Fatigue — S-N curves, Miner's rule — Hinglish notes

3.6.8 · Physics › Spacecraft Structures & Systems Engineering

Fatigue KYA hai?

Spacecraft ke liye kyun important hai: Launch = intense random vibration + acoustic loading + hazaron stress cycles ke kuch minute. Orbit par = har ~90 min mein thermal cycling (din/raat), har cycle joints ko strain karta hai. Kuch bhi ek baar load nahi hota — sab kuch baar baar load hota hai.

Ek cycle describe karne wali key quantities:

$σ_{m} = \frac{σ _{ma x} + σ _{min}}{2} (mean), σ_{a} = \frac{σ _{ma x} - σ _{min}}{2} (amplitude)$

S–N Curve (KYA / KAISE)

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

KAISE banta hai: Bahut saare identical specimens lo. Har ek ko fixed amplitude $S_{i}$ par load karo jab tak woh toot na jaye, $N_{i}$ record karo. Points plot karo. Zyada stress → kam cycles survived.

Basquin power law ki derivation (scratch se)

Empirically, $lo g S$ vs $lo g N$ plot karne par high-cycle regime mein ek straight line milti hai. Log–log mein straight line ka matlab hai power law. Ise build karte hain:

$lo g S = lo g σ_{f}^{'} - b lo g N$

Kyun? Ek line $y = c - b x$ jahan $y = lo g S$ , $x = lo g N$ hai. Dono sides exponentiate karo:

$S = σ_{f}^{'} N^{- b} ⟺ N = (\frac{S}{σ _{f}^{'}})^{- 1/ b}$

$σ_{f}^{'}$ = fatigue strength coefficient ( $N = 1$ par stress).
$b$ = Basquin exponent (slope magnitude, small positive number ~0.05–0.12).

Endurance limit

Miner's Rule (cumulative damage)

Asli problem: Real loading amplitudes ka ek mix hota hai ( $n_{1}$ cycles $S_{1}$ par, $n_{2}$ $S_{2}$ par, …). S–N curve sirf $N_{i}$ = life batata hai agar woh amplitude akele act kare. Hum unhe kaise combine karein?

First principles se derivation

Assume karo ki damage linearly accumulate hoti hai aur level $i$ par har cycle $D$ total unit damage ka equal share $\frac{1}{N _{i}}$ contribute karta hai. $n_{i}$ cycles level $i$ par:

$D_{i} = \frac{n _{i}}{N _{i}}$

Total damage sum hai (independent contributions):

$D = \sum_{i} D_{i} = \sum_{i} \frac{n _{i}}{N _{i}}$

Failure criterion: damage pura budget reach kar le.

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall 12-year-old ko samjhao (Feynman)

Ek paperclip ko aage-peeche moro. Ek baar modna = theek hai. Lekin har wiggle ek tiny crack ko thoda aur bada karta hai, aur kaafi wiggles ke baad — snap! — chahe tumne kabhi zyada zyada nahi khींcha. S–N curve ek chart hai jo kehta hai "ITNA zyada wiggle karo → ITNE wiggles tak chalega." Miner's rule ek phone battery ki tarah hai: har hard wiggle zyada battery drain karta hai, har soft wiggle kam. Jab battery khatam ho jaaye (100%), metal toot jaata hai.

Flashcards

Fatigue failure kya hai?

Cyclic loading se progressive damage aur failure jo static ultimate strength se neeche ki stresses par hoti hai.

S–N (Wöhler) curve ke do axes kya hain?

Stress amplitude

S

(y) vs cycles-to-failure

N

log scale par (x).

Basquin's law state karo.

S = σ_{f}^{'} N^{- b}

, yaani log–log mein ek straight line;

b

Basquin exponent hai.

Endurance limit kya hai aur kis material mein true endurance limit hoti hai?

Ek stress jiske neeche life effectively infinite hoti hai; well-defined limit mainly plain-carbon/low-alloy steels mein hoti hai. Aluminium aur zyaatar common Ti alloys (Ti-6Al-4V) mein NAHI hoti.

Palmgren–Miner's rule state karo.

Failure tab jab

\sum_{i} n_{i} / N_{i} = 1

; har cycle total life ka

1/ N_{i}

fraction use karta hai.

Miner's rule mein

n_{i} / N_{i}

kya represent karta hai?

Stress level

i

par

n_{i}

cycles dwara consume ki gayi total fatigue life ka fraction.

Mean vs amplitude stress formulas?

σ_{m} = (σ_{ma x} + σ_{min}) /2

σ_{a} = (σ_{ma x} - σ_{min}) /2

Miner's rule sirf approximate kyun hai?

Ye load sequence/interaction ignore karta hai; real failure damage sum ~0.3–3 tak range karta hai, isliye safety factors use kiye jaate hain.

σ_{f}^{'} = 1800

MPa,

b = 0.09

S = 300

MPa diya hai,

N

nikalo.

N = (300/1800)^{- 1/0.09} \approx 4.4 \times 1 0^{8}

cycles.

Spacecraft par cyclic load ke do sources?

Launch vibration/acoustics aur on-orbit thermal cycling (~90-min din/raat).

Connections

Stress and Strain — cyclic stress amplitude driver hai
Random Vibration & PSD — launch loads S–N spectrum feed karte hain
Thermal Cycling on Orbit — main long-term fatigue source
Fracture Mechanics & Crack Growth (Paris' Law) — S–N ke peeche physical mechanism
Safety Factors & Margins of Safety — kyun hum $D < 1$ tak design karte hain
Launch Loads & Environments

Fatigue — S-N curves, Miner's rule