Stress concentrations, fracture and fatigue Piet Schreurs Department of Mechanical Engineering Eindhoven University of Technology http://www.mate.tue.nl/ piet December 1, 2016
Overview Stress concentrations Fracture Fatigue Piet Schreurs (TU/e) 2 / 34
Overview Stress concentrations Fracture Fatigue back to top Piet Schreurs (TU/e) 3 / 34
Circular hole in infinite plate σ y σ θ r x 2a σ rr = σ [(1 a2 2 r 2 σ tt = σ [(1 + a2 2 r 2 σ rt = σ 2 ) + (1 + 3a4 ) [1 3a4 r 4 + 2a2 r 2 r 4 4a2 r 2 (1 + 3a4 r 4 ] sin(2θ) ) cos(2θ) ) ] cos(2θ) ] Piet Schreurs (TU/e) 4 / 34
Special points σ rr (r = a, θ) = σ rt (r = a, θ) = σ rt (r, θ = 0) = 0 σ tt (r = a, θ = π 2 ) = 3σ σ tt (r = a, θ = 0) = σ stress concentration factor K t = σ max σ = 3 [-] K t is independent of hole diameter! Piet Schreurs (TU/e) 5 / 34
Stress concentrations c ρ F, T c ρ F, T ρ 2c F, T K t = σ max c = 1 + α σ nom ρ ; σ nom = F A min F, T force; torque ρ minimum radius of curvature α 0.5 for torsion (and bending); 2.0 for tension ρ 0 σ max failure Piet Schreurs (TU/e) 6 / 34
Overview Stress concentrations Fracture Fatigue back to top Piet Schreurs (TU/e) 7 / 34
Crack loading modes Mode I Mode II Mode III Mode I = opening mode Mode II = sliding mode Mode III = tearing mode Piet Schreurs (TU/e) 8 / 34
Crack crack growth increase surface energy = available energy Piet Schreurs (TU/e) 9 / 34
Crack 2a U a = 4aB γ [Nm = J] B = plate thickness γ = surface energy Piet Schreurs (TU/e) 10 / 34
Crack σ 2a 4a σ U i = 2πa 2 B 1 σ 2 2 E [Nm = J] Piet Schreurs (TU/e) 11 / 34
Crack σ σ U a = 4aB γ ; U i = 2πa 2 B 1 σ 2 2 E du i da = du a da 2πa σ2 E = 4γ [Jm 2 ] [Nm = J] critical stress σ c = 2γE πa ; critical crack length a c = 2γE πσ 2 Piet Schreurs (TU/e) 12 / 34
Energy dissipation σ c σ cexperiments dissipation!! ductile - brittle behavior σ ABS, nylon, PC PE, PTFE 0 10 100 ε (%) C v fcc (hcp) metals low strength bcc metals Be, Zn, ceramics high strength metals Al, Ti alloys T Piet Schreurs (TU/e) 13 / 34
Crack What about crack tip stresses? Piet Schreurs (TU/e) 14 / 34
Crack tip stresses x 2 θ r x 1 σ 11 = σ 22 = σ 12 = K I 2πr [ cos( 1 2 θ){ 1 sin( 1 2 θ)sin(3 2 θ)}] K I 2πr [ cos( 1 2 θ){ 1 + sin( 1 2 θ)sin(3 2 θ)}] K I 2πr [ cos( 1 2 θ)sin(1 2 θ)cos(3 2 θ)] Stress Intensity Factor (SIF) : ( ) K I = lim 2πr σ22 θ=0 r 0 [ m 1 2 N m 2 ] Specific (SIF) : literature / analytical / numerical (FEM) Piet Schreurs (TU/e) 15 / 34
SIF for specified cases : (semi-)analytical/literature σ W 2a K I = σ ( πa sec πa W σ πa ) 1/2 small a W σ K I = σ [ a 1.12 π 0.41 a W + a W 1.12σ πa ( a ) 2 ( a ) 3 18.7 38.48 + W W ( a ) ] 4 53.85 W small a W Piet Schreurs (TU/e) 16 / 34
SIF : Numerical analysis ( ) K I = lim 2πr σ22 θ=0 r 0 extrapolation to crack tip Piet Schreurs (TU/e) 17 / 34
Inc: 0 Time: 0.000e+00 Y Z X job1 1 Piet Schreurs (TU/e) 18 / 34
Inc: 0 Time: 0.000e+00 3.690e+03 3.264e+03 2.837e+03 UVW 2.411e+03 1.985e+03 1.559e+03 1.133e+03 7.065e+02 2.803e+02-1.459e+02-5.721e+02 Y Z X job1 Comp 22 of Stress (Rectangular) 1 Piet Schreurs (TU/e) 19 / 34
Energy dissipation 1 Von Mises plastic zones pl.stress pl.strain 0.5 0 0.5 1 0.5 0 0.5 1 1.5 Piet Schreurs (TU/e) 20 / 34
Crack growth criterion K I = K Ic K Ic = Fracture Toughness σ c and a c experimental determination of K Ic (ASTM E399) Material σ y [MPa] K Ic [MPa m ] steel, carbon 241 220 steel, AISI 4340 1827 47.3 Al 2014-T4 448 28.6 Ti 6Al-4V 1103 38.5 Piet Schreurs (TU/e) 21 / 34
Overview Stress concentrations Fracture Fatigue back to top Piet Schreurs (TU/e) 22 / 34
Fatigue falure clam shell markings striations Piet Schreurs (TU/e) 23 / 34
Fatigue load (stress controlled) σ σ max σ m σ min 0 0 i i + 1 t N σ = σ max σ min ; σ a = 1 2 σ σ m = 1 2 (σ max + σ min ) Piet Schreurs (TU/e) 24 / 34
(S-N)-curve S = σ max σ N f S σ th 0 0 log(n f ) reference : σ m = 0 fatigue life : N f fatigue (endurance) limit : σ th N f = (±10 9 ) Rest life : N r = 1 N N f N f Piet Schreurs (TU/e) 25 / 34
(S-N)-curves 600 550 500 450 steelt1 400 σ max [MPa] 350 300 250 steel1020 200 150 100 Mgalloy Al2024T4 10 4 10 5 10 6 10 7 10 8 10 9 N f Piet Schreurs (TU/e) 26 / 34
Endurance limit Tensile strength endurance limit tensile strength Copper 0.23 Aluminum 0.38 Magnesium 0.38 Steel 0.46-0.54 Wrought iron 0.63 Piet Schreurs (TU/e) 27 / 34
Influence factors stress concentrations surface quality material properties environment loading Piet Schreurs (TU/e) 28 / 34
Crack growth a I II a c III a c a i σ a 1 a f N i N f N I : N < N i - a i = initial fatigue crack II : N i < N < N f - slow stable crack propagation - a 1 = non-destr. inspection detection limit III : N f < N - global instability - a = a c : failure Paris law da dn = C( K)m Piet Schreurs (TU/e) 29 / 34
Paris law parameters da dn = C( K)m material K th [MNm 3/2 ] m[-] C 10 11 [!] mild steel 3.2-6.6 3.3 0.24 structural steel 2.0-5.0 3.85-4.2 0.07-0.11 idem in sea water 1.0-1.5 3.3 1.6 aluminium 1.0-2.0 2.9 4.56 aluminium alloy 1.0-2.0 2.6-3.9 3-19 copper 1.8-2.8 3.9 0.34 titanium 2.0-3.0 4.4 68.8 Piet Schreurs (TU/e) 30 / 34
Load spectrum σ 0 N n 1 n 2 n 3 n 4 Palmgren-Miner (1945) law L i=1 n i N if = 1 life time by piecewise integration da dn f ( K, K max) interaction Palmgren-Miner no longer valid : L i=1 n i N if = 0.6 2.0 Piet Schreurs (TU/e) 31 / 34
Random load σ 0 t cyclic counting procedure : (mean crossing) peak count / range pair (mean) count / rain flow count statistical representation load spectrum Piet Schreurs (TU/e) 32 / 34
Measured load histories Piet Schreurs (TU/e) 33 / 34
Design against fatigue Infinite life design σ < σ th (σ < σ e ) no fatigue damage sometimes economically undesirable Safe life design determine load spectra empirical rules / numerical analysis / laboratory tests fatigue life : (S N)-curves apply safety factors Damage tolerant design determine load spectra periodic inspection (NOT; insp. schedules) monitor cracks calculate safe rest life (Paris law, Miner s rule) repair when necessary Fail safe design design for safety : crack arrest / etc. Search : BS7910:2005 Piet Schreurs (TU/e) 34 / 34