The number of elastic constants for a completely anisotropic elastic material is 21.

Explanation:

Anisotropic materials have different mechanical properties in different directions. In the case of elastic materials, these properties are described by a set of elastic constants that relate stress and strain in different directions.

There are two types of elastic constants: scalar constants and tensor constants. Scalar constants are those that are independent of direction and include Young's modulus (E), shear modulus (G), and Poisson's ratio (ν). These constants describe the material's response to stress and strain in any direction.

On the other hand, tensor constants are direction-dependent and describe the material's response to stress and strain in specific directions. For a completely anisotropic material, the number of tensor constants is determined by the number of independent directions in which stress and strain can occur.

The number of independent directions in three-dimensional space is determined by the number of independent components in a second-order tensor, which is given by the formula:

Number of independent components = n(n+1)/2

where n is the number of dimensions. In the case of three-dimensional space, n=3, so the number of independent components is:

Number of independent components = 3(3+1)/2 = 6

Since stress and strain are symmetric tensors, the number of independent components is reduced to half. Therefore, the number of independent elastic constants for a completely anisotropic material in three-dimensional space is:

Number of independent elastic constants = 6/2 = 3

However, each independent elastic constant has three components (one for each dimension), resulting in a total of:

Total number of elastic constants = 3 × 3 = 9

In addition to these nine constants, there are also twelve off-diagonal components that describe the coupling between different directions. These coupling constants are also independent and contribute to the anisotropic behavior of the material. Therefore, the total number of elastic constants for a completely anisotropic elastic material in three-dimensional space is:

Total number of elastic constants = 9 + 12 = 21

Hence, the correct answer is option C (21).