Mathematical modeling of cement paste microstructure by mosaic pattern: Part I. Formulation
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Mathematical modeling of cement paste microstructure by mosaic pattern: Part I. Formulation Yunping Xi Department of Civil and Architectural Engineering, Drexel University, Philadelphia, Pennsylvania 19104
Paul D. Tennis Department of Civil Engineering Northwestern University, Evanston, Illinois 60208
Hamlin M. Jennings Department of Civil Engineering, and Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208 (Received 19 June 1995; accepted 18 March 1996)
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 2 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.
I. INTRODUCTION
Materials science is based upon relationships between processing, microstructure, and properties. However, the microstructures of cement-based materials are complex, and quantitative analysis and characterization has proven difficult. The microstructure controls most properties, such as strength, fracture toughness, shrinkage, and permeability. Therefore, questions are being raised about which microstructural features are important, and about how they should be described quantitatively and modeled so they may be related to macrostructural properties. During the last decade, a number of models have been developed1–4 that simulate the microstructure of cement-based materials. The first model1 used analogue procedures to keep track of spheres, which represent cement particles reacting to form products that expand into the surrounding pore space. With the development of larger computers, digital models were developed2–4,37 which, within the limits of resolution, are able to keep track of the distribution of products. All of these models start with a distribution of cement particles that react with water in a stepwise fashion. Some models start with a distribution of circular or spherical particles, while others start by encoding information from micrographs of real cement. In this latter case, only two-dimensional shapes are possible. One of the shortcomings of present microstructural models is that an enormous amount of information must be maintained in the computers, and the process of J. Mater. Res., Vol. 11, No. 8, Aug 1996
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hydration must be simulated starting with unreacted cement and water. Furthermore, algorithms for the hydration
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