Elastic Behaviour of Solids — Revision Notes

⚡ 30-Second Revision

Stress ($sigma$) — Force per unit area. $\sigma = F/A$. Unit: Pa or $N/m^2$.
Strain ($epsilon$) — Change in dimension / Original dimension. Dimensionless.
Hooke's Law — $\sigma = E \times \epsilon$ (within elastic limit).
Young's Modulus (Y) — $Y = \frac{\text{Normal Stress}}{\text{Longitudinal Strain}} = \frac{FL}{A\Delta L}$.
Bulk Modulus (B) — $B = \frac{\text{Volumetric Stress}}{\text{Volumetric Strain}} = -\frac{PV}{\Delta V}$.
Shear Modulus (G or $eta$) — $G = \frac{\text{Tangential Stress}}{\text{Shear Strain}} = \frac{F_{\parallel}}{A\theta}$.
**Poisson's Ratio ($

u$)**:$\nu = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}$. Dimensionless.

Elastic Potential Energy (U) — $U = \frac{1}{2}F\Delta L = \frac{1}{2} \frac{YA}{L} (\Delta L)^2$.
Energy Density (u) — $u = \frac{1}{2} \text{Stress} \times \text{Strain} = \frac{1}{2} Y (\text{Strain})^2 = \frac{1}{2} \frac{(\text{Stress})^2}{Y}$.
Stress-Strain Curve — Proportional limit, Elastic limit, Yield point, Ultimate Tensile Strength, Fracture point.

2-Minute Revision

Elastic behaviour is the ability of a solid to return to its original shape after deforming forces are removed. This is quantified by stress (force per unit area, $\sigma = F/A$) and strain (relative deformation, $\epsilon = \Delta L/L$).

Hooke's Law states that within the elastic limit, stress is proportional to strain, with the constant of proportionality being the modulus of elasticity. There are three main moduli: Young's Modulus (Y) for longitudinal deformation, Bulk Modulus (B) for volumetric changes, and Shear Modulus (G) for shape changes.

**Poisson's Ratio ($u$) describes lateral contraction during longitudinal stretching. The stress-strain curve** is vital, illustrating a material's response from elastic deformation (where Hooke's Law holds) through the elastic limit, yield point (onset of plastic deformation), ultimate tensile strength (maximum stress), and finally, the fracture point.

Work done in deforming an elastic body is stored as elastic potential energy, with energy density $u = \frac{1}{2} \text{Stress} \times \text{Strain}$. Remember to use consistent units (SI) for all calculations and pay attention to powers of 10.

5-Minute Revision

To quickly revise Elastic Behaviour of Solids for NEET, focus on these core concepts. Start with **Stress ($sigma$)** as the internal restoring force per unit area ($F/A$) and **Strain ($epsilon$)** as the dimensionless ratio of deformation to original dimension ($\Delta L/L$).

Understand their types: normal, tangential, volumetric for stress; longitudinal, shear, volumetric for strain. The fundamental relationship is Hooke's Law: $\sigma = E\epsilon$, valid up to the proportional limit.

The constant $E$ is the modulus of elasticity.

Key moduli to remember are:

1
Young's Modulus (Y) — For stretching/compression. $Y = \frac{FL}{A\Delta L}$. High Y means stiff material.
2
Bulk Modulus (B) — For volume change under uniform pressure. $B = -\frac{PV}{\Delta V}$. High B means incompressible.
3
Shear Modulus (G) — For shape change under tangential force. $G = \frac{F_{\parallel}}{A\theta}$. High G means rigid.

**Poisson's Ratio ($u$)** is the ratio of lateral strain to longitudinal strain, typically between 0 and 0.5. It's dimensionless.

Elastic Potential Energy (U) stored in a stretched wire is $U = \frac{1}{2}F\Delta L$. The energy density (energy per unit volume) is $u = \frac{1}{2}\sigma\epsilon = \frac{1}{2}Y\epsilon^2 = \frac{1}{2}\frac{\sigma^2}{Y}$. Remember the square dependence on force or extension for energy.

The Stress-Strain Curve is crucial: Identify the proportional limit (Hooke's Law holds), elastic limit (material returns to original shape), yield point (permanent deformation begins), ultimate tensile strength (maximum stress), and fracture point (material breaks). Differentiate between ductile (large plastic region) and brittle (small/no plastic region) materials based on this curve. Elastomers (like rubber) show large, reversible deformations with non-linear curves.

Example: A wire of length $1,\text{m}$, area $10^{-6},\text{m}^2$, $Y = 2 \times 10^{11},\text{Pa}$ is stretched by $200,\text{N}$. Stress $= F/A = 200 / 10^{-6} = 2 \times 10^8,\text{Pa}$. Strain $= \text{Stress}/Y = (2 \times 10^8) / (2 \times 10^{11}) = 10^{-3}$. Extension $\Delta L = \text{Strain} \times L = 10^{-3} \times 1 = 1,\text{mm}$. Energy stored $U = \frac{1}{2}F\Delta L = \frac{1}{2} \times 200 \times (10^{-3}) = 0.1,\text{J}$.

Always ensure consistent units and careful calculation of powers of 10.

Prelims Revision Notes

Elastic Behaviour of Solids: NEET Revision Notes

**1. Stress ($sigma$)**: Internal restoring force per unit area. $\sigma = F/A$. SI unit: Pascal (Pa) or $N/m^2$. Types: * Normal Stress: Perpendicular force. Tensile (pulling) or Compressive (pushing). * Tangential (Shear) Stress: Parallel force, causes shape change. * Volumetric (Hydraulic) Stress: Uniform pressure, causes volume change.

**2. Strain ($epsilon$)**: Ratio of change in dimension to original dimension. Dimensionless. * Longitudinal Strain: $\epsilon_l = \Delta L/L$. * Shear Strain: $\epsilon_s = \Delta x/h = \tan\theta \approx \theta$ (for small angles). * Volumetric Strain: $\epsilon_v = \Delta V/V$.

3. Hooke's Law: For small deformations, $\text{Stress} \propto \text{Strain}$, or $\sigma = E\epsilon$. $E$ is modulus of elasticity.

4. Elastic Moduli (Constants of Elasticity): * Young's Modulus (Y): $Y = \frac{\text{Normal Stress}}{\text{Longitudinal Strain}} = \frac{FL}{A\Delta L}$. Measures resistance to stretching/compression.

Unit: Pa. * Bulk Modulus (B): $B = \frac{\text{Volumetric Stress}}{\text{Volumetric Strain}} = -\frac{P}{\Delta V/V} = -\frac{PV}{\Delta V}$. Measures resistance to volume change. Unit: Pa. * Compressibility (K): $K = 1/B$.

Unit: $Pa^{-1}$. * **Shear Modulus (G or $eta$)**: $G = \frac{\text{Tangential Stress}}{\text{Shear Strain}} = \frac{F_{\parallel}}{A\theta}$. Measures resistance to shape change. Unit: Pa.

**5. Poisson's Ratio ($u$)**: $\nu = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = -\frac{\Delta D/D}{\Delta L/L}$. Dimensionless. Typically $0 < \nu < 0.5$.

6. Stress-Strain Curve: Graph of stress vs. strain. * Proportional Limit: Stress $\propto$ Strain (Hooke's Law valid). * Elastic Limit: Max stress without permanent deformation. * Yield Point: Stress at which permanent deformation begins.

* Ultimate Tensile Strength: Max stress material can withstand. * Fracture Point: Material breaks. * Ductile Materials: Large plastic region (e.g., steel, copper). * Brittle Materials: Small/no plastic region (e.

g., glass, cast iron). * Elastomers: Large, reversible deformations, non-linear curve (e.g., rubber).

7. Elastic Potential Energy (U): Work done in deforming an elastic body. * For a stretched wire: $U = \frac{1}{2}F\Delta L = \frac{1}{2}k(\Delta L)^2 = \frac{1}{2} \frac{YA}{L} (\Delta L)^2$. * Energy Density (u): Energy per unit volume. $u = \frac{U}{\text{Volume}} = \frac{1}{2} \text{Stress} \times \text{Strain} = \frac{1}{2} Y \epsilon^2 = \frac{1}{2} \frac{\sigma^2}{Y}$.

Key Points for NEET:

Units and dimensions are crucial. Strain and Poisson's ratio are dimensionless.
Hooke's Law is only for small deformations.
Understand the physical meaning of each modulus.
Be able to interpret the stress-strain curve for different materials.
Practice numerical problems involving all formulas, especially Young's Modulus and energy density.

Vyyuha Quick Recall

Stress Strain Hooke's Young Bulk Shear Poisson Energy

Stress: Force/Area Strain: Change/Original Hooke's: Stress $\propto$ Strain Young: Length change Bulk: Volume change Shear: Shape change Poisson: Lateral/Longitudinal Energy: $\frac{1}{2}F\Delta L$