Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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As already noted, the PD simulation is completely deterministic. However, the particle configurations that mimick the modeled material are not necessarily associated with an ideal lattice. As an example, in order to describe the deformation of the brittle material with a random microstructure, it is necessary to introduce the quenched disorder into the computer model. This initial disorder can be topological (unequal coordination number), geometric (unequal bond length) or structural (unequal bond stiffness and/or strength). It can be introduced through the probability distributions of strength, stiffness, and missing bonds (which mimics porosity or pre-existing damage). This (so called, quenched) disorder increases with damage evolution (the induced disorder) in the course of deformation process. Therefore, the nature of damage evolution is inheretly stochastic due to the initial stochasticity of the PD model (although each individual physical realization of the given statistics is deterministic). 6.4 Lattices Lattice models are a class of CMD models based on the concept of computation domain discretization with an assembly of one-dimensional elements (springs, trusses, beams) endowed by elementary constitutive rules and rupture criteria. They are closely related to PD, as outlined in the preceding chapter, since every system of particles can be associated with a lattice, especially in the case of solids. (As an example, the mentioned modification of the hybrid potential (6.26), with the attractive part of the potential (6.26) 2 being used in the repulsion domain as well, constitutes the simplest lattice model - the spring network.) The first application of the lattice method is attributed to the Russo-Canadian engineer Alexander Hrennikoff [7], who devised it to solve the plane stress problem of a thin elastic plate loaded with in-plane forces. The method then fell into oblivion and remained dormant until the 1980s when its remarkable capabilities for introducing material disorder and heterogeneities into the computational model in a simple and natural way were noticed. Ever since, simplicity and inherent ability to capture localized failure mechanisms led to rapid development of lattice models. Not surprisingly, these models are of special importance for studies of mechanical fracture of quasi-brittle materials (e.g., concrete), which were necessarily of phenomenological character. Nonetheless, applications for metals, ceramics, polymers, composites, granular materials are available in literature ([65]- [67]). The chronological development and adaptations of lattice models to different types of materials and loads are encapsulated herein in a most concise form. To begin with, it should be noted that the continuum can be discretized by lattice models in various ways (plane or spatial lattices, of regular or irregular (random) geometry, with overlapping or non-overlapping elements). Lattice models can also differ in the number of degrees of freedom per node (truss vs. beam), which has proven to be a source of important distinction when it comes to their ability to realistically reproduce physical phenomena. Over time, a consensus was reached that lattice models of irregular geometry with beam interactions were most suitable for fracture simulations, especially in materials characterized with a distinctly heterogeneous structure.