Time Table. MONDAY (ENS-Cachan) Session 1 : introduction, linear elastic fracture mechanics Session 2 : Exercises. THURSDAY (ENS-Cachan)

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1 Time Table MONDAY (ENS-Cachan) Session 1 : introduction, linear elastic fracture mechanics Session 2 : Exercises TUESDAY (ENS-Cachan) Session 3 : Numerical Practice. Determination of SIF Session 4 : LEFM for 3D problems (loading and geometry) WENESDAY Session 5 : Non-linear fracture mechanics (Ecole polytechnique) Session 6 : Experimental practice (Ecole polytechnique + ENS-Cachan) THURSDAY (ENS-Cachan) Session 7 : Mode I fatigue crack growth Session 8 : Mixed Mode fatigue crack growth FRIDAY (ENS-Cachan) : Conferences Session 9 : Novel computational tools and concept for predicting crack growth Session 9 : Ductile fracture and industrial problems 1

2 Session 7 Fatigue, Mechanisms and modeling, Mode I 1. Fatigue crack growth mechanisms 2. History effects in fatigue 3. Modelling 4. Applications 2

3 Historical perspective Fatigue S-N Versailles rail accident, on May 8, 1842 at Meudon (Versailles). A train returning to Paris crashed at Meudon after the leading locomotive broke an axle At the origin of the Stress-Life approach (S-N) 1829 Albert, effect of repeated loads 1839 Poncelet «fatigue» 1943 Rankine effect of stress concentrations 1860 Wöhler fisrt systematic investigations 1886 Bauschinger (reversed deformations effects) 1910 Basquin endurance limit 3

4 Historical perspective K IC Liberty ships at the origin of fracture mechanics 4

5 Historical perspective da/dn De Havilland Comet, (british) first civil aircracft using turbomachines. Known for being at the origin of a serie of accidents due to fatigue crack growth at the origin of fatigue crack growth analyses 1961 Paris Law (crack growth rate versus stress intensity factor amplitude) 1967 Forman (overload effect in fatigue crack growth plastic zone) 1970 Elber (the significance of fatigue crack closure, overload effect via residual stresses) 1975 Pearson (the behaviour of small fatigue cracks) 5

6 Crack length measurements F Crack length increasing COD J.Petit Potential drop Direct optical measurements Digital image corrlation COD ε 6

7 Crack length measurements a da/dn = f(a) a Load cycle N N F max F op F min R=F min /F max ΔF ΔF eff 7

8 Paris diagram ΔK eff = K IC A threshold regime B Paris regime C unstable fracture ΔK eff ( MPa m ) ΔK > Δ eff K th Subcritical crack growth if ΔK is over the non propagation threshold 8

9 Fatigue - Threshold regime Cyclic propagation mechanism by Neumann [Neumann,1969], 9

10 Fatigue - Threshold regime Titanium alloy TA6V [Le Biavant, 2000]. The fatigue crack grows along slip planes. N18 nickel based superalloy at room temperature, [Pommier,1992]. The crack grows at the intersection between slip planes 10

11 Fatigue - Threshold regime fracture surface psudeo-cleavage facets at the initiation site 11

12 Constant amplitude fatigue Mode I Paris law A threshold regime B Paris regime C unstable fracture da dn = CΔ m K eff Paris law + Elber ΔK eff ( MPa m ) 12

13 Paris regime: crack growth by the striation process TA6V 316L OFHC 13

14 Striations, the Pelloux s mechanism How do fatigue striations form [Laird,1967] [Pelloux, 1965] 14

15 Striations Mean crack growth direction Local crack growth directions Secondary cracks Secondary cracks 15

16 Striation spacing measurements The mean striation spacing is measured on the fracture surface. The ratio between the macroscopic crack growth da/dn and the striation spacing s is plotted versus s [Nedbal, 1989]. 16

17 Summary Intrinsic non propagation threshold in fatigue ΔK eff th Threshold & Paris regime : the process zone is very small compared with the plastic zone, (process zone damaged zone) Threshold regime, cristallographic growth, induced by plastic slip Paris regime, growth by the striation process, induces by plasticity In both cases localized plastic deformation at crack tip causes crack growth. Hypothesis : da/dn= α s = β CTOD opening displacement) (s : pas de strie CTOD crack 17

18 Scale independence LEFM : K-dominant area (r/a <0.1), order of magnitude a few mm 18

19 Scale independence plastic zone (r pc ), order of magnitude a few 100 μm 19

20 Scale independence Process zone Order of magnitude: a few 0.1 μm 20

21 Consequence LEFM quantities (KI, KII, KIII) control the behaviour of the K-dominance area that controls the behaviour of the plastic zone that controls the evolution of the process zone 21

22 Environment assisted fatigue crack growth Fatigue-corrosion Air For the same value of ΔK : da/dn air > da/dn vaccuum Vide threshold air < threshold vaccuum the effect disappears above a given value of da/dn Fatigue crack growth in the alloy Al-4Cu-Mg [ Meyn, 1968] 22

23 Effect of the partial pressure of water vapour da/dn (mm/cycle) a = 4 mm a = 7 mm a =10 mm o o water vapour has a detrimental effect in most metallic alloy Evidence of a threshold value of partial pressure of water vapour Water vapour pressure (Torr) o the effect is higher is ΔK is small Bradshaw and Wheeler (1969): first evidence of the effect of the environment in fatigue. 23

24 Environment assisted fatigue crack growth McEvily et Wei (1972): a) da/dn vs ΔK: fatigue b) da/dt vs ΔK: stress corrosion cracking c) Fatigue-corrosion d) Fatigue and stress corrosion cracking e) Fatigue-corrosion and stress corrosion cracking 24

25 Consequence da dt d ρ dt a t = a ρ d ρ dt + a t Is a measure of the effect of the plastic deformation rate Is a measure of the effect corrosion Caution : this term is not expected to be a constant value 25

26 Session 7 Fatigue, Mechanisms and modeling, Mode I 1. Fatigue crack growth mechanisms 2. History effects in fatigue 3. Modelling 4. Applications 26

27 Overload effect Forman, R.G., Kearney V.E., Engle R.M., 1967, Numerical analysis of crack propagation in cyclic loaded structures. Journal of basic Engineering, 1967, Vol 89, pp J. Willenborg, R.M. Engle et H.A. Wood Crack retardation model using an effective stress concept. AFFDL TM 71 1 FBR, Wheeler, O., 1972, Spectrum loading and crack growth, Journal of Basic Engineering 1972, 94, Elber, W., 1971, The significance of fatigue crack closure. ASTM STP 468, 1971, J. Schijve, Effect of load sequences on crack propagation under random and program loading. Engineering Fracture Mechanics, vol. 5 pp , Skorupa M, 1999, Load interaction effects during fatigue crack growth under variable amplitude loading a literature review. Part II: qualitative interpretation, Fat. Fract. of Engng. Mater. Struct., 22 (10), pp Etc. 27

28 Overload effect Crack length (mm) Constant amplitude fatigue idem + 1 OL (factor 1.5) idem + 1 OL (factor 1.8) Number of cycles CCT samples, 0.48% carbon steel, room temperature [Hamam et al. 2005] 28

29 Residual stresses, from FE analyses 1) material law 1, kinematics hardening 2) material law 2, isotropic hardening 29

30 Residual stresses, from FE analyses 1) material law 1, kinematics hardening 2) material law 2, isotropic hardening 30

31 Overload effect + Material hardening Crack length aol (mm) idem after 1 OL (factor 2) idem after 10 OL (factor 2) Constant amplitude fatigue Number of cycles CT samples, 316L austenitic stainless steel, room temperature [Pommier et al] 31

32 Material hardening : cyclic hardening 316 L 32

33 Underloads? CA fatigue after 5 underloads CA fatigue after 10 underloads 0.38% carbon steel : driven at constant ΔK 33

34 Material hardening : Bauschinger effect 316 L 34

35 Session 7 Fatigue, Mechanisms and modeling, Mode I 1. Fatigue crack growth mechanisms 2. History effects in fatigue 3. Modelling 4. Applications 35

36 Motivations The basic mechanisms are understood since the 80 s but not yet accounted for in models Stress Time Stress versus time at the extrados of an Airbus A320 36

37 Load spectra in train wheels z Bending + Normal Radial + Circumferential Stress (MPa) Number of wheel s revolutions 37

38 Huge number of cycles, multiaxial, non-uniform stresses Stress (MPa) Number of wheel s revolutions Paris-Marseille : wheel s revolutions 38

39 Variable and random σ rr (MPa) alignment curve curve 39

40 Context Finite element method Allows elaborate material constitutive behaviours Precise description of the stress and strain fields A huge mesh refinement is required (computation costs) Real cracks have complex shapes (implemention costs) Millions of cycles have to be simulated Analytical models & empirical approaches Fast and efficient computations Can be included in XFEM or Cohesize Zones models Rough hypotheses on the material behaviour Huge experimental for the identifications 40

41 Approach 1) We use a local approach (FE method) for its advantages It can be adapted to any material behaviour (Bauschinger ) It allows exploring various loading cases It allows reducing the number of experiments to perform Results are obtained at the local scale and have to be interpreted 3D non-linear Elastic plastic Fatigue crack growth FE simulations are long and fastidious!!! 41

42 Approach 1) We use a local approach for its advantages 2) Local FE results are then brought to the global scale we use a multiscale approach tailored for cracks and an automated post-processing routine (python script) to generate global cyclic elastic plastic curves at the global scale identification of an elastic-plastic constitutive model at the global scale empirical model (set of 10 partial derivative equations) 42

43 Approach 1) We use a local approach for its advantages 2) Local FE results are brought to the global scale 3) Fatigue crack growth simulation for complex missions are performed with the global model (no FE anymore) Are therefore cheap in computation costs and time And include plasticity induced history effects 43

44 Method d ε = dt f ( σ,... ) Constitutive model LOCAL Tensile Push pull test Expérimental input n 1 44

45 Nickel base super alloy (450 C) 1 non-linear kinematic hardening term γ=780 C= MPa R o =470 MPa Identification of the cyclic elastic-plastic constitutive model for the material at T = 450 C 45

46 Method d ε = dt f ( σ,... ) Constitutive model LOCAL Scale transition Generation of global evolutions Tensile Push pull test Expérimental input n 1 46

47 Scale transition FE Mesh with a crack u ( x,t ) EF 47

48 The velocity field is partitionned into modes Mode I symmetric PPV P*=F.v* Mode II anti-symmetric 48

49 Each mode is projected onto a basis of spatial fields Basis of spatial fields : Velocity field (for instance from a FE computation) Projection : Intensity factors (caution there are different from the nominal (LEFM) ones) : Condensed measure of plastic deformation rate around the crack tip 49

50 How the reference fields are built? «elastic» de reference fields : mode I or II elastic analyses with : 50

51 How the reference fields are built? Additionnal reference fields : Apply a monotonic loading for each mode up to : Extract the FE displacement field : Project it onto the elastic field : 51

52 How the reference fields are built? Determine the difference : Partition that difference as follows (this is an approximation done using the POD) : Non dimension it : ρi can alternatively be «read» as the CTOD 52

53 Extraction of stress intensity factors in any cases Let consider a velocity field (from EF of from DIC) : Project it onto the basis of reference field defined a priori 53

54 Extraction des facteurs d intensité 54

55 Example of results, a circular loading case 55

56 Errors Velocity field at each time step Approximation using only «elastic» fields 56

57 Errors Velocity field at each time step Approximation using only «elastic» fields & additionnal fields 57

58 Example of results Entre c et d 58

59 Example of results 59

60 u Example of results EF EF ( x,t ) u( x,t ) K ( t ) K ( t ) ~ ~ [ ] u ( x ) + ρ ( t ) ρ ( t ) n+ 1 n I n+ 1 I n e I n+ 1 [ ] u ( x ) I n c Measures plasticity at the global scale 60

61 To summarize d ε = dt f ( σ,... ) Constitutive model LOCAL Scale transition Evolution law GLOBAL Tensile Push pull test d ρ = dt g( K I,...) Expérimental input n 1 Constitutive model GLOBAL 61

62 To summarize d ε dt = f ( σ,... ) Constitutive model LOCAL Scale transition Expérimental input n 2 Evolution law GLOBAL Tensile Push pull test Expérimental input n 1 d ρ = dt da dρ =α dt dt g( K I,...) Fatigue crack growth experiment Crack growth model, including history effects, GLOBAL 62

63 da/dt : rate of production of cracked area per unit length of the crack front da dn ΔCTOD da dt dρ = α dt Adjust the coefficient α using one constant amplitude fatigue crack growth experiment da dt dρ = α dt 63

64 To summarize d ε dt = f ( σ,... ) Constitutive model LOCAL Scale transition Expérimental input n 2 Constitutive model GLOBAL Tensile Push pull test Expérimental input n 1 d ρ = dt da dρ =α dt dt g( K I,...) Fatigue crack growth experiment Crack growth model, including history effects, GLOBAL 64

65 Application: railway transportation Prevent fatigue failure Periodic inspections Possible origins of cracks -Ballast impacts -Metallurgical defects -Corrosion pits ICE, Germany PhD thesis of Rami Hamam 65 65

66 Loading conditions Fatigue crack propagation model variable amplitude fatigue avoid cycle counting propagation lives > 10 7 steel 66

67 Single overload : long range retardation 67

68 Block loading : short range retardation 68

69 Stress ratio (mean stress) effect (R>0) 69

70 Stress ratio (mean stress) effect (R<0) X2 70

71 summary Identification Push pull test + FE analyses Fatigue crack growth experiment Validation Constant amplitude fatigue (R>0) or (R<0) Single overloads (residual stresses) Block loadings (material hardening) d ρ = dt da dρ =α dt dt g( K I,...) The global model can be used 71

72 Random loading Number of blocks 72