Question

If G is the modulus of rigidity, K is the bulk modulus and $\mu$ is the poisson's ratio of a material, then the ratio of modulus of rigidity to bulk modulus is:

• A. $\frac{3(1+2\mu)}{2(1+\mu)}$
• B. $\frac{3(1+2\mu)}{2(1-\mu)}$
• C. $\frac{3(1-2\mu)}{2(1+\mu)}$
• D. $\frac{3(1-2\mu)}{2(1-\mu)}$

Hint:

A useful way to remember these formulas is the "2-1" and "3-2" rule.

• Use 2 with $G$ and $(1 \mathbf{+} \mu)$.

• Use 3 with $K$ and $(1 \mathbf{-} 2\mu)$.
Just remember that rigidity ($G$) is associated with addition ($+$) and bulk volume ($K$) is associated with subtraction ($-$).

Answer 1

The Correct Option is C

1. Fundamental Elastic Relations: There are two primary equations that relate Young's modulus ($E$) with $G$, $K$, and Poisson's ratio ($\mu$):

• Relation 1: $E = 2G(1 + \mu)$

• Relation 2: $E = 3K(1 - 2\mu)$

2. Establishing the Ratio: Since both expressions are equal to $E$, we can set them equal to each other to find the direct relationship between $G$ and $K$: $$2G(1 + \mu) = 3K(1 - 2\mu)$$

3. Algebraic Rearrangement: To find the ratio $\frac{G}{K}$, we rearrange the equation: $$\frac{G}{K} = \frac{3(1 - 2\mu)}{2(1 + \mu)}$$ This derived formula shows that the ratio depends entirely on the Poisson's ratio of the material. For most engineering materials where $\mu \approx 0.3$, this ratio is typically around $0.46$.