Q.What is Poisson's ratio? Why is this a unitless quantity?

Lateral strain is the fractional change in the dimension of a body measured perpendicular (transverse) to the applied (longitudinal) deforming force.

When a longitudinal force is applied to a rod or wire (stretching it lengthwise), the rod's length increases — this fractional change, ΔL/L, is the longitudinal strain. At the same time, the rod's diameter (or width) usually DEcreases slightly, because the material tends to become thinner as it's stretched (this is the basis of Poisson's ratio). The fractional change in this perpendicular dimension,

Lateral strain = ΔD/D (or Δb/b, Δr/r etc., depending on the transverse dimension considered) …

The strain that appears in the direction perpendicular to an applied longitudinal force is called lateral strain — e.g. a wire stretched lengthwise also becomes very slightly thinner.

When a longitudinal force is applied to a rod or wire (say along its length), two kinds of strain result:

1. Longitudinal strain — the fractional change in length, Δl/l, along the direction of the applied force.
2. Lateral strain — the fractional change in the perpendicular (transverse) dimension, e.g. Δd/d for the diameter, in the direction perpendicular to the applied force. …

Poisson's ratio is the (negative) ratio of lateral strain to longitudinal strain within the elastic limit.

When a wire or rod of length L and diameter D is stretched by an applied longitudinal (tensile) force, it not only elongates along its length but also becomes slightly thinner across its diameter. This gives rise to two kinds of strain:

• Longitudinal strain = (change in length) / (original length) = dL/L
• Lateral strain = (change in diameter) / (original diameter) = dD/D

Poisson's ratio (usually denoted sigma or nu) is defined as the ratio of lateral strain to longitudinal strain, within the elastic limit of the material:

sigma = -(lateral strain) / (longitudinal strain) = -(dD/D) / (dL/L)

…

Poisson's ratio is the ratio of lateral strain to longitudinal strain when a wire/rod is stretched; it has no units and typically lies between 0.2 and 0.4.

When a rod or wire of length L and diameter D is stretched along its length by a tensile force, it elongates (longitudinal strain = deltaL/L) but simultaneously becomes slightly thinner (lateral strain = deltaD/D, taken as negative since D decreases while L increases).

Poisson's ratio is defined as sigma = -(lateral strain)/(longitudinal strain) = -(deltaD/D)/(deltaL/L).

Since it is a ratio of two strains (both dimensionless), sigma itself is a dimensionless, unit-less pure number.

…