A unifrom bar of Length `'L'` and cross sectional area `'A'` is subjected to a tesnile load `'F', 'Y'` be the Young's modulus and `'sigma'` be the Poisson's ratio then volumeteric strain is

Options

A. `(F)/(AY) (1-sigma)`
B. `(F)/(AY) (2 - sigma)`
C. `(F)/(AY) (1 - 2sigma)`
D. `(F)/(AY) sigma`

Explanation

To find the volumetric strain of a uniform bar subjected to a tensile load, we can follow these steps:

### Step 1: Understand the relationship between stress and strain
When a tensile load \( F \) is applied to a bar of length \( L \) and cross-sectional area \( A \), the stress \( \sigma \) can be defined as:
\[ \sigma = \frac{F}{A} \]

### Step 2: Relate stress to strain using Young's modulus
Young's modulus \( Y \) relates stress to strain \( \epsilon \) as follows:
\[ Y = \frac{\sigma}{\epsilon} \]
From this, we can express strain \( \epsilon \) as:
\[ \epsilon = \frac{\sigma}{Y} = \frac{F}{AY} \]

### Step 3: Determine the volumetric strain
Volumetric strain \( \Delta V/V \) for a material under uniform stress can be expressed in terms of the linear strain and Poisson's ratio \( \nu \) (also denoted as \( \sigma \) in some contexts). The volumetric strain is given by:
\[ \text{Volumetric Strain} = \Delta V/V = \epsilon_1 + \epsilon_2 + \epsilon_3 \]
For a bar under tensile load, the longitudinal strain \( \epsilon_1 \) is given by:
\[ \epsilon_1 = \frac{F}{AY} \]
The lateral strains \( \epsilon_2 \) and \( \epsilon_3 \) can be expressed using Poisson's ratio:
\[ \epsilon_2 = -\nu \epsilon_1 \quad \text{and} \quad \epsilon_3 = -\nu \epsilon_1 \]
Thus, the total volumetric strain becomes:
\[ \text{Volumetric Strain} = \epsilon_1 - \nu \epsilon_1 - \nu \epsilon_1 = \epsilon_1 (1 - 2\nu) \]

### Step 4: Substitute the expression for strain
Now, substituting \( \epsilon_1 \) into the volumetric strain expression:
\[ \text{Volumetric Strain} = \left(\frac{F}{AY}\right)(1 - 2\nu) \]

### Final Expression
Thus, the volumetric strain for the uniform bar subjected to a tensile load is:
\[ \text{Volumetric Strain} = \frac{F}{AY}(1 - 2\nu) \]

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