The strain energy per unit volume of a stretched wire is

Options

A. `1//2 xx ` stress `xx` strain
B. `1//2 xx ("strain")^(2)xx Y`
C. `1//2 xx ("stress")^(2)//Y`
D. all of these

Answer

To find the strain energy per unit volume of a stretched wire, we can follow these steps:

### Step 1: Understand the Concept of Strain Energy
When a wire is stretched, work is done against the internal restoring forces of the wire. This work done gets stored as potential energy in the wire, known as strain energy.

### Step 2: Define Work Done
The work done (W) to stretch the wire can be expressed as:
\[ W = \text{Force} \times \text{Displacement} \]
For a small element of the wire, the work done \( dW \) can be written as:
\[ dW = F \cdot dl \]
where \( F \) is the force applied and \( dl \) is the small displacement.

### Step 3: Relate Force to Young's Modulus
Using Young's modulus (Y), we know:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta l/L} \]
From this, we can express the force as:
\[ F = Y \cdot \frac{A \cdot \Delta l}{L} \]

### Step 4: Substitute Force into Work Done
Substituting the expression for force into the work done gives:
\[ dW = \left(Y \cdot \frac{A \cdot \Delta l}{L}\right) \cdot dl \]

### Step 5: Integrate to Find Total Work Done
To find the total work done to stretch the entire wire, we integrate:
\[ W = \int dW = \int_0^L \left(Y \cdot \frac{A \cdot \Delta l}{L}\right) \cdot dl \]
Since \( Y \), \( A \), and \( L \) are constants, we can take them out of the integral:
\[ W = Y \cdot \frac{A \cdot \Delta l}{L} \int_0^L dl \]
The integral evaluates to \( L \):
\[ W = Y \cdot \frac{A \cdot \Delta l}{L} \cdot L = Y \cdot A \cdot \Delta l \]

### Step 6: Calculate Strain Energy per Unit Volume
The energy stored per unit volume \( U \) is given by:
\[ U = \frac{W}{V} \]
where \( V = A \cdot L \) is the volume of the wire. Thus:
\[ U = \frac{Y \cdot A \cdot \Delta l}{A \cdot L} = \frac{Y \cdot \Delta l}{L} \]

### Step 7: Relate to Stress and Strain
Using the definitions of stress and strain:
- Stress \( \sigma = \frac{F}{A} \)
- Strain \( \epsilon = \frac{\Delta l}{L} \)

We can express the energy stored per unit volume as:
\[ U = \frac{1}{2} \cdot \sigma \cdot \epsilon \]
Thus, the strain energy per unit volume is:
\[ U = \frac{1}{2} \cdot \text{Stress} \cdot \text{Strain} \]

### Final Answer
The strain energy per unit volume of a stretched wire is:
\[ U = \frac{1}{2} \cdot \text{Stress} \cdot \text{Strain} \]

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