Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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considered meso-scale offshoots of both MD (micro-scale) but also the engineering truss and frame structures well-known from structural mechanics (macro-scale). Lattices were the original systems for modeling discontinuous media - various ideas of application in engineering mechanics date, at least, to Hrennikoff’s pioneering work [7]. The modeling of network structures on a much coarser spatial scale than the atomic one eliminates the obvious need to work with a huge number of degrees of freedom, which could result in both computer congestion and data overload (so-called "data glut"), which would be inevitable if the atomic methods were used for modeling even the smallest structures on the macro-scale. This approach also reduces to a relatively modest level the number of nodes necessary to model the heterogeneity of the material structure. Comprehensive reviews of lattice models were published by Ostoja-Starzewski [3,8].

Figure 6.3: (a) Irregular triangular Delaney lattice dual to Voronoi grain thessalation. (b) Mesostructure of a three-phase composite projected on a regular triangular lattice. (c) Assembly of polygonal particles.

Unlike lattice models in which the basic structural elements are one-dimensional, in DEM (discrete element method) models the basic building blocks are typically of the same dimensionality as the considered problem. For example, planar DEMs include models of discontinuous systems comprised of 2D basic constituent elements such as circles, ellipses, or polygons (Figure 6.3c). These discrete elements are provided with geometric, structural, and contact properties that allow their “assemblies” (conglomerates, agglomerates) to approximate the complex phenomenological response of the subject material. In the most concise terms, DEMs enable the simulation of the motion and interactions of a huge number of discrete objects. It is important to note, that DEM unit blocks are actual geometric objects characterized by their dimension and shape, unlike the MD atoms, lattice nodal points, and particles (of PD) that are essentially material points. The macroscopic behavior of DEM models emerges as a consequent feature of the system derived from a small set of meso-properties of individual elements and their interactions. Contacts among discrete elements are endowed with the proscribed cohesive strength (including zero cohesive strength for non-cohesive, loose, material systems) and the ability to dissipate energy that allows representation of both elastic and inelastic phenomena and the nucleation of cracks and cooperative phenomena among them. It is important to emphasize that the properties of the contacts between discrete elements should be, in principle, identifiable based on the