Compute Young's modulus (E) using two methods: from stress and strain, or from force and geometry. Visualize Hooke's law on an interactive stress‑strain graph. Includes material reference data and practical examples.
| Case | Material / Condition | Young's Modulus (GPa) | Source / Remarks |
|---|---|---|---|
| 1 | Structural steel (Q235) @ 20°C | 200 | ASM Handbook, Vol.1 |
| 2 | Aluminum alloy 6061-T6 @ 20°C | 68.9 | ASTM B209 |
| 3 | Aluminum alloy 6061-T6 @ 200°C | 63.0 | ASM high‑temperature data |
| 4 | Carbon‑fiber reinforced polymer (CFRP, longitudinal) | 150 | Anisotropic, longitudinal modulus |
| 5 | Concrete (C30) compression | 30 | EN 1992-1-1 |
| 6 | Titanium alloy Ti-6Al-4V | 114 | AMS 4911 |
Data compiled from public standards and authoritative handbooks; actual values may vary slightly with composition and processing.
For anisotropic materials (e.g., wood, composites, single crystals), a single Young's modulus is insufficient. The generalized Hooke's law introduces the stiffness tensor \( C_{ijkl} \) or compliance tensor \( S_{ijkl} \). In engineering, orthotropic materials are described by engineering constants: longitudinal modulus \( E_1 \), transverse modulus \( E_2 \), shear modulus \( G_{12} \), and Poisson's ratios \( \nu_{12} \). For example, the compliance matrix for an orthotropic material is:
[ ε₁ ] [ 1/E₁ -ν₂₁/E₂ 0 ] [ σ₁ ] [ ε₂ ] = [ -ν₁₂/E₁ 1/E₂ 0 ] [ σ₂ ] [ γ₁₂] [ 0 0 1/G₁₂ ] [ τ₁₂]
This calculator assumes isotropic behavior; for composites please use specialised laminate analysis tools.
Polymers, biomaterials, and other viscoelastic substances exhibit time‑dependent behavior. Under dynamic loading, the modulus is complex: \( E^* = E' + iE'' \). Storage modulus \( E' \) represents the elastic energy storage, while loss modulus \( E'' \) represents energy dissipation. Their ratio is the loss factor \( \tan\delta = E''/E' \). Storage modulus is measured via Dynamic Mechanical Analysis (DMA) and varies with temperature and frequency. Our calculator gives the static (or low‑frequency) modulus, which approximates the low‑frequency limit of \( E' \).
Further advanced topics (hyperelasticity, piezoelectric elasticity) can be found in specialized texts like "Mechanics of Materials" by Gere & Goodno.
Young's modulus (E), also known as the elastic modulus or modulus of elasticity, is a fundamental mechanical property that measures the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in the linear elastic region of a material, as described by Hooke's law:
σ = E · ε
where σ is tensile/compressive stress, ε is axial strain, and E is Young's modulus.
In SI units, E is expressed in pascals (Pa) – typically gigapascals (GPa) for engineering materials. A high modulus indicates a stiff material (e.g., diamond ~1220 GPa); a low modulus indicates a flexible material (e.g., rubber ~0.01 GPa).
The concept was introduced by the English scientist Thomas Young in his 1807 book A Course of Lectures on Natural Philosophy and the Mechanical Arts. However, the actual modulus was first defined by Leonhard Euler in 1727, and Young himself acknowledged earlier work by Euler and Giordano Riccati. Young's modulus became a cornerstone of elasticity theory, later formalized by Cauchy, Poisson, and Navier. Today it is essential in structural engineering, materials science, and biomechanics.
From stress and strain: \( E = \frac{\sigma}{\epsilon} \). Ensure strain is dimensionless (e.g., mm/mm).
From force and geometry: Stress \( \sigma = \frac{F}{A} \), strain \( \epsilon = \frac{\Delta L}{L_0} \), hence \( E = \frac{F / A}{\Delta L / L_0} = \frac{F \cdot L_0}{A \cdot \Delta L} \).
The calculator uses these formulas directly. All inputs must be in consistent SI units (N, m, Pa). The stress‑strain diagram is a straight line through the origin with slope E. If you provide a specific data point, it is plotted as a red dot; its position relative to the line indicates consistency with Hooke's law (for ideal elastic materials).
| Material | E (GPa) | E (10⁶ psi) | Reference |
|---|---|---|---|
| Structural steel | 200 | 29 | ASTM A36 |
| Aluminum alloy (6061) | 69 | 10 | ASTM B209 |
| Copper | 110 | 16 | ASM Handbook |
| Concrete (compression) | 30 | 4.35 | ACI 318 |
| Wood (pine, along grain) | 10 | 1.45 | Wood Handbook |
| Glass | 70 | 10.2 | MatWeb |
| Nylon | 2-4 | 0.3-0.6 | Manufacturer data |
| Rubber | 0.01-0.1 | 0.0015-0.015 | — |
A suspension bridge uses steel cables of diameter 50 mm (area ≈ 0.00196 m²). Under a maximum tension of 500 kN, the cable stretches 0.15 m over an original length of 50 m. Using the force‑geometry mode: F = 500,000 N, A = 0.00196 m², L₀ = 50 m, ΔL = 0.15 m. The calculator yields E ≈ 200 GPa, confirming the material is indeed steel. The stress‑strain point (σ ≈ 255 MPa, ε = 0.003) lies on the steel line, validating elastic design.