The relation betweent `gamma,eta` and K for a elastic material is

Options

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

Answer

To find the relationship between Young's modulus (γ), modulus of rigidity (η), and bulk modulus (K) for an elastic material, we can follow these steps:

### Step 1: Identify the Moduli
We start by identifying the moduli:
- Let γ = Young's modulus
- Let η = Modulus of rigidity (shear modulus)
- Let K = Bulk modulus

### Step 2: Write the Relationships
We know the following relationships for elastic materials:
1. Young's modulus (γ) can be expressed in terms of bulk modulus (K) and shear modulus (η):
\[ \gamma = 3K(1 - 2\sigma) \]
where σ is the Poisson's ratio.

2. Young's modulus (γ) can also be expressed in terms of shear modulus (η):
\[ \gamma = 2η(1 + α) \]
where α is the coefficient of linear expansion.

### Step 3: Set Up the Equations
From the above relationships, we can set up two equations:
1. \(\frac{\gamma}{3K} = 1 - 2\sigma\) (Equation 1)
2. \(\frac{\gamma}{2η} = 1 + α\) (Equation 2)

### Step 4: Manipulate the Equations
We can manipulate these equations to express σ and α in terms of γ, K, and η.

From Equation 1:
\[ \sigma = \frac{1}{2} \left( 1 - \frac{\gamma}{3K} \right) \]

From Equation 2:
\[ α = \frac{\gamma}{2η} - 1 \]

### Step 5: Substitute and Combine
Now, we can substitute these expressions into one another to eliminate σ and α.

### Step 6: Combine the Equations
Multiply Equation 2 by 2:
\[ \frac{\gamma}{η} = 2 + 2α \]
Substituting for α:
\[ \frac{\gamma}{η} = 2 + 2\left(\frac{\gamma}{2η} - 1\right) \]

### Step 7: Solve for η
Now, we can rearrange this equation to solve for η in terms of γ and K:
\[ \frac{\gamma}{η} = 2 + \frac{\gamma}{η} - 2 \]
This simplifies to:
\[ \frac{1}{η} = \frac{3}{\gamma} + \frac{1}{9K} \]

### Step 8: Final Relationship
Rearranging gives us the final relationship:
\[ 3η + \frac{1}{9K} = \frac{1}{γ} \]

### Conclusion
Thus, the relationship between Young's modulus (γ), modulus of rigidity (η), and bulk modulus (K) is:
\[ 3η + \frac{1}{9K} = \frac{1}{γ} \]