GergelyMolnárand Anthony Gravouil

Numericalmodelingofbrittlefractureusingthephase‐fieldmethod

Gergely Molnár andAnthonyGravouil

Introduction

Wörner, 2001.

FIBERGLASS TRUSS GLASS

ROD

A A

Tensile

 streng

th [N

/mm

2 ]

Effective crack length [mm]

Glass plate

Glass fiber strength

Atomic strength

Macroscopicstrength

Introduction

Macroscopicstrength

A A

σ

σ

FIBERGLASS TRUSS

GLASS ROD

covalent bonds no stress redistribution brittle

Molnár et al., 2012, 2013.

IntroductionGriffiththeoryofbrittlefracture

Griffith,1920

2 cS g ha

Müller,2002

Crackmodes

Force /Energy

Distance a

h

Irwin,1957Erdogan ,2000

IntroductionHowdoweapproximateitwithaphase‐field?

2.Minimizationproblem

1.Brittlefracture Griffith,1920cSa

ga

, cE d g d

u u

Mumford& Shah,1989Francfort &Marigo,1998

221,2 2

c

cc

ld dl

E d g dg d d

u u

3.Crackenergydensity

Ambrosio &Tortorelli,1990Bourdin etal.,2000Amoretal.,2009

Miehe etal.,2010a

0 converges

0cl

d

crackenergydensity‐ γ

IntroductionWhatisdiffusedamage?

Solvingfracturemechanics problemwithPartialDifferentialEquations(PDEs)

IntroductionWhatisdiffusedamage?

Solvingfracturemechanics problemwithPartialDifferentialEquations(PDEs)

theoretical crack (d = 1)

damaged zone (0 < d < 1)

undamaged zone (0 ≈ d)

Phase‐fieldmethodStaggeredscheme

Prof.Dr.‐Ing.ChristianMiehe(1956– †2016)

Miehe etal.,2010b

0 1 0

,

,

if

u

dn

n n

E d

E d H

H H

u

u

Robustness!!!Efficiency?

Phase‐fieldmethod

Molnár&Gravouil,2017

Staggeredscheme

Prof.Dr.‐Ing.ChristianMiehe(1956– †2016)

0 1 0

,

,

if

u

dn

n n

E d

E d H

H H

u

u

OpensourceimplementationABAQUS/UELoption(ABAQUS+FORTRANcompiler)

stiffnessmatrix +residuevector foreveryelement

FORTRANandABAQUSfilesareavailableinboth2D and3D

Visualization

Molnár&Gravouil,2017

Phase‐fieldmethodSingleelementsolution

222

222

y

cy

c

cd g c

l

2221y yd c

22 - (2,2) element of the stiffness matrixc

Molnár&Gravouil,2017

Phase‐fieldmethodSingleelementsolution

analytic

analyticy y

y

Molnár&Gravouil,2017

ParametersHowfineshouldthemeshbe?

Miehe etal.,2010a

ParametersHowfineshouldthemeshbe?

0.5

l d

theoretical

Miehe etal.,2010a

/ 2h l

Solvingfracturemechanics problemwithPartialDifferentialEquations(PDEs)

ParametersHowfineshouldthemeshbe?

Solvingfracturemechanics problemwithPartialDifferentialEquations(PDEs)

ParametersHowfineshouldthemeshbe?

ParametersSinglenotchedspecimenundertension

Molnár&Gravouil,2017

Effectoflength‐scale300% → 10%

ParametersSinglenotchedspecimenundershear

Molnár&Gravouil,2017

ParametersDoublenotchedplate

Molnár&Gravouil,2017

ParametersTimestep

Molnár&Gravouil,2017

ParametersTimestep

Molnár&Gravouil,2017

Deformation is applied until 0.008 mm then stopped 

For details see Tutorial 3: Cracked cylinder in tension on www.molnar‐research.com 

Examples

Molnár&Gravouil,2017

Koivisto etal.,2016Réthoré etal.,2010

Asymmetricdoublenotchedplate

Examples

Molnár&Gravouil,2017

Bi‐materialtension

ExamplesPositiveandnegativeenergydegradation

Molnár&Gravouil,2017

0 0g d

20

2 22 2

1 1 2

+2 1

E tr

E

ε

ExamplesAsymmetricbending

Molnár&Gravouil,2017

Bittencourt etal.,1996

3DExamplesInclinedcrackinbending

Lazarus etal.,2008

3DExamplesInclinedpennyshapecrackintension

Molnár&Gravouil,2017

Gravouiletal.,2002

XFEM

Conclusion

Advantagesanddisadvantages

Versatilitydynamics,shells,nonlinearelasticity,largestrains,coupledproblems,plasticity,anisotropy,etc…

HybridFDEMXFEM/GFEMCohesiveZones

• crackinitiation,propagation• branching,merging• fixedmesh• fully3D

Phase‐field Predefinedcrack

• finemesh• finitecracksize• efficiency/robustness

Microstructure

Wheretofindit?Examplesandtutorials:www.molnar‐research.com

FORTRAN filesINPUT filesTutorials

Molnár&Gravouil,2017

ReferencesL.Ambrosio,V.M.Tortorelli,Comm.PureAppl.Math.43(1990)999‐1036.H.Amora,J.‐J.Marigo,C.Maurini,J.oftheMech.andPhys.ofSolids,57(8)(2009)1209‐1229.T.Bittencourt,P.Wawrzynek,A.Ingraffea,J.Sousa,Eng.Fract.Mech.55(2)(1996)321–334.B.Bourdin,G.A.Francfort,J.‐J.Marigo,J.oftheMech.andPhys.ofSolids,48(4)(2000)797‐826.E.Erdogan,InternationalJournalofSolidsandStructures,37(2000) 171–183.G.A.Francfort,J.‐J.Marigo,J.oftheMech.andPhys.ofSolids,46(8)(1998)1319‐1342.A.Gravouil,N.Moës,T.Belytschko,Int.J.Numer.Meth.Engng53(2002)2569–2586.A.A.Griffith,Phil.Trans.oftheRoyalSoc.ofLondon.SeriesA,221(1920),163‐198.G.Irwin,JournalofAppliedMechanics24(1957) 361–364.J.Koivisto,M.‐J.Dalbe,M.J.Alava,S.Santucci,Sci Rep.6,(2016)32278.V.Lazarus,F.‐G.Buchholz,M.Fulland,J.Wiebesiek,IntJFract.153(2008)141–151.C.Miehe,F.Welschinger,M.Hofacker,Int.J.Numer.MethodsEng.,83(10)(2010a)1273–1311.C.Miehe,M.Hofacker,F.Welschinger,Comput.MethodsAppl.Mech.Eng.,199(45–48)(2010b)2765–2778.G.Molnár,L.M.Molnár,I.Bojtár,MaterialsEngineering,19(2012)71‐81.G.Molnár,I.Bojtár,MechanicsofMaterials,59(2013)1‐13.G.Molnár,A.Gravouil,Finite Element in Analysis andDesign130(2017)27‐38.H.W.Müller,Bruchmechanik,Berlin,2002.D.B.Mumford,J.Shah.,Comm.PureAppl.Math.42(5)(1989)577‐685.J.Réthoré,S.Roux,F.Hild,C.R.Mecanique 338(2010)121–126.J.‐D.Wörner,Glasbau,2001.

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