Stress, Strain and Elasticity

Elastic behaviour describes how a solid resists deformation by storing energy and recovering its original shape once the deforming force is removed. NEET typically frames this around the definitions of stress and strain and asks for simple ratio computations using Hooke's law.

Concept

When a deforming force $F$ acts on a body of cross-section $A$, the body experiences a stress equal to $F/A$. The fractional deformation produced is the strain. Stress carries the dimensions of pressure (Pa, N/m$^2$) while strain is dimensionless.

Three principal pairs are tested:

• Longitudinal stress and longitudinal strain $\Delta L/L$.
• Volumetric stress (uniform pressure) and volumetric strain $\Delta V/V$.
• Shear (tangential) stress and shear strain (angle $\phi$).

Within the elastic limit, stress is proportional to strain (Hooke's law). Beyond it, the material yields and permanent deformation sets in.

Formula Derivation

For a wire of natural length $L$ and area $A$ stretched by $\Delta L$ under load $F$, longitudinal stress and strain are

$\sigma = \frac{F}{A}, \quad \varepsilon = \frac{\Delta L}{L}.$

Hooke's law gives

$\sigma = Y \varepsilon \implies Y = \frac{F L}{A \Delta L},$

where $Y$ is Young's modulus. Rearranging,

$\Delta L = \frac{F L}{A Y}.$

The elastic potential energy per unit volume stored in the stretched wire is

$u = \tfrac{1}{2}\,\sigma\,\varepsilon = \tfrac{1}{2} Y \varepsilon^{2} = \frac{\sigma^{2}}{2 Y}.$

NEET-style Worked Example

A steel wire of length $2\ \text{m}$ and cross-section $1\ \text{mm}^{2}$ is loaded with a $100\ \text{N}$ force. Young's modulus $Y = 2 \times 10^{11}\ \text{N/m}^{2}$. Find the extension.

Step 1. Stress: $\sigma = F/A = 100/(1 \times 10^{-6}) = 10^{8}\ \text{Pa}$.

Step 2. Strain: $\varepsilon = \sigma/Y = 10^{8}/(2 \times 10^{11}) = 5 \times 10^{-4}$.

Step 3. Extension: $\Delta L = \varepsilon L = 5 \times 10^{-4} \times 2 = 10^{-3}\ \text{m} = 1\ \text{mm}$.

Common Confusions

• Stress is not pressure. They share units but stress is internal to a deformed solid.
• Strain has no units, yet it is often quoted as a percentage.
• Elastic limit vs proportional limit. Hooke's law holds only up to the proportional limit; the elastic limit can be slightly higher.
• Reversibility. Below the elastic limit, removing the load returns the body to its natural shape; beyond it, a permanent set remains.

Key Takeaways

• Stress $= F/A$ has SI unit Pa.
• Strain is fractional and dimensionless.
• Within the elastic limit, $\sigma = Y \varepsilon$ (Hooke's law).
• Elastic PE density $u = \tfrac{1}{2}\sigma\varepsilon$.
• $\Delta L \propto F L /(A Y)$.