Poisson's ratio is defined as the ratio of lateral strain to longitudinal strain in a stretched or compressed elastic wire or rod. It is a dimensionless and unitless quantity that characterizes the transverse deformation of an isotropic elastic material.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of lateral and longitudinal strain, formula for Poisson's ratio, limiting values of Poisson's ratio for isotropic elastic materials (-1 to 0.5), and basic numerical problems.
PGT extension
Interrelation between elastic constants (Young's modulus Y, Bulk modulus K, Rigidity modulus eta, and Poisson's ratio sigma), volume conservation under elastic deformation (sigma = 0.5), and theoretical derivations of strain components.
Prerequisites
Hooke's law and elastic moduli
Longitudinal and lateral strain concepts
Stress-strain relationships in three dimensions
Definition and Physical Origin
When a wire is subjected to a longitudinal tensile stress, it elongates in the direction of the applied force (longitudinal strain) and contracts in the transverse or lateral directions (lateral strain).
Poisson's ratio (σ) is mathematically defined as the negative ratio of lateral strain to longitudinal strain.
The negative sign is conventionally introduced to make the numerical value of Poisson's ratio positive for most ordinary materials, since a positive longitudinal strain (extension) produces a negative lateral strain (contraction).
Limiting Values of Poisson's Ratio
For thermodynamically stable isotropic elastic materials, the theoretical lower limit of Poisson's ratio is -1 and the upper limit is 0.5.
A material with $\sigma = 0.5 undergoes no change in volume during elastic deformation (incompressible materials like rubber)$.
Most common metals have Poisson's ratio values ranging between 0.25 and 0.35 (e.g., steel is approximately 0.28 to 0.30, and copper is about 0.33).
Relation Among Elastic Constants
The elastic constants of an isotropic solid are interrelated, meaning any two independent constants completely define the elastic behavior of the material.
The fundamental relations connecting Young's modulus (Y), Bulk modulus (K), Rigidity modulus (η), and Poisson's ratio (σ) are $Y = 3K(1 - 2\sigma)$ and $Y = 2\eta(1 + \sigma)$.
Eliminating Y from these relations yields the compatibility equation connecting K, η, and σ.
Volume Deformation and Lateral Contraction
Consider a rod of length L and radius r. Under a tensile force, the length becomes L + ΔL and the radius becomes r - Δr.
Longitudinal strain is given by ε_{long} = ΔL / L, and lateral strain is given by ε_{lat} = Δr / r.
The fractional change in volume can be expressed in terms of longitudinal strain and Poisson's ratio as ΔV / $V = \epsilon_{long}(1 - 2\sigma)$.
Derivations
Relation between Y, η, and σ
Conditions: Isotropic elastic medium under pure shear stress, resolved into principal tensile and compressive strains at 45 degrees.
Consider a cube subjected to shear stress resulting in principal tensile strain $\epsilon = \tau / \eta along one diagonal$ and compressive strain along the orthogonal diagonal.
Alternatively, express the strain of a diagonal of length L under tensile stress σ_0 as $\epsilon = \sigma_0 / Y$.
Relate the strains along the orthogonal axes using Poisson's ratio: lateral strains contribute via -σε.
Equating the shear strain components to the directional Young's modulus and Poisson's ratio formulations yields the standard elasticity relation.
Result: $Y = 2\eta(1 + \sigma)$
Relation between Y, K, and σ
Conditions: Isotropic material subjected to uniform hydrostatic pressure leading to isotropic volumetric strain.
Consider a cube of side length a subjected to equal tensile stresses P in all three orthogonal directions x, y, and z.
The total strain along any one axis, say x, due to stress in all three directions is given by ε_$x = (P / Y) - (\sigmaP / Y) - (\sigmaP / Y) = (P / Y)(1 - 2\sigma)$.
The volumetric strain is the sum of linear strains along the three axes: ΔV / $V = 3\epsilon_x = (3P / Y)(1 - 2\sigma)$.
Using the definition of Bulk modulus $K = P / (\DeltaV / V)$, substitute the volumetric strain expression to obtain Y in terms of K and σ.
Result: $Y = 3K(1 - 2\sigma)$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Poisson's Ratio Definition
$$\sigma = - \frac{\Delta r / r}{\Delta L / L}$$
\sigma = \text{Poisson's ratio}, \Delta r / r = \text{lateral strain}, \Delta L / L = \text{longitudinal strain} Dimensionless
Draw: A straight line plot of lateral strain (Δr / r) on the y-axis against longitudinal strain (ΔL / L) on the x-axis passing through the origin with a negative slope.
Interpret: The negative slope of the straight line directly gives the value of Poisson's ratio (σ). Linearity indicates obedience to Hooke's law in the elastic limit.
Exam use: Tested in conceptual multiple-choice questions asking to identify the slope or interpret the sign of transverse deformation.
Searle's Apparatus / Optical Lever Method (Contextual)
Draw: Schematic setup showing two wires (experimental and reference) suspended from a rigid support, equipped with a spirit level, micrometer screw gauge, and scale.
Interpret: Used for precise measurement of Young's modulus, which indirectly feeds into calculations involving Poisson's ratio when combined with lateral extensometers.
Exam use: Direct factual questions regarding least count and experimental measurement techniques for elastic constants.
Volume Change vs Poisson's Ratio Indicator
Draw: A bar chart or theoretical curve showing fractional volume change (ΔV/V) as a function of σ from -1 to 0.5 for a fixed longitudinal strain.
Interpret: Clearly demonstrates that at $\sigma = 0.5$, volume change is zero (incompressible); for σ < 0.5, volume increases upon stretching (auxetic behavior for negative values).
Exam use: Conceptual PGT questions on incompressible fluids and anomalous elastic behavior.
Solved numericals
TGT core
A metal wire of length 2 m and radius 1 mm is stretched by a force. If the longitudinal strain produced is 0.001 and Poisson's ratio for the wire is 0.3, calculate the lateral strain and the change in radius.
Calculate change in radius Δ$r = - 0.0003 * 10^-3 m = - 3 * 10^-7 m (- 3.0 * 10^-4 mm)$.
Answer: Lateral strai$n = -0.0003; Change in radius = -3 * 10^-7 m (contraction)$
PGT extension
For a given isotropic material, the Young's modulus is 1.2 * 10^11 N/m² and the Rigidity modulus is 4.5 * 10^10 N/m². Find the Poisson's ratio and the Bulk modulus of the material.
Use the relation connecting Y, η, and σ: $Y = 2\eta(1 + \sigma)$.
Substitute the given values: 1.2 * 10^11 = 2 * (4.5 * 10^10) * (1 + \sigma)$.
The material is assumed to be perfectly homogeneous and isotropic, meaning its elastic properties are identical in all directions.
The deformations considered are strictly within the elastic limit where Hooke's law is valid and strains are very small (ε << 1).
Common misconceptions
Common error: Poisson's ratio can have any positive value greater than zero without restriction.
Correct: Poisson's ratio for stable isotropic solids is theoretically restricted between -1 and 0.5.
Exceeding 0.5 would imply a negative bulk modulus, leading to thermodynamic instability where compression increases volume.
Common error: Poisson's ratio has units of strain (or meters).
Correct: Poisson's ratio is strictly dimensionless.
It is defined as the ratio of two similar quantities (lateral strain to longitudinal strain), both of which are dimensionless ratios.
Common error: A negative Poisson's ratio is physically impossible.
Correct: Materials with negative Poisson's ratio (auxetic materials) expand laterally when stretched.
Thermodynamic limits permit values down to -1, and specific microstructures exhibit auxetic behavior.
Quick revision
Poisson's ratio σ is the negative ratio of lateral strain to longitudinal strain, lying strictly between -1 and 0.5 for isotropic elastic solids.
Poisson's Ratio Definition: $\sigma = - \frac{\Delta r / r}{\Delta L / L}$
Relation 1 (Y, η, σ): $Y = 2\eta(1 + \sigma)$
Must remember
Poisson's ratio ($\sigma$) is defined as the negative ratio of lateral strain to longitudinal strain: $\sigma = - \frac{\Delta r / r}{\Delta L / L}$.
The theoretical limits of Poisson's ratio for thermodynamically stable isotropic elastic materials are $-1 \le \sigma \le 0.5$.
For incompressible materials like rubber, the Poisson's ratio is $0.5$, meaning there is no change in volume during elastic deformation.
The relations connecting the elastic constants are $Y = 3K(1 - 2\sigma)$ and $Y = 2\eta(1 + \sigma)$.
Poisson's ratio is dimensionless as it is the ratio of two strains.
Glossary
Poisson's Ratio ($sigma$)The negative ratio of transverse (lateral) strain to longitudinal strain in a stretched or compressed rod.
Longitudinal StrainThe fractional change in length ($Delta L / L$) along the direction of the applied tensile or compressive force.
Lateral StrainThe fractional change in transverse dimensions such as radius or diameter ($Delta r / r$) perpendicular to the applied force.
Young's Modulus ($Y$)The ratio of longitudinal stress to longitudinal strain within the elastic limit.
Bulk Modulus ($K$)The ratio of hydrostatic stress to volumetric strain.
Rigidity Modulus ($eta$)The ratio of shearing stress to shearing strain.
Auxetic MaterialsMaterials that exhibit a negative Poisson's ratio, expanding laterally when stretched.
Comparison of Elastic Constants and Material Behavior
Property
Ordinary Metals (e.g., Steel, Copper)
Incompressible Materials (e.g., Rubber)
Auxetic Materials
Poisson's Ratio ($\sigma$)
0.25 to 0.35
0.5
Negative (-1 to 0)
Volume Change ($\delta V / V$)
Contracts slightly or expands depending on tension
Zero change ($\delta V = 0$)
Expands laterally upon longitudinal extension
Relation Term $(1 - 2sigma)$
Positive
Zero
Greater than 1
Practice Quiz
Choose an option to reveal the explanation.
1. What is the physical significance of the negative sign in the definition of Poisson's ratio $\sigma = - \frac{\Delta r / r}{\Delta L / L}$?
Explanation: A positive longitudinal strain (extension) results in a negative lateral strain (contraction). The negative sign is introduced so that the numerical value of $\sigma$ remains positive for ordinary materials.
2. What are the theoretical lower and upper limits of Poisson's ratio for thermodynamically stable isotropic elastic materials?
Explanation: Based on thermodynamic stability requirements (ensuring positive bulk and rigidity moduli), the limits for isotropic elastic materials are $-1 \le \sigma \le 0.5$.
3. What is the Poisson's ratio of an ideally incompressible material?
Explanation: For an incompressible material, the fractional volume change $\frac{\Delta V}{V} = \frac{\Delta L}{L}(1 - 2\sigma) = 0$, which gives $1 - 2\sigma = 0$, leading to $\sigma = 0.5$.
4. What are the dimensions of Poisson's ratio?
Explanation: Poisson's ratio is the ratio of lateral strain to longitudinal strain. Since both strains are dimensionless ratios of lengths, Poisson's ratio is strictly dimensionless.
5. Which of the following relations correctly connects Young's modulus ($Y$), Bulk modulus ($K$), and Poisson's ratio ($\sigma$)?
Explanation: The standard elasticity relation connecting Young's modulus, Bulk modulus, and Poisson's ratio is $Y = 3K(1 - 2\sigma)$.
6. Which of the following relations correctly connects Young's modulus ($Y$), Rigidity modulus ($\eta$), and Poisson's ratio ($\sigma$)?
Explanation: The standard elasticity relation connecting Young's modulus, Rigidity modulus, and Poisson's ratio is $Y = 2\eta(1 + \sigma)$.
7. If the fractional change in length of a wire under tension is $\frac{\Delta L}{L}$ and its Poisson's ratio is $\sigma$, what is the fractional change in its radius $\frac{\Delta r}{r}$?
Explanation: From the definition $\sigma = -\frac{\Delta r / r}{\Delta L / L}$, rearranging yields $\frac{\Delta r}{r} = -\sigma \frac{\Delta L}{L}$.
8. What is the typical range of Poisson's ratio for most common engineering metals like steel and copper?
Explanation: Most common metals have Poisson's ratio values ranging between 0.25 and 0.35 (e.g., steel is around 0.28 to 0.30, and copper is about 0.33).
9. What does a negative Poisson's ratio indicate about a material?
Explanation: Materials with a negative Poisson's ratio (auxetic materials) expand laterally when subjected to longitudinal tension, rather than contracting.
10. What is the fractional change in volume $\frac{\Delta V}{V}$ of a cylindrical wire of length $L$ and radius $r$ subjected to a longitudinal strain $\epsilon_{long} = \frac{\Delta L}{L}$ in terms of Poisson's ratio $\sigma$?
13. If a material has a Poisson's ratio of $0.5$, what happens to its volume when it is stretched?
Explanation: When $\sigma = 0.5$, the fractional volume change term $(1 - 2\sigma)$ becomes zero, indicating that the volume remains unchanged during elastic deformation.
14. Which of the following represents the correct expression for Young's modulus ($Y$) obtained by eliminating $\sigma$ from the standard relations involving $K$ and $\eta$?
Explanation: Eliminating $\sigma$ between $Y = 3K(1 - 2\sigma)$ and $Y = 2eta(1 + \sigma)$ gives the well-known formula $Y = \frac{9K\eta}{3K + \eta}$.
15. Why is Poisson's ratio restricted from exceeding 0.5 in isotropic stable materials?
Explanation: If $\sigma > 0.5$, the bulk modulus $K = \frac{Y}{3(1 - 2\sigma)}$ would become negative, implying that compressing the material increases its volume, which violates thermodynamic stability.
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 a material is $0.5$, then the ratio of Young's modulus ($Y$) to Bulk modulus ($K$) i.e., $Y/K$ is:
2. For a given material, the Rigidity modulus is $\eta$ and the Bulk modulus is $K$. Young's modulus $Y$ in terms of $K$ and $\eta$ is given by:
Explanation: From the elimination of $\sigma$ between $Y = 3K(1 - 2\sigma)$ and $Y = 2eta(1 + \sigma)$, we get $Y = \frac{9K\eta}{3K + \eta}$.
3. A uniform metal rod is stretched such that its length increases by $1%$. If its Poisson's ratio is $0.3$, the approximate percentage decrease in its radius is:
Explanation: Using $\frac{\Delta r}{r} = -\sigma \frac{\Delta L}{L} = -0.3 \times 1\% = -0.3\%$. Thus, the radius decreases by $0.3%$.
4. What is the value of Poisson's ratio for a material whose Young's modulus is equal to its Bulk modulus?
5. Which of the following relations between elastic constants $Y$, $K$, and $\eta$ is dimensionally and physically correct?
Explanation: Both $Y = \frac{9K\eta}{3K + \eta}$ and its rearrangement $K = \frac{Y\eta}{9eta - 3Y}$ are correct algebraic forms derived from the standard elastic constant relations.
6. If Poisson's ratio of a wire is $0.25$, the ratio of longitudinal strain to lateral strain is:
Explanation: Poisson's ratio $\sigma = -\frac{\text{lateral strain}}{\text{longitudinal strain}} = 0.25$. Therefore, the ratio of longitudinal strain to lateral strain is $\frac{1}{0.25} = 4$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If Poisson's ratio for a material is $0.5$, then the Rigidity modulus ($\eta$) and Young's modulus ($Y$) are related as:
7. In a cylindrical rod subjected to uniaxial tensile testing, if the longitudinal strain is $0.002$ and Poisson's ratio is $0.3$, what is the lateral strain?
Explanation: Auxetic materials possess a negative Poisson's ratio, meaning they expand laterally when stretched longitudinally.
9. What is the relation between Young's modulus $Y$, Bulk modulus $K$, and Rigidity modulus $\eta$ when Poisson's ratio is eliminated?
Explanation: The standard derived relation is $Y = \frac{9K\eta}{3K + \eta}$.
10. If a material is compressible and undergoes a positive volume change ($\delta V > 0$) under tension, what condition must its Poisson's ratio satisfy?
Explanation: The fractional volume change is given by $\frac{\Delta V}{V} = \frac{\Delta L}{L}(1 - 2\sigma)$. For volume to increase ($\Delta V > 0$) under tension ($\Delta L > 0$), we must have $1 - 2\sigma > 0$, which implies $\sigma < 0.5$.