continuous ceramic composites.
Fig. 4. Covariance functions of x-axis and y-axis directions.
Calculate metallic alloy phase volume fraction; εmetal’the volume fraction of metallic alloy phase; m’the exponent of Ry(u); p(y) is the expression of Gaussian distribution; if p(y)r εmetal Calculate c(y); else Calculate c(y) by another way; end (for if in 5); each exponent 1 to m Calculate Hm(y) and Cm; Substitute Hm(y) and Cm into Eq. (5); end (for loop in 9); Solve the m-order Eq. (5);
Q. Wang et al. / Finite Elements in Analysis and Design 114 (2016) 85–91
89
Fig. 5. Covariance functions before and after revision.
Table 2 Transformed algorithm. Algorithm transformation 1 2 3 4 5 6 7 8 9 10 11 12
S(u)’spectral density; S0 (u)’amplitude spectrum; α’random angle; A’complex Fourier coefficient matrix Y’Fourier coefficient matrix S(u) is the fast Fourier transform of Ry(u); S0 (u) is the radication of S(u); for each 1 to u Choose a random value of α A [0,2π]; Calculate the real part and the imaginary part of A; end (for loop in 8); Y is the inverse fast Fourier transform of A;
Fig. 6. Calculation result after a non-linear filter.
4. Model verification A co-continuous metal/ceramic composites sample, which used the same forming process with the sample for SEM image, was scanned using a micro-CT (Sky-Scan 1172) scanner. Because the contrast between the SiC phase and Al phase was not obvious under the X-ray, the micro-CT model was obtained by scanning a single SiC skeleton phase. Then the Al phase was generated by a Boolean operation. The error between the reconstructed model and the real microstructure, which was caused by SiC skeleton microstructure deformation through the machining and interface change through the vacuum pressure infiltration process, was not concerned. The Simpleware commercial package [19] was used to reconstruct the 3D model from the obtained tomographic images. The micro-CT model is shown in Fig. 8.
In order to verify the generation-based optimization method, covariance functions and volume fractions of two phases were analyzed. The ceramic phase volume fractions of the reconstructed model and micro-CT model were 82.93% and 83.9%. The difference between the two volume fractions was 1.2%. And the covariance functions are shown in Fig. 9. The two parameters are all within an acceptable difference range.
5. Conclusions Reconstruction of a 3D co-continuous ceramic composite solid model is an important prerequisite for numerical simulation. Statistical method for reconstructing 3D solid model of the 3D cocontinuous ceramic composites was proposed, which established
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Fig. 7. 3D co-continuous ceramic composite solid model. (a) Composites model. (b) Metallic alloy phase. (c) Ceramic phase.
Fig. 8. Micro-CT co-continuous ceramic composite model.
Fig. 9. Covariance functions of reconstructed and micro-CT co-continuous ceramic composite model.
the 3D co-continuous ceramic composite solid model based only on a single 2D microstructure SEM image. Statistical distribution characteristics of metallic alloy and ceramic phases in 2D microstructure image were obtained by a mathematical method. A Gaussian distribution random field was constructed. The validity of the 3D reconstruction method was verified. And a micro-CT scanning model was proposed to further verify the reconstructed model. The 3D co-continuous ceramic composite reconstruction solid model retained the same geometric characteristics as those of the original real 2D microstructure SEM image as well as the micro-CT model. The volume fraction difference of the metallic alloy and ceramic phases was very small and within the acceptable difference range. The covariance functions were also very close. We will aim to reduce the error and expand the 3D reconstruction method in our future research.
However, the statistical method proposed by this paper still has some limitations. The volume fraction and covariance function are the main control parameters in the method. In the further study, more parameters would be proposed to describe the spatial characters of the composites. The method can obtain an accurate 3D solid model of co-continuous ceramic composites, but it cannot reconstruct 3D solid models of anisotropy materials as well as multi-phase material (more than two phases).
Acknowledgments The support from the Specialized Research Fund for the Doctoral Program of Higher Education of China (20091101120038) is gratefully acknowledged.
Q. Wang et al. / Finite Elements in Analysis and Design 114 (2016) 85–91
References [1] M.C. Breslin, Ceramic/Metal Composite Materials (C4) Produced, The Ohio State University, Columbus, USA, 1994. [2] F.J. Guild, W. Bonfield, Predictive modelling of hydroxyapatite–polyethylene composite, Biomaterials 14 (1993) 985–993. [3] C.K. Natarajan, M. Karel, H. Philippe, Reconstruction of periodic unit cells of multimodal random particulate composites using genetic algorithms, Comput. Mater. Sci. 42 (2008) 352–367. [4] N. Chawla, V.V. Ganesh, B. Wunsch, Three-dimensional (3D) microstructure visualization and finite element modeling of the mechanical behavior of SiC particle reinforced aluminum composites, Scr. Mater. 51 (2004) 161–165. [5] N. Chawla, Y.L. Shen, Mechanical behavior of particle reinforced metal matrix composites, Adv. Eng. Mater. 3 (2001) 357–370. [6] A. Ayyar, N. Chawla, Microstructure-based modeling of crack growth in particle reinforced composites, Compos. Sci. Technol. 66 (2006) 1980–1994. [7] J. Wang, L.C. Steven, S.G. Mogilevskaya, Numerical modeling of the elastic behavior of fiber-reinforced composites with inhomogeneous interphases, Compos. Sci. Technol. 66 (2006) 1–18. [8] S. Arnold, Reduction of thermal residual stress in advanced metallic composites based up on a compensating/complaint layer concept, J. Compos. Mater. 26 (1992) 1287–1309. [9] E. Folias, On the predication of failure at a fiber/matrix interface in a composites subjected to a transverse tensile load, J. Compos. Mater. 26 (1991) 1287–1309.
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[10] W. Węglewski, K. Bochenek, M. Basista, T. Schubert, U. Jehring, J. Litniewski, S. Mackiewicz, Comparative assessment of Young's modulus measurements of metal–ceramic composites using mechanical and non-destructive tests and micro-CT based computational modeling, Comput. Mater. Sci. 77 (2013) 19–30. [11] H.K. Altaf, A.C. Reaz, Fan-beam geometry based inversion algorithm in computed tomography (CT) for imaging of composite materials, Compos. Struct. 110 (2013) 297–304. [12] M. Joshi, A class of stochastic models for porous materials, University of Kansas, Lawrence, 1974. [13] J.A. Quiblier, A new 3D modeling technique for studying porous media, J. Colloid Interface Sci. 98 (1984) 84–102. [14] R.D. Hazlett, Statistical characterization and stochastic modeling of pore networks in relation to fluid flow, Math. Geol. 29 (1997) 801–822. [15] M.R. Wang, J.K. Wang, N. Pan, S.Y. Chen, Mesoscopic predictions of the effective thermal conductivity for microscale random porous media, Phys. Rev. E 75 (2007) 036702. [16] M.P. Echlin, A. Mottura, M. Wang, P.J. Mignone, D.P. Riley, G.V. Franks, T. M. Pollock, Three-dimensional characterization of the permeability of W–Cu composites using a new “TriBeam” technique, Acta Mater. 64 (2014) 307–315. [17] P.M. Adler, C.G. Jacquin, J.A. Quiblier, Flow in simulated porous media, Int. J. Multiph. Flow 16 (1990) 691–712. [18] E. Pardo-Igdzquiza, M. Chica-Olmo, The fourier integral method: an efficient spectral method for simulation of random fields, Math. Geol. 25 (1993) 180–217. [19] Simpleware Ltd., Exeter, UK. 〈http://www.simpleware.com/〉.
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel
Review
Statistical three-dimensional reconstruction of co-continuous ceramic composites Qingxiang Wang, Hongmei Zhang n, Hongnian Cai, Qunbo Fan, Xu Zhang National Key Laboratory of Science and Technology on Materials under Shock and Impact, School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China
art ic l e i nf o
a b s t r a c t
Article history: Received 13 April 2015 Received in revised form 14 December 2015 Accepted 17 December 2015 Available online 19 March 2016
In this study, a 3D solid model of co-continuous ceramic composites was reconstructed using a single 2D microstructure section image. First, the volume fractions and distribution functions of both metallic alloy and ceramic phases were statistically obtained by analyzing a single 2D microstructure SEM image after binarization. The distribution functions of the two phases were then revised to generate a Gaussian random field. A 3D solid model of co-continuous ceramic composites was obtained through 3D fast Fourier transform. A Micro-CT model was proposed for comparison. Volume fraction and distribution function differences between the reconstructed model and the Micro-CT model were acceptable within the tolerance range. Results confirmed that the reproduced solid model retained the original geometrical characteristics of the real 2D microstructure SEM image as well as the micro-CT model. & 2016 Elsevier B.V. All rights reserved.
Keywords: Co-continuous ceramic composites Mathematical statistics Microstructure image 3D reconstruction
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric characteristics of the statistics of co-continuous ceramic composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Digitalization of metallic alloy and ceramic phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Distribution statistics of metallic alloy and ceramic phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Revision of covariance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Establishment of a co-continuous ceramic composite model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Binarization of 2D microstructure SEM image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Calculation and revision of covariance function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Calculation of spectral function and construction of complex conjugate matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Non-linear filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Visualization of 3D binary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Model verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Three-dimensional co-continuous ceramic composites are ceramic matrix composites consisting of metallic alloy and ceramic phases. Both phases exhibit 3D interpenetrating characteristics.
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Corresponding author. Tel.: þ86 10 68913951 605; fax.: þ86 10 6891 3951 608. E-mail address: [email protected] (H. Zhang).
http://dx.doi.org/10.1016/j.finel.2015.12.010 0168-874X/& 2016 Elsevier B.V. All rights reserved.
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The 3D co-continuous ceramic composites were discovered by Breslin et al. at Ohio University [1]. For different service conditions, the ingredient and composition of the metallic alloy and ceramic phases vary in different kinds of 3D co-continuous ceramic composites. Prior to the application of the 3D co-continuous ceramic composites, experiments should be repeated to adjust the ingredient and composition of the metallic alloy and ceramic phases; the process is usually long and costly. Numerical simulation, an important method in studying the properties of co-continuous
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ceramic composites, can effectively shorten the test cycle, thereby reducing the costs. The 3D solid model construction is the first as well as an important step of the numerical simulation for the accuracy in simulation results. The model construction of cocontinuous ceramic composites is one of the difficult problems because of the complex spatial geometric topology of the composites. Metal–ceramic composites usually include particle-reinforced composites, fiber-reinforced composites, and co-continuous ceramic composites. For particle-reinforced composites, researchers often simplify the solid model into a 3D model of uniform particle distribution with periodic arrangement or completely random distribution. For example, Guild [2] has adopted a cylindrical unit cell model to predict the properties of polyethylene hydroxyapatite composites. Natarajan et al. [3] have used a simulated annealing method to reconstruct the 3D model of particlereinforced composites with random distribution of four-phase particles; the proposed method demonstrated good convergence and repeatability. Chawla et al. [4–6] have established a particlereinforced composite model to predict the Young's modulus and stress–strain behavior by using a serial sectioning method. For fiber-reinforced composites, researchers usually use 2D axisymmetric unit cell models for numerical simulation [7–9] of the mechanical behavior under loads. For 3D co-continuous ceramic composites, a 2D model hardly describes the spatial structure characteristics because the ceramic and metallic alloy phases are continuously distributed in the 3D space and interpenetrated with each other to form a complex interwoven net structure. The microstructure-based 3D co-continuous ceramic composite solid model is more difficult to reconstruct than that with particlereinforced composites. Researchers often construct a 3D solid model based on the real microstructure of 3D co-continuous ceramic composites and other materials which have similar microstructure with the composites. Węglewski et al. [10] and Altaf et al. [11] have built a 3D composite solid model via CT scanning method. Joshi [12] and Quiblier [13] proposed and modified Gaussian method firstly to research porous media reconstruction problem to get a 3D porous media model. Hazlett et al. [14] applied simulated annealing method to reconstruct 3D model of reservoir rock, and flow properties were computed from an accurate depiction of the porosity network in three dimensions. Wang et al. [15] had proposed a quartet structure generation set method based on the stochastic cluster growth theory for generating more realistic microstructures of porous media, and predicted the effective thermal conductivities of porous media with multiphase structure and stochastic complex geometries. According to the above researches, these physical experimental methods subject to be limited by the experimental condition, resolution and accuracy. For serial sectioning method, a time-consuming grind and polish process may change microstructure of the material. Because these automatic serial sectioning instruments [16] are merely a few around the globe, most of the studies are accomplished by manual work. For the CT tomography method such as micro-CT, the scanning result depends on the absorptivity of the sample. If phases of the sample have a close absorptivity, the contrast of the phases is not obvious under X-ray such as the SiC/Al composites. So the physical method is confined in modeling process of co-continuous ceramic composites. Among the numerical methods, the Gaussian statistical method is the most appropriate for the modeling process of co-continuous ceramic composites which can save much time and improve the accuracy. The present study proposed a model reconstruction method, namely, statistical method [17,18] of 3D co-continuous ceramic composites based on a single 2D microstructure SEM image. Distribution functions and covariance functions of metallic alloy and ceramic phases were statistically calculated. A random function
was used to construct the Fourier transform coefficient matrix. After the nonlinear filter process, a 3D solid model was generated in line with Gaussian field distribution. Statistical method can avoid repeated capturing of 2D images and can simply and quickly reconstruct a 3D co-continuous ceramic composite solid model with the same characteristics as those of the real structure of cocontinuous ceramic composites under low-cost condition. The reconstructed process is a part of pre-processing in numerical simulation. The reconstructed model can be imported into the finite element software for a further study of its properties.
2. Geometric characteristics of the statistics of co-continuous ceramic composites Co-continuous ceramic composites have a complex spatial topological geometry microstructure. The metallic and ceramic phases are both constituent and interpenetrating. The special structure is obtained by its forming process. The process is indicated as following: firstly, the ceramic particles are accumulated by a stacking manner. Then the ceramic phase is prepared by sintering to form a skeleton. Interpenetrating ceramic phase is formed as well as the interpenetrating pores. Secondly, melt metal is infiltrated in the pores by vacuum pressure infiltration. As a result, both ceramic and metallic phases are continuous in the composites. For the complex continuous character, the 3D reconstruction of the composites is a bottleneck up to now. In order to reconstruct 3D model of co-continuous ceramic composites in this paper, the first step is to calculate geometric characteristic statistics. SEM image is the basement of the algorithm which is shown in Fig. 1. In this paper, the metallic and ceramic phases of the co-continuous ceramic composites are Al and SiC respectively. 2.1. Digitalization of metallic alloy and ceramic phases Metallic alloy and ceramic phases are interpenetrating in the 3D co-continuous ceramic composites exhibiting complex geometric topology. In the reconstruction process of 3D geometric model, the spatial distribution of the two phases can be represented as: 1 P A metallicalloyphase Z ðP Þ ¼ ð1Þ P A ceramicphase 0 where P represents an arbitrary point P(x, y, z) in space, and Z(P)
Fig. 1. SEM image of co-continuous ceramic composites.
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represents the material property for the point P(x, y, z) of the 3D co-continuous ceramic composites. Eq. (1) shows the spatial distribution of the two phases of any 3D co-continuous ceramic composites. By traversing all the points of co-continuous ceramic composites, the volume fractions of the metallic alloy and ceramic phases can be calculated using a statistical method. Volume fraction of the metallic alloy phase can be obtained as Eq. (2) by calculating the average of Eq. (1). The volume fraction can be calculated by summing up the number of metallic alloy phase points and dividing by the total number of points. Accordingly, the volume fraction of the ceramic phase can be obtained by Eq. (3).
εmetal ¼ Z ðP Þ
ð2Þ
εceramic ¼ 1 εmetal
ð3Þ
2.2. Distribution statistics of metallic alloy and ceramic phases The covariance function is used to characterize the distribution of the metallic alloy and ceramic phases by analyzing the 2D SEM image of 3D co-continuous ceramic composites. The covariance function represented as Rz(u) can be calculated by the following Eq. (4): Rz ðuÞ ¼
½Z ðP Þ ε∙½Z ðP þuÞ ε ðε ε2 Þ
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the covariance function Rz(u) should be calculated to the revised function which is denoted as Ry(u) though mathematics. The mathematical processes [18] are explained in Eqs. (5)–(9): Rz ðuÞ ¼
1 X
ðC m Þ2 ∙Rm y ðuÞ
ð5Þ
m¼0
1 C m ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2π ∙m!
cðyÞ ¼
Z
þ1 1
8 ε1 ffi > < pffiffiffiffiffiffiffiffiffiffiffi εð1 εÞ ε ffi > : pffiffiffiffiffiffiffiffiffiffiffi εð1 εÞ
y2 H m ðyÞdy cðyÞexp 2
pðyÞ r εmetal pðyÞ 4 εmetal
2 m y d y2 H m ðyÞ ¼ ð 1Þm exp m exp 2 dy 2 1 pðyÞ ¼ pffiffiffiffiffiffi 2π
Z
y2 dy exp 2 1 y
ð6Þ
ð7Þ
ð8Þ
ð9Þ
where the covariance function Rz(u) and volume fraction of metallic alloy phase εmetal can be obtained directly from Eqs. (2) and (4). Rz(u) is an m-order polynomial function based on Ry(u). The revised covariance function can be obtained by using Eqs. (5)–(9).
ð4Þ
where u represents a distance, and (P þu) represents the point with a distance u from the point P. Covariance function is the distribution relationship between the local phase particle and the one with a distance u from the local phase particle. This function contains the distribution characteristics of the metallic alloy and ceramic phases. The positive or negative result of the function indicates the correlation of the two particles in different spatial positions. If the value of the function is positive, a positive correlation is indicated, and the two particles belong to the same phase, which is either the metallic alloy or the ceramic phase. Conversely, if the value of the covariance function is negative, a negative correlation is indicated, and the two particles belong to different phases: one is metallic alloy phase, and the other is ceramic phase. This reconstruction method is based on the isotropic and homogeneous distribution of the metallic alloy and ceramic phases of the 3D co-continuous ceramic composites.
3. Establishment of a co-continuous ceramic composite model In this study, a 3D co-continuous ceramic composite solid geometry model was established by a statistical method. The flow chart of the statistical method is illustrated in Fig. 2. 3.1. Binarization of 2D microstructure SEM image The 2D SEM image of the 3D co-continuous ceramic composites is shown in Fig. 1. To facilitate computer recognition and programming, the 2D SEM image was binarized firstly. The result of binarization is shown in Fig. 3. The white area is the metallic alloy phase with volume fraction εmetal of 12.46%, and the black area is the ceramic phase with volume fraction εceramic of 87.54%. After binarization, the 2D gray image with 256 brightness levels was transformed into a binary image of two brightness levels consisting of only 0 and 1.
2.3. Revision of covariance function 3.2. Calculation and revision of covariance function To obtain a nearly similar 3D co-continuous ceramic composite solid model of the real microstructure, Gaussian field is a necessary calculating process. Hence, before the covariance function Rz(u) is applied to the 3D space, the distribution characteristics must meet some specific mathematical requirements. Furthermore,
The covariance functions of the x-axis and y-axis directions are compared in Fig. 4. Notably, the covariance functions of the x-axis and y-axis are the same because the 3D co-continuous ceramic composites are isotropic.
Fig. 2. Flow chart of the reconstruction process. (FFT, fast Fourier transform; iFFT, inverse fast Fourier transform).
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The revision of the covariance function is presented in Table 1. Different values of m were calculated. An m¼ 17 indicates that the result met the accuracy requirements substantially. When m 417, the result suggests that the calculating time was very long, and the result slightly changed. To save computing time and ensure the accuracy of the calculation results, m ¼17 was selected for this study. The revised covariance function after revision is shown in Fig. 5.
non-linear filter of Fourier coefficient matrix A. Non-linear filter process can be described by Eq. (10). After the non-linear filter transformation, 3D binary matrix, which showed hexahedral-like 3D matrix, was consisted of 0 and 1 in geometric space (Fig. 6). ( 1 PðyÞ r εmetal Z¼ ð10Þ 0 PðyÞ 4 εmetal
3.3. Calculation of spectral function and construction of complex conjugate matrix
3.5. Visualization of 3D binary matrix
The spectral density S(u) can be calculated from the covariance function Ry(u) using fast Fourier transform (FFT). The amplitude spectrum S0 (u) of covariance function Ry(u) can then be obtained by the spectral density S(u). Furthermore, amplitude spectrum S0 (u) can be transformed into complex Fourier coefficient matrix A by creating a random angle α. After iFFT, Fourier coefficient matrix Y can be finally obtained. To ensure that Y is a real matrix, the matrix A should meet the requirements of Hermitian matrix [17]. The structure of the algorithm is presented in Table 2. 3.4. Non-linear filter The 3D binary matrix Z, which exhibits the same characteristic functions as those of 2D real SEM image, can be obtained after a
The 3D binary matrix Z is a numerical matrix consisting of 0 and 1. Therefore, the 3D solid model is not a solid model. Binary matrix Z can be transformed into a visual 3D model via Eq. (1) by assigning the material properties to 0 and 1 of the binary matrix. Finally, the 3D solid model of co-continuous ceramic composites was obtained as Fig. 7a, and the solid model of the metallic alloy and ceramic phases is shown in Fig. 7b and c. The volume fractions of the metallic alloy and ceramic phases of the 3D solid model were verified. As a result, the volume fraction of the metallic alloy phase was 13.24%, and the ceramic phase was 86.76%. As comparative figures in the real 3D co-continuous ceramic composites, the volume fraction of the metallic alloy and ceramic phases were 17.07% and 82.93%, respectively. Compared with the reconstructed 3D model and real co-continuous ceramic composites, the volume fraction of the metallic alloy phase error was 4.4%, which is within the acceptable error range.
Table 1 Revised algorithm. Algorithm revision 1 2 3 4 5 6 7 8 9 10 11 12 13 Fig. 3. Binary image of 3D co-continuous ceramic composites.
Fig. 4. Covariance functions of x-axis and y-axis directions.
Calculate metallic alloy phase volume fraction; εmetal’the volume fraction of metallic alloy phase; m’the exponent of Ry(u); p(y) is the expression of Gaussian distribution; if p(y)r εmetal Calculate c(y); else Calculate c(y) by another way; end (for if in 5); each exponent 1 to m Calculate Hm(y) and Cm; Substitute Hm(y) and Cm into Eq. (5); end (for loop in 9); Solve the m-order Eq. (5);
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Fig. 5. Covariance functions before and after revision.
Table 2 Transformed algorithm. Algorithm transformation 1 2 3 4 5 6 7 8 9 10 11 12
S(u)’spectral density; S0 (u)’amplitude spectrum; α’random angle; A’complex Fourier coefficient matrix Y’Fourier coefficient matrix S(u) is the fast Fourier transform of Ry(u); S0 (u) is the radication of S(u); for each 1 to u Choose a random value of α A [0,2π]; Calculate the real part and the imaginary part of A; end (for loop in 8); Y is the inverse fast Fourier transform of A;
Fig. 6. Calculation result after a non-linear filter.
4. Model verification A co-continuous metal/ceramic composites sample, which used the same forming process with the sample for SEM image, was scanned using a micro-CT (Sky-Scan 1172) scanner. Because the contrast between the SiC phase and Al phase was not obvious under the X-ray, the micro-CT model was obtained by scanning a single SiC skeleton phase. Then the Al phase was generated by a Boolean operation. The error between the reconstructed model and the real microstructure, which was caused by SiC skeleton microstructure deformation through the machining and interface change through the vacuum pressure infiltration process, was not concerned. The Simpleware commercial package [19] was used to reconstruct the 3D model from the obtained tomographic images. The micro-CT model is shown in Fig. 8.
In order to verify the generation-based optimization method, covariance functions and volume fractions of two phases were analyzed. The ceramic phase volume fractions of the reconstructed model and micro-CT model were 82.93% and 83.9%. The difference between the two volume fractions was 1.2%. And the covariance functions are shown in Fig. 9. The two parameters are all within an acceptable difference range.
5. Conclusions Reconstruction of a 3D co-continuous ceramic composite solid model is an important prerequisite for numerical simulation. Statistical method for reconstructing 3D solid model of the 3D cocontinuous ceramic composites was proposed, which established
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Fig. 7. 3D co-continuous ceramic composite solid model. (a) Composites model. (b) Metallic alloy phase. (c) Ceramic phase.
Fig. 8. Micro-CT co-continuous ceramic composite model.
Fig. 9. Covariance functions of reconstructed and micro-CT co-continuous ceramic composite model.
the 3D co-continuous ceramic composite solid model based only on a single 2D microstructure SEM image. Statistical distribution characteristics of metallic alloy and ceramic phases in 2D microstructure image were obtained by a mathematical method. A Gaussian distribution random field was constructed. The validity of the 3D reconstruction method was verified. And a micro-CT scanning model was proposed to further verify the reconstructed model. The 3D co-continuous ceramic composite reconstruction solid model retained the same geometric characteristics as those of the original real 2D microstructure SEM image as well as the micro-CT model. The volume fraction difference of the metallic alloy and ceramic phases was very small and within the acceptable difference range. The covariance functions were also very close. We will aim to reduce the error and expand the 3D reconstruction method in our future research.
However, the statistical method proposed by this paper still has some limitations. The volume fraction and covariance function are the main control parameters in the method. In the further study, more parameters would be proposed to describe the spatial characters of the composites. The method can obtain an accurate 3D solid model of co-continuous ceramic composites, but it cannot reconstruct 3D solid models of anisotropy materials as well as multi-phase material (more than two phases).
Acknowledgments The support from the Specialized Research Fund for the Doctoral Program of Higher Education of China (20091101120038) is gratefully acknowledged.
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References [1] M.C. Breslin, Ceramic/Metal Composite Materials (C4) Produced, The Ohio State University, Columbus, USA, 1994. [2] F.J. Guild, W. Bonfield, Predictive modelling of hydroxyapatite–polyethylene composite, Biomaterials 14 (1993) 985–993. [3] C.K. Natarajan, M. Karel, H. Philippe, Reconstruction of periodic unit cells of multimodal random particulate composites using genetic algorithms, Comput. Mater. Sci. 42 (2008) 352–367. [4] N. Chawla, V.V. Ganesh, B. Wunsch, Three-dimensional (3D) microstructure visualization and finite element modeling of the mechanical behavior of SiC particle reinforced aluminum composites, Scr. Mater. 51 (2004) 161–165. [5] N. Chawla, Y.L. Shen, Mechanical behavior of particle reinforced metal matrix composites, Adv. Eng. Mater. 3 (2001) 357–370. [6] A. Ayyar, N. Chawla, Microstructure-based modeling of crack growth in particle reinforced composites, Compos. Sci. Technol. 66 (2006) 1980–1994. [7] J. Wang, L.C. Steven, S.G. Mogilevskaya, Numerical modeling of the elastic behavior of fiber-reinforced composites with inhomogeneous interphases, Compos. Sci. Technol. 66 (2006) 1–18. [8] S. Arnold, Reduction of thermal residual stress in advanced metallic composites based up on a compensating/complaint layer concept, J. Compos. Mater. 26 (1992) 1287–1309. [9] E. Folias, On the predication of failure at a fiber/matrix interface in a composites subjected to a transverse tensile load, J. Compos. Mater. 26 (1991) 1287–1309.
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[10] W. Węglewski, K. Bochenek, M. Basista, T. Schubert, U. Jehring, J. Litniewski, S. Mackiewicz, Comparative assessment of Young's modulus measurements of metal–ceramic composites using mechanical and non-destructive tests and micro-CT based computational modeling, Comput. Mater. Sci. 77 (2013) 19–30. [11] H.K. Altaf, A.C. Reaz, Fan-beam geometry based inversion algorithm in computed tomography (CT) for imaging of composite materials, Compos. Struct. 110 (2013) 297–304. [12] M. Joshi, A class of stochastic models for porous materials, University of Kansas, Lawrence, 1974. [13] J.A. Quiblier, A new 3D modeling technique for studying porous media, J. Colloid Interface Sci. 98 (1984) 84–102. [14] R.D. Hazlett, Statistical characterization and stochastic modeling of pore networks in relation to fluid flow, Math. Geol. 29 (1997) 801–822. [15] M.R. Wang, J.K. Wang, N. Pan, S.Y. Chen, Mesoscopic predictions of the effective thermal conductivity for microscale random porous media, Phys. Rev. E 75 (2007) 036702. [16] M.P. Echlin, A. Mottura, M. Wang, P.J. Mignone, D.P. Riley, G.V. Franks, T. M. Pollock, Three-dimensional characterization of the permeability of W–Cu composites using a new “TriBeam” technique, Acta Mater. 64 (2014) 307–315. [17] P.M. Adler, C.G. Jacquin, J.A. Quiblier, Flow in simulated porous media, Int. J. Multiph. Flow 16 (1990) 691–712. [18] E. Pardo-Igdzquiza, M. Chica-Olmo, The fourier integral method: an efficient spectral method for simulation of random fields, Math. Geol. 25 (1993) 180–217. [19] Simpleware Ltd., Exeter, UK. 〈http://www.simpleware.com/〉.