Random Heterogeneous Materials

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Stochastic multiscale models for fracture analysis of functionally graded materials (2008, background): “Second, an FGM microstructure is inherently stochastic and can, in fact, be viewed as a statistically inhomogeneous random field [12,13], which describes randomness in the density, location, size, and shape characteristics of embedded particles in a matrix.”
Normal approximation for a random elliptic equation (2014, background): “1 Introduction Elliptic partial differential equations of the form −∇ · (a(x)∇u) = f arise in many physical applications where the coefficient a(x) may be modeled best as a random field, due to inherent uncertainty and complexity of the physical medium [23].”
Upscaling for unsaturated flow for non‐Gaussian heterogeneous porous media (2007, background): “The topic is often discussed for pore-scale models, where binary fields (describing pore space and grain material) have to be characterized efficiently [e.g., Torquato, 2002; Hilfer, 2002].”
A numerical investigation of the effect of particle clustering on the mechanical properties of composites (2003, background): “The broken horizontal lines in the plot stand for the lower and upper limits for l in any ergodic ensemble of isotropic packings of identical impenetrable spheres [26].”
Numerical homogenization of hardened cement paste (2008, methodology): “Effective material properties are calculated using numerical homogenization schemes which can be found in [9,20, 25,26].”