Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

216

After introductory considerations, a summary of the basic concepts of traditional methods of CMD is presented. It should be noted that this classification of methods is somewhat subjective since their boundaries are blurred, domains overlap and distinctions are iffy. Be it as it may, the tentative classification is as follows: molecular dynamics (MD) and its coarser-scale offshoot dubbed herein particle dynamics (PD), the lattice methods and the discrete element methods (DEM). Although these (particle-based) methods are now widely used to model different classes of materials and various physical phenomena and industrial processes, the most natural applications seem to be the simulations of deformation and flow, damage, and fracture of systems that have the same topology as the representative model structure. Therefore, the modern, advanced applications advocate a modeling approach where it is insisted upon, as far as possible, the direct correspondence between the experimentally determined material and the structure explicitly represented by the numerical model. All materials have a discontinuous (and heterogeneous) structure on some spatial scale (if not macroscopic, perhaps mesoscopic, microscopic, by definition atomic) as illustrated by Figure 6.1. When this scale is “very small” from the standpoint of engineering appli cations, the materials are considered continuous (and homogeneous) (Figure 6.2). This discontinuity and heterogeneity lead to complex mechanical behavior difficult to reproduce with models based on the classical theory of continuum mechanics since the material sub stance: (i) does not fill entirely the space it occupies, and (ii) the physical and mechanical properties may vary significantly within that space and across various directions. Among these complex phenomena notable is the evolution of damage with nucleation, propagation, branching, mutual interaction and coalescence of (micro-/meso-/macro-scale) cracks and other pre-existing flaws and features of material texture that can lead to appearance of flow, diffuse or localized deformation and damage, fracture, and fragmentation. All CMD methods described herein have in common that they cope with these complexities by establishing a computational domain (approximating the material structure) by a collection of discrete building units that are, or may be, interconnected. These models differ from the computational models of continuum mechanics in the definition of the displacement field only in the finite number of nodal points and, accordingly, in the formulation of the problem using algebraic, instead of partial differential, equations. During the 1960s, researchers and engineers working in various fields of mechanics of materials and materials science noticed that solutions obtained using traditional continuum mechanics often exhibited singularities or yielded results inconsistent with experimental observations. In the decade that followed, the awesome development of computer capa bilities and the accompanying advances of numerical methods enabled the emergence of novel particle-based computational methods that used various distinct structural-building elements (atoms, springs, trusses, beams, particles or various shapes) to model materials. CMD is nowadays firmly established as an integral part of not only the cutting-edge research in various fields (e.g., nanotechnology, stem cell research, biomedical engineer ing, space propulsion) but also industrial processes covering a wide range of different application fields (e.g., mining, machining, pharmaceuticals, civil construction, industrial and systems engineering).