DAMAGE AND FATIGUE Continuum Damage Mechanics modeling for fatigue of materials and structures. Rodrigue Desmorat

Transcription

1 DAMAGE AND FATIGUE Continuum Damage Mechanics modeling for fatigue of materials and structures Rodrigue Desmorat LMT Cachan ENS Cachan, 61 av. du Pt Wilson Cachan Cedex Revue Européenne de Génie Civil, Vol. 10, n 6/7, pp , 2006

2 Fatigue issues Fatigue = failure under repeated (initially cyclic) loading 1-10 cycles: material behavior coupled with damage cycles: very low cycle fatigue cycles: low cycle fatigue cycles: high cycle fatigue >10 8 cycles: gigacycle fatigue 86400s/day so that 10 5 cycles at 1Hz takes around 1 day Objectives of the course Give background on Damage Mechanics applied to fatigue problems Give background to build tools able to handle complex loadings (3D, random, seismic, with temperature variations, with coupling with other physics for instance by use of poromechanics effective stress or by multiscale analyses) Thermodynamics framework should allow more finalized extension to geomaterials (rocks, soils ) Modeling still in progress in the Mechanical / Civil Engineering communities

3 Example: thermo-mechanical random fatigue 3D stresses Loading sequence made of 1000 points Temperature A thermo-hydraulic computation gives the temperature and stresses history. A DAMAGE computation must gives the damage D(t), the location of where damage is maximum (where a crack will initiate) and the time to mesocrack initiation.

4 Damage Mechanics becoming an engineering tool? A book on damage models on engineering applications: ductile, creep, fatigue, creep-fatigue and brittle failures on parameters identification on numerical topics on damage threshold on damage anisotropy on micro-defects closure effect (Springer 2005)

5 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 5

6 I- ELASTO-PLASTICITY / CONTINUUM DAMAGE MECHANICS!! E E " E E(1-D) " " p " e Damage = scalar variable D E D = 1! E

7 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 7

8 Plasticity 1D Strain partition! =! e +! p! plasticité f=0 Elasticity R+X! = E" e Criterion function!y élasticité f<0!y Hardening f =! " X " R "! y R = R(p) X = C! p " #X p ( ) Accumulated plastic strain p = "! p dt " p " e "

9 Thermodynamics framework Thermodynamics variables Thermodynamics potential!" = 1 2 (# $ #p ) : E : (# $ # p ) + G(p) State laws! = " #$ #% = E : (% & %p ) = E : % e (!" = # $% $& p R = " #$ Kp linear #p = G %(p) = * ) Kp 1/M power * + R & (1' e 'bp ) exponential

10 Thermodynamics framework Thermodynamics variables Thermodynamics potential!" = 1 2 (# $ #p ) : E : (# $ # p ) + G(p) State laws! = " #$ #% = E : (% & %p ) = E : % e (!" = # $% $& p R = " #$ Kp linear #p = G %(p) = * ) Kp 1/M power * + R & (1' e 'bp ) exponential Stored (blocked) energy density w s

11 Criterion function f =! eq " R "! y Dissipation potential F = f (associated model for single isotropic hardening) Evolution laws Determination of the plastic multiplier plasticity visco-plasticity! p = " #F #$ = " 3 2 $ D $ eq p =!" #F #R = " = 2 $ 3 p : $ p f = 0, f = 0! " f =! v,! v = K N p 1/ N " # = p = f K N N

12 Case of tension from von Mises plasticity! = "! 0 0% $ ' $ 0 0 0' # $ 0 0 0& '! D =! " 1 3 tr! 1 = # 2 3! 0 0 & % ( % 0 " 1 3! 0 ( $ % 0 0 " 1 3! '(! eq = 3 2!D :! D =!! p = p 3 " D = 2 " eq $ p 0 0 ' & ) & 0 # 1 2 p 0 ) %& 0 0 # 1 2 p ()! " p 11! = p = " p Plastic incompressibiliy Tension curve : # %! =! e +! p = " $ E + p &% " = R(p) + " y ' "(!)

13 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 13

14 Damage and effective stress concept S! = F S S D! = F S = Effective stress F S " S D = F ( ) S 1" S S D VER! =! 1" D! = E" e # D %! = E " & e ' E = E(1$ D) Principle of strain equivalence

15 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 15

16 Elasticity coupled with damage! endommagement Damage f=0 élasticité f<0 E E(1-D) " + D=D(Y) ou D=D(! ˆ )

17 Thermodynamics variables Thermodynamics potential State laws!" = 1 (1# D)$ : E : $ 2! = " #$ = E(1& D) : % #%!Y = " #$ #D! Y = 1 2 " : E : "

18 Marigo damage model Damage criterion function f = Y! "(D) Damage potential F = f (associated model) Damage evolution law D =! "F "Y = D =! "1 (Y Max ) = g(y Max )! détermined from the consistency condition f = 0, f = 0! " " $ Ex: g(y) = # $ % $ Y! Y D S Y! S s Y D

19 Mazars damage model Damage criterion function f = ˆ! " # ˆ! =! + :! + Critère de Mazars Critère de von Mises 2 1! 2! u ! 1! u " ^ (# = 0.2) " ^ (# = 0.3)! eq

20 Different damage evolution in tension and in compression D =! t D traction +! c D compression Damage evolution law in tension D traction = 1! " D(1! A t )! " ˆ Max A t exp B t (ˆ " Max! " D ) [ ] Damage evolution law in compression D compression = 1! " D(1! A c )! " ˆ Max A c exp B c (ˆ " Max! " D ) [ ]

21 Tension / compression for concrete 10 σ (MPa) ε E = E(1 D) Mazars model : 1 set of damage parameters for tension 1 set of damage parameters for compression

22 Anisotropic damage modeling Nooru-Mohamed test (1992) Desmorat, Gatuingt, Ragueneau ( ) classical mesh dependency Local FE Non local FE D 22 field D 22 field

23 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 23

24 Plasticity coupled with damage! plasticité Plasticity et and endomagement damage f=0 élasticité f<0 E E E(1-D) " " p " e + D =! # " Y S s $ & p si % ' p > p D ( ) w s > w D threshold in plastic strain threshold in stored energy

25 Thermodynamics variables Strain partition! =! e +! p Thermodynamics potential!" = 1 2 (# $ #p ) : E(1 $ D) : (# $ # p ) + G(r) Stored (blocked) energy density w s

26 State laws! = " #$ #% R =! "# "r = $!Y = " #$ #D = E(1& D) : %e ' Kr linéaire G (r) = ) ( Kr 1/ M ) * R % (1& e &br ) Strain energy density release rate Y = 1 2!e : E :! e = Triaxiality function puissance exp onentiel " 2 eq R # 2E(1 $ D) 2 = " 2 eqr # 2E $ R! = 2 3(1+!) + 3(1" 2!) # ' H & % # ) eq ( 2

27 Criterion function Dissipation potential (non associated model) Evolution laws Damage evolution law (Lemaitre) f =! eq " R "! y =! eq 1" D " R "! y F = f + F D F D = S (s +1)(1! D)! p = " #F #$ = " 1% D " $ # 3 2 Y S % ' & $ D $ eq s+1 r =!" #F #R = " = p (1! D) D =! "F "Y = # % Y $ S s & ( ' p

28 Determination of the plastic multiplier plasticity visco-plasticity f = 0, f = 0! " f =! v,! v = K N p 1/ N " p = f K N N Norton law Mesocrack initiation when D=D c Damage parameters (to be identified) ε pd, S, s, D c

29 Rupture in monotonic loading Accumulated plastic strain to rupture $ p R =! pd + 2ES ' & % " 2 u R ) # ( s D c Sensitivity analysis 5!p R p R p = S R!T X p TX + S R!" u p "u + S R!S S T X " u S + S p R!E E E +S p R!# # # +S p R!s s s + S!$ p R pd p $ pd + S R!D c Dc $ pd D c T X =! H! eq! u =! y + R " stress triaxiality ultimate stress

30 Stress triaxiality effect on p R "A high stress triaxiality makes materials brittle" p R! " pd " pr! " pd s # H # eq

31 Time to rupture in creep Initial Norton law p = #! eq & % $ K 0 N (1" D) ( ' N 0 D =! # " Y S s $ & p = %! # " ' 2 eq R ( 2ES(1) D) 2 $ & % s! ' eq $ # " K 0 N (1) D) & % N 0 Time to damage initiation Time to rupture # 0 K t D =! N pd % $ " eq & ( ' N 0 t R = t D + 1! (1! D c )2s+N 0+1 2s + N $ & % 2ES " 2 eq R # ' ) ( s $ & % 0 K N " eq ' ) ( N 0

32 Sensitivity analysis !t R t R 1.2 t = S R TX!T X T X t + S R "eq!" eq t + S R!N0 N " eq N + S t R K 0 N !K N t + S R!E E K N E + S t R!S S S +S t R!# # # t + S R! s s s + S!$ 0.1 t R pd t $ pd + S R Dc $ pd!d c D c

33 Partial conclusion for damage models and fatigue Previous elasticity coupled with damage models (with no plasticity) cannot reproduce neither material hysteresis nor fatigue damage Plasticity coupled with damage model suitable for low cycle fatigue of metals but difficulties encountered in damage threshold measurements (loading dependency) High temprerature fatigue of metals (creep-fatigue ) represented Absolute need of kinematic hardening in fatigue of metals (even if only briefly presented): Bauschinger effect Rate form constitutive equations possible for damage: facilities to handle 3D, non proportional loadings, temperature variations, coupling with other physics Still a lot to do for application to (more complex) geomaterials

34 II- AMPLITUDE DAMAGE LAWS Numbers of cycles N σ Δε p Δσ ε 34

35 II- AMPLITUDE DAMAGE LAWS Numbers of cycles N or σ Δσ=2σ M in case of symmetric loading 35

36 Damage from number of cycles measurement Engineering damage for fatigue with N Ri the number of cycles to rupture at strain level i Miner's linear damage accumulation rule Example on two level loading R. DESMORAT

37 Amplitude damage law in terms of stress "D "N = g(d)g # ($#,R # )! R " = " min " Max load ratio R " = #" M =1 symmetric loading " M!!

38 ! Amplitude damage law in terms of stress "D "N = g(d)g # ($#,R # ) Does a nonlinear g(d) function leads! to nonlinear damage accumulation? The answer is NO! R " = " min " Max load ratio R " = #" M =1 symmetric loading " M Integrate over each level i Sum over all the levels i

39 Amplitude damage law in terms of strains "D "N = g(d)g # ($%,R # )! Limitations Link between stress and strain amplitude laws not so clear, at least as long as no rate form damage law allows to recover both Non cyclic loading? Needs of cycles counting methods (rainflow ) Extension to 3D?

40 II- DAMAGE EVOLUTION LAWS FOR FATIGUE Lemaitre's law D = " $ Y # S % ' & s p Strain energy release rate density Y = 1 2 "e :E :" e = # eq 2 R $ 2E! Paas law! D # = Cg(D)" eq " eq Generalized damage law D = " $ Y # S % ' & s (

41 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 41

42 Lemaitre's damage law Damage enhanced by the stress level and the stress triaxiality Damage gouverned by plasticity Damage threshold D =! # " Y S s $ & p si % p > p D D = D c amorçage d une fissure Y = 1 2 "e :E :" e = # 2 eqr $ 2E $ R! = 2 3(1+!) + 3(1" 2!) # ' H & % # ) eq ( " H = 1 3 tr"! hydrostatic stress " eq Elastic strain energy Triaxiality function Stress triaxiality 2 von Mises stress

43 Lemaitre's damage law Damage strength Critical damage D = " $ Y # S Damage exponent s % ' & p if Accumulated plastic strain Damage threshold p > p D D = D c amorçage d une fissure! Y = 1 2 "e :E :" e = # 2 eqr $ 2E $ R! = 2 3(1+!) + 3(1" 2!) # ' H & % # ) eq ( " H = 1 3 tr"! hydrostatic stress " eq Elastic strain energy Triaxiality function 2 von Mises stress

44 Lemaitre's damage law Damage strength Damage exponent Stored energy damage threshold Critical damage D = " $ Y # S s % ' & p if w s > w D D = D c amorçage d une fissure! Y = 1 2 "e :E :" e = # 2 eqr $ 2E $ R! = 2 3(1+!) + 3(1" 2!) # ' H & % # ) eq ( " H = 1 3 tr"! hydrostatic stress " eq Elastic strain energy Triaxiality function 2 von Mises stress

45 ! Lemaitre's damage law in fatigue D = Y s " % $ S ' p # & Y S! " 2 eqmax 2ES 2 a 1/s =! u 2ES R # = a 1/ s $ " 2 eqmax 2 R " # u Maximum von Mises stress (symmetric loading) # D = a!% $ Ultimate stress " eqmax " u & ( ' 2s R )s p

46 ! Lemaitre's damage law in fatigue D = Y s " % $ S ' p # & Y S! " 2 eqmax 2ES 2 a 1/s =! u 2ES R # = a 1/ s $ " 2 eqmax 2 R " # u Maximum von Mises stress (symmetric loading) # D = a!% $ Ultimate stress " eqmax " u & ( ' 2s R )s p 1D symmetric fatigue loadings - no damage threshold + cyclic plasticity law!" =!"(!# p ) so that N R = (8ES)s K q cyc D c 2("#) 2s+q

47 Calculated Wöhler curve + cyclic plasticity law!" =!"(!# p ) 1000! Max! 500! M1=450 [ #$p1=0.027 [ #$p2! M2=340 = #$ p $! Max (MPa) Experiments Model as identified 200! f = 220 MPa! f " =180 MPa cycles N R

48 Lemaitre's damage law Stored energy damage threshold D = " $ Y # S s % ' & p if w s > w D amorçage d une fissure! Y = 1 2 "e :E :" e = # 2 eqr $ 2E $ R! = 2 3(1+!) + 3(1" 2!) # ' H & % # ) eq ( " H = 1 3 tr"! hydrostatic stress " eq Elastic strain energy Triaxiality function 2 von Mises stress

49 Damage threshold in terms of stored energy Monotonic loading D=0 as long as! p <! pd! pd " for metals Damage threshold In tension Fatigue loading D=0 as long as N<N D ou p<p D N D!D!N = N D 2"p = p D! p D Damage threshold in fatigue has large values in fatigue depends on the loading

50 Classical thermodynamics : variables R and p p w s =! R(p)dp = 0( p " 0! eq # " y )dp! stored energy w s w s Classical thermodynamics! y Experiments or correction: variables Q and q " p Correction : variables Q and q w s = p! R(p)z(p)dp =! Q(q)dq 0 p 0 z(p) = A m p1!m m

51 The stored energy depends on the choice on thermodynamics variables Unchanged hardening law: Q(q)=R(p) dq=z(p)dp! variables R and r 0 w s " w s 0 variables Q and q A=0.05, m= p 0! w s "

52 Monotonic loading Fatigue loading w s = A(! u "! y )# p 1/ m w s = A(! eq Max "! y )p 1/ m Damage threshold in stored energy 1/ w s = w D = A(! u "! y )# m pd $ p = p D monotonic p D = " pd creep p D = " pd fatigue % # p D = " u $ # y pd ' &# Max eq $ # y ( * ) m!

53 Modeling a loading dependent damage threshold More acurate case with kinematic hardening

54 Number of cycles to rupture in fatigue N R = N D + D c 2!p $ & % 2ES 2 " eqmax R # ' ) ( s N D =! pd 2"p % # u $ # ( y ' & # eqmax $ # * y ) m Sensitivity analysis !N R N = S R!"p N "p R "p + S N R!T X T X T X + S # N R eqmax!# # eqmax eqmax !# N + S R y N #y + S R!E E # y E + S N R!S S S S s N R! s s N +S R!# u N #u + S R!$ $ # u $ + S N R!m m m + S!% 0.5 N R pd N % pd + S R!D c Dc % pd D c

55 Two level fatigue loading Loading dependency of the ratio N D /N R Nonlinear damage accumulation ! "! 2 "! 1 t "! 1 =0.01 "! 2 = n 1 n 2 0.6!"=0.01 n N D N R !"=0.016 N R2 0.4 "! 1 =0.016 "! 2 =0.01! "! 1 "! 2 t N R 0 0 n 1 n n 1 N R1 (computations performed with ZeBuLon Finite Element code)

56 Non linear creep-fatigue interaction 1! "t t!m = 220MPa! M = 200MPa! M = 180MPa N R / N R F Linear interaction law 0 0 c t R / t R c 1 Computations without damage threshold Computations with damage threshold and kinematic hardening N R F N + t R c R t =1 R N R F N + t R c R t < 1 R

57 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 57

58 Paas approach for fatigue Elasticity coupled with damage following one of the laws Paas damage law Peerlings damage law From the time integration over one cycle

59 Generalized damage law Damage governed by the main disspative mechanism through the introduction of a cumulative measure of the irreversibilties π D =! # " Y S s $ & % ' The previous laws can be rewritten in this form as

60 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 60

61 Rocks or soils - Non incompressible plasticity Hydrostatic irreversible strain Deviatoric irreversible strain rate Equivalent (von Mises) irreversible shear strain Damage evolution law?????

62 Rocks or soils - Non incompressible plasticity Hydrostatic irreversible strain Deviatoric irreversible strain rate Equivalent (von Mises) irreversible shear strain 2 possible extensions of Lemaitre's law (a) # D = Y & % ( $ S " ' s " p (b) # D = Y & % ( $ S " ' s p ) "!!

63 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 63

64 Quasi-unilateral conditions Physical mecanism microcracks and microcavities partially closed in compression ONE state of microcracking = ONE damage variable QUASI-UNILATERAL CONDITIONS Mechanical behavior h=1 σ σ = 1 hd σ h=0.2 Elasticity different in tension and in compression Evolution of damage slower in compression than in tension E Ec Et ε

65 State potential introducing the micro-defects closure parameter Postive part in terms of principal stresses " = " # #" Energy equivalence! no possibility Strain equivalence Key : Gibbs potential can be continuously differentiated Isotropic damage (Ladevèze & Lemaitre, 1984)!" = 1+ # 2E & (' $ 2 1 % D + %$ 2 ) 1% hd* + % # & E ( ' 2 2 3$ H 1% D + %3$ H 1% hd ) + * h: micro-defects closure parameter

66 Elasticity law and damage thermodynamics force State laws

67 Mean stress effect Uniaxial case

68 Mean stress effect Uniaxial case For a non symmetric loading with as load ratio R " = " min " Max! For a given stress amplitude, a larger load ratio (more time spent in tension) gives a lower number of cycles to rupture (feature usually represented as straight lines in Goodman and Haigh diagrams)

69 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 69

70 Damage from post-processing of Finite Element results After an elastic computation (Neuber correction) After an elasto-(visco-)plastic computation Uncoupled approach Damage evolution law (Lemaitre) D =! # " Y S D = D c s $ & p si % Damage gouverned by plasticity p > p D Mesocrack initiation Elastic energy Triaxiality function Y = 1 2!e : E :! e = " 2 eqr # 2E $ R! = 2 3(1+!) + 3(1" 2!) # ' H & % # ) eq ( 2 Stress triaxiality

71 Damage by time integration of the evolution law! eq (t),! H (t) = 1 3 tr!(t) p(t) Y(t) =! 2 eq (t)r " (t) 2E which are computed in elasto(-visco-)plasticity Time to damage initiation: p(t D ) = p D! t D Damage calculation: Time to rupture : D(t) = t! D dt = t D t! t D " Y(t) % $ ' # S & s p (t)dt D(t R ) = D c! t R

72 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 72

73 Jump-in-cycles for periodic by blocks loadings IDEA: Before damage growth, run the computation until a stabilized cycle is reached Assume a linear variation of the damage (with respect to N) accumulated internal sliding (plastic strain) over a cycle

74 Once damage has started, calculate Number of cycles to be jumped Divide by the computation time

75 IV- TOWARD AN UNIFIED APPROACH FOR DAMAGE AND FATIGUE?

76 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 76

77 Generalized damage law Thermodynamics variables Thermodynamics potential!" = (1# D) [ w 1 ($) + w 2 ($ # $ % )] + w s (q,a) Criterion fonctiun f =!" 1 # D # x # Q #! s

78 Generalized damage law Thermodynamics variables Thermodynamics potential [ ] + w s (q,a)!" = (1# D) w 1 ($) + w 2 ($ # $ % ) Stored (blocked) energy density Criterion fonctiun f =!" 1 # D # x # Q #! s

79 Dissipation potential (non associated model) F = f + F D F D = S (s +1)(1! D) " $ # Y S % ' & s+1 Evolution laws (normality rule) Cumulative measure of the internal sliding! = # "! dt Generalized damage evolution law D =! # " Y S s $ & % '

80 Outline I- Elasto-plasticity / Continuum Damage Mechanics 1. Plasticity in thermodynamics framework 2. Damage and effective stress concept 3. Elasticity coupled with damage 4. Von Mises plasticity coupled with damage II- Amplitude damage laws III- Damage evolution laws for fatigue 1. Lemaitre's damage law 2. Quasi-brittle materials 3. Rocks or soils 4. Micro-defects closure effect - Mean stress effect 5. Damage post-processing 6. Jump-in-cycles procedure IV- Toward an unified approach for damage and fatigue? 1. Basis of a thermodynamics framework 2. Application to metals, concrete, elastomers and rocks V- High Cycle Fatigue 80

81 Damage and fatigue of concrete Hysteretic response in compression Calculated fatigue curve from time integration of the generalized damage law Aas-Jackobsen formula

82 Damage and fatigue of elastomers E : Green Lagrange strain tensor S : 2nd Piola-Kirchhoff stress tensor ' = D = ( E ' dt! # " Y $ & S% s '

83 Drucker-Prager plasticity coupled with damage recovers laws (a) and (b) with the relationship

84 IV- HIGH CYCLE FATIGUE Mesoscopic RVE behavior remains elastic Damage by post-processing elastic FE computations DAMAGE_2005 post-processor 84

85 Two scale damage model Initial 3D thermoélastic computation: ε ij (t) ou σ ij (t), T(t) Scale transition law: Eshelby-Kröner law with thermal expansion Plasticity and damage at microscale D(t) 4

86 Localization law for thermomechanical loading (basis) µ ~ 1 µ µ L Real problem: with: E = E(1! D)! = E # :" +! ~ Initial Eshelby problem: µ = E # 1 : µ! " +! µ L* Deviatoric part: Hydrostatic part: D LD D µ µ * " µ LD µld µ p! = +! 1# D 2G D L D µ µ * " H! = +! µ H 1# D 3K L with: with: #! =! µ L = " µ! T 6

87 Localization law for thermomechanical loading Deviatoric part: Hydrostatic part:! 1+! a = 3(1 "! ) " µd 1 = 1# bd "D # b((1# D)" µp #" p ) 1 " µ H = 1# ad " # a (1# D)$ µ H [ #$]%T ( ) ( ) 2(4 " 5! ) 15(1 "! )! " µ 1 $ (a # b)d = " + 1# bd 3(1# ad) " 1+ b((1# ' & kk D)"µp #" p ) % ( ) + a (1# D)* µ #* 1# ad b = [ ] +T1! recovers the law proposed by Sauzay and Desmorat (2000) for isothermal cases Thermal effect if: D = 0 and α µ α D 0 even if α µ = α 7

88 Constitutive equations at microscale Thermo-elasto-plasticity coupled with damage with linear kinematic hardening 12

89 Material parameters identification Parameters at RVE mesoscale Parameters at microscale E, υ, α, C y, σ f S, s, h = 0.2, D c =0.3 Plastic modulus Asymptotic fatigue limit Damage parameters (Lemaitre's law) 2 exp. curves necessary per temperature one tension curve (with plasticity) one Wöhler curve 8

90 Example of identification for 2 temperatures E, υ, α, C y, σ f S, s, h = 0.2, D c =0.3 Parameters at mesoscale Damage Model with difference tension/compression (h=0.2, s=4) Model with difference tension/compression (h=0.2, s=3) 1000 σ Max σ Max N R Low T N R Higher T 8

91 Characteristic effects reproduced Nonlinear damage accumulation Mean stress effect Effect in trension-compression, no effect in shear Biaxial effects Thermal and thermomechanical fatigue Fatigue of structures (3D model) Complex, non proportional or random loading (rate form model) 3

92 Out of phase 3D random thermomechanical fatigue 3D stresses Loading sequence made of 1000 points FATHER structure Temperature A thermo-hydraulic computation gives the temperature and stresses history. DAMAGE post-processor gives D(t) and the time to mesocrack initiation. Here around 200 h in accordance with the observations of micro-cracks initiation

93 FATHER results over a cycle and in terms of crack initiation FATHER first cycle at the most loaded point D p time RESULTATS FATHER 11-déc Time to crack initiation computed with DAMAGE_2005 point NR (amorçage) temps en h C11_m50i ,0 C11_m60i ,0 C11_m70i ,7 C11_m80i ,4 C11_m90i ,8 C14_m50i >1E6 >2780 C14_m60i ,2 C14_m70i >1E6 >2780 C21_m60i >1E6 >2780 C24_m70i >1E6 >2780 paramètres du matériau non poii Initiation observed bewteen 200h and 300 h time

94 CONCLUSION Continuum Damage Mechanics allows for the estimation of the crack initiation conditions in fatigue Post-processing approaches efficient Rate form of damage laws allows to handle complex loadings Anisothermal conditions naturally taken into account Rate form damage laws will be also helfull for coupling with other physics (THM, diffusion problems ) Coupling with non associated plasticity possible by use of the (damage) effective stress concept Many materials, many applications concerned Still a lot to do!

95 References "Mechanics of solid materials", J. Lemaitre and J.L. Chaboche, Oxford University Press, 1991 (in english), Dunod, 1985 (in french) "Modélisation et estimation rapide de la plasticité et de l endommagement", R. Desmorat, Habilitation à Diriger des Recherches de l'université Pierre et Marie Curie, "Two scale damage model for quasi-brittle and fatigue damage", R. Desmorat, J. Lemaitre, Handbook of Materials Behavior Models, chapter Continuous Damage, section 6.15, p , "Thermodynamics modelling of internal friction and hysteresis of elastomers.", S. Cantournet & R. Desmorat, C. R. Mécanique, 331, p , "Phenomenological constitutive damage models", R. Desmorat, chapter VII of the book «Local Approach to Fracture», CNRS Summer School MEALOR 2004, Ed. J. Besson, Presses de l Ecole des Mines de Paris, "Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures", J. Lemaitre et R. Desmorat, Springer, "Continuum Damage Mechanics for hysteresis and fatigue of quasi-brittle materials and structures", R. Desmorat, F. Ragueneau, H. Pham, International Journal of Numerical and Analytical Methods for Geomaterials, in press "Damage and fatigue: Continuum Damage Mechanics modeling for fatigue of materials and structures", R. Desmorat, Revue Européenne de Génie Civil, vol 10, p , 2006.