Crac ack k Pr Propagati opagation on Model odel from om Cou - - PowerPoint PPT Presentation

Som

• mnath

nath Ghosh

• sh

Grad aduate uate St Stud udents: nts: Jia iaxi i Zhan ang g an and Su Subh bhen endu du Chak akrabo rabort rty Depar artm tments ents of Civ ivil il, Mechani anical cal an and Mat ateri rials als Sc Scie ience ce &Engine gineeri ering ng

Jo Johns ns Hopkins kins Un Univ iversity sity

Bal altimore imore, , USA SA

Crac ack k Pr Propagati

• pagation
• n Model
• del from
• m Cou
• upled

pled Atom

• mis

istic tic-Cont Continuu inuum m Simulations ulations

NIST ST Works rkshop hop on Atom tomis isti tic c Si Simul ulat ations ions for r Industri ustrial al Needs Gai aithersburg, ersburg, MD Aug ugust ust 2, 2, 20 2018 18 Ackno nowl wledge dgeme ments nts: : Nat atio iona nal l Sc Scie ience e Found undat ation ion

Mod

• delin

eling Fa Fati tigue gue and nd Fa Failu ilure e in n Crys ystalline talline Materials erials

Ti Ti-624 6242

Williams, Sinha, Mills, Bhattacharjee, (2006)

Al Al- 7075 7075-T65 T651

Hochhalter et al. (2010)

Short

• rt and

nd Lon

• ng

g Crack ack Pr Propagation

• pagation

in Po Polycrystalline ycrystalline Micros crostructure tructures

http://fcp.mechse.illinois.edu

• Plastic

tic zone

• ne at grain

in scale e

• Strong
• ng microstru

ructure e de depe pend ndenc ence, e, grain in boundary,

• undary, di

disloc

• cati

ation n struc ucture, ure, slip p , etc. c.

• Insen

ensiti sitive ve to

• micr

crost

• struc

ructure ture

• Paris’ Law

St Stag age I – Sh Short rt Crac ack k Growt

• wth

St Stag age II – Long ng Crac ack k Grow

• wth

Short

• rt Crack

ack Ev Evol

• luti

tion

• n

Coh

• hesiv

esive e zon

• ne

e mod

• dels

els

Separation between material surfaces resisted by cohesive tractions

• Parameters

ameters of cohesi esive ve potent ential al are e typ ypically y calibr brated ated by y ex exper erimen ments ts

• Inter

eracti action

• n bet

etween een crack ck growth wth and local al plast stici city y ev evolut lution

• n which

h affected ted by y the e local l mi microstruct

• structur

ure e is not include ded

Process Zone

Needleman (1990), Ortiz and Pandolfi (1999), Park Paulino, Roesler (2009) Spearot, McDowell (2004), Yamakov, Saether E, Glaessgen E (2008)

Ph Phase ase-Field Field Mo Model delin ing

Diffused crack s 𝑦 = 𝑓−|𝑦|/𝑚𝑑

𝑚𝑑 𝑚𝑑

(s)

Conti tinu nuous us Aux uxil iliary iary Fie ield ld to Approxim roximate ate Sh Shar arp p Crac ack k Di Discon conti tinu nuit ities. ies.

𝑡 is the phase field variable (order parameter):

𝒕 ∈ [𝟏, 𝟐]; ; 𝒕 = 𝟏 perfect ct soli lid; d; 𝒕 = 𝟐 fully cracked d

Helmhol lmholtz tz stor

• red energy

rgy and d crack k di dissipat pation are mod

• deled

eled wi with h ph phase e field: Stored free energy:

ሶ 𝐸 = 𝑋𝑓𝑦𝑢 − ሶ Ψe + ሶ Ψd + ሶ Ψf

Dissipation rate:

Ψ = Ψ𝑓(𝑭𝑓, 𝑡) + Ψ𝑒(𝜽, 𝑡) + Ψ𝑔(𝑡, 𝛼𝑡)

Energy gy fun unctional tionals s ar are typica ically lly not t roo

• oted

ed in in the at atomis misti tic sour urce e of the frac acture ture regi gion

• n

Clayton and Knap (2015), Miehe, Hofacker, and Welschinger (2010), Ambati, Gerasimov, and Lorenzis, (2015).

Objec jective tive of

• f th

this s Stu tudy dy

 Upscaling Upscaling of

• f varia

variabl bles fro from ato atomi mic-sc scale ale molecul molecular ar dyn ynam amics ics si simu mula lati tion

• ns in

in a se self lf-con consiste sistent nt mo model el.  De Develop velop phys ysics ics-base based, d, integrated integrated fra frame mewor work of

• f crack

ack evolut volution ion and nd de defor formation mation mode models ls for for cr cryst ystal alline line ma materi terial als that that can an be be used sed in in co conjunc njuncti tion wit ith cr crys ystal tal pla last stici icity ty fin init ite ele lemen ment mo models els.

Seq equence ence of

• f Ste

teps s in n Buil uildin ing th the e Mod

• del

el

• Char

arac acterizin erizing g me mechan anisms isms in at atomi mist stic ic si simu mulation ations

(Zhang, Ghosh JMPS, 2013)

• Self-con

consi sistent stent coupled pled at atomi mistic stic-con conti tinu nuum m mo model el (Zhang, Chakraborty, Ghosh IJMCE 2017, Ghosh Zhang IJF 2017)

• Crack propa

paga gatio tion n using coupled led model

• Hyperdy

rdynam namics ics for time me-sca scale le ac accelerati leration

• n

(Chakraborty, Zhang, Ghosh CMS 2016, Chakraborty, CMS Ghosh 2018)

• Extracting

ting crack growth th mo models els, , e.g. phas ase e fie ield ld ene nergies gies

(Ghosh Zhang IJF 2017, Chakraborty, CMS Ghosh 2018)

• I. Atomistic
• mistic Si

Simu mulations lations wi with h Me Mech chan anism ism Chara racteriz cterization ation and Quanti tificat ication ion

micro twinning dislocation structure evolution Crack propagation

MD D sim imul ulat ation ion prov

• vid

ides es tool

• ls

s to cap apture ure defor

• rmati

ation

• n mechan

anism isms s whic ich domin minate ate crac ack k tip ip pla lasti ticit ity an and af affect t crac ack k propag

• pagati

ation

• n proc
• cess

ess.

• J. Zhang

hang and d S. Gho hosh, , JMPS PS , Vol

• l. 61, 1670–1690,

690, 2013

I.

• I. Charact

aracterization erization and d Qu Quanti antification fication of

• f

Mec echan hanisms isms in Molecu

• lecular

lar Simul mulation ion

Dislocati

• cation
• n Extrac

racti tion

• n (DX

DXA) A)

Deformatio rmation n gradien ent for

• r twi

wins Crack ck sur urface ace

Di Dislo locat cation ion CNA, , DX DXA Di Dislo locat cation ion densit ity, , Bur urge gers rs vector tor Twin in De Defor

• rma

mation tion gr grad adient ient Twin volume ume fractio tion Crac ack k sur urface face Equi uiva valen lent t ell llip ipse Crack k length, h, openi ning ng

• Nick

ckel l Singl gle e Crys ystal tal: : MEAM M po potential ntial

• NPT

T ensem emble ble ~1K 100 nm m x 60nm nm x 25nm(10 m(10 million n atom

• ms)

s) Periodic dic bou

• und

ndary ary con

• ndi

dition tion Initial al small ll cr crack ck in the he ce center er Tensil nsile e loa

• adi

ding, ng, strain in con

• ntrolle

led d Strain in-rate rate ~ 107 s-1

A MD MD Mo Model el to to St Study dy Ev Evol

• luti

tion

• n of
• f

Crac ack k and d Ass ssoc

• ciat

iated ed Mec echanisms hanisms

• A. Ev

Evol

• luti

tion

• n of
• f De

Deform

• rmation

ation Mec echan hanisms isms

Dislocation segments colored by magnitude of Burgers vector

1 : 112 { 111 } 6 1 : 110 { 111 } 6 1 : 100 { 111 } 3 gree r n e b b b bl e d u   

Stabilized Dislocation Structure at 2.7% Strain.

• After

er cr critica cal stress, s, pa partial al di disloc

• cati

ation emiss ssio ion n from

• m crack-tip

p slip caused ed by dislocat ation

• n gliding

ng blunt unts s crack k tip an p and d redu duces s stress ss con

• ncentrat

entration

• No
• brittle

e cr crack ck pr prop

• paga

agation n by bond

• nd cl

cleava vage ge

• Crack

ck evol

• luti

ution n for

• r thi

his or

• rienta

ntation n is gov

• verne

erned by hy hydr drost

• stat

atic strain n and d slip, p, resemb embli ling ng voi

• id

d growt

• wth
• For
• rmatio

mation n of

• f di

disloc

• catio

ation n jun unct ctio ions, ns, jun unct ctio ion n length gth take kes s 30% % of

• f tot
• tal lengt

gth

𝑐𝑚𝑣𝑓: 𝒄𝟐 =

1 6 112 111

𝑠𝑓𝑒: 𝒄𝟑 =

1 6 110 111

𝑕𝑠𝑓𝑓𝑜: 𝒄𝟒 =

1 3 100 111

• Deformatio

ation mechanism sms s divided d into two categor gorie ies, s, (i) twin n partials ials contri tributi buting ng to slip (ii) disloca

• catio

tion n motion n contrib tribut uting ng to slip.

• Tw

Twins s forme med at at crac ack k tip by sequential tial lead ading par artial ial dislocatio cations ns nucleate eated d on adjace cent nt {111} plane

• All twin

n partial ials s are edge partials als with h no cross ss-sl slip ip

• At ∼ 3.3%

3% tensil ile strain, n, dislocatio cation n loop starts rts to emit from the crack k tip, , gliding g in the {111} plane.

• Tw

Twin bounda daries ries imp mpede disloca

• catio

tion n mo motion, n, simi milar ar to dislocati cation

• n jun

unction

• B. Ev

Evol

• lut

ution ion of

• f De

Deform

• rmation

ation Me Mechan hanisms isms

Dislocations interact with twin boundary forming stair-rod dislocations

Dislocation segments colored by magnitude of Burgers vector

Decomposition

• mposition of Ene

nergy y Associ ssociated ated wit ith Deforma rmati tion

• n Mechanisms

anisms

En Ener ergy gy Pa Partitio titionin ning g Ass ssoc

• ciat

iated ed with th Def efor

• rmati

mation

• n Mec

echanisms hanisms

Disloc

• cation

ation do domina inated ed Twi win n first the hen di disloc

• cation

tion Energy rgy evol

• lution

n pr prof

• file

e ca can sug ugges gest t the he de deform

• rmation

n mechanism! anism!

𝒆𝑿 = 𝒆𝑽𝐟𝐦 + 𝒆𝑽𝒋𝒐𝒇𝒎 + 𝒆𝑹

Heat transfer coefficient 𝜈 ≜

𝑅 𝑅+𝑉𝑗𝑜𝑓𝑚

• II. A C

Cou

• upled

pled Con

• ncurre

urrent nt Mo Model el

Fin init ite e te temp mperatu rature re Non-equ equil ilibr ibrium ium si simula latio tion Sh Should uld han andle le dis islo location cation tr tran ansf sfer

Zhang, Chakraborty and Ghosh (2017). Ghosh and Zhang (2017).

Nume merical rical Impleme plementation ntation

Boundary atoms external force on atom 𝑘

𝑔

𝑘 ext = 𝑥 𝑘𝑆𝐽 𝐷

Boundary nodes displacement on node 𝐽 𝑣𝐽

𝐷 = ෍ 𝑘 𝑛𝐽

𝑥

𝑘⟨𝑣𝑘 𝐵⟩

Handshake region

Quasi-static equilibrium configuration Solve for reaction force of any boundary node 𝐽 with displacement B.C.

𝑆𝐽

FEM

Velocity Verlet and NVE ensemble (LAMMPS) Solve for displacement field and its time average for any atom 𝑘 in handshake region

𝑣𝑗 𝑢 , 𝑣𝑗 𝑢

Molecular Dynamics

Concur current rent Formulati ulation

• n

Equi uili libri brium um equa uations ions for r the qua uasi si-st stat atic ic probl blem m are obta tained ned by min inim imiz izing ing the total al potentia ential l energy gy fun unctio ctiona nal l of the syste stem. m. Compa pati tibil bilit ity Constra nstraint int

Concur current rent Formulati ulation

• n

Equi uili libri brium um configura figuration tion of the coup uple led-co concu curr rrent ent syst stem em Compa pati tibil bilit ity Constra nstraint int

Nume merical rical Impl pleme ementati ntation

Conti tinu nuum um Model MD D Model el

In addition to inter-atomic interactions, atoms also experience forces due to atom-node interaction. Damping is applied to this region for maintaining temperature and elastic waves are suppressed by using a Langevin thermostat.

Non-li linear near in inter-atomic atomic in interactions actions

Near r cr crack ck tip, p, the he stress s and strain in to

• evalu

luate ate the he length gth scale e pa paramete ameter

Φ 𝜗 = 1 2! 𝑑𝑗𝑘𝑙𝑚𝜗𝑗𝑘𝜗𝑙𝑚 + 1 3! 𝑑𝑗𝑘𝑙𝑚𝑛𝑜𝜗𝑗𝑘𝜗𝑙𝑚𝜗𝑛𝑜

Sel elf-Consist Consistency ency  Non

• n-line

linearit arity y and nd Non

• n-locality

locality

Non-loc local al in inter-atom atomic c interact actions ions

𝜏𝑗𝑘 = 𝑑𝑗𝑘𝑙𝑚 𝜗𝑙𝑚 − 𝑚2𝛼2𝜗𝑙𝑚

Com

• mparing

paring average rage de deformat rmatio ion n gradien ent at interface rface of

• f atom
• misti

istic and con

• ntinu

nuum

Slope corresponds to 𝑚2, it is found for Nickel with EAM potential used, 𝑚 ≈ 0.5Å

[1] S. Hara and J. Li, PRB 82, 184114 (2010)

III.

• I. Time

me Ac Accele elerati ration

• n of
• f At

Atomi

• mistic

stic Dom

• main

ain Us Using ng St Strain ain Bo Boos

• st Hy

Hyper erdyn dynamics amics

     

     

max 1 2 max max max

Biased system potential, Boost potential, 1 = 0

b

b Mises N Mises i i i b Mises Mises Mises c c

V V r V r F V r V N F if q q      



             



max 2 max

1 Boost Factor exp

Mises c Mises i i c b

if q V V q V K T                           

• ηi

Mises is Von-Mises strain of atom ‘i’, calculated from least square

based atomic deformation gradient.

• Nb is total number of atoms to be boosted.
• Vmax and qc are material parameters.

Det eter ermination mination of

• f Strain

ain Boo

• ost

st Hy Hyperdy perdyna namics mics Pa Paramet rameters ers

• “qc” corresponds to critical value of

“ηmax” at the onset of transition.

• Vmax scales with the energy barrier.
• MD simulation is conducted for a small

sample to calibrate “𝑟𝑑 ” and “Vmax”.

2 max 1 Mises i i c

V V q                 

Determination termination of

• f Boo
• ost

st Region gion

• Extra computational cost scales with number of atoms in boost region.
• Boost region should include all the critical atoms from where nucleation is

likely to happen.

• New atoms are tagged adaptively as ‘to be boosted’ during the simulation.

Simulati mulation

• n Res

esult lts s – Orientation ientation 1 1

• Less number of dislocations nucleate at low strain rate.
• Reduction in number of dislocation increases the free path

for dislocation to glide before it interacts with other dislocation forming immobile junction (stair-rod dislocation).

Conventional MD with ሶ 𝜁 = 2 ∗ 107/𝑡𝑓𝑑 Hyperdynamics with ሶ 𝜁 = 104/𝑡𝑓𝑑

Lattice Orientation 1 : [110] [111] [112] x y z

 

  

Simulati mulation

• n Res

esult lts s – Orientation ientation 2 2

• At high strain rate nucleation of

successive leading partials in parallel slip plane is preferred forming a micro twin band.

• At low strain rate leading partial is

followed is by a trailing partial in the same slip plane forming a full dislocation.

Lattice Orientation 2 : [112] [111] [110] x y z

 

  

• Ref. S. Chakraborty, J. Zhang and S.

Ghosh, Computational Material Science, 121:23-34(2016).

Ti Time me-Scale Scale Bridging ridging in n Cou

• upled

pled Con

• ncurrent

current Simulation ulation

∆𝑢𝐵 = ∆𝑢𝑁𝐸 ∗ exp ∆𝑊 𝐿𝑐𝑈

Ref: “Concurrent Atomistic-Continuum Model using Hyperdynamics Accelerated MD Simulations.” S. Chakraborty and S. Ghosh, Comp. Mat.

• Sci. (Accepted).

𝑤 = ሶ 𝑏 = ቊ𝑑. 𝐿𝐽 − 𝐿𝐽𝐷 1/2, 𝑗𝑔 𝐿𝐽 > 𝐿𝐽𝐷 0, 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓

Crack ack Pr Propagation

• pagation Rate

te at t Low

• w Str

train ain Rates es

• 2L=200nm.
• R=32nm.
• 2aO=20nm.

ሶ 𝜁 = 107 ሶ 𝜁 = 106 ሶ 𝜁 = 104

Ef Effec ect t of

• f Tem

Temperat perature ure and nd Str train ain Rate te

• n Crac

ack k Propagat pagation ion Rate

• Parameters corresponding to the evolution of crack have

both strain rate and temperature effect on it.

• Temperature effect is more significant than strain rate for

the particular orientation used in the present study.

\ \

2 . ; 0; 1 : ( ) : ( ) 2 2

c A A A I I c C C I I

el inel s C inel p p p p s el

dW dQ dU dU dA dQ dU dW t u dA dQ dU dU dV r dV r dA dU dU      

      

                           

    

: ncremental work-potential. :generated heat. :incremental recoverable elastic strain-energy. :energy due to defects. :surface energy density. : increment in crack surface area.

el inel s

dW i dQ dU dU dA 

Th Thermodynam ermodynamics ics of

• f De

Deform

• rmati

tion

• n

Mec echan hanisms isms

ρ0Ψ = 1 2 ෨ 𝐅𝐟: 𝐃𝐟: ෨ 𝐅𝐟 + gc 2lc s2 + lc

2∆Xs. ∆Xs

෩ Ee = 1 2 Je 2/3 − 1 g1 s − g2 s 𝚱 + 1 2 g2 s 𝐆𝐟𝐔𝐆𝐟 − 𝚱 𝑕1 𝑡 = ቊ 1 , 𝐾𝑓 < 1 1 − 𝑡 , 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 𝑕2 𝑡 = 1 − 𝑡

gc = 4.6 Joules/m2

Calibration libration of

• f El

Elastic stic Ph Phase ase-Field Field Mod

• del

el Pa Paramet rameter er

Tr Transfer ansfer of

• f Di

Disl slocat

• catio

ion n from

• m Ato

tomisti istic c to C

• Con
• ntinuu

inuum m Dom

• main

ain

• DXA[1] is used to

extract dislocation from atomic data.

• The dislocation is

converted into equivalent density form.

• The dislocation is

transferred into the continuum in density form.

[1] A. Stukowski and K. Able, Modelling

• Simul. Mater. Sci. Eng. 18:825-

847(2010). Dislocation in di

discrete

form Dislocation in de

dens nsity

form

Non

• nlocal

local Fo Formulation mulation for

• r th

the e Ev Evol

• luti

tion

• n

Law w of

• f the

e Incoming

• ming Disl

slocati

• cation
• n Fl

Flux. x.

 

_

. . . . 1 1 . 2 1

1 max 0, . H min 0, .

ji

i i i Flux FluxGeneration FluxDepletion nNeighbor i j i i FluxGeneration Flux ji pass d j a nNeighbor i i i FluxDepletion Flux ji j

c d v e c d v         

     

                            

 

 

_

1 2 ji i

1 H c and c are material parameter. d :distance between i'th Gauss point and it's j'th neighbor. v : is the dislocation velocity.

ji

i pass d a

e               

Characterization and quantification of dislocations Conversion from discrete to density form Evolution of Incoming dislocation in continuum

 Bui uild lding ing a C a Compr prehe hensi nsive ve Se Self lf-Con Consi siste stent nt Fram amework ework Coup upli ling ng CPFEM EM wit ith MD D for r Crac ack k Evolution lution models els in in CPFEM  De Develo lope ped d robu bust st char arac acte teri rizat zation ion an and qua uantif ntification ication method hods s for r defor

• rma

mation tion mechani anism sm an and crac ack k evoluti lution

• n for

r at atom

• mic

ic sim imul ulat ations ions for r transf ansfer er of dis islo loca cati tion

• n rela

lated ed var aria iable bles. s.  Hyperd perdyna ynamics mics has as be been us used to br brid idge ge the tim ime scal ale dif iffere rence nce be betwee een n at atomis misti tic c an and the conti tinu nuum um domai ain.  Extra racting cting phas ase-fie field ld energie gies s from

• m the self

lf consiste sistent nt model

Summary mmary

Ongoing ing:

• Develop

elopmen ment of

• f information

rmation pa passing ing on

• n pl

plastici icity ty wh when n di disloc

• cation

ation reaches es the he interface rface of

• f Ωatomistic and

nd Ωcontinuum and nd bui uildin ing g ph phase e field ener ergi gies es for

• r de

defect ct and d fracture e sur urfaces aces