Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E: Question bank

3.6.3 · D5 · Physics › Spacecraft Structures & Systems Engineering › Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E

Shuru karne se pehle, ek glossary taaki koi bhi symbol bina samjhe use na ho:

$F$ = applied pulling (ya pushing) force, newtons (N) mein.
$A$ = bar ka cross-sectional area — wo face ka size jo tumhe dikhta agar tum seedha usme se kaat lo — m² mein.
$L$ = bar ki original (unloaded) length, m mein; $Δ L$ = loading ke baad length mein change, m mein.
$σ$ (stress) $= F / A$ , force divided by cross-section area — "pull kitna crowded hai", pascals (Pa = N/m²) mein.
$ε$ (strain) $= Δ L / L$ , stretch divided by original length — ek pure ratio, koi units nahi.
$E$ (Young's modulus) $= σ / ε$ , stress–strain graph ke straight part ka slope — "material kitna stubborn hai", Pa mein.
$σ_{yield}$ (yield stress) = stress ka ceiling jahan material spring back karna band kar deta hai aur permanently deform hone lagta hai. Iske neeche material elastic hai (shape mein wapas aata hai); iske upar, plastic (bent reh jaata hai). Dekho Yield Strength and Plastic Deformation.

Neeche ke do figures wo visual backbone hain jinka traps baar baar reference karte hain: Figure 1 dikhata hai ki real curve par "slope $E$ " versus "ceiling $σ_{yield}$ " ka kya matlab hai; Figure 2 dikhata hai ki kyun deflection formula $Δ L = F L / (A E)$ mein $A$ aur $E$ neeche hote hain.

Figure — Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E

True or false — justify

TF1 — "Ek thick bar aur ek thin wire, dono same aluminium ke, unki stiffness

E

same hoti hai."

True.

E

ek material property hai; ise part ki shape se koi matlab nahi. Part stiffness (force per stretch) alag hogi, lekin

E

identical hai.

TF2 — "Cross-sectional area double karne se fixed force ke liye stress double ho jaata hai."

False. Stress

F / A

hai, toh

A

double karne se stress aadha ho jaata hai — wahi force zyada material par share ho jaata hai.

TF3 — "Strain metres mein measure hota hai."

False. Strain

Δ L / L

hai, length over length, isliye units cancel ho jaate hain — ye dimensionless hai (kabhi kabhi microstrain mein quote hota hai,

1 μ ε = 1 0^{- 6}

TF4 — "High Young's modulus guarantee karta hai ki material strong hai."

False.

E

slope hai (stiffness); strength wo stress ceiling

σ_{yield}

hai (Figure 1) jahan ye fail hota hai. Cast iron jaisa stiff material brittle ho sakta hai aur jaldi fail ho sakta hai. Dekho Yield Strength and Plastic Deformation.

TF5 — "Yield point ke neeche, stress aur strain proportional hote hain."

True. Wo linear region (Figure 1 mein

σ_{yield}

ke left mein) wahi jagah hai jahan Hooke's Law

σ = E ε

kaam karta hai; proportionality ka constant

E

hai.

TF6 — "Bar ko 2 mm kheeenchne ka matlab hamesha same strain hota hai."

False. Strain original length par depend karta hai: 2 m strut par 2 mm

1 0^{- 3}

hai, lekin 20 mm sample par 2 mm

0.1

hai — sau guna zyada.

TF7 — "Agar do struts same force carry kar rahe hain lekin ek zyada lamba hai, toh lamba wala zyada stretch karta hai."

True (same

A

E

F

Δ L = F L / (A E)

se, stretch seedha length

L

ke proportion mein badhta hai.

TF8 — "Kam

E

wala material same stress mein zyada stretch karta hai."

True. Strain

= σ / E

hai, isliye chhota

E

zyada strain deta hai — isliye aluminium (

70

GPa) steel (

200

GPa) se equal stress par ~2.9× zyada strain karta hai.

TF9 — "Bar ke andar stress is par depend karta hai ki tum kaun sa imaginary cut choose karte ho."

False (uniform axial bar ke liye). Newton ke 3rd law se har cut par same internal force

F

hona chahiye, isliye

σ = F / A

har cross-section par same hai.

Spot the error

SE1 — "Band 3 mm stretch hua, isliye uska strain 3 mm hai."

Error: strain ek ratio hai,

Δ L / L

, kabhi bhi raw length nahi. Iska koi matlab banne se pehle tumhe 3 mm ko original length se divide karna hoga.

SE2 — "Stress 70 hai, toh main seedha 70 ko

Δ L = F L / (A E)

mein dalunga."

Error: units. "70" likely MPa hai; formula ko SI (Pa aur m) chahiye. Pehle

70 MPa = 7 \times 1 0^{7}

Pa convert karo, warna answer

1 0^{6}

se off hoga.

SE3 — "Steel aluminium se zyada stiff hai, isliye steel break hone se pehle zyada stress carry kar sakta hai."

Error: stiffness ko strength ke saath confuse kiya. Stiffness (

E

) set karta hai ki wo kitna bend karta hai, fail kab hoga nahi. Ek soft aluminium alloy kuch steels se zyada strong ho sakti hai; actual

σ_{yield}

check karo.

SE4 — "Maine

σ > σ_{yield}

paaya, toh main

Δ L = F L / (A E)

use karta rahunga."

Error:

Δ L = F L / (A E)

Hooke's Law hai, sirf elastic region mein valid hai (Figure 1 mein ceiling ke left mein). Yield ke baad material permanently deform hota hai aur ye formula bahut zyada overshoot karta hai. Dekho Stress-Strain Curve.

SE5 — "Strain ke koi units nahi hain, isliye main ise ek length mein add kar sakta hoon."

Error: dimensionless ka matlab 'ek length' nahi hota. Tum sirf strain ko

L

se multiply karke length recover kar sakte ho (

Δ L = ε L

), add karke nahi.

SE6 — "Strut stiff hai (

E

high), isliye ye kabhi fail nahi ho sakta."

Error: stiffness sirf deflection limit karta hai; ek stiff-but-brittle strut achanak snap ho sakta hai jab stress uski (possibly low) ceiling

σ_{yield}

hit karta hai. Stiffness aur failure ke against margin alag sawal hain — doosre ke liye ek Factor of Safety chahiye.

SE7 — "Is composite ka

E = 150

GPa hai, isliye ye aluminium ke

70

GPa se zyada stretch karta hai."

Error: ulta logic. Zyada

E

ka matlab same stress mein kam stretch hai, kyunki

ε = σ / E

mein

E

badhne se ye shrink hota hai (Figure 1 mein steeper slope).

Why questions

WHY1 — Force ko area se divide kyun karte hain instead of sirf force use karne ke?

Kyunki failure aur stretching is par depend karte hain ki force kitna concentrated hai, uski raw size par nahi. 1000 N pull ek baal ke liye bahut crowd hai lekin ek girder ke liye kuch nahi —

A

se divide karne par part size strip ho jaata hai.

WHY2 — Stretch ko original length se divide kyun karte hain?

Material describe karne ke liye, kisi particular part ko nahi. Wahi fractional stretch ek meaningful quantity hai; ek fixed 1 mm ek short aur ek long bar par bahut alag cheezein mean karta hai.

WHY3 —

E

ke units stress (Pa) jaisi kyun hain?

Kyunki

E = σ / ε

hai aur

ε

dimensionless hai — pascals ko ek pure number se divide karne par pascals hi milte hain.

WHY4 —

E

ko slope aur strength ko ceiling kyun kehte hain?

Stress–strain graph (Figure 1) par,

E

initial straight line ka gradient hai (

ε

mein rise per

σ

mein rise); strength

σ_{yield}

wo height hai jahan line straight rehna band kar deti hai. Ek slope aur ek height alag cheezein hain.

WHY5 — Deflection formula

Δ L = F L / (A E)

mein

A

aur

E

neeche kyun hain?

Zyada area ya zyada stiffness dono stretching ko resist karte hain (Figure 2 mein parallel springs), isliye dono

Δ L

reduce karte hain — ye denominator mein belong karte hain. Zyada length strain ko zyada distance par accumulate karne deta hai, isliye

L

upar hai.

WHY6 — Do materials same stress par fail kyun ho sakte hain lekin pehle wildly different amounts stretch kar sakte hain?

Kyunki failure stress (ceiling) aur

E

(uski slope) independent hain. Spring steel aur ek stiff polymer ek

σ_{yield}

share kar sakte hain lekin bahut alag

E

rakh sakte hain, isliye failure par bahut alag strain.

WHY7 — Hooke's Law trust karne se pehle

σ < σ_{yield}

check kyun karna chahiye?

Kyunki

σ = E ε

sirf real curve ka linear part hai (Figure 1).

σ_{yield}

ke upar line permanent deformation mein bend over ho jaati hai aur simple proportionality toot jaati hai. Dekho Yield Strength and Plastic Deformation.

Edge cases

EC1 — Agar applied force zero ho toh stress kya hai?

σ = 0/ A = 0

. Koi load nahi, koi internal crowding nahi — aur Hooke's Law se strain bhi zero hai, isliye bar apni natural length par rehta hai.

EC2 — Jab area

A

zero ki taraf shrink hota hai (ek sharp notch ya thinning), stress ka kya hota hai?

σ = F / A

infinity ki taraf blow up karta hai jab

A \to 0

— isliye cracks aur notches dangerous hain: ye locally area starve kar dete hain aur stress spike kar deta hai.

EC3 — Kya strain negative ho sakta hai, aur physically iska kya matlab hai?

Haan. Compression ke under bar chota hota hai,

Δ L < 0

, isliye

ε < 0

(hamara sign convention). Stress

σ

tab compressive hota hai (

< 0

); Hooke's Law

σ = E ε

elastic range mein matching sign ke saath phir bhi hold karta hai.

EC4 — Ek perfectly rigid (infinitely stiff) material ke liye strain kya hai?

Jab

E \to \infty

ε = σ / E \to 0

kisi bhi finite stress ke liye — ye load carry karta hai bina bilkul stretch kiye. Koi bhi real material truly rigid nahi hota, lekin bahut high

E

iske paas aata hai.

EC5 — Kya axial stretching cross-section change karta hai, aur kya hamara

σ = F / A

notice karta hai?

Physically haan — laterally kheenchne se bar sideways thin ho jaata hai (Poisson's Ratio: ek taraf stretch karo, perpendicular taraf shrink). Lekin engineering stress original

A

use karta hai, isliye ye is chhote change ko ignore karta hai; sirf true stress shrinking area account karta hai.

EC6 — Ek strut mein stress kya hoga jise dono ends clamped karke heat kiya gaya ho, koi external force na ho?

Ye bada aur compressive ho sakta hai

F_{external} = 0

ke saath bhi: material expand karna chahta hai lekin kar nahi sakta, isliye clamps push back karte hain. Ye Thermal Stress dikhata hai ki stress ke liye koi applied external pull zaroori nahi.

EC7 — Ek bahut lambe strut ki limit mein, uske total stretch mein kya dominant hota hai?

Length khud:

Δ L = F L / (A E)

L

ke saath linearly badhta hai, isliye ek lamba tie-rod proportionally zyada stretch karta hai chahe uska strain (per-metre) same rahe. Ye Spacecraft Load Paths and Struts mein strut sizing drive karta hai.

Recall Traps ka ek-sentence summary

Lagbhag har trap ek ratio (stress, strain, $E$ ) ko ek raw amount (force, stretch) ke saath confuse karne se aata hai, ya ek slope ( $E$ ) ko ek ceiling ( $σ_{yield}$ ) ke saath — har symbol ko "per unit area", "per unit length", ya "slope of the graph" se anchor karo aur traps dissolve ho jaate hain.