Question

The relationship between Young’s modulus (E), Bulk modulus (K) and Poisson’s ratio ($\mu$) is given by

• A. $E = 2K(1-2\mu)$
• B. $E = 3K(1-2\mu)$
• C. $E = 3K(1-3\mu)$
• D. $E = 2K(1-3\mu)$

Hint:

Remember the three important elastic relations: $E = 2G(1+\mu)$, $E = 3K(1-2\mu)$, and $K = \dfrac{E}{3(1-2\mu)}$.

Answer 1

The Correct Option is B

Step 1: Start from the standard three-constant identity \( E = \dfrac{9KG}{3K+G} \), which relates E, K and the shear modulus G.
Step 2: Replace G using \( G = \dfrac{E}{2(1+\mu)} \) and simplify: \( E\left[3K+\dfrac{E}{2(1+\mu)}\right] = \dfrac{9KE}{2(1+\mu)} \).
Step 3: Multiply through by \(2(1+\mu)\), then divide by \(E\): \( 6K(1+\mu) + E = 9K \).
Step 4: Solve for E: \( E = 9K - 6K(1+\mu) = 3K - 6K\mu \).
\[ \boxed{E = 3K(1-2\mu)} \]

Answer 2

Another way to check this relation is to test the four candidate formulas against the approximate, well-known elastic properties of a real material, such as mild steel, for which \(E \approx 200\ \text{GPa}\), \(K \approx 160\ \text{GPa}\) and \(\mu \approx 0.3\).

1. E = 2K(1-2mu): Substituting the steel values gives \(2(160)(1-0.6) = 2(160)(0.4) = 128\ \text{GPa}\), far below the actual \(E \approx 200\ \text{GPa}\), so this formula underestimates E and can be ruled out.
2. E = 3K(1-2mu): Substituting gives \(3(160)(1-0.6) = 3(160)(0.4) = 192\ \text{GPa}\), which is very close to the actual value of \(E \approx 200\ \text{GPa}\) for steel (the small gap is only due to using rounded textbook values of K and mu). This formula is consistent with real material behavior.
3. E = 3K(1-3mu): Substituting gives \(3(160)(1-0.9) = 3(160)(0.1) = 48\ \text{GPa}\), which is nowhere close to the actual \(E \approx 200\ \text{GPa}\), so this option is ruled out.
4. E = 2K(1-3mu): Substituting gives \(2(160)(0.1) = 32\ \text{GPa}\), again far too small to match steel's actual Young's modulus, so this option is also ruled out.

Only the formula with a coefficient of 3 outside and \((1-2\mu)\) inside the bracket reproduces a value close to the known Young's modulus of steel.

Therefore, the correct answer is E = 3K(1-2μ).