Poissons Ratio
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Describe and define Poisson’s ratio.
Consider a rod under an axial tensile load P such that the material is within the elastic limit. The normal stress on x plane is \(\sigma_{xx} = \dfrac{P}{A}\) and the associated longitudinal strain in the x direction can be found out from \(\epsilon_x = \dfrac{\sigma_{xx}}{E}\). As the material elongates in the x direction due to the load P, it also contracts in the other two mutually perpendicular directions, i.e., y and z directions. Hence, despite the absence of normal stresses in y and z directions, strains do exist in those directions and they are called lateral strains.
The ratio between the lateral strain and the axial/longitudinal strain for a given material is always a constant within the elastic limit and this constant is referred to as Poisson’s ratio.
It is denoted by \(nu\). Since the axial and lateral strains are opposite in sign, a negative sign is introduced in the definition to make \(\nu\) positive. \[\nu = - \dfrac{\mathrm{lateral\ strain}}{\mathrm{longitudinal\ strain}}\]
Poisson’s ratio can be as low as 0.1 for concrete and as high as 0.5 for rubber. In general, it varies from 0.25 to 0.35 and for steel it is about 0.3.
Young’s modulus = E = \(\dfrac{\mathrm{tensile\ stress}}{\mathrm{tensile\ strain}}\)
Young’s modulus = E = \(\dfrac{\mathrm{compressive\ stress}}{\mathrm{compressive\ strain}}\)
Shear modulus = G = \(\dfrac{\mathrm{shear\ stress}}{\mathrm{shear\ strain}}\)
Bulk modulus = K = \(\dfrac{\mathrm{Volumetric\ stress}}{\mathrm{volumetric\ strain}}\)
K= \(\dfrac{\mathrm{pressure}}{\mathrm{volumetric\ strain}}\) = \(\dfrac{p}{(\delta V/ V)}\)