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: Understanding the Question:
The problem provides the axial strain (\(\epsilon_z\)) on a rod under uniaxial stress and the material's Poisson's ratio (\(\nu\)). We need to calculate the resulting volumetric strain (\(\epsilon_V\)).
Step 2: Key Formula or Approach:
The volumetric strain, \(\epsilon_V\), is the sum of the strains in three mutually perpendicular directions: \[ \epsilon_V = \epsilon_x + \epsilon_y + \epsilon_z \] For a uniaxial stress applied along the z-axis, the strain in that direction is the axial strain, \(\epsilon_z\). The strains in the transverse (lateral) directions, \(\epsilon_x\) and \(\epsilon_y\), are related to the axial strain by the Poisson's ratio, \(\nu\): \[ \epsilon_x = \epsilon_y = -\nu \epsilon_z \] Substituting the lateral strains into the volumetric strain equation gives a simplified formula for uniaxial loading: \[ \epsilon_V = (-\nu \epsilon_z) + (-\nu \epsilon_z) + \epsilon_z = \epsilon_z(1 - 2\nu) \] Step 3: Detailed Explanation:
We are given the following values:

• Axial strain, \(\epsilon_z = 0.02\)
• Poisson's ratio, \(\nu = 0.3\)

Now, we substitute these values into the derived formula: \[ \epsilon_V = 0.02 \times (1 - 2 \times 0.3) \] \[ \epsilon_V = 0.02 \times (1 - 0.6) \] \[ \epsilon_V = 0.02 \times (0.4) \] \[ \epsilon_V = 0.008 \] Step 4: Final Answer:
The volumetric strain is 0.008. The answer is already in three decimal places.