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� 3D characterization and modelling of smallfatigue cracks in polycrystalline materialsHenry Proudhon – 2016 IRSP conferenceMINES ParisTech, PSL – research university, Centre des Matériaux, UMR CNRS 7633

May 31, 2016

Fatigue in structural (polycrystalline) materials : amultiscaled problem

structure

σ

material coupon microstructure crystal lattice

∼ 10 m ∼ 0.1 m ∼ 1 mm ∼ 1 nm

Fatigue in structural (polycrystalline) materials : amultiscaled problem

structure

σ

material coupon microstructure crystal lattice

∼ 10 m ∼ 0.1 m ∼ 1 mm ∼ 1 nm

3D propagation of fatigue cracks in opaque materials → need forX-ray microtomographyPlastic deformation results in the motion of lots of defects in thecrystal lattice (dislocations) → continuum scale (� μm)Local values are crucial when studying deformation and fracture→ what can we learn from crystal plasticity FEM simulations ?

Contents

1 Early stages of fatigue fracture

2 In situ 3D observation of fatigue crack propagation in Al alloys

3 Simulation of short fatigue crack propagation in polycrystals

4 Summary

Coworkers :N. Guéninchault (Ph. D.student)E. Nizery (Ph. D. student,now at Constellium)J. Li (Ph. D. student, nowat Areva)

J.-Y. Buffière & W. Ludwig(INSA Lyon/ESRF)S. Forest (MINESParisTech)

Contents

1 Early stages of fatigue fractureLiterature review : where we arePresent work strategy : where we are going

2 In situ 3D observation of fatigue crack propagation in Al alloys

3 Simulation of short fatigue crack propagation in polycrystalsTomographic fatigue experimentCPFE model for short fatigue crack propagationFatigue crack propagation results in the experimentalpolycrystal

4 Summary

Microsructurally short cracks

Long crack(LEFM)

Long crack threshold

Constant-amplitude loadingR=constant

short cracks

Short crackfrom notch

Suresh

etal.,

Int.M

etalReview

s,1984

Complex problem (inherentlythree dimensional). . .

[Herbig et al., 2011]The grain microstructure crossed by the fatigue crack has a stronginfluence on the crack path and growth rate : (i) single vsmultiple slip, (ii) effect of grain boundaries.

Influence of grain boundariesDo grain boundaries slow down cracks ? Tilt/twist business ?

Observations [Schaef et al., 2011]Ni superalloy

Fib tomography

modeling [Zhai et al., 2000]

Tilt/Twist geometric model

Grain boundary crossing mechanism

Present work strategy

Experiments Use in situ fatigue testing combined with X-raymicrotomography and diffraction with sub-micronspatial resolution to observe 3D crack propagationmechanisms.

Simulations Use crystal plasticity based Finite Elementsimulations to study the effect of the microstructureon crack initiation and growth.

→ Identify the governing mechanisms for short crackpropagation at the grain scale.

Contents

1 Early stages of fatigue fractureLiterature review : where we arePresent work strategy : where we are going

2 In situ 3D observation of fatigue crack propagation in Al alloys

3 Simulation of short fatigue crack propagation in polycrystalsTomographic fatigue experimentCPFE model for short fatigue crack propagationFatigue crack propagation results in the experimentalpolycrystal

4 Summary

Computed X-ray Micro-Tomography

X Rays

specimen

� radiographsreconstructed

3D object

European Synchrotron Synchrotron X-ray tomography

Parallel beam → no magnificationSample size limited by CCD size,typically ∼ 1 mmMonochromatic coherent beam(phase contrast)Relatively low availability →laboratory sources

Specimens for in-situ synchrotron fatigue machinep y

Experimental fatigue test details

Ex situ observation every ∼ 500cycles (optical microscopy,SEM)Ex situ EBSD orientation mapIn situ ID19 ESRF (may 2014)→ ∼ 90 �= cracks and 500tomographic volumes (needautomation) [Buffière et al., 2006]

3D Fatigue crack propagation

22 000 cycles 26 000 cycles

29 000 cycles 30 500 cycles

3D Fatigue crack propagation

22 000 cycles 26 000 cycles

29 000 cycles 30 500 cycles

crystallographicbranch

Crack growth measurements

Outcome of crack growth measurements

FCGR (μm/cycle) 22000-26700 26700-29000 29000-30500Zone 1 0.0034 0.0067 0.0105Zone 4 0.0046 0.0111 0.0156Zone 6 0.0054 0.0116 0.0143

Measured fatigue crack growth rates (FCGR) from automatedtomographic imaging [Kentheswaran, 2015].

Mechanisms and local crack growthStatistical processing of 2D+3D data lead to the followingconclusions :

The local crack growth rate of a crystallographic plane kinkedat 55◦ is 30% lower than without bifurcation ;A net life saving of 2 000 fatigue cycles (on a 50 000 totallife) is systematically obtained when such a bifurcation isobserved.

Study of 3D short crack propagation[Proudhon et al., 2012]

Measurements on another 2xxx Al alloy (grain size �100 μm)

ouverture charge min (�m) ouverture charge max (�m)

σ100 �m

COD

Study of 3D short crack propagation[Proudhon et al., 2012]

Measurements on another 2xxx Al alloy (grain size �100 μm)

ouverture charge min (�m) ouverture charge max (�m)

σ100 �m

COD

3D characteri-sation → inputfor modeling

0

0.5

1.0

1.5

2.0

0 0.5 1.0 1.5 2.0 2.5 3.0

N=63000

N=75000

N=80000

ΔK (MPa.m1/2)

da/dN

(�

m/cycle)

10-4

10-3

10-2

10-1

100

101

102

10-1 100 101 102

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Contents

1 Early stages of fatigue fractureLiterature review : where we arePresent work strategy : where we are going

2 In situ 3D observation of fatigue crack propagation in Al alloys

3 Simulation of short fatigue crack propagation in polycrystalsTomographic fatigue experimentCPFE model for short fatigue crack propagationFatigue crack propagation results in the experimentalpolycrystal

4 Summary

Diffraction contrast tomographyDeveloped at ESRF by W. Ludwig et al. → Non destructivecharacterization of 3D grain microstructures (plasticallyundeformed, monophase materials)[Ludwig et al., 2008, Johnson et al., 2008]

2D sample illumination with monochromatic beam (Δλ/λ = 10−4)Continuous rotation over 360◦, integration over 0.05◦

Simultaneous acquisition of transmitted and diffracted beamAcquisition time : ∼ 0.1 − 10 h

In-situ synchrotron fatigue experiment

ANR CRYSTAL

in-situ fatigue testing at ID11 (ESRF) Grenoble, similar to[Herbig et al., 2011]

1 Diffraction Contrast Tomography (DCT) → 3D microstructure2 Phase Contrast Tomography (PCT) → 3D crack shape

Tomographic fatiguesample, Ti-β alloy

Fatigue machine[Buffière et al., 2006]

In-situ synchrotron fatigue experiment

ANR CRYSTAL

in-situ fatigue testing at ID11 (ESRF) Grenoble, similar to[Herbig et al., 2011]

1 Diffraction Contrast Tomography (DCT) → 3D microstructure2 Phase Contrast Tomography (PCT) → 3D crack shape

Tomographic fatiguesample, Ti-β alloy

387 grains sample

Fatigue machine[Buffière et al., 2006]

In-situ synchrotron fatigue experiment

ANR CRYSTAL

in-situ fatigue testing at ID11 (ESRF) Grenoble, similar to[Herbig et al., 2011]

1 Diffraction Contrast Tomography (DCT) → 3D microstructure2 Phase Contrast Tomography (PCT) → 3D crack shape

Tomographic fatiguesample, Ti-β alloy

387 grains sample

Fatigue machine[Buffière et al., 2006]

100 μm

3D crack propagation

Data processing

Reconstruction DCT & PCT, pre-alignment,resampling

Data processing

Reconstruction DCT & PCT, pre-alignment,resampling

Alignment New automatic procedure based onthe ITK library (C++)http://www.itk.org

Data processing

Reconstruction DCT & PCT, pre-alignment,resampling

Alignment New automatic procedure based onthe ITK library (C++)http://www.itk.org

Crack segmentation Manual and Automatedbased on ITK library

Data processing

Reconstruction DCT & PCT, pre-alignment,resampling

Alignment New automatic procedure based onthe ITK library (C++)http://www.itk.org

Crack segmentation Manual and Automatedbased on ITK library

Meshing Based on VTK libraryhttp://www.vtk.org

Data processing

Reconstruction DCT & PCT, pre-alignment,resampling

Alignment New automatic procedure based onthe ITK library (C++)http://www.itk.org

Crack segmentation Manual and Automatedbased on ITK library

Meshing Based on VTK libraryhttp://www.vtk.org

Matching Matching the crack with the grainmicrostructure, many user functionsfor crack analysis

Evolution of the crack during the fatigue testCrack growth rate decrease at first grains boundariesBifurcation at grain boundaries but also inside grains

Crack propagation modelcyclic loading

cracked body CPFE computationof N loading cycles

[MPa]

Evaluation of the damage indicator Dnear the crack front

0

10

20

(MPa)

0°

60°

θ=61°120°

180°

240° 300°

θ=299°

Determination of local crackpropagation direction and length

Crack propagation through remeshingand plastic field transfer

[Li et al., 2014] Comp. Mat. Sci., [Proudhon et al., 2016] IJF

Driving force for fatigue crack propagation

σ

Driving force for fatigue crack propagation

σ

Contributions to crack propagation

Resolved shear stress τ s

Slip rate γs

Normal stress on slip plane σsn = σ∼ : n s ⊗ n s

Driving force for fatigue crack propagation

σ

Contributions to crack propagation

Resolved shear stress τ s

Slip rate γs

Normal stress on slip plane σsn = σ∼ : n s ⊗ n s

Damage indicator

D = maxs |γs | (|τ s | + k〈σsn〉)

k controls the sensitivity to the normal stress ;local crack propagation direction can bedetermined by analysing the field of D ;D cumulates through time which allows to usea threshold for crack propagation.

Growing the crackInterpolation of the damage indicator field (D) to the circles ofinterest centered at control points in planes normal to the crackfront :

R0 : radius of thecircles of interest,here R0 ∼ 1 μm

0

10

20

30

0°

60°120°

180°

240° 300°

Damage indicator (D)

Extension of the crack surfaceby remeshing

Meshing real microstructures

image data FEM mesh

Small strain crystal plasticity [Meric and Cailletaud, 1991]

Material : Ti55531Near beta titanium alloyBCC crystal structureWell recrystallizedmicrostructure,grain size ∼ 65 μm

200 μm

Cubic elasticity σ∼ = C∼∼ : ε∼e

Resolved shear stress τ s = σ∼ : m∼ s

with (s = {110}〈111〉)Norton law γs = sign(τ s)

⟨ |τ s |−r s

K

⟩n

Plastic strain rate ε∼p =

∑ns=1 γsm∼ s

Orientation tensorm∼ s = 1

2(l s ⊗ n s + n s ⊗ l s)

Propagation in the experimental microstructureFatigue test : Fmax = 35 N, R=0.1Realistic 3D meshing of the polycrystal from the tomographicimage + initial notchCrystal plasticity, Parallel computing (32 nodes)

U3 = 5 μm

3D crack3D crack

U3 = 0

fatigue load

F

t Key figures of the calculation

5 millions DoF∼ 100 Gb memory neededsequential resolution time →2 cycles/weekMUMPS parallel solverFETI algorithm for domaindecomposition

→ 2 cycles/day

Propagation in the experimental microstructure

111

101001

Contents

1 Early stages of fatigue fractureLiterature review : where we arePresent work strategy : where we are going

2 In situ 3D observation of fatigue crack propagation in Al alloys

3 Simulation of short fatigue crack propagation in polycrystalsTomographic fatigue experimentCPFE model for short fatigue crack propagationFatigue crack propagation results in the experimentalpolycrystal

4 Summary

Summary and outlookComputed X-ray microtomography is a powerful tool to investigatedamage and microstructure organisation within a wide range of materials

Key figures for X-ray microtomography

Spatial resolution routinely ≈ 0.5 μm

Phase, absorption and diffraction contrasts

in situ experiments to study damage evolution in a knownmicrostructure

Laboratory tomography now highly available

Crystal plasticity FEM calculations

Ideal to simulate your experiment and test/validate a materialbehaviour

Towards the study of damage evolution in a known microstructure

limited to continuum mechanics although higher order models exist

NANOX a new in situ stress rig for DCT

controled load

Reconstruction of individual grainswith elastic strains information

ANR CRYSTAL

W. Ludwig

Outlook : imaging individual grains by topotomography

G

Rotation

Base tilt T0

low tilt

up tilt

rotation axis // G

incoming beam

di racted beam

High resolution detector

Principle : align the scattering vectorGhkl and the rotation axis ω

W. Ludwig et al. J. Appl. Cryst. 2001N. Guéninchault et al. JSR, under

review

Outlook : imaging individual grains by topotomography

G

Rotation

Base tilt T0

low tilt

up tilt

rotation axis // G

incoming beam

di racted beam

High resolution detector

Principle : align the scattering vectorGhkl and the rotation axis ω

W. Ludwig et al. J. Appl. Cryst. 2001N. Guéninchault et al. JSR, under

review

Current focus :Early plastic activity in individualgrains (Al2.5%Li)

in situ testing (Nanox)grain size ∼ 150 μm

Outlook : imaging individual grains by topotomography

G

Rotation

Base tilt T0

low tilt

up tilt

rotation axis // G

incoming beam

di racted beam

High resolution detector

Principle : align the scattering vectorGhkl and the rotation axis ω

W. Ludwig et al. J. Appl. Cryst. 2001N. Guéninchault et al. JSR, under

review

Current focus :Early plastic activity in individualgrains (Al2.5%Li)

in situ testing (Nanox)grain size ∼ 150 μm

unlo

aded

0.2%

stra

in

Buffière, J., Ferrié, E., Proudhon, H., and Ludwig, W. (2006).3D visualisation of fatigue cracks in metals using high resolution synchrotron.Materials Science and Technology, 22(9) :1019–1024.

Herbig, M., King, A., Reischig, P., Proudhon, H., Lauridsen, E. M., Marrow, J., Buffière, J.-Y., and Ludwig,W. (2011).3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combineddiffraction and phase-contrast X-ray tomography.Acta Materialia, 59(2) :590–601.

Johnson, G., King, A., Honnicke, M. G., Marrow, J., and Ludwig, W. (2008).X-ray diffraction contrast tomography : a novel technique for three-dimensional grain mapping ofpolycrystals. II. The combined case.Journal of Applied Crystallography, 41(2) :310–318.

Li, J., Proudhon, H., Roos, A., Chiaruttini, V., and Forest, S. (2014).Crystal plasticity finite element computation of crack growth in single crystals.Computational Material Science, 94 :191–197.IWCMM23 Special Issue.

Ludwig, W., Schmidt, S., Lauridsen, E. M., and Poulsen, H. F. (2008).X-ray diffraction contrast tomography : a novel technique for three-dimensional grain mapping ofpolycrystals. I. Direct beam case.Journal of Applied Crystallography, 41(2) :302–309.

Meric, L. and Cailletaud, G. (1991).Single crystal modeling for structural calculations : Part 2—finite element implementation.Journal of Engineering Materials and Technology, 113(1) :171–182.

Proudhon, H., Li, J., Wang, F., Roos, A., Chiaruttini, V., and Forest, S. (2016).3D simulation of short fatigue crack propagation by finite element crystal plasticity and remeshing.International Journal of Fatigue, 82, Part 2 :238–246.10th Fatigue Damage of Structural Materials Conference.

Proudhon, H., Moffat, A., Sinclair, I., and Buffiere, J.-Y. (2012).Three-dimensional characterisation and modelling of small fatigue corner cracks in high strength Al-alloys.Comptes Rendus Physique, 13(3) :316–327.

Schaef, W., Marx, M., Vehoff, H., Heckl, A., and Randelzhofer, P. (2011).A 3-D view on the mechanisms of short fatigue cracks interacting with grain boundaries.Acta Materialia, 59(5) :1849–1861.

Zhai, T., Wilkinson, A. J., and Martin, J. W. (2000).A crystallographic mechanism for fatigue crack propagation through grain boundaries.Acta Materialia, 48(20) :4917–27.

Diffraction contrast tomographyDeveloped at ESRF by W. Ludwig et al. → Non destructivecharacterization of 3D grain microstructures (plasticallyundeformed, monophase materials)[Ludwig et al., 2008, Johnson et al., 2008]

2D sample illumination with monochromatic beam (Δλ/λ = 10−4)Continuous rotation over 360◦, integration over 0.05◦

Simultaneous acquisition of transmitted and diffracted beamAcquisition time : ∼ 0.1 − 10 h

NANOX a new in situ stress rig for DCT

controled load

Reconstruction of individual grainswith elastic strains information

ANR CRYSTAL

W. Ludwig

→ Ph.D. thesis of Nicolas Guéninchault

Meshing the polycrystalline specimen

1- raw data from DCT

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Meshing the polycrystalline specimen

1- raw data from DCT 2- mesh grain boundaries 3- full 3D mesh

beg go end

Small strain crystal plasticity [Meric and Cailletaud, 1991]

Material : Ti55531Near beta titanium alloyBCC crystal structureWell recrystallizedmicrostructure,grain size ∼ 65 μm

200 μm

Cubic elasticity σ∼ = C∼∼ : ε∼e

Resolved shear stress τ s = σ∼ : m∼ s

with (s = {110}〈111〉)Norton law γs = sign(τ s)

⟨ |τ s |−r s

K

⟩n

Plastic strain rate ε∼p =

∑ns=1 γsm∼ s

Orientation tensorm∼ s = 1

2(l s ⊗ n s + n s ⊗ l s)

Computation under tension

x

y

grain orientationsmeasured by DCT

({111} stereo projection)

von Mises equivalent stress (MPa)300 400 500 600 700 800

Averaged strains/stresses per grain

Full fieldsolution

0

200

400

600

800

1000

1200

0 0.005 0.01 0.015 0.02 0.025 0.03

Axi

al s

tress

(MP

a)

Axial strain (mm/mm)

all grains

Averaged stresses per grain

Grain by grain comparaison of axial strain ε33 (pure α-Ti)

DCT measurements FE predictions

larger errors in big grains