6. Deformation of solids

Written by: Adhulan Rajkamal
Formatted by: Adhulan Rajkamal

Index

Deformation is caused by tensile or compressive forces:
- Tensile → Causes extension
- Compressive → Causes compression
Load → The external force applied to a material, causing it to deform; measured in newtons (N)
- Weight of the load on a spring → tensile force acting on the spring
Extension → The increase in length of a material when subjected to a tensile load
- \( \text{Extension} = \text{Stretched Length} – \text{Original Length} \)
Compression → The decrease in length of a material when subjected to a compressive load (pushing force)
Hooke’s Law → Provided that the limit of proportionality is not exceeded, the extension of an object is proportional to the applied load
- The law can be expressed as: \( F \propto e \)
- Introducing a constant: \( F = k e \)
- \( k \) is the spring constant, defined as force per unit extension: \( k = \frac{F}{e} \)
Limit of proportionality → The maximum point up to which the material obeys Hooke’s Law; beyond this point, deformation becomes non-linear

Stress (\( \sigma \)) → The force applied per unit cross-sectional area; measured in pascals (Pa): \[ \sigma = \frac{F}{A} \]
- F → Force applied (N)
- A → Cross-sectional area (m²)
Strain (\( \varepsilon \)) → The ratio of extension (or compression) to the original length of a material: \[ \varepsilon = \frac{e}{L_0} \]
- e → Extension or compression (m)
- L₀ → Original length (m)
- Strain has no units
Young Modulus (\( E \)) → A measure of the stiffness of a material, defined as the ratio of stress to strain within the proportionality limit

The following setup can be used to perform a simple experiment to measure the Young modulus of a wire.

- Set-up: A wire is clamped at one end with a load carrier attached to the other; a paper flag marks the reference point.
- Measurements:
  - Original length (\( L_0 \)) is measured from the clamp to the reference point.
  - Diameter (\( d \)) is measured using a micrometer, and the cross-sectional area is calculated as: \[ A = \frac{1}{4} \pi d^2 \]
  - Extension (\( e \)) is measured as masses (\( m \)) are added to the carrier.
- Calculation:
  - Load (\( F \)) is found using: \[ F = mg \]
  - A graph of \( F \) (y-axis) against \( e \) (x-axis) is plotted (Force-extension graph).
  - The Young modulus (\( E \)) is determined from the gradient (\( \frac{F}{e} \)) as: \[ E = \frac{F L_0}{e A} = \frac{F}{e} \times \frac{L_0}{A} = \textit{gradient} \times \frac{L_0}{A} \]
Note: Ensure the wire does not exceed the limit of proportionality for valid results.

Elastic limit → maximum stress or force that a material can withstand and still return to its original shape and size when the load is removed
Elastic deformation → Object returns to its original shape and size when the force on it is removed; occurs within elastic limit
Plastic deformation → Object does not return to its original shape and size when force is removed; occurs when the material is stressed beyond the elastic limit
Area under force-extension graph represents the work done
- Elastic potential energy of a deformed material = work done to deform the material (within its limit of proportionality)

Equations to calculate the elastic potential energy of a deformed material:
- \( E_p = \frac{1}{2} F e \) → Related to the area under the force-extension graph.
  - \( F \) → Force
  - \( e \) → Extension
- \( E_p = \frac{1}{2} k e^2 \)
  Where:
  - \( k \) → Spring constant
  - \( e \) → Extension
Derivation of the second equation:
- From the first equation: \( E_p = \frac{1}{2} F e \) and Hooke’s Law: \( F = k e \)
- Substituting \( F = k e \) into the first equation:
  \( E_p = \frac{1}{2} (k e) e = \frac{1}{2} k e^2 \)