Ultra?Large Scale Simulations of dynamic materials failure

Ultra?Large Scale Simulations of dynamic materials failure

(Parte 1 de 8)

CHAPTER 14

Ultra–Large Scale Simulations of Dynamic Materials Failure

Markus J. Buehler, Huajian Gao Max Planck Institute for Metals Research, Stuttgart, Germany

1. Introduction2
Dynamics of Materials Failure4

2. Motivation for the Atomistic Viewpoint: Nanoscale Governs

Potentials and Applications5
4. Empirical Interatomic Potentials8
4.1. Pair Potentials8
4.2. Multibody Potentials9
5. Physical and Mechanical Properties of Solids9
5.1. Unstable Stacking Fault Energy9
5.2. Fracture Surface Energy10
Wave Velocities, Virial Stress, and Strain10
6. Simulation Techniques1
6.1. Classical Molecular Dynamics1
6.2. Advanced Molecular Dynamics Methods12
6.3. Concurrent and Hierarchical Multiscale Methods14
Atomistic Information18
6.5. Discussion18

3. Classical Molecular Dynamics versus ab initio Methods: 5.3. Mechanical Properties of Crystals: Elastic Constants, 6.4. Continuum Approaches Incorporating

Supercomputers19
8. Analysis Techniques: Visualization and Data Processing20
8.1. Visualization Techniques20

7. Classical Molecular Dynamics Implemented on

ISBN: 1-58883-042-X/$35.0 Copyright © 2005 by American Scientific Publishers All rights of reproduction in any form reserved.

Handbook of Theoretical and Computational Nanotechnology

Edited by Michael Rieth and Wolfram Schommers Volume X: Pages (1–41)

2 Ultra–Large Scale Simulations of Dynamic Materials Failure

Visualization of Crystal Defects21

8.2. Postprocessing of Atomistic Simulation Data:

Mechanisms of Deformation23
9.1. Brittle Fracture and Defect Dynamics23
9.2. Ductile Failure28
Geometric Confinement31
Modeling of Nanostructures36
10. Conclusions and Discussion37
References38

9. Using Very Large Simulations to Study Different 9.3. Deformation Mechanisms in Materials under 9.4. Materials Science–Biology Interactions and

1. INTRODUCTION

When materials are deformed, they display a small regime in which deformation is reversible. This is referred to as the elastic regime. Once the forces on the material are increased, deformation becomes irreversible, and the deformation of a body caused by an applied stress remains after the stress is removed. This regime is referred to as the plastic regime. The study of plastic deformation using large-scale computer simulations will be the focus of this chapter.

In the classical picture, materials failure can be divided into two generic types: brittle and ductile. In the brittle case, atomic bonds are broken as material separates along a crack front. The type of failure of such materials is often characterized by the simultaneous motion of thousands of small cracks, as observed when glass shatters. This type of failure usually happens rapidly, as following a large impacts cracks propagate at velocities close the speed of sound [18, 4–46]. An enormous amount of research has been carried out over the last hundred years or so and has been summarized in recent books [16, 4]. The origin of fracture research dates back to the early 20th century in studies by Griffith [56] and Irwin [76]. The Griffith criterion provides a quantitative estimate of the condition under which material fails, and it is based on simple energetic and thermodynamic arguments. The Griffith criterion states that materials fail when the mechanical elastic energy released by crack propagation equals the fracture surface energy 2 s:

where G is the mechanical energy release rate [4]. This thermodynamic view of fracture was the foundation for the field of fracture mechanics. The continuum mechanics theory of fracture is a relatively well-established framework. In the continuum theory, the stress field in the vicinity of the crack tip is given by the asymptotic solution [12, 4, 157] and exhibits a universal character independent of the details of the applied loading. The loading of cracks can be separated into three different modes. Mode I is opening loading, mode I is shear loading, and mode I is antiplane shear loading.

In ductile failure, a catastrophic event such as rapid propagation of thousands of cracks does not occur. Tough materials like metals do not shatter: They bend easily because plastic deformation occurs through the motion of dislocations in the material. Ductility is sometimes defined as the property of a metal that allows it to be drawn into wires or filaments. Since their discovery in the early 1930s, dislocations have helped to explain many of the perplexing physical and mechanical properties of metals, some of which remained mysterious even until this date [128]. One of the topics that were discussed controversially is that the resistance of materials to shear is significantly less than the theoretical strength [54, 89, 128]. This phenomenon can only be explained by the existence of dislocations and their motion. The behavior of dislocations in crystals is very complex and involves multiple mechanisms for generation and annihilation, as it is summarized in [67]. Collective events may occur through interaction among many dislocations or between dislocations and other defects such as grain boundaries. Dislocation motion can be envisioned as similar to the movement of a caterpillar.

Ultra–Large Scale Simulations of Dynamic Materials Failure 3

Figure 1 shows the mechanism of dislocation motion through crystals. Moving the dislocation through the crystal is energetically more favorable than shearing the crystal as a whole. Similarly, it is easier to push a ripple through a carpet than to pull the carpet at one end over a sticky floor.

The tendency of materials to be ductile or brittle depends on their atomic microstructure.

The face-centered-cubic (fcc) packing is known to have a strong propensity toward ductility; body-centered-cubic (bcc) is much less so. Glasses do not have extended crystallinity because atoms are randomly packed. They have no slip-planes and mostly exhibit brittle failure with little ductility. Although atomic bonds are broken by stretching the solid in brittle fracture, the sliding between planes is achieved by shearing the solid in ductile failure. The ease of the atomic slip depends on the atomic arrangement of the slip planes. The more compact, and consequently less bumpy, planes slip better. Ductile versus brittle failure is schematically summarized in Fig. 2. Figure 2a shows brittle materials failure by propagation of cracks, and Fig. 2b depicts ductile failure by generation of dislocations at a crack tip. Although brittle and ductile failure have both been studied extensively, for a long time it remained unclear what separate ductile from brittle failure.

What is the origin of such fundamentally different behaviors? It was established that the origin of brittle versus ductile behavior is at the atomic scale. Studies by Rice and Thomson [1] revealed that there exists a competition between ductile (dislocation emission) and brittle (cleavage) mechanisms at the tip of a microcrack. Imperfections such as microcracks are considered the seeds for failure and exist in real materials. The model by Rice and Thomson has been recently extended to include a new material parameter, the unstable stacking fault energy us [109, 110]. The unstable stacking fault energy describes the resistance of the material to motion of dislocations, whereas the fracture surface energy describes the resistance of materials to fracture. At a crack tip, the unstable stacking fault energy competes with the fracture surface energy s [4] [see Eq. (1)]. Once these material parameters are known, it is often possible to quantitatively predict material behavior.

Recent research results indicate that dislocation-based processes and cleavage are not the only mechanisms for deformation of materials. Materials under geometrical confinement are also referred to as materials in small dimensions. The behavior of these materials is characterized by the interplay of interfaces (e.g., grain boundaries), constraints (e.g., substrates), and free surfaces. Examples for such materials are nanocrystalline materials [152, 153] or ultrathin submicron films [23, 24]. It was shown by computer simulation that in such materials, with grain sizes of tens of nanometers and below, deformation can be completely dominated by grain boundary processes. Even though such material behavior is ductile (as materials can be bent without cleavage), no dislocation motion is required. Because of the small sizes of the grains, dislocations can not be generated, because, for instance, Frank– Read sources are too large to fit within a grain, or because dislocations are energetically very expensive under very small geometrical confinement, as shown in earlier publications [43, 102, 103]. This leads to unexpected mechanisms of deformation, such as motion of partial dislocations.

Figure 1. Mechanism of motion of dislocations. The crystal is sheared apart by motion of an extra half plane from the left to the right. Once the half plane has propagated through the crystal, it has slipped by one atomic plane.

4 Ultra–Large Scale Simulations of Dynamic Materials Failure brittle crack propagation ductile dislocation emission

(a) (b)

Figure 2. (a) Brittle versus (b) ductile materials failure. In brittle failure a large number of cracks propagate through the material, breaking it apart. In ductile failure, such catastrophic event does not occur and the material bends through the motion of dislocations.

This chapter is organized as follows: We present an overview of today’s modeling of materials failure from a very fundamental, atomistic points of view. Such research often involves resource demanding computer simulations. Because ultralarge scale simulations heavily rely on supercomputers, we review the state-of-the-art computing and the associated programming techniques. We show that classical molecular dynamics with empirical potentials are, at this time, the only feasible approach to model the large length scale associated with the plasticity of materials. We continue with a brief discussion of the simulation tools, data processing, and visualization techniques, as these are key to the analysis of simulation results. We discuss a broad selection of research activities, highlight several topics, and provide a more in-depth discussion of selected areas. Specific focus is given to brittle and ductile behavior of materials and to recent progress in plasticity of materials in small dimensions and in particular nanostructured materials.

2. MOTIVATION FOR THE ATOMISTIC VIEWPOINT: NANOSCALE GOVERNS DYNAMICS OF MATERIALS FAILURE

Historically, the classical physics of continuum has been the basis for most theoretical and computational tools of engineers. In early stages of computational plasticity, dislocations and cracks were often treated using linear continuum mechanics theory, relying on numerous phenomenological assumptions. Over the last decades, there has been a new realization that understanding nanoscale behavior is required for understanding how materials fail (e.g., [4, 89]). This is partly because of the increasing trend to miniaturization as relevant length scales of materials approach several nanometers in modern technology. Once the dimensions of materials reach submicron length scales, the continuum description of materials is questionable and the full atomistic information is necessary to study materials phenomena.

Ultra–Large Scale Simulations of Dynamic Materials Failure 5

Atomistic simulations have proved to be a unique and powerful way to investigate the complex behavior of dislocations, cracks, and grain boundary processes at a very fundamental level. Atomistic methods are often the core in modern materials modeling. One of the strengths and the reason for the great success of atomistic methods is its very fundamental viewpoint of materials phenomena. The only physical law that is put into the simulations is Newton’s law and a definition of how atoms interact with each other. Despite this very simple basis, very complex phenomena can be simulated. Unlike many continuum mechanics approaches, atomistic techniques require no a priori assumption on the defect dynamics. Drawbacks of atomistic simulations are the difficulty of analyzing results and the large computational resources necessary to perform the simulations. This becomes more evident as the simulation sizes increase to systems with billions of atoms [8].

Once the atomic interactions are chosen, the complete material behavior is determined.

Although in some cases it is difficult to find the correct potential for a specific material, atomic interactions can often be chosen such that generic properties common to a large class of materials are incorporated (e.g., ductile materials). This allows us to design “model materials” to study specific materials phenomena. Despite the fact that model building has been in practice in fluid mechanics for many years, the concept of “model materials” in materials science is relatively new [8]. However, atomic interactions can be calculated very accurately for a specific atomic interaction, using quantum mechanics methods such as the density functional theory [123]. Richard Feynman says in his famous Feynman’s Lectures in Physics [40]:

If in some cataclysm all scientific knowledge were to be destroyed and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words?

I believe it is the atomic hypothesis that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see there is an enormous amount of information about the world, if just a little imagination and thinking are applied.

This underlines the that a natural choice for studying materials failure at a fundamental level are atomistic simulations. The atomistic level provides a most fundamental, sometimes referred to as the ab initio, description of the failure processes [4]. Many materials phenomena are multiscale phenomena. For a fundamental understanding, simulations should ideally capture the elementary physics of single atoms and reach length scales of thousands of atomic layers at the same time.

Recently, an increasing number of researchers have grown to consider the computer as a tool to do science, similar to how experimentalists use their lab to perform experiments. Computer simulations have been sometimes referred to as “computer experiments.” Designing smart computer experiments is the key to a successful simulation.

3. CLASSICAL MOLECULAR DYNAMICS VERSUS AB INITIO METHODS: POTENTIALS AND APPLICATIONS

Adapting the atomistic viewpoint, a fundamental description of the materials can be obtained. However, characterization of the interatomic interactions remains an important issue, as these are the core of atomistic modeling and simulation methods. The major differences between various atomistic methods are how atomic interactions are calculated.

With the expression for the potential energy i of a particle given by the chosen potential, the total energy of the system Etot can be obtained by summing over all particles. The force vector f for each particle is obtained by the gradient of the total energy with respect to a given particle location in space:

6 Ultra–Large Scale Simulations of Dynamic Materials Failure and N is the total number of particles. During the last decades, numerous potentials with different levels of accuracy have been proposed, each having its problems and strengths. The approaches range from accurate quantum-mechanics based treatments (e.g., first-principle density functional theory methods, or tight-binding potentials) [29], to multibody potentials (e.g., embedded atom approaches as proposed in [42]) to the most simple and computationally least-expensive pair potentials (e.g., Lennard–Jones [LJ]) [9, 32]. The first molecular dynamics study was a LJ model of Argon 1964 [108].

In density functional theory and related methods, the full quantum mechanical equations are solved to calculate the force on particles and are therefore numerically most expensive. Because the full quantum mechanical information is incorporated, the complete chemistry of atoms can be modeled (e.g., chemical reactions). Multibody potentials are often constructed based on quantum mechanical understanding of the binding, which is then devised into an empirical equation (e.g., electrons are not treated explicitly in embedded atom potentials (EAM) potentials but appear as electron density instead). Pair potentials assume that the force between atoms only depends on the distance between neighboring atoms. One of the recent developments is ab initio molecular dynamics, as reported by the group around Parrinello (Car–Parinello molecular dynamics) [29, 91]. In this method, only valence electrons are treated explicitly, and the interaction with the core electrons is treated based on pseudopotentials. Most quantum mechanics methods scale as O N3 or worse (the Car– Parinello method, depending on the algorithm chosen, can scale slightly better), whereas molecular dynamics methods based on empirical potentials scale linearly with the number of particles as O N . Any scaling other than linear is a severe computational burden and basically inhibits usage of the method for very large simulations.

An overview over the most prominent materials simulation techniques is shown in Fig. 3.

In the plot we indicate which length- and timescale quantum mechanics–based methods, classical molecular dynamics methods, and numerical continuum mechanics methods can reach. Quantum mechanics–based treatments are still limited to very short time- and length scales, on the order of a few nanometers and picoseconds. Once empirical interactions are assumed in classical molecular dynamics schemes, the length- and timescales achieved are dramatically increased, approaching micrometers and nanoseconds [8]. Continuum mechanics–based simulation tools can treat virtually any length scale, but they lack a proper description at small scales and are therefore often not suitable to describe materials failure processes in full detail (see discussion in Section 2). Figure 4 shows typical length and timescales for time sec

Quantum mechanics based methods

Classical molecular dynamics

Mesoscopic theories

Classical continuum theories

Figure 3. Overview of different atomistic simulation tools. The plot summarizes quantum mechanics–based simulation approaches (e.g., [29]), classical molecular dynamics [9] as well as continuum mechanics based methods (e.g., Refs. [14, 80]).

(Parte 1 de 8)

Comentários