Elastic constants — identify the relation that does NOT hold For an isotropic, linear-elastic material, E is Young's modulus, C is the shear modulus (G), K is the bulk modulus, and m is Poisson's ratio. Which of the following relations is NOT correct?
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AE = 2 C (1 + m)
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BE = 3 K (1 - 2 m)
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CK = E / [3 (1 - 2 m)]
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DC = E / [2 (1 + m)]
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EE = 3 K (1 + 2 m)
Answer
Correct Answer: E = 3 K (1 + 2 m)
Explanation
Introduction / Context:In isotropic elasticity, only two independent constants are needed; the rest are linked by standard identities. Spotting an incorrect relation is a quick test of fundamentals.
Given Data / Assumptions:
- Material is homogeneous, isotropic, and linear-elastic.
- Symbols: E (Young's modulus), C or G (shear modulus), K (bulk modulus), m (Poisson's ratio).
Concept / Approach:Correct identities include:E = 2 G (1 + m)E = 3 K (1 - 2 m)G = E / [2 (1 + m)]K = E / [3 (1 - 2 m)]Any expression that contradicts these, such as replacing (1 - 2 m) with (1 + 2 m), is incorrect.
Step-by-Step Solution:
Check option (a): matches E = 2 G (1 + m) ⇒ correct.Check option (b): matches E = 3 K (1 - 2 m) ⇒ correct.Option (c): rearrangement of (b) ⇒ correct.Option (d): rearrangement of (a) ⇒ correct.Option (e): uses (1 + 2 m) instead of (1 - 2 m) ⇒ not correct.Verification / Alternative check:For m = 0.25, compute both sides numerically to see that (e) fails while others match.
Why Other Options Are Wrong:
- (a)–(d) are standard and internally consistent for isotropic materials.
Common Pitfalls:Sign error in the bulk modulus relation; mixing symbols G and C for shear modulus.
Final Answer:E = 3 K (1 + 2 m)