Elastic constants – E and K: Select the correct relationship between Young's modulus (E) and bulk modulus (K) for an isotropic material in terms of Poisson's ratio v.

Introduction / Context:In isotropic linear elasticity, the elastic constants E (Young’s modulus), G (shear modulus), K (bulk modulus), and v (Poisson’s ratio) are interrelated. Knowing one pair allows computation of the others, which is especially useful when test data report different constants than those needed for analysis.

Given Data / Assumptions:

• Material is homogeneous and isotropic.
• Small strains and linear elastic behavior.
• Standard definitions of elastic constants apply.

Concept / Approach:The classical relationships among elastic constants are: E = 2 * G * (1 + v) and K = E / (3 * (1 - 2v)). Rearranging the second yields E = 3 * K * (1 - 2v). These formulas ensure consistency between volumetric and deviatoric responses in isotropic media.

Step-by-Step Solution:

Verification / Alternative check:Dimensional consistency is satisfied. For typical metals with v ≈ 0.3, the factor (1 - 2v) ≈ 0.4, giving E ≈ 1.2K, which aligns with typical ranges.

Why Other Options Are Wrong:

• E = 2 * K * (1 + v): confuses K with G; the coefficient 2 belongs with G.
• E = K / (1 - 2v): missing factor 3; overestimates E.
• E = 9 * K * v / (1 + v): not a standard relation between E and K.

Common Pitfalls:Using these isotropic relations for anisotropic materials (e.g., composites) where they do not hold.

Final Answer:E = 3 * K * (1 - 2v)