Multiaxial Fatigue. Professor Darrell F. Socie. Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign
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1 Multiaxial Fatigue Professor Darrell F. Socie Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Darrell Socie, All Rights Reserved
2 Contact Information Darrell Socie Mechanical Science and Engineering 1206 West Green Urbana, Illinois USA Tel: Fax: Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 125
3 Outline State of Stress ( Chapter 1 ) Fatigue Mechanisms ( Chapter 3 ) Stress Based Models ( Chapter 5 ) Strain Based Models ( Chapter 6 ) Fracture Mechanics Models ( Chapter 7 ) Nonproportional Loading ( Chapter 8 ) Notches ( Chapter 9 ) Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 2 of 125
4 State of Stress Stress components Common states of stress Shear stresses Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 125
5 Stress Components Z σ z τ zy τ zx τ xz τ yz τ xy τ yx σ y σ x Y X Six stresses and six strains Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 125
6 Stresses Acting on a Plane Z Z X σ x τ x z Y φ τ x y θ Y X Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 125
7 Principal Stresses τ 13 σ 3 τ12 τ 23 σ 2 σ 1 σ 3 - σ 2 ( σ X + σ Y + σ Z ) + σ(σ X σ Y + σ Y σ Z σ X σ Z -τ 2 XY - τ2 YZ -τ2 XZ ) - (σ X σ Y σ Z + 2τ XY τ YZ τ XZ - σ X τ 2 YZ - σ Y τ2 ZX - σ Z τ2 XY ) = 0 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 125
8 Stress and Strain Distributions 100 % of applied stress θ Stresses are nearly the same over a 10 range of angles Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 125
9 Tension σ 3 τ γ/2 σ x σ 1 σ 2 σ 1 σ ε ε 1 σ 2 = σ 3 = 0 ε 2 = ε 3 = νε 1 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 125
10 Torsion σ 2 τ γ/2 σ 3 σ 1 σ τ xy ε 1 ε 3 ε σ 3 X σ 1 Y σ 2 ε 2 σ 1 = τ xy Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 125
11 Biaxial Tension σ 3 τ γ / 2 σ 3 σ ε σ 1 = σ x σ 1 = σ 2 ε 1 = ε 2 σ 2 = σ y ε 3 2ν = 1 ν ε Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 125
12 Shear Stresses σ 3 σ 3 σ 2 σ 1 Maximum shear stress σ 2 σ 1 Octahedral shear stress σ σ Mises: σ = 3 3 τ oct τoct = τ13 = τ τ 13= τ ( ) ( ) ( ) 2 oct= σ1 σ3 + σ1 σ 2 + σ2 σ3 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 125
13 State of Stress Summary Stresses acting on a plane Principal stress Maximum shear stress Octahedral shear stress Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 12 of 125
14 Fatigue Mechanisms Crack nucleation Fracture modes Crack growth State of stress effects Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 125
15 Crack Nucleation Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 14 of 125
16 Slip Bands Extrusion Undeformed material Intrusion Loading Unloading Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 15 of 125
17 Slip Bands Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 16 of 125
18 Mode I, Mode II, and Mode III Mode I opening Mode II in-plane shear Mode III out-of-plane shear Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 125
19 Stage I and Stage II loading direction free surface Stage I Stage II Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 18 of 125
20 Case A and Case B Growth along the surface Growth into the surface Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 19 of 125
21 Mode I Growth crack growth direction 5 µm Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 20 of 125
22 Mode II Growth shear stress slip bands 10 µm crack growth direction Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 21 of 125
23 304 Stainless Steel - Torsion 1.0 Damage Fraction N/N f Shear Tension 0 Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 22 of 125
24 Stainless Steel - Tension Damage Fraction N/N f Tension Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 23 of 125
25 Inconel Torsion 1.0 Tension Damage Fraction N/N f Shear Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 24 of 125
26 Inconel Tension Damage Fraction N/N f Tension Shear Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 25 of 125
27 1045 Steel - Torsion 1.0 Tension Damage Fraction N/N f f Shear 0 Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 26 of 125
28 Steel - Tension Damage Fraction N/N f Shear Tension Nucleation Fatigue Life, 2N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 27 of 125
29 Fatigue Mechanisms Summary Fatigue cracks nucleate in shear Fatigue cracks grow in either shear or tension depending on material and state of stress Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 28 of 125
30 Stress Based Models Sines Findley Dang Van Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 29 of 125
31 Bending Torsion Correlation Shear stress in bending 1/2 Bending fatigue limit Shear stress Octahedral stress Principal stress Shear stress in torsion 1/2 Bending fatigue limit Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 125
32 Test Results Cyclic tension with static tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 125
33 Cyclic Tension with Static Tension 1.5 Axial stress Fatigue strength Mean stress Yield strength 1.5 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 32 of 125
34 Cyclic Torsion with Static Torsion 1.5 Shear Stress Amplitude Shear Fatigue Strength Maximum Shear Stress Shear Yield Strength Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of
35 Cyclic Tension with Static Torsion 1.5 Bending Stress Bending Fatigue Strength Static Torsion Stress Torsion Yield Strength 3.0 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 125
36 Cyclic Torsion with Static Tension 1.5 Torsion shear stress Shear fatigue strength Axial mean stress 1.5 Yield strength Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 125
37 Conclusions Tension mean stress affects both tension and torsion Torsion mean stress does not affect tension or torsion Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 125
38 Sines τ 2 oct + α(3σ h ) =β 1 6 ( σ x α( σ σ mean x y ) 2 + σ + ( σ mean y x + σ σ z mean z ) 2 ) = β + ( σ y σ z ) 2 + 6( τ 2 xy + τ 2 xz + τ 2 yz ) + Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 125
39 Findley τ 2 + k σ n = max f tension torsion Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 125
40 Bending Torsion Correlation Shear stress in bending 1/2 Bending fatigue limit Shear stress Octahedral stress Principal stress Shear stress in torsion 1/2 Bending fatigue limit Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 125
41 Dang Van Σ ij (M,t) E ij (M,t) τ( t) + aσ ( t) = b h m σ ij (m,t) ε ij (m,t) V(M) Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 125
42 Dang Van ( continued ) τ τ(t) + aσ h (t) = b τ Failure predicted σ h σ h Loading path Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 125
43 Stress Based Models Summary Sines: Findley: τ 2 oct τ α(3σ h ) =β kσ n = max f Dang Van: τ σ ( t) + a ( t) = b h Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 42 of 125
44 Strain Based Models Plastic Work Brown and Miller Fatemi and Socie Smith Watson and Topper Liu Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 125
45 Octahedral Shear Strain 0.1 Plastic octahedral shear strain range 0.01 Tension Torsion Cycles to failure Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 125
46 Plastic Work Plastic Work per Cycle, MJ/m A T T A A T T A T Torsion A Axial T A T A T Fatigue Life, N f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 125
47 Brown and Miller Fatigue Life, Cycles 2 x x x γ = Normal Strain Amplitude, ε n Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 125
48 Case A and B Growth along the surface Growth into the surface Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 125
49 Brown and Miller ( continued ) Uniaxial Equibiaxial Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 125
50 Brown and Miller ( continued ) 1 α α α ( ) γ = γ max + S ε n γ max 2 + S ε = A n σ ' f 2σ E n, mean b ' f + ε f ( 2N ) B ( 2N ) f c Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 125
51 Fatemi and Socie Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 125
52 γ / 3 Loading Histories ε C F G H I J Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 51 of 125
53 Crack Length Observations 2.5 F-495 H-491 Crack Length, mm J-603 I-471 C-399 G Cycles Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 52 of 125
54 Fatemi and Socie γ 2 1+ k σ n,max σ y = ' τf (2N f ) G bo + γ ' f (2N ) f co Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 125
55 Smith Watson Topper Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 125
56 SWT σ ' 2 ε1 σf 2b ' ' b+ c n = (2N f ) + σfεf (2Nf ) 2 E Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 125
57 Liu Virtual strain energy for both mode I and mode II cracking W I = ( σ n ε n ) max + ( τ γ) W ' 2 ' ' b+ c 4σf 2b I = 4σfεf(2N f ) + (2Nf ) E W II = ( σ n ε n ) + ( τ γ) max W ' 2 ' ' bo+ co 4τf 2bo II = 4τfγ f(2n f ) + (2Nf ) G Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 125
58 Cyclic Torsion Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 125
59 Cyclic Torsion with Static Tension Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Tension Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 125
60 Cyclic Torsion with Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 125
61 Cyclic Torsion with Tension and Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Compression Hoop Tension Shear Damage Tensile Damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 125
62 Test Results Load Case γ/2 σ hoop MPa σ axial MPa N f Torsion ,200 with tension ,300 with compression ,000 with tension and compression ,200 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 61 of 125
63 Conclusions All critical plane models correctly predict these results Hydrostatic stress models can not predict these results Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 62 of 125
64 Loading History Shear strain Axial strain Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 63 of 125
65 Model Comparison Summary of calculated fatigue lives Model Equation Life Epsilon ,060 Garud 6.7 5,210 Ellyin ,450 Brown-Miller ,980 SWT ,930 Liu I ,280 Liu II ,420 Chu ,040 Gamma 26,775 Fatemi-Socie ,350 Glinka ,220 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 125
66 Strain Based Models Summary Two separate models are needed, one for tensile growth and one for shear growth Cyclic plasticity governs stress and strain ranges Mean stress effects are a result of crack closure on the critical plane Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 125
67 Separate Tensile and Shear Models τ σ 3 σ 1 σ σ 2 σ 1 = τ xy Inconel 1045 steel stainless steel Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 125
68 Cyclic Plasticity ε γ ε p γ p ε σ γ τ ε p σ γ p τ Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 125
69 γ 2 Mean Stresses ' σ f σ mean ε eq = W = I E f b ' f ( 2N ) + ε ( 2N ) ' max f n b ' c + S ε ) n= ( S) (2N f ) + ( S) εf (2N f γ 2 σ σ 1+ k σ n,max y [( σ ε ) + ( τ γ) ] n σ n 2σ E ' τf = (2N f ) G max bo + γ ' f f (2N ) ' 2 ε1 σf 2b ' ' b+ c n = (2N f ) + σfεf (2Nf ) 2 E f 2 1 R c co Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 125
70 Fracture Mechanics Models Mode I growth Torsion Mode II growth Mode III growth Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 125
71 Mode I, Mode II, and Mode III Mode I opening Mode II in-plane shear Mode III out-of-plane shear Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 125
72 Mode I and Mode II Surface Cracks σ Mode II σ σ Mode I σ Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 71 of 125
73 Biaxial Mode I Growth 10-3 σ = 193 MPa λ = -1 0 λ = σ = 386 MPa λ = -1 0 λ = 1 da/dn mm/cycle da/dn mm/cycle K, MPa m K, MPa m Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 72 of 125
74 Surface Cracks in Torsion Mode II Mode III Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 125
75 Failure Modes in Torsion Transverse Longitudinal Spiral Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 125
76 Fracture Mechanism Map 60 No Cracks Yield Strength, MPa Spiral Cracks Longitudinal Transverse Cracks Hardness Rc Shear Stress Amplitude, MPa Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 125
77 Mode I and Mode III Growth 10-3 da/dn, mm/cycle K I K III K I, K III MPa m Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 125
78 Mode I and Mode II Growth T6 Aluminum Mode I R = 0 Mode II R = -1 da/dn, mm/cycle SNCM Steel Mode I R = -1 Mode II R = K I, K II MPa m Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 125
79 Fracture Mechanics Models K da dn = C ( ) m K eq [ ] K + 8 K + 8 K (1 ν eq = I II III ) K K eq eq [ ] K + K + (1+ν ) K 0. 5 = I II [ ] 2 2 K + K K + K 0. 5 = I I II III II K eq ( ε) = (F K eq II E γ) 2(1 + ν) 2 + (FE ε) σ I 2 n,max ( ε) = FG γ 1+ k πa σ ys 0.5 πa Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 78 of 125
80 Fracture Surfaces Bending Torsion Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 79 of 125
81 Mode III Growth Crack growth rate, mm/cycle K = 14.9 K = 12.0 K = 11.0 K = 10.0 K = Crack length, mm Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 80 of 125
82 Fracture Mechanics Models Summary Multiaxial loading has little effect in Mode I Crack closure makes Mode II and Mode III calculations difficult Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 81 of 125
83 Nonproportional Loading In and Out-of-phase loading Nonproportional cyclic hardening Variable amplitude Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 82 of 125
84 In and Out-of-Phase Loading γ ε x ε x t ε x = ε o sin(ωt) 1 + ν ε γ xy ε y γ xy ε x γ xy γ xy = (1+ν)ε o sin(ωt) t In-phase ε x = ε o cos(ωt) t γ xy = (1+ν)ε o sin(ωt) t γ γ 1 + ν ε Out-of-phase γ Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 83 of 125
85 In-Phase and Out-of-Phase γ xy 2 2 τ xy γ xy 2 ε x 2τ xy ε 1 θ σ x Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 84 of 125 ε x σ x
86 Loading Histories out-of-phase γ xy /2 γ xy /2 diamond ε x ε x square γ xy /2 cross γ xy /2 ε x ε x Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 85 of 125
87 Loading Histories in-phase out-of-phase diamond square cross Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 86 of 125
88 Findley Model Results τ/2 MPa σ MPa n,max τ/ σ n,max N/N ip in - phase out - of - phase diamond square cross - tension cycle cross - torsion cycle out-of-phase γ xy /2 diamond γ xy /2 square γ xy /2 cross γ xy /2 ε x ε x ε x ε x in-phase Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 87 of 125
89 Nonproportional Hardening ε x t ε x = ε o sin(ωt) γ xy t γ xy = (1+ν)ε o sin(ωt) In-phase ε x γ xy t t ε x = ε o cos(ωt) γ xy = (1+ν)ε o sin(ωt) Out-of-phase Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 88 of 125
90 In-Phase Axial Shear Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 89 of 125
91 90 Out-of-Phase Axial 600 Shear Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 90 of 125
92 Critical Plane Proportional N f = 38,500 N f = 310, Out-of-phase N f = 3,500 N f = 40, Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 91 of 125
93 Loading Histories Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 92 of 125
94 Stress-Strain Response Case Case Case 3 Shear Stress ( MPa ) Case 4 Case 5 Case Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 125
95 Stress-Strain Response (continued) Case 7 Case 9 Case Shear Stress ( MPa ) Case 11 Case 12 Case Axial Stress ( MPa ) Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 94 of 125
96 Maximum Stress 2000 All tests have the same strain ranges Equivalent Stress, MPa Fatigue Life, N f Nonproportional hardening results in lower fatigue lives Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 95 of 125
97 Nonproportional Example Case A Case B Case C Case D σ x σ x σ x σ x σ y σ y σ y σ y Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 96 of 125
98 Shear Stresses Case A Case B Case C Case D τxy τxy τxy τxy Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 97 of 125
99 Simple Variable Amplitude History ε x σ x Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 98 of 125
100 Stress-Strain on 0 Plane 300 τ xy ε x γ xy Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 99 of 125
101 Stress-Strain on 30 and 60 Planes 30º plane º plane ε ε Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 100 of 125
102 Stress-Strain on 120 and 150 Planes 120º plane º plane ε 120 ε Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 101 of 125
103 Shear Strain History on Critical Plane Shear strain, γ time Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 102 of 125
104 Fatigue Calculations Load or strain history Cyclic plasticity model Stress and strain tensor Search for critical plane Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 103 of 125
105 Nonproportional Loading Summary Nonproportional cyclic hardening increases stress levels Critical plane models are used to assess fatigue damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 104 of 125
106 Notches Stress and strain concentrations Nonproportional loading and stressing Fatigue notch factors Cracks at notches Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 105 of 125
107 Notched Shaft Loading M T M X P M Y τ θ z σ θ σ z Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 106 of 125
108 Stress Concentration Factors 6 Stress Concentration Factor Torsion Bending d ρ D D/d Notch Root Radius, ρ/d Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 107 of 125
109 Hole in a Plate λσ σ θ a τ rθ σ θ σ σ r σ r r τ rθ σ θ λσ Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 108 of 125
110 Stresses at the Hole λ = 1 σ θ σ Angle λ = -1 λ = 0 Stress concentration factor depends on type of loading Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 109 of 125
111 Shear Stresses during Torsion 1.5 τ rθ σ r a Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 110 of 125
112 Torsion Experiments Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 111 of 125
113 Multiaxial Loading Uniaxial loading that produces multiaxial stresses at notches Multiaxial loading that produces uniaxial stresses at notches Multiaxial loading that produces multiaxial stresses at notches Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 112 of 125
114 Thickness Effects Longitudinal Tensile Strain Transverse Compression Strain Thickness 50 mm 30 mm 15 mm 7 mm y z x 100 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 113 of 125
115 Applied Bending Moments M X M X M Y M Y Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 114 of 125
116 Bending Moments on the Shaft M Y A M X B D C B C A A B D C Location D Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 115 of 125
117 Bending Moments M A B C D M 5 = M 5 A B C D M Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 116 of 125
118 Torsion Loading M X t 1 σ T σ 1 t 3 M T σ z σ 1 = σ z M X σ T σ T σ 1 = σ T t 2 t 4 M T σ z t 1 t 2 t 3 t 4 σ T σ 1 σ T Out-of-phase shear loading is needed to produce nonproportional stressing Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 117 of 125
119 Plate and Shell Structures K t = 3 K t = 4 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 118 of 125
120 Fatigue Notch Factors 6 Stress concentration factor K f Torsion d ρ K f Bending K t Bending K t Torsion ρ Notch root radius, d D D d = 2.2 Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 119 of 125
121 Fatigue Notch Factors ( continued ) Calculated K f bending torsion conservative non-conservative Peterson s Equation K f K T 1 = 1 + a 1 + r Experimental K f Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 120 of 125
122 Fracture Surfaces in Torsion Circumferencial Notch Shoulder Fillet Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 121 of 125
123 Stress Intensity Factors λ = 1 λ = F 0.75 λ = 1 σ R 0.50 λσ λσ a 0.25 σ a R da dn ( ) m = C = F σ πa K eq K I Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 122 of 125
124 Crack Growth From a Hole 100 λ = -1 λ = 0 λ = 1 Crack Length, mm x x x x 10 6 Cycles Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 123 of 125
125 Notches Summary Uniaxial loading can produce multiaxial stresses at notches Multiaxial loading can produce uniaxial stresses at notches Multiaxial stresses are not very important in thin plate and shell structures Multiaxial stresses are not very important in crack growth Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 124 of 125
126 Final Summary Fatigue is a planar process involving the growth of cracks on many size scales Critical plane models provide reasonable estimates of fatigue damage Multiaxial Fatigue Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 125 of 125
127 Multiaxial Fatigue University of Illinois at Urbana-Champaign
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