Fatigue — S-N curves, Miner's rule: Visual walkthrough

3.6.8 · D2 · Physics › Spacecraft Structures & Systems Engineering › Fatigue — S-N curves, Miner's rule

Step 1 — "Stress ka ek cycle" actually kaisa dikhta hai

KYA HAI. Koi bhi formula aane se pehle, ek metal ki single bar dekho jo bar bar kheenchi (stretch) aur dabayi (squeeze) ja rahi hai. Hum jo quantity track karte hain woh hai stress — material ko cross-section area ke per unit par kitna kheeencha ja raha hai. Ise "metal ke andar tightness" samjho. Hum ise $σ$ (Greek letter sigma, stress ka hamaara symbol) kehte hain.

KYUN. Fatigue repeated loading ke baare mein hai, isliye sabse pehli cheez jo hum draw karenge woh hai ek repeat kaisa dikhta hai. Agar hum ek wiggle ke top, bottom, aur middle ko naam nahi de sakte, toh baad mein "damage per wiggle" ka matlab nahi bata sakte.

PICTURE. Figure mein, stress $σ$ time ke saath wave ki tarah upar-neeche hota hai. Teeno heights padhein:

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

$σ_{ma x}$ — wave ka sabse ooncha point (metal sabse zyada kheeencha hua).
$σ_{min}$ — sabse neecha point (metal dabaaya hua, ya sabse kam kheeencha hua).
Midline aur swing ki half-height ke apne names hain, jo aage aate hain.

$σ_{m} = average of top and bottom \frac{σ _{ma x} + σ _{min}}{2}, σ_{a} = half of the full up-down swing \frac{σ _{ma x} - σ _{min}}{2}$

$σ_{m}$ (mean) — midline ki height. Yeh batata hai ki poori wave kitni upar shift hai. Geometrically yeh do dashed lines ka average hai.
$σ_{a}$ (amplitude) — top aur bottom ke beech vertical distance ka aadha. Yeh swing kitni badi hai measure karta hai, chahe midline kahaan bhi ho.

Ab hum $σ_{a}$ (swing) ko end tak follow karenge — yeh main character hai. Mean $σ_{m}$ apne khaas step (Step 6) mein wapas aayega, jahan hum exactly dekhenge ki shifted midline life ko kaise change karti hai.

Step 2 — Ek wiggle ek crack badhati hai: amplitude kyun king hai

KYA HAI. Metal ki surface pe zoom karo. Hamesha ek tiny flaw hota hai — ek scratch, ek grain boundary, ek machining mark. Jab bar kheenchi jaati hai, woh flaw thoda sa khul jaata hai; jab dabaaya jaata hai, band ho jaata hai. Har open–close ek cycle hai, aur har cycle crack ko thoda aur lamba kar deta hai.

KYUN. Yahi physical reason hai ki ek bar ek bada pull survive kar sakta hai lekin hazaar chhote pulls se toot jaata hai — aur yeh batata hai ki kaun sa number damage ko control karta hai. Swing $σ_{a}$ hi crack ko open aur close karti hai, isliye $σ_{a}$ driver hai. Mean $σ_{m}$ sirf crack ko thoda ajar rakhhta hai taaki har swing thodi aur gehri kaati.

PICTURE. Figure mein dekho crack tip kaise bahar ki taraf ratchet karta hai, ek notch per cycle. Kitni tez ek crack per cycle lambi hoti hai iska deeper physics Paris' Law hai; yahan hum sirf yeh jaante hain ki "zyada swing ⇒ per cycle zyada growth".

Step 3 — Data collect karo: raw stress vs life

KYA HAI. Bahut saari identical bars lo. Har ek ko ek alag, fixed amplitude do aur use tab tak wiggle karo jab tak woh toot na jaaye, record karo ki woh kitne cycles $N$ chala. Har bar ke liye ek dot plot karo: swing size upar, life across.

KYUN. Hum swing size aur life ko connect karne wala ek law chahte hain. Law ka andaaza nahi lagaya ja sakta — pehle data ki shape dekhni padti hai. Yeh step pure experiment hai: abhi koi formula nahi.

PICTURE. Ordinary (linear) axes par dots left side par steep neeche girte hain aur phir floor se chipak jaate hain — ek itna tedha curve ki fit karna bekar hai. Dhyan do ki dots tens of cycles se hundreds of millions tak phailey hain: $x$ -values kai powers of ten span karte hain.

Vertical axis: $S = σ_{a}$ , hamari chosen swing size.
Horizontal axis: $N$ , measured life.
Left side par steep bend woh problem hai jo hum aage fix karte hain.

Step 4 — Logarithm kyun lete hain (aur log hota kya hai)

KYA HAI. Logarithm yeh sawal ka jawab deta hai "ten to the what gives this number?" — $lo g_{10} 1000 = 3$ likhte hain kyunki $1 0^{3} = 1000$ . Toh $lo g_{10}$ kisi number ki powers of ten mein size ko ek plain chhote number mein badal deta hai: $10, 1 0^{2}, 1 0^{3}$ ban jaate hain $1, 2, 3$ , evenly spaced.

YEH TOOL KYUN. Hamare $N$ values $\sim 10$ se $\sim 1 0^{8}$ tak hain — aath powers of ten. Linear axes chhote numbers ko origin par ek smear mein daba dete hain. $lo g_{10} N$ lena us smear ko evenly stretch kar deta hai taaki hum actually structure dekh sakein. $lo g_{10} S$ bhi lete hain, symmetry ke liye. Yahi ek tool hai jo "many orders of magnitude mein spread" ko "evenly spread" mein badal deta hai.

PICTURE. Unhi dots ko $lo g_{10} N$ across aur $lo g_{10} S$ upar ke saath re-plot karo. Tedha curve snap karke ek straight line ban jaata hai (high-cycle region mein). Straight line ek gift hai: straight lines ka ek simple equation hota hai.

Step 5 — Straight line se Basquin's law tak

KYA HAI. Straight line hai $y = c - b x$ . $y = lo g_{10} S$ aur $x = lo g_{10} N$ rakhein:

$height lo g_{10} S = c : intercept (value at l o g_{10} N = 0) lo g_{10} σ_{f}^{'} - slope magnitude b across lo g_{10} N$

KYUN. Step 4 mein humne ek line find ki; ab hum uske do numbers ko naam dete hain taaki us se compute kar sakein. Har line ko exactly do cheezein chahiye: kahaan shuru hoti hai (intercept) aur kitni steep hai (slope).

PICTURE. Figure mein, intercept line ki height hai $lo g_{10} N = 0$ par, yaani $N = 1$ (ek single cycle) par. Slope $- b$ hai "har 1 across ke liye $b$ neeche". Dono marked hain.

$σ_{f}^{'}$ — fatigue strength coefficient: woh stress jo line $N = 1$ par predict karti hai. Yeh line ki height set karta hai.
$b$ — Basquin exponent: swing badhne par life kitni tezi se girti hai (chhota, $\sim 0.05$ – $0.12$ ). Minus sign ka matlab life girti hai jaise stress badhta hai.

Ab logarithm undo karo dono sides par $10$ raise karke — Step 4 ka exact inverse move, usi base 10 ka use karke jo humne wahan choose kiya tha:

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

Left form: life $N$ do, woh swing $S$ milegi jo woh le sakti hai.
Right form: swing $S$ do, life $N$ milegi. (Humne dono sides ko $- 1/ b$ power par raise karke $N$ ke liye solve kiya.)

Step 6 — Mean stress $σ_{m}$ curve ko kya karta hai

KYA HAI. Ab tak sab kuch yeh assume karta tha ki wave symmetrically zero ke around swing karti hai, yaani $σ_{m} = 0$ . Lekin ek bolt jo tension mein pre-loaded hai, ya ek pressurised tank, ka midline upar shift hota hai: $σ_{m} > 0$ . Woh shift matter karta hai, kyunki ek crack jo positive mean ki wajah se pehle se ajar hai, har swing par aur zyada khulta hai aur tezi se badhta hai. Toh wahi swing $S$ shorter life deti hai jab $σ_{m}$ positive ho.

YEH TOOL KYUN. Basquin's law sirf swing $S$ jaanta hai; woh midline se blind hai. Mean ko fold karne ke liye hum ek real $(σ_{a}, σ_{m})$ pair ko ek equivalent zero-mean amplitude $σ_{a r}$ mein convert karte hain — woh swing jo, bina kisi mean ke, same damage karti. Do standard conversions yeh karte hain, material ki ultimate strength $σ_{U T S}$ use karke (woh stress jo ek pull mein todi hai, Stress and Strain se):

$σ_{a r} = Goodman (straight line) \frac{σ _{a}}{1 - \frac{σ _{m}}{σ _{U T S}}}, σ_{a r} = Gerber (parabola) \frac{σ _{a}}{1 - ( \frac{σ _{m}}{σ _{U T S}} ) ^{2}}$

$σ_{a}$ — real swing jo tumne apply ki.
$σ_{m}$ — real midline height; agar $σ_{m} = 0$ ho toh dono formulas collapse hokar $σ_{a r} = σ_{a}$ dete hain aur kuch nahi badalta.
$σ_{U T S}$ — ek pull mein breaking stress; jaise $σ_{m} \to σ_{U T S}$ hota hai, denominator $\to 0$ hota hai aur $σ_{a r} \to \infty$ (breaking stress ke paas mean ka matlab hai part pehle se almost gone hai — zero life).
$σ_{a r}$ ko $S$ ki jagah Basquin mein feed karo mean-corrected life padhne ke liye.

PICTURE. Figure allowable amplitude $σ_{a}$ ko mean $σ_{m}$ ke against plot karta hai. $σ_{m} = 0$ par tumhe full endurance amplitude milti hai; Goodman line seedhi neeche $σ_{U T S}$ par zero tak jaati hai, Gerber parabola uske upar bulge karti hai. Curve ke neeche koi bhi $(σ_{m}, σ_{a})$ point survive karta hai; us par ya usse upar fail karta hai.

Step 7 — Edge case: kya line kabhi girnaa band karti hai?

KYA HAI. Basquin ki straight line, agar hamesha trust ki jaaye, toh kehti hai ki koi bhi swing eventually bar ko tod degi (life sirf astronomically badi ho jaati hai). Real materials yahan do families mein split hoti hain.

KYUN. Ek derivation ko sab cases cover karne chahiye. Line sirf humara jo data hai uska fit hai; bahut kam stress par real curve flat bend ho sakta hai — ya nahi bhi ho sakta. Kya hota hai yeh material par depend karta hai, aur yeh galat samajhna ek classic, dangerous mistake hai.

PICTURE. Ek plot par do curves:

Steel (plain-carbon / low-alloy): curve flat ho jaata hai ek stress par jise endurance limit $S_{e}$ kehte hain. $S_{e}$ se neeche swing ⇒ effectively infinite life. Basquin line ki jagah ek horizontal floor aa jaata hai.
Aluminium & most titanium (e.g. Ti-6Al-4V): koi flat floor nahi — line $1 0^{7}$ cycles ke baad bhi neeche slope karti rehti hai. Inke liye tumhe ek stated life par strength quote karni padegi (maan lo $N = 5 \times 1 0^{8}$ ), kabhi "infinite life" nahi.

Step 8 — Real life ek mix hai: damage-fraction idea

KYA HAI. Real launch aur orbit loading ek steady swing nahi hoti — yeh $n_{1}$ cycles swing $S_{1}$ par hoti hai, phir $n_{2}$ $S_{2}$ par, aur aise aage. S–N curve sirf jaanta hai ki har swing akele kya karta hai ( $N_{1}$ , $N_{2}$ , …). Hum inhe combine karna chahte hain.

KYUN. Random vibration aur thermal cycling kai amplitudes ke messy spectra deliver karte hain. Combining rule ke bina S–N curve real spacecraft ke liye bekar hai.

PICTURE. "Fatigue life" ke liye ek fuel gauge socho, jo full shuru hoti hai. Level $i$ par ek cycle tank ka $1/ N_{i}$ fraction drain karta hai (kyunki $N_{i}$ aise cycles usse exactly empty kar denge). Toh $n_{i}$ cycles drain karte hain

$D_{i} = cycles done \div cycles it would take alone \frac{n _{i}}{N _{i}}$

$n_{i}$ — cycles jo tumne actually level $i$ par apply kiye.
$N_{i}$ — level $i$ par cycles-to-failure, S–N curve se padha (Step 5, Step 6 se mean-corrected agar zaroorat ho).
$D_{i}$ — tank ka woh dimensionless slice jo level $i$ ne use kiya.

Step 9 — Slices ko sum karna: Palmgren–Miner rule

KYA HAI. Assume karo ki slices independent hain aur bas add ho jaate hain. Total damage sabhi fractions ka sum hai; tank empty hota hai (part fail hota hai) jab total 1 (100%) tak pahunche.

KYUN. "Linear, independent accumulation" sabse simple honest guess hai — ise S–N curve ke alawa koi extra data nahi chahiye. Yeh nature ka law nahi balki ek average model hai, aur yehi exact reason hai ki hum baad mein safety factors multiply karte hain.

PICTURE. Fraction-bars ko end to end stack karo "1" marked line ki taraf. Line cross karo ⇒ failure.

Worked check (numbers jo parent se match karte hain)

Ek-picture summary

Sab kuch ek canvas par: ek wiggling stress wave (Steps 1–2) log–log S–N line ko feed karti hai (Steps 3–5), mean-stress correction allowable swing ko bend karti hai (Step 6) aur steel-vs-Ti fork low-stress tail set karta hai (Step 7); us line se lives padhna ek fatigue tank bharta hai jiske slices 1 tak sum hote hain (Steps 8–9).

Recall Poore walkthrough ki Feynman retelling

Socho ek metal bar ko squeeze aur stretch kar rahe ho. Ek squeeze-and-stretch ek cycle hai; swing kitni badi hai, woh amplitude hai — hum ise $S$ kehte hain. Har swing ek tiny built-in scratch ko thoda aur wide kheenchti hai, toh enough swings ke baad — snap. Humne bahut saari bars ko alag alag swings par todaa aur record kiya ki har ek kitne swings survive kari ( $N$ ). Numbers millions mein phailey the, toh humne inhe base-ten logarithms se squish kiya (jo sirf powers of ten count karte hain), aur dots straight line mein aa gaye. Log-paper par straight line ka matlab power law hai, aur woh hai Basquin's $S = σ_{f}^{'} N^{- b}$ : ek swing do, ek life lo. Agar bar average mein stretched bhi hai (ek positive mean stress), toh crack ajar baith jaata hai aur jaldi mar jaata hai — Goodman/Gerber correction allowed swing ko shrink karta hai iske account ke liye. Steel ka ek safe swing hai jiske neeche woh hamesha ke liye chalti hai (endurance limit); aluminium aur titanium mein nahi hota, toh kabhi unhe infinite life promise mat karo. Finally, real rockets ek mix of swings se shake karte hain. Fatigue ko battery ki tarah treat karo: har swing iska $1/ N$ drain karta hai. Sabhi drains add karo — jab total 1 hit kare, part dead hai. Woh sum Miner's rule hai, aur kyunki real metals order-of-loading par thoda cheat karte hain, hum hamesha ek safety margin rakhte hain.

Active Recall

S–N ko log–log axes par kyun plot karte hain?

N

kai powers of ten span karta hai; base-ten logs inhe evenly space karte hain aur power-law data ko straight line mein badal dete hain.

Kya

S

ek nayi quantity hai?

Nahi —

S

sirf Step 1 ke stress amplitude

σ_{a}

ka S–N-chart naam hai.

Positive mean stress

σ_{m}

fatigue life ko kya karta hai?

Woh crack ko ajar rakhta hai jisse woh tezi se badhta hai — life choti hoti hai; Goodman/Gerber corrections

(σ_{a}, σ_{m})

ko ek equivalent zero-mean amplitude mein convert karte hain.

S–N line par

N = 1

par intercept kya represent karta hai?

Fatigue strength coefficient

σ_{f}^{'}

, woh stress jo fit ek single cycle ke liye predict karta hai.

Miner's rule mein level

i

par ek cycle ka cost kya hai?

Total fatigue life ka fraction

1/ N_{i}

;

n_{i}

cycles ka cost

n_{i} / N_{i}

hai.

Kin materials mein true endurance limit nahi hota?

Aluminium aur most titanium alloys (e.g. Ti-6Al-4V) — inke S–N curves neeche slope karte rehte hain.