Relation between E (Modulus of elasticity) and G (Shear modulus)

The correct answer is option 'B': E = 2G (1 - µ).

Explanation:
The modulus of elasticity (E) and the shear modulus (G) are two important mechanical properties that describe the behavior of a material under stress. They are related to each other through a mathematical equation.

Poisson's ratio (µ):
Before understanding the relation between E and G, it is important to know about Poisson's ratio (µ). Poisson's ratio is a measure of the transverse deformation of a material when it is stretched in one direction. It is defined as the ratio of lateral strain to the longitudinal strain.

Relation between E and G:
The correct relation between E and G is given by E = 2G (1 - µ). Let's understand the components of this equation in detail:

1. E: Modulus of elasticity (also known as Young's modulus) is a measure of a material's stiffness or rigidity. It describes the material's ability to resist deformation under an applied load.

2. G: Shear modulus (also known as the modulus of rigidity) measures a material's resistance to shear stress. It describes the material's ability to resist deformation caused by shear forces.

3. µ: Poisson's ratio is denoted by the Greek letter µ (mu). It is the ratio of lateral strain (strain in the transverse direction) to the longitudinal strain (strain in the axial direction) when a material is under tension.

The relation E = 2G (1 - µ) shows that the modulus of elasticity (E) is directly proportional to the shear modulus (G) multiplied by a factor of (1 - µ).

Significance of the relation:
This relation is important because it helps in understanding the relationship between different mechanical properties of a material. It provides insights into how a material will behave under different types of stresses and strains.

By knowing the values of E, G, and µ for a particular material, we can determine its mechanical behavior, such as its ability to withstand loads, its resistance to deformation, and its response to external forces.

Overall, the relation E = 2G (1 - µ) gives us a quantitative understanding of the interplay between the modulus of elasticity and the shear modulus, enabling engineers and scientists to design and analyze materials and structures effectively.