Young's Modulus

The relation

For elastic deformation:

σ = E · ε

where σ = F/A (stress, Pa) and ε = ΔL/L (strain, unitless).

For a rod of length L, cross-section A, applied force F, elongation ΔL:

F = (E · A / L) · ΔL

(Equivalent to Hooke's law for the rod, with effective k = E·A/L.)

Young's modulus of common materials

Material	E (GPa)
Diamond	~1050
Carbon nanotube (axial)	~1000
Tungsten	411
Steel	~200
Brass	100
Aluminum	69
Glass	50-90
Concrete	~30
Wood (along grain)	~10
Bone	~14
Cartilage	~0.025
Rubber (natural)	0.001-0.1
Soft tissue (skin)	~0.001

Sound speed in solids

Longitudinal sound speed:

v_long = √(E / ρ)

Material	E (GPa)	ρ (kg/m³)	v (m/s)
Aluminum	69	2700	~5060
Steel	200	7850	~5050
Concrete	30	2400	~3540
Wood	10	500	~4470
Glass	70	2500	~5290
Diamond	1050	3500	~17,300

JavaScript — Young's modulus calculations

// Stress and strain
function stress(force_N, area_m2) { return force_N / area_m2; }
function strain(deltaL, L) { return deltaL / L; }

// Young's modulus from stress-strain
function youngsModulus(stress_Pa, strain) { return stress_Pa / strain; }

// Elongation from F, A, L, E
function elongation(force_N, area_m2, length_m, E_Pa) {
  return force_N * length_m / (area_m2 * E_Pa);
}

// Steel rod, 1 m long, 1 cm² cross-section, 1000 N force
const E_steel = 200e9;  // Pa
console.log(`Steel rod 1m × 1cm², 1000N: ${(elongation(1000, 1e-4, 1, E_steel) * 1e6).toFixed(2)} µm`);
// 50 µm = 50 microns — small but measurable

// Spring constant of rod
function rodSpringConstant(E_Pa, area_m2, length_m) {
  return E_Pa * area_m2 / length_m;
}

console.log(`Steel rod k: ${rodSpringConstant(200e9, 1e-4, 1).toFixed(0)} N/m`);
// 2 × 10⁷ N/m — very stiff

// Sound speed
function soundSpeedSolid(E_Pa, density_kgm3) {
  return Math.sqrt(E_Pa / density_kgm3);
}

console.log(`Steel: v = ${soundSpeedSolid(200e9, 7850).toFixed(0)} m/s`);
console.log(`Aluminum: v = ${soundSpeedSolid(69e9, 2700).toFixed(0)} m/s`);

// Maximum stress before yield (yield strength varies widely; mild steel ~250 MPa)
const yield_steel = 250e6;
console.log(`Steel yield strain: ${(yield_steel / E_steel).toExponential(2)}`);
// 0.125% — yield at very small deformation

// Bone in osteopathic terms
function boneFracture(force_kN, area_cm2) {
  // Strain to fracture ~0.01 (1%); E ~ 14 GPa
  const E_bone = 14e9;
  const stress_Pa = force_kN * 1000 / (area_cm2 * 1e-4);
  return stress_Pa / E_bone;
}

console.log(`Bone strain at 1 kN over 1 cm²: ${boneFracture(1, 1).toExponential(2)}`);

// Composite material rule of mixtures (parallel)
function compositeYoungs(volume_fractions, E_values) {
  return volume_fractions.reduce((sum, f, i) => sum + f * E_values[i], 0);
}

// Carbon fiber (E~250 GPa) at 60%, epoxy (E~3 GPa) at 40% (axial direction)
console.log(`CF/epoxy 60/40: ${compositeYoungs([0.6, 0.4], [250e9, 3e9]) / 1e9} GPa`);

Where Young's modulus matters

• Civil engineering. Buildings, bridges, dams — material selection based on E and yield strength.
• Aerospace. Aircraft (Al, composites), spacecraft (Ti, composites). Need lightweight + stiff.
• Mechanical engineering. Springs, shafts, bearings — designed for specific stiffness.
• Sports. Golf clubs, tennis rackets, vaulting poles, bicycles. Tuning E for performance.
• Medical. Bone vs implants — stiffness mismatches can cause stress shielding (bone resorption).
• Geology. Earth's crust elasticity; seismic wave propagation.
• Manufacturing. Metal forming — deformation limits set by E and yield.

Common mistakes

• Confusing stiffness with strength. Stiffness (E) — resistance to deformation. Strength — resistance to breaking. Different properties; some materials high E but low strength (cast iron) or vice versa (rubber).
• Using Young's modulus beyond elastic limit. σ = E·ε holds in elastic regime. Beyond yield, plastic deformation occurs; relationships nonlinear.
• Treating Young's modulus as direction-independent. Anisotropic materials (wood, fiber composites) have different E along different directions. Must specify direction.
• Forgetting Poisson's ratio. Stretching one direction usually contracts perpendicular directions (Poisson). For 3D problems, need shear modulus too.
• Confusing units. E in Pa or GPa. 1 GPa = 10⁹ N/m². psi conversion: 1 GPa ≈ 145,000 psi.
• Treating temperature as irrelevant. E varies with T (slowly for metals; significantly for polymers). High-T applications need separate analysis.