Question

The relationship between modulus of elasticity(E), Poisson's ratio(\(\mu\)) and bulk modulus(K) is given by

• A. \(K = \frac{E}{3(1-2\mu)}\)
• B. \(K = \frac{E}{3(1+2\mu)}\)
• C. \(G = \frac{E}{2(1+\mu)}\)
• D. \(G = \frac{E}{2(1-\mu)}\)

Hint:

Remember the two key relationships: $E = 3K(1-2\mu)$ and $E = 2G(1+\mu)$.
These formulas are fundamental to solving elasticity problems in solid mechanics.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the correct mathematical relationship between three fundamental elastic constants of an isotropic material: Young's Modulus (\(E\)), Bulk Modulus (\(K\)), and Poisson's Ratio (\(\mu\)).

Step 2: Key Formula or Approach: This is a standard relationship in elasticity theory. For a homogeneous and isotropic material, the three main elastic moduli (\(E\), \(G\), and \(K\)) are related to Poisson's ratio by standard equations:
\[ E = 3K(1 - 2\mu) \]
\[ E = 2G(1 + \mu) \]

Step 3: Detailed Explanation:

• Young's Modulus (\(E\)) measures tensile/compressive elasticity, Bulk Modulus (\(K\)) measures volumetric elasticity, and Poisson's Ratio (\(\mu\)) relates lateral strain to longitudinal strain.

• The relationship between Young's Modulus and Bulk Modulus is:
\[ E = 3K(1 - 2\mu) \]

• Rearranging this equation to solve for the Bulk Modulus \(K\) yields:
\[ K = \frac{E}{3(1 - 2\mu)} \]

• This rearranged equation matches Option A.

• Option C represents the correct relationship for the Shear Modulus \(G\), but it is written as \(G = \frac{E}{2(1+\mu)}\), which is correct but the question specifically asks for the relationship involving the bulk modulus.

Step 4: Final Answer:
The correct relationship is \(K = \frac{E}{3(1-2\mu)}\).