Mechanical Properties of Solids

• Fundamental Concepts

  • Deforming Force: External force changing shape or size
  • Restoring Force: Internal force opposing deformation
  • Elasticity: Property to regain original shape after force removal
    • Example: Steel (more elastic than rubber)
  • Plasticity: Inability to regain original shape (permanent deformation)
    • Example: Putty, Clay

• Stress ($\sigma$)

  • Definition: Restoring Force per unit Area ($F/A$)
  • SI Unit: $N/m^2$ or Pascal ($Pa$)
  • Dimensional Formula: $[M L^{-1} T^{-2}]$
  • Types of Stress
    • Longitudinal Stress: Force perpendicular to cross-sectional area
      • Tensile Stress: Increases length
      • Compressive Stress: Decreases length
    • Tangential / Shearing Stress: Force parallel to surface area
    • Hydraulic / Volumetric Stress: Uniform liquid pressure acting normally from all sides

• Strain ($\epsilon$)

  • Definition: Change in dimension / Original dimension
  • Nature: Dimensionless quantity (No units)
  • Types of Strain
    • Longitudinal Strain: $\Delta L / L$
    • Shearing Strain: Angular displacement $\theta$ (in radians)
    • Volume Strain: $\Delta V / V$

• Hooke's Law & Stress-Strain Curve

  • Hooke's Law: Within elastic limit, $\text{Stress} \propto \text{Strain}$
  • Stress-Strain Curve Regions
    • Proportional Limit: Hooke's law strictly holds
    • Elastic Limit / Yield Point: Max stress with zero permanent set
    • Permanent Set: Strain remaining after removing stress
    • Ultimate Tensile Strength: Maximum stress material can withstand
    • Fracture Point: Point where wire breaks
  • Material Classification
    • Ductile Materials: Large plastic region (e.g., Copper, Iron)
    • Brittle Materials: Breaks immediately after elastic limit (e.g., Glass, Cast Iron)
    • Elastomers: Large strain for small stress, no Hooke's law region (e.g., Rubber, Aorta)

• Moduli of Elasticity

  • Young's Modulus ($Y$)
    • Formula: $Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{F \cdot L}{A \cdot \Delta L}$
    • Applies to: Solids only
  • Shear Modulus / Modulus of Rigidity ($G$ or $\eta$)
    • Formula: $G = \frac{\text{Shearing Stress}}{\text{Shearing Strain}} = \frac{F}{A \cdot \theta}$
    • Applies to: Solids only
  • Bulk Modulus ($B$)
    • Formula: $B = -\frac{\text{Hydraulic Stress}}{\text{Volume Strain}} = -\frac{P \cdot V}{\Delta V}$
    • Applies to: Solids, Liquids, and Gases
    • Compressibility ($K$): Reciprocal of Bulk Modulus ($K = 1/B$)

• Poisson's Ratio ($\sigma$ or $\nu$)

  • Definition: Ratio of Lateral Strain to Longitudinal Strain
  • Formula: $\sigma = -\frac{\Delta d / d}{\Delta L / L}$
  • Values:
    • Theoretical Range: $-1$ to $0.5$
    • Practical Range: $0$ to $0.5$

• Elastic Potential Energy in a Stretched Wire

  • Definition: Work done against internal restoring forces
  • Total Energy ($U$): $U = \frac{1}{2} \times \text{Load} \times \text{Extension} = \frac{1}{2} F \Delta L$
  • Energy Density ($u$): Energy per unit volume
    • Formula: $u = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2} Y (\text{Strain})^2$

• Practical Applications

  • Metallic Ropes in Cranes: Made of braided thin wires to increase flexibility and load strength
  • I-Shaped Beams: Minimizes bending ($\delta = \frac{W L^3}{4 Y b d^3}$) and prevents buckling
  • Maximum Height of Mountains: Limited to $\approx 10\text{ km}$ due to elastic limit of rocks