Bulk Modulus and Poisson's Ratio

In Section (STRESS AND STRAIN), we have seen that when a body is submerged in a fluid, it undergoes a hydraulic stress (equal in magnitude to the hydraulic pressure). This leads to the decrease in the volume of the body thus producing a strain called volume strain [Eq. (Volume strain =$\frac{\Delta V}{V}$)]. The ratio of hydraulic stress to the corresponding hydraulic strain is called bulk modulus. It is denoted by symbol B.

$B=-\frac{p}{\Delta V/V}$

The negative sign indicates the fact that with an increase in pressure, a decrease in volume occurs. That is, if p is positive, ∆V is negative. Thus for a system in equilibrium, the value of bulk modulus B is always positive. SI unit of bulk modulus is the same as that of pressure i.e., N m$^{–2}$ or Pa. The bulk moduli of a few common materials are given in Table 1.

The reciprocal of the bulk modulus is called compressibility and is denoted by k. It is defined as the fractional change in volume per unit increase in pressure.

$k=\frac{1}{B}=-\frac{1}{\Delta p} \times \frac{\Delta V}{V}$

It can be seen from the data given in Table 1 that the bulk moduli for solids are much larger than for liquids, which are again much larger than the bulk modulus for gases (air).

Table 1: Bulk moduli (B) of some common Materials

Thus, solids are the least compressible, whereas, gases are the most compressible. Gases are about a million times more compressible than solids!

Gases have large compressibilities, which vary with pressure and temperature. The incompressibility of the solids is primarily due to the tight coupling between the neighbouring atoms. The molecules in liquids are also bound with their neighbours but not as strong as in solids. Molecules in gases are very poorly coupled to their neighbours. 

Table 2 shows the various types of stress, strain, elastic moduli, and the applicable state of matter at a glance.

Table 2 Stress, strain and various elastic moduli

Example 1 

The average depth of Indian Ocean is about 3000 m. Calculate the fractional compression, ∆V/V, of water at the bottom of the ocean, given that the bulk modulus of water is 2.2 × 10$^9$  N m$^{–2}$. (Take g = 10 m.s$^{–2}$)

Answer 

The pressure exerted by a 3000 m column of water on the bottom layer 

p = hρg 

= 3000 m × 1000 kg.m$^{–3}$ × 10 m.s$^{–2}$ 

= 3 × 10$^7$  kg m$^{–1}$ s$^{–2}$ 

= 3 × 10$^7$  N.m$^{–2}$ 

Fractional compression ∆V/V, is 

∆V/V = stress/B = (3 × 10$^7$  N.m$^{–2}$)/(2.2 × 10$^9$  N.m$^{–2}$) 

= 1.36 × 10$^{–2}$ or 1.36 % 

POISSON’S RATIO 

Careful observations with the Young’s modulus experiment (explained in section 9.6.2), show that there is also a slight reduction in the cross-section (or in the diameter) of the wire. The strain perpendicular to the applied force is called lateral strain. Simon Poisson pointed out that within the elastic limit, lateral strain is directly proportional to the longitudinal strain. 

The ratio of the lateral strain to the longitudinal strain in a stretched wire is called Poisson’s ratio. If the original diameter of the wire is d and the contraction of the diameter under stress is ∆d, the lateral strain is ∆d/d. If the original length of the wire is L and the elongation under stress is ∆L, the longitudinal strain is ∆L/L. 

Poisson’s ratio is then (∆d/d)/(∆L/L) or (∆d/∆L) × (L/d). Poisson’s ratio is a ratio of two strains; it is a pure number and has no dimensions or units. Its value depends only on the nature of material. For steels the value is between 0.28 and 0.30, and for aluminium alloys it is about 0.33.

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