Step 1: Definition:
Elastic energy density represents the elastic potential energy stored per unit volume within a deformed substance. It's equivalent to the work performed per unit volume to deform the material.
Step 2: Formula and Concepts:
The work (\(dW\)) required to stretch a wire is \(dW = F \, dx\), where \(F\) is the force applied and \(dx\) is the resulting elongation.
Energy density (\(u\)) is work per unit volume (\(V\)). For elastic deformation, it equals the area under the stress-strain curve.
Stress (\(\sigma\)) is \(F/A\), and strain (\(\epsilon\)) is \(x/L\).
For materials obeying Hooke's Law, stress is proportional to strain (\(\sigma = E\epsilon\)), resulting in a linear stress-strain relationship passing through the origin.
Step 3: Detailed Derivation:
Energy density \(u\) is the area under the stress-strain curve, forming a triangle:
\[ u = \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
On the stress-strain graph, the base corresponds to strain (\(\epsilon\)), and the height represents the corresponding stress (\(\sigma\)).
Therefore, energy density is expressed as:
\[ u = \frac{1}{2} \times \sigma \times \epsilon \]
\[ u = \frac{1}{2} \times \text{stress} \times \text{strain} \]
Step 4: Conclusion:
The elastic energy density in a stretched wire is \( \frac{1}{2} \times \text{stress} \times \text{strain} \).