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

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 00-014 Darrell Socie, All Rights Reserved 1 of 63

Stress Concentration 00-014 Darrell Socie, All Rights Reserved 4 of 63 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 Stress Concentration 00-014 Darrell Socie, All Rights Reserved 6 of 63

Stress Concentration 00-014 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

Stress Concentration 00-014 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 =

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 00-014 Darrell Socie, All Rights Reserved 15 of 63

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 00-014 Darrell Socie, All Rights Reserved 19 of 63

SN Materials Data Stress Amplitude, MPa 10000 1000 100 93 steels 17 aluminums 10 1 10 10 10 3 10 4 10 5 10 6 10 7 Fatigue Life, Reversals Stress Concentration 00-014 Darrell Socie, All Rights Reserved 0 of 63

εn Materials Data Strain Ampliture 10 1 0.1 10-93 steels 17 aluminums 10-3 10-4 1 10 10 10 3 10 4 10 5 10 6 10 7 Fatigue Life, Reversals Stress Concentration 00-014 Darrell Socie, All Rights Reserved 1 of 63

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 00-014 Darrell Socie, All Rights Reserved 5 of 63

K f 4 3 1 Peterson s Equation 070MPa α = 0.05 u K f 1.8 K t 1 = 1 + α 1+ ρ mm 0 10-4 10-3 10-0.1 1 10 10 10 3 α ρ No effect when ρ << α Full effect when ρ >> α Stress Concentration 00-014 Darrell Socie, All Rights Reserved 3 of 63

Notched SN Curve 10000 Stress Amplitude, MPa 1000 Notched specimen Smooth specimen data K f 100 10 0 10 1 10 10 3 10 4 10 5 10 6 10 7 Cycles Stress concentrations are not very important at short lives Stress Concentration 00-014 Darrell Socie, All Rights Reserved 36 of 63

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 > 1.1 πa π 10-1 1 10 100 K,MPa m K TH Stress Concentration 00-014 Darrell Socie, All Rights Reserved 43 of 63

Frost Data 1.0 0.8 Rotating bending Notched bar Notched plate nucleation fracture S nominal S fatigue limit 0.6 0.4 0. 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 Stress Concentration 00-014 Darrell Socie, All Rights Reserved 44 of 63

Test Results Nominal Stress Amplitude 1000 100 Strength Limited Crack Growth Dominated K t = 10.7 K t =.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 Stress Concentration 00-014 Darrell Socie, All Rights Reserved 47 of 63

Stress Concentration