| Summary: | Thesis: Ph. D., Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, 2016. === Cataloged from PDF version of thesis. === Includes bibliographical references (pages 191-198). === In this thesis we obtain analytical approximations to the probability distribution of the elastic tensor and fracture strengths of material models with random heterogeneities. We start by investigating the effective elastic properties of one-, two-, and three-dimensional rectangular blocks whose Young's modulus varies spatially as a lognormal random field. We decompose the spatial fluctuations of the Young's log-modulus F = In E into first- and higher-order terms and find the joint distribution of the effective elastic tensor by multiplicatively combining the term-specific effects. Through parametric analysis of the analytical solutions, we gain insight into the effective elastic properties of this class of heterogeneous materials. Building on this analysis we find analytical approximations to the probability distribution of fracture properties of one-dimensional rods and thin two-dimensional plates for systems in which: only the Young's modulus varies spatially as an isotropic lognormal field and more generally, both the Young's modulus and the local material strength vary spatially as possibly correlated lognormal fields. The properties considered are the elongation, strength, and toughness modulus at fracture initiation and at ultimate failure. For all quantities at fracture initiation our approach is analytical in I D and semi-analytical in 2D. For ultimate failure, we quantify the random effects of fracture propagation and crack arrest by fitting regression models to simulation data and combine the regressions with the distributions at fracture initiation. Through parametric analysis, we gain insight into the strengthening/weakening roles of the Euclidean dimension, size of the specimen, and the correlation, variance and correlation function of the random fields. Finally, we extend the approach to investigate the elasticity of non-lognormal random heterogeneous materials. First we investigate the elastic bulk stiffness of two-dimensional checkerboard specimens in which square tiles are randomly assigned to one of two component phases. This is a model system for multi-phase polycrystalline materials such as granitic rocks and many ceramics. We study how the bulk stiffness is affected by different characteristics of the specimen and obtain analytical approximations to the probability distribution of the effective stiffness. In particular we examine the role of percolation of the soft and stiff phases. In small specimens, we find that the onset of percolation causes significant discontinuities in the effective modulus, whereas in large specimens the influence of percolation is smaller and gradual. Secondly we study the effective stiffness of multi-phase composite systems in which the Young's modulus varies as a filtered Poisson point process and find that the homogenization approach initially developed for lognormal systems produces accurate results also for this class of non-lognormal systems. === by Leon Sokratis Scheie Dimas. === Ph. D.
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