Fatigue and Fracture
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1 Fatigue and Fracture Static Strength and Fracture Stress Concentration Factors Professor Darrell F. Socie Mechanical Science and Engineering University of Illinois Darrell Socie, All Rights Reserved
2 Static Strength and Fracture Stress Concentration Factors Fracture Mechanics Approximate Stress Intensity Factors Ductile vs. Brittle Fracture Static Strength and Fracture Darrell Socie, All Rights Reserved 1 of 106
3 Stress Concentration Load flow lines Static Strength and Fracture Darrell Socie, All Rights Reserved of 106
4 Circular Notch λ r θ τ rθ θ r r a τ rθ λ θ Static Strength and Fracture Darrell Socie, All Rights Reserved 3 of 106
5 Static Strength and Fracture Darrell Socie, All Rights Reserved 4 of 106 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 Static Strength and Fracture Darrell Socie, All Rights Reserved 5 of 106
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 Static Strength and Fracture Darrell Socie, All Rights Reserved 6 of 106
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 Static Strength and Fracture Darrell Socie, All Rights Reserved 7 of 106
9 Plane Stress Plane Strain thick plate plane strain z = 0 ε z = 0 thin plate plane stress z = 0 ε z = 0 Static Strength and Fracture Darrell Socie, All Rights Reserved 8 of 106
10 Thick or Thin? Plane strain Plane stress Static Strength and Fracture Darrell Socie, All Rights Reserved 9 of 106
11 Transverse Strains Longitudinal Tensile Strain Transverse Compression Strain ν = 0.5 Thickness 40 mm 0 mm 10 mm ν = mm y z x 5 D 100 Static Strength and Fracture Darrell Socie, All Rights Reserved 10 of 106
12 Notch Stresses y z x t ε x ε z x z Static Strength and Fracture Darrell Socie, All Rights Reserved 11 of 106
13 Fracture Surfaces Static Strength and Fracture Darrell Socie, All Rights Reserved 1 of 106
14 Fracture Surfaces Static Strength and Fracture Darrell Socie, All Rights Reserved 13 of 106
15 Stress or Strain Control? elastic material plastic zone Elastic material surrounding the plastic zone forces the displacements to be compatible, I.e. no gaps form in the structure. Boundary conditions acting on the plastic zone boundary are displacements. Strains are the first derivative of displacement Static Strength and Fracture Darrell Socie, All Rights Reserved 14 of 106
16 Define K and K ε 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 Static Strength and Fracture Darrell Socie, All Rights Reserved 15 of 106
17 K and K ε K T S Stress (MPa) ε K K ε = = S ε e K T e Strain Static Strength and Fracture Darrell Socie, All Rights Reserved 16 of 106
18 Stress and Strain Concentration Stress/Strain Concentration First yielding K ε K T K T K 1 Nominal Stress Static Strength and Fracture Darrell Socie, All Rights Reserved 17 of 106
19 Notched Plate Experiments Materials: 1018 Hot Rolled Steel 7075-T6 Aluminum /4 thick 1 1 Static Strength and Fracture Darrell Socie, All Rights Reserved 18 of 106
20 1018 Stress-Strain Curve true fracture strain, ε f = 0.86 true fracture stress, f = 770 MPa Strain Static Strength and Fracture Darrell Socie, All Rights Reserved 19 of 106
21 7075-T6 Stress-Strain Curve true fracture strain, ε f = 0.3 true fracture stress, f = 70 MPa Strain Static Strength and Fracture Darrell Socie, All Rights Reserved 0 of 106
22 1018 Steel Test Data load, kn edge diamond slot hole displacement, mm Static Strength and Fracture Darrell Socie, All Rights Reserved 1 of 106
23 7075-T6 Test Data 100 load, kn edge diamond slot hole displacement, mm Static Strength and Fracture Darrell Socie, All Rights Reserved of 106
24 Failure of a Notched Plate Static Strength and Fracture Darrell Socie, All Rights Reserved 3 of 106
25 Failures from Stress Concentrations Net section stresses must be below the flow stress Notch strains must be below the fracture strain Static Strength and Fracture Darrell Socie, All Rights Reserved 4 of 106
26 D vs 3D Plates and shells D stress state Solids 3D stress state Static Strength and Fracture Darrell Socie, All Rights Reserved 5 of 106
27 Stress Concentration in a Bar 5 4 o 3 1 Axial, z Tangential, θ Radial, r r ρ Static Strength and Fracture Darrell Socie, All Rights Reserved 6 of 106
28 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 Static Strength and Fracture Darrell Socie, All Rights Reserved 7 of 106
29 Static Strength and Fracture Darrell Socie, All Rights Reserved 8 of 106 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
30 Constraint Factors a /ρ CF P max = A net flow CF Static Strength and Fracture Darrell Socie, All Rights Reserved 9 of 106
31 Effect of Constraint Stress, z Increasing constraint Strain, ε z Higher strength and lower ductility Static Strength and Fracture Darrell Socie, All Rights Reserved 30 of 106
32 Notched Bars 1 8 R 1 16 R 1 R V notch Static Strength and Fracture Darrell Socie, All Rights Reserved 31 of 106
33 1018 Steel Test Data V displacement, mm Static Strength and Fracture Darrell Socie, All Rights Reserved 3 of 106
34 7075-T6 Test Data V displacement, mm Static Strength and Fracture Darrell Socie, All Rights Reserved 33 of 106
35 Conclusion Net section area, state of stress and material strength control the failure load in a structure only in ductile materials. In brittle materials, cracks will form before the maximum load capacity of the structure is reached. Static Strength and Fracture Darrell Socie, All Rights Reserved 34 of 106
36 Static Strength and Fracture Stress Concentration Factors Fracture Mechanics Approximate Stress Intensity Factors Ductile vs. Brittle Fracture Static Strength and Fracture Darrell Socie, All Rights Reserved 35 of 106
37 Fractures Static Strength and Fracture Darrell Socie, All Rights Reserved 36 of 106
38 Stress Concentration a ρ K T = 1+ a ρ for a crack a ~ 10-3 ρ ~ 10-9 K T ~ 000 Static Strength and Fracture Darrell Socie, All Rights Reserved 37 of 106
39 Fracture Mechanics Parameters G strain energy release rate K stress intensity factor J J-integral R crack growth resistance Static Strength and Fracture Darrell Socie, All Rights Reserved 38 of 106
40 Strain Energy Release Rate U = 0 U = Γ a a strain energy surface energy Static Strength and Fracture Darrell Socie, All Rights Reserved 39 of 106
41 Plastic Energy Term U = Γ + P a strain energy surface energy + plastic energy Static Strength and Fracture Darrell Socie, All Rights Reserved 40 of 106
42 Strain Energy Release Rate, G F x internal strain energy = external work a U= 1 Fx F U G= a x Static Strength and Fracture Darrell Socie, All Rights Reserved 41 of 106
43 Strain Energy Release Rate, G F x F a U a + a a + a t G= x 1 U t a G is the energy per unit crack area needed to extend a crack Static Strength and Fracture Darrell Socie, All Rights Reserved 4 of 106
44 Static Strength and Fracture Darrell Socie, All Rights Reserved 43 of 106 Stress Intensity Factor, K r θ θ θ θ π = 3 sin sin 1 cos r K x θ θ + θ π = 3 sin sin 1 cos r K y 3 cos sin cos r K xy θ θ θ π = τ θ + + ν ν θ π = sin cos r G K u θ + + ν ν θ π = cos sin r G K v x u x = ε x v y = ε x v y u xy + = γ
45 Design Philosophy Stress < Strength < y Stress Intensity < Fracture Toughness K < K Ic Two cracks with the same K will have the same behavior Static Strength and Fracture Darrell Socie, All Rights Reserved 44 of 106
46 Fracture Toughness K IC = πa f a w fracture toughness flaw shape flaw size operating stresses Static Strength and Fracture Darrell Socie, All Rights Reserved 45 of 106
47 K and G G= K G= E ( 1 ν ) E K plane stress plane strain Static Strength and Fracture Darrell Socie, All Rights Reserved 46 of 106
48 Elastic-Plastic Stress Field y small scale yielding r r* plastic zone r p y = K πr * cos θ 1+ sin θ 3θ sin Static Strength and Fracture Darrell Socie, All Rights Reserved 47 of 106
49 Critical Crack Sizes What would the critical crack size be in a standard tensile test? a critical = 1 K π IC f Static Strength and Fracture Darrell Socie, All Rights Reserved 48 of 106
50 Fracture Toughness From M F Ashby, Materials Selection in Mechanical Design, 1999, pg 431 Static Strength and Fracture Darrell Socie, All Rights Reserved 49 of 106
51 Measuring Fracture Toughness force displacement Static Strength and Fracture Darrell Socie, All Rights Reserved 50 of 106
52 Thickness Effects From Wilhem Fracture Mechanics Guidelines for Aircraft Structural Applications AFFDL-TR Static Strength and Fracture Darrell Socie, All Rights Reserved 51 of 106
53 Static Strength and Fracture Darrell Socie, All Rights Reserved 5 of 106 Plastic Zone Size plane stress plane strain ys p K 1 r π = ys p K 6 1 r π =
54 3D Plastic Zone plane stress plane strain Static Strength and Fracture Darrell Socie, All Rights Reserved 53 of 106
55 Fracture Surfaces Static Strength and Fracture Darrell Socie, All Rights Reserved 54 of 106
56 Thickness Requirements fracture toughness, MPa m K Ic 40 K c t >.5 0 ys thickness, mm Static Strength and Fracture Darrell Socie, All Rights Reserved 55 of 106
57 Size Requirements t,w a,a.5 K ys 50r ys K Ic t, mm 04- T T Ti-6Al-4V Ti-6Al-4V PH p Static Strength and Fracture Darrell Socie, All Rights Reserved 56 of 106
58 Strength, Toughness, Flaw Size Fracture Toughness, MPa m C-Ni-Cr-Mo steel Yield Strength, MPa = ys Static Strength and Fracture Darrell Socie, All Rights Reserved 57 of = Flaw Size, mm ys
59 Silver Bridge Collapse, Wearne, P. TV Books, NY 1999 Static Strength and Fracture Darrell Socie, All Rights Reserved 58 of 106
60 Source Static Strength and Fracture Darrell Socie, All Rights Reserved 59 of 106
61 Eyebar 1 wide thickness 1 6 Static Strength and Fracture Darrell Socie, All Rights Reserved 60 of 106
62 Material Properties u = 100 ksi y = 75 ksi Working stress 50 ksi CVN =.6 ft-lb at 3 F CVN = 8.6 ft-lb at 165 F Static Strength and Fracture Darrell Socie, All Rights Reserved 61 of 106
63 Fracture Toughness Barsom-Rolfe K 3 IC E = (CVN) (psi in,ft lb) Corten-Sailors K IC = 15.5 CVN (ft lb) Roberts-Newton K IC = 9.35CVN 1.65 (ft lb) Barsom-Rolfe Corten-Sailors Roberts-Newton K IC =15.9ksi K IC = 5.0ksi K IC = 45.ksi in in in Average 8.7 Static Strength and Fracture Darrell Socie, All Rights Reserved 6 of 106
64 Critical Crack Size Assume a corner crack K IC = π ( 1.1) πa Let = y Let = 50 a ~ inches a ~ inches Static Strength and Fracture Darrell Socie, All Rights Reserved 63 of 106
65 Modern Aircraft Materials Bucci et. al., Need for New Materials in Aging Aircraft Structures Journa of Aircraft, Vol. 37, 000, 1-19 Static Strength and Fracture Darrell Socie, All Rights Reserved 64 of 106
66 Summary Both stress and flaw size govern fracture K Ic > πa f a W Static Strength and Fracture Darrell Socie, All Rights Reserved 65 of 106
67 Static Strength and Fracture Stress Concentration Factors Fracture Mechanics Approximate Stress Intensity Factors Ductile vs. Brittle Fracture Static Strength and Fracture Darrell Socie, All Rights Reserved 66 of 106
68 Stress Intensity Factors Analytical Theory of elasticity Numerical Finite element Experimental Compliance Handbook Approximate Static Strength and Fracture Darrell Socie, All Rights Reserved 67 of 106
69 Stress Intensity Factors K = MsM Φ t πa M s free surface effects M t back surface effects Φ crack shape effects Static Strength and Fracture Darrell Socie, All Rights Reserved 68 of 106
70 M s free surface effects c a M s = 1.1 a b Static Strength and Fracture Darrell Socie, All Rights Reserved 69 of 106
71 M t back surface effects a b M t = b πa tan πa b 5 4 M t a / b Static Strength and Fracture Darrell Socie, All Rights Reserved 70 of 106
72 Φ crack shape effects Q 3 1 c a c a π Φ = k Q Q π / = 0 = 1 a c 1 k Q a c sin 1.65 β dβ a c Static Strength and Fracture Darrell Socie, All Rights Reserved 71 of 106
73 Edge Cracked Plate in Tension F a b = b πa tan πa b a.0 b (1 sin πa cos b πa ) b 3 Static Strength and Fracture Darrell Socie, All Rights Reserved 7 of 106
74 Edge Cracked Plate in Bending F a b = b πa tan πa b (1 sin πa cos b πa ) b 4 Static Strength and Fracture Darrell Socie, All Rights Reserved 73 of 106
75 Tension and Bending 10 8 a F b 6 tension 4 bending a / b Static Strength and Fracture Darrell Socie, All Rights Reserved 74 of 106
76 Boundary Conditions 4 uniform stress 3 a F b 1 uniform displacement a / b Static Strength and Fracture Darrell Socie, All Rights Reserved 75 of 106
77 Handbook Static Strength and Fracture Darrell Socie, All Rights Reserved 76 of 106
78 Handbook Stress Intensity Factors Handbook Y. Murakami Editor, Pergamon Press Static Strength and Fracture Darrell Socie, All Rights Reserved 77 of 106
79 Useful approximations Corner crack Two free edges Semicircular shape a a K = (1.1) π πa Static Strength and Fracture Darrell Socie, All Rights Reserved 78 of 106
80 Crack Shape Through crack K = 1.1 πa Semielliptical crack K = 1.1 π πa = 0.71 πa a Static Strength and Fracture Darrell Socie, All Rights Reserved 79 of 106
81 Superposition Crack tip stresses: ij ij K = K = tension πr tension f ij bending ( θ) + f ( θ) + K πr K bending πr f ij ij K total ( θ) = f ( θ) πr ij K total = K tension + K bending tension + bending Static Strength and Fracture Darrell Socie, All Rights Reserved 80 of 106
82 Stress Gradients Nominal Stress x K average + 3 crack tip πa Static Strength and Fracture Darrell Socie, All Rights Reserved 81 of 106
83 Pressure Vessel Pr = t P Pr K = t K = P πa + P r t + 1 πa πa Pr = t Static Strength and Fracture Darrell Socie, All Rights Reserved 8 of 106
84 Cracks at Notches S K t S S D a a D + a a << D a >> D Static Strength and Fracture Darrell Socie, All Rights Reserved 83 of 106
85 Stress Intensity Factors 3.0 K = K ts πa S K D ( D a ) K = S π a D Static Strength and Fracture Darrell Socie, All Rights Reserved 84 of 106
86 Cracks at Holes Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack Static Strength and Fracture Darrell Socie, All Rights Reserved 85 of 106
87 Static Strength and Fracture Darrell Socie, All Rights Reserved 86 of 106 Summary π > W a a f K Ic ~ ys / a ~ 0.1 mm mm 1 ~ W a f
88 Static Strength and Fracture Stress Concentration Factors Fracture Mechanics Approximate Stress Intensity Factors Ductile vs. Brittle Fracture Static Strength and Fracture Darrell Socie, All Rights Reserved 87 of 106
89 Fracture Behavior LEFM EPFM Collapse Static Strength and Fracture Darrell Socie, All Rights Reserved 88 of 106
90 Failure Analysis Diagram K K Ic 1 LEFM EPFM Collapse 1 f low Static Strength and Fracture Darrell Socie, All Rights Reserved 89 of 106
91 Plastic Collapse P ( P W a )B < f low a B S = P W B Nominal stress W S = flow 1 a W P Static Strength and Fracture Darrell Socie, All Rights Reserved 90 of 106
92 Fracture a W S = B P W B S a S πa f = W = π a W K Ic W f K Ic a W S Static Strength and Fracture Darrell Socie, All Rights Reserved 91 of 106
93 Fracture vs. Collapse Fracture and collapse equally likely flow a W = π a w K 1 Ic W f a W K Ic flow = 1 a W π a w W f a W Static Strength and Fracture Darrell Socie, All Rights Reserved 9 of 106
94 Material Properties ys K Ic K Ic ys T T Ti-6Al-4V Ti-6Al-4V PH Static Strength and Fracture Darrell Socie, All Rights Reserved 93 of 106
95 Fracture Diagram K Ic = f low 1 m W = 1m S f low K Ic = f low 0.5 m K Ic = f low 0.1 m a / W Static Strength and Fracture Darrell Socie, All Rights Reserved 94 of 106
96 Fracture Diagram S 1.5 W = 1m W = 0.1m K Ic = f low 0.5 m f low a / W Static Strength and Fracture Darrell Socie, All Rights Reserved 95 of 106
97 Fracture vs. Collapse K flow Ic W = 1 a W π a W W πa πa tan W Static Strength and Fracture Darrell Socie, All Rights Reserved 96 of 106
98 Fracture Diagram Plastic Collapse K f low c W Fracture small flaws high stresses a / W large flaws low stresses Static Strength and Fracture Darrell Socie, All Rights Reserved 97 of 106
99 Typical Stress Concentration K T K T elastic failure Stress distribution Strain distribution Static Strength and Fracture Darrell Socie, All Rights Reserved 98 of 106
100 Stress and Strain Concentration Stress/Strain Concentration First yielding K ε K T K T K 1 Nominal Stress Static Strength and Fracture Darrell Socie, All Rights Reserved 99 of 106
101 Failure Diagram Ductile Material P ε f is large max = A ε net f KT E ε f = K ε T ys fully plastic strength limited P = A max flow net 0 a/w 1 Strength limit is reached before cracking at the notch Static Strength and Fracture Darrell Socie, All Rights Reserved 100 of 106
102 Failure Diagram Brittle Material P max = flow A net ε f = K ε T ys elastic P ε f is small max = A ε K net T ductility limited f E 0 a/w 1 Cracks form at the notch before the limit load is reached Static Strength and Fracture Darrell Socie, All Rights Reserved 101 of 106
103 Cracks at Notches S K t S S D a a D + a a << D a >> D Static Strength and Fracture Darrell Socie, All Rights Reserved 10 of 106
104 Cracks at Holes Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack Static Strength and Fracture Darrell Socie, All Rights Reserved 103 of 106
105 Notch Fracture What ratio of strength to toughness is needed to avoid fracture? K c = ys 1.1 K T π0.1r For K T = 3 and r = 10 mm K c > ys 0.18 to avoid fracture from the notch Static Strength and Fracture Darrell Socie, All Rights Reserved 104 of 106
106 Material Properties ys K Ic K Ic ys T T Ti-6Al-4V Ti-6Al-4V PH Static Strength and Fracture Darrell Socie, All Rights Reserved 105 of 106
107 Summary Fracture is a likely failure mode for all higher strength materials Fracture is even more likely at stress concentrators Static Strength and Fracture Darrell Socie, All Rights Reserved 106 of 106
108 Fatigue and Fracture
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