The tensorial form of Hooke's law for the strain-stress relationship in a crystal is (in the Voigt notation):
where $\sigma$ is the strain, $\epsilon$ is the stress and C is the stiffness tensor:
For a crystalline system of the cubic symmetry class, the stiffness tensor reduces to:
The Born criterion for the stability of an unstrained crystal is that free energy must be represented by a positive defined quadratic form. In the case of a cubic crystal, it is known that this is equivalent to the following three conditions on the elastic constants:
$$C_{11} – C_{12} > 0$$
$$ C_{44} > 0$$
$$ C_{11} + 2 C_{12} > 0$$
But what about lower symmetry classes? What is the generic Born criterion for stability of a crystal? I have quite convinced myself that all the eigenvalues of $C$ must be positive, but I cannot find confirmation of that anywhere. Is it right? Is there a reference on that topic?
Best Answer
I've found a good analysis of the stability conditions for a crystal's elastic constants, both unstrained and under stress, in:
To quote them:
So, almost two years later, after realizing a lot of people had trouble with this question (and there are mistakes made in some of the literature on the topic), we have published a short pedagogical reference on the issue: F. Mouhat and F.-X. Coudert, Phys. Rev. B, 90, 224104 (2014) (also on arXiv).