For the bar referred to in Question 1, if it is known that the metal’s shear modulus G = 72.3 GPa, calculate Poisson’s ratio, ν, for the metal.

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Begin with the fundamental relation for isotropic linear-elastic materials: G = E / [2(1 + ν)], where G is the shear modulus, E is Young's modulus, and ν is Poisson's ratio. From this, Poisson’s ratio can be expressed as ν = (E / (2G)) − 1. Since only G is provided (G = 72.3 GPa) and E is not given in the prompt, ν cannot be uniquely determined without an additional property (such as E or K). The problem, however, lists several candidate ν values, so a study of what E would have to be for each ν helps evaluate plausibility.

Option-by-option analysis:
- Option 0.301: If ν = 0.301, then E would be E = 2G(1 + ν) = 2 × 72.3 × (1 + 0.301) ≈ 144.6 × 1.301 ≈ 188.1 GPa. This implies a Young’s modulus around 188 GPa, which lies within the common range for many metals (roughly 180–200 GPa for steel, aluminum alloys, etc.). The result is physically plausible for a metal with the given shear modulus.
- Option 0.285: If ν = 0.285, then E ≈ 144.6 × (1 + 0.285) ≈ 144.6 × 1.285 ≈ 185.0 GPa. This also falls in a plausible range for metals, suggesting this ν value is compatible with a reasonable E in the same ballpark.
- Option 0.246: If ν = 0.246, then E ≈ 144.6 × (1 + 0.246) ≈ 144.6 × 1.246 ≈ 180.0 GPa. An E around 180 GPa is characteristic of many softer metals or certain alloys, so this is still physically reasonable, though slightly lower than typical high‑strength steels.
- Option 0.320: If ν = 0.320, then E ≈ 144.6 × (1 + 0.320) ≈ 144.6 × 1.320 ≈ 191.0 GPa. This also yields a plausible Young’s modulus in the upper part of the common metal range, so this option remains consistent with a metal having G = 72.3 GPa.

Overall, each candidate ν value can be reconciled with a corresponding E in the broad metal range through the relation E = 2G(1 + ν). Since only G is specified, there isn’t a unique ν without additional data (such as a given E, K, or another mechanical property). In many metals, Poisson’s ratio tends to be around 0.30, so ν values near 0.30 (such as 0.301 or 0.320) are more typical, with 0.301 being very close to the conventional 0.30 target. The other values yield reasonable E estimates but slightly deviate from a canonical near-0.30 metal value.

Thus, the reasoning shows how each ν option maps to a corresponding E and demonstrates the dependency on an extra property to fix ν definitively.

INU1127 (24/25) Knowledge Check: E, G and ν

For the bar referred to in Question 1, if it is known that the metal’s shear modulus G = 72.3 GPa, calculate Poisson’s ratio, ν, for the metal.

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