Isotropic, Homogeneous, and Linearly Elastic Material:

An isotropic material is one that has the same mechanical properties in all directions. In other words, the material behaves the same way regardless of the direction in which it is loaded.

A homogeneous material is one that has uniform properties throughout its volume. This means that the material's properties, such as density, elasticity, and strength, are constant at every point within the material.

A linearly elastic material is one that obeys Hooke's law, which states that the stress in a material is proportional to the strain applied to it. This relationship is expressed as σ = Eε, where σ is the stress, E is the elastic modulus (Young's modulus), and ε is the strain.

Hooke's Law and Elastic Constants:

Hooke's law relates stress and strain in a linearly elastic material. However, the relationship between stress and strain depends on the material's behavior in different directions. For an isotropic material, the elastic constants can be defined to describe this relationship.

The elastic constants that describe the linear elastic behavior of an isotropic material are as follows:

1. Young's Modulus (E): Young's modulus is a measure of the stiffness of a material. It relates stress and strain in the axial direction of a material. For an isotropic material, there is only one Young's modulus.

2. Poisson's Ratio (ν): Poisson's ratio describes the lateral strain that occurs when a material is stretched or compressed in the axial direction. It relates the strain in the lateral direction to the strain in the axial direction. For an isotropic material, there is only one Poisson's ratio.

Number of Independent Elastic Constants:

For an isotropic, homogeneous, and linearly elastic material, the number of independent elastic constants can be determined as follows:

1. Young's modulus is a scalar quantity that describes the material's behavior in the axial direction. Therefore, there is only one independent elastic constant associated with Young's modulus.

2. Poisson's ratio is also a scalar quantity that relates the lateral strain to the axial strain. Therefore, there is only one independent elastic constant associated with Poisson's ratio.

Hence, the total number of independent elastic constants for an isotropic, homogeneous, and linearly elastic material is 2.

Therefore, the correct answer is option B, which states that there are 2 independent elastic constants.