6. Fatigue strength
Fatigue fracture What is fatigue fracture? Under a constant cyclic loading or irregular cyclic loading, Component fractures How to fracture Under cyclic loading, component suddenly fractures Stress level Under low stress level below yield stress, fatigue fracture happens Fracture cause 8-9% of fracture is by fatigue Corrosion, Rupture 3% Delayed fracture Stress corrosion 5% Thermal fatigue Corrosion fatigue Fretting fatigue Low cycle fatigue Static fracture 3% % Fatigue 8% 6%
Fatigue and fatigue fracture surface Fatigue fracture ()Origin ()Crack growth Surface of component Stress concentration (Notch, Key, Inclusion) After crack initiation, along max. stress plane Smooth surface, macroscopically few plastic deformation (3)Macroscopic (4)Microscopic Beach mark (under irregular cyclic stress) Striation (5)Other When crack grows, cross section decreases Ductile fracture Rough surface
Fatigue fracture factor Main factor Stress Compression(-) Tension(+) time other causes Stress concentration Environment Combined stress () Max. tensile stress Over loading () Cyclic stress Large Residual stress (3) Number of cyclic stress Metallurgical
Cyclic stress I max a m min m : Mean stress max + min m = a : Stress amplitude max min a = min R : Stress ratio R = max Stress Tension(+) Compression(-) m = R = m < a < R < m = a R = m > a R > Time (a)alternating (b)partial alter (c)pulsating Definition of cyclic stress (d)partial Pulsating
Cyclic stress test Rotating fatigue tester Stress Tension(+) Compression(-) m = R = Time Alternating A point on surface of sample cycle Upper Sid e Lower Side Upper Compression Tension Compression Rotating fatigue tester Specimen
S ー N curve I Fundamental diagram to evaluate fatigue property Cyclic stress(stress amplitude a )- Cyclic number to failure Stress amplitude a [MPa] 3 Nominal stress P Failure Non failure N Mild steel Aluminum alloy Rotaing bending m = (R= ) 5 6 7 8 Fatigue life log N f What does P mean a = cyclically applies Specimen fails N = N Cyclic number to failure N S-N curves of almimum alloy and mild steel
S ー N curve II Stress aplitude a [MPa] 3 Failure Non failure Mild steel Aluminum alloy Rotating bending m = (R= ) Fatigue strength at 7 Fatigue limit 5 6 7 Cyclic number to failure N 8 (S-N curves aluminum alloy and mild steel) Mild steel Fatigue limit = 6MPa Aluminum alloy 7 fatigue strength = 35MPa Fatigue limit 7 cycles fatigue strength Clear knee point (Mild steel, Titanium, Carbon steel) Over 7 cycle, fatigue life = (Super long life region, a decreases) Not clear knee point(non iron metals) Fatigue life is not at N= 7 cycles, fatigue strength
S ー N curve Ⅲ (Extremely Low Cycle Fatigue) (Low Cycle Fatigue) (High Cycle Fatigue) Hysteresis loop a ; High level Plastic deformation a 3 4 5 6 7 Cyclic number to failure N f Elastic regio a ; Elastic stress
P ー S ー N curve Unevenness of fatigue life Material, geometry, stress ratio, stress amplitude are constant Life differs times Fatigue probability Stress amplitude a % 5% P =.5(S N curve) P=.99 P=.9 P=.5 P=. P=. P N ; Fracture probability at N P N N N N Cyclic number to failure log N P-S-N curve N Area ; Fracture probability till N
Fatigue limit and mechanical prpertiesⅠ Mechanical properties Yield strength S Tensile strength B Vickers hardness Probe Diamond probe Sample d Brinnel H B Vickers H V Fatigue limit w Iron and steel Aluminum alloy Cupper alloy Load P W HV = = cos Surfacearea dd W =.854 [ kgf / mm ] d d (Rotating bending ). 5 w, B w (.6 ±.) HV w. 33 B w. 5 B d
Fatigue limit and mechanical propertiesⅡ Fatigue limits and static strength of iron and steel Alternating torsion Alternating tension compression Fatigue limits and static strength of iron and steel Ratio of tensile strength (/B) Ratio of Vickers (/HV).3..43.5 Pulsating tension.33. Plane bending.5.8
Low cycle fatigue (Extremely Low Cycle Fatigue) (Low Cycle Fatigue) Hysteresis loop a ; High level (Plastic deformation) Short fatigue life a Under high temperature, Machine 3 4 5 6 7 Cyclic number to failure N f Cyclic thermal strain Nuclear vessel Steam turbine
Hysteresis loopⅠ Tensile strain High loading B ε m E A Yielding a Unloading Stress m Strain ε D ε pa ε a Δε p ε pa ε ea ε a C a Δ Compression Yielding Δε Compression (Bauschinger effect ) Hysteresis loop
Hysteresis loopⅡ B Stress m Strainε ε m E ε pa A Δε p ε pa C ε ea a a Δ Δε t ; Total strain range ε ea ; Elastic strain ε pa ; Plastic strain ε m ; Mean strain Δε e ; Elastic strain range Δε p ; Plastic strain range D ε a Δε Hysteresis loop ε a Δ Δε t = Δεe + Δεp = + Δεp E Area of hysteresis loop = Plastic work/ volume Low cycle fatigue Transfer to heat Low speed
Hysteresis loop Ⅲ Static stress-strain curve Stress Stress range changes with increasing N Annealed steel Δ increase Cold rolling steel Δdecrease Till 5% of life Shape of hysteresis loop saturates Δε Strain Cyclic stress-strain curve Δ Cyclic stress-strain curve Δ = Δε K ' n' Δ ; Cyclic Stress K ; n ; Cyclic hardening index ( General n.5)
Strain range and fatigue life Relation between Δε p and N f of low cycle fatigue Plastic strain range Δε p - - -3 - ε f b.5 N f = 4 Manson-Coffin relation Δε N b p f = C 3 4 Cyclic number to failure N f Manson-Coffin law 5 Δε C ( 式 6.5) N b p f = b,c ; Constant ε f (For many materials,b.5) A = ln Af = ln φ A ; Cross section before A ; Cross section after φ; Reduction of area ε f ; Failure ductile N f =/4 cycle, Δε p =ε f
Microscopic fracture appearance Crack initiation, First stage of crack growth Enlargement Cyclic stress Surface Slip band Intrusion Surface Extrusion (Ⅰ) First stage of crack growth Aluminum alloy Crack initiation continuously relates to growth Steel, Titanium Crack size is similar to grain size
Microscopic fracture appearance Ⅱ Stage IIa of crack growth process Cyclic stress Direction of crack growth Small crack Grow in grain (along slip plane) 試験片表面 Stress concentration gives rise to damage at crack tip Continuous Surface (Ⅰ) (Ⅱa) Stage II crack growth Crack growth rate Crackgrowt hrate = da dn (a ; crack length N ; Cyclic number Crack tip Granular High Intergranular Delay
Microscopic fracture appearance Ⅲ Stage IIb of crack growth process Cyclic stress Crack growth direction Microscopic structure effect transfer 試験片表面 Mechanics factor (Stress intensity factor) (Striation) Surface da dn = 分の数 μm / cycle Pure Titanium (Ⅰ) (Ⅱa) (Ⅱb) Stage II of crack growth Striation spacing crack growth rate
Microscopic fracture appearance Ⅳ Stage IIc of crack growth process Crack growth direction Striation Cyclic stress High crack growth rate (High strength steel Cleavage, intergranular cracking ) Final fracture Surface Ductile fracture (Ⅰ) (Ⅱa) (Ⅱb) (Ⅱc) Stage II of crack growth
Crack growth lawⅠ Linear fracture mechanics K Ⅰ = K Ⅰ For different crack length, the same stress intensity factor Elastic stress and elastic plastic stress becomes the same a a At crack tip, the same fracture happens (a)same elastic stress field Small yielding condition Application to fatigue crack ρ= Plastic zone (b)same Plastic elastic stress Crack growth properies Stress intensity factor range ΔK ΔK K max = K K = Δ πa F max min = πa F max, K min = πa min F
Crack growth rateⅡ Stress range Δ Δ Stress ratio = Δ ; Cyclic stress,a ; (a) Time t ΔK a き裂長さ a For long a, S.I.F.range ΔK き裂長さ a (b) ΔK =ΔK Time t Stress intensity factor range,δk, change S. I. F. K Driving force Δ >Δ Δ = a Δ a Δ K = Δ K Large
Crack growth rate log(da/dn) Crack growth rate Ⅲ(Paris law) Threshold Steady growth m Final failure Paris law da dn = C ( ΔK ) m C, m ; material constant For many materilas, m = ~7 Resistance of crack growth Fatigue life estimation Stress intensity factor log(δk) ΔK th ; Threshold S.I.F. Lower limit of crack growth ΔK decrease da/dn Before fractureδk ( R) ΔK = K fc (R ; stress ratio,k fc ; Fatigue fracture toughness)
Notch effectⅠ(notch) (Notch) Cross section suddenly changes Hole Screw Key Defect etc. Origin of crack Stress concentration at notch root Fast crack growth Fracture How to evaluate stress concentration FEM Notched component few data of fatigue 凹凸 Decrease of fatigue strength
Notch effectⅡ(fatigue limit of notched material) ρ Fatigue limit of notched material Fatigue limit w A w Branch ρ=ρ B Fatigue strength w Rotating bend Non propagating crack Stress concentration Crack strength w K t Fatigue limit of notched material D C Fatigue limit (Two types) Fatigue strength w For Smooth specimen, Limit stress not to initiate crack Crack strength w Fracture stress to occur non-propagating crack Crack initiates, but not fracture
Notch effectⅢ(fatigue notch factor K f ) ρ Branch point B Material constant Dependence on ρ Fatigue limit w A w Branchρ=ρ B Fatigue strength w Rotating bending Crack strength D w Non-propagting crack C Fatigue strength w ρ>ρ ; No non-propagating crack Crack strength w ρ<ρ ; Non-propagating crack Fatigue notch factor K f How much is Fatigue limit decreased by the notch Stress concentration K t Fatigue limit of notched specimen Fatigue limit of smooth specimen w w w K f =, K f = w w
Fatigue limit and stress concentration. Fatigue limit w / w, w / w.5 K t =K f S3C Notch depth t =.mm t =.5mm d t ρ w w w w w Fatigue strength w w w Crack strength w w Branch point Different notch depth K t of branch point differs Rotating bending(d=5mm) 3 4 Different diameter Kt of branch point differs Stress concentration K t
/ρand K t /K f Ⅰ K t / K f, K t / K f.. S3C ρ.5mm Notch depth t =.mm t =.5mm K t / K f K t / K f /ρ [mm - ] Relation between, K t / K t f / and K f /ρ If Elastic Max. stress and notch radii is the same, Fatigue limit is the same.
Fatigue limit and non-propagating crackⅠ mac =+ a x Micro-non-propagating crack min =- a Non-propagation of micro-crack δ (δ: Crack opening displacement) Plastic zone l Plastic zone Fatigue limit of steel After initiated crack grows, Limited stress which non-propagates (Threshold stress which crack does not grow.) δ; very small at crack tip like closing Size of non-propagating crack Size of inclusion and defect No opening at crack tip Non-propagation Effect on fatigue limit
Fatigue limit and non-propagating crackⅡ No notched specimen Fatigue limit of smooth and notched specimen of steel Nonpropagation
Notch sensitivity K t / K f, K t / K f.. η = S3C ρ.5mm K K f t Increase of notch sensitivity K t / K f K t / K f K t = K f (Max. notch sensitivity) Notch depth t =.mm t =.5mm /ρ [mm - ] K t =.67 and ρ=mm Notch sensitivity factor Relation between, K t / K f t / とK / f and ρ (<η<) Pure Ti SC Al alloy η=.8 η=.69 η=.88 Insensibility Sensitivity
Size effect ρ Rotating bending Two main factors ρ Rotating bending Size effect For the same materials, Big Size Strength decrease Stress gradient Similar size of specimens /ρ increases K t K t, K K increases f f Large Surface area (statistical factor) Dangerous cross section increases Probability of existing microcrack increases Decrease of fatigue strength For the same K t, K f and K f decrease w K f =, K w f = w w Σ w and w increase
Mean stress effectⅠ Diagram of fatigue limit F E w Stress amplitude A Alternating Push-pull H 45 S G a C Pulsating Push-pull 45 D Large plastic deformation happens S ; Yield stress Effect of mean stress on fatigue limit B A ; Smooth specimen w B ; True facture stress T - S m S Diagram of fatigue limit Mean stress T Area FECD ; Possible area of Safety use
Surface effectⅠ (Effect of surface roughness) Effect of surface roughness Large Surface roughness Fatigue strength m = w / w..5 Al alloy Bending fatigue 凹凸 Ti alloy Annealed steel Steel. Surface roughness H max [μm] Decrease of fatigue limit
Surface effect
Estimation of fatigue lifeⅠ (Low cycle fatigue) Strain range Δε p, Δε t Low cycle fatigue - - Manson-coffin law Δε N b p f = C Steel Total strain range-life curve.53 ( Δε t.5) N =. 3 Plastic strain range-life curve.55 p N f εf Δε = b.55 f C = ε f -3 3 4 5 Number of cycle to failure Strain range fatigue life curve N f Practical aspect Important total strain Range-fatigue life curve
Estimation of fatigue lifeⅡ (Crack growth life) Crack growth life Paris law da dn = C ( ΔK ) m C, m ; Material constant For many materials,m = ~7 Integration N C = N C dn = a i ; Initial crack length a c ; Critical crack length a a i c C ( ΔK ) m da N C = = a a i c C da = ( ) m ( ) m ( ) a Δ πa F CΔ π F a { ( ) ( ) } m m a a i ( m ) C( Δ π F ) m c a i c m da
Linear cumulative damage lawⅠ Under Fluctuating stress, estimation of fatigue life Stress Stress range changes during cycle For Stress, Fatigue life N f = N Time For Stress, Fatigue life N f = N Stress (a) Time Miner law) After cycles n (n <N ), cycles n n n D = + = ( N N (D;cumulative damage) (b) Fatigue damage step stage fluctuating stress
Linear cumulative damage lawⅡ Under Fluctuating stress, estimation of fatigue life Stress Miner law After cycles n (n <N ), Stress (a) Time D = cycles n n N + = n N (When D =, fatigue fracture) Actually, Time (a) Cyclic stress High Low (b)cyclic stress Low High (b) D< D> step stage fluctuating stress (For some case, D=.~ must modify)
Linear cumulative damage lawⅢ Stress amplitude a 3 n N n N n 3 Miner law Σ(n i /N i )= Modified miner law W N 3 * N 3 = Number of cycles N Linear cumulative damage law
平均応力 残留応力の影響 Ⅱ( 疲労限度線図 ) w 平均応力の影響 ( 教科書 P84 図 6.6(b)) 疲労限度 a ゲルバー線修正グッドマン線図ゾーダーベルク線平均応力 ; m S B 疲労限度 = a w m B n B ; 引張り強さ m ; 平均応力 w ; 平滑材の疲労限度 n = 直線 n = 放物線 n = S に置き換えた 残留応力の影響 圧縮残留応力 圧縮の平均応力が作用する 引張り残留応力 引張りの平均応力が作用する に対応する
切欠材の疲労限度の推定 切欠材の疲労限度の推定 ( 教科書 P84 図 6.6(b)) 縦軸 ; w K f w ; 平滑材の疲労限度 K f ; 切欠係数 疲労限度 a ゲルバー線 修正グッドマン線図 横軸 ; 引張り強さ B w w / K f ゾーダーベルク線平均応力 ; m 切欠材の疲労限度として評価 S 図. 疲労限度線図 B
組み合わせ応力下の疲労強度 Ⅰ ( 教科書 P86 図 6.7) M b a c d z M y x M T b a c d z T M y x b a c d z T y x y τ y τ y τ max = (a) 曲げ き裂発生領域 x 45 き裂発生領域き裂発生領域 τ τ x x τ τ θ τ τ max = τ τ = τ τ + max (b) 曲げ+ねじり (c) ねじり 図. 曲げ ねじり組み合わせ応力における最大せん断応力
組み合わせ応力下の疲労強度 Ⅱ 疲労き裂 主として せん断応力の繰り返しで発生 max max = + τ τ τ = + w τ w τ max τ = w max w τ = ( 曲げのとき ) ( ねじりのとき ) 近似 ( ) w τ = + ( 式 6.7) w w τ = と仮定する max τ τ + = y M M T T x z a b c d ( 注意 ) 曲げとねじりの応力サイクルの位相差がある時 ない時に比べて τ max が小さい 切欠材では 曲げとねじりで K t および ρ が異なる
表面効果と疲労限度 Ⅲ ( 環境の影響 ) 塩水などによる腐食 腐食ピットの発生 応力腐食割れ 疲労限度低下 高温環境 表面層の酸化 軟化 疲労限度低下
表面効果と疲労限度 Ⅱ ( 表面処理の影響 ) 鋼材の焼きなまし 焼きならし 脱炭 浸炭 窒化 3 表面圧延 4ショットピーニング 5メッキ C 含有量の減少により軟化 疲労き裂が発生しやすい引張り残留応力の発生 疲労限度低下 硬度が増す圧縮残留応力の発生 疲労限度上昇 加工硬化 圧縮残留応力の発生 疲労限度上昇 メッキ層中の微細な割れ 疲労限度低下 6 コーティング イオン注入 強化層の形成 ( 研究中 ) 疲労限度上昇
/ρ と K t /K f による整理 Ⅱ 表. 材料の引張り強さ 疲労強度 ρ と近似式の係数 + + + = ρ ρ ρ 3 3 3 A A A K K f t ( 式 6.) (ρ>ρ ) A, B ; 係数 ρ ; 切欠半径 + + + = ρ ρ ρ 3 3 3 B B B K K f t ( 式 6.) (ρ<ρ ) ( 教科書 P8 表 6.)