Contact Information. Multiaxial Fatigue. When is Multiaxial Fatigue Important? Uniaxial Stress. Proportional Biaxial. Nonproportional Multiaxial

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1 Multiaxial Fatigue Introduction Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Contact Inormation Darrell Socie Mechanical Engineering 16 West Green Urbana, Illinois 6181 Oice: 315 Mechanical Engineering Laboratory d-socie@uiuc.edu Tel: Fax: Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14 When is Multiaxial Fatigue Important? Uniaxial Stress Complex state o stress Complex out o phase loading Z one principal stress one direction Y X Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 3 o 14 Proportional Biaxial Nonproportional Multiaxial Z Y principal stresses vary proportionally but do not rotate Z Y Principal stresses may vary nonproportionally and/or change direction 1 = α = β 3 X X Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 4 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 5 o 14

2 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 6 o 14 Crankshat Shear and Normal Strains º 9º γ γ.5 Microstrain Time Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 7 o 14 Shear and Normal Strains 3D stresses.5 45º 135º γ.5 γ Transverse Compression Strain Longitudinal Tensile Strain Thickness 5 mm 3 mm 15 mm 7 mm x y z 1 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 8 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 9 o 14 Notch Stresses Multiaxial Fatigue y z x t x z x z State o Stress Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14

3 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14 Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations State o Stress Stress components Common states o stress Shear stresses Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 13 o 14 Stress Components Stresses Acting on a Plane Z z Z τ zx τ zy Z X τ xz τ xy τ yx τ yz y x τ x z Y x Y φ τ x y X θ Y Six stresses and six strains X Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 14 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 15 o 14 Principal Stresses Stress and Strain Distributions 3 τ 3 τ ( X + Y + Z ) + ( X Y + Y Z X Z -τ XY - τ YZ -τ XZ ) -( X Y Z + τ XY τ YZ τ XZ - X τ YZ - Y τ ZX - Z τ XY ) = 1 τ1 % o applied stress θ Stresses are nearly the same over a 1 range o angles Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 16 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 17 o 14

4 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 18 o 14 Tension Torsion 3 τ γ/ τ γ/ x 1 1 τ 3 xy = 3 = = 3 = ν 1 3 X 1 Y 1 = τ xy Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 19 o 14 Biaxial Tension Shear Stresses τ γ/ = y 1 = x 3 1 = 1 = ν 3 = 1 ν 1 Maximum shear stress 1 Octahedral shear stress 1 3 Mises: = 3 3 τ oct τoct = τ13 =. 94 τ τ 13= τ ( ) ( ) ( ) oct= Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14 Shear Stress Inluence State o Stress Summary Stresses acting on a plane Principal stress Maximum shear stress Octahedral shear stress τ Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 3 o 14

5 Multiaxial Fatigue Stress Strain Relationships Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 5 o 14 Jenkin 19 Elastic Stress Strain Relationships About six months ago I wrote a paper, knowing that I should be very busy in the autumn and made a model to illustrate a point in it. But as I played with the model to learn how to use it, it grew too strong or me and took command and or the last six months I have been its obedient slave --- or the model explained the whole o my subject Fatigue. Fatigue in Metals, The Engineer, Dec. 8, 19 x y z = τ xy τ yz τ xz E ( 1+ ν)( 1 ν) 1 ν ν ν ν 1 ν ν x ν ν ν 1 y 1 ν z γ xy 1 ν γ yz 1 ν γ xz Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 6 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 7 o 14 Yield Suraces Equations 1 Tresca yield surace von Mises yield surace 3τ xy X Tresca Mises F = = F or F ys = = ys = = ys F x y x y τ xy ys = = Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 8 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 9 o 14

6 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 3 o 14 Mises Yield Suraces Isotropic Hardening yield cylinder axis 3 π - plane Yield surace ater loading 3τ 3 ys 1 1 Initial yield surace Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 31 o 14 Cyclic Loading Kinematic Hardening B B 3τ stabilized stress range A C = A y B y α Strain Control Stress Control Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 3 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 33 o 14 Ratcheting Cyclic Mean Stress Relaxation γ Cyclic torsion with a mean tension stress Cyclic strain with mean stress Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 34 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 35 o 14

7 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 36 o 14 Cyclic Creep Plasticity Models 3τ 3τ 3τ c b a b c a 3τ 3τ 3τ d d e e Cyclic stress with a mean stress Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 37 o 14 Why is modeling needed? Flow and Hardening Rules * A B C D time B D C A * 3τ n φ ij dαij α ij d ij Flow rule d p ij F = dλ Hardening rule p F = ( ij) + g( ) = ij Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 38 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 39 o 14 Multiaxial Example Multiaxial Example γ xy.6 τ xy 5 x 5 τ xy 5 x x x γ xy Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 4 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 41 o 14

8 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 4 o 14 Summary Isotropic Hardening Kinematic Hardening Cyclic creep or ratcheting Mean stress relaxation Nonproportional hardening Multiaxial Fatigue Fatigue Mechanisms Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Size Scale or Studying Fatigue Atoms Dislocations Crystals Specimens Structures Understand the physics on this scale Model the physics on this scale Use the models on this scale Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 44 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 45 o 14 The Fatigue Process Ewing and Humrey Crack nucleation Small crack growth in an elastic-plastic stress ield Macroscopic crack growth in a nominally elastic stress ield Final racture N = 1, N =, Cyclic deormation leads to the development o slip bands and atigue cracks Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 46 o 14 N = 1, N = 4, N = 17, Ewing, J.A. and Humrey, J.C. The racture o metals under repeated alterations o stress, Philosophical Transactions o the Royal Society, Vol. A, 193, 41-5 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 47 o 14

9 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 48 o 14 Crack Nucleation Slip Band in Copper Polak, J. Cyclic Plasticity and Low Cycle Fatigue Lie o Metals, Elsevier, 1991 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 49 o 14 Slip Band Formation Slip Bands Extrusion Undeormed material Intrusion Loading Unloading Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 5 o 14 Ma, B-T and Laird C. Overview o atigue behavior in copper sinle crystals II Population, size, distribution and growth Kinetics o stage I cracks or tests at constant strain amplitude, Acta Metallurgica, Vol 37, 1989, Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 51 o 14 Crack Initiation at Inclusions Subsurace Crack Initiation Langord and Kusenberger, Initiation o Fatigue Cracks in 434 Steel, Metallurgical Transactions, Vol 4, 1977, Y. Murakami, Metal Fatigue: Eects o Small Deects and Nonmetallic Inclusions, Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 5 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 53 o 14

10 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 54 o 14 Fatigue Limit and Strength Correlation Crack Nucleation Summary Highly localized plastic deormation Surace phenomena Stochastic process Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 55 o 14 Surace Damage Stage I and Stage II surace surace bulk loading direction 1 µm 1 µm ree surace -5 austenitic steel in symmetrical push-pull atigue ( C, p /= ±.4%) : short cracks on the surace and in the bulk From Jacques Stolarz, Ecole Nationale Superieure des Mines Presented at LCF 5 in Berlin, 3 Stage I Stage II Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 56 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 57 o 14 Stage I Crack Growth Small Cracks at Notches S notch plastic zone Single primary slip system notch stress ield S individual grain near - tip plastic zone Stage I crack is strongly aected by slip characteristics, microstructure dimensions, stress level, extent o near tip plasticity crack tip plastic zone D a Crack growth controlled by the notch plastic strains Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 58 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 59 o 14

11 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 6 o 14 Small Crack Growth Crack Length Observations 1. mm N = 9 Inconel 718 =. N = 936 Crack Length, mm.5 F-495 H-491 J-63 I-471 C-399 G Cycles Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 61 o 14 Crack - Microstructure Interactions Strain-Lie Data 1µm 1 1mm 1 µm racture Crack size da/dn, mm/cycle F 1-6 E 1-7 D A B C Crack Length, mm Akiniwa, Y., Tanaka, K., and Matsui, E., Statistical Characteristics o Propagation o Small Fatigue Cracks in Smooth Specimens o Aluminum Alloy 4-T3, Materials Science and Engineering, Vol. A14, 1988, Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 6 o 14 Strain Amplitude Reversals, N Most o the lie is spent in microcrack growth in the plastic strain dominated region Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 63 o 14 Stage II Crack Growth Long Crack Growth Locally, the crack grows in shear Macroscopically it grows in tension Plastic zone size is much larger than the material microstructure so that the microstructure does not play such an important role. Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 64 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 65 o 14

12 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 66 o 14 Crack Growth Rates o Metals Stresses Around a Crack Maximum Load Material strength does not play a major role in atigue crack growth monotonic plastic zone Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 67 o 14 Stresses Around a Crack (continued) Crack Closure Minimum Load a b c cyclic plastic zone S = S = 175 S = 5 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 68 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 69 o 14 Crack Opening Load Damaging portion o loading history Mode I, Mode II, and Mode III Mode I opening Mode II in-plane shear Mode III out-o-plane shear Opening load Nondamaging portion o loading history Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 7 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 71 o 14

13 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 7 o 14 Mode I Growth Mode II Growth shear stress slip bands crack growth direction 5 µm 1 µm crack growth direction Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 73 o 14 Damage Fraction N/N 145 Steel - Tension Tension.4 Shear. Nucleation 1 µm crack Fatigue Lie, N Damage Fraction N/N 145 Steel - Torsion 1. Tension Shear Nucleation Fatigue Lie, N Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 74 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 75 o Stainless Steel - Torsion Stainless Steel - Tension Damage Fraction N/N Shear Tension Damage Fraction N/N Tension Nucleation Nucleation Fatigue Lie, N Fatigue Lie, N Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 76 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 77 o 14

14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 78 o 14 Damage Fraction N/N Inconel Torsion 1. Tension.8.6 Shear.4. Nucleation Fatigue Lie, N Damage Fraction N/N Inconel Tension 1. Tension Shear. Nucleation Fatigue Lie, N Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 79 o 14 Multiaxial Fatigue Stress Based Models Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 81 o 14 Fatigue Mechanisms Summary Fatigue cracks nucleate in shear Fatigue cracks grow in either shear or tension depending on material and state o stress Stress Based Models Sines Findley Dang Van Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 8 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 83 o 14

15 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 84 o 14 Shear stress in bending Bending Torsion Correlation 1/ Bending atigue limit 1..5 Shear stress Octahedral stress Principal stress Shear stress in torsion 1/ Bending atigue limit Test Results Cyclic tension with static tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 85 o 14 Cyclic Tension with Static Tension 1.5 Cyclic Torsion with Static Torsion 1.5 Axial stress Fatigue strength 1..5 Shear Stress Amplitude Shear Fatigue Strength Mean stress Yield strength Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 86 o Maximum Shear Stress Shear Yield Strength Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 87 o Cyclic Tension with Static Torsion Cyclic Torsion with Static Tension 1.5 Bending Stress Bending Fatigue Strength 1..5 Torsion shear stress Shear atigue strength Static Torsion Stress Torsion Yield Strength Axial mean stress Yield strength 1.5 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 88 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 89 o 14

16 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 9 o 14 Conclusions Tension mean stress aects both tension and torsion Torsion mean stress does not aect tension or torsion 1 6 Sines α( + + τ oct + α(3 ) =β ( ) + ( ) + ( ) + 6( τ + τ + τ ) + x mean x y mean y x mean z z ) = β y h z xy xz yz Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 91 o 14 Findley τ + k n = max Shear stress in bending Bending Torsion Correlation 1/ Bending atigue limit 1..5 Shear stress Octahedral stress Principal stress Shear stress in torsion 1/ Bending atigue limit tension torsion Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 9 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 93 o 14 Dang Van Isotropic Hardening Σ ij (M,t) E ij (M,t) m τ() t + a () t = b h ij (m,t) ij (m,t) V(M) stabilized stress range Failure occurs when the stress range is not elastic Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 94 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 95 o 14

17 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 96 o 14 Multiaxial Kinematic and Isotropic Dang Van ( continued ) a) b) 3 3 C o Yield domain expands and translates τ τ(t) + a h (t) = b τ Failure predicted O o O o R o 1 1 c) d) C L 3 3 Loading path R L h Loading path h O L O o O o ρ 1 1 ρ* stabilized residual stress Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 97 o 14 Stress Based Models Summary Multiaxial Fatigue Sines: Findley: τ oct + α(3 ) =β τ + k n = max h Strain Based Models Dang Van: τ() t + a () t = b h Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 98 o 14 Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Strain Based Models Plastic Work Brown and Miller Fatemi and Socie Smith Watson and Topper Liu Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 11 o 14

18 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14 Octahedral Shear Strain Plastic Work Plastic octahedral shear strain range Torsion Tension Cycles to ailure Plastic Work per Cycle, MJ/m A T T A A T T A Fatigue Lie, N T A T A T A Torsion Axial o 9 o T 18 o 135 o 45 o 3 o Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 13 o 14 Brown and Miller Case A and B Fatigue Lie, Cycles x x1 x1 1. γ = Growth along the surace Growth into the surace Normal Strain Amplitude, n Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 14 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 15 o 14 Brown and Miller ( continued ) Brown and Miller ( continued ) Uniaxial 1 α α $ ( max ) α γ = γ +S n Equibiaxial γ max ' + S = A n E n, mean b ' + ( N ) B ( N ) c Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 16 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 17 o 14

19 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 18 o 14 Fatemi and Socie γ / 3 Loading Histories C F G H I J Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 19 o 14 Crack Length Observations Fatemi and Socie Crack Length, mm F-495 H-491 J-63 I-471 C-399 G-34 γ 1+ k n,max y ' τ bo ' = (N ) + γ (N ) G co Cycles Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 11 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 111 o 14 Smith Watson Topper SWT ' 1 b ' ' b+ c n = (N ) + (N ) E Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 11 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 113 o 14

20 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 114 o 14 Liu Cyclic Torsion Virtual strain energy or both mode I and mode II cracking W I = ( n n ) max + ( τ γ) W ' ' ' b+ c 4 b I = 4(N ) + (N ) E Cyclic Shear Strain Cyclic Tensile Strain W II = ( n n ) + ( τ γ) max W ' ' ' bo+ co 4τ bo II = 4τγ (N ) + (N ) G Cyclic Torsion Shear Damage Tensile Damage Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 115 o 14 Cyclic Torsion with Static Tension Cyclic Torsion with Compression Cyclic Shear Strain Cyclic Tensile Strain Cyclic Shear Strain Cyclic Tensile Strain Cyclic Torsion Static Tension Shear Damage Tensile Damage Cyclic Torsion Static Compression Shear Damage Tensile Damage Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 116 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 117 o 14 Cyclic Torsion with Tension and Compression Test Results Cyclic Shear Strain Cyclic Tensile Strain Load Case γ/ hoop MPa axial MPa N Torsion.54 45, with tension ,3 with compression , with tension and compression , Cyclic Torsion Static Compression Hoop Tension Shear Damage Tensile Damage Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 118 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 119 o 14

21 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14 Conclusions Loading History All critical plane models correctly predict these results Hydrostatic stress models can not predict these results.3 Shear strain Axial strain Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 11 o 14 Model Comparison Summary o calculated atigue lives Model Equation Lie Epsilon ,6 Garud 6.7 5,1 Ellyin ,45 Brown-Miller 6. 3,98 SWT 6.4 9,93 Liu I ,8 Liu II 6.4 5,4 Chu ,4 Gamma 6,775 Fatemi-Socie 6.3 1,35 Glinka , Strain Based Models Summary Two separate models are needed, one or tensile growth and one or shear growth Cyclic plasticity governs stress and strain ranges Mean stress eects are a result o crack closure on the critical plane Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 1 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 13 o 14 Separate Tensile and Shear Models Cyclic Plasticity 3 τ 1 = τ xy 1 Inconel 145 steel stainless steel γ p γ p γ τ p γ p τ Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 14 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 15 o 14

22 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 16 o 14 γ Mean Stresses ' mean eq = ( N ) + ( N) E ' n b ' + S n= ( S) (N ) + ( S) (N E ' γ n,max τ bo ' co 1+ k = (N ) + γ (N ) y G max ) c W = I b ' ' 1 b ' ' b+ c n = (N ) + (N ) E 1 R [( ) + ( τ γ) ] n n max c Multiaxial Fatigue Fracture Mechanics Models Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Fracture Mechanics Models Mode I growth Torsion Mode II growth Mode III growth Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 18 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 19 o 14 Mode I, Mode II, and Mode III Mode I opening Mode II in-plane shear Mode III out-o-plane shear Mode I and Mode II Surace Cracks Mode II Mode I Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 13 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 131 o 14

23 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 13 o Biaxial Mode I Growth = 193 MPa λ = -1 λ = = 386 MPa λ = -1 λ = 1 Surace Cracks in Torsion Mode II da/dn mm/cycle da/dn mm/cycle Mode III K, MPa m K, MPa m Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 133 o 14 Failure Modes in Torsion Fracture Mechanism Map 6 No Cracks Transverse Longitudinal Spiral Yield Strength, MPa 15 1 Spiral Cracks Longitudinal Transverse Cracks Hardness Rc Shear Stress Amplitude, MPa Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 134 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 135 o 14 Mode I and Mode III Growth 1-3 Mode I and Mode II Growth T6 Aluminum Mode I R = Mode II R = -1 da/dn, mm/cycle K I K III da/dn, mm/cycle SNCM Steel Mode I R = -1 Mode II R = K I, K III MPa m Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 136 o K I, K II MPa m Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 137 o 14

24 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 138 o 14 Fracture Mechanics Models K da = C dn ( ) m K eq [ K ]. 5 I + 8 KII + 8 KIII (1 ν [ ]. 5 eq= KI + KII + (1+ν ) KIII = [ K + K K + K ]. 5 eq = ) K K eq I I II II.5 E K eq( ) = (FII γ) + (FE I ) (1 ) + ν n,max K eq() = FG γ 1+ k πa ys πa Fracture Suraces Bending Torsion Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 139 o 14 Mode III Growth Fracture Mechanics Models Summary Crack growth rate, mm/cycle K = 14.9 K = 1. K = 11. K = 1. K = 8. Multiaxial loading has little eect in Mode I Crack closure makes Mode II and Mode III calculations diicult Crack length, mm Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 14 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 141 o 14 Multiaxial Fatigue Nonproportional Loading Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 143 o 14

25 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 144 o 14 Nonproportional Loading In and Out-o-phase loading Nonproportional cyclic hardening Variable amplitude In and Out-o-Phase Loading x x = o sin(ωt) t x γ xy γ xy γ xy = (1+ν) o sin(ωt) t In-phase y x x = o cos(ωt) t γ xy γ xy = (1+ν) o sin(ωt) t Out-o-phase γ 1+ ν γ γ 1+ ν γ Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 145 o 14 In-Phase and Out-o-Phase Loading Histories γ xy τ xy out-o-phase γ xy / γ xy / diamond x x γ xy τ xy 1 θ x x square γ xy / cross γ xy / x x x x Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 146 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 147 o 14 Loading Histories Findley Model Results out-o-phase diamond in-phase out-o-phase τ/ MPa n,max MPa τ/ +.3 n,max N/N ip in-phase out-o-phase diamond square cross - tension cycle cross - torsion cycle γ xy/ diamond γ xy/ square γ xy/ cross γ xy/ square x x x x cross in-phase Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 148 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 149 o 14

26 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 15 o 14 Nonproportional Hardening In-Phase x γ xy t t x = o sin(ωt) γ xy = (1+ν) o sin(ωt) Axial 6 Shear 3 In-phase x t x = o cos(ωt) γ xy t γ xy = (1+ν) o sin(ωt) Out-o-phase -6-3 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 151 o 14 9 Out-o-Phase Critical Plane Axial 6 Shear 3 Proportional N = 38,5 N = 31, 6 Out-o-phase N = 3,5 N = 4, Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 15 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 153 o 14 Loading Histories Stress-Strain Response 3 15 Case Case 3 15 Case Shear Stress ( MPa ) Case 4 Case 5 Case Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 154 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 155 o 14

27 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 156 o 14 Shear Stress ( MPa ) Stress-Strain Response (continued) Case 7 Case 9 Case Case 11 Case 1 Case Equivalent Stress, MPa Maximum Stress All tests have the same strain ranges Axial Stress ( MPa ) Fatigue Lie, N Nonproportional hardening results in lower atigue lives Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 157 o 14 Nonproportional Example Shear Stresses Case A Case B Case C Case D x x x x Case A Case B Case C Case D τxy τxy τxy τxy y y y y Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 158 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 159 o 14 Simple Variable Amplitude History Stress-Strain on Plane γ xy.6 τ xy 15 x 3 τ xy 15 x x x γ xy Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 16 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 161 o 14

28 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 16 o 14 Stress-Strain on 3 and 6 Planes Stress-Strain on 1 and 15 Planes 3º plane 3 6º plane º plane º plane Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 163 o 14 Shear Strain History on Critical Plane Fatigue Calculations.5 Load or strain history Shear strain, γ time Cyclic plasticity model Stress and strain tensor -.5 Search or critical plane Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 164 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 165 o 14 An Example Biaxial and Uniaxial Solution Analysis model Single event 16 input channels 4 elements Damage Uniaxial solution Signed principal stress Critical plane solution 1-7 From Khosrovaneh, Pattu and Schnaidt Discussion o Fatigue Analysis Techniques or Automotive Applications Presented at SAE Element rank based on critical plane damage Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 166 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 167 o 14

29 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 168 o 14 Nonproportional Loading Summary Nonproportional cyclic hardening increases stress levels Critical plane models are used to assess atigue damage Multiaxial Fatigue Stress Concentrations Proessor Darrell F. Socie University o Illinois at Urbana-Champaign 3 Darrell Socie, All Rights Reserved Outline State o Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations Notches Stress and strain concentrations Nonproportional loading and stressing Fatigue notch actors Cracks at notches Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 17 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 171 o 14 Notched Shat Loading Stress Concentration Factors 6 τ θz θ z M T M X P M Y Stress Concentration Factor Torsion Bending d ρ D D/d Notch Root Radius, ρ/d Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 17 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 173 o 14

30 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 174 o 14 Hole in a Plate Stresses at the Hole 4 λ θ 3 λ = 1 a τ rθ 1 λ = r θ λ r τ rθ θ r θ λ = -1 Angle -4 Stress concentration actor depends on type o loading Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 175 o 14 Shear Stresses during Torsion Torsion Experiments τ rθ.5 1 r a Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 176 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 177 o 14 Multiaxial Loading Thickness Eects Uniaxial loading that produces multiaxial stresses at notches Multiaxial loading that produces uniaxial stresses at notches Multiaxial loading that produces multiaxial stresses at notches Transverse Compression Strain Longitudinal Tensile Strain Thickness 5 mm 3 mm 15 mm 7 mm x y z 1 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 178 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 179 o 14

31 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 18 o 14 Applied Bending Moments Bending Moments on the Shat A M Y M X M X M X M Y M Y B D C B C A A B D C Location D Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 181 o 14 Bending Moments Torsion Loading M A B C D M X M T M X t 1 T z T T 1 t 3 1 = z 1 = T 5 5 M= M M T t z t 4 A B C D M t 1 t t 3 t 4 Out-o-phase shear loading is needed to produce nonproportional stressing T 1 T Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 18 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 183 o 14 Plate and Shell Structures Fatigue Notch Factors K t = 3 K t = 4 Stress concentration actor K Torsion d ρ K Bending K t Bending K t Torsion ρ.1.15 Notch root radius, d D D d =. Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 184 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 185 o 14

32 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 186 o 14 Fatigue Notch Factors ( continued ) Fracture Suraces in Torsion Calculated K bending torsion conservative non-conservative Peterson s Equation K T 1 K = 1 + a 1 + r Circumerencial Notch Experimental K Shoulder Fillet Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 187 o 14 Stress Intensity Factors 1.5 λ = λ = 1. F.75 λ = 1 R.5.5 λ a λ a R da = C ( K ) m eq dn K I = F πa Crack Growth From a Hole Crack Length, mm 1 λ = -1 λ = λ = x x x x 1 6 Cycles Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 188 o 14 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 189 o 14 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 Multiaxial Fatigue - Lecture 3 Darrell Socie, University o Illinois at Urbana-Champaign, All Rights Reserved 19 o 14