A steel wire of unloaded length m and cross-sectional area

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Fluids and Continuum Physics, Ch Mechanical properties of materials: the Young modulus, plasticity and fracture

Problem

A steel wire of unloaded length 2.002.00 m and cross-sectional area 1.0 mm21.0\ \text{mm}^2, with Young modulus E=200E = 200 GPa and yield stress σY=300\sigma_Y = 300 MPa, hangs vertically and carries a load of 200200 N. The wire is of uniform cross-section, its own weight is ignored, and it is loaded slowly along the straight part of its stress-strain curve. A student writes the following solution for the elastic energy stored. σ=F/A0=(200 N)/(1.0×10−6 m2)=2.0×108\sigma = F/A_0 = (200\ \text{N})/(1.0\times10^{-6}\ \text{m}^2) = 2.0\times10^{8} Pa. ϵ=σ/E=(2.0×108)/(2.0×1011)=1.0×10−3\epsilon = \sigma/E = (2.0\times10^{8})/(2.0\times10^{11}) = 1.0\times10^{-3}. Energy per unit volume u=σϵ=2.0×108×1.0×10−3=2.0×105 J m−3u = \sigma\epsilon = 2.0\times10^{8}\times 1.0\times10^{-3} = 2.0\times10^{5}\ \text{J m}^{-3}. Volume V=A0L0=1.0×10−6×2.00=2.0×10−6 m3V = A_0 L_0 = 1.0\times10^{-6}\times 2.00 = 2.0\times10^{-6}\ \text{m}^3, so the stored energy is U=uV=0.40U = uV = 0.40 J. ((a)) Identify the single error in the student's working and say, in one sentence, what the correct expression for uu is and why. ((b)) Give the correct energy per unit volume and the correct total stored energy.

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