Question

Total number of elastic constant of an orthotropic material are

Answer 1

If you consider the most generalized Hooke's law applied to an anisotropic elastic solid:
σij=Cijkl∗εklσij=Cijkl∗εkl
where CijklCijkl is the elastic stiffness tensor, which is a fourth rank tensor.

Upon expanding the above equation we would get nine equations, each with nine terms. Hence there are 81 constants in all.

But the stress and strain tensors are symmetric; which leads to:
Cijkl=CjiklCijkl=Cjikl and Cijkl=CijlkCijkl=Cijlk

Hence 36 terms of the elastic stiffness tensor are independent and distinct. But further reduction in the number of independent constants is possible.

Generally, Cij=CjiCij=Cji

Which means that of the 36 constants, there are 6 constants where i=ji=j. This leaves 30 constants, but only half of them are independent. Therefore, for the general anisotropic linear elastic solid there are (30/2) + 6 = 21 independentelastic constants.

For an isotropic elastic material there are only 2 independent elastic constants.

You can refer to this book by G.E. Dieter for better understanding (Mechanical metallurgy)