Stress Concentration. Professor Darrell F. Socie Darrell Socie, All Rights Reserved

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1 Stress Concentration Professor Darrell F. Socie Darrell Socie, All Rights Reserved

2 Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and K f 6. Small Crack Growth 7. Small Notches Stress Concentration Darrell Socie, All Rights Reserved 1 of 63

3 Stress Concentration Factor local = applied 1+ a ρ Inglis Solution 1910 ρ a Applied stress Local stress Stress Concentration Darrell Socie, All Rights Reserved of 63

4 Circular Notch λ r θ τ rθ θ r r a τ rθ λ θ Stress Concentration Darrell Socie, All Rights Reserved 3 of 63

5 Stress Concentration Darrell Socie, All Rights Reserved 4 of 63 Stresses θ + λ + +λ = cos a r a r a r r θ + λ + +λ = θ cos a r a r θ + λ = τ θ sin a r a r r Independent of size, dependant only on r/a

6 Stress Distribution 3 For tension λ = 0 and θ = 90 λ r θ θ a stress λ 1 r a 3 4 r Stress Concentration Darrell Socie, All Rights Reserved 5 of 63

7 Stress Ratio Effects λ 4 3 λ = 1 1 λ = 0 λ θ Angle λ = -1-4 stresses around the circumference of a hole Stress Concentration Darrell Socie, All Rights Reserved 6 of 63

8 Elliptical Notches stress K T K T a b = 1 + ρ= ρ a K T Sharp Notch: high K T high gradient b Blunt Notch: low K T low gradient a Stress Concentration Darrell Socie, All Rights Reserved 7 of 63

9 Stress Concentration in a Bar 5 4 o 3 1 Axial, z Tangential, θ Radial, r r ρ Stress Concentration Darrell Socie, All Rights Reserved 8 of 63

10 Bridgeman Analysis (1943) r a z r ρ ys z r a r ρ ys Elastic stress distribution Plastic stress distribution τ= z r = cons tant Stress Concentration Darrell Socie, All Rights Reserved 9 of 63

11 Stress Concentration Darrell Socie, All Rights Reserved 10 of 63 Stresses ρ ρ + + = a r a a ln 1 o z π = a 0 z z dr r P ρ + ρ + = π a 1 ln a 1 a P flow max P max = A net flow CF CF constraint factor

12 Constraint Factors a /ρ CF P max = A net flow CF Stress Concentration Darrell Socie, All Rights Reserved 11 of 63

13 Effect of Constraint Stress, z Increasing constraint Strain, ε z Higher strength and lower ductility Stress Concentration Darrell Socie, All Rights Reserved 1 of 63

14 Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and K f 6. Small Crack Growth 7. Small Notches Stress Concentration Darrell Socie, All Rights Reserved 13 of 63

15 Stress Concentration Darrell Socie, All Rights Reserved 14 of 63 Notch Rules Neuber Glinka Seeger n 1 t K E Se K + = ε = n 1 t K n 1 1 E d Se K + + = ε = S K K S * K S K K E e S K p t y t Limit p n 1 * * p = = + = ε =

16 Define K and K ε after Yielding S, e, ε Define: nominal stress, S nominal strain, e notch stress, notch strain, ε Stress concentration Strain concentration K K ε = S ε = e Stress Concentration Darrell Socie, All Rights Reserved 15 of 63

17 K and K ε K t S Stress (MPa) ε K K ε = = S ε e K t e Strain Stress Concentration Darrell Socie, All Rights Reserved 16 of 63

18 Stress and Strain Concentration Stress/Strain Concentration First yielding K K t ε K t K 1 Nominal Stress Stress Concentration Darrell Socie, All Rights Reserved 17 of 63

19 Neuber s Rule K t S Stress (MPa) ε K t S K t Actual stress e = ε Stress calculated with elastic assumptions K t e Strain Stress Concentration Darrell Socie, All Rights Reserved 18 of 63

20 Neuber s Rule for Fatigue Stress and strain amplitudes Elastic nominal stress Substitute and rearrange K t S K t e K t e S = = = S E E ε ε The product of stress times strain controls fatigue life Stress Concentration Darrell Socie, All Rights Reserved 19 of 63

21 SN Materials Data Stress Amplitude, MPa steels 17 aluminums Fatigue Life, Reversals Stress Concentration Darrell Socie, All Rights Reserved 0 of 63

22 εn Materials Data Strain Ampliture steels 17 aluminums Fatigue Life, Reversals Stress Concentration Darrell Socie, All Rights Reserved 1 of 63

23 10 5 E ε ε E or S 10 K t Fatigue Life, Reversals Stress Concentration Darrell Socie, All Rights Reserved of 63

24 Glinka s Rule K t S Add in the missing strain energy Stress (MPa) ε K t S K t e = dε K t e Strain Stress Concentration Darrell Socie, All Rights Reserved 3 of 63

25 Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and K f 6. Small Crack Growth 7. Small Notches Stress Concentration Darrell Socie, All Rights Reserved 4 of 63

26 A Dilemma Stress analysis and stress concentration factors are independent of size and are related only to the ratio of the geometric dimensions to the loads Fatigue is a size dependant phenomena How do you put the two together? Stress Concentration Darrell Socie, All Rights Reserved 5 of 63

27 Similitude Stress Concentration Darrell Socie, All Rights Reserved 6 of 63

28 Fatigue of Notches From Dowling, Mechanical Behavior of Materials, 1999 Stress Concentration Darrell Socie, All Rights Reserved 7 of 63

29 Notch Size K t K f K t K f Large Notch Small Notch Stress Concentration Darrell Socie, All Rights Reserved 8 of 63

30 Microstructure Size K t K f K t Kf Low Strength High Strength Stress Concentration Darrell Socie, All Rights Reserved 9 of 63

31 Stress Gradient K t K t K f K f Low K t High K t Stress Concentration Darrell Socie, All Rights Reserved 30 of 63

32 K t vs K f 10 8 K f = K t K f 6 4 Experiments K t Stress Concentration Darrell Socie, All Rights Reserved 31 of 63

33 K f Peterson s Equation 070MPa α = 0.05 u K f 1.8 K t 1 = 1 + α 1+ ρ mm α ρ No effect when ρ << α Full effect when ρ >> α Stress Concentration Darrell Socie, All Rights Reserved 3 of 63

34 Pererson s Constant α, mm Ultimate Strength, MPa Stress Concentration Darrell Socie, All Rights Reserved 33 of 63

35 Static Strength hole K t =.5 slot K t = 5 diamond K t = 0 edge K t = 0 Stress Concentration Darrell Socie, All Rights Reserved 34 of 63

36 1018 Steel Test Data load, kn edge diamond slot hole displacement, mm Stress Concentration Darrell Socie, All Rights Reserved 35 of 63

37 Notched SN Curve Stress Amplitude, MPa 1000 Notched specimen Smooth specimen data K f Cycles Stress concentrations are not very important at short lives Stress Concentration Darrell Socie, All Rights Reserved 36 of 63

38 Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and K f 6. Small Crack Growth 7. Small Notches Stress Concentration Darrell Socie, All Rights Reserved 37 of 63

39 Smith - Miller Long crack l > 0.13 Dρ K = πa Short crack l < 0.13 Dρ K = D ρ πl K fatigue Stress Concentration Darrell Socie, All Rights Reserved 38 of 63

40 Cracks at Notches S K t S S D a a D + a a << D K K t S πa a >> D = K = S π( D+ a ) Stress Concentration Darrell Socie, All Rights Reserved 39 of 63

41 Stress Intensity Factors 3.0 K = K ts πa S K D ( D a ) K = S π a D Stress Concentration Darrell Socie, All Rights Reserved 40 of 63

42 Cracks at Holes Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack Stress Concentration Darrell Socie, All Rights Reserved 41 of 63

43 Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and K f 6. Small Crack Growth 7. Small Notches Stress Concentration Darrell Socie, All Rights Reserved 4 of 63

44 Crack Growth Data 10-6 Crack Growth Rate, m/cycle da dn K c = C K m Nonpropagating cracks K TH > 1.1 πa π K,MPa m K TH Stress Concentration Darrell Socie, All Rights Reserved 43 of 63

45 Frost Data Rotating bending Notched bar Notched plate nucleation fracture S nominal S fatigue limit nonpropagating cracks K t K t Frost, A Relation Between the Critical Alternating Propagation Stress and Crack Length for Mild Steel Proceedings of the Institute for Mechanical Engineers, Vol. 173, No. 35, 1959, Stress Concentration Darrell Socie, All Rights Reserved 44 of 63

46 Significance For K t > 4, the notch acts like a crack with a depth D S fl = K th πd K t does not play a role for sharp notches! Stress Concentration Darrell Socie, All Rights Reserved 45 of 63

47 Specimens with Similar Geometry K t = 10.7 K t = Ultimate Strength 780 MPa Yield Strength 660 MPa Stress Concentration Darrell Socie, All Rights Reserved 46 of 63

48 Test Results Nominal Stress Amplitude Strength Limited Crack Growth Dominated K t = 10.7 K t =.4 Fatigue Strength Dominated Threshold Stress Intensity Dominated Total Fatigue Life, Cycles Stress Concentration Darrell Socie, All Rights Reserved 47 of 63

49 Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and K f 6. Small Crack Growth 7. Small Notches Stress Concentration Darrell Socie, All Rights Reserved 48 of 63

50 Small Crack Growth Stress Concentration Darrell Socie, All Rights Reserved 49 of 63

51 Threshold Stress Concentration Darrell Socie, All Rights Reserved 50 of 63

52 Normalized Thresholds Stress Concentration Darrell Socie, All Rights Reserved 51 of 63

53 Growth from Notches Stress Concentration Darrell Socie, All Rights Reserved 5 of 63

54 Cracks at Notches notch plastic zone notch stress field D a crack tip plastic zone Stress Concentration Darrell Socie, All Rights Reserved 53 of 63

55 Crack Growth Crack growth rate Short cracks Notch plasticity controls Long cracks Crack tip plasticity controls K or a Stress Concentration Darrell Socie, All Rights Reserved 54 of 63

56 Closure Observations a b b ε c d a c 1 st subcycle d 500 th subcycle 106 steel ε 1 / = ε / = Stress Concentration Darrell Socie, All Rights Reserved 55 of 63

57 Closure Correlation Stress Concentration Darrell Socie, All Rights Reserved 56 of 63

58 Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and K f 6. Small Crack Growth 7. Small Notches Stress Concentration Darrell Socie, All Rights Reserved 57 of 63

59 area Stress Concentration Darrell Socie, All Rights Reserved 58 of 63

60 Small Notches Stress Concentration Darrell Socie, All Rights Reserved 59 of 63

61 Threshold Stress Intensity Stress Concentration Darrell Socie, All Rights Reserved 60 of 63

62 Hardness Corelation Stress Concentration Darrell Socie, All Rights Reserved 61 of 63

63 Flaw Sensitivity Stress Concentration Darrell Socie, All Rights Reserved 6 of 63

64 Fatigue Limit Stress Concentration Darrell Socie, All Rights Reserved 63 of 63

65 Stress Concentration

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