Short Answer:

Hooke’s Law states that within the elastic limit of a material, the stress developed in it is directly proportional to the strain produced. This means that if you double the stress, the strain also doubles, as long as the material does not go beyond its elastic limit.

Mathematically, it is written as:

where is the modulus of elasticity of the material, which shows how stiff or flexible a material is.

Detailed Explanation :

Hooke’s Law

Hooke’s Law is one of the most important principles in the study of elasticity and material mechanics. It was proposed by the British scientist Robert Hooke in 1660. The law explains how materials deform (change in shape or size) when forces are applied to them, and how they return to their original shape when the forces are removed.

According to Hooke’s Law, when a material is loaded within its elastic limit, the stress induced in the material is directly proportional to the strain produced. The elastic limit is the maximum stress that a material can bear without undergoing permanent deformation. Once the stress exceeds this limit, the material will not return to its original shape after the removal of the force.

The relationship is expressed as:

where,

Stress = Force per unit area (N/m² or Pascal),
Strain = Change in length / Original length (dimensionless),
E = Young’s Modulus of Elasticity (N/m²).

This formula indicates that the ratio of stress to strain remains constant for a given material within its elastic limit. This constant is known as the Modulus of Elasticity or Elastic Constant.

Explanation of the Law

When an external force is applied to a material, it causes deformation. For example, if you stretch a wire, it elongates slightly. The elongation (strain) is proportional to the applied force (stress) as long as the force is not too large. If you remove the force, the wire returns to its original length—this is the elastic behavior described by Hooke’s Law.

However, if you continue increasing the load beyond a certain limit, the wire will not return to its original length after removing the force. At this stage, it enters the plastic region, and Hooke’s Law no longer applies.

In the initial stage of loading, the stress-strain curve is a straight line, showing a direct proportional relationship. The slope of this straight line represents Young’s Modulus (E), which measures the stiffness of the material. A higher modulus means the material is stiffer and less deformable, like steel; a lower modulus means the material is more flexible, like rubber.

Mathematical Expression

Let the applied force be , the original length of the material be , the cross-sectional area be , and the extension produced be .
Then,

and

According to Hooke’s Law,

This equation is used to calculate the deformation or extension in materials under different loads.

Graphical Representation

When stress is plotted against strain, a straight line is obtained in the initial part of the curve, known as the linear elastic region. The slope of this line represents the modulus of elasticity (E). The point beyond which the line deviates from straightness is called the elastic limit. Beyond this point, permanent deformation occurs.

The linear region obeys Hooke’s Law, while the non-linear region does not.

Applications of Hooke’s Law

Hooke’s Law is widely used in mechanical engineering, structural design, and material testing. Some key applications include:

Design of Springs: Springs operate based on Hooke’s Law, where force is proportional to extension or compression.
Stress and Strain Measurement: It is used to determine the modulus of elasticity of materials by performing tensile tests.
Building and Bridge Design: Engineers use the law to ensure structures can bear loads safely without permanent deformation.
Machine Components: Shafts, beams, and other parts are designed using Hooke’s Law to predict elastic deformation under working loads.
Elastic Material Behavior: Helps in studying the behavior of metals, alloys, and other elastic materials under various forces.

Limitations of Hooke’s Law

It is valid only within the elastic limit of a material.
It cannot be applied to materials that do not have a clear elastic region, such as plastics and rubber.
It does not consider time-dependent effects such as creep or fatigue.
It applies only for small deformations, not for very large strains.

Conclusion:

Hooke’s Law explains the linear relationship between stress and strain within the elastic limit of a material. It forms the foundation of elasticity and material mechanics, allowing engineers to predict how materials will behave under different loads. This principle is essential in designing structures, machines, and mechanical components that can safely withstand applied forces without permanent deformation.