3.021J / 1.021J / J / J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008

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1 MIT OpenCourseWare J / 1.021J / J / J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008 For information about citing these materials or our Terms of Use, visit:

2 1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation Part II - lecture 6 Atomistic and molecular methods 1

3 Content overview I. Continuum methods 1. Discrete modeling of simple physical systems: Equilibrium, Dynamic, Eigenvalue problems 2. Continuum modeling approaches, Weighted residual (Galerkin) methods, Variational formulations 3. Linear elasticity: Review of basic equations, Weak formulation: the principle of virtual work, Numerical discretization: the finite element method II. Atomistic and molecular methods 1. Introduction to molecular dynamics 2. Basic statistical mechanics, molecular dynamics, Monte Carlo 3. Interatomic potentials 4. Visualization, examples 5. Thermodynamics as bridge between the scales 6. Mechanical properties how things fail 7. Multi-scale modeling 8. Biological systems (simulation in biophysics) how proteins work and how to model them III. Quantum mechanical methods 1. It s A Quantum World: The Theory of Quantum Mechanics 2. Quantum Mechanics: Practice Makes Perfect 3. The Many-Body Problem: From Many-Body to Single-Particle 4. Quantum modeling of materials 5. From Atoms to Solids 6. Basic properties of materials 7. Advanced properties of materials 8. What else can we do? Lectures 2-10 February/March Lectures March/April Lectures April/May 2

4 Overview: Material covered Lecture 1: Introduction to atomistic modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion) Lecture 2: Basic statistical mechanics (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function, solution techniques: Monte Carlo and molecular dynamics) Lecture 3: Basic molecular dynamics (advanced property calculation, chemical interactions) Lecture 4: Interatomic potential and force field (pair potentials, fitting procedure, force calculation, multi-body potentials-metals/eam & applications, neighbor lists, periodic BCs, how to apply BCs) Lecture 5: Interatomic potential and force field (cont d) (organic force fields, bond order force fields-chemical reactions, additional algorithms (NVT, NPT), application: mechanical properties basic introduction) Lecture 6: Application to mechanics of materials-brittle materials (significance of fractures/flaws, brittle versus ductile behavior [motivating example], basic deformation mechanisms in brittle fracture, theory, case study: supersonic fracture (example for model building); case study: fracture of silicon (hybrid model), modeling approaches: brittle-pair potential/reaxff, applied to silicon Lecture 7: Application to mechanics of materials-ductile materials (dislocation mechanisms), case study: failure of copper nanocrystal Lecture 8: Review session Lecture 9: QUIZ 3

5 II. Atomistic and molecular methods Lecture 6: Application to mechanics of materials-brittle materials Outline: 1. Brittle versus ductile materials 2. Basic deformation mechanism in brittle materials: crack extension 3. Atomistic modeling studies of fracture 3.1 Physical properties of atomic lattices 3.2 Supersonic/subsonic dynamic fracture 3.3 Reactive potential/hybrid models (introduction) Goal of today s lecture: Learn basics in mechanics of materials, focused on deformation & fracture & understand the role of flaws, defects in controlling material properties Learn basics in fracture of brittle materials Learn atomistic modeling approaches, include concept of model materials and reactive force field, multi-paradigm modeling approach Today s lecture prepares you to carry out MD simulations of fracture using two modeling approaches (problem set 2) 4

6 1. Brittle versus ductile materials 5

7 Tensile test of a wire Brittle Ductile Stress Necking Strain Brittle Ductile Figure by MIT OpenCourseWare. Figure by MIT OpenCourseWare. 6

8 Ductile versus brittle materials BRITTLE DUCTILE Glass Polymers Ice... Copper, Gold Shear load Difficult to deform, breaks easily Figure by MIT OpenCourseWare. Easy to deform hard to break 7

9 Deformation of materials: Nothing is perfect, and flaws or cracks matter Failure of materials initiates at cracks Griffith, Irwine and others: Failure initiates at defects, such as cracks, or grain boundaries with reduced traction, nano-voids, other imperfections 8

10 SEM picture of material: nothing is perfect 9

11 Significance of material flaws Image removed due to copyright restrictions. Please see: Fig. 1.3 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Stress concentrators: local stress >> global stress 10

12 Deformation of materials: Nothing is perfect, and flaws or cracks matter Macro, global σ (r) Micro (nano), local r Failure of materials initiates at cracks Griffith, Irwine and others: Failure initiates at defects, such as cracks, or grain boundaries with reduced traction, nano-voids, other imperfections 11

13 Cracks feature a singular stress field, with singularity at the tip of the crack stress tensor y 1 σ ( r) ~ r σ yy σ xy σrθ σ xx σ rr r θ σ θθ x Figure by MIT OpenCourseWare. K I : Stress intensity factor (function of geometry) 12

14 Crack extension: brittle response Image removed due to copyright restrictions. Please see: Fig. 1.4 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Large stresses lead to rupture of chemical bonds between atoms Thus, crack extends Figure by MIT OpenCourseWare. 13

15 Lattice shearing: ductile response Instead of crack extension, induce shearing of atomic lattice Due to large shear stresses at crack tip Lecture 7 τ τ τ τ Figures by MIT OpenCourseWare. 14

16 Brittle vs. ductile material behavior Whether a material is ductile or brittle depends on the material s propensity to undergo shear at the crack tip, or to break atomic bonds that leads to crack extension Intimately linked to the atomic structure and atomic bonding Related to temperature (activated process); some mechanism are easier accessible under higher/lower temperature Many materials show a propensity towards brittleness at low temperature Molecular dynamics is a quite suitable tool to study these mechanisms, that is, to find out what makes materials brittle or ductile 15

17 Historical example: significance of brittle vs. ductile fracture Liberty ships: cargo ships built in the U.S. during World War II (during 1930s and 40s) Eighteen U.S. shipyards built 2,751 Liberties between 1941 and 1945 Early Liberty ships suffered hull and deck cracks, and several were lost to such structural defects Twelve ships, including three of the 2710 Liberties built, broke in half without warning, including the SS John P. Gaines (sank 24 November 1943) Constance Tipper of Cambridge University demonstrated that the fractures were initiated by the grade of steel used which suffered from embrittlement. She discovered that the ships in the North Atlantic were exposed to temperatures that could fall below a critical point when the mechanism of failure changed from ductile to brittle, and thus the hull could fracture relatively easily. 16

18 Liberty ships: brittle failure Courtesy of the U.S. Navy. 17

19 2. Basic deformation mechanism in brittle materials: crack extension 18

20 Introduction: brittle fracture Materials: glass, silicon, many ceramics, rocks At large loads, rather than accommodating a shape change, materials break Image removed due to copyright restrictions. Photo of broken glass. 19

21 Science of fracture: model geometry Typically consider a single crack in a crystal Remotely applied mechanical load Following discussion focused on single cracks and their behavior remote load a remote load Figure by MIT OpenCourseWare. 20

22 Brittle fracture loading conditions Commonly consider a single crack in a material geometry, under three types of loading: mode I, mode II and mode III Mode I Mode II Mode III Tensile load, focus of this lecture Figure by MIT OpenCourseWare. Tensile load, focus of this lecture 21

23 Brittle fracture mechanisms: fracture is a multiscale phenomenon, from nano to macro Image removed due to copyright restrictions. Please see: Fig. 6.2 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

24 Focus of this part Basic fracture process: dissipation of elastic energy Fracture initiation, that is, at what applied load to fractures initiate Fracture dynamics, that is, how fast can fracture propagate in material 23

25 Basic fracture process: dissipation of elastic energy Image removed due to copyright restrictions. Please see: Fig. 6.5 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Undeformed Stretching=store elastic energy Release elastic energy dissipated into breaking chemical bonds 24

26 Elasticity = reversible deformation Please also see: Fig. 3.1 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

27 Continuum description of fracture Fracture is a dissipative process in which elastic energy is dissipated to break bonds (and to heat at large crack speeds) Energy to break bonds = surface energy γ s (energy necessary to create new surface, dimensions: energy/area, Nm/m 2 ) "Relaxed" element a ~ σ δa "Strained" element a ~ a ~ ξ σ ε (2) (2) a = 1 σ 2 ξab ~ 2 E σ (1) (1)! = 2γ Ba~ Figure by MIT OpenCourseWare. s G 2 1 σ 2 E 2 σ ξ = = 2γ s 2E change of elastic energy energy to create surfaces σ = 4γ E s 26 ξ

28 Griffith condition for fracture initiation Energy release rate, that is, the elastic energy released per unit crack advance must be equal or larger than the energy necessary to create new surfaces Provides criterion to predict failure initiation Calculation of G can be complex, but straightforward for thin strips as shown above Approach to calculate G based on stress intensity factor (see further literature, e.g. Broberg, Anderson, Freund, Tada) 27

29 Brittle fracture mechanisms Once nucleated, cracks in brittle materials spread rapidly, on the order of sound speeds Sound speeds in materials (=wave speeds): Rayleigh-wave speed c R (speed of surface waves) shear wave speed c s (speed of shear waves) longitudinal wave speed c l (speed of longitudinal waves) Maximum speeds of cracks is given by sound speeds, depending on mode of loading (mode I, II, III) Linear elastic continuum theory 28

30 Sound speeds in materials: overview Wave speeds are calculated based on elastic properties of material μ = shear modulus E = 8/ 3μ μ = 3/8E 29

31 Limiting speeds of cracks: linear elastic continuum theory Images removed due to copyright restrictions. Please see: Fig and 6.90 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Cracks can not exceed the limiting speed given by the corresponding wave speeds unless material behavior is nonlinear Cracks that exceed limiting speed would produce energy (physically impossible - linear elastic continuum theory) 30

32 Physical reason for crack limiting speed Physical (mathematical) reason for the limiting speed is that it becomes increasingly difficult to increase the speed of the crack by adding a larger load When the crack approaches the limiting speed, the resistance to fracture diverges to infinity Image removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

33 Linear versus nonlinear elasticity=hyperelasticity Images removed due to copyright restrictions. Please see: Fig. 6.35b in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Linear elasticity: Young s modulus (stiffness) does not change with deformation Nonlinear elasticity = hyperelasticity: Young s modulus (stiffness) changes with deformation 32

34 Subsonic and supersonic fracture Under certain conditions, material nonlinearities (that is, the behavior of materials under large deformation = hyperelasticity) becomes important This can lead to different limiting speeds than described by the model introduced above σ ( r) ~ 1 r Deformation field near a crack Images removed due to copyright restrictions. Please see: Fig. 6.35a and 6.49 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

35 Limiting speeds of cracks Images removed due to copyright restrictions. Please see: Fig and 6.90 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Under presence of hyperelastic effects, cracks can exceed the conventional barrier given by the wave speeds This is a local effect due to enhancement of energy flux 34 Subsonic fracture due to local softening, that is, reduction of energy flux

36 Stiffening vs. softening behavior Images removed due to copyright restrictions. Please see: Fig. 6.35b in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Increased/decreased wave speed 35

37 Energy flux reduction/enhancement Images removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Energy flux related to wave speed: high local wave speed, high energy flux, crack can move faster (and reverse for low local wave speed) Buehler et al., Nature,

38 Summary: atomistic mechanisms of brittle fracture Brittle fracture rapid spreading of a small initial crack Cracks initiate based on Griffith condition G = 2γ s Cracks spread on the order of sound speeds (km/sec for many brittle materials) Cracks have a maximum speed, which is given by characteristic sound speeds for different loading conditions) Maximum speed can be altered if material is strongly nonlinear, leading to supersonic or subsonic fracture 37

39 3. Atomistic modeling studies of fracture 38

40 Mike Ashby (Cambridge University): Model building - review A model is an idealization. Its relationship to the real problem is like that of the map of the London tube trains to the real tube systems: a gross simplification, but one that captures certain essentials. Image removed due to copyright restrictions. Map of the real Boston subway system and surrounding areas. Image removed due to copyright restrictions. Simplified map of the subway system used at MBTA.com. Physical situation Model 39

41 Model building - review Mike Ashby (Cambridge University): The map misrepresents distances and directions, but it elegantly displays the connectivity. The quality or usefulness in a model is measured by its ability to capture the governing physical features of the problem. All successful models unashamedly distort the inessentials in order to capture the features that really matter. At worst, a model is a concise description of a body of data. At best, it captures the essential physics of the problem, it illuminates the principles that underline the key observations, and it predicts behavior under conditions which have not yet been studied. 40

42 1 φ = k 2 A simple atomistic model: geometry 2 0 ( r r0 ) elasticity (store energy) stable configuration of pair potential 2D triangular (hexagonal) lattice Y V X I x a r 3 1/2 r o Weak fracture layer I y bond can break Pair potential to describe atomic interactions Confine crack to a 1D path (weak fracture layer): Define a pair potential whose bonds never break (bulk) and a potential whose bonds break φ = k k 0 0 ( r r ) ( r break 0 2 r ) 0 r 2 < r r break 41 r break

43 Harmonic and harmonic bond snapping potential 1 φ = k ( ) r r 0 Image removed due to copyright restrictions. Please see: Fig. 4.6 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, k 2 (r r ) r < r 0 0 break 2 φ = 1 k ( ) 2 2 r 0 break r 0 r rbreak 42

44 3.1 Physical properties of atomic lattices How to calculate elastic properties and fracture surface energy parameters to link with continuum theory of fracture 43

45 Energy approach to elasticity 1 st law of thermodynamics (TD) 2 nd law of TD Change in entropy is always greater or equal than the entropy supplied in form of heat; difference is due to internal dissipation Dissipation rate Dissipation rate after considering 1 st law of TD: or 44

46 Energy approach to elasticity Elastic deformation: Assume only internal energy change Expand equation du/dt = du/dx dx/dt Therefore: With strain energy density: 45

47 Cauchy-Born rule Concept: Express potential energy U for a atomistic representative volume element (RVE) as a function of applied strain 1 Φ( ε ij ) = ( ε ij ) dω Ω 0U 0 Ω U( ε ij ) : potential energy of RVE as function of εij Ω0 : volume of RVE Impose macroscopic deformation gradient on atomistic volume element, then calculate atomic stress this corresponds to the macroscopic stress Strictly valid only far away from defects in periodic lattice (homogeneous deformation, perfect lattice, amorphous solid-average) Allows direct link of potential to macroscopic continuum elasticity 46

48 1D example: Cauchy-Born rule Impose homogeneous strain field on 1D string of atoms σ = Then obtain from that ij cijklε kl Image removed due to copyright restrictions. Please see: Fig. 4.1 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, r = ( 1+ ε ) r0 1 1 Φ( ε ) = φ( r) = φ((1 + ε ) r0 ) r D r D 0 0 Strain energy density function interatomic potential out-of-plane area r D 0 atomic volume 47

49 2D hexagonal lattice [010] [100] r 2 r 2 r 2 r 2 r 1 r 1 r 1 r 1 y r 2 r 2 r 2 r 2 x Figure by MIT OpenCourseWare. Φ = 3 8 ( ε + 2ε ε + 3ε + ( ε ε ) ) '' φ xx xx yy yy yx + xy 1 Φ( ε ij ) = ( ε ij ) dω Ω 0U 0 Ω 48

50 2D triangular lattice, LJ potential Strain in [ 100] (x) -direction S train in [ 010] (y) -direction σ 5 4 σ 5 4 Stress Stress [010] Strain ε xx Strain ε yy [100] r 1 r 2 r 2 r 1 r 1 r 2 r 2 r 1 LJ Solid Harmonic Solid Poisson ratio LJ solid y x r 2 r 2 r 2 r 2 80 Tangent modulus E xx 80 Tangent modulus E yy Figure by MIT OpenCourseWare ε = 1 σ = Strain ε xx Strain ε yy 12:6 LJ potential Figure by MIT OpenCourseWare. 49

51 2D triangular lattice, harmonic potential σ xx, σ yy, and ν x, ν y Stress vs. Strain and Poisson Ratio Tangent Modulus 50 Stress σ xx Poisson ratio ν x Stress σ 45 yy Poisson ratio ν y E x, E y E x E y Strain Strain Elastic properties of the triangular lattice with harmonic interactions, stress versus strain (left) and tangent moduli E x and E y (right). The stress state is uniaxial tension, that is the stress in the direction orthogonal to the loading is relaxed and zero. Figure by MIT OpenCourseWare.

52 Elastic properties Enables to calculate wave speeds: 51

53 Surface energy calculation y fracture plane x fracture plane r o 3 1/2 r o Harmonic potential with bond snapping distance r break Figure by MIT OpenCourseWare. Note: out-of-plane unity thickness 52

54 3.2 Atomistic modeling of supersonic/subsonic dynamic fracture Focus: effects of material nonlinearities (reflected in choice of model) Please see: Buehler, Markus J., Farid F. Abraham, and Huajian Gao. "Hyperelasticity Governs Dynamic Fracture at a Critical Length Scale." Nature 426 (November 13, 2003): , and 53 Buehler, Markus J., and Huajian Gao. "Dynamical Fracture Instabilities Due To Hyperelasticity at Crack Tips." Nature 439 (January 19, 2006):

55 Coordinate system and atomistic model Y V X I x a r 3 1/2 r o Weak fracture layer I y Figure by MIT OpenCourseWare. Pair potential to describe atomic interactions Confine crack to a 1D path (weak fracture layer) 54

56 MD model development: biharmonic potential Polymers, rubber Metals Stiffness change under deformation, with different strength Atomic bonds break at critical atomic separation Want: simple set of parameters that control these properties (as few as possible, to gain generic insight) 1

57 Biharmonic potential control parameters Images removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

58 Biharmonic potential definition 57

59 Energy filtering: visualization approach Only plot atoms associated with higher energy Enables determination of crack tip (atoms at surface have higher energy, as they have fewer neighbors than atoms in the bulk) Energy of atom i N U i = φ( r ij ) j= 1 Image removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

60 MD simulation results: confirms linear continuum theory Relative crack extension a a (t) Tip speed V t Nondimensional time t crack speed =time derivative of a v(t)=da/dt Figure by MIT OpenCourseWare. 59

61 Virial stress field around a crack Images removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

62 Virial stress field around a crack 0.6 v/c r = v/c r = v/c r = σ xx 0.2 σ yy 0.2 σ XY θ θ θ Continuum theory MD modeling v/c r = v/c r = Hoop or opening stress σ θ 0.2 σ θ Maximum principal stress Figures by MIT OpenCourseWare θ 61

63 Biharmonic potential bilinear elasticity Images removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

64 Subsonic fracture Softening material behavior leads to subsonic fracture, that is, the crack can never attain its theoretical limiting speed Materials: metals, ceramics < 3.4 = c R 63

65 Supersonic fracture Stiffening material behavior leads to subsonic fracture, that is, the crack can exceed its theoretical limiting speed Materials: polymers > 3.4 = c R 64

66 Harmonic system 65

67 Physical basis for subsonic/supersonic fracture Changes in energy flow at the crack tip due to changes in local wave speed (energy flux higher in materials with higher wave speed) Controlled by a characteristic length scale χ Image removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

68 Reduced crack 1.7 speed c/c R Different ratios of spring constants Stiffening Intersonic fracture k2/k1=1/4 k2/k1=1/2 k2/k1=2 k2/k1=3 k2/k1=4 1.1 Shear wave speed 0.9 Sub-Rayleigh fracture Softening r on Stiffening and softening effect: Increase or reduction of crack speed 67

69 Supersonic fracture: mode II (shear) 68

70 Hyperelasticity and dynamic fracture Concept of speed of sound in dynamic fracture must be carefully defined Sound speed under zero load only relevant under certain conditions Sound speed under large load (due to hyperelastic effects, that is, a change in elastic stiffness) may become relevant: Governing effect of local properties at crack tip, leads to subsonic/supersonic fracture Buehler, M. J., F. F. Abraham, H. Gao. "Hyperelasticity Governs Dynamic Fracture at a Critical Length Scale." Nature 426 (2003):

71 3.3 Reactive potential/hybrid models (introduction) Focus: model particular fracture properties of silicon (chemically complex material) 70

72 Concept: concurrent multi-paradigm simulations Crack tips, defects (dislocations) FE (continuum) Organic phase nonreactive atomistic ReaxFF Multi-paradigm approach: combine different computational methods (different resolution, accuracy..) in a single computational domain Decomposition of domain based on suitability of different approaches Inorganic phase Interfaces (oxidation, grain boundaries,..) nonreactive atomistic Example: concurrent FEatomistic-ReaxFF scheme in a crack problem (crack tip treated by ReaxFF) and an interface problem (interface treated by ReaxFF). 71

73 Concurrent multi-paradigm simulations: link nanoscale to macroscale 72

74 Simulation Geometry: Cracking in Silicon Image removed due to copyright restrictions. Please see: Fig. 2 in Buehler, Markus J, et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field." Physical Review Letters 96 (2006): or Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Consider a crack in a silicon crystal under mode I loading. Periodic boundary conditions in the z-direction (corresponding to a plane strain case). 73

75 Fracture of silicon single crystals Use multi-paradigm scheme that combines the Tersoff potential and ReaxFF ReaxFF region is moving with the crack tip (110) crack surface, 10 % strain Reactive region is moving with crack tip Image removed due to copyright restrictions. Please see: Fig. 4a in Buehler, Markus J, et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field." Physical Review Letters 96 (2006): or Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, ReaxFF Tersoff 74

76 Cracking in Silicon: Hybrid model versus Tersoff based model Image removed due to copyright restrictions. Please see: Fig. 3a,b in Buehler, Markus J, et al. "Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field." Physical Review Letters 96 (2006): or Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Conclusion: Pure Tersoff can not describe correct crack dynamics 75

77 Quantitative comparison w/ experiment Image removed due to copyright restrictions. Please see: Fig. 1c in Buehler, M., et al. "Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals." Physical Review Letters 99 (2007): or Fig. 6.99b in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

78 Crack dynamics Image removed due to copyright restrictions. Please see: Fig. 2 in Buehler, M., et al. "Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals." Physical Review Letters 99 (2007): or Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

79 Atomistic fracture mechanism Image removed due to copyright restrictions. Please see: Fig. 3 in Buehler, M., et al. "Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals." Physical Review Letters 99 (2007): or Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

80 Fracture initiation and instabilities 79

81 Fracture mechanism: tensile vs. shear loading Image removed due to copyright restrictions. Please see: Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, Shear (mode II) loading: Crack branching Tensile (mode I) loading: Straight cracking Mode I Mode II Images removed due to copyright restrictions. Please see figures in Buehler, M.J., A. Cohen, and D. Sen. Multi-paradigm modeling of fracture of a silicon single crystal under mode II shear loading. Journal of Algorithms and Computational Technology 2 (2008):