Question

A prismatic bar of volume v is subjected to a compressive force in the longitudinal direction. If the poissons ratio of the bar is m and the longitudinal strain is , then the final volume of the bar will be

Answer 1

Final volume =  V + (1+m)$\frac{\delta L}{L}$ × V

Given:

A prismatic bar of volume v is subjected to a comprehensive force in the longitudinal direction.

The poisons ratio of the bar is m and the longitudinal strain is e

To find:

Final volume of the bar.

Explanation:

This is a two dimensional stress system.

The compression load is applied will decrease length and increase diameter.

So that Volumetric strain = linear strain + lateral strain

Volumetric strain = $\frac{\delta V}{V}$

Linear strain = $\frac{\delta L}{L}$

poisons ratio = $\frac{lateral \ strain}{linear \ strain}$

lateral strain = m×linear strain

lateral strain = m× $\frac{\delta L}{L}$

Volumetric strain = linear strain + lateral strain

Volumetric strain = $\frac{\delta L}{L}$ + m× $\frac{\delta L}{L}$

Volumetric strain =  (1+m)$\frac{\delta L}{L}$

                      $\frac{\delta V}{V}$ = (1+m)$\frac{\delta L}{L}$

                      $\delta V$ =   (1+m)$\frac{\delta L}{L}$ × V

Final volume = initial volume + change in volume

Final volume = V + $\delta V$ = V + (1+m)$\frac{\delta L}{L}$ × V

Final volume =  V + (1+m)$\frac{\delta L}{L}$ × V

To learn more...

1)Volumetric strain produced in sphere is how much time the strain in its diamete.

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2)If volumetric strain is zero what is poisson's ratio .

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