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Mechanics

Relation Between Stress and Strain

Stress is defined as the internal restoring force per unit area set up inside a deformed body, while strain is the fractional change in dimension resulting from that stress. Hooke's law establishes a linear proportionality between stress and strain within the elastic limit of a material.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of stress and strain, types of stress (longitudinal, shearing, volumetric), Hooke's law, elastic moduli (Young's modulus, bulk modulus, modulus of rigidity), and basic numerical problems.

PGT extension

Inter-relations between elastic constants (Y, K, η, σ), Poisson's ratio limits (-1 to 0.5), stored elastic potential energy per unit volume, strain energy in twisted cylinders, and thermal stresses.

Prerequisites

Dimensions and dimensional formulas of mechanical quantities
Forces in equilibrium and Newton's laws of motion
Basic vector and tensor concepts for advanced PGT stress analysis

Concept of Stress

Stress is a tensor quantity measured as force per unit cross-sectional area, F/A.
Normal stress acts perpendicular to the surface (causing elongation or compression), whereas tangential or shearing stress acts parallel to the surface.
SI unit of stress is N/m² or Pascal (Pa) with dimensional formula [ML^-1T^-2].

Concept of Strain

Strain is a dimensionless and unitless quantity representing the relative deformation of a body.
Longitudinal strain is ΔL/L, volumetric strain is ΔV/V, and shearing strain is θ or tan θ ≈ Δx/l.
It expresses how much a given body is deformed relative to its original configuration.

Hooke's Law and Elastic Moduli

Within the elastic limit, stress is directly proportional to strain, meaning Stres$s = E \times Strain$, where E is the modulus of elasticity.
Young's modulus (Y) governs longitudinal deformation, Bulk modulus (K) governs volumetric compression, and Rigidity modulus (η) governs shape change under shear.
The reciprocal of bulk modulus is known as compressibility.

Poisson's Ratio and Elastic Constants Interrelation

Poisson's ratio (σ) is the ratio of lateral strain to longitudinal strain under uniaxial stress.
Theoretical isotropic limits for Poisson's ratio are -1 to 0.5, while practical materials exhibit values between 0 and 0.5.
Inter-relations include $Y = 3K(1 - 2\sigma)$ and $Y = 2\eta(1 + \sigma)$.

Derivations

Derivation of Elastic Potential Energy Density

Conditions: Isotropic, linearly elastic material stressed within the proportional limit

Consider a wire of original length L and cross-sectional area A subjected to a variable stretching force F.
The incremental work done in stretching the wire by an infinitesimal length dl is dW = F dl.
Substitute $F = (YA/L)l into the work integral from l = 0$ to $l = \DeltaL$.
Integrate to get $W = (1/2) \times (YA/L) (\DeltaL)^{2}$.
Divide by volume (V = AL) to obtain energy density $U = (1/2) \times Stress \times Strain$.

Result: $U = \frac{1}{2} \times \text{Stress} \times \text{Strain} = \frac{1}{2} Y \epsilon^2$

Relation between Young's Modulus, Bulk Modulus, and Modulus of Rigidity

Conditions: Homogeneous, isotropic elastic solid

Write the independent constitutive relations linking Young's modulus Y, Bulk modulus K, and Rigidity modulus η.
Use Poisson's ratio σ to eliminate lateral strain components from volumetric and shear strain equations.
Apply the standard identities: $\sigma = \frac{3K - 2\eta}{6K + 2\eta}$.
Substitute σ back into either $Y = 3K(1 - 2\sigma) or Y = 2\eta(1 + \sigma)$.
Obtain the master relation connecting all three moduli: \frac{9}{Y} = \frac{1}{K} + \frac{3}{\eta}.

Result: \frac{9}{Y} = \frac{1}{K} + \frac{3}{\eta}

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Hooke's Law	$$\sigma = E \epsilon$$	$\sigma = Stress$, E = Elastic Modulus, $\epsilon = Strain$ Pa or N/m²	Within elastic limit
Young's Modulus	$$Y = \frac{F/A}{\Delta L / L} = \frac{FL}{A \Delta L}$$	F = Force, A = Area, L = Length, ΔL = Elongation N/m²	Uniform wire under tensile/compressive load
Bulk Modulus	$$K = -V \frac{\Delta P}{\Delta V}$$	V = Initial Volume, ΔP = Pressure change, ΔV = Volume change N/m²	Uniform hydrostatic compression
Modulus of Rigidity	$$\eta = \frac{F/A}{\theta}$$	F/A = Shearing stress, $\theta = Shearing angle in radians$ N/m²	Tangential loading causing shape change
Poisson's Ratio	$$\sigma = -\frac{\Delta D / D}{\Delta L / L}$$	D = Diameter, L = Length, negative sign indicates opposite lateral/longitudinal change Dimensionless	Uniaxial tensile stress
Elastic Potential Energy Density	$$u = \frac{1}{2} \text{Stress} \times \text{Strain} = \frac{\sigma^2}{2Y}$$	u = Energy per unit volume, $\sigma = Stress$, Y = Young's modulus J/m³	Linear elastic deformation
Inter-relation 1 (Y, K, σ)	$$Y = 3K(1 - 2\sigma)$$	Y = Young's modulus, K = Bulk modulus, $\sigma = Poisson's ratio$ N/m²	Isotropic material
Inter-relation 2 (Y, η, σ)	$$Y = 2\eta(1 + \sigma)$$	Y = Young's modulus, $\eta = Rigidity modulus$, $\sigma = Poisson's ratio$ N/m²	Isotropic material

Graphs and experiments

Stress-Strain Curve

Draw: A plot of Stress (y-axis) versus Strain (x-axis) showing proportional limit, elastic limit, yield point, ultimate tensile strength, and fracture point.

Interpret: The linear region represents Hooke's law validity. Beyond the yield point, permanent plastic deformation occurs leading up to necking and fracture.

Exam use: Direct identification questions on ductile vs. brittle materials and the exact definitions of yield and breaking stress.

Searle's Apparatus Experiment

Draw: Experimental setup featuring two long parallel wires (experimental and reference), a spirit level, a micrometer screw gauge, and added dead weights.

Interpret: Used to determine Young's modulus of a metallic wire accurately by measuring small elongations for known loads.

Exam use: Questions on least count, error propagation, and formula application for Y.

Compressibility Measurement via Bulk Modulus

Draw: Graph of fractional volume change (ΔV/V) versus applied uniform pressure (ΔP).

Interpret: The slope of the linear graph yields the inverse of the bulk modulus, which is the compressibility factor.

Exam use: Conceptual questions on fluids and solids under high hydrostatic pressure.

Solved numericals

TGT core

A steel wire of length 2 m and cross-sectional area 1 mm² is pulled by a force of 200 N. If Young's modulus of steel is 2 × 10¹¹ N/m², calculate the elongation produced in the wire.

Identify given values: L = 2 m, $A = 1 mm^{2} = 1 \times 10^{-6} m^{2}$, F = 200 N, $Y = 2 \times 10^{11} N/m^{2}$.
Rearrange Young's modulus formula for elongation: ΔL = \frac{FL}{YA}.
Substitute values: Δ$L = \frac{200 \times 2}{2 \times 10^{11} \times 1 \times 10^{-6}}$.
Simplify the arithmetic: Δ$L = \frac{400}{2 \times 10^5} = 2 \times 10^{-3} m$.

Answer: 2 mm (or 2 × 10&sup-;³ m)

PGT extension

For a given isotropic material, the Young's modulus is equal to its bulk modulus. Find the Poisson's ratio of the material.

State the relation connecting Young's modulus, bulk modulus, and Poisson's ratio: $Y = 3K(1 - 2\sigma)$.
Substitute the given condition $Y = K into the equation: K = 3K(1 - 2\sigma)$.
Divide both sides by K (since K ≠ 0): $1 = 3(1 - 2\sigma)$.
Solve for σ: 1/$3 = 1 - 2\sigma => 2\sigma = 1 - 1/3 = 2/3 => \sigma = 1/3$.

Answer: 1/3 (or 0.33)

PGT extension

Calculate the elastic potential energy stored per unit volume in a copper wire subject to a longitudinal stress of 4 × 10&sup7; N/m². Take Young's modulus of copper as 1.2 × 10¹¹ N/m².

Identify given values: Stress $\sigma = 4 \times 10^{7} N/m^{2}$, $Y = 1.2 \times 10^{11} N/m^{2}$.
Use the energy density formula in terms of stress and Young's modulus: $u = \frac{\sigma^2}{2Y}$.
Substitute the numerical values: $u = \frac{(4 \times 10^7)^2}{2 \times 1.2 \times 10^{11}}$.
Calculate numerator: 16 × 10¹&sup4;.
Calculate denominator: 2.4 × 10¹¹.
Perform division: $u = \frac{16 \times 10^{14}}{2.4 \times 10^{11}} = 6.67 \times 10^{2} J/m^{3}$.

Answer: 6.67 × 10² J/m³

Assumptions and limitations

Hooke's law is strictly valid only within the proportional limit of the stress-strain curve.
Materials are assumed to be perfectly homogeneous and isotropic for standard elastic constant inter-relations.
Deformations are assumed to be small so that higher-order nonlinear strain terms can be neglected.

Common misconceptions

Common error: Strain has SI units of percentage or ratio dimensions like meters.

Correct: Strain is strictly a dimensionless quantity.

It is the ratio of two like quantities (change in dimension to original dimension).

Common error: Poisson's ratio can take any positive value up to infinity.

Correct: Poisson's ratio for stable isotropic materials lies strictly between -1 and 0.5.

Thermodynamic stability requirements dictate that bulk and shear moduli must remain positive.

Common error: A stiffer material has a lower Young's modulus.

Correct: A higher Young's modulus indicates greater stiffness.

Modulus is the slope of the stress-strain curve; higher slope means less deformation for the same applied stress.

Quick revision

Stress is restoring force per unit area (N/m²), while strain is fractional deformation (dimensionless).
Hooke's law states Stres$s = E \times Strain within the elastic limit$.
Young's modulus (Y), Bulk modulus (K), and Rigidity modulus (η) characterize tensile, volumetric, and shear elasticity respectively.
The inter-relation master formula is 9/$Y = 1/K + 3/\eta$.
Poisson's ratio σ relates lateral strain to longitudinal strain with theoretical bounds of -1 to 0.5.

Must remember

Stress is a tensor quantity measured as force per unit cross-sectional area ($F/A$), with SI unit $N/m^2$ or Pascal (Pa) and dimensional formula $[ML^{-1}T^{-2}]$.
Strain is a dimensionless and unitless quantity representing relative deformation: longitudinal ($\Delta L/L$), volumetric ($\Delta V/V$), and shearing ($\theta$).
Hooke's law states that within the elastic limit, stress is directly proportional to strain: $\sigma = E \epsilon$.
Young's modulus ($Y$), Bulk modulus ($K$), and Rigidity modulus ($\eta$) govern longitudinal, volumetric, and shape changes respectively.
The reciprocal of bulk modulus is known as compressibility.
Poisson's ratio ($\sigma$) is the ratio of lateral strain to longitudinal strain, with theoretical isotropic limits of $-1$ to $0.5$.
Inter-relations between elastic constants are given by $Y = 3K(1 - 2\sigma)$ and $Y = 2\eta(1 + \sigma)$.
Elastic potential energy density is expressed as $u = \frac{1}{2} \text{Stress} \times \text{Strain} = \frac{\sigma^2}{2Y}$.

Glossary

StressRestoring internal force per unit area, measured in $N/m^2$ or Pa.

StrainDimensionless ratio of change in dimension to the original dimension.

Young's Modulus ($Y$)Ratio of longitudinal stress to longitudinal strain.

Bulk Modulus ($K$)Ratio of normal stress to volumetric strain under hydrostatic pressure.

Modulus of Rigidity ($\eta$)Ratio of shearing stress to shearing strain.

Poisson's Ratio ($\sigma$)Ratio of lateral strain to longitudinal strain.

CompressibilityReciprocal of the bulk modulus ($1/K$).

Comparison of Elastic Moduli

Modulus	Governing Deformation	Associated Stress & Strain
Young's Modulus ($Y$)	Longitudinal elongation/compression	Normal stress & Longitudinal strain
Bulk Modulus ($K$)	Volumetric compression/expansion	Hydrostatic pressure & Volumetric strain
Rigidity Modulus ($\eta$)	Shape change / Shear	Tangential stress & Shearing strain

Practice Quiz

Choose an option to reveal the explanation.

1. What is the dimensional formula of stress?

Explanation: Stress is force per unit area: $[MLT^{-2}] / [L^2] = [ML^{-1}T^{-2}]$.

2. Which of the following quantities is dimensionless and unitless?

Explanation: Strain is the ratio of two like quantities (change in dimension to original dimension), making it dimensionless and unitless. (Note: Poisson's ratio is also dimensionless, but strain is the baseline dimensionless deformation quantity).

3. Hooke's law is valid strictly:

Explanation: Hooke's law states that stress is proportional to strain only within the elastic limit.

4. What is the reciprocal of the bulk modulus called?

Explanation: Compressibility is defined as the inverse of the bulk modulus ($1/K$).

5. The theoretical isotropic limits for Poisson's ratio are:

Explanation: Thermodynamic stability requires isotropic materials to have Poisson's ratio between $-1$ and $0.5$.

6. Which modulus of elasticity is involved in changing the volume of a body without changing its shape?

Explanation: Bulk modulus ($K$) governs volumetric compression or expansion under uniform hydrostatic pressure.

7. What is the SI unit of stress?

Explanation: Stress is force per unit area, measured in $N/m^2$ or Pascals (Pa).

8. If a wire is stretched to double its length, the longitudinal strain is:

Explanation: Longitudinal strain is $\Delta L / L$. If final length is $2L$, $\Delta L = L$, so strain = $L / L = 1.0$.

9. The elastic potential energy density stored in a strained wire is given by:

Explanation: Energy density $u = \frac{1}{2} \times \text{Stress} \times \text{Strain}$.

10. Which inter-relation correctly links Young's modulus ($Y$), Bulk modulus ($K$), and Poisson's ratio ($\sigma$)?

Explanation: The standard inter-relation is $Y = 3K(1 - 2\sigma)$.

11. Tangential or shearing stress causes:

Explanation: Shearing stress acts parallel to the surface, causing angular deformation or change in shape.

12. A higher Young's modulus for a material indicates:

Explanation: A higher Young's modulus means less deformation for a given stress, denoting greater stiffness.

13. What is the relation between Young's modulus ($Y$), Rigidity modulus ($\eta$), and Poisson's ratio ($\sigma$)?

Explanation: The correct inter-relation is $Y = 2\eta(1 + \sigma)$.

14. What is the nature of stress when it acts perpendicular to the surface, causing compression?

Explanation: Normal stress acts perpendicular to the surface; when it causes compression, it is normal compressive stress.

15. Why can Poisson's ratio for practical stable materials not exceed $0.5$?

Explanation: If $\sigma = 0.5$, the bulk modulus becomes infinite and volume change becomes zero ($\Delta V/V = 0$). Exceeding $0.5$ would imply a physically impossible negative volume change under tension.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. If the length of a wire is doubled and radius is also doubled, what happens to its Young's modulus?

Explanation: Young's modulus is a characteristic material property independent of the dimensions of the wire.

2. For a given isotropic material, if the Poisson's ratio is $0.25$, the ratio of Young's modulus to rigidity modulus ($Y/\eta$) is:

Explanation: Using $Y = 2\eta(1 + \sigma)$, we get $Y/\eta = 2(1 + 0.25) = 2(1.25) = 2.5$.

3. A wire of length $L$ and cross-sectional area $A$ is stretched by force $F$. The work done per unit volume is:

Explanation: Stored elastic potential energy density (work done per unit volume) is $\frac{1}{2} \text{Stress} \times \text{Strain}$.

4. The bulk modulus of an incompressible liquid is:

Explanation: For an incompressible liquid, volume change $\Delta V = 0$ for any pressure change, making $K = -V(\Delta P / \Delta V) = \infty$.

5. If Poisson's ratio of a material is $0.5$, then the material is:

Explanation: When $\sigma = 0.5$, $Y = 3K(0) = 0$ or $K$ is infinite, indicating that volumetric strain is zero (incompressible).

6. A force $F$ is applied to a wire of radius $r$ and length $L$, producing an elongation $\Delta L$. If the same force is applied to a wire of the same material with radius $2r$ and length $2L$, the elongation will be:

Explanation: Elongation $\Delta L = \frac{FL}{A Y} = \frac{FL}{\pi r^2 Y}$. For new wire, $\Delta L' = \frac{F(2L)}{\pi (2r)^2 Y} = \frac{2FL}{4\pi r^2 Y} = \frac{\Delta L}{2}$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which of the following is the correct dimensional formula for Young's modulus?

Explanation: Young's modulus has the same dimensions as stress (force per unit area), which is $[ML^{-1}T^{-2}]$..

2. What is the value of Poisson's ratio if $Y = 2\eta$?

Explanation: From $Y = 2\eta(1 + \sigma)$, if $Y = 2\eta$, then $2\eta = 2\eta(1 + \sigma) \implies 1 + \sigma = 1 \implies \sigma = 0$.

3. A cube of side $a$ is subjected to a uniform hydrostatic pressure $P$. If the bulk modulus is $K$, the fractional change in volume ($\Delta V / V$) is:

Explanation: Bulk modulus $K = -V(\Delta P / \Delta V)$. Thus, fractional change $\Delta V / V = -\Delta P / K = -P / K$.

4. The potential energy per unit volume of a stretched wire is proportional to:

Explanation: Energy density $u = \frac{\sigma^2}{2Y}$, which is proportional to the square of the stress.

5. When a solid cylindrical rod is twisted, the deformation involved is governed by:

Explanation: Twisting a cylinder produces shear strain and shape change, which is governed by the modulus of rigidity ($\eta$).

6. If the ratio of radii of two wires of the same material is $2:1$ and lengths are equal, the ratio of their Young's moduli is:

Explanation: Young's modulus is an intrinsic property of the material and does not depend on dimensions; hence the ratio is $1:1$.

7. What does the slope of the linear region in a stress-strain curve represent?

Explanation: Within the elastic limit, stress = $E \times$ strain, so the slope of the stress-strain curve is the elastic modulus $E$.

8. For an isotropic elastic material, the relationship between $Y$, $K$, and $\eta$ is:

Explanation: Eliminating $\sigma$ from $Y = 3K(1 - 2\sigma)$ and $Y = 2\eta(1 + \sigma)$ yields $Y = \frac{9K\eta}{3K + \eta}$.

9. Longitudinal strain is defined as:

Explanation: Longitudinal strain is $\Delta L / L$, representing the relative change in length.

10. Which of the following materials generally exhibits the highest Young's modulus?

Explanation: Steel is much stiffer than rubber, wood, or glass, resulting in a significantly higher Young's modulus.