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Diffusion behaviors of heterogeneous materials are of paramount importance in many engineering problems. Numerical models that take into account the internal structure of such materials are robust but computationally very expensive. This burden can be partially decreased by using discrete models, however even then the practical application is limited to relatively small material volumes. This paper formulates a homogenization scheme for discrete diffusion models. Asymptotic expansion homogenization is applied to distinguish between (i) the continuous macroscale description approximated by the standard finite element method and (ii) the fully resolved discrete mesoscale description in a local representative volume element (RVE) of material. Both transient and steady‐state variants with nonlinear constitutive relations are discussed. In all the cases, the resulting discrete RVE problem becomes a simple linear steady‐state problem that can be easily pre‐computed. The scale separation provides a significant reduction of computational time allowing the solution of practical problems with a negligible error introduced mainly by the finite element discretization at the macroscale. [ABSTRACT FROM AUTHOR]

Static discrete models are advantageously used for the simulation of fracture in quasibrittle heterogeneous materials. In order to correctly capture strain localization during the fracture process, it is often necessary to represent material heterogeneity in the model directly via its discrete geometry. Depending on the specimen size and the size of the heterogeneities, these simulations are typically extremely computationally demanding. The contribution aims to reduce this computational cost via the implementation of adaptivity in the construction of the discrete model geometry. The simulation starts with coarse discretization, which provides correct elastic behavior and is then adaptively refined during the simulation in regions that suffer high stresses that induce cracking and strain localization. The technique is applied in deterministic and probabilistic simulations and demonstrated on several examples. [ABSTRACT FROM AUTHOR]

Discrete models of solids have been motivated, in large part, by the discontinuous and heterogeneous nature of material structure and its breakdown under loading. The capabilities of discrete models have evolved over the past several decades, offering novel means for investigating material structure–property relationships. However, lack of understanding of both the utilities and disadvantages of discrete models limits their further development and applications. This paper reviews relevant features of discrete approaches applied to modeling the mechanical behavior of geomaterials, concrete materials in particular. The discrete models are classified according to their form and abilities to represent elastic and fracture behaviors in the presence of large-scale material heterogeneity. Discretization of the material domain plays a large role in this respect. Emphasis is placed on particle-based lattice models. The relative merits of various strategies for introducing reinforcing components, which are essential for many applications, are outlined. Recent advances are highlighted, including the use of discrete models for coupled, multi-field analysis. The merits of discrete approaches are summarized in the conclusions. • Discrete methods for simulating concrete fracture are critiqued. • Model capabilities depend on the choice of physical or non-physical forms of discretization. • The topics of stress oscillation, Poisson's ratio and volumetric effects are examined. • Dual-lattice models are suited to modeling hygro-mechanical couplings in cracked media. • Application examples demonstrate the utilities and expanding use of discrete methods. [ABSTRACT FROM AUTHOR]