Fatigue — S-N curves, Miner's rule: Question bank

3.6.8 · D5 · Physics › Spacecraft Structures & Systems Engineering › Fatigue — S-N curves, Miner's rule

Symbol & concept refresher (traps se pehle padho)

Traps dhundne se pehle tumhe exactly pata hona chahiye ki har symbol ka matlab kya hai aur usse picture karna aana chahiye. Neeche kuch bhi use nahi hoga jab tak yahan define na ho.

Definition Is page par use hone wale symbols

$S$ = stress amplitude = ek cycle mein peak-to-peak stress swing ka aadha hissa, $S = σ_{a} = \frac{σ _{ma x} - σ _{min}}{2}$ . Stress-vs-time graph mein wiggle ki height picture karo.
$N$ = cycles to failure = amplitude $S$ par kitne identical wiggles ke baad part snap karta hai.
$σ_{m}$ = mean stress = woh centre-line jiske aas-paas wiggle oscillate karta hai, $σ_{m} = \frac{σ _{ma x} + σ _{min}}{2}$ .
$R$ = stress ratio $= σ_{min} / σ_{ma x}$ . Yeh cycle ki shape bataata hai: $R = - 1$ = fully reversed (mean zero), $R = 0$ = zero-to-tension, $R \to 1$ = bade mean ke aas-paas chhoti si swing.
$σ_{f}^{'}$ = fatigue strength coefficient = woh stress jo Basquin line $N = 1$ par predict karta hai (ek curve-fit intercept, $σ_{U T S}$ nahi).
$b$ = Basquin exponent = $lo g S$ – $lo g N$ line ke slope ka magnitude, ek chhota number ( $\approx 0.05$ – $0.12$ ).
$S_{e}$ = endurance limit = woh flat stress floor jiske neeche kuch steels effectively hamesha ke liye chalti hain.
$n_{i}$ = level $i$ par apply kiye gaye cycles ki sankhya; $n_{i} / N_{i}$ = damage fraction (us level par use ki gayi life ka fraction).
$D$ = cumulative damage $= \sum_{i} n_{i} / N_{i}$ , ek dimensionless "battery drain" jo failure par $1$ ho jaata hai.
$K_{t}$ = stress-concentration factor = nominal stress se kitne guna zyaada local stress ek notch/hole par hota hai.

Figure s01 dekho: time axis par ek stress cycle draw hai, jisme $σ_{ma x}$ , $σ_{min}$ , mean line $σ_{m}$ , aur amplitude arrow $S = σ_{a}$ sab label hain. Amplitude vertical swing hai; mean woh jagah hai jahan swing centred hai.

Figure — Fatigue — S-N curves, Miner's rule

Figure s02 dekho — log–log axes par S–N curve. Isme teen regions dikhte hain jo tumhe pehchaanne chahiye: left mein low-cycle (plastic) knee jahan Basquin ki straight line bend ho ke toot jaati hai, beech mein straight high-cycle Basquin line, aur right mein endurance floor $S_{e}$ (steel ke liye present, Al/Ti ke liye absent). "Kaun sa regime?" wale har trap ka reference is picture se hai.

Figure s03 dekho — Goodman / Gerber mean-stress diagram. Amplitude $S$ vertical axis par hai, mean $σ_{m}$ horizontal par. Line ke andar ka design point safe hai; wahi amplitude unsafe ho jaata hai jab mean $σ_{U T S}$ ki taraf right mein shift hota hai. Isi tarah nonzero-mean data ko S–N curve mein enter karne se pehle correct kiya jaata hai.

Figure s04 dekho — ek notch local stress ko raise karta hai: nominal field uniform hai, lekin hole par streamlines crowd ho jaate hain isliye peak local stress nominal se $K_{t}$ guna zyaada hoti hai. Wahi local peak, nominal nahi, crack initiation drive karta hai — isliye surface finish aur stress concentration real fatigue mein dominant hote hain.

True or false — justify karo

Yield se neeche safely rakha static stress guarantee karta hai ki part kabhi nahi tutega.

False — fatigue cracks fully elastic range mein bhi cyclic loading ke under nucleate aur grow karte hain, isliye ek static load par safe part phir bhi kaafi cycles ke baad snap ho sakta hai.

Mean stress

σ_{m}

ka fatigue life par koi effect nahi, sirf amplitude matter karta hai.

False — amplitude dominant hai, lekin positive (tensile) mean crack ko thoda khula rakhta hai isliye har swing "zyaada kaatati hai", life chhoti ho jaati hai; figure s03 ki Goodman line exactly isi cheez ko correct karti hai.

Ek true endurance limit ka matlab hai aluminium struts ko infinite life ke liye design kiya ja sakta hai.

False — true endurance limit mainly plain-carbon / low-alloy steel ki khaasiyat hai; aluminium aur common Ti alloys

1 0^{7}

cycles ke baad bhi slope karte rehte hain (figure s02 mein koi floor nahi), isliye ek specified high-cycle strength par design kiya jaata hai.

S–N curve par zyaada stress amplitude ka matlab hai zyaada cycles to failure.

False — ulta hai: zyaada amplitude

S

ka matlab kam cycles

N

, kyunki har harsh cycle crack ko zyaada kholti hai; curve right ki taraf neeche slope karta hai.

Miner's rule kehta hai failure exactly tab hoti hai jab

D = 1

har baar.

False —

D = 1

sirf average linear-damage criterion hai; real tests mein roughly

0.3

–

3

tak scatter hota hai kyunki loading order aur overloads physics badal dete hain, isliye engineers safety factors rakhte hain.

Basquin's law

S = σ_{f}^{'} N^{- b}

ek fundamental physical law hai jo atoms se derive hua hai.

False — yeh high-cycle regime mein

lo g S

lo g N

ka ek empirical straight-line fit hai (figure s02 ka beech wala segment); power law bas wahi hai jo log–log coordinates mein ek line dikhti hai.

Orbit par thermal cycling fatigue nahi kar sakti kyunki spacecraft vibrate nahi kar raha.

False — har ~90-minute ka day/night thermal cycle joints ko strain karta hai kyunki materials expand aur contract karte hain, isliye woh mechanical loading ki tarah thousands of stress cycles deliver karta hai.

Basquin ki straight line

N = 1

tak fatigue life describe karti hai.

False — kam

N

par material plastically yield karta hai aur curve knee mein bend ho jaata hai (figure s02 ka left); is low-cycle regime ko strain-based (Coffin–Manson) behaviour govern karta hai, high-cycle Basquin line nahi.

Spot the error

"Do load levels combine karne ke liye, stresses add karo:

S_{1} + S_{2}

, phir curve se ek

N

padho."

Galat tool — tum damage fractions

n_{i} / N_{i}

(dimensionless life-usages) sum karte ho, stresses nahi; pehle har level ke liye separately S–N curve se har

N_{i}

nikalo.

"Basquin exponent

b

ek bada number hai jaise 5, kyunki fatigue stress ke liye bahut sensitive hai."

Slope magnitude

b

chhota hota hai (

\approx 0.05

–

0.12

); uska reciprocal

m = 1/ b

(~8–20) bada hota hai, isliye

S

mein thodi si drop

N

ko enormously multiply kar deti hai.

"Kyunki

D_{1} = 0.10

aur

D_{2} = 0.40

dete hain

D = 0.50

, humne aadha stress use kar liya hai."

Humne aadha life budget (damage

D

) use kiya hai, stress nahi;

D

tolerable cycles ki total sankhya ka ek dimensionless fraction hai, kisi bhi stress magnitude se unrelated.

σ_{f}^{'}

ultimate tensile strength

σ_{U T S}

hai."

Nahi —

σ_{f}^{'}

fatigue strength coefficient hai, woh stress jo Basquin line

N = 1

par predict karta hai; yeh ek curve-fit intercept hai, generally static

σ_{U T S}

se alag.

D = 0.99

par ek part basically theek hai, kaafi margin bachi hai."

Usne apni 99% fatigue life kharch kar di hai sirf 1% budget baaki hai; yeh brink par hai, aur

D_{f ai l}

mein scatter ko dekho toh woh already fail ho sakta hai.

"Ek smooth polished bar aur bolt-hole wale bar ki fatigue life equal nominal stress par same hoti hai."

Galat — hole local stress ko

K_{t}

factor se raise karta hai (figure s04), isliye cracks bahut pehle initiate hote hain; surface finish aur stress concentration pehle-order ke fatigue drivers hain, details nahi.

"Random launch vibration fatigue nahi hai, yeh bas ek bada load hai."

Random vibration minutes mein varying amplitude ke thousands of stress cycles deliver karta hai, bilkul wahi mixed-amplitude spectrum jiske liye Miner's rule banaya gaya tha.

Why questions

S–N curve ka

N

-axis logarithmic scale par kyun plot kiya jaata hai?

Kyunki

N

kai orders of magnitude span karta hai (

1 0^{3}

1 0^{8}

+); log axis power-law relationship ko figure s02 mein dikhne wali readable straight line mein convert kar deta hai aur saare regimes ek saath visible rehte hain.

Amplitude

σ_{a}

damage kyun drive karta hai sirf peak stress

σ_{ma x}

se zyaada?

Crack baar baar khulne aur release hone se badhta hai; yeh swing

S = σ_{a}

(figure s01) hai jo crack tip ko cycle karta hai, jabki bina swing ke ek steady peak use baar baar work nahi karta.

Miner's rule

D_{f ai l} > 1

kyun de sakta hai (part prediction se zyaada chalti hai)?

Ek bada tensile overload crack tip par compressive residual stress chhod sakta hai jo use band karke baad mein crack growth slow karta hai, part ko

D = 1

ke baad bhi survive karne deta hai.

Agar Miner's rule already failure predict karta hai toh hum safety factors kyun rakhte hain?

Kyunki

D = 1

ek average hai real-world scatter ke saath (~0.3–3) order effects aur material variability se; margin is uncertainty ko cover karta hai taaki design conservative rahe.

lo g S

–

lo g N

plot par ek straight line kyun power law imply karta hai?

Ek line

lo g S = lo g σ_{f}^{'} - b lo g N

exponentiate karne par

S = σ_{f}^{'} N^{- b}

ban jaati hai; log ke andar multiply karna bahar power correspond karta hai, isliye linear-in-logs hamesha power law hota hai.

Jab static stress margin positive ho tab bhi fatigue life kyun check karte hain?

Static margin sirf guarantee karta hai ki turant yield ya rupture nahi hoga; fatigue ek time/cycle failure mode hai jise static check dekh nahi sakta, isliye dono ko independently satisfy karna padta hai.

Goodman correction allowable amplitude ko

σ_{m}

badhne par kyun lower karta hai?

Kyunki tensile mean stress crack ko khula rakhta hai, isliye kam swing tolerable hai; figure s03 ki Goodman line safe amplitude ko neeche slide kar deti hai jab mean

σ_{U T S}

ki taraf jaata hai.

Edge cases

Agar ek load level ka stress amplitude endurance limit

S_{e}

se neeche hai (steel ke liye), toh uska damage contribution kya hai?

Effectively zero — woh cycles figure s02 ke flat floor par baithte hain jahan

N \to \infty

, isliye

n_{i} / N_{i} \to 0

aur woh negligible life budget consume karte hain.

Jab applied amplitude

S

σ_{f}^{'}

ke paas aata hai toh predicted life

N

ka kya hota hai?

N

1 ke paas aata hai, kyunki Basquin's line

N = 1

par

S = σ_{f}^{'}

se guzarti hai; lekin wahan material actually plastic low-cycle knee mein hota hai, isliye Basquin sirf nominal hai.

Aluminium part ke liye, tum legitimately "infinite-life" stress kabhi quote kyun nahi kar sakte?

Aluminium ka S–N curve

1 0^{7}

cycles ke baad bhi koi flat floor ke bina slope karta rehta hai, isliye koi bhi stress eventually failure cause karega; tum ek fixed bade

N

(jaise

5 \times 1 0^{8}

) par strength quote karte ho.

Stress ratio

R

same amplitude

S

ke liye fatigue life kaise change karta hai?

R

ko 1 ki taraf raise karne se mean stress

σ_{m}

badhta hai (crack zyaada khula rehta hai), life chhoti ho jaati hai;

R = - 1

(fully reversed, zero mean) woh benign baseline hai jiske against S–N data usually quote hota hai.

Extreme ratios jaise

R < - 1

(compression-dominated) ya

R > 1

ke baare mein kya?

R < - 1

ka matlab hai cycle mostly compressive hai ek chhoti si tensile blip ke saath, isliye crack driving weak hai aur life lambi hai;

R > 1

fully compressive cycling mein hota hai jahan crack faces dabke band rehte hain aur bulk stress se initiation strongly suppress hota hai.

Kya environment (corrosion, vacuum, temperature) S–N curve shift karta hai?

Haan — ek corrosive medium koi bhi true endurance limit hata deta hai aur poori curve ko lower karta hai (corrosion-fatigue), jabki high temperature creep add karta hai; isliye laboratory S–N data ko on-orbit environment ke liye de-rate karna padta hai.

Agar do identical spectra apply kiye jaate hain lekin ulte order mein, toh kya Miner's rule alag

D

predict karta hai?

Nahi — sum

\sum n_{i} / N_{i}

order-independent hai, yahi Miner ka blind spot hai; real life mein differ hota hai kyunki early overload residual stress chhod deta hai jo baad ke crack growth ko change karta hai.

D = 0

physically kya represent karta hai, aur kya damage kabhi negative ho sakta hai?

D = 0

ek pristine part hai jisme zero cycles apply hue hain; Miner's model mein damage negative nahi ja sakta — yeh sirf failure budget ki taraf upar accumulate hota hai, kabhi heal nahi hota.

Ek part sirf compressive cyclic stress dekhta hai zero tension ke saath — kya fatigue cracking likely hai?

Bahut kam likely, kyunki crack faces ek saath dabbe rehte hain rather than khiche jaate hain; tensile swings hi cracks propagate karti hain, isliye pure compression comparatively benign hai.

Repeated cycles se aaya crack jab bada ho jaata hai, kya S–N framework final failure describe karta hai?

Achhe se nahi — ek baar dominant crack exist karne par, growth Paris' law aur stress intensity se govern hota hai, total-life S–N curve se nahi, jo mainly crack initiation cover karta hai.