Physics: Elasticity Concepts and Formulas

1. Definitions

• Rigid Body: A body whose size and shape cannot be changed, regardless of the force applied. No perfectly rigid body exists in nature.
• Deformation Force: A force that changes the size or shape of a body without moving it as a whole.
• Restoring Force: A force that restores a body’s size and shape when the deformation force is removed. It is equal in magnitude but opposite in direction to the deformation force. This force is critical for the body's elastic nature.
• Elasticity: The ability of a material to regain its original size and shape after deformation force is removed. E.g., Steel, Rubber. Quartz is nearly perfectly elastic.
• Plasticity: The inability of a material to regain its original size and shape after deformation force is removed. E.g., Putty, Chewing gum.

2. Stress and Strain

• Stress: The restoring force per unit area.

  • Formula: Stress = Restoring Force / Area
  • Unit: N/m² or Pascal
  • Types:
    • Normal Stress: Stress perpendicular to the surface.
    • Tangential Stress: Stress parallel to the surface.
    • Volume Stress/Bulk Stress: Stress that changes the body’s volume.
    • Longitudinal Stress: Can be tensile (increasing length) or compressive (decreasing length).

• Strain: The change produced per unit dimension.

  • Types:
    • Longitudinal Strain: Change in length/original length.
    • Shearing Strain: Relative displacement between two layers divided by distance between them.
    • Volumetric/Bulk Strain: Change in volume/original volume.

3. Elastic Limit and Hooke’s Law

• Elastic Limit: The maximum stress that can be applied to a body such that it regains its original size and shape.
• Hooke’s Law: Within the elastic limit, stress is directly proportional to strain, forming a linear stress-strain graph.

4. Moduli of Elasticity

• Young’s Modulus (Y): Ratio of longitudinal stress to longitudinal strain.
• Rigidity Modulus: Ratio of tangential stress to shearing strain.
• Bulk Modulus (B): Ratio of volume stress to bulk strain.
  • Compressibility: The reciprocal of bulk modulus.

5. Poisson’s Ratio (σ)

• Ratio of lateral contraction strain to longitudinal elongation strain.
• It is dimensionless and has no units.
• Theoretical and practical limits range from 0 to 0.5.

6. Thermal Force and Stress

• Thermal Force: Force exerted by a metal bar fixed between supports when heated.
• Thermal Stress: Linear compressive stress caused by thermal force.

7. Energy in Elasticity

• Work Done/Elastic Potential Energy:
  • Formula: ( W = \frac{1}{2} \times \text{Stress} \times \text{Strain} \times \text{Volume} )
• Strain Energy per Unit Volume (Energy Density):
  • Formula: ( E = \frac{1}{2} \times \text{Stress}^2 / Y ), where Y is Young's modulus.