In short: Does anybody know if there exists a compendium, a document, a book or a stone tablet listing the independent elements of the elastic stiffness and compliance tensors ( that is, naming the elements, e.g., $C_{1111},C_{2222},\ldots$ that are independent and giving the values of the dependent elements as functions of the independent ones ) for all space groups? Thanks very much!!

In long: Recently I was doing some calculations that involved handling the elastic stiffness tensor from elasticity theory, namely $C_{ijkl}$, for materials with quite specific symmetries. As you probably know, $C_{ijkl}$ has, in general, 21 independent elements. But, depending on the symmetries of the crystal you are regarding, this can change dramatically, down to 2 independent elements for isotropic materials.

I thought to my-self: Well, there is only a finite number of space groups out there, so somebody should have one day calculated the independent elements of $C_{ijkl}$ for all space groups that exist. And, while we are at it, also for its inverse, the compliance tensor $S_{ijkl}$. But I was not succesfull in finding such a compendium anywhere and had to do the calculation on my own. Poor me. Perhaps we can help future generations with that information?


2 Answers 2


You might want to have a look at the figure at the bottom of:


this also suggests that tetragonal cases can be split into two subgroups with slightly different symmetries.


I am not sure if I understand your question. Knowing the group symmetry is not enough to know the value of the elements of the stiffness/compliance tensor. Although, the group symmetry tells you how many independent parameters you have and what coefficients are equal to each other.

That being said, yes there are references that I know is:

A. E. H. Love (1906). Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, second edition.

Based on the way you formulated your questions I would suggest:

Nye, J. F. (1985). Physical properties of crystals: their representation by tensors and matrices. Oxford University Press.

The list of different groups for stiffness/compliance tensors is the following:

  • A monoclinic material present one plane of symmetry. They have 13 coefficients.

  • A trigonal material is a monoclonic material with aditional symmetry with respect to rotations of 60 degrees. They have 7 independent coefficients.

  • An orthotropic material presents three (orthogonal) planes of symmetry. They have 9 independent coefficients.

  • A tetragonal material is an orthotropic material with a further 45 degree. They present 7 independent coefficients.

  • A transverse isotropic material is an orthotropic material where one of those planes of symmetry presents isotropy. They have 5 independent coefficients.

  • A cubic material is an orthotropic material with symmetries respect to 90 degrees rotations. They have 3 independent coefficients.

  • An isotropic material is symmetric with respect to all rotations. They have 2 independent coefficients.

  • $\begingroup$ Hi! Thanks for your contribution. I did not want to imply the symmetry sets the values of the strain tensor elements, I really wanted to have, as you figured, a list that tells me for a given material symmetry which elements of the stiffness tensor are independent and which value the rest of the stiffness tensor elements have. That being said, I think the list you gave is incomplete, actually. I had to handle a material of tetragonal crystal structure that had additional symmetries of its unit cell ( space group I(-)42d ) , which resulted in 6 independent elements, not 7... $\endgroup$ May 22, 2020 at 11:02
  • $\begingroup$ @Stefanowitschko, that can happen. But it is not considered a different type of material. I suggest that you check Nye's book. $\endgroup$
    – nicoguaro
    May 22, 2020 at 11:50

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