Abstract
This paper develops a mathematical model using mosaic patterns to characterize structural features of complex, multiphase, and multidimensional microstructures, such as those for cement paste. A multiphase microstructure can be characterized by m independent parameters; the first m– 1 parameters are equivalent to the volume fractions of the phases, while the final parameter describes the grain size, and thus, the spatial arrangement of the microstructure. An evaluation procedure for the parameters is given; they can be evaluated based on a 2D image, and then the 3D microstructure can be simulated by the present model. The relationship among the model parameters and material parameters, such as water-to-cement ratio and particle size distribution, are also established.
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References
H. M. Jennings and S. K. Johnson, J. Am. Ceram. Soc. 69, 790–795 (1986).
D. P. Bentz and E.J. Garboczi, Ceramic Transactions, Vol. 16, Advances in Cementitious Materials, edited by S. Mindess (The American Ceramic Society, Inc., Westerville, OH, 1990), pp. 211–226.
D. P. Bentz, E. J. Garboczi, D. Gingold, C. J. Lobb, and H. M. Jennings, Ceramic Transactions, Vol. 16, Advances in Cementitious Materials, edited by S. Mindess (The American Ceramic Society, Inc., Westerville, OH, 1990), pp. 227–235.
D. P. Bentz and E.J. Garboczi, ACI Mater. J. 88 (5), September–October, 518–529 (1991).
T. Naganuma, G. Deodatis, M. Shinozuka, and E. Samaras, Proc. 4th Int. Conf. on Structural Safety and Reliability, I, 251–260 (1985).
M. Shinozuka, Engineering Mechanics Division, ASCE 98 (6), 1433–1451 (1972).
F. Yamazaki and M. Shinozuka, Structural Eng./Earthquake Eng. 5 (2), Japan Society of Civil Engineers (Proc. of ASCE No. 398/I-10), 313s–323s (1988).
Z. P. Bazant and Y. Xi, in Materials Research in Concrete (Vom Werkstoff Zur Konstruktion, Karlsruhe, 1990), pp. 249–267.
Z. P. Bazant and Y. Xi, J. Eng. Mechanics Division, ASCE 119 (1), 301–322 (1993).
J. P. Delhomme, Adv. Water Resources 1 (5), 251–266 (1978).
G. de Marsily, in Fundamentals of Transport Phenomena in Porous Media, edited by J. Bear and M. Y. Corapcioglu (NATO ASI Series, Series E: Applied Sciences–No. 82, Martinus Nijhoff Publishers, New York, 1984), pp. 721–764.
L. Smith and R. A. Freeze, Water Resources Research 15 (6) Dec., 1543–1559 (1979).
J. P. Delhomme, Water Resources Research 15 (2), April 269–280 (1979).
P. Legendre and M. J. Fortin, Vegetatio (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989), pp. 107–138.
M.R. T. Dale and D.A. MacISAAC, J. Ecology 77, 78–91 (1989).
M. S. Bartlett, Adv. Appl. Prob. 6, 336–358 (1974).
E. C. Pielou, Biometrics, March, 156–167 (1964).
E. C. Pielou, Mathematical Ecology (John Wiley & Sons, New York, 1977).
M. Schon, J. Hirnforsch 27 (4), 401–408 (1986).
L. C. Cole, Ecology 30, 411–424 (1949).
H. A. David, Order Statistics (Wiley, New York, 1970).
M. G. Kendall and P. A. P. Moran, Geometrical Probability (Hafner Publishing Company, New York, 1963).
R. T. Coverdale, E.J. Garboczi, H. M. Jennings, B. J. Christensen, and T.O. Mason, J. Am. Ceram. Soc. (1993).
J. G. Kemeny and J. L. Snell, Finite Markov Chain (van Nostrand, New York, 1960).
I. Saxl, Stereology of Objects with Internal Structure (Elsevier, Amsterdam, 1989).
E. E. Underwood, Quantitative Stereology (Addison-Wesley Publishing Company, London, 1970).
J. Serra, Image Analysis and Mathematical Morphology, Vol. 1 (Academic Press, London, 1982).
J. Serra, Image Analysis and Mathematical Morphology, Vol. 2: Theoretical Advances (Academic Press, Harcourt Brace Jovanovich, Publishers, London, 1988).
E. K. Attiogbe, ACI Mater. J. 90 (2), March–April, 174–181 (1993).
H. Solomon, CBMS-NSF, Regional Conf. Series in Applied Mathematics (Soc. for Industrial and Applied Math., Philadelphia, PA, 1978).
M. G. Kendall and A. Stuart, The Advanced Theory of Statistics 2, 2nd ed. (Griffin, London, 1967).
A. Fellous, J. Granara, and K. Krickeberg, Lecture Notes in Biomathematics, 23, Managing Editor S. Levin, Geometrical Probability and Biological Structures: Buffon’s 200th Anniversary, edited by R.E. Miles and J. Serra (Springer-Verlag, Berlin, 1978).
E. E. Underwood, Stereology and Quantitative Metallography (1972).
S. Goudsmit, Rev. Mod. Phys. 17, 321–322 (1945).
A. Bezjak, Cement and Concrete Research 16, 260–264 (1986).
T. Knudsen, Cement and Concrete Research 14, 622–630 (1984).
D. P. Bentz, P. V. Coveney, E.J. Garboczi, M. F. Kleyn, and P.E. Stutzman, Modelling and Stimulation in Materials Science and Engineering 2, 783–808 (1994).
P. D. Tennis, H.M. Jennings, and Y. Xi, J. Mater. Res. (1996, submitted).
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Xi, Y., Tennis, P.D. & Jennings, H.M. Mathematical modeling of cement paste microstructure by mosaic pattern: Part I. Formulation. Journal of Materials Research 11, 1943–1952 (1996). https://doi.org/10.1557/JMR.1996.0245
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DOI: https://doi.org/10.1557/JMR.1996.0245