ELASTIC PROPERTIES OF SOLID

Elastic properties refer to the ability of a material to deform under the application of a force and return to its original shape when the force is removed. Hooke’s law, Young’s modulus, and work done in springs and strings are important concepts related to the elastic properties of solids.

1. Hooke’s Law: Hooke’s law states that the deformation of an elastic material is directly proportional to the applied force, as long as the material remains within its elastic limit. Mathematically, this can be expressed as F = kx, where F is the applied force, x is the deformation, and k is the spring constant of the material. Hooke’s law applies to both compression and tension forces.
2. Young’s Modulus: Young’s modulus is a measure of a material’s stiffness or resistance to deformation under an applied force. It is defined as the ratio of stress to strain, where stress is the force applied per unit area and strain is the resulting deformation per unit length. Mathematically, this can be expressed as E = (F/A)/(ΔL/L), where E is the Young’s modulus, F is the applied force, A is the cross-sectional area of the material, ΔL is the change in length, and L is the original length.
3. Work Done in Springs and Strings: When a force is applied to a spring or a string, work is done on the material, and it stores potential energy in the form of elastic potential energy. The amount of work done is equal to the product of the force and the displacement of the material. Mathematically, this can be expressed as W = (1/2)kx^2, where W is the work done, k is the spring constant of the material, and x is the displacement.

In summary, Hooke’s law describes the relationship between force and deformation, Young’s modulus measures a material’s stiffness, and work done in springs and strings relates to the potential energy stored in elastic materials.

 

Behaviour of elastic materials under stress

Elastic materials, such as springs and rubber bands, exhibit specific behaviors when subjected to stress. When a force is applied to an elastic material, it deforms in response, and the material will return to its original shape when the force is removed.

When an elastic material is subjected to stress, its behavior is characterized by several features related to the applied load. These features can be observed on a stress-strain curve or an extension graph.

1. Elastic Limit: The elastic limit is the point on the graph where the material starts to deform permanently. It is the maximum stress that a material can withstand without undergoing permanent deformation. If the stress applied to the material is below the elastic limit, the material will return to its original shape once the load is removed.
2. Yield Point: The yield point is the point on the graph where the material starts to undergo plastic deformation. Beyond this point, the material will not return to its original shape once the load is removed. The yield point marks the transition from elastic to plastic behavior.
3. Maximum Load: The maximum load is the point on the graph where the material reaches its maximum stress. At this point, the material may continue to deform, but it will no longer be able to support any additional load.
4. Breaking Point: The breaking point is the point on the graph where the material fractures or breaks apart. Beyond this point, the material can no longer withstand any stress and fails completely.

In an extension graph, the features described above can be observed in the form of a curve. The initial part of the curve is linear, representing the elastic region where the material deforms elastically. The slope of this linear region is the material’s modulus of elasticity. The curve then becomes nonlinear, representing the plastic region where the material undergoes permanent deformation. The yield point is the point at which the curve deviates from the linear region. The curve continues to rise until it reaches the maximum load, beyond which it drops sharply, indicating failure at the breaking point.

Overall, the behavior of elastic materials under stress is complex and depends on various factors such as the type of material, the applied load, and the duration of the stress. Understanding the features of load described above is crucial for designing and engineering materials that can withstand specific stress conditions.

 

Simple calculations on Hooke’s law and Young’s modulus

Hooke’s Law:

Hooke’s law states that the deformation of an object is directly proportional to the force applied to it, provided that the material is within its elastic limit. This relationship can be expressed mathematically as:

F = kx

Where F is the force applied, x is the displacement or deformation of the object, and k is the spring constant or stiffness of the object. The spring constant represents the amount of force required to produce a unit displacement in the object.

 

1) Let’s say you have a spring with a spring constant of 10 N/m, and you apply a force of 5 N to it. Using Hook’s Law, we can calculate the amount of deformation of the spring:

F = kx

5 N = 10 N/m * x

x = 0.5 m

So the spring will deform by 0.5 meters when a force of 5 N is applied to it.

 

2) Suppose a spring with a spring constant of 100 N/m is stretched by a distance of 0.1 m. Using Hooke’s law, we can calculate the force applied as:

F = kx = (100 N/m)(0.1 m) = 10 N

 

Young’s Modulus:

Young’s modulus is a measure of the stiffness of a material. It is defined as the ratio of the stress (force per unit area) applied to a material to the strain (deformation per unit length) produced in the material. The mathematical expression for Young’s modulus is:

E = stress/strain = (F/A)/(∆L/L)

Where E is the Young’s modulus, F is the force applied, A is the cross-sectional area of the material, ∆L is the change in length of the material, and L is the original length of the material.

 

1) Now suppose we have a steel rod with a cross-sectional area of 0.01 m² and an original length of 1 m. If a force of 5000 N is applied to the rod, causing it to stretch by 0.01 m, we can use Young’s modulus to calculate the stiffness of the material as:

E = stress/strain = (F/A)/(∆L/L) = (5000 N/0.01 m²)/(0.01 m/1 m) = 5 × 10¹¹ N/m²

This tells us that steel is a very stiff material, requiring a large amount of force to produce a small amount of deformation.