Relationship between modulus of elasticity E, modulus of rigidity C (G), and Poisson’s ratio μ: select the correct formula.

Introduction / Context:
In isotropic linear elasticity, only two independent elastic constants are required. The commonly used relations connect Young’s modulus E, shear modulus C (also denoted G), bulk modulus K, and Poisson’s ratio μ.

Given Data / Assumptions:

• Material is homogeneous and isotropic.
• Linear elastic response.
• Standard definitions of E, C (G), μ.

Concept / Approach:
The fundamental isotropic elasticity relations include E = 2 C (1 + μ) and E = 3 K (1 − 2 μ). These allow conversion between sets of elastic constants based on measured data.

Step-by-Step Solution:

Verification / Alternative check:
Cross-check with E = 3 K (1 − 2 μ) and the identity K = E / 3 (1 − 2 μ) to ensure consistency.

Why Other Options Are Wrong:
E = 3 C (1 − 2 μ) confuses K with C.C = E (1 − 2 μ) / 2 (1 + μ) is algebraically equivalent to the correct formula, but the question asks for E as a function of C and μ; the simplest correct expression is E = 2 C (1 + μ).Other forms given are incorrect dimensionally or miss the 1 + μ factor.

Common Pitfalls:
Interchanging K and C; omitting μ; using plastic-range values where these linear relations no longer hold.

Final Answer:

E = 2 C (1 + μ)