Young's Modulus
Young's modulus (also called the modulus of elasticity) is the ratio of engineering stress to engineering strain in the linear-elastic region of a material's stress-strain curve. It is a fundamental mechanical property that quantifies the stiffness of a solid material — a higher value means the material resists deformation more effectively. Young's modulus is essential in structural design for calculating deflections, natural frequencies, and load-bearing capacity of components.
Key Formula
E = sigma / epsilon = (F / A0) / (delta_L / L0)
LaTeX: E = \frac{\sigma}{\varepsilon} = \frac{F / A_0}{\Delta L / L_0}
| Symbol | Meaning | Unit |
|---|---|---|
| E | Young's modulus (modulus of elasticity) | Pa (GPa) |
| σ | Engineering stress | Pa |
| ε | Engineering strain | dimensionless |
| F | Applied force | N |
| A₀ | Original cross-sectional area | m² |
| ΔL | Elongation | m |
| L₀ | Original length | m |
Worked Example
Problem
A copper wire 2 m long and 1.5 mm in diameter stretches by 0.8 mm under a tensile load of 80 N. Calculate the Young's modulus.
Solution
Step 1: Compute cross-sectional area. A₀ = π × (0.0015/2)² = π × (7.5 × 10⁻⁴)² = 1.7671 × 10⁻⁶ m² Step 2: Compute engineering stress. σ = F / A₀ = 80 / 1.7671 × 10⁻⁶ = 4.527 × 10⁷ Pa Step 3: Compute engineering strain. ε = ΔL / L₀ = 0.0008 / 2.0 = 4.0 × 10⁻⁴ Step 4: Calculate Young's modulus. E = σ / ε = 4.527 × 10⁷ / 4.0 × 10⁻⁴ = 1.132 × 10¹¹ Pa ≈ 113 GPa
Answer
Young's modulus E ≈ 113 GPa (literature value for copper: ~110–128 GPa)
Young's Modulus of Common Engineering Materials
| Material | Young's Modulus (GPa) | Density (kg/m³) | Typical Use |
|---|---|---|---|
| Steel (structural) | 200 | 7 850 | Bridges, buildings |
| Aluminium alloy | 69 | 2 700 | Aircraft structures |
| Copper | 117 | 8 960 | Electrical wiring |
| Concrete | 30 | 2 400 | Foundations, slabs |
| Carbon fibre composite | 70–200 | 1 600 | Aerospace, sports |
| Rubber (natural) | 0.01–0.1 | 920 | Seals, tyres |
Interactive Tools
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Related Terms
Engineering Stress
Engineering stress is defined as the applied force divided by the original cross-sectional area of a specimen, regardless of any deformation that occurs during loading. It is the conventional measure used in materials testing and structural analysis because the original dimensions are easily measured before the test begins. Engineering stress is widely used in design calculations, material data sheets, and stress-strain curves to characterise material behaviour under uniaxial loading.
Engineering Strain
Engineering strain is the ratio of the change in length of a specimen to its original length when subjected to axial loading, expressed as a dimensionless number or percentage. It quantifies how much a material deforms relative to its initial size and is the conventional measure plotted alongside engineering stress to produce stress-strain curves. Engineering strain assumes uniform deformation and uses the original gauge length, making it straightforward to measure experimentally.
Shear Stress
Shear stress is the component of stress that acts parallel (tangential) to a cross-sectional surface, as opposed to normal stress which acts perpendicular to it. It arises when equal and opposite forces act along parallel planes in a material, causing layers to slide relative to one another. Shear stress is critical in the design of bolts, welds, shafts, beams, and adhesive joints, where failure along a plane is the governing mode.
Named after Thomas Young (1773–1829), a British polymath who described the coefficient of elasticity in his 1807 lectures. The concept had been explored earlier by Leonhard Euler and others, but Young provided a clear quantitative definition. 'Modulus' derives from Latin 'modus' (measure).