Elastic constants – Relationship among E (Young’s modulus), C (shear modulus), and K (bulk modulus)\nSelect the correct formula that relates Young’s modulus E, shear modulus C, and bulk modulus K for an isotropic elastic material.

Introduction / Context:
For isotropic, linear elastic materials, only two independent elastic constants are required. Various equivalent relationships link Young’s modulus E, shear modulus C (often denoted G), bulk modulus K, and Poisson’s ratio ν.

Given Data / Assumptions:

• Material is homogeneous, isotropic, and linearly elastic.
• Standard elasticity relations apply.

Concept / Approach:
The well-known relations are E = 2C(1 + ν) and E = 3K(1 − 2ν). Eliminating ν between these gives a direct relation among E, C, and K: E = 9KC / (3K + C).

Step-by-Step Derivation:
Start with E = 2C(1 + ν) and E = 3K(1 − 2ν).Solve the pair for ν and eliminate it.Algebra yields E = 9KC / (3K + C).

Verification / Alternative check:
Another equivalent form is 1/E = 1/(9K) + 1/(3C), which rearranges to E = 9KC/(3K + C).

Why Other Options Are Wrong:
Option (a) lists two separate relations involving ν, not a direct three-constant relation; option (c) is correct but different in form (reciprocal); (d) is not valid for isotropic elasticity; (e) misses the factor 3 in the numerator.

Common Pitfalls:
Confusing symbols C and G; misapplying relations for anisotropic materials.

Final Answer:
E = 9 * K * C / (3 * K + C)