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Exercises · 8.2

Q.Figure 8.9 shows the strain-stress curve for a given material. What are

(a) Young's modulus and
(b) approximate yield strength for this material?
Fig 8.9 -- Strain-stress curve for a given material, used to read off Young's modulus and the approximate yield strength
Figure 8.9
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✓ Free question

Young's modulus is the slope of the straight (proportional) part of the stress-strain graph, and the yield strength is the stress at which the curve stops being linear and begins to level off. Reading the graph gives Y≈7.5×1010 N m−2Y\approx 7.5\times10^{10}\ \text{N m}^{-2} and a yield strength of about 3×108 N m−23\times10^{8}\ \text{N m}^{-2}.

Concept

In the initial straight portion of a stress-strain curve, stress is proportional to strain (Hooke's law). The constant of proportionality is Young's modulus, Y=stressstrainY=\dfrac{\text{stress}}{\text{strain}}, which equals the slope of that straight line. The yield strength is the stress at the point where the curve departs from the straight line and the material begins to deform permanently.

(a) Young's modulus

Take a point on the straight region: at strain ε=0.002\varepsilon=0.002 the stress is σ=150×106 N m−2\sigma=150\times10^{6}\ \text{N m}^{-2}.

Y=σε=150×1060.002=7.5×1010 N m−2.Y=\frac{\sigma}{\varepsilon}=\frac{150\times10^{6}}{0.002}=7.5\times10^{10}\ \text{N m}^{-2}.

Any other point on the line gives the same value, e.g. 225×106/0.003=7.5×1010225\times10^{6}/0.003=7.5\times10^{10}.

(b) Yield strength

The curve stays straight up to a strain of about 0.0030.003 and then bends over, flattening near a maximum stress of about 300×106 N m−2300\times10^{6}\ \text{N m}^{-2}. The stress at which this non-linear (plastic) behaviour sets in is the yield strength:

σy≈300×106=3×108 N m−2.\sigma_y\approx 300\times10^{6}=3\times10^{8}\ \text{N m}^{-2}.

✓Final answer

  1. Y≈7.5×1010 N m−2Y\approx 7.5\times10^{10}\ \text{N m}^{-2}.
  2. Approximate yield strength ≈3×108 N m−2\approx 3\times10^{8}\ \text{N m}^{-2} (300×106 N m−2300\times10^{6}\ \text{N m}^{-2}).

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