Concept of Elastic Body, Stress, Strain, Stress-Strain Diagram, and Hooke’s Law

Definition

An elastic body is a material that undergoes deformation when subjected to an external force but returns to its original shape and size once the force is removed. Stress is the internal resistance offered by a body to an external deforming force (Force per unit Area), while Strain is the ratio of change in dimension to the original dimension of the body.

Main Content

1. Concept of Stress and Strain

Stress ($\sigma$): Mathematically, stress is defined as Force ($F$) divided by the cross-sectional area ($A$). It is measured in Pascals ($N/m^2$).
Strain ($\epsilon$): It is a dimensionless quantity representing the geometric deformation. For example, linear strain is the change in length ($\Delta L$) divided by the original length ($L$).

2. Hooke’s Law

Hooke’s Law states that within the elastic limit, the stress applied to a material is directly proportional to the strain produced.
The constant of proportionality is known as the Modulus of Elasticity ($E$ or Young’s Modulus). The formula is $\sigma = E \times \epsilon$.

3. Stress-Strain Diagram

This diagram represents the behavior of a material under increasing load.
It typically tracks the progression from the Proportional Limit to the Elastic Limit, Yield Point, Ultimate Strength, and finally, the Fracture Point.

Stress (σ)
  |      B (Elastic Limit)
  |     /
  |    /  C (Yield Point)
  |   /  _-- D (Ultimate Strength)
  |  /_-'
  | /  '-- E (Fracture)
  |/
  +------------------------- Strain (ε)
  (Diagram: Typical Stress-Strain curve for Mild Steel)

Working / Process

1. Applying Tensile Force

A specimen is placed in a Universal Testing Machine (UTM).
A gradual axial pull is applied to measure how the material resists deformation.

2. Measuring Deformation

Sensors or extensometers track the minute changes in the length of the specimen.
The load is recorded simultaneously to map the data points for the stress-strain curve.

3. Calculating Elastic Constants

By identifying the slope of the linear portion of the graph (the region where Hooke's law holds), we calculate the Young’s Modulus ($E$).
This value helps engineers predict how a specific material will behave under structural loads.

Advantages / Applications

Structural Safety: Engineers use these concepts to ensure that bridges, buildings, and vehicles do not cross their elastic limit and suffer permanent deformation.
Material Selection: Helps in choosing the right material (e.g., steel vs. rubber) based on how much stress it can withstand before failing.
Manufacturing: Essential in metal forming and machining processes where understanding the yield strength is critical for success.

Summary

The study of stress and strain allows us to understand how materials deform and recover under load. Hooke’s Law provides the foundation for elastic behavior, while the stress-strain diagram maps the journey of a material from its initial state to final rupture. Key terms include Young’s Modulus (measure of stiffness), Elastic Limit (maximum stress before permanent deformation), and Yield Point (start of plastic flow).