 Myself, Mr. Akshay Kumar Suvde, Assistant Professor, Department of Mechanical Engineering. Dear students, today we are going to study strain energy and impact loading. Learning outcome, at the end of the session, students will be able to understand the concept of strain energy and estimate the strain energy stored due to gradual load. So, various terms involved in strain energy. So, first of all, we will study what is mean by strain energy. Whenever a body is strained, the energy is absorbed in the body. The energy absorbed in the body due to stretching effect is known as strain energy. The strain energy stored in the body is also equal to work done by the applied load in stretching the body. For example, if you take a rubber band and if you apply the pull, then the body will get extended. So, when body gets extended, the energy will be stored in the body. That is nothing but it is called as strain energy. The next term, resilience. When a body is subjected to external load within the elastic limit, the body gets deformed. On the removal of external load, the body will get its original shape. This will happen under the load when it is within the elastic limit. From Hooke's law, stress is directly proportional to strain. And as we know, stress is nothing but load per unit area, whereas strain is ratio of change in length to the original length. So, from this, as the load increases, the deformation will also increases and vice versa, where load should be within the elastic limit. Therefore, total strain energy stored in the body is known as resilience. Also, the resilience is defined as capacity of a strained body for doing a work on removal of force. That means, when the force is removed, the body will regain its original size and shape. Hence, for getting the original size and shape, the work has to be done by the body, that is resilience. Proof resilience. The maximum amount of strain energy that can be stored in the body up to the elastic limit is known as proof resilience. The strain energy stored in the body will be maximum when the body is stressed up to the elastic limit. So, we can see the graph between stress and strain. And when the body is stressed within the elastic limit, as you see, the stress is directly proportional to strain. And hence, within the elastic limit, when the body is stressed corresponding to its elastic limit, maximum amount of strain energy will be stored in the body, that is, proof resilience. Modulus of resilience. Mathematically, modulus of resilience can be defined as it is the ratio of proof resilience to the volume of the body. And hence, proof resilience to the volume of the body is equal to modulus of resilience, which is given by sigma square upon 2 e. Units of resilience is joule per meter cube, where v is the volume of the body, sigma is the tensile stress or compressive stress, e, Young's modulus of the material of the body. Now, we will study the strain energy stored due to gradually applied load. So, at this moment, we will pause for a certain time and think what is mean by gradually applied load and what are the examples of gradually applied load. So, gradually applied load is nothing but the load which is gradually increased from 0 to certain value within certain steps. For example, if I apply 100 kilo Newton load in a step of 5 kilo Newton, so load will increase from 0 to 5, then 10 kilo Newton, 15 kilo Newton and so on. This is what is the example of gradually applied load. So, as you see the load extension diagram for a body which is subjected to tensile load up to its elastic limit. So, load extension diagram shows the body is subjected to tensile load up to its elastic limit and hence within the elastic limit, the stress is directly proportional to strain and hence the graph is straight line which is represented by AC. Sigma is the stress developed in the body, E is Young's modulus of elasticity, A cross sectional area of the body, P gradually applied load which is increasing gradually up to its elastic limit from value 0 to P as shown in the diagram. Therefore, load is equal to stress into area and therefore, P is equal to sigma into A where X is the deformation of the extension of the body which is also increasing from 0 to X as shown in the load extension diagram. L it is the length of the body, V volume of the body which is given by area of the area into length, U it is the strain energy stored in the body. When the body will be loaded within the elastic limit, the work done by the load in deforming the body will be equal to strain energy stored in the body and therefore, strain energy stored in the body is equal to work done by the load in deforming the body. Also, strain energy stored in the body that can be determined from the load extension diagram and that is equal to the area of the triangle S B C. So, area of the body is equal to the area of the triangle A B C will give us the strain energy stored in the body and therefore, area will be equal to one half into A B into B C and therefore, strain energy U is equal to one half into X multiplied by P. Let us the value of P which we will be have calculated sigma into A in the above equation that value we will put. So, U will be equal to one half into sigma into A, this is equation number 1. According to the Hooke's law, stress is directly proportional to strain and hence stress is equal to Young's modulus into strain and therefore, sigma is equal to Young's modulus into strain, but in this case strain is equal to deformation to the length. So, put the value of strain that is equal to X by L and therefore, stress sigma is equal to E into X upon L and therefore, the extension of deformation X is equal to sigma multiplied by L upon E, this is equation number 2. And now, let us use the value of the extension of deformation X in strain energy equation and we will have U is equal to one half sigma into L divided by E multiplied by sigma into A. So, by solving this equation you will get U is equal to one half multiplied by sigma square upon E into A into L. Where A into L is nothing, but it is the volume of the body and therefore, strain energy stored in the body due to gradually applied load is equal to sigma square upon 2 E into volume of the body. So, it is given by sigma square upon 2 E into V. This material is referred from a book of Strength of Materials by Dr. R. K. Bansal and S.S. Bhavikatti. Thank you.