Elastic Behaviour of Solids

Stress and its Types

Stress is defined as the restoring force acting per unit area of a deformed body. Mathematically, stress = restoring force / area = F / A. The SI unit of stress is newton per square meter (N m⁻²) and its dimensional formula is [ML⁻¹T⁻²]. Stress is a tensor quantity and normal stress has three types:

• Longitudinal stress: Stress normal to the surface causing change in length. It can be tensile (increase in length) or compressive (decrease in length).
• Volumetric stress: Stress causing change in volume under normal force acting on all surfaces.
• Tangential stress (Shearing stress): Stress due to force applied tangentially, causing change in shape without volume change.

Strain and its Types

Strain is the ratio of change in configuration of a body due to deforming force to its original configuration. It is dimensionless and has no units.

• Longitudinal strain: Change in length / Original length = Δl / l
• Volumetric strain: Change in volume / Original volume = ΔV / V
• Shearing strain: When force is applied parallel to surface, causing shape change without volume change. Defined as the angle θ through which a vertical line rotates under tangential force, θ = ΔL / L, where ΔL is displacement and L is perpendicular distance from fixed surface.

Hooke's Law

Within the elastic limit, stress is directly proportional to strain, i.e., Stress ∝ Strain.

Key Terms

• Deforming force: Force that changes the configuration of a body.
• Elasticity: Property of a body to regain original shape after removal of deforming force.
• Perfectly elastic body: Regains original form completely, e.g., quartz.
• Plastic body: Does not regain original form after deforming force removal, e.g., putty.
• Elastic limit: Maximum deforming force up to which body regains original form completely.

Key Formulae

• Normal stress, S = F / A
• Breaking force = Breaking stress × Area
• Longitudinal strain = Δl / l
• Volumetric strain = ΔV / V
• Shearing strain, θ = ΔL / L

Solved Examples

Example 1: A wire of length 2 m and cross-sectional area 1 mm² is stretched by a force of 10 N. Calculate the longitudinal stress and strain if the extension produced is 0.5 mm.

Solution:

Given: Length, l = 2 m = 2000 mm; Area, A = 1 mm²; Force, F = 10 N; Extension, Δl = 0.5 mm

Longitudinal stress, S = F / A = 10 N / 1 mm² = 10 N/mm²

Longitudinal strain = Δl / l = 0.5 mm / 2000 mm = 0.00025

Thus, stress = 10 N/mm², strain = 0.00025 (dimensionless).

Practice Set

• Level 1: Define stress and strain. What are their SI units?
• Level 2: Explain the difference between longitudinal stress and tangential stress.
• Level 3: A wire of length 1.5 m and cross-sectional area 2 mm² is stretched by a force of 20 N. If the extension produced is 0.3 mm, calculate the longitudinal stress and strain.

Answer Key

• Level 1: Stress is force per unit area (N/m²), strain is change in length/original length (dimensionless).
• Level 2: Longitudinal stress acts normal to surface causing length change; tangential stress acts parallel causing shape change without volume change.
• Level 3: Stress = 20 N / 2 mm² = 10 N/mm²; Strain = 0.3 mm / 1500 mm = 0.0002.

Modulus of Elasticity

Definition and Types

Modulus of elasticity (E) is the ratio of stress to corresponding strain within the elastic limit, E = Stress / Strain.

Types of modulus of elasticity:

• Young's modulus (Y): Ratio of longitudinal stress to longitudinal strain, Y = (F / A) / (Δl / l) = (F / A) × (l / Δl). It measures stiffness of solid materials.
• Bulk modulus (K): Ratio of volume stress to volumetric strain, K = − (P V) / ΔV. It applies to solids, liquids, and gases.
• Modulus of rigidity (η): Ratio of tangential stress to shearing strain, η = (F / A) / θ = F / (Aθ). It applies to solids only.

Poisson's Ratio

Poisson's ratio (σ) is the ratio of lateral strain to longitudinal strain, σ = (ΔD / D) / (Δl / l) = − (ΔD · l) / (D · Δl). Its practical value lies between 0 and 0.5.

Key Terms

• Compressibility: Reciprocal of bulk modulus, c = 1 / K = − ΔV / (P V).
• Elastic fatigue: Loss of strength due to repeated alternating strains.
• Yield strength: Maximum stress a material can sustain without permanent deformation.

Key Formulae

• Young’s modulus, Y = Fℓ / (AΔℓ)
• Bulk modulus, K = − (F / A) / (ΔV / V) = − P V / ΔV
• Modulus of rigidity, η = Fℓ / (AΔℓ)
• Poisson’s ratio, σ = − (ΔD / D) / (Δℓ / ℓ)
• Relations among elastic constants:
  Y = 3K(1 − 2σ)
  Y = 2η(1 + σ)
  σ = (3K − 2η) / (2η + 6K)
  9 / Y = 1 / K + 3 / η
• Elastic potential energy, U = ½ × Stress × Strain × Volume
• Elastic potential energy per unit volume, u = ½ × Stress × Strain = ½ Y × (Strain)²
• Work done in stretching wire, W = ½ × Load × Extension

Solved Examples

Example 1: A wire of length 2 m and cross-sectional area 1 mm² is stretched by a force of 10 N producing an extension of 0.5 mm. Calculate Young's modulus.

Solution:

Given: l = 2 m = 2000 mm, A = 1 mm², F = 10 N, Δl = 0.5 mm

Young's modulus, Y = (F / A) × (l / Δl) = (10 / 1) × (2000 / 0.5) = 10 × 4000 = 40000 N/mm² = 4 × 10¹⁰ N/m²

Thus, Young's modulus is 4 × 10¹⁰ pascal.

Practice Set

• Level 1: Define Young's modulus and bulk modulus.
• Level 2: Explain Poisson's ratio and its practical range.
• Level 3: A wire of length 1 m and cross-sectional area 0.5 mm² is stretched by 5 N force producing an extension of 0.2 mm. Calculate Young's modulus.

Answer Key

• Level 1: Young's modulus is ratio of longitudinal stress to strain; bulk modulus is ratio of volume stress to volumetric strain.
• Level 2: Poisson's ratio is lateral strain divided by longitudinal strain; practical values lie between 0 and 0.5.
• Level 3: Y = (F / A) × (l / Δl) = (5 / 0.5) × (1000 / 0.2) = 10 × 5000 = 50000 N/mm² = 5 × 10¹⁰ Pa.

Quick Reference Table

• Stress: Force applied per unit area on a body, measured in N/m².
• Strain: Ratio of change in dimension to original dimension, dimensionless.
• Longitudinal Stress and Strain: Related to change in length of a body.
• Volumetric Stress and Strain: Related to change in volume of a body.
• Shearing Stress and Strain: Related to change in shape without volume change.
• Hooke's Law: Stress is proportional to strain within elastic limit.
• Young's Modulus (Y): Ratio of longitudinal stress to longitudinal strain, indicates stiffness.
• Bulk Modulus (K): Ratio of volume stress to volumetric strain, indicates compressibility.
• Modulus of Rigidity (η): Ratio of tangential stress to shearing strain, indicates rigidity.
• Poisson's Ratio (σ): Ratio of lateral strain to longitudinal strain, typically between 0 and 0.5.
• Elastic Limit: Maximum stress up to which a material regains original shape.
• Elastic Potential Energy: Energy stored in a stretched or compressed body.
• Relations among Elastic Constants: Mathematical relations connecting Y, K, η, and σ.

Common Mistakes and Misconceptions

• Confusing stress with force: Stress is force per unit area, not just force.
• Mixing up strain types: Longitudinal strain changes length, shearing strain changes shape.
• Ignoring elastic limit: Assuming materials always return to original shape beyond elastic limit.
• Incorrect units: Stress has units of N/m², strain is dimensionless.
• Misinterpreting Poisson's ratio: It cannot be negative beyond -1 or exceed 0.5.
• Applying Hooke's law beyond elastic limit: Stress and strain are proportional only within elastic limit.
• Forgetting that modulus of rigidity applies only to solids, not liquids or gases.
• Confusing bulk modulus with Young's modulus: Bulk modulus relates to volume change, Young's modulus to length change.

Glossary

• Stress: Force per unit area acting on a material.
• Strain: Measure of deformation representing displacement between particles.
• Elasticity: Ability of a material to regain original shape after deformation.
• Elastic Limit: Maximum stress a material can withstand without permanent deformation.
• Hooke's Law: Principle stating stress is proportional to strain within elastic limit.
• Young's Modulus: Measure of stiffness of a solid material.
• Bulk Modulus: Measure of a material's resistance to uniform compression.
• Modulus of Rigidity: Measure of a material's resistance to shear deformation.
• Poisson's Ratio: Ratio of lateral strain to longitudinal strain.
• Elastic Potential Energy: Energy stored in a deformed elastic body.
• Deforming Force: Force causing change in shape or size of a body.
• Perfectly Elastic Body: Material that completely recovers original shape after deformation.
• Plastic Body: Material that does not recover original shape after deformation.
• Compressibility: Reciprocal of bulk modulus, indicating how compressible a material is.
• Elastic Fatigue: Weakening of material due to repeated stress cycles.
• Yield Strength: Maximum stress before permanent deformation occurs.