III Fatigue Models 1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow?
Outline Sources of knowledge Modeling Crack nucleation Non propagating cracks Crack growth AM 11/03 2
Source of knowledge The intrinsic fatigue properties of the component s material are determined using small, smooth or cracked specimens. AM 11/03 3
Laboratory fatigue test specimens Smooth specimens Crack growth specimen This information can be used to predict, to compute the answer to. AM 11/03 4
Outline Sources of knowledge Modeling Crack nucleation Non propagating cracks Crack growth AM 11/03 5
The three BIG fatigue questions 1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow? AM 11/03 6
Crack nucleation 1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow? AM 11/03 7
Cyclic behavior of materials? ε monotonic cyclic Bauschinger effect: monotonic and cyclic stressstrain different! AM 11/03 8
Cyclic Deformation Hysteresis loops? ε AM 11/03 9
MODEL: Cyclic stress-strain curve? ε σ Cyclic stressstrain curve ε = σ E + σ K' 1/n' ε ε = σ E +2 σ 2K' 1/n' Hysteresis loops for different levels of applied strain AM 11/03 10
MODEL: Cyclic stress-strain curve ε = σ E + σ K' 1/n' ε = σ E + σ K 1/n AM 11/03 11
MODEL USES K t σ, ε S(t) Simulate local cyclic stress-strain behavior at a notch root. May need to include cyclic hardening and softening behavior and mean stress relaxation effects. Notched component AM 11/03 12
Planar and wavy slip materials Wavy slip materials Planar slip materials AM 11/03 13
Concept of fatigue damage 1954 - Coffin and Manson Plastic strains cause the accumulation of fatigue damage. AM 11/03 14
MODEL: Strain-life curve.1.01 ε ' f c ²e total strain ampltude elastic strain ampltude plastic strain ampltude Fatigue test data from straincontrolled tests on smooth specimens..001 σ' f E b Elastic component of strain,? ε e.0001 10 3 10 4 10 5 Reversals (2Nf) 10 6 Plastic component of strain,? ε p ε t = ε e + ε p = σ' f E ( 2N f ) b + ε' f 2N f ( ) c AM 11/03 15
Transition Fatigue Life 2N tr = ε f ' E σ f ' 1 b c.1.01 ε ' f c ²e total strain ampltude elastic strain ampltude plastic strain ampltude.001 σ' f E b.0001 10 3 10 4 10 5 Reversals (2Nf) 10 6 Transition fatigue life, N tr Elastic strains = plastic strains AM 11/03 16
100 10 1 10 4 MODEL: Linear cumulative damage n i N i 10 5 10 6 Fatigue Life ( N, cycles) 2,000,000 cycles Endurance Limit 10 7 Miner's Rule of Cumulative Fatigue Damage is the best and simplest tool available to estimate the likelihood of fatigue failure under variable load histories. Miner's rule hypothesizes that fatigue failure will occur when the sum of the cycle ratios (n i /N i ) is equal to or greater than 1 n i 1 N.0 i i n i = Number of cycles experienced at a given stress or load level. N i = Expected fatigue life at that stress or load level. AM 11/03 17
Mean stress effects Effects the growth of (small) fatigue cracks. Good! log? ε/2 Compressive mean stress Zero mean stress Tensile mean stress log 2N f Bad! AM 11/03 18
MODEL: Morrow s mean stress correction σ f /E log? ε/2 σ m /E Zero mean stress Tensile mean stress log 2N f ε 2 = σ ' f σ m E ( 2N f ) b ' +ε f ( 2N f ) c AM 11/03 19
MODEL: Mean stress effects 0.1 Morrow ε 2 = σ ' f σ m E ( 2N f ) b ' +ε f ( 2N f ) c SWT -200 MPa 0.01 Morrow -200 MPa 0.001 Morrow +200 MPa SWT +200MPa σ max ε 2 0.0001 1E+02 1E+03 1E+04 1E+05 1E+06 = σ ' 2 f E ( 2N f) 2b ' ' +σ f ε f ( 2N f ) b +c Cycles Smith-Watson-Topper AM 11/03 20
MODEL USES.1.01.001 ε ' f c 0.002 σ' f E b ²e total strain ampltude elastic strain ampltude plastic strain ampltude A strain gage mounted in a high stress region of a component indicates a cyclic strain of 0.002. The number of reversals to nucleate a crack (2N I ) is estimated to be 86,000 or 43,000 cycles..0001 10 3 10 4 10 5 86,000 reversals 10 6 Reversals (2Nf) ε t = ε e + ε p = σ' f E ( 2N f ) b + ε' f 2N f ( ) c AM 11/03 21
Outline Sources of knowledge Modelling Crack nucleation Non propagating cracks Crack growth AM 11/03 22
The three BIG fatigue questions 1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow? AM 11/03 23
Non-propagating cracks Sharp notches may nucleate cracks but the remote stress may not be large enough to allow the crack to leave the notch stress field. AM 11/03 24
Threshold Stress Intensity The forces driving a crack forward are related to the stress intensity factor K = Y S πa? S a At or below the threshold value of? K, the crack doesn t grow. AM 11/03 25
MODEL: Non-propagating cracks Endurance limit for smooth specimens Stress range,? S Non-propagating cracks Failure a th = 1 π K th S 2 Crack length AM 11/03 26
Tensile-Shear Spot Weld Groove Welded Butt Joint? S W K = Y 2 K = S πa P /B πa + S πa? K? P? K? P =? S W a a W 8 initial crack length Cycles, N P2 Cycles, N AM 11/03 27 P2
MODEL USES a a th = 1 π? S K th S 2 Will the defect of length a grow under the applied stress range?s? How sensitive should our NDT techniques be to ensure safety? AM 11/03 28
Outline Fatigue mechanisms Sources of knowledge Modelling Crack nucleation Non propagating cracks Crack growth AM 11/03 29
The three BIG fatigue questions 1. Will a crack nucleate? 2. Will it grow? 3. How fast will it grow? AM 11/03 30
How fast will it grow??? Measuring Crack Growth - Everything we know about crack growth begins here! Cracks grow faster as their length increases. AM 11/03 31
MODEL: Paris Power Law Growth of fatigue cracks is correlated with the range in stress intensity factor:? K. K = Y S πa da dn = C( K)n AM 11/03 32
Fatigue fracture surface Scanning electron microscope image - striations clearly visible Schematic drawing of a fatigue fracture surface AM 11/03 33
MODEL: Crack growth rate versus?k Crack growth rate (da/dn) is related to the crack tip stress field and is thus strongly correlated with the range of stress intensity factor: (? K=Y? Svpa). da dn = C( K)n Paris power law AM 11/03 34
Crack Growth Data All ferritic-pearlitic, that is, plain carbon steels behave about the same irrespective of strength or grain size! AM 11/03 35
Crack growth rates of metals K E The fatigue crack growth rates for Al and Ti are much more rapid than steel for a given? K. However, when normalized by Young s Modulus all metals exhibit about the same behavior. AM 11/03 36
MODEL: Mean stresses (R ratio) da dn = C K m 1 R ( )K c K Forman s equation which includes the effects of mean stress. da dn = C K m 1 R ( ) γ Walker s equation which includes the effect of mean stress. AM 11/03 37
PROBLEM: Variable load histories? While some applications are actually constant amplitude, many or most applications involve variable amplitude loads (VAL) or variable load histories (VLH). Aircraft load histories AM 11/03 38
FIX: Root mean cube method K = m ( K i ) m n i n i AM 11/03 39
PROBLEM: Overloads, under-loads? Overloads retard crack growth, under-loads accelerate crack growth. AM 11/03 40
PROBLEM: Periodic overloads? Effects of Periodic Overloads Infrequent overloads help, but more frequent may be even better. Too frequent overloads very damaging. AM 11/03 41
FIX: MODEL for crack tip stress fields Monotonic plastic zone size Cyclic plastic zone size r y = 1 βπ K σ y ß is 2 for (a) plane stress and 6 for (b) AM 11/03 42
MODEL: Overload plastic zone size Crack growth retardation ceases when the plastic zone of ith cycle reaches the boundary of the prior overload plastic zone. r y = 1 βπ K σ y ß is 2 for plane stress and 6 for plain strain. AM 11/03 43
MODEL: Retarded crack growth Wheeler model da dn VA = β da dn CA β = 2 r y a p a i k r y = 1 2 π K σ ys 2 AM 11/03 44
Retarded crack growth Effects of Periodic Overloads Infrequent overloads help, but more frequent may be even better. Too frequent overloads very damaging. AM 11/03 45
PROBLEM: Periodic overloads? Effects of Periodic Overloads Infrequent overloads help, but more frequent may be even better. Too frequent overloads very damaging. AM 11/03 46
Sequence Effects Last one controls! Compressive overloads accelerate cdrack growth by reducing roughness of fracture surfaces. Compressive followed by tensile overloads retardation! Tensile followed by compressive, little retardation AM 11/03 47
Sequence effects Overloads retard crack growth, under-loads accelerate crack growth. AM 11/03 48
PROBLEMS: Small Crack Growth? Small cracks don t behave like long cracks! Why? AM 11/03 49
PROBLEMS: Environment? A. Dissolution of crack tip. A B B. Dissolution plus H+ acceleration. C. H+ acceleration C D. Corrosion products may retard crack growth at low? K. AM 11/03 50 D
FIX: Crack closure Crack open S Remote Stress, S S = S max Time, t S max S open, S close S S = 0 Crack closed Remote Stress, S S max S open, S close Time, t Plastic wake New plastic deformation AM 11/03 51
CAUSE: Plastic Wake Short cracks can t do this so they behave differently! AM 11/03 52
Effective part of load history S Damaging portion of loading history S S Opening load Nondamaging portion of loading hist S AM 11/03 53
MODEL: Crack closure da dn = C K ( )m K max ( ) p Extrinsic Intrinsic AM 11/03 54
Other sources of crack closure S max S open Roughness induced crack closure (RICC) (a) S max S open Oxide induced crack closure (OICC) (b) AM 11/03 55
MODEL: Crack closure? K eff = U? K U = K eff K = S max S open S max S min = 1 1 R 1 S open S max U 0.5 + 0.4R (sometimes) Plasticity induced crack closure (PICC) A, A', A'' Crack tip positions A A' A'' Plastic zones for crack positions A...A Initial crack length Plastic wake AM 11/03 56
MODEL USE:?K effective da dn = C( K)n The use of the effective stress intensity factor accounts for the effects of mean stress. da dn = C' ( K eff) n = C' ( U K) n AM 11/03 57