Fatigue and Fracture

Fatigue and Fracture Static Strength and Fracture Stress Concentration Factors Professor Darrell F. Socie Mechanical Science and Engineering University of Illinois 004-013 Darrell Socie, All Rights Reserved

Static Strength and Fracture Stress Concentration Factors Fracture Mechanics Approximate Stress Intensity Factors Ductile vs. Brittle Fracture Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 1 of 106

Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 4 of 106 Stresses θ + λ + +λ = cos 4 3 1 1 1 1 4 a r a r a r r θ + λ + +λ = θ cos 3 1 1 1 1 4 a r a r θ + λ = τ θ sin 3 1 1 4 a r a r r Independent of size, dependant only on r/a

Stress Ratio Effects λ 4 3 λ = 1 1 λ = 0 λ θ 0-1 - -3 30 60 90 10 150 180 Angle λ = -1-4 stresses around the circumference of a hole Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 6 of 106

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 004-013 Darrell Socie, All Rights Reserved 7 of 106

Transverse Strains Longitudinal Tensile Strain Transverse Compression Strain 0.001 0.00 0.003 0.004 0.005 0 0.00 0.004 0.006 0.008 0.010 0.01 ν = 0.5 Thickness 40 mm 0 mm 10 mm ν = 0.3 5 mm y z x 5 D 100 Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 10 of 106

Notch Stresses y z x t ε x ε z x z 7 0.01-0.005 63.5 0 15 0.01-0.003 70.6 14.1 30 0.01-0.00 73.0 1.8 50 0.01-0.001 75.1 9.3 Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 11 of 106

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 004-013 Darrell Socie, All Rights Reserved 14 of 106

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 004-013 Darrell Socie, All Rights Reserved 15 of 106

1018 Stress-Strain Curve 500 400 300 00 100 true fracture strain, ε f = 0.86 true fracture stress, f = 770 MPa 0 0 0.1 0. 0.3 0.4 0.5 Strain Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 19 of 106

7075-T6 Stress-Strain Curve 700 600 500 400 300 00 true fracture strain, ε f = 0.3 true fracture stress, f = 70 MPa 100 0 0 0.05 0.1 0.15 0. Strain Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 0 of 106

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 004-013 Darrell Socie, All Rights Reserved 4 of 106

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 004-013 Darrell Socie, All Rights Reserved 7 of 106

Static Strength and Fracture 004-013 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

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 004-013 Darrell Socie, All Rights Reserved 34 of 106

Fracture Mechanics Parameters G strain energy release rate K stress intensity factor J J-integral R crack growth resistance Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 38 of 106

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 004-013 Darrell Socie, All Rights Reserved 4 of 106

Static Strength and Fracture 004-013 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 1 1 3 cos r G K u θ + + ν ν θ π = cos 1 1 3 sin r G K v x u x = ε x v y = ε x v y u xy + = γ

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 004-013 Darrell Socie, All Rights Reserved 44 of 106

Size Requirements t,w a,a.5 K ys 50r ys K Ic t, mm 04- T3 345 44 40.7 7075 -T6 495 5 6.4 Ti-6Al-4V 910 105 33.3 Ti-6Al-4V 1035 55 7.1 4340 860 99 33.1 4340 1510 60 3.9 17-7 PH 1435 77 7. 5100 070 14 0.1 p Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 56 of 106

Strength, Toughness, Flaw Size Fracture Toughness, MPa m 80 70 60 50 40 30 0 10 0 0.5C-Ni-Cr-Mo steel 1400 1500 1600 1700 1800 1900 000 100 Yield Strength, MPa = ys Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 57 of 106 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 = 3..8.4.0 1.6 1. 0.8 0.4 Flaw Size, mm ys

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 004-013 Darrell Socie, All Rights Reserved 6 of 106

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 004-013 Darrell Socie, All Rights Reserved 64 of 106

Stress Intensity Factors Analytical Theory of elasticity Numerical Finite element Experimental Compliance Handbook Approximate Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 67 of 106

Φ crack shape effects Q 3 1 c 0 0 0.1 0. 0.3 0.4 0.5 a c a π Φ = k Q Q π / = 0 = 1 a c 1 k Q 1+ 1.464 a c sin 1.65 β dβ a c Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 71 of 106

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 004-013 Darrell Socie, All Rights Reserved 80 of 106

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 004-013 Darrell Socie, All Rights Reserved 9 of 106

Material Properties ys K Ic K Ic ys 100 50 00 0.800 04-T3 345 44 0.18 7075-T6 495 5 0.051 Ti-6Al-4V 910 105 0.115 Ti-6Al-4V 1035 55 0.053 4340 860 99 0.115 4340 1510 60 0.040 17-7 PH 1435 77 0.054 5100 070 14 0.007 Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 93 of 106

Fracture Diagram K Ic = f low 1 m W = 1m S f low 1.5 1 0.5 K Ic = f low 0.5 m K Ic = f low 0.1 m 0 0 0. 0.4 0.6 0.8 1 a / W Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 94 of 106

Fracture Diagram 1 0.8 Plastic Collapse K f low c W 0.6 0.4 Fracture 0. 0 0 0. 0.4 0.6 0.8 1 small flaws high stresses a / W large flaws low stresses Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 97 of 106

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 004-013 Darrell Socie, All Rights Reserved 100 of 106

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 004-013 Darrell Socie, All Rights Reserved 101 of 106

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 004-013 Darrell Socie, All Rights Reserved 104 of 106

Material Properties ys K Ic K Ic ys 100 50 00 0.800 04-T3 345 44 0.18 7075-T6 495 5 0.051 Ti-6Al-4V 910 105 0.115 Ti-6Al-4V 1035 55 0.053 4340 860 99 0.115 4340 1510 60 0.040 17-7 PH 1435 77 0.054 5100 070 14 0.007 Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 105 of 106

Summary Fracture is a likely failure mode for all higher strength materials Fracture is even more likely at stress concentrators Static Strength and Fracture 004-013 Darrell Socie, All Rights Reserved 106 of 106

Fatigue and Fracture