The relation between Y. `eta` and B is

Options

A. `(1)/(Y) = (1)/(3eta) +(1)/(9B)`
B. `(9)/(Y)=(1)/(eta) +(3)/(B)`
C. `(1)/(eta) =(1)/(B) +(1)/(Y)`
D. `(9)/(Y) = (3)/(eta) +(1)/(B)`

Explanation

To find the relation between Young's modulus (Y), rigidity modulus (η), and bulk modulus (B), we can follow these steps:

### Step 1: Write down the definitions
Young's modulus (Y), rigidity modulus (η), and bulk modulus (B) are defined as follows:
- Young's modulus (Y) is defined as the ratio of tensile stress to tensile strain.
- Rigidity modulus (η) is defined as the ratio of shear stress to shear strain.
- Bulk modulus (B) is defined as the ratio of volumetric stress to the change in volume strain.

### Step 2: Use the relationships between the moduli
From material mechanics, we have the following relationships:
1. \( Y = 2η(1 + u) \) (where \( u \) is Poisson's ratio)
2. \( Y = 3B(1 - 2u) \)

### Step 3: Rearrange the equations
We can rearrange these equations to express them in terms of Y, η, and B.
From the first equation, we can express η:
\[ η = \frac{Y}{2(1 + u)} \quad \text{(Equation 1)} \]
From the second equation, we can express B:
\[ B = \frac{Y}{3(1 - 2u)} \quad \text{(Equation 2)} \]

### Step 4: Substitute the expressions into a single equation
To find a relation involving all three moduli, we can substitute the expressions for η and B into a single equation. We can manipulate the equations as follows:
1. Substitute η from Equation 1 into the relation \( \frac{9}{Y} = \frac{3}{η} + \frac{1}{B} \):
\[ \frac{9}{Y} = \frac{3(2(1 + u))}{Y} + \frac{1}{\frac{Y}{3(1 - 2u)}} \]
2. Simplifying this gives:
\[ \frac{9}{Y} = \frac{6(1 + u)}{Y} + \frac{3(1 - 2u)}{Y} \]

### Step 5: Combine the terms
Now, combine the terms on the right-hand side:
\[ \frac{9}{Y} = \frac{6(1 + u) + 3(1 - 2u)}{Y} \]
This simplifies to:
\[ \frac{9}{Y} = \frac{9 - 3u}{Y} \]

### Step 6: Final relation
Thus, we can conclude that:
\[ \frac{9}{Y} = \frac{3}{η} + \frac{1}{B} \]

### Final Answer
The relation between Young's modulus (Y), rigidity modulus (η), and bulk modulus (B) is:
\[ \frac{9}{Y} = \frac{3}{η} + \frac{1}{B} \]