The relationship between Young’s Modulus (E), Modulus of Rigidity (G) and Poisson’s ratio (µ) is given by the expression
- AE = 2G(1 + µ)Correct
- BE = 3G(1 – 2µ)
- CG = 2E(1 + µ)
- DG = 3E(1 – 2µ)
Solution
Concept
Elastic Constants Relationship
Formula
E = 2G(1 + µ)
Step-by-step solution
1. Identify the given elastic constants: Young’s Modulus (E), Modulus of Rigidity (G), and Poisson’s ratio (µ). 2. Recall the fundamental elastic constant relation in mechanics of materials: E = 2G(1 + µ). 3. Compare the relation with the given options and select Option (a) as the correct answer.
Explanation
The correct relationship connecting Young’s Modulus (E), Modulus of Rigidity (G), and Poisson’s ratio (µ) is E = 2G(1 + µ). Young’s Modulus measures tensile or compressive stiffness, Modulus of Rigidity measures shear stiffness, and Poisson's ratio relates transverse strain to axial strain. This standard formula allows engineers to calculate elastic material properties when two of the constants are known.
Shortcut
To easily remember the relationship E = 2G(1 + µ): - 2G Connection: Associate "G" (Modulus of Rigidity) with "2G" network technology to remember the multiplier 2. - Sign Rule: Rigidity (G) adds strength, so it uses a PLUS sign: (1 + µ). - Comparison Trick: Bulk Modulus (K) uses 3 and a minus sign: E = 3K(1 - 2µ).
Common mistake
A common mistake is confusing the relation between Young's Modulus (E) and Shear Modulus (G) with the relation involving Bulk Modulus (K), which is E = 3K(1 - 2µ), leading students to pick options like B or D by incorrectly swapping K with G. Another frequent error is swapping E and G in the formula, incorrectly choosing option C instead of the correct relation E = 2G(1 + µ).
Correct answer: A) E = 2G(1 + µ)