Question

The linear strain in $x,\,y$ and $z$ directions are ${e_x}$ , ${e_y}$ and ${e_z}$ respectively. Then the volumetric strain is given by
(A) ${e_x}{e_y}{e_z}$
(B) ${e_x} + {e_y} + {e_z}$
(C) ${e_z} = {e_x}{e_y}$
(D) ${e_z} = \dfrac{{{e_x} + {e_y}}}{2}$

Answer 1

Hint:The strain is defined as the deformation of the body after giving its some force greater than its elasticity. The strain resembles the movement of the particles in the body to the other position and it does not return back to the original position as the complete elastic body does.

Complete step by step solution:
The linear strain in the direction $x$ is ${e_x}$
The linear strain in the direction $y$ is ${e_y}$
The linear strain in the direction $z$ is ${e_z}$
The linear strain is the term also specified by the transverse strain and it is obtained by the ratio of the original length to the deformed length. It mainly occurs in the body which is subjected to the longitudinal stress. At this time, the body has the increased dimensions in the longitudinal region and decreased dimension in the lateral regions. Hence there will be a decrease in the lateral region causing the lateral areas to contract. In this question, in three directions, the lateral strain happens, and so the total lateral strain is the sum of all the three strains. Hence the volumetric strain is equal to ${e_x} + {e_y} + {e_z}$ .

Thus the option (B) is correct.

Note:Even though the volumetric strain is the ratio of the difference in the volume to that of the original volume, the addition of the lateral strain that acts in the three directions provides the answer for it. Since the dimension of the object increased three dimensional means, in other ways the volume increased.