Investigation of Dynamic Behavior of Brittle Solids by Discrete Systems

Abstract

With the remarkably rapid growth of computer capabilities and corresponding advance of numerical algorithms as background, the landscape of the engineering and scientific research is continually changing. In the pre-computer era, the physical sciences were characterized by interplay between theory and experiment, both necessarily simplified to eliminate the complexities that rendered the phisical phenomena impossible to tackle. In the last two-three decades, the computer technology surge altered substantially this relationship by enriching the research by a new element: the computer experiment. The change is far reaching indeed. The computer simulations significantly pushed the envelope of “solvable” problems. When they, by constantly demanding more accurate inputs from both theorists and experimentalists, manage to come very close to capturing the reality of a phenomenon, they become an extremely powerful tool indispensible in interpretation of the experimental results at the spatial and temporal scales beyond the reach of everdeveloping experimental techniques. More than that: the virtual laboratory enables us to perform experiments that are beyond imagination in the actual laboratory, with practically unlimited level of control. The objective of the research efforts reviewed in this monograph was threefold. First, to identify patterns and dominant aspects of dynamic response of idealized brittle solids subjected to high strain rates. Second, to propose simple, approximate, design oriented models needed to capture some of those salient features. Third, to investigate universal trends in which disorder and strain rate influence that dynamic response. The outlined approach leads to rational estimates of the radial tractions required to expand dynamically a cylindrical cavity and, consequently, the forces on the projectile nose resisting penetration; and eventually, the corresponding penetration depth of the superior-strength projectile through the infinite brittle medium. The models are based on the micromechanics of deformation and damage evolution in the generic brittle material with random microstructure and inferior tensile strength. The considered problems are, in addition to ballistics, also frequently encountered in mining, metalworking, transportation, etc. The proposed models accounts for the prominent properties of brittle materials: a random microstructure, pressure-dependent shear strength, inferior tensile strength and presence of process induced micro-defects; and the characteristic deformation features such as the rate-dependent fracture pattern, granular flow, and deterioration of the effective material properties. The effect of “pre-existing” material imperfections on the material properties and their change in the course of deformation is an indication that a rational model should be sought within the scope of damage mechanics. The strategy selected in this study is to use the virtual laboratory experiments to reduce the dependence on actual tests (often too difficult and/or expensive, if not impossible, to perform) and provide a reasonably detailed picture of the state of material and mechanical fields in the infinite brittle target. Due to the extreme loading conditions, the concepts of strain, stiffness tensor, damage and temperature are, at best, conditionally acceptable since the corresponding deformation process is nonlocal, non-linear and non-equilibrated. To eliminate large spatial and temporal fluctuations the “local” values of these fields are determined by volume averaging. As a consequence, the model resolution is rather coarse and the results are primarily directed to the estimates of the global parameters of the problem which are, fortunately, most important for engineering purposes. The approximation of a solid by an ensemble of interacting particles (so called, lattice, spring-network, or discrete models) is selected for at least three reasons. First, the introduction of morphological and structural disorder is straightforward. Second, the selection of the constitutive relations is not arbitrary since it can be, in principle, inferred from the molecular dynamics simulations on the sub-meso scales. Finally, there is no need for developing time-consuming numerical techniques to track the material interfaces. The rationale for the selected strategy is fully supported by the simulation results. This monograph grew out of my dissertation “Dynamic Loading of Brittle Materials with Random Microstructure” presented at Arizona State University (1997) in partial fulfillement of the requirements for the degree Doctor of Philosophy. The offsprings of that dissertation were numerous research papers prepared in collaboration with the late Professor Dusan Krajcinovic. The monograph is organized in five chapters describing the major subject areas; the chapters are divided into sections and the sections into thematic subsections. It is important to note that this book is not a review monograph of the discrete numerical methods, but a brief treatise on a narrow multidisciplinary research area. Thus, many valuable references in the topical areas are surely inadverantly left out. The material in Section 2.1 is a brief survey of standard background information on conventional molecular dynamics. The introduction to particle dynamics models presented in Section 2.2 is based mostly on Krajcinovic and Mastilovic (1999) and Mastilovic and Krajcinovic (1999a). The core of the monograph, which comprises Chapters 3 and 4 describing the numerical simulations and the corresponding analytical modelling, is founded primarily on the following publications: Mastilovic and Krajcinovic (1999b) (Sections 3.1 through 3.3); Mastilovic and Krajcinovic (1999a) (Section 3.4 and 4.1); Krajcinovic and Mastilovic (2001a) (Section 3.4); and Mastilovic et al. (2008) (Subsection 3.3.1). Section 4.2, concerned with the penetration depth modelling, follows closely analysis in Mastilovic and Krajcinovic (1999c). The topics covered in Appendices A and D are originally published in Mastilovic (2008) and Mastilovic (accepted). Finally, the mentioned advent of high productivity computing takes its toll on many results presented in this study. Many numerical models that pushed the envelope of PC performace 10-15 years ago are orders of magnitude behind the current state-ofthe- art, which is an unavoidable drawback of postponed publication in rapidly developing research areas. Therefore, as I prepared the manuscript, I had to resist a strong urge to re-do many simulations presented herein; successfully, I am happy to admit. Last but not least, I am indepted to many for help and support during the years of work shaped in this monograph; no one is named here but no one is forgotten.