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

3.6.8 · D1 · Physics › Spacecraft Structures & Systems Engineering › Fatigue — S-N curves, Miner's rule

Yeh page — bilkul zero se — har symbol, word, aur picture build karta hai jinpar parent note Fatigue topic rely karta hai. Upar se neeche padho; koi bhi cheez tab tak nahi aayegi jab tak earn nahi ho jaati.

1. Force, area, aur stress $\sigma$

"Loading" ki baat karne se pehle, humein ek fair tarika chahiye yeh kehne ka ki material ko kitna squeeze ya pull kiya ja raha hai. 10 N se moti rope kheenchna gentle hai; usi 10 N se baal kheenchna toot jaata hai. Toh sirf force kaafi nahi — humein force ko us area par share karna hoga jis par woh act karta hai.

Figure dekho: wahi force $F$ (orange arrows) do bars ko pull kar rahi hai. Patli bar ka area chhota hai, isliye force material ke thode "columns" mein crammed hai — high stress (red). Moti bar usi force ko bahut saare columns par spread kar deti hai — low stress (green). Stress woh hai jo material actually feel karta hai, chahe part ka size kuch bhi ho.

Greek letter $\sigma$ (sigma) sirf stress ka traditional naam hai. Jab bhi dekho, samjho "material ko per unit area kitna squeeze kiya ja raha hai". Gehri foundations Stress and Strain mein hain.

2. Tension, compression, aur sign

Pull aur push opposites hain, isliye hum stress ko ek sign dete hain.

Fatigue ke liye sign kyun matter karta hai? Kyunki cracks tension se khulti hain aur compression se dab jaati hain. Ek cycle jo strong tension mein jaata hai woh bahut zyada dangerous hai us se jo sirf compression mein wiggle karta hai. Sign rakhne se hum track kar sakte hain ki load crack ko kis taraf dhakel raha hai.

3. ${\sigma }_{UTS}$ — ultimate strength ("ek bade pull" ki limit)

Fatigue ka poora surprise ek sentence mein capture hota hai: parts repeated loading se fail hote hain ${\sigma }_{UTS}$ se bahut kam stress par. Isliye ${\sigma }_{UTS}$ woh benchmark hai jis se hum compare karte hain aur woh surprise feel karte hain. (${\sigma }_{UTS}$ ke against design margins Safety Factors & Margins of Safety mein handle kiye jaate hain.)

4. Ek "cycle" — stress ka ek upar-neeche jaana

Launch ek part ko shake karta hai; orbit use heat aur cool karta hai. Dono cases mein stress baar baar badhta aur ghatata hai. Ek poora badhna-ghatna ek cycle hai.

Wavy blue line dekho — stress time ke against plot hua hai. Iska ek top aur ek bottom hai:

In dono se hum do numbers banate hain jo actually ek cycle describe karte hain:

Figure par yeh kya hain: mean ${\sigma }_m$ (gray dashed line) woh centre hai jiske around wave hilti hai; amplitude ${\sigma }_a$ (orange arrow) woh hai kitna upar (ya neeche) us centre se swing karta hai. Ise alag kyun karte hain? Kyunki crack swing se khulti hai aur centre se ajar rakhti hai — isliye topic ko dono numbers alag chahiye. Peak aur valley add karke half karne se middle milta hai; subtract karke half karne se half-swing milta hai. Bas yahi do formulas hain: "do numbers ka middle" aur "unka aadha gap".

5. $S$ — stress amplitude, renamed

Parent note amplitude ko $S$ likhta hai. Yahan kuch naya nahi hai:

6. $N$ — cycles ki ginti, aur kyun powers of ten mein count karte hain

Yahan scale problem hai. Zyada hard load pe part $N=1000$ cycles mein mar sakta hai; gently load karne par $N=500,000,000$ survive kar sakta hai. Ye numbers aadhe million ke factor se differ karte hain — dono ko ek ruler-style axis par fit nahi kar sakte. Isliye hum logarithmic scale use karte hain.

Do number lines dekho. Ordinary (linear) line par, $10$, $100$, $1000$ almost far left mein stacked hain — dikhai nahi dete. Log line par evenly spaced hain. Topic ko logs kyun chahiye: fatigue lives bahut badi range of cycle counts mein failti hai, aur — agla section dikhayega — data sirf log–log form mein ek straight line banta hai. Straight lines fit aur read karna aasan hai.

7. Power laws aur log–log mein straight line

Parent note kehta hai "log–log mein straight line matlab power law hai". Yeh earn karte hain.

$y=c{x}^k$ ka log lo. Log rules use karo — "product ka log add hota hai" aur "power ka log front mein aa jaata hai": $\log y=\log c+k\log x.$ Ab $Y=\log y$ aur $X=\log x$ rakho: yeh padhta hai $Y=\log c+kX$ — slope $k$ wali straight line ki equation. Toh:

8. Fraction, sum ($\sum$), aur "the budget"

Miner's rule chhote pieces add karta hai. Do notation bits:

9. Yeh foundations topic mein kahan feed karte hain

Ise ek river ki tarah padho: stress cycle ke idea ko feed karta hai; cycle mean aur amplitude deta hai; amplitude plus log/power-law tools S–N curve banate hain; curve har level ko ek $N$ deta hai; un $N$'s ke fractions Miner's rule mein $1$ tak sum hote hain. Neighboring topics jo fatigue mein pour karte hain woh hain Random Vibration & PSD aur Launch Loads & Environments (yeh cycles supply karte hain), Thermal Cycling on Orbit (slow orbital cycles), aur Fracture Mechanics & Crack Growth (Paris' Law) (badhti crack ki actual physics).

Equipment checklist

Khud test karo — jab tum bina dekhe yeh sab answer kar sako tab parent note ke liye ready ho.