Define Hooke’s law.

Short Answer:

Hooke’s Law states that, within the elastic limit, the stress developed in a material is directly proportional to the strain produced in it. In simple terms, if the applied load is doubled, the deformation (strain) also doubles, provided the material remains within its elastic limit.

Mathematically, it is expressed as:

where  = stress,  = strain, and  = modulus of elasticity. This law is fundamental for understanding material behavior under load.

Detailed Explanation:

Hooke’s Law

In the study of mechanics of materials, Hooke’s Law is one of the most basic and important principles that describe the relationship between stress and strain in an elastic material. The law was proposed by the English scientist Robert Hooke in 1676. It forms the foundation of the concept of elasticity, which is the property of a material to regain its original shape and size after the removal of the external load.

Hooke’s Law states that, within the elastic limit of a material, the strain produced is directly proportional to the stress applied. That means the deformation in a material increases proportionally with the applied force, as long as the material has not reached its yield point (the point beyond which permanent deformation occurs).

Mathematically,

where,

  •  = stress (N/m² or Pa)
  •  = strain (dimensionless)
  •  = Young’s modulus or modulus of elasticity (N/m² or Pa)

This relationship is linear and forms the basis for the stress-strain diagram in materials engineering.

Explanation of Hooke’s Law

When an external force is applied to a body, it tends to deform. If the force is small, the deformation is also small and temporary — that is, the body returns to its original shape after the removal of the force. This is called the elastic behavior of the material.

In this elastic range, the deformation (strain) is directly proportional to the applied stress. But when the applied force exceeds the elastic limit, the material no longer follows Hooke’s Law, and permanent deformation occurs.

The proportional relationship between stress and strain can be represented by the equation:

where  is a constant of proportionality known as Young’s Modulus or Modulus of Elasticity.

The higher the value of , the stiffer the material is — meaning it undergoes smaller deformation for the same amount of stress.

Graphical Representation of Hooke’s Law

If we plot a graph between stress (σ) on the Y-axis and strain (ε) on the X-axis, we get a straight line in the elastic region of the material.

  • The linear portion of the curve shows that stress is directly proportional to strain.
  • The slope of this straight line gives the modulus of elasticity (E).
  • The end of the straight portion marks the limit of proportionality or elastic limit. Beyond this point, the material no longer obeys Hooke’s Law.

Thus, Hooke’s Law is valid only up to this elastic limit.

Mathematical Form of Hooke’s Law

  1. Proportional Relationship:

or

  1. Where:
    •  = Stress in N/m²
    •  = Strain (no unit)
    •  = Young’s modulus in N/m²
  2. Rearranging the Equation:

This shows that Young’s modulus is the ratio of stress to strain within the elastic limit.

Importance of Hooke’s Law

  1. Foundation of Elasticity:
    Hooke’s Law forms the basis for understanding elastic behavior in materials such as steel, aluminum, copper, and rubber within their elastic range.
  2. Design of Structures and Machines:
    Engineers use Hooke’s Law to calculate the stresses and deformations in beams, columns, shafts, and springs.
  3. Determination of Material Properties:
    Using this law, the modulus of elasticity (E) can be experimentally determined for various materials.
  4. Prediction of Deformation:
    It helps predict how much a material will stretch or compress under a given load.
  5. Stress-Strain Relationship:
    The linear relationship between stress and strain helps in constructing stress-strain diagrams, which are used in material testing and design.

Limitations of Hooke’s Law

While Hooke’s Law is very useful, it has certain limitations:

  1. Valid Only Within Elastic Limit:
    The law is applicable only up to the elastic limit of the material. Beyond this limit, the material enters the plastic region, where deformation becomes permanent.
  2. Not Suitable for Nonlinear Materials:
    Materials like rubber, plastics, and some composites do not follow a perfectly linear stress-strain relationship.
  3. Temperature Effect:
    The relationship between stress and strain can change with temperature. At higher temperatures, materials may not obey Hooke’s Law.
  4. Assumes Homogeneous Material:
    The law assumes that the material is uniform and isotropic, which may not always be true in real-world materials.

Despite these limitations, Hooke’s Law remains an essential tool in mechanical and civil engineering because it accurately describes the behavior of most metallic materials under normal working conditions.

Practical Example

Consider a steel wire of 2 m length and 2 mm diameter subjected to a tensile load of 100 N.

  • Cross-sectional area .
  • Stress .

If the Young’s Modulus of steel ,
then,

This means the strain produced in the wire is 1.592 × 10⁻⁴, showing a very small deformation — which is typical within the elastic range.

Conclusion

Hooke’s Law states that within the elastic limit, stress is directly proportional to strain. It helps in determining the elastic properties of materials and predicting their behavior under small deformations. The law provides a linear relationship between stress and strain and forms the foundation of elasticity and material mechanics. Although it is limited to the elastic range, Hooke’s Law is widely used in engineering applications for designing safe and efficient structures, machines, and components.