Elastic Constants are the constants that express the elastic behaviour of a given material. There are three elastic constants.
1. Young’s Modulus
2. Shear Modulus
3. Bulk Modulus
Sometimes, people add Poisson’s ratio to the list thereby making it Four Elastic Constants.
We have already learnt about Young’s Modulus and its significance in one of our previous post. We shall now learn about the remaining two elastic constants.
Let’s try to understand a few fundamental concepts before we learn about the remaining elastic constants.
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SHEAR FORCE
Consider a cubical object and glue its one side to a rigid surface. When a force is applied on the body in the direction parallel to the rigid surface as shown, then the body tries to move in the same direction.
Since the body is fixed on one surface it cannot move. A reaction force is developed on the fixed surface which is opposite to the direction of applied force. But the other end is free and it starts to move in the direction of the force.
On increasing the magnitude of the force, it slices the body along the direction of applied force. In simple words, the force which tears/cuts a body is called a Shear Force.
The best example of the application of shear force is Scissors. Two opposite forces acting on a paper cut the paper.
In the case of normal stress, the surface perpendicular to the applied load resists the load. But shear force is resisted by the surface parallel to the force.
For easy understanding, consider a pile of books on a table. If you apply a force on one of the books, the book tries to move away. But the friction caused by the adjacent books prevents it from falling. Shear force causes Shear Stress which is the ratio of Shear force to that of shearing Area.
SHEAR STRAIN (γ)
In the case of a Normal Strain, there would be an increase or decrease in the length which depends on the force applied (either Tension or Compression). But in the above-discussed example, as the bottom surface is fixed, there would be a change in the angular deflection.
The angular distortion between two mutually perpendicular planes is known as shear strain. If Φ is the angle between them, then shear strain can be calculated as Tan Φ.
SHEAR MODULUS
Shear modulus is also called as Modulus of Rigidity or Rigidity Modulus. It is denoted by G. It is the ability of a material to withstand the shear force with minimum deformation. The value of ‘G’ denotes the rigidity of the body.
Mathematically, G can be written as
$$Shear\>Modulus \>(G) = \frac{Shear\>stress}{Shear\>strain}$$
The higher value of G implies that the body is more rigid i.e., it is very hard to deform (sideways) and vice versa.
Shear modulus of fluids is ZERO. They can’t take up any shear force, that’s why water bodies have the tendency to reduce the effect of an earthquake on surrounding buildings.
BULK MODULUS
Bulk modulus is a volumetric property. It is generally denoted by K. It is the ability of a material to withstand the all-round pressure without a reduction in volume or in other words with less volumetric strain.
To understand this in a better way, let us consider a cube made of water or soil. Usually, soil and fluid are always subjected to force in all directions. Imagine a compressive force acting on the cube in all directions. This causes a reduction in the volume as the cube starts shrinking. The resistance to the applied compression force is known as Bulk Modulus.
Mathematically, ‘K’ can be written as
$$Bulk\>Modulus\>(K) = \frac{Change\>in\>pressure}{Volumetric\>strain}$$
Volumetric Strain is defined as the ratio of change in volume to the original volume.
In simple words, ‘K’ denotes the compressibility and density of the material. It is inversely proportional to compressibility and directly proportional to its density. It implies more the ‘K’ value, less is the compressibility and more is the density.
Diamond has more value of ‘K’. Hence it has a high density and it is less compressible.
SIGNIFICANCE OF ELASTIC CONSTANTS
The three elastic Constants describes a material response to stress or strain.
SHEAR MODULUS
1. Shear Modulus is used to calculate torsional stiffness (GJ) and shear stiffness (GA).
2. It identifies the elastic behaviour of a material due to shear force.
BULK MODULUS
Bulk modulus has more significance in soil and fluid mechanics. For example,
1. Compressibility and density of soil are the basic factors required for many geotechnical calculations.
2. Based on the compressibility of fluid, the pressure at which the fluid can be stored in a container is calculated.
RELATIONSHIP BETWEEN ELASTIC CONSTANTS
A relationship can be established among the three elastic constants using Poisson’s ratio(µ) as they all are dependent on Hooke’s law. Here, E is Young’s Modulus, G is Shear Modulus and K is Bulk Modulus.
The relation between E and G is given by
$$E = 2G (1+µ)$$
The relation between E and K is given by
$$E = 3K (1-2µ)$$
The relation between all three elastic constants is given by
$$E = \frac{9KG}{(3K+G)}$$
POINTS TO REMEMBER
1. Elastic constants can be determined by using the Stress-Strain Curves. If a curve is drawn for Normal Stress to Normal strain, then the slope of that curve is Young’s Modulus (E) while the slope of Shear Stress to Shear Strain curve gives Shear Modulus (G).
2. Any Modulus is a measure of resistance to deformation.
3. All three elastic constants hold good as long as Hooke’s Law is Valid i.e., up to proportionality limit.
4. For an Anisotropic Materials, the elastic constants cannot completely describe its behaviour and hence generalized Hooke’s law has to be established.