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APEX CLASS ESTD 2018

IITJEE: Stress-Strain Curve, Young’s Modulus (Y) & Thermal Stress Visualizer

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Ashish Kumar Bahekar

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Apex Class • IITJEE Mechanical Properties of Solids Lab

Stress-Strain Curve, Young’s Modulus (Y) & Thermal Stress Visualizer

Master NCERT Stress-Strain Checkpoints (A-B-C-D-E), Elastic Energy Density (½ Yε²), Self-Weight Stretch, and Thermal Stress!

🔬 Hooke’s Law: Stress (F/A) = Y × Strain (ΔL/L). Select Concept:
Stress (σ) Strain (ε = ΔL / L) → O’ (Permanent Set) A (Prop. Limit) B (Yield σ_y) C ★ D (Ultimate Strength σ_u) E (Fracture ✕) Slope = Y 🏆 NCERT Checkpoints (O → E) • O to A: Hooke’s Law strictly obeyed (Slope = Y) • A to B: Not linear, but STILL Elastic (Returns to O!) • Point B: Elastic Limit / Yield Point (Yield Strength σ_y) • B to D: Plastic Deformation (Permanent Set OO’) • Point D: Ultimate Tensile Strength (Max Stress!) • D to E: Necking / Thinning occurs even as load drops! 🎯 #1 NCERT Trap: Between A and B, Stress is NOT proportional to Strain, yet the wire still returns to original length when unloaded! 1. Brittle (Glass, Cast Iron) D & E Close! Snaps with almost NO plastic region! 2. Ductile (Steel, Copper, Al) Large D–E Distance! Can be drawn into thin wires 3. Elastomer (Rubber, Aorta) NO Hooke’s Law region! (Hysteresis) 💡 Classic Conceptual Question: Steel is MORE ELASTIC than Rubber because Young’s Modulus Y_steel (2×10¹¹ Pa) >> Y_rubber! Area = Energy Density (u) u = ½ × Stress × Strain Stress σ Strain ε Elastic Potential Energy Formulas 1. Energy Density (u = U / Volume, in J/m³): u = ½ (Stress)(Strain) = ½ Y (Strain)² = Stress² / (2Y) 2. Total Stored Energy in Wire (Volume = A·L): U = ½ F · ΔL = ½ (YA / L) ΔL² 3. Hanging Block m Drops by ΔL (JEE Trap!): Gravity Work = mgΔL → ½mgΔL Stored + ½mgΔL Heat! ⚡ Notice the exact parallel with Capacitors: Gravity does W = F·ΔL work, stores U = ½F·ΔL in the wire, and loses ½F·ΔL as heat! 1. Poisson’s Ratio (σ) & Volume Change σ = – (Δr / r) / (ΔL / L) [Dimensionless] • Theoretical Limits: -1 < σ < 0.5 • Practical Limits (Metals): 0 < σ < 0.5 (~0.3) • Fractional Volume Change when Stretched: ΔV / V = (1 – 2σ) (ΔL / L) [If σ=0.5 → ΔV = 0!] 2. Bulk (B), Shear (η) & Inter-Relations • Bulk Modulus: B = – ΔP / (ΔV / V) = 1 / Compressibility K • Shear Modulus (Rigidity): η = (F_∥ / A) / θ • Relations Connecting Y, B, η, and σ: Y = 3 B (1 – 2σ) | Y = 2 η (1 + σ) 9 / Y = 3 / η + 1 / B 🌊 Ideal Liquid & Rigid Body Facts: For an Ideal Liquid, Shear Modulus η = 0; for a Perfectly Rigid Body, Y = B = η = ∞!
Young’s Modulus (Y)
Y = (F · L) / (A · ΔL)
Depends ONLY on Material & Temp!
Energy Density (u = U/V)
u = ½ × Stress × Strain
Also u = ½ Y ε² = σ² / (2Y)
Compressibility (K)
K = 1 / B = -ΔV / (V·ΔP)
SI Unit: Pa⁻¹ or m²/N
Volume Change on Stretch
ΔV / V = (1 – 2σ)(ΔL / L)
Derived from V = π r² L
🔩 Equivalent Spring Constant of a Rod/Wire: k_eq = Y·A / L. Select Problem Type:
Load F ≡ Load F k_wire = Y · A / L ⇒ SHM Period T = 2π √(mL / YA)! Elongation Scaling & Constant Volume Trap 1. Standard Formula: ΔL = (F · L) / (π r² Y) ⇒ ΔL ∝ L / r² • If radius is halved (r → r/2) & length doubled (L → 2L): ΔL_new = [ 2 / (1/2)² ] ΔL = 8 × ΔL! 2. Constant Mass / Volume Wire Trap (V = A·L = Const): ΔL = (F · L²) / (V · Y) ⇒ ΔL ∝ L² and ΔL ∝ 1 / r⁴! ⚠️ Watch for the words “A given mass of wire is stretched…”: When Volume V is constant, ΔL ∝ L² and ΔL ∝ 1/r⁴! COM at L / 2! Entire Weight Mg acts at L / 2 Self-Weight Elongation Formulas ΔL_self = (M g L) / (2 A Y) = (ρ g L²) / (2 Y) • Exactly HALF the elongation of an end load Mg! • Notice ΔL_self = ρgL²/(2Y) is INDEPENDENT of Area A! • Maximum Length of Vertical Wire Before Breaking under its own weight: L_max = σ_breaking / (ρ · g)! 🪨 Elastic Energy Stored due to Self-Weight: Integrate ½Yε² dV from 0 to L → U_self = (M² g² L) / (6 A Y) (Notice the 1/6 factor)! Heated Rod (Y, A, α, +ΔT) Walls Prevent Expansion ΔL = L α ΔT! Thermal Strain: ε = ΔL / L = α · ΔT 🏆 Thermal Stress & Force Equations Thermal Stress: σ_th = Y × Strain = Y · α · ΔT Thermal Force: F = Y · A · α · ΔT • JEE Trap #1: Thermal Force F is INDEPENDENT of Length L! • JEE Trap #2: If the rod is FREE to expand (not clamped), Thermal Stress inside the rod is ZERO! 🔥 Heating a clamped rod creates COMPRESSIVE stress; cooling a clamped wire creates TENSILE tension F = YAαΔT! 1. Depression (Sagging δ) of a Loaded Beam δ = (W · L³) / (4 Y · b · d³) • L = Span Length, b = Breadth, d = Vertical Depth • To minimize sagging δ: Increase DEPTH (d) because δ ∝ 1 / d³ (Doubling depth reduces sag by 8×!) • I-Shaped Girders give large ‘d’ while saving weight! 2. Breaking Stress vs. Breaking Force Trap • Breaking Stress (σ_max): Property of MATERIAL only! Cutting a cable to half length does NOT change σ_max or F_max! • Breaking Force (Load): F_max = σ_max × Area (π r²) Doubling radius (r → 2r) quadruples max load (4 F_max)! • Crane Rope Design: A_min = m(g + a_max) / σ_yield 🌉 Why Mountain Height on Earth is Limited to ~10 km: At H_max ≈ σ_shear / (ρ_rock × g) ≈ 10 km, the base rock flows plastically!
Equivalent Spring (k_eq)
k = Y · A / L
Restoring Force F = (YA/L) ΔL
Self-Weight Stretch
ΔL = MgL / (2AY) = ρgL²/(2Y)
Half of End-Load Elongation
Thermal Force (Clamped)
F = Y · A · α · ΔT
Independent of Rod Length L!
Torsional Shear Twist
r · θ = L · φ (τ = πηr⁴θ/2L)
Hollow Shafts are Stronger than Solid!
🔥 2 High-Yield JEE Elasticity Shortcuts:
1. Three Tensions Length Formula: If a wire has length L₁ under tension T₁ and length L₂ under tension T₂, its natural unstretched length is L₀ = (L₁T₂ – L₂T₁) / (T₂ – T₁)!
2. Bulk Modulus of Ideal Gas: B_isothermal = P  |  B_adiabatic = γ P  |  B_polytropic (PVⁿ = const) = n P!
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