Question

The relation of three elastic constants of a metal in terms of Young's Modulus (E), Modulus of rigidity (G) and Bulk Modulus (K) is expressed as

• A. \( E = \frac{3\text{KG}}{(3\text{K}+\text{G})} \)
• B. \( E = \frac{6\text{KG}}{(3\text{K}+\text{G})} \)
• C. \( E = \frac{8\text{KG}}{(3\text{K}+\text{G})} \)
• D. \( E = \frac{9\text{KG}}{(3\text{K}+\text{G})} \)

Hint:

An easy way to remember this formula is to memorize its reciprocal form: \( \frac{9}{E} = \frac{3}{G} + \frac{1}{K} \). Finding a common denominator and flipping this reciprocal equation gives the standard formula directly!

Answer 1

The Correct Option is D

Step 1: Start from the reciprocal elastic constants relation.
Instead of eliminating Poisson's ratio from the two separate formulas, we can use the well known reciprocal relation that links all three constants directly: \[ \frac{1}{E} = \frac{1}{3G} + \frac{1}{9K} \]

Step 2: Combine the fractions on the right.
Taking the LCM as \( 9KG \): \[ \frac{1}{E} = \frac{3K + G}{9KG} \]

Step 3: Invert both sides to get E.
\[ \boxed{E = \frac{9KG}{3K+G}} \] This matches option (D).