Generic Born stability criteria 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?
 A: I've found a good analysis of the stability conditions for a crystal's elastic constants, both unstrained and under stress, in:

J. W. Morris Jr and C. R. Krenn, Philos. Mag. A 2000, 12, 2827–2840

To quote them:

In the linear elastic limit the conditions of internal stability reduce to the familiar
  condition that the 6 x 6 matrix $C_{ij}$ of elastic moduli have no negative eigenvalues.


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).
A: While I can't find a reference for this, I think your criterion is correct.  Here's the argument I would use - if it's the same as yours, then maybe it's right!  The elastic energy is (http://ciks.cbt.nist.gov/garbocz/manual/node8.html)
$$E = \frac{1}{2}\int d^d r \epsilon_i C_{ij} \epsilon_j$$ 
where $C_{ij}$ is symmetric.  Because $C_{ij}$ is symmetric and real, it can be diagonalized, and its eigenvectors are complete and orthogonal (https://en.wikipedia.org/wiki/Hermitian_matrix).  Then you can expand $\epsilon_j$ in terms of the eigenvectors of $C_{ij}$, $\bf{v}^{(k)}$, i.e. $C_{ij} v_j^{(k)} = \lambda_k v_i^{(k)}$.
$$\epsilon_j = \sum_k (\bf{v}^{(k)} \cdot \bf{\epsilon}) v^{(k)}_j$$
(This assumes that $\bf{v}^{(k)}$ is orthonormal, which we can always choose without loss of generality.)  We then find that
$$E = \frac{1}{2} \int d^d r \lambda_k (\bf{v}^{(k)} \cdot \bf{\epsilon})^2$$.
Stability means that there are no modes that will lead to an unbounded decrease in energy, and no marginal modes - i.e. that $\lambda_k > 0$ for all $k$ - your positive definite quadratic form.  
A: This derivation is correct for cubic crystals without external pressure. 
A fresh review can be found in Rev. Mod. Phys. 84, 945 (2012) Lattice instabilities in metallic elements 
http://rmp.aps.org/abstract/RMP/v84/i2/p945_1
A: A detailed analysis may be found also in arXiv:1104.0173 [astro-ph.SR], D. A. Baiko "Shear modulus of neutron star crust", Eqs. (29)-(33).
http://arxiv.org/abs/1104.0173
