Elasticity describes how materials deform under stress and return to original shape when stress is removed. Stress (σ) = force per area. Strain (ε) = fractional deformation. Elastic moduli relate stress to strain. Beyond elastic limit, permanent deformation occurs.
Elasticity calculators
Stress, strain, moduli, Poisson ratio, energy
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| Concept | Formula | Notes / Units / Conditions |
|---|---|---|
| Stress | σ = F / A | N/m² or Pascal (Pa) |
| Strain (longitudinal) | ε = ΔL / L₀ | Dimensionless |
| Strain (volumetric) | ε_v = ΔV / V₀ | Dimensionless |
| Strain (shear) | γ = Δx / L or tan θ ≈ θ | Dimensionless (small angle) |
| Young's Modulus | Y = (F/A) / (ΔL/L₀) | For wires, rods (elastic) |
| Bulk Modulus | B = -P / (ΔV/V₀) | Negative sign → volume decrease |
| Rigidity / Shear Modulus | G or η = (F/A) / γ | For twisting, shearing |
| Poisson's Ratio | ν = - (Δd/d₀) / (ΔL/L₀) | Usually 0.2–0.5; negative sign for convention |
| Elastic Potential Energy | U = ½ Y A (ΔL)² / L₀ | Energy stored in deformed solid |
| Relation between moduli | Y = 2G(1 + ν) Y = 3B(1 − 2ν) |
Important relations |
About Elasticity
Elasticity describes how materials deform under applied forces and return to their original shape when forces are removed. The key concepts are stress (force per unit area) and strain (fractional deformation).
Stress and Strain
Stress (σ) = Force per unit area = F/A. Units: Pascal (Pa = N/m²). Stress causes deformation.
Strain (ε) = Fractional change in dimension = ΔL/L₀ (longitudinal) or ΔV/V₀ (volumetric). Strain is dimensionless.
Elastic Moduli
Young's modulus (Y) = stress/strain for longitudinal deformation. Measures stiffness: higher Y = stiffer material. Steel: ~200 GPa, rubber: ~0.01 GPa.
Bulk modulus (B) = -pressure/volumetric strain. Measures resistance to volume change. Water: ~2.2 GPa, steel: ~160 GPa.
Shear modulus (G) = shear stress/shear strain. Measures resistance to shape change (twisting, shearing).
Poisson's Ratio
Poisson's ratio (ν) = -lateral strain/longitudinal strain. When stretched, materials contract laterally. Typical range: 0.2–0.5. Rubber: ~0.5 (nearly incompressible), cork: ~0 (no lateral contraction).
Elastic Limit and Plastic Deformation
Below the elastic limit, deformation is reversible. Beyond it, permanent (plastic) deformation occurs. The stress-strain curve shows: elastic limit → yield point → ultimate stress → breaking stress.