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

3.6.3 · Physics › Spacecraft Structures & Systems Engineering

Hume stress aur strain ki zaroorat KYUN hai?

PROBLEM kya hai? Ek spacecraft engineer ko guarantee deni hoti hai ki koi strut launch loads ke under na toote aur na hi zyada deform ho. Akela force ($F$ newtons mein) kaafi information nahi hai — 1000 N ka pull ek baal tod deta hai lekin ek girder par kuch nahi karta.

Normalize KYUN karein? Kyunki failure aur stretching concentration aur proportion par depend karti hai, na ki raw force par. Cross-section area ko double karne se force ki internal "bheed" aadhi ho jaati hai. Isliye hum aisi quantities banate hain jo part ki size hata deti hain aur sirf material ka behavior rakhti hain.

Stress — σ = F/A derive karna

Hum isse KAISE banate hain (scratch se):

1. Ek bar ko force $F$ se kheencho. Equilibrium mein, bar ke across koi bhi imaginary cut mein material ko dono sides se same $F$ se wapas kheenchna chahiye (Newton's 3rd law).
2. Woh internal force cut area $A$ par spread hoti hai.
3. Force ki intensity define karo = force ÷ area:

$\sigma =\frac{F}{A}$

Yeh step kyun? Hum $A$ se isliye divide karte hain kyunki bade area mein same force "kam bheed" wali hoti hai — har bond ek chhota sa share carry karta hai.

Strain — ε = ΔL/L derive karna

Hum isse KAISE banate hain:

1. Original length $L$. Load ke under yeh $L+\Delta L$ ho jaati hai.
2. 2 m ki bar jo 1 mm stretch hoti hai woh "kam strained" hai ek 1 mm ki bar se jo 1 mm stretch hoti hai. Isliye hum stretch ko original length se compare karte hain:

$\epsilon =\frac{\Delta L}{L}$

Dimensionless kyun? Length ÷ length se units cancel ho jaate hain. Strain ek pure ratio hai — stretch ka ek percentage.

Young's Modulus E — dono ko jodna

Hum $E$ ko first principles se KAISE nikalte hain:

1. Observed linearity: $\sigma \propto \epsilon$.
2. Ek proportionality constant $E$ introduce karo:

$\boxed{\sigma =E\epsilon }$

3. Definitions substitute karo "engineer ka stretch formula" dekhne ke liye:

$\frac{F}{A}=E\frac{\Delta L}{L}⟹\Delta L=\frac{FL}{AE}$

Stiff hona achha AUR bura KYUN hota hai? High $E$ = diye gaye load ke liye chhota deflection (telescope point karne ke liye achha). Lekin stiffness ≠ strength — ek stiff material phir bhi brittle ho sakta hai. $E$ ek slope hai, strength ek ceiling hai.

Worked examples

Common mistakes (Steel-manned)

Flashcards

Connections

• Hooke's Law — spring-scale version, $F=kx$, jise yeh materials tak generalize karta hai.
• Yield Strength and Plastic Deformation — jahan $\sigma =E\epsilon$ kaam karna band kar deta hai.
• Poisson's Ratio — axial strain ke saath aane wala sideways strain.
• Stress-Strain Curve — poora graph; $E$ uska initial slope hai.
• Spacecraft Load Paths and Struts — jahan yeh numbers real hardware size karte hain.
• Thermal Stress — force ki jagah temperature se $\Delta L$ ($\alpha \Delta T$).
• Factor of Safety — working stress aur yield stress ke beech design margin.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps