Definitions:

1. Poisson's Ratio (ν): Poisson's ratio is the ratio of lateral strain to longitudinal strain in a material when subjected to axial stress. When a material is stretched, it tends to become narrower in the perpendicular direction. Mathematically: $\nu = -\frac{\text{Lateral strain}}{\text{Axial strain}}$ It is a dimensionless constant, typically denoted by the symbol $\nu$.

2. Bulk Modulus (K): The bulk modulus measures a material's resistance to uniform compression. It is defined as the ratio of volumetric stress to the corresponding volumetric strain: $K = \frac{\text{Pressure (change in stress)}}{\text{Volumetric strain}}$ Its unit is Pascals (Pa), the same as pressure.

3. Modulus of Rigidity (G) or Shear Modulus: The modulus of rigidity (also known as shear modulus, $G$) is a measure of a material's rigidity or stiffness when subjected to shear stress. It is defined as: $G = \frac{\text{Shear stress}}{\text{Shear strain}}$ It also has the unit of Pascals (Pa).

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Derivation of the formula $E = 3K(1 - 2\nu)$

Symbols:

• $E$: Young’s Modulus
• $K$: Bulk Modulus
• $\nu$: Poisson’s Ratio

Starting Point:

In the theory of elasticity, the relationship between Young's modulus $E$, Poisson's ratio $\nu$, and the bulk modulus $K$ can be derived using the basic definitions of these constants.

1. Bulk Modulus Relation:
   The bulk modulus $K$ is related to the volumetric strain caused by stress. It can be linked to Young's modulus $E$ and Poisson’s ratio $\nu$ through the following relationship: $K = \frac{E}{3(1 - 2\nu)}$

2. Rearranging the above equation for $E$: $E = 3K(1 - 2\nu)$

This is the required formula.

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Numerical Problem:

Let’s solve the numerical example. Assume we are given:

• $K = 5 \times 10^9 \, \text{Pa}$ (Bulk Modulus)
• $\nu = 0.3$ (Poisson's Ratio)

Using the derived formula $E = 3K(1 - 2\nu)$, we can calculate Young's modulus $E$.

1. Substituting the given values: $E = 3 \times 5 \times 10^9 \times (1 - 2 \times 0.3)$

2. Simplify the expression: $E = 15 \times 10^9 \times (1 - 0.6)$ $E = 15 \times 10^9 \times 0.4$ $E = 6 \times 10^9 \, \text{Pa}$

Thus, the Young's modulus $E$ is $6 \times 10^9 \, \text{Pa}$ or $6 \, \text{GPa}$.

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Do you want more details or have any questions?

Here are 5 related questions to explore further:

1. What is the relationship between Young’s modulus and the modulus of rigidity?
2. How does Poisson's ratio affect the behavior of materials under stress?
3. What are typical values of Poisson’s ratio for different materials?
4. Can you derive the formula for $E$ in terms of $G$ and $\nu$?
5. How does the bulk modulus influence the compressibility of a material?

Tip: When working with material properties, always check the units to ensure consistency, especially when dealing with Pa, GPa, or MPa.