For an isotropic elastic material with Young's modulus E, shear modulus N (rigidity), bulk modulus K, and Poisson's ratio ν = 1/m, identify the correct relationship(s) among E, N, K, and m.

Introduction / Context:In linear elasticity for isotropic materials, the four elastic constants (E, N, K, ν) are interrelated; any two determine the others. Many exam questions test recognition of these canonical identities, including notational variants where ν = 1/m.

Given Data / Assumptions:

• E: Young's modulus; N: modulus of rigidity (shear modulus G); K: bulk modulus.
• Poisson's ratio ν = 1/m.
• Material is homogeneous and isotropic; small-strain elasticity applies.

Concept / Approach:Standard relationships are: E = 2G(1 + ν), E = 3K(1 − 2ν), and E = 9KG/(3K + G). Substituting G = N and ν = 1/m yields the given forms. These identities are dimensionally consistent and widely tabulated.

Step-by-Step Solution:1) From shear relation: E = 2N(1 + ν) → using ν = 1/m → E = 2N(1 + 1/m).2) From volumetric relation: E = 3K(1 − 2ν) → E = 3K(1 − 2/m).3) Combined relation: E = 9 K N / (3K + N) (independent of ν explicitly).

Verification / Alternative check:Pick E, ν and compute N and K using the first two formulas; then back-substitute into the third to confirm equality. All three are consistent for isotropic materials (ν between −1 and 0.5 for stability).

Why Other Options Are Wrong:

• Each of A–C is correct; omitting any would be incomplete.

Common Pitfalls:

• Confusing m with ν (remember ν = 1/m here).
• Using E = 3K(1 − ν) (incorrect; the factor is 1 − 2ν).

Final Answer:All of the above.