Elastic Potential Energy

Question

Elastic energy density for linear stress-strain (stress σ, strain ε) equals?

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1. **Definition**: Energy density is the work done in deforming a material per unit volume.
2. **Work Link**: Work $= \int F \, dx = 	ext{Area under Force-Extension curve}$.
3. **Volume-Normalized**: For a linear elastic material (Hooke's Law), the stress-strain plot is a straight line through the origin.
4. **Geometric Calculation**: The area under the triangle is $\frac{1}{2} 	imes 	ext{base} 	imes 	ext{height} = \frac{1}{2} 	imes 	ext{strain} (\epsilon) 	imes 	ext{stress} (\sigma)$.
5. **Result**: **$u = \frac{1}{2} \sigma \epsilon$**. Also writable as $\frac{1}{2} Y \epsilon^2$ since $\sigma = Y \epsilon$.

Elastic Potential Energy

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