Module V: Elasticity (Phy)

1. Elasticity and Elastic Constants

• Elasticity: The property of a body by virtue of which it regains its original shape and size after the removal of deforming forces.

• Stress: Internal restoring force per unit area.
  $\text{Stress} = \frac{F}{A}$

• Strain: Relative change in dimension produced in a body.
  $\text{Strain} = \frac{\Delta L}{L}$

• Hooke's Law: Within the elastic limit, stress is directly proportional to strain.
  $\text{Stress} = E \times \text{Strain}$

• Elastic Moduli:

  • Young's Modulus ($Y$): Ratio of longitudinal stress to longitudinal strain.
    $Y = \frac{F L}{A \Delta L}$

  • Bulk Modulus ($K$): Ratio of normal stress to volumetric strain.
    $K = -\frac{P V}{\Delta V}$

  • Modulus of Rigidity ($\eta$): Ratio of shearing stress to shearing strain.
    $\eta = \frac{F}{A \theta}$

  • Poisson's Ratio ($\sigma$): Ratio of lateral strain to longitudinal strain.
    $\sigma = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}$

2. Relations Among Elastic Constants

• Interrelations derived from isotropic elasticity theory:

  1. Relation between $Y$, $K$, and $\sigma$:
     $Y = 3 K (1 - 2 \sigma)$

  2. Relation between $Y$, $\eta$, and $\sigma$:
     $Y = 2 \eta (1 + \sigma)$

  3. Eliminating $Y$ yields Poisson's ratio in terms of $K$ and $\eta$:
     $\sigma = \frac{3 K - 2 \eta}{6 K + 2 \eta}$

  4. Relation connecting $Y$, $K$, and $\eta$:
     $\frac{9}{Y} = \frac{1}{K} + \frac{3}{\eta}$

• Theoretical Limits of Poisson's Ratio ($\sigma$):

  • Theoretical range: $-1 \le \sigma \le 0.5$

  • Practical range for isotropic materials: $0 \le \sigma \le 0.5$

3. Twisting Couple in a Wire

• Consider a cylindrical wire of length $L$ and radius $R$, fixed at one end and twisted at the free end through an angle $\theta$.

• Torsional Rigidity ($C$): Restoring couple per unit twist angle.
  $C = \frac{\pi \eta R^4}{2 L}$

• Total Restoring Twisting Couple ($T$):
  $T = C \theta = \frac{\pi \eta R^4 \theta}{2 L}$

• Work Done in Twisting the Wire:
  $W = \frac{1}{2} C \theta^2 = \frac{\pi \eta R^4 \theta^2}{4 L}$

4. Bending of Beams with Symmetric Cross Section

• Beam: A homogeneous, isotropic bar of uniform cross section whose length is large compared to its transverse dimensions.

• Neutral Surface and Neutral Axis: The layer of the beam that undergoes no strain during bending is the neutral surface, and its intersection with the transverse cross section is the neutral axis.

• Bending Moment ($M$): The moment of internal restoring forces about the neutral axis.
  $M = \frac{Y I<em>g}{R</em>c}$
  where $I<em>g$ is the geometrical moment of inertia and $R</em>c$ is the radius of curvature of the neutral axis.

• Geometrical Moment of Inertia ($I_g$):

  • Rectangular cross section of breadth $b$ and depth $d$:
    $I_g = \frac{b d^3}{12}$

  • Circular cross section of radius $r$:
    $I_g = \frac{\pi r^4}{4}$

5. Cantilever

• Definition: A beam fixed horizontally at one end and loaded with a weight at the free end.

• Depression at the Free End ($\delta$):

  • When the mass of the beam is negligible, the depression $\delta$ produced by load $W$ at the free end of length $L$ is:
    $\delta = \frac{W L^3}{3 Y I_g}$

• Expressions for Specific Cross Sections:

  • Rectangular cross section:
    $\delta = \frac{4 W L^3}{Y b d^3}$

  • Circular cross section:
    $\delta = \frac{4 W L^3}{3 \pi Y r^4}$