Madhava Publications
Introduction
While studying rotational motion, we restricted ourselves to simpler situations of rigid bodies. A rigid body generally means a hard solid object having a definite shape and size. But in reality, bodies can be stretched, compressed and bent. Even the appreciably rigid steel bar can be deformed when a sufficiently large external force is applied to it. This means that solid bodies are not perfectly rigid. A solid has a definite shape and size. In order to change (or deform) the shape or size of a body, a force is required
When we pull a helical spring by hanging a mass with it, it will get stretched and hence change its length and shape. As soon as we remove the hanging mass, the spring will instantly regain its original shape and size.
The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed, is known as elasticity and the deformation caused is known as elastic deformation.
What are the Mechanical Properties of Solids?
Mechanical properties of solids elaborate the characteristics such as the resistance to deformation and their strength. Strength is the ability of an object to withstand the applied stress, to what extent can it bear the stress. Resistance to deformation is how resistant an object is to the change of shape. If the resistance to deformation is less, the object can easily change its shape and vice versa. Therefore, some of the mechanical properties of solids include:
- Elasticity: When we stretch an object, it changes its shape and when we leave, it regains its shape. Or we can say it is the property to regain the original shape once the external force is removed. Example: Spring
- Plasticity: When an object changes its shape and never comes back to its original shape even when an external force is removed. It is the property of permanent deformation. Example: Plastic materials.
- Ductility: When an object can be pulled in thin sheets, wires, or plates, it has ductile properties. It is the property of being drawn into thin wires/sheets/plates. Example: Gold or Silver
- Strength: The ability to withstand an applied stress without failure. Many categories of objects have higher strength than others.
In this chapter, we will study elasticity in detail
Elastic body and Elasticity
A body that returns to its original shape and size on the removal of the deforming force (when deformed within a certain limit) is called an elastic body.
The property of matter by virtue of which it regains its original shape and size when the deforming force has been removed is called elasticity. For example: If we stretch a spring and release it, it will regain its original size.
Now let us understand how it happens?
Whenever a load is attached to a thin hanging wire, it elongates and the load moves downward. The amount by which the wire elongates depends on the amount of load and the nature of the material.
The cohesive force between the molecules of hanging mass offers resistance against the deformation and the force of resistance increases with the deformation.
As soon as the deformation force is removed, the wire tries to regain its original shape. Thus we may conclude that if some external force is applied to a body, it has two effects on it.
- Deformation of the body.
- Internal resistance (restoring) forces are developed.
Stress
Whenever an external force is applied to a body, at each cross-section of the body, an internal restoring force is developed which tends to restore the body to its original state. The internal restoring force per unit area of the cross-section of the deformed body is called stress.
It is usually denoted by the σ (sigma)
S.I unit 
Its dimensional formula is
Depending on the ways the deforming force are applied to a body, there are three types of stress
- Longitudinal stress / Normal stress
- Shearing stress
- Volume stress
Normal stress
It is defined as the restoring force per unit area acting perpendicular to the surface of the body. It is of two types
Longitudinal or Tensile stress
When two equal and opposite forces are applied at the end of a circular rod as shown in the figure and increase its length, a restoring force equal to the applied force F normal to the cross-section of the rod comes into existence. This restoring force per unit area of cross-section is known as tensile stress.
Tensile stress = F/A
Consider a rod of length l, Force F area applied on it due to which the final length of the rod becomes
When two equal and opposite forces are applied at the ends of a rod as shown in the figure to decrease in length or compress it, then again restoring force equal to the applied force F comes into existence. This restoring force per unit area of the cross-section of the rod is known as compressive stress.
Compressive stress = F/A
Consider the length of the rod is L, when two equal force is applied, the final length of the rod becomes
Under tensile stress or compressive stress, the net force acting on an object is zero but the object is deformed. Tensile stress and compressive stress are also terms as longitudinal stress.
Shearing or Tangential stress
When a deforming force acts tangentially to the surface of a body, it produces a change in the shape of the body without any change in volume. This tangential force applied per unit area is equal to the tangential stress.
Tangential stress = F/A
In the case of tangential stress, the deforming force F is applied on a top surface of the cubical body in a tangential direction due to which the upper face is deformed by an angle
Volumetric Stress or Bulk Stress
If a body is subjected to a uniform force from all sides, then the corresponding stress is called volumetric stress. There is a change in the volume of the body but not a change in a geometric shape.
Bulk / volumetric stress = F/A
In the case of hydraulic stress, the force applied is perpendicular to every point on the surface due to which a change in volume
Strain
When a deforming force acts on a body, the body undergoes a change in its shape and size. The ratio of the change in configuration of the body to the original configuration is called strain.
Strain = 
If there is a change in any of the configuration of the body due to applied deforming force on it, then the body is said to be stained or deformed.
Since strain is the ratio of two like quantities, it is dimensionless and has no units.
According to a change in configuration, change in length, volume, or shape of the body, the strain can be classified as
- Longitudinal strain
- Volumetric strain
- Shear strain
- Longitudinal strain: It is defined as the change in length per unit of original length when the body is deformed by external forces.
- Volumetric strain: It is defined as the change in volume per unit of original volume when the body is deformed by external forces.
- Shear strain: It is the deforming forces that produce a change in the shape of the body, then the strain is called shear strain. It is defined as the angle in radians through which a plane perpendicular to the fixed surface of the cubical body gets turned under the effect of tangential force.
The angle θ is called the angle of shear.
Hooke's Law is a principle of physics that states that the force needed to extend or compress a spring by some distance is proportional to that distance. The law is named after 17th-century British physicist Robert Hooke, who sought to demonstrate the relationship between the forces applied to a spring and its elasticity.
Hooke’s law states that the extension produced in the wire is directly proportional to the load applied within the defined limit of elasticity.
Extension produced α load applied
Later on, it was found that this law is applicable to all types of deformation such as compression, bending, twisting etc. and thus a modified form of Hooke’s law was given as,
Within the elastic limit, the stress developed is directly proportional to the strain produced in a body.
Or,
For example: when we apply force on a spring, it gets compressed/ elongated and a restoring force is produced opposite to its displacement.
- Hooke’s law is valid only in the linear portion of the stress-strain curve. This law is not valid for large values of strains.
- Modulus of elasticity ‘E’ depends on the nature of the material of the body and is independent of its dimensions like length, area, volume etc.
Stress-strain Relationship
When a wire is stretched by an applied force, then a typical graph is obtained (especially in the case of metals) as shown below.
- OA is a straight line showing that the material follows Hooke’s law. Point A is called the proportionality limit (
- Up to B, if the load is removed, the material will regain its original shape. So material shows elastic behavior up to point B.
Thus curve OB represents the elastic curve. Point B is called yield point and corresponding stress is called yield strength
- Beyond point B, the strain increases rapidly even for a small change in stress or load. And when the load is removed between points B and D, the body does not regain its original dimensions. So, even when the stress (or load) is zero, there remains some strain on the material. The deformation is called plastic deformation.
- Point D on the graph represents the ultimate tensile strength (
Modulus of Elasticity
The modulus of elasticity or coefficient of elasticity of a body is defined as the ratio of stress to the corresponding strain within the elastic limit.
S.I. unit of Modulus of Elasticity is
There are three types of modulus of elasticity:
- Young’s Modulus ( Y)
- Bulk modulus ( B)
- Modulus of rigidity or shear modulus (G)
1. Young’s Modulus
Within the elastic limit, the ratio of longitudinal stress to the longitudinal strain is called Young’s Modulus of the material of the wire.
Young's modulus (Y) is a property of the material that tells us how easy it can stretch and deform and is defined as the ratio of tensile stress (σ) to tensile strain (ε). Where stress is the amount of force applied per unit area (σ = F/A) and strain is extension per unit length (ε = dl/l).
,
Where F= load, A= area of cross-section, L= Length of rod
For metals Young’s moduli are large. Therefore, these materials require a large force to produce a small change in length.
Steel has a larger value of young’s modulus than copper, brass and aluminum. Steel is more elastic than copper, brass and aluminum. It is for this reason that steel is preferred in heavy-duty machines and in structural designs. Wood, bone, concrete and glass have rather small Young’s moduli.
If the extension is produced by a load of mass ‘m’ then, F= mg.
Here the wire has a circular cross-section. So area of cross-section
So the formula for young’s modulus can be written as
2. Bulk Modulus of Elasticity
Within the elastic limit, the ratio of normal stress to the volumetric strain is called the bulk modulus of elasticity.
So,
What is meant by bulk modulus?
Sometimes referred to as incompressibility, the bulk modulus is a measure of the ability of a substance to withstand changes in volume when under compression on all sides. It is equal to the quotient of the applied pressure divided by the relative deformation.
The units for the bulk modulus are Pascal’s (Pa) or newtons per square meter (N/m2) in the metric system.
Compressibility: The reciprocal of the bulk modulus of a material is called its compressibility.
Bulk modulus is used to measure how incompressible a solid is. Besides, the more the value of B for a material, the higher is its nature to be incompressible. For example, the value of B for steel is
3. Modulus of rigidity or shear modulus
Within the elastic limit, the ratio of tangential stress (shear stress) to shear strain is called the modulus of rigidity of shear modulus. It is denoted by G or η.
To measure the stiffness of materials, the shear modulus is one of many quantities. The deformation of a solid is concerned with the shear modulus when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force.
Let us consider a cube whose lower face is fixed and a tangent force F acts on the upper face whose area is A, as shown in the figure.
Let the vertical sides of the cube shit through an angle θ, called shear strain
Therefore, the Modulus of rigidity is given by 
According to the diagram,
Poisson’s Ratio
When a material is stretched in one direction, it tends to compress in the direction perpendicular to that of force application and vice versa. The measure of this phenomenon is given in terms of Poisson’s ratio. For example, a rubber band tends to become thinner when stretched.
What is the position ratio?
Poisson's ratio is “the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.” Here, Compressive deformation is considered negative. Tensile deformation is considered positive.
Therefore Poisson’s ratio σ is given by,
Poisson's ratio is the ratio of lateral strain to longitudinal strain. It has no units.
Relation between various modulus of Elasticity
The elastic moduli of a material, like Young’s Modulus, Bulk Modulus, and Shear Modulus are specific forms of Hooke’s law, which states that for an
elastic material, the strain experienced by the corresponding stress applied is proportional to that stress. Thus, we can write the relation between elastic constants by the following equation:
Where,
- G is the Shear Modulus
- Y is the Young’s Modulus
- B is the Bulk Modulus
- σ is Poisson’s Ratio
We can derive the elastic constant’s relation by combining the mathematical expressions relating to terms individually.
- Young modulus can be expressed using Bulk modulus and Poisson’s ratio as –
- Similarly, Young’s modulus can also be expressed using rigidity modulus and Poisson’s ratio as-
- Combining the above two-equation and solving them to eliminate Poisson’s ratio we can get a relation between Young’s modulus and bulk modulus B and modulus of rigidity as -
A fun thing to try: Virtual lab
Below is the link to the simulation of Hooke’s law
In this simulation, we have three parts: Intro, systems, and energy
- Intro: In the Intro, we have a spring whose one end is fixed and the other end can be pulled.
- We can fix the spring constant of the spring and then by changing the applied force we can see how much displacement it is producing in the spring.
- We can then change the spring constant of the spring and do the same process again.
- We will come to know that it is easier to stretch a spring with lower spring constant.
- System: In the system we have the option of parallel combination of spring and a series combination of spring. We can choose any of them.
- We can then fix the spring constant of the springs and apply force and see how the displacement is changing with the applied force.
- Now we can change the spring constant and do the same process.
- This should be done with both combinations of spring: Series and parallel.
- Energy: In this part we can see the value of potential energy of the spring.
- We can fix the spring constant of the spring and then change the displacement and see how the potential energy is varying.
- We can also fix the displacement and vary spring constant and see how the potential energy of the spring is changing.
