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?

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:

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)