Mechanical Properties of Solids
How solids deform under stress — elasticity, stress-strain curves, and engineering applications
Table of Contents
8.1 Introduction
A rigid body generally means a hard solid object having a definite shape and size. But in reality, bodies can be stretched, compressed and bent. Even the appreciably rigid steel bar can be deformed when a sufficiently large external force is applied on it.
The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed, is known as elasticity. The deformation caused is known as elastic deformation. Substances that get permanently deformed are called plastic and this property is called plasticity.
8.2 Stress and Strain
When a body is subjected to a deforming force, a restoring force is developed in the body. The restoring force per unit area is known as stress.
Types of Stress and Strain
Longitudinal (Tensile/Compressive)
Force perpendicular to cross-section. Changes length. ε = ΔL/L
Shearing
Force parallel to cross-section. Changes shape. ε = tan θ ≈ θ
Hydraulic (Bulk)
Force from all directions. Changes volume. ε = ΔV/V
Strain
Dimensionless quantity — ratio of deformation to original dimension
8.3 Hooke's Law
Robert Hooke discovered that for small deformations, stress is directly proportional to strain:
Within the elastic limit, the ratio of stress to strain is a constant for a given material. This constant is called the modulus of elasticity. Hooke's law is the foundation of the theory of elasticity and is valid only for small deformations.
8.4 Stress-Strain Curve
The stress-strain curve for a material provides information about its mechanical properties, including its elastic behavior, strength, and ductility.
Key Points on the Curve
- Proportional Limit (A): Up to this point, stress ∝ strain (Hooke's law valid). The graph is a straight line.
- Elastic Limit (B): Beyond this point, permanent (plastic) deformation begins. Material won't return to original shape.
- Yield Point (C): Material begins to deform significantly without increase in stress. Important for engineering design.
- Ultimate Stress (D): Maximum stress the material can withstand before fracture.
- Fracture Point (E): Material breaks.
In the elastic region (up to point B), the material returns to its original shape when the load is removed. In the plastic region (B to E), permanent deformation occurs. Engineering design typically keeps stress well below the elastic limit.
8.5 Elastic Moduli
The modulus of elasticity measures a material's resistance to deformation. There are three types corresponding to three types of stress:
Young's Modulus (Y)
Shear Modulus (G)
Bulk Modulus (B)
Compressibility
Steel: Y ≈ 2 × 10¹¹ Pa, G ≈ 8 × 10¹⁰ Pa, B ≈ 1.6 × 10¹¹ Pa. Rubber: Y ≈ 10⁶ Pa (much more elastic than steel!). Diamond: B ≈ 4.4 × 10¹¹ Pa (most incompressible material known).
8.6 Applications of Elastic Behaviour
The elastic properties of materials play an important role in engineering design. Understanding stress, strain, and elastic moduli helps engineers design safe and efficient structures.
Bridge Design
Bridges must support their own weight plus traffic. Engineers use materials with high Young's modulus (like steel) and design I-shaped beams to maximize strength while minimizing weight.
Building Construction
Columns and beams must not deform beyond elastic limits under load. Knowledge of elastic moduli ensures structural safety during earthquakes and high winds.
Aircraft Design
Materials must be lightweight yet strong. Aluminum alloys and composites are chosen for their high strength-to-weight ratio.
Artificial Limbs
Modern prosthetics use materials that are both light and strong, balancing durability with comfort for the user.
Railway tracks have an I-shaped cross-section because this shape provides maximum resistance to bending (maximum second moment of area) for a given amount of material. This makes the track both strong and economical.