Similar to Young’s Modulus, Poisson’s Ratio is a material specific and independent constant. It describes how different materials deforms under loading. Before we learn about Poisson’s ratio, let us learn about types of Strains.
TYPES OF STRAINS
We gave an overview of Strain in our previous post. To revise pretty quickly, Strain is a dimensionless quantity which is defined as the ratio of change in dimension to its original dimension. Not to confuse, It is a ratio between quantities of same dimension.
There are FOUR different types of strains. They are:
i. LONGITUDINAL STRAIN
It is the ratio of change in dimension to its original dimension in the direction of force.
ii. LATERAL STRAIN
It is the ratio of change in dimension to its original dimension in the direction perpendicular to force.
iii. VOLUMETRIC STRAIN
It is the ratio of change in Volume to the original Volume. It can be described as the sum of linear strains in all the three mutually perpendicular directions.
iv. SHEAR STRAIN
It is a strain accompanying a shearing action and we shall learn more about it later.
WHAT IS POISSON’S RATIO
Poisson, a French mathematician, determined that when a body is pulled in one direction, it gets compressed in the other perpendicular directions.
Poisson’s ratio is defined as the ratio of the lateral strain to the longitudinal strain. it is generally denoted by the symbol µ. This is more like a mathematical formula. But What exactly it explains?
$$µ = (- \frac{Lateral\;Strain}{Longitudinal\;Strain})$$
It explains the relation between linear strain and lateral strain. It can be illustrated using a rubber band. Whenever a rubber band is pulled/stretched, it can be observed that the length of the rubber band increases and at the same time, it becomes thinner which can be observed in the image below.
Increase of length gives Linear Strain while a decrease in the thickness gives Lateral Strain. Poisson’s ratio can tell us how much the band gets thinner. Therefore, It is the parameter which can tell us how much a body deforms in the lateral direction when the load is applied in the longitudinal direction.
DIFFERENT VALUES OF POISSON’S RATIO
Various materials have different values of Poisson’s ratio. But it is limited to the range of 0.5 to -1. Engineering Materials has a range of 0 to 0.5.
For a material whose µ is in between 0 and 0.5, a pull/tension in the longitudinal axis creates compression in the other two perpendicular directions. Whereas for materials whose µ value falls under -1 and 0, a tension/pull in one axis, creates tension in the other two perpendicular directions, thereby increasing the volume of the body.
µ value for Steel is generally taken as 0.3 while concrete is taken between 0.1 to 0.2.
For a material whose µ value is 0, there is no deformation in the lateral directions when the load is applied in the longitudinal direction. Cork exhibits such behaviour. This is one of the primary reason for the usage of cork in wine bottle as it does not deform laterally when compressed.
GENERALIZED HOOKE’S LAW
One of the major application of Poisson’s ratio is extending the concept of Hooke’s law. Hooke’s law can be applied to determine the strain in the loading direction within the elastic limit.
$$ε_x = \frac{σ_x}{E}\;;\;\; ε_y = \frac{σ_y}{E}\;;\;\; ε_z = \frac{σ_z}{E}$$
The above equations are applicable only for uniaxial loading in their respective directions. But practically, all structures are three dimensional and they experience stresses in all the three directions.
In such triaxial loading cases, the stresses in each of the three directions create strains in the other two lateral directions. This cannot be obtained by Hooke’s law.
Poisson’s effect states that the load develops strain, not only in the loading direction but also in the lateral directions. When a load is applied in only X-direction, by Poisson’s ratio,
$$ ε_y = {-µ}\times{ε_x}\;;\;\;
ε_z = {-µ}\times{ε_x}$$
Therefore,
$$ε_x = \frac{σ_x}{E}\;;\;\;
ε_y = (\frac{-µ\times σ_x}{E})\;;\;\;
ε_z = (\frac{-µ\times σ_x}{E})$$
Now strain in the X-direction is given by,
$$ε_x = \frac{σ_x}{E} – \frac{µ \times σ_y}{E} -\frac{µ \times σ_z}{E}$$
Similarly, strains in different directions in a triaxial loading is given by,
$$ε_y = \frac{σ_y}{E} – \frac{µ \times σ_z}{E} -\frac{µ \times σ_x}{E}$$
$$ε_z = \frac{σ_z}{E} – \frac{µ \times σ_x}{E} -\frac{µ \times σ_y}{E}$$
Sum of all these three strains gives Volumetric Strain, which is given by,
$$ε_v = \frac{(σ_x + σ_y + σ_z)\times(1-2µ)}{E}$$
For a material whose µ value is 0.5, εv value becomes 0, such materials are called Incompressible Materials. Rubber has µ value close to 0.5.
Important to remember that, whatever we have learnt here, is valid only for the ISOTROPIC Materials under Linear Elastic Region. It gets complex for Anisotropic and Orthotropic Materials.
IMPORTANCE OF POISSONS RATIO
1. It explains how a body responds to any applied load.
2. It explains the ability of a material to be strained.
3. It is used in relating different Elastic Constants.
4. It plays a significant part in Generalizing Hooke’s Law.
FOOD FOR THOUGHT
Why is there a limit of 0.5 in Poisson’s ratio value? Why the negative sign is used in the formula to calculate Poisson’s ratio?
Do think and tell us in the comments.