Physics · Ch 6 — Mechanical Properties of Solids

Poisson's ratio

6.5.4

Poisson's ratio

When a wire, fixed at one end, is stretched by a force applied at its free end, two things happen simultaneously: the wire's length increases, and its diameter decreases slightly — the wire becomes both longer and thinner (Fig. 6.6(a)). The reverse happens under compression: if equal and opposite forces are applied inward along a wire's length, its length decreases while its diameter simultaneously increases, so the wire becomes both shorter and thicker (Fig. 6.6(b)).

This pairing of a change along the direction of the applied force with an opposite-sense change perpendicular to it is described using two kinds of strain. The ratio of the change in dimension to the original dimension, measured in the direction of the applied force, is called the linear strain (this is the same longitudinal strain from section 6.5.1). The ratio of the change in dimension to the original dimension, measured perpendicular to the applied force, is called the lateral strain. Within the elastic limit, the ratio of lateral strain to linear strain is called Poisson's ratio, $\sigma$ :

\sigma = \dfrac{\text{Lateral strain}}{\text{Linear strain}}

If $l$ is a wire's original length, $\Delta l$ the increase or decrease in its length, $D$ its original diameter, and $d$ the corresponding change in diameter, then

\sigma = \dfrac{d/D}{\Delta l/l} = \dfrac{d\,L}{D\,\Delta l}

Poisson's ratio, being a ratio of two strains (both already dimensionless), has no unit and no dimensions. …

Figure 6.6(a)Wire under stretching: length increases, diameter decreases

What this figure shows. A cylindrical wire is shown in two states, one above the other or side by side for comparison: the original wire (shorter, thicker, of diameter D) and the same wire after a stretching force is applied at its free end while the other end is fixed (now longer, and visibly thinner, of a slightly smaller diameter). Arrows indicate the applied force pulling the free end outward along the wire's length. The figure demonstrates that stretching a wire simultaneously increases its length (linear strain, in the direction of the force) and decreases its diameter (lateral strain, perpendicular to the force) — the two strains whose ra …

Figure 6.6(b)Wire under compression: length decreases, diameter increases

What this figure shows. A cylindrical wire (or short rod) is shown in two states for comparison: the original wire, and the same wire after equal and opposite forces are applied inward along its length, compressing it. In the compressed state the wire is visibly shorter along its length and visibly thicker (larger diameter) than the original. This is the mirror-image companion to Fig. 6.6(a): where stretching lengthens-and-thins a wire, compressing it shortens-and-thickens it — in both cases the dimension along the applied force and the dimension perpendicular to it change in opposite senses, which is the general …

Table 6.4Poisson ratio σ of some familiar materials, all falling within the typical 0.25-0.35 metal range or close to it

Material | Poisson ratio σ

Glass (crown) | 0.2

Steel | 0.28

Aluminium | 0.36

Brass | 0.37

Copper | 0.37 …

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