20 COMPUTING IN SCIENCE & ENGINEERING

How materials fracture is one of the most funda-mental problems in materials science and engi-neering. Typically, the stress fields near the cracktip are highly nonlinear, and stress-field decayfar from the tip is very slow, making fracture a

difficult problem. Progress in large-scale scien-tific computing during the last decade has helpedresearchers successfully study dynamic fracture.

Dynamic fracture involves multiple scales oflength and time. When a material with a notchis stretched, a crack starts to propagate. Initially,the crack propagates slowly and leaves behindsmooth, mirror-like fracture surfaces. The crackthen accelerates to a certain critical speed atwhich the propagation becomes unstable. Theresulting fracture surfaces are very rough andsometimes involve multiple branches (see Fig-ure 1). Crack branching is also a highly nonlin-ear and complex phenomenon.

A special type of fractal geometry called self-affine fractal helps us describe rough fracturesurfaces. When we view a fracture surface at dif-ferent length scales—for example, using a mi-croscope with continuously varying magnifica-tion—the root-mean square roughness of thesurface varies as Lζ. The roughness exponent, ζ,denotes how the roughness scales with thelength, L. Researchers have experimentally mea-sured fracture-surface roughness with optical,atomic-force, and scanning-tunneling micro-scopes for a wide variety of materials (such asmetals, alloys, semiconductors, and ceramics).The scaling law for fracture-surface roughnessis L0.8. Surprisingly, this scaling law of surfaceroughness (ζ = 0.8) is independent of the mate-rial or fracture mode.

From an engineering point of view, designingmicrostructures to increase materials’ fracturetoughness (or resistance to fracture) is becomingincreasingly important. For example, ceramicsare extremely brittle in conventional forms, butthey can be made ductile when we assemblethem by consolidating nanometer-sized pow-ders. In ceramic matrix composites, the com-posite’s toughness is often much larger than the

PRIYA VASHISHTA AND AIICHIRO NAKANO

Louisiana State University

1521-9615/99/$10.00 © 1999 IEEE

G U E S T E D I T O R S ’I N T R O D U C T I O N

DYNAMIC FRACTURE ANALYSIS

SEPTEMBER/OCTOBER 1999 21

average of its constituents’ toughness values. Youcan understand this synergistic effect only byconsidering atomistic processes such as thestructure and frictional motion at the interfaces.

Simulation methods

Molecular-dynamics (MD) methods—whichphysicists, chemists, biologists, and engineersalike regularly use now—have introduced greatdegrees of cross-fertilization in various fields.Availability of efficient, scalable, and portableparallel-simulation algorithms coupled with sus-tained teraflop computing speeds on parallelcomputers are making it possible to performreal-materials simulations.

The ambitious goal of realistic-materials sim-ulations is to study synthesis, processing, andproperties of new materials before experimentalsynthesis and to test their behavior in extremeenvironments of temperature, pressure, and uni-axial stresses. However, in an age where we havea trillion-dollar economy of manmade materialsin energy, transportation, aerospace, electronics,and defense technologies, such an approach willclearly yield enormous dividends.

MD is a powerful tool for understanding dy-namic fracture that involves long-range stress-mediated phenomena and mechanical nano-structure failure. MD simulation providesphase-space trajectories—positions and veloci-ties of all atoms at all times—which are then an-alyzed using classical statistical mechanics. Forrealistic modeling of these systems, however, youmust extend the scope of simulations to largesystem sizes and long simulated times.

The most prohibitive computational problemin these simulations is associated with the calcu-lation of long-range parts of the interatomic po-tentials—the Coulomb interaction betweencharged atoms in the material. To overcome thisproblem, various researchers have designedspace-time multiresolution algorithms. Theseinclude the computation of the Coulomb inter-action with the Fast Multipole Method, whichreduces the computation from O(N2) to O(N) foran N-atom system. A multiple time-scale ap-proach exploits disparate time scales associatedwith slowly and rapidly varying parts of inter-atomic interactions.

Understanding dynamic fracture in metal, ce-ramic, or polymer components requires micro-scopic examination of plasticity. This is due todislocation emission and the interaction ofcracks with defects such as grain boundaries. In

this context, it is also important to know thestress inhomogeneities within the system.

Environmental effects are also important indynamic fracture. Oxidation is one of the majorcauses of damage, especially at high temperaturesand under stress. For example, oxidation embrit-tlement of ceramic-matrix composites involvesingress of oxygen through matrix cracks in thecomposite, and it drastically changes structuralperformance. Most metals, polymers, and someceramics are not stable against oxidation. Designand lifetime prediction of materials depend cru-cially on understanding oxidation’s effects.

Recent developments in parallel-computingtechnologies have made it possible to performatomistic simulations of dynamic fracture con-taining up to 100 million atoms. A number ofissues mentioned earlier—dynamic crack insta-bility, roughness of fracture surfaces, andtoughness of nanophase materials and compos-ites—are under investigation in various exper-iments. Effects of oxidation and other reactiveenvironments are also being attempted. How-ever, computing at this scale requires innova-tive parallel algorithms. These include space-time multiresolution schemes, load balancingfor parallel processing, data compression forscalable I/O, and visualization of and knowl-edge discovery from very large data sets to

Figure 1. An example of crack branching in dynamic fracture. (a)Molecular-dynamics simulation of dynamic fracture in a graphitesheet at a high strain rate, which we and our colleagues performedat the Concurrent Computing Laboratory for Materials Simulationsat Louisiana State University. (b) Krishnaswamy Ravi-Chandar’s ex-perimental observation of crack branching in glass. Note that thelength scales in (a) and (b) differ by a factor of 105, yet at these twowidely different length scales, the nature of the crack branching isremarkably similar.

22 COMPUTING IN SCIENCE & ENGINEERING

Matching length scalesA full understanding of dynamic fracture

requires an understanding of defects and

microstructures as well as the influence

of chemical agents and radiation on me-

chanical performance at all length and

energy scales.

However, researchers have extensively

used continuum methods based on sim-

ple constitutive relations (such as Hooke’s

law) to deal with dynamic fracture prob-

lems. Clearly, simple constitutive rela-

tions 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 be-

havior of materials under extreme condi-

tions of temperature, pressure, and uni-

axial stresses (see Figure A).

At the lowest level, such connections

between the two scales of representa-

tion—atomistic simulations at or near

the crack tip and continuum-

mechanics-based simulations

in the outer regions farther

from the crack tip—might in-

volve 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 mul-

tiscale dynamic fracture simu-

lation.

The hybrid electronic-

atomistic-continuum method

will link density-functional-

theory-based electronic struc-

ture methods with atomistic

molecular-dynamics simula-

tions and continuum thermo-

dynamic approaches. First-

principles calculations based on the den-

sity-functional theory and the local den-

sity approximation using pseudopoten-

tials have become the preferred method

of calculating energies and forces. How-

ever, these methods are computationally

demanding. The problem of seamlessly

matching the electronic-atomistic and

atomistic-continuum boundary regions

(the handshaking regions) is a very chal-

lenging algorithmic problem because

three very different types of differential

equations govern the three regions.

Therefore, a seamless multilevel simula-

tion of dynamic fracture is one of the

most challenging problems in the entire

field of simulation. The electronic-atom-

istic-continuum hybrid simulation tech-

niques will cover time scales from a frac-

tion of femto seconds to microseconds

and length scales from angstroms to mi-

crons. We believe during next three to

five years, with parallel machines having

sustained multiteraflop performance and

efficient hybrid algorithms, it will be fea-

sible to embed electronic structure calcu-

lations of roughly 104 atoms into 109

atom MD simulations that match seam-

lessly 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 calcula-

tions, in atomistic simulation with the

MD method, and a continuum approach

based on the finite-element method.1

A critical aspect of large-scale simula-

tion is the ability to represent informa-

tion contained in massive amounts of

data in a form and over media that en-

hances both understanding and visual

appreciation of the scientific content. To-

ward this objective, use of an Immersa-

Desk 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 technol-

ogy, we will be able to control super-

computing simulations interactively and

collaboratively, using high-performance

network connections.

Reference1. F.F. Abraham et al., “Spanning the Length

Scales in Dynamic Simulation,” Computers in

Physics, Vol. 12, No. 6, 1998, pp. 538Ð546.

Figure A. Multiscale 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.

(1) (2) (3)

SEPTEMBER/OCTOBER 1999 23

make the simulation results comprehensive tohumans.

Although atomistic processes are essential tounderstanding dynamic fracture, the stress fieldassociated with a crack is very long-ranged. Ac-cordingly, it is impossible to separate boundaryconditions from crack dynamics. Coupling oflength scales is therefore essential for an under-standing of macroscopic fracture phenomena.Hybrid schemes combining finite-element ap-proaches based on linear elasticity and atomisticsimulations using MD method and electronic-structure calculations have a promising future(see the “Matching length scales” sidebar). How-ever, the complexity of these hybrid schemesposes an unprecedented challenge in scientificsoftware development. Dynamic fracture thusprovides an exciting testbed for multidisciplinaryresearch between materials science and com-puter science.

The articles

This issue on dynamic fracture analysis featuresfive articles that survey important developmentsin the field and speculate about future researchdirections. The survey is by no means a com-prehensive one.

This issue combines the confluence of ideasand expertise on fracture from diverse points ofview, ranging from high-performance comput-ing and communications to atomistic simula-tions. The first article, written by KrishnaswanyRavi-Chandar and Wolfgang Knauss, introducesexperimental issues in fracture and how multi-scale computational schemes can answer some ofthe outstanding problems. The article by Elis-abeth Bouchaud and Florin Paun deals withcombined experimental and computational ap-proaches to the fascinating problem of fracture-surface roughness, which has puzzled numerousmathematicians, computer scientists, physicists,materials scientists, and engineers. The articleby Aiichiro Nakano, Rajiv K. Kalia, and PriyaVashishta reviews computer science techniquesthat enable large-scale atomistic simulations offracture, including multiresolution approachesto parallel computing, data management, and vi-sualization. Michael Marder’s article also usesboth computer simulations and laboratory ex-periments to understand the complex, nonlineardynamics of fracture. Finally, Priya Vashishta,Rajiv K. Kalia, and Aiichiro Nakano explain cru-cial roles played by large-scale atomistic simula-tions to understand a number of materials issues

in dynamic fracture—nanostructural design ofnovel high-fracture-toughness materials, envi-ronmental effects on fracture, and interfacial andnanoindentation fractures.

A word about computational science educa-tion: From the undergraduate and graduate stu-dent point of view, the standard university-levelphysics, chemistry, materials science, or engi-neering curricula seldom cultivate the ability toperform large-scale computer simulations. Suchan effort requires an integration of several disci-plines covered in disjointed courses. We hope tosomeday see courses on simulation methods thatintegrate these disparate disciplines under oneunified point of view.1,2

References1. J. Langer, “Computing in Physics: Are We Taking It Too Seri-

ously? Or Not Seriously Enough?” Physics Today, Vol. 52, No. 7,1999, pp. 11Ð12.

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 computa-

tional sciences in the Department of Physics and Astronomy

and the Department of Computer Science at Louisiana State

University. He cofounded the Louisiana State University Con-

current Computing Laboratory for Materials Simulations. His

research interests include very large-scale atomistic simula-

tions of synthesis, processing, and properties of novel mate-

rials, 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 ob-

tained his PhD in theoretical physics from University of Tokyo,

Japan. His research interests include parallel multilevel algo-

rithms; 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 Sci-

ence, Coates Hall, Louisiana State Univ., Baton Rouge, LA

70803-4020; nakano@bit.csc.lsu.edu; http://www.cclms.

lsu.edu.