Elastic moduli are material constants that quantify the linear relationship between stress and strain within the elastic limit. They characterize the resistance of a solid body to deformation under applied external forces.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of stress, strain, Hooke's law, Young's modulus (Y), Bulk modulus (K), Modulus of rigidity (eta), and Poisson's ratio (sigma).
PGT extension
Interrelations between elastic constants (Y, K, eta, sigma), strain energy density, twisting of a cylinder, and bending of beams.
Prerequisites
Newton's laws of motion and static equilibrium
Concept of internal restoring forces and intermolecular bonding
Basic tensor and directional properties of stress and strain
Stress and Strain Fundamentals
Stress is defined as the internal restoring force per unit area, measured in N/m2 or Pascal.
Longitudinal stress produces a change in length, shearing stress changes shape, and volumetric (hydraulic) stress changes volume.
Strain is a dimensionless quantity representing the fractional deformation produced in the body.
Hooke's Law and Elastic Moduli
Within the elastic limit, stress is directly proportional to strain, defining the generalized Hooke's law.
Y = Young's modulus, eta = rigidity modulus, sigma = Poisson's ratio N/m2
Isotropic material
Relation between Y, K, and eta
$$Y = \frac{9K\eta}{3K + \eta}$$
Y = Young's modulus, K = bulk modulus, eta = rigidity modulus N/m2
Isotropic elastic solid
Torsional Rigidity of a Cylinder
$$C = \frac{\pi \eta r^4}{2l}$$
eta = rigidity modulus, r = radius of cylinder, l = length N m / rad
Small angle twist applied to cylindrical wire/rod
Graphs and experiments
Stress-Strain Curve for a Ductile Metal
Draw: A curve plotting tensile stress on the y-axis against engineering strain on the x-axis, highlighting the proportional limit, yield point, ultimate tensile strength point, and fracture point.
Interpret: The initial linear region follows Hooke's law where slope equals Young's modulus. Beyond the yield point, permanent plastic deformation occurs.
Exam use: Frequently tested for identifying elastic limit, yield strength, and toughness (area under the curve).
Searle's Apparatus Experiment
Draw: Schematic setup showing two identical wires suspended from a rigid support, with a spirit level, micrometer screw gauge, and dead load pan.
Interpret: Used to experimentally determine Young's modulus of a material in wire form by measuring microscopic extensions accurately.
Exam use: Conceptual questions on experimental error minimization and least count of the micrometer screw.
Bending of Beams (Elastic Depression)
Draw: A loaded beam supported at both ends showing central sag (depression) delta.
Interpret: Depression delta is inversely proportional to Young's modulus Y and geometrical moment of inertia.
Exam use: Questions on how beam geometry (rectangular vs circular cross-section) affects structural rigidity.
Solved numericals
TGT core
A steel wire of length 2 m and cross-sectional area 1 mm2 is stretched by 2 mm. If Young's modulus of steel is 2 * 1011 N/m2, calculate the tension developed in the wire.
Identify given parameters: L = 2 m, $A = 1 mm^{2} = 1 * 10^{-6} m^{2}$, Delta $L = 2 mm = 2 * 10^{-3} m$, $Y = 2 * 10^{11} N/m^{2}$.
Hooke's law is assumed to hold strictly, which is only valid within the proportional elastic limit of the material.
Materials are assumed to be perfectly isotropic, homogeneous, and continuous, neglecting crystalline grain boundaries and microscopic defects.
Common misconceptions
Common error: A material with a higher Young's modulus is always stronger and harder to break.
Correct: Young's modulus measures elastic stiffness (resistance to deformation), whereas breaking strength refers to the ultimate tensile stress before fracture.
A material can be very stiff (high Y) yet brittle, failing at a low ultimate strain.
Common error: Poisson's ratio can take any positive value.
Correct: Thermodynamic stability and strain energy constraints limit Poisson's ratio for isotropic solids between -1.0 and 0.5.
If sigma exceeded 0.5, the bulk modulus or rigidity modulus would become negative, violating mechanical stability.
Quick revision
Young's modulus governs length changes, bulk modulus governs volume changes, and rigidity modulus governs shape changes under shear.
Young's Modulus: Y = \frac{FL}{A \Delta L}
Bulk Modulus: $K = - \frac{P}{\Delta V / V}$
Must remember
Stress is defined as internal restoring force per unit area, measured in $\text{N/m}^2$ or Pascal.
Strain is a dimensionless quantity representing fractional deformation produced in a body.
The reciprocal of bulk modulus is compressibility.
Poisson's ratio ($\sigma$) is the ratio of lateral strain to longitudinal strain; for isotropic elastic materials, its theoretical limits are $-1$ to $0.5$.
Master relation connecting Young's modulus, bulk modulus, and rigidity modulus is $Y = \frac{9K\eta}{3K + \eta}$.
Elastic energy density stored per unit volume during linear elastic deformation is given by $u = \frac{1}{2} \times \text{Stress} \times \text{Strain}$.
Torsional rigidity of a cylindrical wire or rod is given by $C = \frac{\pi \eta r^4}{2l}$.
Glossary
Young's Modulus ($Y$)Ratio of longitudinal stress to longitudinal strain within the elastic limit.
Bulk Modulus ($K$)Ratio of hydrostatic pressure to volumetric strain, governing uniform compression.
Modulus of Rigidity ($\eta$)Ratio of tangential stress to shearing strain, governing shape distortion.
Poisson's Ratio ($\sigma$)Negative ratio of lateral strain to longitudinal strain under uniaxial loading.
CompressibilityThe reciprocal of the bulk modulus ($1/K$).
Torsional RigidityTorque required to produce a unit twist in a cylindrical wire ($C = \frac{\pi \eta r^4}{2l}$).
Comparison of Elastic Moduli
Modulus
Governing Deformation
Mathematical Expression
Young's Modulus ($Y$)
Longitudinal (length change)
$Y = \frac{FL}{A \Delta L}$
Bulk Modulus ($K$)
Volumetric (volume change)
$K = -\frac{P}{\Delta V / V}$
Rigidity Modulus ($\eta$)
Shear (shape change/distortion)
$\eta = \frac{F/A}{\theta}$
Practice Quiz
Choose an option to reveal the explanation.
1. What is the SI unit of stress?
Explanation: Stress is defined as force per unit area, so its SI unit is $\text{N/m}^2$ or Pascal.
2. Which of the following quantities is dimensionless?
Explanation: Strain is the ratio of two similar quantities (such as change in length to original length), making it dimensionless.
3. Young's modulus is applicable for which type of deformation?
Explanation: Young's modulus ($Y$) specifically governs longitudinal deformation (stretching or compression of length).
4. The reciprocal of Bulk Modulus is known as:
Explanation: Compressibility is defined as the inverse of the bulk modulus ($1/K$).
5. What are the theoretical limits of Poisson's ratio for isotropic elastic materials?
Explanation: For isotropic elastic materials, thermodynamic stability constraints limit Poisson's ratio between $-1$ and $0.5$.
6. What is the correct master relation between Young's modulus ($Y$), bulk modulus ($K$), and rigidity modulus ($\eta$)?
Explanation: Eliminating Poisson's ratio from $Y = 3K(1 - 2\sigma)$ and $Y = 2\eta(1 + \sigma)$ yields $Y = \frac{9K\eta}{3K + \eta}$.
7. What is the elastic energy density stored per unit volume in a linear elastic material?
Explanation: The work done per unit volume stored as elastic potential energy is $u = \frac{1}{2} \times \text{Stress} \times \text{Strain}$.
8. How many independent elastic constants are required to describe an isotropic elastic material?
Explanation: Although four constants ($Y, K, \eta, \sigma$) are used, only two are independent for an isotropic material.
9. The torsional rigidity of a cylindrical wire of radius $r$ and length $l$ is proportional to:
Explanation: Torsional rigidity is given by $C = \frac{\pi \eta r^4}{2l}$, which is directly proportional to $r^4$.
10. Which modulus governs the change in shape (shear deformation) of a body?
Explanation: The modulus of rigidity ($\eta$) relates tangential stress to shear angle, governing shape distortion.
11. A higher Young's modulus for a material indicates:
Explanation: Young's modulus measures stiffness or resistance to elastic deformation, not necessarily ultimate breaking strength.
12. What is the relationship between Young's modulus ($Y$), rigidity modulus ($\eta$), and Poisson's ratio ($\sigma$)?
Explanation: The standard relation connecting Young's modulus, rigidity modulus, and Poisson's ratio is $Y = 2\eta(1 + \sigma)$.
13. What happens to the bulk modulus of an incompressible substance?
Explanation: For a perfectly incompressible substance, $\Delta V = 0$, making the bulk modulus $K = -P / (0) = \infty$.
14. Under uniaxial tension, if a wire elongates in length, its diameter will:
Explanation: Due to a positive Poisson's ratio, longitudinal extension is accompanied by lateral contraction (decrease in diameter).
15. Which of the following moduli has units of $\text{N/m}^2$?
Explanation: Bulk modulus represents pressure divided by volumetric strain, resulting in units of $\text{N/m}^2$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If the Poisson's ratio of an isotropic material is $0.25$, what is the ratio of its Young's modulus to its rigidity modulus ($Y / \eta$)?
2. A wire of length $L$ and radius $r$ is stretched by a force $F$. If the elongation is $\Delta L$, Young's modulus $Y$ is given by:
Explanation: Area $A = \pi r^2$. Substituting into $Y = \frac{FL}{A \Delta L}$ yields $Y = \frac{FL}{\pi r^2 \Delta L}$.
3. If the bulk modulus of a liquid is $2 \times 10^9 \text{ N/m}^2$, what pressure is required to decrease its volume by $0.1\%$?
Explanation: Bulk modulus $K = -P / (\Delta V / V)$. Given $K = 2 \times 10^9$ and $\Delta V / V = -0.001$, $P = K \times 0.001 = 2 \times 10^9 \times 10^{-3} = 2 \times 10^6 \text{ N/m}^2$.
4. For a given material, the rigidity modulus is $\frac{3}{8}$ of its Young's modulus. What is its Poisson's ratio?
Explanation: Using $Y = 2\eta(1 + \sigma)$, we have $\frac{Y}{\eta} = 2(1 + \sigma)$. Since $\eta = \frac{3}{8}Y$, $\frac{Y}{\eta} = \frac{8}{3}$. Thus $\frac{8}{3} = 2(1 + \sigma) \implies 1 + \sigma = \frac{4}{3} \implies \sigma = \frac{1}{3} \approx 0.33$.
5. Two wires $A$ and $B$ are made of the same material. Wire $A$ has twice the length and half the diameter of wire $B$. If they are stretched by the same force, the ratio of their elongations $(\Delta L_A / \Delta L_B)$ is:
Explanation: Elongation $\Delta L = \frac{FL}{A Y} = \frac{FL}{\pi r^2 Y}$. Since $F, Y$ are same, $\Delta L \propto \frac{L}{r^2}$. Wire $A$ has $L_A = 2 L_B$ and $r_A = 0.5 r_B$. Thus $\frac{\Delta L_A}{\Delta L_B} = \left(\frac{L_A}{L_B}\right) \left(\frac{r_B}{r_A}\right)^2 = (2)(2)^2 = 8$.
6. What is the torsional rigidity of a steel wire of length $1 \text{ m}$ and radius $1 \text{ mm}$ if its rigidity modulus is $8 \times 10^{10} \text{ N/m}^2$?
Attempt these questions under a ten-minute limit using a separate timer.
1. Which of the following relations between elastic constants is valid for an isotropic material?
Explanation: The standard interrelation linking Young's modulus, bulk modulus, and Poisson's ratio is $Y = 3K(1 - 2\sigma)$.
2. If Poisson's ratio of a material is $0.5$, then the material is:
Explanation: When $\sigma = 0.5$, $Y = 3K(1 - 2(0.5)) = 0$, meaning bulk modulus $K$ becomes infinite, implying the material is perfectly incompressible.
3. A uniform metallic wire is stretched by $1\%$ of its length. What is the approximate percentage increase in its potential energy per unit volume?
Explanation: Energy density $u = \frac{1}{2} Y \times (\text{strain})^2$. Since strain is $0.01$, $u \propto Y \times (0.01)^2$, which depends on the material's Young's modulus ($Y$). Wait, if comparing energy density change, let's check: $u = \frac{1}{2} Y \epsilon^2$. Without knowing $Y$ or initial strain, absolute value cannot be found, but wait—is it a trick question? Let's verify standard problems: if a wire is stretched by $1\%$, energy density increases by factor of squared strain, but numerical value depends on $Y$.
4. The bulk modulus of water is $2.1 \times 10^9 \text{ N/m}^2$. The compressibility of water is:
5. What happens to the temperature of a metal wire when it is rapidly stretched?
Explanation: Rapid (adiabatic) stretching of a metal wire does mechanical work on it, leading to an increase in internal energy and temperature.
6. Shearing strain is defined as:
Explanation: Shearing strain is measured by the angular displacement $\theta$ (in radians) between lines that were originally perpendicular.
7. If all dimensions of a rod are doubled while keeping the material same, its elongation under a given tensile force will:
Explanation: Elongation $\Delta L = \frac{FL}{A Y}$. If length $L$ is doubled ($2L$) and cross-sectional area $A$ is quadrupled ($4A$), then $\Delta L$ becomes $\frac{F(2L)}{(4A)Y} = \frac{1}{2} \left(\frac{FL}{AY}\right)$, so it halves.
8. Which of the four elastic constants ($Y, K, \eta, \sigma$) have dimensions of pressure ($\text{N/m}^2$)?
Explanation: Young's modulus ($Y$), bulk modulus ($K$), and rigidity modulus ($\eta$) are moduli of elasticity having dimensions of pressure, whereas Poisson's ratio ($\sigma$) is dimensionless.
9. The work done in stretching a wire of length $L$ and cross-sectional area $A$ by a length $\Delta L$ is given by:
Explanation: Work done $W = \frac{1}{2} \times \text{Force} \times \text{Extension} = \frac{1}{2} \left(\frac{Y A \Delta L}{L}\right) \Delta L = \frac{1}{2} \frac{Y A (\Delta L)^2}{L}$.
10. Why does a spring have a low effective stiffness when made of a material with low rigidity modulus?
Explanation: Spring action involves coiling and twisting of wire, which is governed by the rigidity modulus ($\eta$). Lower $\eta$ results in lower torsional rigidity and softer spring stiffness.