Hooke’s Law and Poisson’s Ratio

Hooke’s Law

Robert Hooke, an English mathematician conducted several experiments and concluded that stress is proportional to strain up to elastic limit. This is called Hooke’s law. Thus Hooke’s law states that ‘stress is proportional to strain up to elastic limit.’

σ ∝ ϵ

where ′σ′ is stress and ′ϵ′ is strain

Hence,

σ = E ϵ

Where ‘E’ is the constant of proportionality of the material, known as Modulus of Elasticity or Young’s modulus, named after the English scientist Thomas Young (1773–1829). 

The present day sophisticated experiments have shown that for mild steel the Hooke’s law holds good up to the proportionality limit which is very close to the elastic limit. For other materials, Hooke’s law does not hold good. However, in the range of working stresses, assuming Hooke’s law to hold good, the relationship does not deviate considerably from actual behaviour. Accepting Hooke’s law to hold good, simplifies the analysis and design procedure considerably. Hence Hooke’s law is widely accepted. The analysis procedure accepting Hooke’s law is known as Linear Analysis and the design procedure is known as the working stress method.

Poisson’s Ratio (μ)

When a material undergoes changes in length, it undergoes changes of opposite nature in lateral directions. For example, if a rectangular bar is subjected to direct tension in its axial direction it elongates and at the same time its sides contract as shown in Fig.1.

Fig. 1 Changes in Axial and Lateral Directions due to Tensile Force

If we define the ratio of change in axial direction to original length as linear strain and change in lateral direction to the original lateral dimension as lateral strain, it is found that within elastic limit there is a constant ratio between lateral strain and linear strain. This constant ratio is called Poisson’s ratio. 

Thus,

It is denoted by 1/m or μ.

For most of metals its value is between 0.25 to 0.33. Its value for steel is 0.3 and for concrete 0.15.