Learning Objectives
- Understand stress, strain, and their types
- Learn Hooke's law and the stress-strain curve
- Define and differentiate between Young's modulus, bulk modulus, and shear modulus
- Understand elastic potential energy stored in a deformed body
- Study applications of elasticity in daily life and engineering
Key Concepts
Stress
Restoring force per unit area developed within a body when an external force is applied.
Stress = F/A (SI unit: Pa or N/m²)
Types: Tensile stress (stretching), Compressive stress (compression), Shearing stress (tangential force), Hydraulic stress (uniform pressure from all sides).
Strain
Ratio of change in dimension to original dimension. It is dimensionless.
Longitudinal strain: ΔL/L (change in length / original length)
Volumetric strain: ΔV/V (change in volume / original volume)
Shearing strain: tan φ ≈ φ = Δx/L (angle of shear)
Hooke's Law
Within the elastic limit, stress is directly proportional to strain.
Stress = E × Strain, where E is the modulus of elasticity.
Elastic Moduli
Young's Modulus (Y): Y = (F/A)/(ΔL/L) = FL/AΔL (for longitudinal deformation)
Bulk Modulus (B): B = -P/(ΔV/V) = -PV/ΔV (for volume change under uniform pressure). Compressibility = 1/B.
Shear Modulus / Rigidity Modulus (G or η): G = (F/A)/φ (for shearing deformation)
For most materials: Y > G (typically Y ≈ 2-3 times G).
Stress-Strain Curve
Key regions on the stress-strain curve for a ductile material (like mild steel):
- Proportional limit: Up to this point, stress ∝ strain (Hooke's law valid).
- Elastic limit: Maximum stress for which body returns to original shape.
- Yield point: Beyond this, permanent deformation begins (plastic deformation).
- Ultimate tensile strength: Maximum stress the material can withstand.
- Fracture point: Material breaks.
Ductile materials (e.g., copper, aluminium): large plastic deformation before fracture.
Brittle materials (e.g., glass, cast iron): little or no plastic deformation; fracture soon after elastic limit.
Elastomers (e.g., rubber): large strain for small stress, but return to original shape.
Elastic Potential Energy
Energy stored per unit volume: u = ½ × stress × strain = ½ × (stress)²/E = ½ × E × (strain)²
Total energy stored: U = ½ × F × ΔL = ½ × (F²L)/(AY)
Poisson's Ratio
σ = -(Lateral strain)/(Longitudinal strain) = -(Δd/d)/(ΔL/L)
Theoretical range: -1 to 0.5. For most materials: 0.2 to 0.4.
Summary
When external forces deform a solid, internal restoring forces develop (stress). The deformation relative to original dimensions is strain. Hooke's law states that stress is proportional to strain within the elastic limit. Young's modulus, bulk modulus, and shear modulus quantify material response to different types of deformation. The stress-strain curve reveals key mechanical properties. Elastic potential energy is stored in deformed bodies.
Important Terms
- Elasticity: Property of a body to regain its original shape after removal of deforming force
- Plasticity: Property of permanent deformation under stress
- Young's Modulus: Ratio of longitudinal stress to longitudinal strain
- Bulk Modulus: Ratio of hydraulic stress to volumetric strain
- Shear Modulus: Ratio of shearing stress to shearing strain
- Poisson's Ratio: Ratio of lateral strain to longitudinal strain
Quick Revision
- Stress = F/A; Strain = ΔL/L (dimensionless)
- Hooke's law: Stress ∝ Strain (within elastic limit)
- Y = FL/AΔL; B = PV/ΔV; G = (F/A)/φ
- Elastic PE per unit volume = ½ × stress × strain
- Stress-strain curve: proportional limit → elastic limit → yield → UTS → fracture
- Steel is more elastic than rubber (higher Young's modulus)