What Young's modulus is

Young's modulus (or longitudinal elastic modulus), denoted E, is the constant of proportionality linking stress and strain in the elastic region of a material under uniaxial tension or compression. It was formalised by Thomas Young in 1807, though the linear relation σ = E·ε had been known as Hooke's law since 1676 ("ut tensio, sic vis"). E measures the intrinsic stiffness of a material: how much a specimen elongates under a given stress. It has pressure units (Pa, MPa or GPa) and for common structural materials it spans four orders of magnitude: from ~200 GPa for steel down to ~0.01 GPa (10 MPa) for vulcanised rubber. It is the key parameter for the design of beams, ties, slabs and any element under static loads in the elastic regime.

Hooke's law: σ = E·ε

Hooke's law states that in a linear elastic material stress σ is proportional to strain ε:

σ = E · ε   with   σ = F/A₀   and   ε = ΔL/L₀

where F is the applied axial force (N), A₀ the initial specimen cross-section (mm²), ΔL the measured elongation (mm) and L₀ the initial gauge length. Stress σ is in MPa (N/mm²) and strain ε is dimensionless (often expressed in % or μm/m). Young's modulus is obtained by inversion: E = σ/ε. Example: steel bar F = 50 kN, φ = 14 mm, L₀ = 200 mm, ΔL = 0.31 mm → A₀ = 153.94 mm², σ = 324.8 MPa, ε = 0.155%, E = 209.6 GPa — consistent with structural steel (E ≈ 200-215 GPa). The law holds only up to the proportional limit, beyond which the σ-ε curve loses linearity and yielding starts.

The uniaxial tensile test ISO 6892 / ASTM E8

The tensile test is the most common mechanical characterisation of structural materials. Reference standards: ISO 6892-1 (metals, ambient conditions), ISO 527 (plastics), ASTM E8/E8M (metals US), EN 12390-13 (concrete, secant modulus). A standardised specimen with circular or rectangular section is gripped between two jaws of a universal testing machine (electromechanical or servo-hydraulic); an extensometer (analogue, electrical or video) measures elongation over the gauge length, while a load cell records force. The machine pulls the specimen at controlled speed (typically 5-25 mm/min for steel, 1 mm/min for concrete) up to failure, plotting the full σ-ε curve. From the curve one extracts: Young's modulus E (elastic slope), yield strength R_p0.2 (stress at 0.2% residual plastic strain), tensile strength R_m (peak stress), elongation at fracture A%, reduction of area Z%.

The stress-strain curve: 4 characteristic zones

The σ-ε curve of a ductile metal has four zones:

1. Linear elastic zone (0 → σ_p): strain is proportional to stress (Hooke), reversible: upon unloading the specimen returns to its initial length. E is measured here. For carbon steels σ_p ≈ 180-250 MPa.
2. Yield plastic zone (σ_p → R_eH): decreasing slope, appearance of Lüders bands (45° slips), permanent deformation. For mild steels yielding is sharp (R_eH and R_eL); for alloyed steels transition is gradual and the conventional R_p0.2 is used.
3. Strain-hardening zone (R_eL → R_m): material "hardens" through plastic deformation, requires increasing stress to keep deforming. R_m is the peak stress at which necking begins.
4. Necking and rupture (R_m → break): the section locally shrinks (necking), engineering stress drops while true stress keeps rising, until ductile cup-cone fracture for structural steels.

Brittle materials (cast iron, ceramics, glass, concrete in tension) show instead sudden fracture without appreciable plastic phase: their curve is nearly linear up to collapse.

E of common materials (in GPa)

Young's modulus spans 5 orders of magnitude. Typical values at 20 °C:

• Diamond: 1050 GPa (stiffest natural material).
• Tungsten carbide (WC): 550-700 GPa.
• Tungsten: 411 GPa.
• Molybdenum: 329 GPa.
• Carbon steels: 200-215 GPa (independent of strength).
• Grey cast iron: 100-130 GPa.
• Copper: 110-128 GPa; brass: 100-125.
• Titanium (Ti-6Al-4V): 114 GPa.
• Glass: 60-90 GPa (silica ~72).
• Aluminium: 69-79 GPa (regardless of alloy).
• Concrete: 25-40 GPa (secant, depending on class).
• Bone: 14-25 GPa.
• Hardwood parallel to grain: 8-15 GPa (oak ~12, beech ~11).
• Nylon: 2-4 GPa; polycarbonate: 2.2 GPa.
• HDPE polyethylene: 0.8-1.4 GPa.
• Vulcanised natural rubber: 0.01-0.1 GPa.

Specific modulus E/ρ is crucial in aerospace: carbon fibre has E/ρ ≈ 100 MN·m/kg vs steel 26 and aluminium 25, reason for its heavy use in advanced aeronautics.

Shear modulus G and Poisson's ratio ν

An isotropic material is characterised by only two independent elastic constants. Besides E, we use Poisson's ratio ν, defined as the ratio of transverse contraction to longitudinal elongation (ν = −ε_trans/ε_long). Typical values: steel 0.27-0.30; aluminium 0.33; copper 0.34; concrete 0.15-0.20; rubber 0.49 (nearly incompressible). Shear modulus G is derived from E and ν:

G = E / (2·(1 + ν))

For steels with ν = 0.30: G = 210/2.6 ≈ 80 GPa. For aluminium with ν = 0.33: G = 70/2.66 ≈ 26 GPa. G is the elastic constant in shear deformation (τ = G·γ, with γ shear strain) and is essential for torsional shafts, shear bolts, resin connections. This calculator estimates G assuming ν = 0.30 (editable default); for detailed design measure G directly by torsion tests or use tabulated values for the specific material.

Sources of error in E measurement

• ΔL measurement errors: using a contact extensometer (accuracy ±0.001 mm) is essential. Measuring ΔL from machine crosshead travel includes machine elasticity (acting as a series spring), inflating ΔL by 20-50% and underestimating E.
• Grip slippage: if the specimen slips from the grips (especially for hardened steels or hard materials), ΔL is overestimated and E underestimated. Wedge or threaded grips fix this.
• Initial non-linearity: the first part of the curve (< 20 MPa) can be non-linear due to specimen/grip settling; measure E in the central part of the elastic range (30-70% of R_p0.2).
• Load misalignment: eccentric loading generates parasitic bending that biases ε. Spherical joints or optical aligners correct this.
• Temperature: E drops ~5% per +100 °C in steel; +10% at −196 °C. Test at controlled temperature (20 ± 2 °C for ISO 6892 ambient conditions).
• Strain rate: E is nearly rate-independent for metals up to 10^−3 s^−1; for polymers and viscoelastics the dependence is strong (E increases with rate).

How to use the calculator

Enter the force F applied to the specimen in kN (load cell reading during test, taken in the linear elastic region; typically 20-100 kN for structural steel in standard specimens). Enter the diameter φ in mm (initial calliper measurement, typical sections 6, 10, 14 mm for steels; M20 for concrete). Enter the initial gauge length L₀ in mm (typically 5·φ for ISO 6892 type 5 = 70 mm for φ14, or 50 mm ASTM). Enter the elongation ΔL measured with the extensometer (mm; steels in elastic zone typically 0.05-0.5 mm; plastics up to tens of mm). Optionally specify the Poisson's ratio ν for better G estimate (default 0.30 for metals). The calculator returns section A₀, stress σ, strain ε in %, Young's modulus E in GPa (with comparison to theoretical values of common materials), axial stiffness k, estimated shear modulus G and a qualitative material identification.