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How does Poisson’s ratio describe the sideways narrowing of a stretched metal bar?

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mechanics of solids Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).

In the mechanics of solids, Poisson’s ratio ( $ν$ ) quantifies the relationship between the axial stretching of a material and its corresponding transverse contraction. When a metal bar is subjected to axial tension, it elongates in the direction of the applied force while simultaneously narrowing in the perpendicular (transverse) direction.1

Poisson’s ratio is defined as the negative ratio of the transverse strain ( $ϵ_{t r a n s}$ ) to the axial strain ( $ϵ_{a x i a l}$ ):

$ν = - \frac{ϵ_{t r a n s}}{ϵ_{a x i a l}}$

The negative sign is included in the formula to ensure that the resulting ratio is a positive value for most common materials, as the transverse strain is typically opposite in sign to the axial strain. For metals like aluminum or stainless steel, this ratio is typically between 0.2 and 0.35, indicating a predictable degree of sideways narrowing as the bar is stretched. This property is a fundamental aspect of how materials respond to stress and is essential for understanding the deformation of solids.12