A wire gives (strain, stress in MPa) readings (0, 0), (0.001, 80), (0.002, 160), (0.003, 240), (0.004, 300), (0.006, 330). Original length is 2.0 m, area 1.5 × 10⁻⁶ m² and volume 3.0 × 10⁻⁶ m³. (a) Find Young modulus. (b) Find force and extension at strain 0.003. (c) Find energy density and energy. (d) Identify departure from proportionality. (e) Describe a safe experiment and distinguish proportional from elastic limit.

Question

Accepted Answer

E = 8.0 × 10¹⁰ Pa; force 360 N; extension 6.0 mm; energy density 3.6 × 10⁵ J/m³; energy 1.08 J. Initial Young modulus E = stress/strain = 240 × 10⁶/0.003 = 8.0 × 10¹⁰ Pa; earlier points give the same ratio. Force F = σA = 240 × 10⁶ × 1.5 × 10⁻⁶ = 360 N. Extension = strain × length = 0.003 × 2.0 = 0.0060 m. Linear-region energy density is triangular graph area ½σε = ½ × 240 × 10⁶ × 0.003 = 3.6 × 10⁵ J/m³. Total energy is 1.08 J. At strain 0.006, linear extrapolation predicts 480 MPa but observed stress is 330 MPa, so direct proportionality has ended. Clamp behind a shield, add small loads and measure extension with a long gauge length. The proportional limit is loss of linearity; the elastic limit requires unloading to test return to original length.

Stress–strain energy and elastic behaviour

What a complete response should do

Metal stress–strain data

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Metal stress–strain data — supplied values
Strain	Stress / MPa
0	0
0.001	80
0.002	160
0.003	240
0.004	300
0.006	330