Question

A rod is elastically deformed by a uniaxial stress resulting in a strain of 0.02. If the Poisson's ratio is 0.3, the volumetric strain is _______________ (answer up to three decimal places)

Hint:

For uniaxial stress problems, the relationship \(\epsilon_V = \epsilon_{axial}(1 - 2\nu)\) is a direct and quick way to find volumetric strain from axial strain and Poisson's ratio.

Answer 1

Step 1: Definition of Volumetric Strain:
Volumetric strain ($\epsilon_V$) is defined as the ratio of change in volume to the original volume and can be expressed as the sum of linear strains in three perpendicular directions: \[ \epsilon_V = \epsilon_x + \epsilon_y + \epsilon_z \]
Step 2: Strain Under Uniaxial Loading:
When a rod is subjected to uniaxial stress along the axial direction (z-axis), the axial strain is: \[ \epsilon_z = 0.02 \] Due to Poisson’s effect, lateral strains in x and y directions are: \[ \epsilon_x = \epsilon_y = -\nu \epsilon_z \] where $\nu = 0.3$.
Step 3: Substitution and Calculation: \[ \epsilon_V = \epsilon_z + \epsilon_x + \epsilon_y \] \[ \epsilon_V = \epsilon_z - \nu\epsilon_z - \nu\epsilon_z = \epsilon_z(1 - 2\nu) \] \[ \epsilon_V = 0.02 \times (1 - 2 \times 0.3) \] \[ \epsilon_V = 0.02 \times (1 - 0.6) = 0.02 \times 0.4 = 0.008 \] Final Answer:
The volumetric strain is $0.008$.