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AnopensourceABAQUS/UELimplementationofthephase‐fieldmethodtosimulatebrittlefracture
Gergely Molnár andAnthonyGravouil
ContentIntroductiontophase‐fields
1. Phase-field concept2. Strain energy degradation3. Staggered scheme
Parameters1. Finite element mesh2. Time step3. Length-scale parameter
ABAQUS/UELimplementationConclusion
IntroductionWhatisdiffusedamage?
Solvingfracturemechanics problemwithPartialDifferentialEquations(PDEs)
theoretical crack (d = 1)
damaged zone (0 < d < 1)
undamaged zone (0 = d)
IntroductionWhatisdiffusedamage?
Solvingfracturemechanics problemwithPartialDifferentialEquations(PDEs)
IntroductionHowdoweapproximateitwithaphase‐field?
2.Minimizationproblem
1.Brittlefracture Griffith,1921cga
, 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
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
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?
0.5
l d
theoretical
Miehe etal.,2010a
/ 2h l
ParametersDoublenotchedspecimenundertension
Molnár&Gravouil,2017
EffectofFEmesh
ParametersDoublenotchedspecimenundertension
Molnár&Gravouil,2017
Effectoftimestepsize
ParametersSinglenotchedspecimenundertension
Molnár&Gravouil,2017
Effectoflength‐scale
ExamplesCrackbranching
Molnár&Gravouil,2017
OptionsPositiveandnegativeenergydegradation
Molnár&Gravouil,2017
0 0g d
20
2 22 2
1 1 2
+2 1
E tr
E
ε
OpensourceimplementationABAQUS/UELoption(ABAQUS+FORTRANcompiler)
stiffnessmatrix +residuevector foreveryelement
FORTRANandABAQUSfilesareavailableinboth2D and3D
Visualization
Molnár&Gravouil,2017
OpensourceimplementationExamplesandtutorials:www.molnar‐research.com
FORTRAN filesINPUT filesTutorials
Molnár&Gravouil,2017
Conclusion
Advantagesanddisadvantages
Versatilitydynamics,shells,nonlinearelasticity,largestrains,coupledproblems,plasticity,anisotropy,etc…
XFEM/GFEMCohesiveZones
• crackinitiation,propagation• branching,merging• fixedmesh• fully3D
Phase‐field Predefinedcrack
• finemesh• finitecracksize• efficiency/robustness
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.B.Bourdin,G.A.Francfort,J.‐J.Marigo,J.oftheMech.andPhys.ofSolids,48(4)(2000)
797‐826.G.A.Francfort,J.‐J.Marigo,J.oftheMech.andPhys.ofSolids,46(8)(1998)1319‐1342.A.A.Griffith,Phil.Trans.oftheRoyalSoc.ofLondon.SeriesA,221(1921),163‐198.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,A.Gravouil,Finite Element in Analysis andDesign130(2017)27‐38.D.B.Mumford,J.Shah.,Comm.PureAppl.Math.42(5)(1989)577‐685.
