Young's Modulus Calculator
Enter force, original length, cross-sectional area and extension to compute Young's modulus E=FL/(A·ΔL). F=100 N, L=1 m, A=1e-4 m², ΔL=1e-3 m → E=1e9 Pa (1 GPa).
Input Data
Results
At a glance:Young's modulus (Thomas Young, 1807, English physicist) measures a material's stiffness in its linear-elastic range (where Hooke's law holds): the ratio of normal stress σ to normal strain ε, E=σ/ε=(F/A)/(ΔL/L)=F·L/(A·ΔL). Physical meaning: (1) E characterises stiffness — larger E means harder to deform (rigid), smaller E means easier (soft); (2) stress σ=F/A (Pa, force per area), strain ε=ΔL/L (dimensionless, relative deformation); (3) Hooke's law σ=Eε is the linear-elastic constitutive relation, valid for small strain (ε<0.1%); (4) beyond the yield strength the material enters plastic deformation (permanent), and Hooke's law fails; (5) E is an intrinsic material property, independent of shape and temperature-dependent. Typical values: rubber 0.01–0.1 GPa (very soft); polymers 0.1–5 GPa; wood 5–15 GPa; concrete 20–40 GPa; glass 50–80 GPa; aluminium 69 GPa; copper 117 GPa; steel 200 GPa; tungsten 400 GPa; diamond 1000 GPa (stiffest). History: Young proposed the modulus concept in 1807; Cauchy systematised stress–strain tensors in the 1820s. Applications: (1) structural engineering (stiffness of beams, columns, trusses); (2) material selection (aerospace uses titanium for high E/ρ); (3) building mechanics (earthquake, wind loads); (4) mechanical design (shafts, springs, bearings); (5) civil engineering (concrete, rebar, foundations); (6) biomechanics (bone E≈20 GPa, blood vessel E≈1 MPa).
Formula
Young's modulus: E = σ/ε = (F/A)/(ΔL/L) = F·L/(A·ΔL)
Stress: σ = F/A
Strain: ε = ΔL/L
Hooke's law: σ = E·ε
Elastic energy: U = ½·E·ε²·V = ½·F·ΔL
$$E = \frac{\sigma}{\varepsilon} = \frac{F/A}{\Delta L/L} = \frac{F L}{A\,\Delta L}, \quad \sigma = \frac{F}{A}, \quad \varepsilon = \frac{\Delta L}{L}, \quad \sigma = E\varepsilon$$How to Use
- Enter applied force F (N).
- Enter original length L (m).
- Enter cross-sectional area A (m²).
- Enter extension ΔL (m).
- The tool computes Young's modulus E.
Case Studies
Structural engineering and material choice
Hong Kong's Bank of China Tower (70 floors) uses a steel frame (E=200 GPa) with concrete (E=30 GPa). Steel resists tension, concrete compression — exploiting each material's strength; each floor carries thousands of tonnes, steel beams extend <1 mm.
Tsing Ma Bridge main cable: E=200 GPa, length 1377 m, area 0.6 m². Under 2000 t load, ΔL=F·L/(E·A)=2e7×1377/(2e11×0.6)≈0.23 m (23 cm), buffering wind load.
Material choice: aerospace titanium E=110 GPa, density 4500 kg/m³, E/ρ≈24 (comparable to steel's 25). Polymers have low E but are light, used for non-load parts.
Biomechanics and medical use
Bone E≈20 GPa (hard), cartilage E≈1 MPa (soft), blood vessel E≈1 MPa, skin E≈0.5 MPa. Tissues differ by ~4 orders of magnitude — bone bears load, cartilage cushions, vessels pulse.
Orthopaedic implants: titanium E≈110 GPa, 5× bone's 20 GPa. Stress shielding causes osteoporosis (lack of stress stimulus). New biodegradable magnesium alloys E≈45 GPa closer to bone.
Hong Kong University medical-school biomechanics research. Arteriosclerosis raises vessel E, lowers elasticity, raises blood pressure. Measuring pulse wave velocity (PWV) infers vessel E to diagnose cardiovascular disease.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.