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Microstructural Description of Composites, Statistical Methods

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Mechanics of Microstructured Materials

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 464))

Abstract

For the purpose of mechanical modeling the behaviour of composite materials it is necessary to identify the descriptors that in a best way characterize the special dispersion of fillers. Different morphological measures are discussed along with the description of relevant experimental techniques and simulation methods.

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References

  • J. Aboudi. Mechanics of Composite Materials. Elsevier, Amsterdam, 1991.

    MATH  Google Scholar 

  • N.D. Alberola, G. Merle, and K. Bendarti. Unidirectional fibre-reinforced polymers: Analytical morphology approach and mechanical modelling based on the percolation concept. Polymer, 40:315–328, 1999.

    Google Scholar 

  • B. Altan, Z. Misirli, and H. Yorucu. The determination of the homogeneity in multiphase mixtures. Z. Metall, 81:221–228, 1990.

    Google Scholar 

  • G. Ausias, IF. Agassant, M. Vincent, P.G. Lafleur, P.A. Lavoie, and Carreau P J. Rheology of short glass fiber reinforced polypropylene. J. Rheol, 36:525–542, 1992.

    Google Scholar 

  • M. Axelsen and R. Pyrz. Correlation between fracture toughness and the microstructure morphology in transversely loaded unidirectional composites. In R. Pyrz, editor, Microstructure-Property Interactions in Composite Materials, pages 15–26. Kluwer Academic Publishers, Dordrecht, 1995a.

    Google Scholar 

  • M. Axelsen and R. Pyrz. Effect of microcrack formation in transversely loaded unidirectional composites. In N. Sottos, editor, Durability and Damage of Composite Materials, MD-Vol. 69–1, pages 239–248. ASME, New York, 1995b.

    Google Scholar 

  • I. Balberg, C.H. Anderson, S. Alexander, and N. Wagner. Excluded volume and its relation to the onset of percolation. Phys. Rev. B, 30:3933–3943, 1984.

    Google Scholar 

  • P.P. Bansal and A J. Ardell. Average nearest-neighbor distances between uniformly distributed finite particles. Metallography, 5:97–111, 1972.

    Google Scholar 

  • R.S. Bay and C. L. Tucker. Fibre orientation in simple injection mouldings. Part II: Experimental results. Polym. Engng. Sci., 32:240–253, 1992.

    Google Scholar 

  • M.L. Becraft and A.B. Metzner. The rheology, fiber orientation, and processing behavior of fiber-filled fluids. Rheol,, 36:143–174, 1992.

    Google Scholar 

  • LG. Berryman. Random close packing of hard spheres and disks. Phys. Rev. A, 27:1053–1061,1983.

    Google Scholar 

  • LG. Berryman. Measurement of spatial correlation functions using image processing techniques. J. Appl. Phys., 57:2374–2384, 1985.

    Google Scholar 

  • IG. Berryman and G.W. Milton. Microgeometry of random composites and porous media. J. Phys. D, 21:87–94, 1988.

    Google Scholar 

  • B. Billia, H. Jamgotchian, and H. Nguyen Thi. Statistical analysis of the disorder of two-dimensional cellular arrays in directional solidification. Metall. Trans. A, 22:3041–3050, 1991.

    Google Scholar 

  • B. Bochenek and R. Pyrz. Identification of clustered distributions. Submitted to Comp. Sci. Techn., 2003a.

    Google Scholar 

  • B. Bochenek and R. Pyrz. (2003). Reconstruction of random microstructures — a stochastic optimization problem. Submitted to Comput. Mat. Sci., 2003b.

    Google Scholar 

  • J. Boselli, P.D. Pitcher, P J. Gregson, and I. Sinclair. Random close packing of hard spheres and disks. J. Microsc., 195:104–112, 1999.

    Google Scholar 

  • N.J. Bozarth, J. Gillespie, and R.L. McCullough. Fiber orientation and its effect upon thermoelastic of short carbon fiber reinforced poly(etheretherketone). Polym. Compos., 8:74–81, 1987.

    Google Scholar 

  • J.R. Brockenbrough, W.H. Hunt, and O. Richmond. A reinforced material model using actual microstructural geometry. Scripta Metall. Mater., 27:385–390, 1992.

    Google Scholar 

  • LS. Chappel, T.A. Ring, and LD. Birchall. Particle-size distribution effects on sintering rates. J. Appl. Phys., 60:383–391, 1986.

    Google Scholar 

  • T. Chen, G.L Dvorak, and Y. Benveniste. Mori-Tanaka estimates of the overall elastic moduli of certain composite materials. J. Appl. Meek, 59:539–546, 1992.

    MATH  Google Scholar 

  • S.N. Chiu. Aboav-Weaire’s and Lewis’ laws — A review. Mat. Char., 34:149–165, 1995.

    Google Scholar 

  • D.B. Corson. Correlation functions for predicting properties of heterogeneous materials. J. Appl. Phys., 45:3159–3170, 1974.

    Google Scholar 

  • C.W. Corti, P. Cotterill, and G.A. Fitzpatrick. The evolution of the interparticle spacing in dispersion alloys. Int. Metall. Rev., 19:77–88, 1974.

    Google Scholar 

  • L.M. Cruz-Orive and E.R. Weibel. Sampling designs for stereology. J. Microsc., 122:235–257, 1989

    Google Scholar 

  • D. Cule and S. Torquato. Generating random media from limited microstructural information via stochastic optimization. J. Appl. Phys., 86:3428–3437, 1999.

    Google Scholar 

  • A.C. Curtis, P.S. Hope, and I.M. Ward. Modulus development in oriented short-glass-fiber-reinforced polymer composites. Polymer Composites, 3:138–145, 1982.

    Google Scholar 

  • P.L Davy and F.L Guild. The distribution of interparticle distance and its application in finite-element modelling of composite materials. Proc. R. Soc. Lond. A, 418:95–112, 1988.

    Google Scholar 

  • B. Derby. Microstructure and fracture behavior of particle-reinforced metal-matrix composites. J. Microsc, 177:357–368, 1995.

    Google Scholar 

  • P.L Diggle, J. Besag, and LT. Gleaves. Statistical analysis of spatial point patterns by means of distance methods. Biometrics, 32:659–667, 1976.

    MATH  Google Scholar 

  • P.L Diggle. Statistical Analysis of Spatial Point Pattern, Academic Press, London, 1983; p. 70.

    Google Scholar 

  • P. L Diggle and A. G. Chetwynd. Second-order analysis of spatial clustering for inhomogeneous populations. Biometrics, 47:1155–1163, 1999

    Google Scholar 

  • W.L Drugan and LR. Willis. A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids, 44:497–524, 1996.

    MathSciNet  MATH  Google Scholar 

  • G.L Dvorak. Transformation field analysis of inelastic composite materials. Proc. R. Soc. Lond. A, 437:311–327,1992.

    MathSciNet  MATH  Google Scholar 

  • K.E. Evans and A.G. Gibson. Prediction of the maximum packing fraction achievable in randomly oriented short-fibre composites. Compos. Sci. Techn., 25:149–162, 1986.

    Google Scholar 

  • R.K. Everett and LH. Chu. Modeling of non-uniform composite microstructures. J. Comp. Mat., 27:1128 – 1144,1993.

    Google Scholar 

  • R.K. Everett. Quantification of random fiber arrangements using a radial distribution function approach. J. Comp. Mat., 30:748–758, 1996.

    Google Scholar 

  • J.J. Finney. Random packings and the structure of simple liquids. Proc. R. Soc. Lond. A, 319:479–493, 1970.

    Google Scholar 

  • G. Fischer and P. Eyerer. Measuring spatial orientation of short fibre reinforced thermoplastics by image analysis. Polym. Comp., 9:297–304, 1988.

    Google Scholar 

  • F. Folgar and C.L. Tucker. Orientation behavior of fibers in concentrated suspensions. J. Reinf. Plast. Comp., 3:98–119, 1984.

    Google Scholar 

  • P.S. From and R. Pyrz. Computer assisted stereology of short fibre composites: (I) determination of fibre length distribution. Sci. Engng. Comp. Mat., 8:143–157, 1999.

    Google Scholar 

  • A. Getis and B. Boots. Models of Spatial Processes. Cambridge University Press, Cambridge, 1978.

    MATH  Google Scholar 

  • S. Ghosh, Z. Nowak, and K. Lee. Quantitative characterization and modeling of composite microstructures by Voronoi cells. Acta Mat, 45:2215–2234, 1997a.

    Google Scholar 

  • S. Ghosh, Z. Nowak, and K. Lee. Tesselation-based computational methods for the characterization and analysis of heterogeneous microstructures. Comp. Sci. Techn., 57:1187–1210, 1997b.

    Google Scholar 

  • S. Ghosh and S. Moorthy. Particle fracture simulation in non-uniform microstructures of metal-matrix composites. Acta Mat, 46:965–982, 1998.

    Google Scholar 

  • J.-F. Gouyet. Physics and Fractal Structures, Springer, New York, 1996.

    Google Scholar 

  • H.J.G. Gundersen and E.B. Jensen. The efficiency of systematic sampling in stereology and its prediction. J. Microsc, 147:229–263, 1987.

    Google Scholar 

  • H. J.G. Gundersen, E.B. Jensen, K. Kiêu, and J. Nielsen. The efficiency of systematic sampling in stereology reconsidered. J. Microsc., 193:199–211, 1999.

    Google Scholar 

  • A.A. Gusev. Representative volume element size for elastic composites: A numerical study. J. Mech. Phys. Solids, 45:1449–1459, 1997.

    MATH  Google Scholar 

  • H. Hermann. Stochastic Models of Heterogeneous Materials, Trans Tech Publications, Zurich, 1991.

    Google Scholar 

  • J.R. Isasi, R. G. Alamo and L. A. Mandelkern. A study of the dilation of the unit cell of metallocene isotactic poly (propylenes): The monoclinic form. J. Polym. Sci. B, 35:2511–2522, 1997.

    Google Scholar 

  • W.C. Jackson, S.G. Advani, and C.L. Tucker. Predicting the orientation of short fibers in thin compression mouldings. J. Comp. Mat, 20:539–557, 1986.

    Google Scholar 

  • J.W. Ju and T.M. Chen. Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. J. Microsc 103:123–144, 1994.

    MathSciNet  MATH  Google Scholar 

  • J. W. Ju and K.H. Tseng. Effective elastoplastic behavior of two-phase ductile matrix composites: A micromechanical framework. Int. J. Solids Struct, 33:4267–4291, 1996.

    MATH  Google Scholar 

  • L.M. Karlsson and A. Liljeberg. Second-order stereology for pores in translucent alumina studied by confocal scanning laser microscopy. J. Microsc, 175:186–194, 1994.

    Google Scholar 

  • P.A. Kamezis, G. Durrant, and B. Cantor. Characterization of reinforcement distribution in cast Al-alloy SiCp composites. Mat Char., 40:97–109, 1998.

    Google Scholar 

  • D. König, S. Carvajal-Gonzales, A.M. Downs, J. Vassy, and LP. Rigaut. Modelling and analysis of 3-D arrangements of particles by point processes with examples of application to biological data obtained by confocal scanning light microscopy. J. Microsc., 161:405–433, 1991.

    Google Scholar 

  • S. Kunz-Douglass, P. W. R. Beaumont, and M. F. Ashby. A model for the toughness of epoxy-mbber particulate composites. J. Mater. Set, 15:1109–1123, 1980.

    Google Scholar 

  • Y.W. Kwon and C.T. Liu. Effects o non-uniform particle distributions on damage evolution in pre-cracked particulate composite specimens. Polym. & Polym. Comp., 6:387–397, 1998.

    Google Scholar 

  • B.J. Lee and M.E. Mear. Effect of inclusion shape on the stiffness of non-linear two phase composites. J. Mech. Phys. Solids, 39:627–649, 1991.

    Google Scholar 

  • J. Lemaitre, A. Gervois, LP. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau. Arrangement of cells in Voronoi tessellations of monosize packing of disks. Phil. Mag. B, 67:347–362, 1993.

    Google Scholar 

  • H.W. Lotwick and B.W. Silverman. Methods for analyzing spatial processes of several types of points. J. Roy. Statist. Soc. B, 44:406–413, 1982.

    MathSciNet  Google Scholar 

  • M. Manoharan, C. Liu, and L.L. Lewandowski. Microstructure and particle size effects on fracture in aluminum metal matrix composites. In K. Salama et al., editors, Advances in Fracture Research, pages 2977–2985. Pergamon Press, Oxford, 1989.

    Google Scholar 

  • T. Mattfeldt, H. Frey, and C. Rose. Second order stereology of benign and malignant alterations of the human mammary gland. J. Microsc., 171:143–151, 1993.

    Google Scholar 

  • M.N. Miller. Bounds for effective electrical, thermal, and magnetic properties of heterogeneous materials. J. Math. Phys., 10:1988–2004, 1969.

    Google Scholar 

  • S.J. Milne, M. Patel, and E. Dickinson. Experimental studies of particle packing and sintering behavior of monosize spherical silica powders. J. European Ceram. Soc, 11:1–7, 1993.

    Google Scholar 

  • G.W. Milton. Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids, 30:177–191, 1982.

    MathSciNet  MATH  Google Scholar 

  • B. Moginger and P. Eyerer. Detennination of the weighting function g(βi,r,vf) for fibre orientation analysis of short fibre-reinforced composites. Composites, 22:39–4–399, 1991.

    Google Scholar 

  • T. Mura. Micromechanics of Defects in Solids. Kluwer, Dordrecht, 1987.

    Google Scholar 

  • Y. Nakamura S. Okabe, N. Yoshimoto, and T. Iida. Effect of particle shape on the mechanical properties of particle-filled PVC. Polym. &Polym. Comp., 6:407–14, 1998.

    Google Scholar 

  • Y. Nakamura, S. Okabe, and T. Iida, T. Effects of particle shape, size and interfacial adhesion on the fracture strength of silica-filled epoxy resin. Polym. & Polym. Comp., 7:177–186, 1999.

    Google Scholar 

  • S. Nemat-Nasser and M. Hori. Micromechanics: Overall Properties of Heterogeneous Materials, North- Holland, Amsterdam, 1993.

    MATH  Google Scholar 

  • A.S. Nielsen and R. Pyrz. In-situ observation of thermal residual strains in carbon/thermoplastic micro- composites using Raman spectroscopy. Polym. & Polym. Comp., 5:245–257, 1997.

    Google Scholar 

  • A.S. Nielsen and R. Pyrz. Study of the influence of thermal history on the load transfer efficiency and fibe failure in carbon/polypropylene microcomposites using Raman spectroscopy. Composite Interfaces, 6:467–482, 1999.

    Google Scholar 

  • A.S. Nielsen and R. Pyrz. A novel approach to measure local strains in polymer matrix systems using polarized Raman microscopy. Comp. Sci. Techn., 62:2219–2227, 2002.

    Google Scholar 

  • A.S. Nielsen, D.N. Batchelder, and R. Pyrz. Estimation of crystallinity of isotactic polypropylene using Raman spectroscopy. J. Polymer, 43:2671–2676, 2001.

    Google Scholar 

  • H. Okumura and F. Yonezawa. New expression of the bulk viscosity. Physica A, 321:207–219, 2003.

    Google Scholar 

  • M. Ostoja-Starzewski, P.Y. Sheng, and I. Jasiuk. Influence of random geometry on effective properties and damage formation in composite materials. J. Engng. Mat. Tech., 116:384–391, 1994.

    Google Scholar 

  • P.G. Parkhouse and A. Kelly. The random packing of fibres in three dimensions. Proc. R. Soc. London A, 451:737–746, 1995.

    Google Scholar 

  • P. Ponte Castaneda and LR. Willis. The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Solids, 43:1919–1951, 1995.

    MathSciNet  MATH  Google Scholar 

  • M.L Powell. Computer-simulated random packing of spheres. Powder Techn., 25:45–52, 1980.

    Google Scholar 

  • R. Pyrz. Quantitative description of the microstructure of composites. Part I: Morphology of unidirectional composite systems. Comp. Sci. Techn., 50:197–208, 1994a.

    Google Scholar 

  • R. Pyrz. Correlation of microstructure variability and local stress field in two-phase materials. Mater. Sci. Engng. A, 177:253–259, 1994b.

    Google Scholar 

  • R. Pyrz and B. Bochenek. Statistical model of fracture in materials with disordered microstructure. Sci. Engng. Comp. Mat., 3:95–109, 1994.

    Google Scholar 

  • R. Pyrz. Pattern analysis and topological effects in transversely loaded unidirectional composites. In W. Tzuchizing and T.W. Chou, editors, Proceedings of the International Conference Progress in Advanced Materials and Mechanics, pages 581–586. Peking University Press, Beijing, 1996a.

    Google Scholar 

  • R. Pyrz. Disorder and fracture model for transversely loaded composite materials. In H. Nisitani et al., editors, Computer Aided Assessment and Control of Localized Damage, pages 385–392. Computational Mechanics Publications, Southhampton, 1996b.

    Google Scholar 

  • R. Pyrz. Fractal characterization of second phase dispersion in composite materials. Sci. Engng. Comp. Mat., 6:141–150, 1997a.

    Google Scholar 

  • R. Pyrz, Recent trends in morphological characterization of microstructures in polymer composites. In S/I/ Andersen et al., editors, Proceedings of the 18 th Rise International Symposium on Materials Science, page 81. Riso National Laboratory, Roskilde, 1997b.

    Google Scholar 

  • R. Pyrz and B. Bochenek. Topological disorder of microstmcture and its relation to the stress field. Int. J. Solids Struct., 35:2413–2427, 1998.

    MATH  Google Scholar 

  • R. Pyrz. Application of X-ray microtomography to the study of polymer composites. In Proceedings of the 12 th International Conference on Composite Materials, paper no. 909. Paris, 1999.

    Google Scholar 

  • R. Pyrz. X-ray microtomography of composite materials. In S. Bandyopahyay et al., editors, Proc. ACUN- 3 International Conference on Technology Convergence in Composite Applications, pages 185–190. Sydney, 2001.

    Google Scholar 

  • R. Pyrz. Calculation of local stresses using molecular dynamic simulations. Submitted to Comp. Sci. Techn., 2003.

    Google Scholar 

  • J. Quintanilla. Microstmcture and properties of random heterogeneous materials: A review of theoretical results. Polym. Engng. Sci., 39:559–585, 1999.

    Google Scholar 

  • N. Rivier. Statistical crystallography. Structure of random cellular networks. Phil. Mag. B, 52:795–819, 1985.

    Google Scholar 

  • B.D. Ripley. Modelling spatial patterns. J. Roy. Statist. Soc. B, 39:172–212, 1977.

    MathSciNet  Google Scholar 

  • K. Saito, S. Araki, and T. Nakamura. Stereological modeling and micromechanical analysis of rubber- particle reinforced epoxy composite materials. Mech. Comp. Mater., 32:317–329, 1996.

    Google Scholar 

  • K. Sandau and H. Kurz. Measuring fractal dimension and complexity — An alternative approach with an application.J. Microsc, 1997, 186, 164–176, 1997.

    Google Scholar 

  • A. Sasov and D. van Dyck. Desktop X-ray microscopy and microtomography. J. Microsc, 191, 151–158, 1998.

    Google Scholar 

  • J. Schjødt-Thomsen and R. Pyrz. Overall creep modelling of short fibre reinforced composites with weakened interfaces and complex fiber orientation distributions. Mech. Mater., 32:349–359, 2000.

    Google Scholar 

  • J. Schjødt-Thomsen and R. Pyrz. The influence of stochastic interfacial parameter distribution on the stiffness of composites with complex fiber orientation. Comp. Sci. Techn., 61:697–704,2001.

    Google Scholar 

  • J. Schjødt-Thomsen and R. Pyrz. Stress fields in heterogeneous solids with spatially dispersed inclusions. In R. Pyrz et al., editors, New Challenges in Mesomechanics, pages 75–82. Aalborg University, 2002.

    Google Scholar 

  • J. Serra. Image Analysis and Mathematical Morphology, Academic Press, London, 1982; p. 271.

    MATH  Google Scholar 

  • R. Sibson. The Dirichlet tessellation as an aid in data analysis. Scand. J. Statist., 7:1.4–20, 1980.

    MathSciNet  Google Scholar 

  • P. Soille and J.F. Rivest. On the validity of fractal dimension measurements in image analysis. J. Visual Comm. Image Represent., 7:217–229, 1996.

    Google Scholar 

  • W.A. Spitzig, J.F. Kelly, and O. Richmond. Quantitative characterization of second-phase populations. Metallography, 18:235–261, 1985.

    Google Scholar 

  • D. Srolovitz, T. Egami, and V. Vitek. Radial distribution function and structural relaxation in amorphous solids. Phys. Rev. B, 24:6936–6944,1981.

    Google Scholar 

  • D. Stauffer and A. Aharoni. Introduction to Percolation Theory. Taylor & Francis, Washington, 1992.

    Google Scholar 

  • J.H. Steele, A stereological analysis of ductile fracture by microvoid coalescence. In K. Salama et al., editors, Advances in Fracture Research, pages 3449–3456. Pergamon Press, Oxford, 1989.

    Google Scholar 

  • D. Stoyan. On correlations of marked point processes. Math. Nachr., 116:197–204, 1984.

    MathSciNet  MATH  Google Scholar 

  • B J. Super and A.C. Bovik. Localized measurement of image fractal dimension using Gabor filters. J. Visual Comm. Image Represent, 1991, 2, 114–128, 1991.

    Google Scholar 

  • Y. Takao, T.W. Chou, and M. Taya. Effective longitudinal Young’s modulus of misoriented short fiber composites. J. Appl. Mech., 49:536–540, 1982.

    MATH  Google Scholar 

  • G.P. Tandon and G J. Weng. The effect of aspect ratio of inclusions on the elastic properties of unidirec-tionally aligned composites. Polymer Composites, 5:327–333, 1984.

    Google Scholar 

  • M. Taya, K. Muramatsu, DJ. Lloyd, and R. Watanabe. Determination of distribution patterns of fillers in composites by micromorphological parameters. JSME Int. J. I, 34:198–206, 1991.

    Google Scholar 

  • A. Tewari, M. Dighe, and A.M. Gokhale. Quantitative characterization of spatial arrangement of micropores in cast microstructures. Mat. Char., 40:119–132, 1998.

    Google Scholar 

  • S. Toll. (1998). Packing mechanics of fibre reinforcements. Polym. Engng. Sci., 38:1337–1350.

    Google Scholar 

  • S. Torquato. Microstructure characterization and bulk properties of disordered two-phase media. J. Stat. Phys., 45:843–873, 1986.

    MathSciNet  Google Scholar 

  • S. Torquato. Random heterogeneous media: Microstructure and improved bounds on effective properties. Appl. Mech. Rev., 44:37–76, 1991.

    MathSciNet  Google Scholar 

  • S. Torquato. Unified methodology to quantify the morphology and properties of inhomogeneous media. Physica A, 201:19–91, 1994.

    Google Scholar 

  • E.M. Tory, N.A. Cochrane, and S.R. Wadell. Anisotropy in simulated random packing of equal spheres. Nature, 220:1023–1024, 1968.

    Google Scholar 

  • W.M. Visscher and M. Bolsterli. Random packing of equal and unequal spheres in two and three dimensions. Nature, 239:504–507, 1972.

    Google Scholar 

  • Y.M. Wang and G.J. Weng. The influence of inclusion shape on the overall viscoelastic behavior of composites. J. Appl. Mech., 59:510–518, 1992.

    MATH  Google Scholar 

  • D. Weaire and N. Rivier. Soap, cells and statistics — Random patterns in two dimensions. Contemp. Phys., 25:59–99, 1984.

    Google Scholar 

  • E.R. Weibel. Stereological Methods. Academic Press, London, 1980.

    Google Scholar 

  • K. Wiencek and D. Stoyan. Spatial correlations in metal structures and their analysis. Mat. Char., 31:47–53,1993.

    Google Scholar 

  • A.S. Wimolkiatisak, J.P. Bell, D.A. Scola, and J. Chang. Assessment of fiber arrangement and contiguity in composite materials by image analysis. Polym. Comp., 11:274–279, 1990.

    Google Scholar 

  • CL. Yeong and S. Torqato. Reconstructing random media. Phys.Rev. E, 57:495–506, 1998a.

    MathSciNet  Google Scholar 

  • C. L. Yeong and S. Torqato. Reconstructing random media. II. Three-dimensional media from two-dimensional cuts. Phys.Rev. E, 58:224–233, 1998b.

    MathSciNet  Google Scholar 

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  1. Institute of Mechanical Engineering, Aalborg University, Aalborg, Denmark

    Ryszard Pyrz

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  1. Vienna University of Technology, Austria

    Helmut J. Böhm

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Pyrz, R. (2004). Microstructural Description of Composites, Statistical Methods. In: Böhm, H.J. (eds) Mechanics of Microstructured Materials. International Centre for Mechanical Sciences, vol 464. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2776-6_6

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  • DOI: https://doi.org/10.1007/978-3-7091-2776-6_6

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