Bulk Modulus of Elasticity

A rigid body is usually defined as a hard solid object with a definite shape and size. However, in reality, bodies can be compressed, stretched, and bent. Also, the appreciably rigid steel bar can be deformed with the application of a sufficiently large external force on it. It implies that solid bodies are not perfectly rigid. A solid body has a definite shape and size. For changing or deforming the shape or size of a body, an external force is needed. By stretching a helical spring or gently pulling its ends, the length increases slightly. Furthermore, when you leave the ends of the spring, it comes back to its original shape and size. The property of a body by which it tends to regain its original size and shape when the applied force is removed is what we refer to as elasticity, and the deformation caused is called elastic deformation. However, if we apply an external force to a lump of putty or mud, they won't regain their previous shape as they do not tend to do so, and get permanently deformed. Such kinds of substances are called plastic, and the property is known as plasticity.

The elastic behaviour of materials or the property of elasticity plays a significant role in engineering design. For instance, while designing a building, having some knowledge of the elastic properties of materials like concrete, steel, etc is essential. The same holds for the design of bridges, automobiles, rope ways, etc.


Stress-strain Curve

The relationship between stress and strain for a given material under tensile stress can be proved with the help of an experiment. In a standard test of tensile properties, a test cylinder or wire is stretched by an external force. The fractional change in length or strain and the applied external force needed to cause the strain are recorded. The applied external force is gradually increased in steps, and the change in length is also recorded. A graph is plotted between the stress (which is equal in magnitude to the applied external force per unit surface area) and the strain produced. A typical graph for metal is shown below.

The stress-strain curves vary from one material to the other. The stress-strain curves help us in understanding how a given material deforms with an increase in the load. From the graph, we can see and find that in the region between points O and A, the curve is linear. In this region, Hooke's law is followed and obeyed. The body regains its original dimensions when the applied external force is removed. In this region, the solid body behaves as an elastic body. Moving ahead, in the region between points A and B, stress and strain are not directly proportional. Nevertheless, the body still tends to return to its original dimension when the load is removed. The point B in the curve is called yield point (also referred to as the elastic limit), and the corresponding stress is called yield strength (σy) of the material. If the load increases further, the stress shall also exceed the yield strength, and the strain increases rapidly, even for very little change in the stress. The portion of the curve between points B and D shows the same.

When the load is removed, say at some point C, between B and D, the body does not regain its original shape and size. In such a case, even when the stress is 0, the strain is not 0. The material is said to then have a permanent set. The deformation is known as plastic deformation. The point D on the graph refers to the ultimate tensile strength of the material. Beyond point D, additional strain is produced even by a reduced applied external force, and fracture occurs at point E. If the ultimate strength and fracture between points D and E are close, the material is referred to as brittle. If they are quite far apart, the material is referred to as ductile. The stress-strain behaviour varies from one material to another. For instance, rubber can be pulled several times, and it shall still return to its original shape. Although the elastic region is large enough, the material does not follow Hooke’s law in most of the regions. Secondly, there is no well-defined plastic region. The substances like rubber, which can be stretched to cause large strains, are known as elastomers.


Elastic Moduli 

The proportional region, within the elastic limit of the stress-strain curve (region OA in the above figure), is of utmost importance for both structural and manufacturing engineering designs. The ratio of stress and strain, known as modulus of elasticity, is found to be a significant characteristic or property of the material.


Bulk Modulus 

We already know and have seen as well that when a body is submerged in a fluid, it undergoes or experiences hydraulic stress, which is equal in magnitude to the hydraulic pressure. The same leads to a decrease in the volume of the body and produces a strain known as volume strain.

Bulk Modulus is defined as the ratio of hydraulic stress to the corresponding hydraulic strain. It is denoted by symbol B, and can be expressed as: 

B = \[\frac{-p}{(\frac{∆V}{V})}\]

The negative sign in the formula indicates that as the pressure increases, the volume decreases. To be specific, if p or pressure is positive, then ∆V or the change in volume is negative. Hence, for a system in equilibrium, the value of bulk modulus or B is always positive. The SI unit of the bulk modulus is the same as that of pressure that is N m2 or Pa. The bulk moduli of a few common materials are specified in the table below:


State 

Material

B(GPa)


Solids 

Aluminium

72

Brass

61

Copper

140

Glass

37

Iron

100

Nickel

260

Steel

160




Liquids 

Water

2.2

Ethanol

0.9

Carbon Disulphide

1.56

Glycerin

4.76

Mercury

25

Gases 

Air( at STP )

1.0 x 10⁻⁴


The reciprocal of the bulk modulus or 'B' is known as compressibility and is denoted by the symbol k. It is defined or expressed as the fractional change in volume per unit increase in pressure. 

k = (\[\frac{1}{B}\]) = - (\[\frac{1}{∆p}\]) × (\[\frac{∆V}{V}\])

It can be seen from the data given in the table that the bulk moduli for solid bodies are much larger than it is for liquids, which are again much larger than the bulk modulus for gases like air.

Therefore, the solid bodies are the least compressible, whereas gases are the most compressible. Also, gases are somewhere around a million times more compressible than solids. 

Gases have large compressibility, which vary with both 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 the case of solids. Molecules in gases are very poorly coupled to their neighbouring atoms.