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: Recall standard elastic relations.
In the theory of elasticity, Young’s modulus, bulk modulus, and Poisson’s ratio are interrelated material properties for isotropic and homogeneous materials.
Step 2: Write the known relation.
The standard formula connecting Young’s modulus ($E$), bulk modulus ($K$), and Poisson’s ratio ($\mu$) is given by:
\[ E = 3K(1 - 2\mu) \]
Step 3: Match with the options.
Comparing the derived relation with the given options, it directly matches option (B).
Step 4: Conclusion.
Hence, the correct relationship is $E = 3K(1 - 2\mu)$.

Answer 2

Instead of recalling the relation directly, we can derive it by analyzing a small cubic element subjected to equal normal stress \(p\) on all three mutually perpendicular faces (a state of hydrostatic stress, which is exactly the condition used to define the bulk modulus \(K\)). Using this derivation we can check which of the four options is dimensionally and physically consistent.

Under a hydrostatic stress \(p\) acting simultaneously along the x, y and z directions, the strain along any one direction (say x) is obtained by superposing the direct strain due to the stress in that direction and the Poisson contraction due to the stresses in the other two directions:

\[ \varepsilon_x = \frac{p}{E} - \mu\frac{p}{E} - \mu\frac{p}{E} = \frac{p}{E}(1-2\mu) \]

By symmetry, \(\varepsilon_y = \varepsilon_z = \varepsilon_x\), so the volumetric strain is

\[ \frac{\Delta V}{V} = \varepsilon_x + \varepsilon_y + \varepsilon_z = \frac{3p}{E}(1-2\mu) \]

By the definition of the bulk modulus, \( \dfrac{\Delta V}{V} = \dfrac{p}{K} \). Equating the two expressions for the volumetric strain gives

\[ \frac{p}{K} = \frac{3p}{E}(1-2\mu) \quad\Rightarrow\quad E = 3K(1-2\mu) \]

With this derived relation in hand, let us check each option:

1. E = 2K(1-2mu): The coefficient here is 2, but the derivation above shows the coefficient must be 3 (it comes from summing the strain contribution along all three axes of the cube). This option is incorrect.
2. E = 3K(1-2mu): This matches exactly what was derived from the hydrostatic strain analysis above, so this option is correct.
3. E = 3K(1-3mu): The coefficient 3 on K is right, but the bracket should contain \(1-2\mu\), not \(1-3\mu\); the factor of 2 arises from the two Poisson-contraction terms in \(\varepsilon_x\), not three. This option is incorrect.
4. E = 2K(1-3mu): This has both the coefficient of K and the coefficient inside the bracket wrong, so it cannot be correct.

Only the relation with a coefficient of 3 outside and \((1-2\mu)\) inside the bracket satisfies the equilibrium of a hydrostatically stressed cube.

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