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1.021, 3.021, 10.333, 22.00 Introduction to Modeling and SimulationSpring 2011

Part I – Continuum and particle methods

Markus J. BuehlerLaboratory for Atomistic and Molecular MechanicsDepartment of Civil and Environmental EngineeringMassachusetts Institute of Technology

Application to modeling brittle materialsLecture 7

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Content overview

I. Particle and continuum methods1. Atoms, molecules, chemistry2. Continuum modeling approaches and solution approaches 3. Statistical mechanics4. Molecular dynamics, Monte Carlo5. Visualization and data analysis 6. Mechanical properties – application: how things fail (and

how to prevent it)7. Multi-scale modeling paradigm8. Biological systems (simulation in biophysics) – how

proteins work and how to model them

II. 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?

Lectures 2-13

Lectures 14-26

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Overview: Material covered so far…Lecture 1: Broad introduction to IM/S

Lecture 2: Introduction to atomistic and continuum modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)

Lecture 3: Basic statistical mechanics – property calculation I(property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function)

Lecture 4: Property calculation II (Monte Carlo, advanced property calculation, introduction to chemical interactions)

Lecture 5: How to model chemical interactions I (example: movie of copper deformation/dislocations, etc.)

Lecture 6: How to model chemical interactions II (pair potentials, fracture – introduction)

Lecture 7: Application – MD simulation of materials failure

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Lecture 7: Application to modeling brittle materials

Outline:1. Basic deformation mechanism in brittle materials - crack extension2. Atomistic modeling of fracture

2.1 Physical properties of atomic lattices 2.2 Application

3. Bond order force fields - how to model chemical reactions

Goal of today’s lecture: Apply our tools to model a particular material phenomena – brittle fracture (useful for pset #2)Learn basics in fracture of brittle materialsLearn how to build a model to describe brittle fracture (from scratch)

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1. Basic deformation mechanism in brittle materials - crack extension

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Brittle fracture

Materials: glass, silicon, many ceramics, rocks

At large loads, rather than accommodating a shape change, materials break

Image courtesy of quinn.anya. License: CC-BY.

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Basic fracture process: dissipation of elastic energy

δa

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

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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=

energy to create surfaces

ξγσ Es4

=

E

2

21 σ

σ

ε

change of elastic (potential) energy = G

(1)(2)

a

δa

~

a~

"Relaxed"element

"Strained"element

ξ

a

σ

σ

Image by MIT OpenCourseWare.

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Griffith condition for fracture initiationEnergy release rate G, 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)

syE

G 221:

2

== ξσ

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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

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

Image by MIT OpenCourseWare.

Mode I Mode II Mode III

Tensile load, focus of this lecture

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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)

Image by MIT OpenCourseWare.

Linear Nonlinear

Mode I

Mode II

Mode III

Limiting speed v

Limiting speed v

Cr Cs

Cs

Cl

Cl

Subsonic Supersonic

SupersonicIntersonicSub-RayleighSuper-

Rayleigh

Mother-daughter mechanism

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Sound speeds in materials: overview

Wave speeds are calculated based on elastic properties of material

μ = shear modulus μ3/8=E E8/3=μ

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

Image removed due to copyright restrictions. Fig. 6.2 in Buehler, Markus J. Atomistic Modeling of Materials Failure. Springer, 2008.

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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 (=dynamic fracture toughness)

divergence

Driving force (applied load)

EG

vyfE

G s

2

2

~

),(21

σ

ξσ==

Image removed due to copyright restrictions.Please see: Fig. 6.15 in Buehler, Markus J. Atomistic Modeling of Materials Failure. Springer, 2008.

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2. Atomistic modeling of fracture

What is a model?Mike 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”© Massachusetts Bay Transportation Authority. All rights

© Google, Inc. All rights reserved. This content is reserved. This content is excluded from our Creative Commons excluded from our Creative Commons license. For more license. For more information, see http://ocw.mit.edu/fairuse.information, see http://ocw.mit.edu/fairuse. 17

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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

stable configuration of pair potential

A “simple” atomistic model: geometry2

00 )(21 rrk −=φ

⎪⎪⎩

⎪⎪⎨

⎧

≥−

<−=

break2

0break0

break2

00

)(21

)(21

rrrrk

rrrrkφ

bond can break

2D triangular (hexagonal) latticeelasticity (store energy)

Image by MIT OpenCourseWare.

X

YV

31/2rora

Iy

Ix

Weak fracture layer

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Harmonic and harmonic bond snapping potential

200 )(

21 rrk −=φ

φ

φ

r

r

breakr

⎪⎪⎩

⎪⎪⎨

⎧

≥−

<−=

break2

0break0

break2

00

)(21

)(21

rrrrk

rrrrkφ

'φ

0r

0r

~ force

~ force

'φ

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2.1 Physical properties of atomic lattices

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

theory of fracture

εεψσ

∂∂

=)(

2

2 )(ε

εψ∂

∂=EStress Young’s modulus

free energy density (energy per unit volume)

εσ E=

21

1D example – “Cauchy-Born rule”

Impose homogeneous strain field on 1D string of atomsThen obtain from that

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

rDr

rDr

⋅+⋅

=⋅

= εφφεψ

(1+ε) r01 32

r0

Dr ⋅0 atomic volume

Strain energy densityfunction

interatomicpotential

out-of-plane area

0)1( rr ⋅+= ε

εεψσ

∂∂

=)(

2

2 )(ε

εψ∂

∂=E

εσ E=

22

2D hexagonal lattice

( )21221

2222211

211

'' )(32383)( εεεεεεφεψ ++++=ij

ij

ijij ε

εψσ

∂∂

=)(

klij

ijijklc

εεεψ

∂∂∂

=)(2

k=''φ Image by MIT OpenCourseWare.

r2

[010]

[100]

y

x

r2 r2

r2

r1r1

r2

r2 r2

r2

r1r1

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2D triangular lattice, LJ potential

ε = 1σ = 1

12:6 LJ potential

Image by MIT OpenCourseWare.

Image by MIT OpenCourseWare.

r2

[010]

[100]

y

x

r2 r2

r2

r1r1

r2

r2 r2

r2

r1r1

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

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2D triangular lattice, harmonic potential

Image by MIT OpenCourseWare.

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.

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Elastic properties

Enables to calculate wave speeds:

k=''φ

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Surface energy calculation

Note: out-of-plane unity thickness

Harmonic potential with bond snapping distance rbreak

Image by MIT OpenCourseWare.

31/2ro

ro

fracture plane

fracture plane

x

y

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2.2 Application

Focus: effects of material nonlinearities (reflected in choice of model)

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Coordinate system and atomistic model

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

Image by MIT OpenCourseWare.

Y

X

V

31/2ro

Ixa r

Weak fracture layer

Iy

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Linear versus nonlinear elasticity=hyperelasticity

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

Image by MIT OpenCourseWare.

Str

ess

Strain

Stiffen

ing

Softening

Hyperelasticity

Line

ar the

ory

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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

large deformationnonlinear zone

“singularity”

Deformation field near a crack

small deformation

rr 1~)(σ

Image by MIT OpenCourseWare.

Strain

Str

ess

Line

ar e

last

ic

clas

sica

l the

ories

Nonlinear real

materials

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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

Image by MIT OpenCourseWare.

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Stiffening vs. softening behavior

Increased/decreased wave speed

E

2

21 σ

σ

ε

“linear elasticity”

real materials

Image by MIT OpenCourseWare.

Str

ess

Strain

Stiffen

ing

Softening

HyperelasticityLine

ar the

ory

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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)

34

Biharmonic potential – control parameters

breakε

0k

1k

01ratio / kkk =

breakronr r 0

0

rrr −

=ε

'φ

35

Biharmonic potential definition

The biharmonic potential is defined as:

where is the critical atomic separation for the onset of the hyperelastic effect, and

and

are found by continuity conditions of the potential at . The values and refer to the small- and large-strain spring constants.

36

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

Image by MIT OpenCourseWare.

a

x-coordinate

z-co

ord

inat

e

200

180

160

140

120

100

80

60

40

20

050 100 150 200 250 300 350 4000

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a (t)

v(t)=da/dt

crack speed=time derivative of a

MD simulation results: confirms linear continuum theory

Image by MIT OpenCourseWare.

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

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Virial stress field around a crack

large stressesat crack tip - induce bond failure

Images removed due to copyright restrictions.Please see: Fig. 6.30 in Buehler, Markus J. Atomistic Modeling of Materials Failure. Springer, 2008.

39

Biharmonic potential – bilinear elasticity

Image by MIT OpenCourseWare.

StiffeningSoftening

2

1.5

1

0.5

00 0.01 0.02 0.03

Str

ess

(σ)

Strain (ε)

Et=66

Et=33

Harm

onic

solid

Hypere

lastic

solid

2

1.5

1

0.5

00 0.01 0.02 0.03

Str

ess

(σ)

Strain (ε)

Et=66

Et=33

Harmon

ic soli

dHype

relastic solid

40

Harmonic system

41

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

< 3.4 = cR

42

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

> 3.4 = cR

43

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

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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

Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Buehler, M., F. Abraham, and H. Gao. "Hyperelasticity Governs Dynamic Fracture at a Critical Length Scale." Nature 426 (2003): 141-6. © 2003.

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Supersonic fracture: mode II (shear)

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): 141-6.

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Theoretical concept: 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)

Image by MIT OpenCourseWare.

Classical

New

Fracture process zoneHyperelastic zone

K dominance zone Characteristic energy length

Energy Flux Related to Wave Speed: High local wave speed, high energy flux, crack can move faster (and reversefor low local wave speed).

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3. Bond order force fields - how to model chemical reactions

48

Challenge: chemical reactions

CHARMM-type potential can not describe chemical reactions

Energy

C-C distance

Transition point ???

sp3sp2 r

stretchφsp3

sp2

49

Why can not model chemical reactions with |spring-like potentials?

20stretchstretch )(

21 rrk −=φ

20bendbend )(

21 θθφ −= k

Set of parameters only valid for particularmolecule type / type of chemical bond

Reactive potentials or reactive force fields overcome these limitations

32 stretch,stretch, spspkk ≠

50

How can one accurately describe the transition energies during chemical reactions?

Use computationally more efficient descriptions than relying on purely quantum mechanical (QM) methods (see part II, methods limited to 100 atoms)

22

H

HCCHHCCH2

−=−→−≡−+

involves processeswith electrons

Key features of reactive potentials

q q

q q

q

q

qq q

A

B

A--B

A

B

A--B

??

51

Molecular model that is capable of describing chemical reactions

Continuous energy landscape during reactions (key to enable integration of equations)

No typing necessary, that is, atoms can be sp, sp2, sp3… w/o further “tags” – only element types

Computationally efficient (that is, should involve finite range interactions), so that large systems can be treated (> 10,000 atoms)

Parameters with physical meaning (such as for the LJ potential)

Key features of reactive potentials

52

Theoretical basis: bond order potential

Modulate strength of attractive part(e.g. by coordination, or “bond order”)

Abell, Tersoff

Changes in spring constant as function of bond orderContinuous change possible= continuous energy landscape during chemical reactions

Concept: Use pair potential that depends on atomic environment(similar to EAM, here applied to covalent bonds)

Image by MIT OpenCourseWare.

5

0

-5

-100.5 1 1.5 2 2.5 3 3.5

Triple

Double

Single

S.C.

F.C.C.

Pote

ntial

ener

gy

(eV)

Distance (A)o

Effective pair-interactions for various C-C (Carbon) bonds

53

Theoretical basis: bond order potential

Image by MIT OpenCourseWare.

5

0

-5

-100.5 1 1.5 2 2.5 3 3.5

Triple

Double

Single

S.C.

F.C.C.

Pote

ntial

ener

gy

(eV)

Distance (A)o

Effective pair-interactions for various C-C (Carbon) bonds

54

Concept of bond order (BO)

sp3

sp2

sp

r

BO

1

2

3

55

Bond order based energy landscape

Bond length

Bond order

Energy

Bond length

Energy

Bond order potentialAllows for a more general description of chemistryAll energy terms dependent on bond order

Conventional potential(e.g. LJ, Morse)

Pauling

56

Historical perspective of reactive bond order potentials

1985: Abell: General expression for binding energy as a sum of near nieghbor pair interactions moderated by local atomic environment

1990s: Tersoff, Brenner: Use Abell formalism applied to silicon (successful for various solid state structures)

2000: Stuart et al.: Reactive potential for hydrocarbons

2001: Duin, Godddard et al.: Reactive potential for hydrocarbons “ReaxFF”

2002: Brenner et al.: Second generation “REBO” potential for hydrocarbons

2003-2005: Extension of ReaxFF to various materials including metals, ceramics, silicon, polymers and more in Goddard‘s group

57

Example: ReaxFF reactive force field

William A. Goddard IIICalifornia Institute of Technology

Adri C.T. v. DuinCalifornia Institute of Technology

Courtesy of Bill Goddard. Used with permission.

58

ReaxFF: A reactive force field

underover

torsanglevalCoulombvdWaalsbondsystem

EE

EEEEEE

++

++++= ,

2-body

multi-body

3-body 4-body

Total energy is expressed as the sum of various terms describingindividual chemical bonds

All expressions in terms of bond order

All interactions calculated between ALL atoms in system…

No more atom typing: Atom type = chemical element

59

Example: Calculation of bond energy

Bond energy between atoms i and j does not depend on bond distance

Instead, it depends on bond order

underovertorsanglevalCoulombvdWaalsbondsystem EEEEEEEE ++++++= ,

( )be,1bond e be,1BO exp 1 BO p

ij ijE D p⎡ ⎤= − ⋅ ⋅ −⎣ ⎦

60

Bond order functions

(1)

All energy terms are expressed as a function of bond orders

(2) (3)

0 0 0

BO exp exp expij ij ijij

r r rr r r

β ββ π ππσ π ππ

σ π ππα α α⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎢ ⎥ ⎢ ⎥⎢ ⎥= ⋅ + ⋅ + ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦

BO goes smoothly from 3-2-1-0

Characteristic bond distance

Fig. 2.21c in Buehler, Markus J. Atomistic Modelingof Materials Failure. Springer, 2008. © Springer. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.

(2)(3)

(1)

61

Illustration: Bond energy

Image removed due to copyright restrictions.Please see slide 10 in van Duin, Adri. "Dishing Out the Dirt on ReaxFF."http://www.wag.caltech.edu/home/duin/FFgroup/Dirt.ppt.

62

underovertorsanglevalCoulombvdW aalsbondsystem EEEEEEEE ++++++= ,

vdW interactions

Accounts for short distance repulsion (Pauli principle orthogonalization) and attraction energies at large distances (dispersion)Included for all atoms with shielding at small distances

Image removed due to copyright restrictions.Please see slide 11 in van Duin, Adri. "Dishing Out the Dirt on ReaxFF."http://www.wag.caltech.edu/home/duin/FFgroup/Dirt.ppt.

63

Resulting energy landscape

Contribution of Ebond and vdW energy

Source: van Duin, C. T. Adri, et al. "ReaxFF: A Reactive ForceField for Hydrocarbons." Journal of Physical Chemistry A 105 (2001). © American Chemical Society. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.

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LaAc

: not currently described by ReaxFF

Current development status of ReaxFF

A

B

A--B

Allows to interface metals, ceramics with organic chemistry: Key for complex materials, specifically biological materials

Periodic table courtesy of Wikimedia Commons.

65

Mg-water interaction: How to make fire with water

http://video.google.com/videoplay?docid=4697996292949045921&q=magnesium+water&total=46&start=0&num=50&so=0&type=search&plindex=0

Mg

Video stills removed due to copyright restrictions; watch the video now: http://www.youtube.com/watch?v=QTKivMVUcqE.

66

Mg – water interaction – ReaxFF MD simulation

MIT OpenCourseWarehttp://ocw.mit.edu

3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and SimulationSpring 2012

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