Why the components of elasticity tensor are 21?

It's known that the elasticity tensor is such that $$C_{ijkl}=C_{jikl}=C_{ijlk}=C_{klij}.$$

The first two equalities imply that we have a $$6 \times 6$$ symmetric matrix. So far so good.

I can't understand why the last condition implies exactly 21 components. I've seen this answer, but I cannot really understand why they are precisely 21.

A $$6 \times 6$$ matrix has 36 different components. When you reduce it to a symmetric case it has $$1 + 2 + 3 + 4 + 5 + 6 = 21$$, where we are summing the number of entries without double counting.

• But why does $C_{ijkl}=C_{klij}$ implies that we go from 36 to 21? It seems like if you consider {i,j} and {k,l} as a "unique" index Commented Jun 6, 2021 at 21:55
• @bobinthebox A tensor with the other symmetries is isomorphic to a $6 \times 6$ matrix. Thus, you can represent is as one using Voigt or Mandel notation. After that, you are restricting your matrix to be symmetric as well, and that takes you from 36 to 21. Commented Jun 6, 2021 at 22:59

The question and current accepted answer presuppose a certain symmetry in the elastic stiffness (and compliance) tensor such that Voigt notation produces a diagonally symmetric matrix. One might still ask: Whence this symmetry?

If we slowly deform an elastic object, the volumetric strain energy $$U$$ it gains depends on the work done:

$$dU=\sigma_i\,d\varepsilon_i.$$

In addition, linear elasticity specifies that

$$\sigma_i=C_{ij}\varepsilon_j,$$

where $$C$$ is the stiffness matrix. So we have

$$\frac{\partial}{\partial\varepsilon_j}\left(\frac{\partial U}{\partial\varepsilon_i}\right)=C_{ij}.$$

"But since [the strain energy] is a function only of the state of the body," as Nye notes in Physical Properties of Crystals, "the order of differentiation is immaterial, and the left-hand side of this equation is symmetrical with respect to $$i$$ and $$j$$":

$$\frac{\partial^2 U}{\partial\varepsilon_j\varepsilon_i}=\frac{\partial^2 U}{\partial\varepsilon_i\varepsilon_j},$$

implying that $$C_{ij}=C_{ji}$$.