3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to …dspace.mit.edu/.../contents/lecture-notes/p2_l6.pdf · 2019. 9. 12. · 2. Basic deformation mechanism in brittle materials:

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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008

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1

1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation

Part II - lecture 6

Atomistic and molecular methods

2

Content overviewLectures 2-10February/March

Lectures 11-19March/April

Lectures 20-27April/May

I. Continuum methods1. Discrete modeling of simple physical systems:

Equilibrium, Dynamic, Eigenvalue problems2. Continuum modeling approaches, Weighted residual (Galerkin) methods,

Variational formulations3. Linear elasticity: Review of basic equations,

Weak formulation: the principle of virtual work, Numerical discretization: the finite element method

II. Atomistic and molecular methods1. Introduction to molecular dynamics2. Basic statistical mechanics, molecular dynamics, Monte Carlo3. Interatomic potentials4. Visualization, examples5. Thermodynamics as bridge between the scales6. Mechanical properties – how things fail7. Multi-scale modeling8. Biological systems (simulation in biophysics) – how proteins work and

how to model them

III. Quantum mechanical methods1. It’s A Quantum World: The Theory of Quantum Mechanics2. Quantum Mechanics: Practice Makes Perfect3. The Many-Body Problem: From Many-Body to Single-Particle4. Quantum modeling of materials5. From Atoms to Solids6. Basic properties of materials7. Advanced properties of materials8. What else can we do?

3

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

4

II. Atomistic and molecular methods

Lecture 6: Application to mechanics of materials-brittle materials

Outline:1. Brittle versus ductile materials2. Basic deformation mechanism in brittle materials: crack extension3. Atomistic modeling studies of fracture

3.1 Physical properties of atomic lattices 3.2 Supersonic/subsonic dynamic fracture3.3 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 materialsLearn atomistic modeling approaches, include concept of “model materials” and reactive force field, multi-paradigm modeling approachToday’s lecture prepares you to carry out MD simulations of fracture using two modeling approaches (problem set 2)

5

1. Brittle versus ductile materials

6

Tensile test of a wire

Brittle Ductile

Strain

Stre

ss

BrittleDuctile

Necking

Figure by MIT OpenCourseWare. Figure by MIT OpenCourseWare.

7

Ductile versus brittle materials

Difficultto deform,breaks easily

Easy to deformhard to break

BRITTLE DUCTILE

Glass Polymers Ice...

Shear load

Copper, Gold

Figure by MIT OpenCourseWare.

8Griffith, Irwine and others: Failure initiates at defects, such as cracks, or grain boundaries with reduced traction, nano-voids, other imperfections

Deformation of materials:Nothing is perfect, and flaws or cracks matter

Failure of materials initiates at cracks

9

SEM picture of material: nothing is perfect

10

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, 2008.

Stress concentrators: local stress >> global stress

11

“Macro, global”

“Micro (nano), local”

)(rσ r

Deformation of materials:Nothing is perfect, and flaws or cracks matter

Griffith, Irwine and others: Failure initiates at defects, such as cracks, or grain boundaries with reduced traction, nano-voids, other imperfections

Failure of materials initiates at cracks

12

Cracks feature a singular stress field, with singularity at the tip of the crack

rr 1~)(σ

:IK

stress tensor

Stress intensity factor (function of geometry)

σxy

σrθσrr

σθθ

σxx

σyy

r

x

y

θ


13

Crack extension: brittle response

Large stresses lead to rupture of chemicalbonds between atoms

Thus, crack extends



14

Lattice shearing: ductile response

Instead of crack extension, induce shearing of atomic latticeDue to large shear stresses at crack tipLecture 7

Figures by MIT OpenCourseWare.

τ

τ

τ

τ

15

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 orductile

16

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 1945Early 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.

17

Liberty ships: brittle failure

Courtesy of the U.S. Navy.

18

2. Basic deformation mechanism in brittle materials: crack extension

19

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.

20

Science of fracture: model geometry

Typically consider a single crack in a crystalRemotely applied mechanical loadFollowing discussion focused on single cracks and their behavior

remote load

remote load

a


21

Brittle fracture loading conditionsCommonly consider a single crack in a material geometry, under three types of loading: mode I, mode II and mode III

Tensile load, focus of this lecture

Mode I Mode II Mode III

Tensile load, focus of this lecture


22

Brittle fracture mechanisms: fracture is a multi-scale phenomenon, from nano to macro


23

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

Basic fracture process: dissipation of elastic energy


Undeformed Stretching=store elastic energy Release elastic energydissipated into breaking chemical bonds 24

25

Elasticity = reversible deformation

Please also see: Fig. 3.1 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

26

Continuum description of fractureFracture 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 γs (energy necessary to create new surface, dimensions: energy/area, Nm/m2)

aBs~2γ=

!Ba

E~

21 2

ξσ=

change of elastic energy energy to create surfaces

sEG γξσ 2

2

2

==

ξγσ Es4

=

E

2

21 σ

σ

ε


a~

(1)(2)

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ (1)(2)

27

Griffith condition for fracture initiationEnergy 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)

28

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 cR (speed of surface waves)shear wave speed cs (speed of shear waves)longitudinal wave speed cl (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

29

Sound speeds in materials: overview

Wave speeds are calculated based on elastic properties of material

μ = shear modulus μ3/8=E E8/3=μ

30

Limiting speeds of cracks: linear elastic continuum theory

• 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)

Images removed due to copyright restrictions.Please see: Fig. 6.56 and 6.90 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

31

Physical reason for crack limiting speedPhysical (mathematical) reason for the limiting speed is that it becomes increasingly difficult to increase the speed of the crack by adding a larger loadWhen the crack approaches the limiting speed, the resistance to fracture diverges to infinity


32

Linear versus nonlinear elasticity=hyperelasticity

Linear elasticity: Young’s modulus (stiffness) does not changewith deformationNonlinear elasticity = hyperelasticity: Young’s modulus (stiffness) changes with deformation

Images removed due to copyright restrictions.Please see: Fig. 6.35b in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

33

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

Deformation field near a crack

rr 1~)(σ

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, 2008.

34

Limiting speeds of cracks

• 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• Subsonic fracture due to local softening, that is, reduction of energy flux

Images removed due to copyright restrictions.Please see: Fig. 6.56 and 6.90 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

35

Stiffening vs. softening behavior

Increased/decreased wave speed

Images removed due to copyright restrictions.Please see: Fig. 6.35b in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

36

Energy flux reduction/enhancement

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, 2003

Images removed due to copyright restrictions.Please see: Fig. 6.109 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

37

Summary: atomistic mechanisms of brittle fracture

Brittle fracture – rapid spreading of a small initial crack

Cracks initiate based on Griffith condition G = 2γs

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

38

3. Atomistic modeling studies of fracture

39

Model building - reviewMike Ashby (Cambridge University):

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.

“Physical situation” “Model”

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.

40

Model building - review

Mike Ashby (Cambridge University):

The map misrepresents distances and directions, but it elegantlydisplays 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.

41

A “simple” atomistic model: geometry

Pair potential to describe atomic interactionsConfine crack to a 1D path (weak fracture layer): Define a pair potentialwhose bonds never break (bulk) and a potential whose bonds break

200 )(

21 rrk −=φ

⎪⎪⎩

⎪⎪⎨

⎧

≥−

<−=

break2

0break0

break2

00

)(21

)(21

rrrrk

rrrrkφ

X

YV

31/2rora

Iy

Ix

Weak fracture layer

bond can

y (store energy) 2D triangular (hexagonal) lattice

break

elasticit

stable configuration of pair potential

42

Harmonic and harmonic bond snapping potential


1φ = k ( )2

2 0 r − r0

⎧1 k 2

⎪⎪ 0(r − r0 ) r < r2 break

φ = ⎨⎪1 k ( )2

⎩2 0 rbreak − r0 r ≥ r⎪ break

43

3.1 Physical properties of atomic lattices

How to calculate elastic properties and fracture surface energy – parameters to link with continuum

theory of fracture

44

Energy approach to elasticity

1st law of thermodynamics (TD)

2nd law of TDChange 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 1st law of TD:

or

45

Energy approach to elasticity

Elastic deformation:

Assume only internal energy change

Expand equation dU/dt = dU/dx dx/dt

Therefore:

With strain energy density:

46

Cauchy-Born rule

Concept: Express potential energy U for a atomistic representative volume element (RVE) as a function of applied strain

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

ΩΩ

=Φ ∫Ω

dU ijij

0

)(1)(0

εε

:)( ijU ε potential energy of RVE as function of ijε:0Ω volume of RVE

47

1D example: Cauchy-Born rule

Impose homogeneous strain field on 1D string of atomsThen obtain from thatklijklij c εσ =

))1((1)(1)( 000

rDr

rDr

⋅+⋅

=⋅

=Φ εφφε

Dr ⋅0 atomic volume

Strain energy densityfunction

interatomicpotential

out-of-plane area

0)1( rr ⋅+= ε


48

2D hexagonal lattice

( )222'' )(32383

xyyxyyyyxxxx εεεεεεφ ++++=Φ ΩΩ

=Φ ∫Ω

dU ijij

0

)(1)(0

εε

r2

[010]

[100]

y

x

r2 r2

r2

r1r1

r2

r2 r2

r2

r1r1


49

2D triangular lattice, LJ potential

ε = 1σ = 1

12:6 LJ potential

5

4

3

2

1

0 0

00

5

4

3

2

1

0 0

�

� �

�

00

�

�

[ ] (x) S [ ] (y)

0.1 0.2

0.1 0.2

0.1 0.2

20

40

60

80

Strain xx

Stre

ss

Stre

ss

Strain xx

0.1 0.2

20

40

60

80

Strain yy

Strain yy

Tangent modulus Exx

Strain in 100 -direction train in 010 -direction

Tangent modulus Eyy

LJ Solid Harmonic Solid Poisson ratio LJ solid


r2

[010]

[100]

y

x

r2 r2

r2

r1r1

r2

r2 r2

r2

r1r1


2D triangular lattice, harmonic potential


00

σ xx,

σyy

, and

νx,

νy

1

2

3

4

0.02 0.04Strain

Stress vs. Strain and Poisson Ratio

0.06 0.08

Stress σxxPoisson ratio νx

Poisson ratio νy

Stress σyy

025

E x, E

y

30

35

40

45

50

0.02 0.04Strain

Tangent Modulus

0.06 0.08

ExEy

Elastic properties of the triangular lattice with harmonic interactions, stress versus strain (left) and tangent moduli Ex and Ey (right). The stress state is uniaxial tension, that is the stress in the direction orthogonal to the loading is relaxed and zero.

51

Elastic properties

Enables to calculate wave speeds:

52

Surface energy calculation

Note: out-of-plane unity thickness

Harmonic potential with bond snapping distance rbreak

31/2ro

ro

fracture plane

fracture plane

x

y


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): 141-146, and 53Buehler, Markus J., and Huajian Gao. "Dynamical Fracture Instabilities Due To Hyperelasticity at Crack Tips." Nature 439 (January 19, 2006): 307-310.

54

Coordinate system and atomistic model

Pair potential to describe atomic interactionsConfine crack to a 1D path (weak fracture layer)

X

YV

31/2rora

Iy

Ix

Weak fracture layer


1

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)

56

Biharmonic potential – control parameters


57

Biharmonic potential definition

58

Energy filtering: visualization approach

Only plot atoms associated with higher energyEnables determination of crack tip (atoms at surface have higherenergy, as they have fewer neighbors than atoms in the bulk)

∑=

=N

jiji rU

1

)( φEnergy of atom i


59

a (t)

v(t)=da/dt

crack speed=time derivative of a

MD simulation results: confirms linear continuum theory

0 100 200 300 400 500 600

500

1000

1500

2000

2500

0Rel

ativ

e cr

ack

exte

nsio

n ∆

a

0 100 200 300 400 500 600

1

2

3

4

5

6

0

Tip

spee

d V

t

Nondimensional time t


60

Virial stress field around a crack


61

Virial stress field around a crack

Hoop or opening stress

Maximum principal stress

σ 1

0.5

1

1.5

00 50

θ100 150

v/cr = 0

0.2σ xx

0.4

0.6v/cr = 0

0

0 50θ

100 150

Continuum theoryMD modeling

σ XY

v/cr = 0

0-0.2

-0.1

0.1

0.2

0

50θ

100 150

0.2σ yy

0.4

0.6

v/cr = 0

0

0 50θ

100 150

σ θ

0.2

0.4

0.6

v/cr = 0

0

0 50 100 150θ

Figures by MIT OpenCourseWare.

62

Biharmonic potential – bilinear elasticity


63

Subsonic fractureSoftening material behavior leads to subsonic fracture, that is, the crack can never attain its theoretical limiting speedMaterials: metals, ceramics

< 3.4 = cR

64

Supersonic fractureStiffening material behavior leads to subsonic fracture, that is, the crack can exceed its theoretical limiting speedMaterials: polymers

> 3.4 = cR

65

Harmonic system

66

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


67

0.7

0.9

1.1

1.3

1.5

1.7

1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

k2/k1=1/4k2/k1=1/2k2/k1=2k2/k1=3k2/k1=4

Different ratios of spring constants

Stiffening and softening effect: Increase or reduction of crack speed

ron

Shear wave speed

Intersonic fracture

Sub-Rayleigh fracture

Reduced crack speed c/cR

Softening

Stiffening

68

Supersonic fracture: mode II (shear)

69

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): 141-146.

70

3.3 Reactive potential/hybrid models (introduction)

Focus: model particular fracture properties of silicon (chemically complex material)

71

Concept: concurrent multi-paradigm simulations

ReaxFF

FE (c

ontin

uum

)

Organic phase

Inorganic phase

nonreactiveatomistic

nonreactiveatomistic

• Multi-paradigm approach: combine different computational methods (different resolution, accuracy..) in a single computational domain

• Decomposition of domainbased on suitability of different approaches

• Example: concurrent FE-atomistic-ReaxFF scheme in a crack problem (crack tip treated by ReaxFF) and an interface problem (interface treated by ReaxFF).

Interfaces (oxidation, grain boundaries,..)

Crack tips, defects (dislocations)

72

Concurrent multi-paradigm simulations:link nanoscale to macroscale

73

Simulation Geometry: Cracking in Silicon

• Consider a crack in a silicon crystal under mode I loading.

• Periodic boundary conditions in the z-direction (corresponding to a plane strain case).

Image removed due to copyright restrictions.Please see: Fig. 2 in Buehler, Markus J, et al. "Multiparadigm Modeling of Dynamical Crack Propagationin Silicon Using a Reactive Force Field." Physical Review Letters 96 (2006): 095505.orFig. 6.95 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

74TersoffReaxFF

• Use multi-paradigm scheme that combines the Tersoff potential and ReaxFF

• ReaxFF region is moving with the crack tip

Fracture of silicon single crystals

Reactive region is moving with crack tip(110) crack surface, 10 % strain

Image removed due to copyright restrictions.Please see: Fig. 4a in Buehler, Markus J, et al. "Multiparadigm Modeling of Dynamical Crack Propagationin Silicon Using a Reactive Force Field." Physical Review Letters 96 (2006): 095505.orFig. 6.97 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

75

Cracking in Silicon: Hybrid model versusTersoff based model

Conclusion: Pure Tersoff can not describe correct crack dynamics

Image removed due to copyright restrictions.Please see: Fig. 3a,b in Buehler, Markus J, et al. "Multiparadigm Modeling of Dynamical Crack Propagationin Silicon Using a Reactive Force Field." Physical Review Letters 96 (2006): 095505.orFig. 6.96 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

76

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): 165502.orFig. 6.99b in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

77

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): 165502.orFig. 6.102 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

78

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): 165502.orFig. 6.104 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

79

Fracture initiation and instabilities

80

Shear (mode II) loading: Crack branchingTensile (mode I) loading: Straight cracking

Fracture mechanism: tensile vs. shear loading

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): 203-221.

Image removed due to copyright restrictions. Please see: Fig. 6.106 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to …dspace.mit.edu/.../contents/lecture-notes/p2_l6.pdf · 2019. 9. 12. · 2. Basic deformation mechanism in brittle materials: