, Louisiana State University

Pages: pp. 20-23

How materials fracture is one of the most fundamental problems in materials science and engineering. Typically, the stress fields near the crack tip are highly nonlinear, and stress-field decay far from the tip is very slow, making fracture a difficult problem. Progress in large-scale scientific computing during the last decade has helped researchers successfully study dynamic fracture.

Dynamic fracture involves multiple scales of length and time. When a material with a notch is stretched, a crack starts to propagate. Initially, the crack propagates slowly and leaves behind smooth, mirror-like fracture surfaces. The crack then accelerates to a certain critical speed at which the propagation becomes unstable. The resulting fracture surfaces are very rough and sometimes involve multiple branches (see Figure 1). Crack branching is also a highly nonlinear and complex phenomenon.

Figure 1 An example of crack branching in dynamic fracture. (a) Molecular-dynamics simulation of dynamic fracture in a graphite sheet at a high strain rate, which we and our colleagues performed at the Concurrent Computing Laboratory for Materials Simulations at Louisiana State University. (b) Krishnaswamy Ravi-Chandar's experimental observation of crack branching in glass. Note that the length scales in (a) and (b) differ by a factor of 10 ^{5}, yet at these two widely different length scales, the nature of the crack branching is remarkably similar.

A special type of fractal geometry called self-affine fractal helps us describe rough fracture surfaces. When we view a fracture surface at different length scales—for example, using a microscope with continuously varying magnification—the root-mean square roughness of the surface varies as *L*^{ζ}. The roughness exponent, ζ, denotes how the roughness scales with the length, *L*. Researchers have experimentally measured fracture-surface roughness with optical, atomic-force, and scanning-tunneling microscopes for a wide variety of materials (such as metals, alloys, semiconductors, and ceramics). The scaling law for fracture-surface roughness is *L*^{0.8}. Surprisingly, this scaling law of surface roughness (ζ = 0.8) is independent of the material or fracture mode.

From an engineering point of view, designing microstructures to increase materials' fracture toughness (or resistance to fracture) is becoming increasingly important. For example, ceramics are extremely brittle in conventional forms, but they can be made ductile when we assemble them by consolidating nanometer-sized powders. In ceramic matrix composites, the composite's toughness is often much larger than the average of its constituents' toughness values. You can understand this synergistic effect only by considering atomistic processes such as the structure and frictional motion at the interfaces.

Molecular-dynamics (MD) methods—which physicists, chemists, biologists, and engineers alike regularly use now—have introduced great degrees of cross-fertilization in various fields. Availability of efficient, scalable, and portable parallel-simulation algorithms coupled with sustained teraflop computing speeds on parallel computers are making it possible to perform real-materials simulations.

The ambitious goal of realistic-materials simulations is to study synthesis, processing, and properties of new materials before experimental synthesis and to test their behavior in extreme environments of temperature, pressure, and uniaxial stresses. However, in an age where we have a trillion-dollar economy of manmade materials in energy, transportation, aerospace, electronics, and defense technologies, such an approach will clearly yield enormous dividends.

MD is a powerful tool for understanding dynamic fracture that involves long-range stress-mediated phenomena and mechanical nanostructure failure. MD simulation provides phase-space trajectories—positions and velocities of all atoms at all times—which are then analyzed using classical statistical mechanics. For realistic modeling of these systems, however, you must extend the scope of simulations to large system sizes and long simulated times.

The most prohibitive computational problem in these simulations is associated with the calculation of long-range parts of the interatomic potentials—the Coulomb interaction between charged atoms in the material. To overcome this problem, various researchers have designed space-time multiresolution algorithms. These include the computation of the Coulomb interaction with the Fast Multipole Method, which reduces the computation from *O*( *N*^{2}) to *O*( *N*) for an *N*-atom system. A multiple time-scale approach exploits disparate time scales associated with slowly and rapidly varying parts of interatomic interactions.

Understanding dynamic fracture in metal, ceramic, or polymer components requires microscopic examination of plasticity. This is due to dislocation emission and the interaction of cracks with defects such as grain boundaries. In this context, it is also important to know the stress inhomogeneities within the system.

Environmental effects are also important in dynamic fracture. Oxidation is one of the major causes of damage, especially at high temperatures and under stress. For example, oxidation embrittlement of ceramic-matrix composites involves ingress of oxygen through matrix cracks in the composite, and it drastically changes structural performance. Most metals, polymers, and some ceramics are not stable against oxidation. Design and lifetime prediction of materials depend crucially on understanding oxidation's effects.

Recent developments in parallel-computing technologies have made it possible to perform atomistic simulations of dynamic fracture containing up to 100 million atoms. A number of issues mentioned earlier—dynamic crack instability, roughness of fracture surfaces, and toughness of nanophase materials and composites—are under investigation in various experiments. Effects of oxidation and other reactive environments are also being attempted. However, computing at this scale requires innovative parallel algorithms. These include space-time multiresolution schemes, load balancing for parallel processing, data compression for scalable I/O, and visualization of and knowledge discovery from very large data sets to make the simulation results comprehensive to humans.

Although atomistic processes are essential to understanding dynamic fracture, the stress field associated with a crack is very long-ranged. Accordingly, it is impossible to separate boundary conditions from crack dynamics. Coupling of length scales is therefore essential for an understanding of macroscopic fracture phenomena. Hybrid schemes combining finite-element approaches based on linear elasticity and atomistic simulations using MD method and electronic-structure calculations have a promising future (see the " Matching length scales" sidebar). However, the complexity of these hybrid schemes poses an unprecedented challenge in scientific software development. Dynamic fracture thus provides an exciting testbed for multidisciplinary research between materials science and computer science.

This issue on dynamic fracture analysis features five articles that survey important developments in the field and speculate about future research directions. The survey is by no means a comprehensive one.

This issue combines the confluence of ideas and expertise on fracture from diverse points of view, ranging from high-performance computing and communications to atomistic simulations. The first article, written by Krishnaswany Ravi-Chandar and Wolfgang Knauss, introduces experimental issues in fracture and how multi-scale computational schemes can answer some of the outstanding problems. The article by Elisabeth Bouchaud and Florin Paun deals with combined experimental and computational approaches to the fascinating problem of fracture-surface roughness, which has puzzled numerous mathematicians, computer scientists, physicists, materials scientists, and engineers. The article by Aiichiro Nakano, Rajiv K. Kalia, and Priya Vashishta reviews computer science techniques that enable large-scale atomistic simulations of fracture, including multiresolution approaches to parallel computing, data management, and visualization. Michael Marder's article also uses both computer simulations and laboratory experiments to understand the complex, nonlinear dynamics of fracture. Finally, Priya Vashishta, Rajiv K. Kalia, and Aiichiro Nakano explain crucial roles played by large-scale atomistic simulations to understand a number of materials issues in dynamic fracture—nanostructural design of novel high-fracture-toughness materials, environmental effects on fracture, and interfacial and nanoindentation fractures.

A word about computational science education: From the undergraduate and graduate student point of view, the standard university-level physics, chemistry, materials science, or engineering curricula seldom cultivate the ability to perform large-scale computer simulations. Such an effort requires an integration of several disciplines covered in disjointed courses. We hope to someday see courses on simulation methods that integrate these disparate disciplines under one unified point of view. ^{1,}^{2}

A full understanding of dynamic fracture requires an understanding of defects and microstructures as well as the influence of chemical agents and radiation on mechanical performance at all length and energy scales.

However, researchers have extensively used continuum methods based on simple constitutive relations (such as Hooke's law) to deal with dynamic fracture problems. Clearly, simple constitutive relations are invalid in nonlinear regions close to cracks or dislocation cores while atomistic simulations are ideally suited to handle such nonlinearities. However, a complete atomistic simulation on a mesoscopic scale demands too much computationally and isn't warranted for small deformations, where finite-element methods can do a good job at a fraction of the full atomistic simulation's cost. A solution to this stalemate is the atomistic-continuum hybrid simulation approach, which covers the multiple length scales involved in studying the mechanical behavior of materials under extreme conditions of temperature, pressure, and uniaxial stresses (see Figure A).

AMultiscale simulation and visualization approaches combining (1) the finite-element, (2) molecular-dynamics, and (3) quantum-mechanical methods. The finite-element (represented by meshes) and MD (blue and red spheres) regions are coupled through a handshake region (cyan and magenta). Atoms near the crack (yellow and green) are treated quantum mechanically.At the lowest level, such connections between the two scales of representation—atomistic simulations at or near the crack tip and continuum-mechanics-based simulations in the outer regions farther from the crack tip—might involve a simple parameter passing from the atomic scale results to the continuum code. From an aesthetic and practical point of view, this is not a satisfactory solution to the problem of seamless multiscale dynamic fracture simulation.

The hybrid electronic-atomistic-continuum method will link density-functional-theory-based electronic structure methods with atomistic molecular-dynamics simulations and continuum thermodynamic approaches. First-principles calculations based on the density-functional theory and the local density approximation using pseudopotentials have become the preferred method of calculating energies and forces. However, these methods are computationally demanding. The problem of seamlessly matching the electronic-atomistic and atomistic-continuum boundary regions (the handshaking regions) is a very challenging algorithmic problem because three very different types of differential equations govern the three regions. Therefore, a seamless multilevel simulation of dynamic fracture is one of the most challenging problems in the entire field of simulation. The electronic-atomistic-continuum hybrid simulation techniques will cover time scales from a fraction of femto seconds to microseconds and length scales from angstroms to microns. We believe during the next three to five years, with parallel machines having sustained multiteraflop performance and efficient hybrid algorithms, it will be feasible to embed electronic structure calculations of roughly 10 ^{4} atoms into 10 ^{9} atom MD simulations that match seamlessly into the finite-element region. A hybrid approach has been developed by Farid Abraham, Jeremy Broughton, Norm Bernstein, and Efthimios Kaxiras, which couples an empirical tight-binding method for electronic-structure calculations, in atomistic simulation with the MD method, and a continuum approach based on the finite-element method. ^{1}

A critical aspect of large-scale simulation is the ability to represent information contained in massive amounts of data in a form and over media that enhances both understanding and visual appreciation of the scientific content. Toward this objective, use of an ImmersaDesk or a CAVE—a fully immersive and interactive multiviewer environment that links human perception (audio, visual, and tactile) to the simulated world on parallel machines—is highly desirable. By using the ImmersaDesk or CAVE technology, we will be able to control supercomputing simulations interactively and collaboratively, using high-performance network connections.

ReferenceF.F.Abrahamet al.,"Spanning the Length Scales in Dynamic Simulation,"Computers in Physics,Vol. 12,No. 6,1998,pp. 538-546.- 1. J. Langer, "Computing in Physics: Are We Taking It Too Seriously? Or Not Seriously Enough?"
*Physics Today,*Vol. 52, No. 7, 1999, pp. 11-12. - 2. "Physics Departments Explore Innovative Curriculum Approaches,"
*APS News,*Series II, Vol. 6, No. 8, Aug./Sept. 1997, p. 2.

Priya Vashishta is the Cray Research Professor of computational sciences in the Department of Physics and Astronomy and the Department of Computer Science at Louisiana State University. He cofounded the Louisiana State University Concurrent Computing Laboratory for Materials Simulations. His research interests include very large-scale atomistic simulations of synthesis, processing, and properties of novel materials, nanoscale devices, and dynamic fracture on massively parallel and distributed computers. Contact him at the CCLMS, Nicholson Hall, Louisiana State Univ., Baton Rouge, LA 70803-4001; priyav@bit.csc.lsu.edu; http://www.cclms.lsu.edu.

Aiichiro Nakano is an associate professor of computer science and a member of the Concurrent Computing Laboratory for Materials Simulations at Louisiana State University. He obtained his PhD in theoretical physics from University of Tokyo, Japan. His research interests include parallel multilevel algorithms; visualization, data-management, and networking technologies for computational sciences; computer-aided nanomaterials design; computational nanoelectronics; and dynamic fracture. Contact him at the Dept. of Computer Science, Coates Hall, Louisiana State Univ., Baton Rouge, LA 70803-4020; nakano@bit.csc.lsu.edu; http://www.cclms.lsu.edu.