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Every solid material has a certain property of elasticity. It can either be completely elastic, partially elastic or plastic. Thus, different solids also have different modulus of elasticity depending upon the properties of the solid.
Internal forces are produced in the body to counter the external forces when a body is deformed. The work done by the body against these external forces is stored in the body as elastic potential energy or elastic strain energy.
In this article, we shall learn more about elasticity in solids, elastic strain energy and the derivation of the formula for elastic strain energy. But before we move further, let us learn the different properties of solid materials in detail.
Elasticity
When an external force is applied to a solid body, it may get deformed or may become out of proportion. When a body is deformed, then internal forces develop inside that material to oppose the outside deforming force. The internal force tries to restore the body of the material to its original shape.
Thus, elasticity can be defined as the tendency of a body to get its original shape back after the deforming force is removed. The elasticity of every material differs. The more easily a body gains its original shape after the deforming force is removed, the more elastic that body is.
Perfectly Elastic Body
A perfectly elastic body can regain its natural shape after the deforming forces are removed. An example is Quartz.
Partially Elastic
Partially Elastic bodies are those which only partially regain their original shape after the removal of the deforming forces. Neither are they fully elastic nor fully plastic.
Plasticity
A plastic body stays in its deformed state even after the external force is removed. The original shape is unable to be restored even partially. This property of a solid is known as plasticity.
Stress
Stress can be defined as the ratio of the restorative internal force developed to oppose the deforming force to the area on which the force is applied.
In equilibrium, the magnitude of Stress produced is equal to the magnitude of the deforming force applied on a body.
Stress can be represented as:
Stress = ForceArea = FA
The SI unit of Stress is given as N/m2
The unit of Stress in the CGS unit is Dyne cm-2
The dimensional formula is
Types of Stress
Stress can be of two types:
- Normal Stress
Normal Stress can be defined as the ratio of the restoring force to the area perpendicular to the surface of the body. It is further classified into
Tensile Stress
Compressive Stress
- Tangential Stress
Tangential Stress is when the restorative force is parallel to the surface area of the body.
Strain
Strain can be simply defined as the ratio of the change in shape or size of the body to the original shape or size of the body after the deforming force is applied. It does not have a unit.
Types of Strain
Strain is classified into three types:
- Longitudinal Strain
When the change in the shape of the body occurs along the length of the body, then it is known as longitudinal Strain. It is the ratio of the change in length of the body to the original length of the body.
Longitudinal Strain = change in length original length =
- Volumetric strain
It is the ratio of change in the volume of a body to the original volume. It can be represented as
Volumetric Strain =
Where V = original volume
= change in volume
- Shear strain
Shear strain can be defined as the angular tilt in a body due to Tangential Stress.
Now that we know what the different forms of Stress and Strain are, let us understand the relation between Stress and Strain.
Hooke’s Law
This law states that within the limits of elasticity, the ratio of Stress to Strain is constant. This constant is known as the Modulus of Elasticity. It can be represented as:
Modulus of elasticity = StressStrain
As we know that Strain does not have any SI unit, the modulus of elasticity takes the SI unit of Stress. The SI unit of Stress is given as N/m2
The unit in the CGS unit is Dyne cm-2
The dimensional formula is
Young’s Modulus
Young’s Modulus is defined as the ratio of Longitudinal Stress to Longitudinal Strain, given the elasticity limit is maintained.
Thus, Young’s Modulus can be represented as
Young’s modulus, Y = = =
It has the same unit as that of the longitudinal stress as N/m2
It can also have its SI unit as Pa.
Bulk’s Modulus
It can be simply defined as the ratio of Longitudinal Stress to Volumetric Strain, within the limits of elasticity. It is represented as:
Bulk’s Modulus, K = =
The negative sign here represents that pressure variation and volume variation will always be negative to each other.
The opposite of Bulk’s Modulus is also known as Compressibility.
Shear Modulus or Modulus of Rigidity
Shear Modulus can be represented as the ratio of Tangential Stress to Shear strain.
Modulus of Rigidity can be given as:
Modulus of rigidity = Tangential StressShear Strain =
Elastic Strain Energy
In its natural state, a body’s potential energy is equal to its molecular forces. Hence, we can consider the potential energy to be equal to zero. But when a body is deformed, internal forces develop to restore it to its original shape. Thus, work is being done against external forces. This work is stored as potential energy or elastic strain energy in the body.
Within the elastic limit, the amount of work done by the external forces will be equal to the elastic strain energy stored in the body. This energy can also be calculated by calculating the area under the Stress vs strain graph.
The SI unit of elastic strain energy is Nm or joules (J)
Derivation Of Elastic Strain Energy
Before we can derive the formula for elastic strain energy, there are a few assumptions that we must keep in mind. These are
The material should be completely elastic
Stress that is developed in the body should be within the proportional limit
The load should be applied slowly to the body
Now, let us suppose we have a wire having a length L and area A. If this wire is fixed at one end, and force is applied from the other end, the wire will get elongated by x metres length.
According to Hooke’s Law,
StressStrain = Young’s Modulus (Y)
This can be written as:
Stress = E x Y
The force being applied is variable; hence, the deformation also gradually increases. For small deformation dx, the differential work done is dW, then we will get,
Thus, the potential energy or elastic strain energy is
Thus, Elastic strain energy = 12 x Stress x Strain x Volume
Conclusion
When a body gets deformed, internal forces develop in the body to restore it to its original shape. Thus, work is being done against external forces. This work is stored in the body as potential energy called elastic strain energy. This energy depends on several characteristics of a solid material. These characteristics include elasticity, plasticity, Stress, and Strain.
