Professor Darrell F. Socie University of Illinois at Urbana-Champaign 2002 Darrell Socie, All Rights Reserved

Outline 1. Notch Rules 2. Fatigue Notch Factor 3. Stress Intensity Factors for Notches 4. Frost Data and K f 5. Small Crack Growth 6. Small Notches Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 50

Stress Concentration Factor σ local = σ applied 1+ 2 a ρ Inglis Solution 1910 ρ 2a Applied stress Local stress Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 2 of 50

Notch Rules K 2 t K 2 t Se K K 2 p p = Se S = Neuber σ σε = E Glinka σ = σ dε = E Seeger * S e * Limit σ y = K t σε = S * + σ E + σ K 1 1+ n p 1 n σ K 1 n σ + K K t = S K 1 n Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 50

Define K σ and K ε after Yielding S, e σ, ε Define: nominal stress, S nominal strain, e notch stress, σ notch strain, ε Stress concentration Strain concentration K K σ ε σ = S ε = e Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 50

Stress and Strain Concentration Stress/Strain Concentration First yielding 2 K ε K t K t K 1 σ Nominal Stress Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 50

Neuber s Rule K t S Stress (MPa) σ ε K t S K t Actual stress e = σε Stress calculated with elastic assumptions K t e Strain Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 50

Neuber s Rule for Fatigue Stress and strain amplitudes Elastic nominal stress Substitute and rearrange K t S K t e 2 2 K t e 2 S 2 = = = S 2E E σ 2 σ ε 2 2 ε 2 The product of stress times strain controls fatigue life Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 50

SN Materials Data Stress Amplitude, MPa 10000 1000 100 93 steels 17 aluminums 10 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, Reversals Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 50

εn Materials Data Strain Ampliture 10 1 0.1 10-2 93 steels 17 aluminums 10-3 10-4 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, Reversals Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 50

10 5 E σ 2 ε 2 S σ ε K t or E 2 2 2 10 4 10 3 10 2 10 1 1 10 10 2 10 3 10 4 10 5 10 6 10 7 Fatigue Life, Reversals Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 50

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? Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 50

K f 4 3 2 1 Peterson s Equation 2070MPa α= 0.025 σu K f 1.8 K t 1 = 1 + α 1+ ρ mm 0 10-4 10-3 10-2 0.1 1 10 10 2 10 3 α ρ No effect when ρ << α Full effect when ρ >> α Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 20 of 50

1018 Steel Test Data 100 80 load, kn 60 40 edge diamond slot hole 20 0 0 2 4 6 8 10 displacement, mm Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 23 of 50

Notched SN Curve 10000 Stress Amplitude, MPa 1000 Notched specimen Smooth specimen data K f 100 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Cycles Stress concentrations are not very important at short lives Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 24 of 50

Stress Intensity Factors 3.0 K = K ts πa S K D 2.0 1.0 ( D a ) K = S π + 0 0 0.1 0.2 0.3 0.4 0.5 0.6 a D Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 27 of 50

Cracks at Holes 1 10 1 12 Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 28 of 50

Crack Growth Data 10-6 Crack Growth Rate, m/cycle 10-7 10-8 10-9 10-10 10-11 da dn K c = C K m Nonpropagating cracks K TH 2 > σ1.12 πa π 10-12 1 10 100 K,MPa m K TH Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 50

Frost Data 1.0 0.8 Rotating bending Notched bar Notched plate nucleation fracture S nominal S fatigue limit 0.6 0.4 0.2 0 1 nonpropagating cracks K t 1 3 5 7 9 11 13 15 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, 811-836 Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 50

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! Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 32 of 50

Specimens with Similar Geometry K t = 10.7 K t = 2.4 25 25 2.5 2.5 Ultimate Strength 780 MPa Yield Strength 660 MPa Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 50

Test Results Nominal Stress Amplitude 1000 100 Strength Limited Crack Growth Dominated K t = 10.7 K t = 2.4 Fatigue Strength Dominated Threshold Stress Intensity Dominated 1 1 10 100 10 3 10 4 10 5 10 6 10 7 Total Fatigue Life, Cycles Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 50

Crack Growth Crack growth rate Short cracks Notch plasticity controls Long cracks Crack tip plasticity controls K or a Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 50

Closure Observations a b σ b ε c d a c 1 st subcycle d 500 th subcycle 1026 steel ε 1 /2 = 0.005 ε 2 /2 = 0.001 Fatigue Seminar 2002 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 42 of 50

Notches in Fatigue