ONERA Fatigue Model. Z-set group. March 14, Mines ParisTech, CNRS UMR 7633 Centre des Matériaux BP 87, Evry cedex, France

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1 ONERA Fatigue Model Z-set group Mines ParisTech, CNRS UMR 7633 Centre des Matériaux BP 87, Evry cedex, France March 14, 2013

2 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

3 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

4 ONERA 4 / 56 Fatigue Model F1 : Uniaxial fatigue curves Woehler (N f, σ a = σ ) curve from symmetric fatigue tests (R = 1) 2 F2 : Loading ratio / mean-stress dependence Woehler curves with loading ratios R > 1, Haigh diagram F3 : Non-linear cumulation dependence of the loading path history on the damage results F4 : Temperature Effect F5 : Multiaxiality ONERA Fatigue Model 4/56

5 Selected 5 / 56 fatigue model: ONERA model (Chaboche) σ cycle period T σ max σ = σmax+σmin 2 σ a = σ 2 N f = 1 (β + 1) [1 α(σ a, σ)]» σa β M( σ) σ min with α = 1 a σa σ l ( σ) σ u σ max time N f = σ u σ max a (β + 1) σ a σ l ( σ)» σa β M( σ) Mean stress dependence: σ l ( σ) = σ l b 1 σ, M( σ) = M b 2 σ Coefficients : σ u, σ l0, β, M 0, b 1, b 2 ONERA Fatigue Model 5/56

6 F1: 6 / 56Calibration of symmetric fatigue results σ a = σ 2 σ u N f = [ σu σmax σa a (β+1) σa σl 0 M0 ] β increasing β decreasing M 0 σ l e+06 1e+08 1e+10 Nf Active coefficients on symmetric fatigue tests (R σ = 1) UTS : σ u fatigue limit : σ l0 slope : β position along N f axis : M 0 ONERA Fatigue Model 6/56

7 F1: 7 / 56Calibration of symmetric fatigue results demo reset open terminal ONERA Fatigue Model 7/56

8 F2: 8 / 56Mean stress dependence Haigh diagram σa = σ Woehler curve σa = σ 2 R = 1 Nf = 10 6 R = mean stress σ e+06 1e+08 1e+10 Nf b 1 : decrease of σ l depending on mean stress σ σ l0 σ l ( σ) = 1 + b 1 σ b 2 : translation to smaller N f values M 0 M( σ) = 1 + b 2 σ demo reset open terminal ONERA Fatigue Model 8/56

9 F2: 9 / 56Mean stress Effect demo reset open terminal ONERA Fatigue Model 9/56

10 F3: 10 / 56 Non-linear cumulation (1/3) For complex loading paths σ load sequence Linear cumulation (Miner s rule) is not conservative sequence effect: midly damaging cycles have a greater effect if they occur after strongly damaging one cycles with amplitude below the fatigue limit do have an effect if prior fatigue damage occurred Needs a multi-axial rainflow procedure (see tools) time ONERA Fatigue Model 10/56

11 F3: 11 / 56 Non-linear cumulation (2/3) Fatigue damage evolution equation: δd = h 1 (1 D) β+1i» α(σ β a, σ) σ a δn M( σ) (1 D) with α = 1 a σa σ l ( σ) σ u σ max integration of D between 0 and 1 N f = 1 (β + 1) [1 α(σ a, σ)]» σa β M( σ) ONERA Fatigue Model 11/56

12 F3: 12 / 56 Non-linear cumulation (3/3) σ D i 1 cycle i D i σ σ load sequence time integration between D i 1 and D i above the fatigue limit: σ a > σ l ( σ), α < 1 [ 1 (1 Di ) β+1] 1 α [ 1 (1 Di 1 ) β+1] 1 α = 1 N f ( σ, σ, σ max ) below: σ a σ l ( σ), α = 1 [ 1 (1 Di ) β+1 ] ln 1 (1 D i 1 ) β+1 ( ) σ a β = (β + 1) M( σ) ONERA Fatigue Model 12/56

13 F3: 13 / 56 Non-linear cumulation demo reset open terminal ONERA Fatigue Model 13/56

14 F4: 14 / 56 Temperature Effect N f = 1 S max a (β + 1) S 2 σ l( S) " # S β 2 where S(T ) = σ(t ) M( S) σ u Normalized coefficients : σ l0, M, b 1, b 2 demo reset open terminal ONERA Fatigue Model 14/56

15 F5: 15 / 56 Multiaxiality ONERA Fatigue Model 15/56

16 F5: 16 / 56 Multiaxiality Multiaxial stress path : P = σ i, i = 1, n Cycle critical quantities in the multiaxial case Maximum stress : { } σ max = max σ i I, σ i i I maximum principal stress of σ i P Mean [ stress : { σ = 1 i 2 tra(σ ) } { + min i tra(σ ) } ] i P max i P Amplitude : σ a = σ 2 = radius of the SEH for points in P (see tools) Loading ratio : R = σ σa σ+σ a Multiaxial properties close to the classical Sines criterion ONERA Fatigue Model 16/56

17 Fatigue 17 / 56 models summary Inputs required by the fatigue model: From the constitutive model used in FE analysis: cyclic hardening modeling (σ a ) mean stress relaxation modeling ( σ) Dedicated post-processing tools: calculation of the amplitude of a multiaxial stress path multiaxial rainflow algorithm ONERA Fatigue Model 17/56

18 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

19 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

20 Basic Tools 20/56 Multiaxial 20 / 56 loading path amplitude (1/2) solve the Smallest Enscribed Hypersphere problem (SEH) j n o ff i J(σ X ) min X R(X ) = max i generic geometrical problem with applications in many fields (pattern recognition, protein analysis, political science, ray-tracing...) still, efficient algorithms are fairly recent (Bend Gartner, "Fast and Robust Smallest Enclosing Balls" 1999)

21 Basic Tools 21/56 Multiaxial 21 / 56 loading path amplitude (2/2) Iteratively solve optimization problems SEH(S) on subsets S of P At each iteration, solution of SEH(S) is found by solving a linear system with size the number of points in S Fast convergence to the solution of SEH(P)

22 Basic Tools 22/56 SEH 22 / 56example Initialization: first point σ 1 in the loading path P σ 1

23 Basic Tools 23/56 SEH 23 / 56example Farthest of σ 1 is Fst(σ 1 ) = σ 2 σ 1 Fst(σ 1 ) = σ 2

24 Basic Tools 24/56 SEH 24 / 56example Solve SEH(σ 1, σ 2 ) : solution is H 1 with center X 1 σ 1 X 1 σ 2

25 Basic Tools 25/56 SEH 25 / 56example Farthest of X 1 is Fst(X 1 ) = σ 3 σ 1 Fst(X 1 ) = σ 3 X 1 σ 2

26 Basic Tools 26/56 SEH 26 / 56example Solve SEH(σ 1, σ 2, σ 3 ) : solution is H 2 with center X 2 σ 1 σ 3 X 2 σ 2

27 Basic Tools 27/56 SEH 27 / 56example Farthest of X 2 is Fst(X 2 ) = σ 4 σ 1 σ 3 Fst(X 2 ) = σ 4 X 2 σ 2

28 Basic Tools 28/56 SEH 28 / 56example Solve SEH(σ 1, σ 2, σ 3, σ 4 ) : solution is H 3 with center X 3 all points are in H 3, end σ 1 σ 4 X 3 σ 3 σ 2

29 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

30 Basic Tools 30/56 Multiaxial 30 / 56 rainflow algorithm (1/4) extract sub-cycles from a complex stress path for (non-)linear damage cumulation σ generalize the 1D rainflow method to multiaxial stress states (6 independent components) use the concept of active surface used in some plasticity models (Melnikov, Semenov) time

31 Basic Tools 31/56 Multiaxial 31 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B from O to A time

32 Basic Tools 32/56 Multiaxial 32 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B from O to A time

33 Basic Tools 33/56 Multiaxial 33 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B from O to A time

34 Basic Tools 34/56 Multiaxial 34 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B at point A time

35 Basic Tools 35/56 Multiaxial 35 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D B O time unloading from A to B

36 Basic Tools 36/56 Multiaxial 36 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D B O time unloading from A to B

37 Basic Tools 37/56 Multiaxial 37 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B at point B time

38 Basic Tools 38/56 Multiaxial 38 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B from B to C time

39 Basic Tools 39/56 Multiaxial 39 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B from B to C time

40 Basic Tools 40/56 Multiaxial 40 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D O B at point C time

41 Basic Tools 41/56 Multiaxial 41 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D B O time extra loading from C to D

42 Basic Tools 42/56 Multiaxial 42 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D B O time extra loading from C to D

43 Basic Tools 43/56 Multiaxial 43 / 56 rainflow algorithm (2/4) follow the stress path and recursively builds active surfaces included in one another a cycle is detected when the size of the current surface exceeds the previous one σ A C D B O time extra loading from C to D

44 Basic Tools 44/56 Multiaxial 44 / 56 rainflow algorithm (3/4) An active surface s is defined by: U s its closing point created when unloading occurs X s its center Update of center: X s(s i) intersection of the mediatrix of (U s 1, S i) with (X s 1, U s 1) X s 1 X s (S i ) U s 1 i S i 1 2 (U s 1 + S i )

45 Basic Tools 45/56 Multiaxial 45 / 56 rainflow algorithm (3/4) An active surface s is defined by: U s its closing point created when unloading occurs X s its center Update of center: X s(s i+1) intersection of the mediatrix of (U s 1, S i+1) with (X s 1, U s 1) X s 1 X s (S i+1 ) U s 1 S i S i (U s 1 + S i+1 )

46 Basic Tools 46/56 Multiaxial 46 / 56 rainflow algorithm (4/4) Initialization: X 0 = X SEH center of the SEH calculated for the whole loading path U 0 = 0 to avoid residuals reorganize the path to begin by the maximum point σ m such that: J(σ m X SEH) = max J(σ i X i P SEH) σ σ m σ σ m time time

47 Basic Tools 47/56 Multiaxial 47 / 56 rainflow algorithm (4/4) Initialization: X 0 = X SEH center of the SEH calculated for the whole loading path U 0 = 0 to avoid residuals reorganize the path to begin by the maximum point σ m such that: J(σ m X SEH) = max J(σ i X i P SEH) σ σ m σ σ m time time

48 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

49 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

50 **process 50 / 56 onera (multiaxial) rainflow sub-cycles extraction (nonlinear) cumulation **output_number... **material_file mat **process onera [*cycle beg1-end1/rep1... begn-endn/repn] [*preload beg1-end1/rep1... begn-endn/repn] *mode NLC_ONERA LC *fatigue fatigue_rainflow 2 [*creep creep 2] [*reverse 3] % material file ***post_processing_data **process onera a 0.1 % (0<a<1) ***return Zpost input commands 50/56

51 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

52 **process 52 / 56 fatigue_rainflow number of cycles to failure according to the ONERA fatigue model **material_file mat **process fatigue_rainflow *var sig [*mean_stress (standard variant)] [*mode with_a] [*normalized_coeff] [*reverse 3] % material file ***post_processing_data **process fatigue_rainflow sigma_u sigma_l 0.2 M 40.0 beta 2.0 [ a 0.1 ] [ b1 0.1 b2 0.1 sigma_n sigma_p 0.01 ] ***return Zpost input commands 52/56

53 Plan 1 ONERA Fatigue Model 2 Basic Tools Multiaxial stress amplitude (SEH) Multiaxial rainflow 3 Zpost input commands process onera process fatigue_rainflow Calibration scripts

54 Stress-controlled 54 / 56 script Input file (file stress): Command: *frequency 20.0 *temperature *material endo.mat *stress *load_factor -1.0 [*creep] [*delay] Zrun -zp stress_control.z7p -s ZLanguage.args stress Output (file stress.out): # nr dsig/2 dsig sig_max sig_min nf nc e e e e e e e... Use in SimOpt: demo terminal Zpost input commands 54/56

55 Strain-controlled 55 / 56 script Input file (file strain): Command: *frequency 20.0 *temperature *material endo.mat *behavior comportement.mat *strain 3.e-3 1.e-2 5.e-4 *load_factor -1.0 [*creep] [*delay] Zrun -zp strain_control.z7p -s ZLanguage.args strain Output (file strain.out): e e e e e e e... Use in SimOpt: demo terminal Zpost input commands 55/56

56 Goodman 56 / 56 diagram script Input file (file goodman): Command: *frequency 1. *target 1.e7 *temperature *material mat.mat *steady_stress [*creep] [*delay] Zrun -zp goodman.z7p -s ZLanguage.args goodman Output (file goodman.out): # ss sa nr e e e e e e Use in SimOpt: demo terminal Zpost input commands 56/56