I am Sotrush Gainkar, today I am teaching a mechanics of material in that one. A volumetric strain due to three mutually perpendicular stresses. Generally we call it as a generalized Hooke's law. We are considering a rectangular block and this is the stress sigma x sigma x sigma y sigma y sigma z sigma z at a length x y z z. Consider a rectangular, rectangular block subjected to 3, 10 side stresses. In the matrix, sigma y, sigma z in 3 mutually perpendicular directions, 3 mutually perpendicular direction as in the figure. Now the initial volume, v by handset. Now taking log on both side, log v is equal to log x plus log y plus log z. If you take a log and v v becomes log v and the product becomes additions here and further differentiate 1 by v is equal to dv 1 by x dx plus 1 by y dy plus 1 by z dz. If you differentiate the log, it becomes 1 by v by dv and x it becomes 1 by x into delta x and so on. Delta v by v is equal to delta x by x plus delta v delta y by y plus delta z by z. Delta v by v is equal to e v delta x by x is nothing but e x. These are the three parts of the strain, change in length to original length. Delta v by v delta x by z is equal to e v delta x by x dx plus 1 by y dz. What is e? Strain is nothing but delta y by n. We call it as strain. Either you can light epsilon or small. Now as per the diagram, from diagram, diagram what happens? If a tension is applied in x direction, the length increases but in the case of y and z, reduction occurs. If a tension is applied in the x direction, reduction in the case of y and z occurs. Then the Poisson's ratio comes in picture. Therefore strain in the x direction, e x is nothing but we can write as sigma divided by e, sigma divided by n. Now there is a reduction in the y and z, therefore 1. Sigma x, sorry sigma, this is the sigma x, sigma y will be m e. 1 by m is nothing but Poisson's ratio. Similarly sigma z divided by m e. Reduction occurs in the case of y and z, y and z. This is in case of x direction. If it is in case of y direction, sigma y divided by e. And reduction occurs in the case of x and z. Minus sigma x divided by m e minus sigma z divided by m e. m is 1 by m is nothing Poisson's ratio. Because here the forces are acting mutually perpendicular to each other. In last case, there was a single force acting over the faces. Here, multi forces are acting over the face. And they are mutually perpendicular to each other. And sigma z is the e z is nothing but sigma z divided by e minus sigma x divided by m e minus sigma y divided by m e. These are the strains, strains in the medial, strain in the medial, as per the diagram, as per the diagram. As the tension is applied in z direction, reduction in x and y take place, reduction in x and y take place. Now, substitute these values in this equation. We have to substitute this value in this equation, then what happens? e is equal to sigma x divided by e minus sigma y divided by m e minus sigma z divided by m e. Next, this one plus sigma y divided by e minus sigma x divided by m e minus sigma z divided by m e plus this one. Sigma z divided by e minus sigma x divided by m e minus sigma y divided by m e. First, I will consider sigma x, sigma y and sigma z, sigma x divided by e plus sigma y divided by e plus sigma z divided by e. Next, x terms, this is the one next term, this is the other next term. Then it becomes minus 2 sigma x divided by m e. Next, y terms, y, one more y is here, minus 2 sigma y divided by m e and last will be minus 2 sigma z divided by m e, m e and this part. Next, I will take a common here, this and this, make common. Sigma x divided by e, 1 minus 2 by m, next I will take from the y common plus sigma y divided by e, 1 minus 2 divided by m and sigma z and sigma z. I will take sigma z by e common, 1 minus 2 m among these three, 1 minus 2 m is common. Sigma x divided by e plus sigma y divided by e plus sigma z divided by e, 1 minus 2 by m. 1 minus 2 by m, further it can be written as sigma x plus sigma y plus sigma z divided by e into 1 minus 2 by m, 2 by m. This is the formula for volumetric strain due to three mutually perpendicular stresses. I am Satush Gai Dhankar, thank you very much.