Elastic Energy and Energy Density
For quasistatic linear elastic loading from zero, stored strain energy is U = one-half F Delta L, and energy density is u = one-half sigma epsilon.
Why this shows up in the exam
Energy in stretched wires · Comparing energy densities · Reading area under stress-strain graphs
Learn the idea
Linear elastic energy is one-half force times extension or one-half stress times strain per volume. Because force rises from zero to its final value during gradual stretching, the average force is half the final force and the stored energy is the triangular area under the graph.
🧠 Memory hook: Elastic energy is the triangle under the loading line.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- U = (1/2)F Delta L = F² L/(2AY) — energy stored in a uniform linearly stretched wire
- u = U/V = (1/2)sigma epsilon — elastic energy per unit volume
- u = sigma²/(2Y) = (1/2)Y epsilon² — equivalent uniaxial forms
How to approach it
- 1Confirm loading is linear and quasistatic
- 2Choose force-extension or stress-strain form
- 3Check whether the requested quantity is total energy or energy density
Common slip-ups that cost marks
- •Using F Delta L without the one-half
- •Comparing total energies when energy density is asked
- •Using nonlinear unloading paths with the linear formula
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A wire 1 m long and cross-sectional area 2 mm^2 extends by 1 mm under a 200 N load. Find Young modulus.
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