Hooke's Law - Stress And Strain

When force is applied to a material, we know that it either stretches or compresses in response to the applied force. In mechanics, the force applied per unit area is known as stress and is denoted by the symbol σ. The extent to which the material compresses or stretches is known as strain. Different materials respond differently to applied stress.  This information is necessary for engineers while selecting materials for their structures.

In 19th-century, while studying springs and elasticity, English scientist Robert Hooke noticed that many materials exhibited a similar property when the stress-strain relationship was studied. There was a linear region where the force required to stretch the material was proportional to the extension of the material. This is known as Hooke’s Law. In this article, let us discuss Hooke’s law in detail.

What is Hooke’s Law?

Hooke’s law states that

the strain of the material is proportional to the applied stress within the elastic limit of that material.

When the elastic materials are stretched, the atoms and molecules deform until stress is been applied and when the stress is removed they return to their initial state.

Mathematically, Hooke’s law is commonly expressed as:

F = –k.x


  • F is the force
  • x is the extension length
  • k is the constant of proportionality known as spring constant in N/m

Interested to learn more about spring? Below are the links:

Hooke’s Law Experiment

Consider a spring with load application as shown in the figure.

Hooke’s Law

The figure shows the stable condition of the spring when no load is applied, the condition of the spring when elongated to an amount x under the load of 1 N, the condition of the spring elongated to 2x under the influence of load 2 N.

Different springs, depending on the material, will have different spring constants. This can be calculated. The constant calculated empirically can be used further. The figure shows us three instances, the stable condition of the spring, the spring elongated to an amount x under a load of 1 N, and the spring elongated to 2x under a load of 2 N. If we plug in these values into the equation above we get the spring constant for the material in consideration.

σ = Eε


  • σ is the stress
  • E is the modulus of elasticity also known as Young’s modulus
  • ε is the strain

When the stress is removed from the material, there are two types of deformation that can take place: plastic deformation and elastic deformation

Hooke’s Law Graph

The figure below shows the stress-strain curve for low carbon steel.

Hooke’s Law

The material exhibhits elastic behaviour up to the yield strength point, after which the material loses elasticity and exhibits plasticity.

From the origin till the proportional limit nearing yield strength, the straight line implies that the material follows Hooke’s law. Beyond the elastic limit between proportional limit and yield strength, the material loses its elastic nature and starts exhibiting plasticity. The area under the curve from origin to the proportional limit falls under the elastic range. The area under the curve from a proportional limit to the rupture/fracture point falls under the plastic range.

The ultimate strength of a material is defined based on the maximum ordinate value given by the stress-strain curve (from origin to rupture). The rupture strength is given by the value at a point of rupture.

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