Mathematical Physics - Volume II - Numerical Methods

6.1 Introduction

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It cannot be overemphasized that this classification is somewhat arbitrary, the bound aries between the models are hard to draw, and the associated length scales are subject to definition for each material separately. For example, the spatial scale corresponding to the meso-structure of concrete (of the order of centimeters or even decimeters as illustrated in Figure 6.1d) or rock massifs (of the order of decimeters or meters, Figure 6.1i) exceeds that of silicon carbide (SiC) by several orders of magnitude although they can all be classified the quasi-brittle systems. With reference to Figure 6.2, on the nano-scale (e.g., the crystal lattice), it is possible to use the atomic models based on quantum mechanics. These rigorous methods 1 are based on the Schrödinger wave equation and semiempirical effective potentials which approximate quantum effects [1]. In contrast, classical MD relies on Newton’s equations of motion and empirical potentials. Therefore, the traditional MD sacrifices the quantum mechanics rigor for the benefit of a much larger spatio-temporal modeling range. Observation scales and corresponding numerical models of mechanics of discontinua (conditionally divided into the three broad, intertwined categories: lattices, PD and DEM, as discussed above) correspond to concrete (Figure 6.1d) (inspired by [2]). The MD refers to models where the basic building object is a point mass which may represent an atom, a molecule, a nanocluster, as well as a planet in a galaxy. In this short introduction, we always consider atoms (thus, the length scale in Figure 6.2) but, in general, it could be any of the above. From now on, the term MD, unless specified otherwise, is used for the traditional (classic) MD where each atom is treated as a point mass m i (and, generally, a fixed charge q i ). The term particle dynamics (PD), as used herein, designates "a coarse scale cousin to molecular dynamics" [3] sometimes also called the quasi-MD [4] to emphasize this kinship. It is developed to simulate phenomena on coarser spatial scales—the dynamic response of a material, either solid or fluid—based on a generalization of the MD modeling approach. The role of an atom is taken over by aggregates of atoms or molecules, represented by a material point called a " continuum particle " or a "quasi-particle". Depending on a particular application, this entity can represent, for example, a nanocluster, a ceramic grain, a concrete aggregate, a composite particle, a clastic rock granule and can, therefore, cover a wide range of spatial scales (up to the above-mentioned cosmological scales). Particle models use tried and tested MD techniques to directly confront various challenges of extremely complex physics. A critical step in the PD modeling is the transition from an adopted atomic potential to an interparticle potential (bottom-up approach) or a definition of an interparticle potential (a set of constitutive rules) on macro-scale (top-down), which is a common theme in all CMD numerical approaches. Traditional references for particle modeling are [4], [5], while [6] can be consulted for a review of recent developments. Lattices are arguably the simplest CMD models (specifically, the spring-networks among them), comprised of one-dimensional discrete structural units such as springs, trusses or beams (Figure 6.3a,b). These elementary building units are assigned both geo metric and structural properties and fracture characteristics that allow them to mimic elastic and inelastic deformation and fracture of the abstracted material. Lattice models could be 1 So-called, "ab initio (first principle) MD", also known as "Born-Oppenheimer" or "Carr Parrinello" MD.