Hooke's Law - Stress And Strain

On the application of stress on an elastic body, the strain experienced by the body (amount of deformation) is directly proportional to the stress applied.

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

Hooke’s Law

What is Hooke’s Law?

As per Hooke’s Law, the force required to compress or elongate (cause deformation) the spring by a small distance is proportional to that distance. The relation is as given below

\(\sigma , \alpha , \epsilon\)

\(\sigma = k, \epsilon\)

where,

\(\sigma\) is the applied force

\(\epsilon\) is the strain

k is the constant

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 1N, and the spring elongated to 2x under a load of 2N. If we plug in these values into the equation above we get the spring constant for the material in consideration.

Hooke’s Law and Stress-Strain Curve

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

Hooke’s Law

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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Practise This Question

A metal rod, fixed at both ends, has a length l,area of cross section A and youngs modulus Y and coefficient of linear expansion α is heated through t0 C. The work done in heating the rod is